MATH 624 Group 2 Final Project
Collin Hoskins, Meiren Jiang, Cordell Jones, Samuel Harold
12/13/2021
Abstract
For this analysis, we developed a statistical model for the global prediction of new COVID deaths for the top 10 countries based on the highest amount of deaths. A variety of supervised and unsupervised learning methods were used.
1. Intro
The COVID-19 virus has caused a global pandemic which has detrimentally affected the world causing loss of population and extreme use of various resources. Medical facilities resources have been exacerbated, labor forces have dwindled, and businesses have needed to adapt to wildly changing markets. The purpose of this project was to develop a statistical model for the global prediction of new COVID deaths for the top ten leading countries for benefit of global preparations and accommodations of physical, medical, and populous resources. Model evaluation was based on MSE comparison between models for prediction accuracy. Work for this group was divided equally among colleagues. Work was divided into four parts, data discovery and visualization, tree based methods, linear methods, and unsupervised methods. A number of models were tested and compared with the mentioned methods before deciding on the final working model. Our initial presumption is that tree methods will likely be the most accurate for this analysis. We are testing multiple models to see if other models may be more accurate than our initial thought. This may also prove to be incorrect, as a more complex model does not always mean more accurate predictions. There also is potential for certain models to over-fit due to variables tha directly impact new deaths included in models.
2. Method
The methodology in this analysis consisted of diagnosing a world issue, exploring data based around the issue, building various models to compare accuracy, and finally analyze model results for insights.
2.1 Data
Data was collected from one source using the code below to extract a csv file at the link “https://raw.githubusercontent.com/owid/covid-19-data/master/public/data/owid-covid-data.csv”. This resource can be found through a browser at with the link “https://github.com/owid/covid-19-data”. The input data was filtered to the top 10 leading countries in deaths. Variables that were not scaled to thousands were appropriately changed to this scaling and variables that were scaled in another fashion which had a thousand counterpart were removed to reduce duplicated variables and model noise. Some other variables with unneeded information, such as excess_mortality_rate, were also removed for noise reduction.
Reading the initial data set and filtering
require(dplyr)
dataset <- read.csv("https://raw.githubusercontent.com/owid/covid-19-data/master/public/data/owid-covid-data.csv", stringsAsFactors = T) %>% select(-c(continent, location, weekly_icu_admissions, weekly_icu_admissions_per_million, weekly_hosp_admissions,
weekly_hosp_admissions_per_million, hosp_patients_per_million, total_boosters_per_hundred, continent,
location, weekly_icu_admissions, weekly_icu_admissions_per_million, weekly_hosp_admissions,
weekly_hosp_admissions_per_million, icu_patients_per_million, total_deaths_per_million,
new_deaths_per_million, new_tests, excess_mortality_cumulative, total_vaccinations_per_hundred,
people_fully_vaccinated_per_hundred, new_vaccinations_smoothed_per_million, total_cases_per_million,
new_people_vaccinated_smoothed_per_hundred, excess_mortality_cumulative_absolute, new_cases_per_million,
excess_mortality_cumulative_per_million, excess_mortality_cumulative, excess_mortality, total_tests,
people_vaccinated_per_hundred, new_deaths_smoothed_per_million, new_cases_smoothed_per_million,
new_tests_smoothed, new_deaths_smoothed, reproduction_rate,
new_tests_smoothed_per_thousand, tests_per_case, new_vaccinations_smoothed, tests_units,
new_people_vaccinated_smoothed, new_cases_smoothed)) %>%
filter(iso_code %in% c("USA", "IND", "BRA", "RUS", "GBR", "FRA", "TUR", "ITA", "COL", "DEU"))
## Creating "by thousands" variable
dataset$total_cases_thousand = dataset$total_cases / 1000
dataset$new_cases_thousand = dataset$new_cases / 1000
dataset$new_deaths_thousand = dataset$new_deaths / 1000
dataset$icu_patients_thousand = dataset$icu_patients / 1000
dataset$hosp_patients_thousand = dataset$hosp_patients / 1000
dataset$total_vaccinations_thousand = dataset$total_vaccinations / 1000
dataset$people_vaccinated_thousand = dataset$people_vaccinated / 1000
dataset$people_fully_vaccinated_thousand = dataset$people_fully_vaccinated / 1000
dataset$total_boosters_thousand = dataset$total_boosters / 1000
dataset$new_vaccinations_thousand = dataset$new_vaccinations / 1000
dataset$population_thousand = dataset$population / 1000
dataset$total_deaths_thousand = dataset$total_deaths / 1000
# Remove old scales
dataset = subset(dataset, select = -c(total_cases, new_cases, new_deaths, icu_patients, hosp_patients,
total_vaccinations, people_vaccinated, people_fully_vaccinated,
total_boosters, new_vaccinations, population, total_deaths)) Dates
dataset$date <- as.Date(dataset$date)
## Creating day of week variable and
## Creating season variable (using astronimical start dates)
dataset <- mutate(dataset, day_of_week = as.factor(weekdays(date)),
season = as.factor(case_when(months(date) %in% c("March", "April", "May") ~ "Spring",
months(date) %in% c("June", "July", "August") ~ "Summer",
months(date) %in% c("September", "October", "November") ~ "Fall",
months(date) %in% c("December", "January", "February") ~ "Winter")))
dataset$year <- factor(lubridate::year(dataset$date))
dataset$month <- factor(lubridate::month(dataset$date))
dataset <- dataset %>% replace(is.na(.), 0)
dataset$iso_code <- factor(dataset$iso_code)
no_date <- select(dataset, -date)
require(caTools)
require(rsample)
## Splitting the data
set.seed(101)
split <- rsample::initial_split(dataset, prop = 0.8, strata = "new_deaths_thousand")
train.dat <- rsample::training(split)
test.dat <- rsample::testing(split)2.2 Data Visualization
We created an interactive geographical shiny app for exploratory information related to average daily cases, average new vaccinations, average daily deaths, population density, and HDI. Additional time series charts are available here as well for our late used methods. This shiny app is available to view via internet browser at “https://choskins.shinyapps.io/shiny/”.
The figure below depicts the training and testing split of the response variable (new daily deaths by thousands) for the time series.
require(ggplot2)
require(ggpubr)
ggplot(data = train.dat, aes(x = date, y = new_deaths_thousand)) + geom_point(color = "blue") +
geom_point(data = test.dat, aes(x = date, y = new_deaths_thousand), color = "orange") + theme_classic()
There is no clear trend, other than the potential of month in the number of COVID-19 deaths. The blue points in the figure above indicate the observations in the training set, and the orange points compose the observations in the testing set. The distribution with time is fairly consistent.
A <- ggplot(dataset) + geom_boxplot(aes(x = iso_code, y = new_deaths_thousand, fill = iso_code)) + theme_classic()
B <- ggplot(dataset) + geom_boxplot(aes(x = season, y = new_deaths_thousand, fill = season)) + theme_classic()
C <- ggplot(dataset) + geom_point(aes(x = new_vaccinations_thousand, y = new_deaths_thousand)) + theme_classic() +
facet_wrap(~season)
D <- ggplot(dataset) + geom_histogram(aes(x = new_deaths_thousand, fill = month)) + facet_wrap(~month)
A
The new daily deaths vary significantly depending on the country. Notice that the United States (USA) has the highest median new daily deaths, but India (IND) has the highest single value of new daily deaths.
B
As expected the Winter and Spring seasons have the highest median values for new daily deaths. However, the Summer season has the highest single value of new daily deaths, which occurred, which occurred in late June.
C
The relationship between new daily vaccinations and new daily deaths is interesting to observe. There are bundles of observations near 0 vaccinations in which there were many deaths, which is due to the lag between vaccine availability and the pandemic. The Spring season has a very unique behavior, as all observations are bundled closely together.
D
Each month has a similar distribution of new daily deaths. Each month has the highest counts of new daily deaths near 0, which is likely due to our data preparation, in which we scaled the new daily deaths.
2.3 Models
Multiple models were used for this analysis. They included forward and backward subset selection, multinomial linear models, random forest, ridge and lasso regression, bagging, and generalized boosted regression models.
2.3.1 Decision Tree
To demonstrate a comparative analysis with regression tree, bagging, random forests, and boosting we constructed a regression tree for new_deaths_thousand on the rest of the predictors and calculated the test prediction error.
require(tree)
tree.deaths = tree(new_deaths_thousand ~.-date, data = train.dat)
cv.deaths = cv.tree(tree.deaths)
plot(cv.deaths$size, cv.deaths$dev, type = 'b')
prune.deaths = prune.tree(tree.deaths, best = 11)
## prediction error
yhat = predict(prune.deaths, newdata = test.dat)
tree_mse <- mean((yhat - test.dat$new_deaths_thousand)^2)
tree_mse## [1] 0.10403132.3.2 Bagged Tree
The main difference between random forest (RF) a and boostrapped aggregated decision tree (Bagged Tree) is that in Bagged Trees, the number of variables selected at each node is not restricted. In RF, the number of variables selected is usually decided using the length of the predictors divided by 3 (for numeric responses). We assume that this model will predict accurately, but has possibility to be inaccurate comparatively due to complexity and over-fitting.
require(randomForest)
# bag.deaths
bag.deaths = randomForest(new_deaths_thousand ~.-date, data = train.dat, mtry = 34, importance = TRUE)
## prediction error
yhat.bag = predict(bag.deaths, newdata = test.dat)
bag_mse <- mean((yhat.bag - test.dat$new_deaths_thousand)^2)
bag_mse## [1] 0.031192072.3.3 Bagged Tree
We assume that this model will predict accurately as well, but has possibility to be inaccurate comparatively for similar reasons.
library(gbm)
boost.deaths = gbm(new_deaths_thousand ~.-date, data = train.dat, distribution = "gaussian", n.trees = 500,
interaction.depth = 3)
summary(boost.deaths)
## var rel.inf
## new_cases_thousand new_cases_thousand 55.02597430
## total_deaths_thousand total_deaths_thousand 8.88267446
## month month 8.48560545
## stringency_index stringency_index 4.26941788
## iso_code iso_code 4.21446789
## total_cases_thousand total_cases_thousand 2.83350436
## people_fully_vaccinated_thousand people_fully_vaccinated_thousand 2.36773427
## population_density population_density 2.22477511
## new_vaccinations_thousand new_vaccinations_thousand 2.13787079
## day_of_week day_of_week 2.04988095
## icu_patients_thousand icu_patients_thousand 1.62896489
## total_tests_per_thousand total_tests_per_thousand 1.26361074
## positive_rate positive_rate 0.85478346
## new_tests_per_thousand new_tests_per_thousand 0.82848913
## total_vaccinations_thousand total_vaccinations_thousand 0.80571454
## hosp_patients_thousand hosp_patients_thousand 0.79957341
## people_vaccinated_thousand people_vaccinated_thousand 0.61423046
## population_thousand population_thousand 0.25941600
## season season 0.09215162
## diabetes_prevalence diabetes_prevalence 0.07611676
## total_boosters_thousand total_boosters_thousand 0.07323791
## median_age median_age 0.05707419
## gdp_per_capita gdp_per_capita 0.04791565
## cardiovasc_death_rate cardiovasc_death_rate 0.03618752
## extreme_poverty extreme_poverty 0.02686811
## hospital_beds_per_thousand hospital_beds_per_thousand 0.02580326
## male_smokers male_smokers 0.01795690
## aged_65_older aged_65_older 0.00000000
## aged_70_older aged_70_older 0.00000000
## female_smokers female_smokers 0.00000000
## handwashing_facilities handwashing_facilities 0.00000000
## life_expectancy life_expectancy 0.00000000
## human_development_index human_development_index 0.00000000
## year year 0.00000000## prediction error
yhat.boost = predict(boost.deaths, newdata = test.dat, n.trees = 500)
boost_mse <- mean((yhat.boost - test.dat$new_deaths_thousand)^2)
boost_mse## [1] 0.030358492.3.4 Random Forest
We assume that this model will predict accurately, but has possibility to be inaccurate comparatively for similar reasons. Our hopes is for this to be more accurate than bagging or boosting due to its composition consisting of aspects of both bagging and boosting. Once again this may actually decrease accuracy due to complexity.
rf <- randomForest(new_deaths_thousand ~.-date, data = train.dat, mtry = (35/3), importance = T)
rf_predictions <- predict(rf, test.dat[-23])
## prediction error
rf_mse <- mean((rf_predictions - test.dat$new_deaths_thousand)^2)
rf_mse## [1] 0.02707872.3.5 Multiple Linear Regression (MLR)
Multiple linear regression was used in attempt to find meaningful results from a less complex model. As a simple and easy to use model it was favorable for quick and easy to interpreted results. A minimum error of 0.164 was achieved through this model type. We continued after to test other models to see if we could produce more accurate results with a lower error. We anticipate this method may possibly have the highest error rate, since the data we are analyzing are composed from time-series data in which the trends with time have been removed. We do not assume this will be as accurate as the more complex models due to the large amount of variables in each model.
model1 = glm(new_deaths_thousand ~ ., data=train.dat)
results <- summary(model1)
## Extracting variables with low p-values from the generalized multiple linear regression
pvals <- data.frame(results$coefficients)
pvals <- filter(pvals, pvals$Pr...t.. < 0.05)
print(rownames(pvals))## [1] "iso_codeCOL" "iso_codeDEU"
## [3] "iso_codeFRA" "iso_codeGBR"
## [5] "iso_codeIND" "iso_codeITA"
## [7] "iso_codeRUS" "iso_codeTUR"
## [9] "iso_codeUSA" "total_tests_per_thousand"
## [11] "new_tests_per_thousand" "positive_rate"
## [13] "stringency_index" "total_cases_thousand"
## [15] "new_cases_thousand" "icu_patients_thousand"
## [17] "hosp_patients_thousand" "total_vaccinations_thousand"
## [19] "people_vaccinated_thousand" "people_fully_vaccinated_thousand"
## [21] "total_boosters_thousand" "total_deaths_thousand"
## [23] "day_of_weekMonday" "day_of_weekSaturday"
## [25] "day_of_weekSunday" "day_of_weekTuesday"
## [27] "day_of_weekWednesday" "month3"
## [29] "month6"A MLR model is built based on the coefficients from the model that have low p-values.
model2 <- glm(new_deaths_thousand ~ iso_code + total_tests_per_thousand + stringency_index +
icu_patients_thousand + total_deaths_thousand + day_of_week +
month + new_tests_per_thousand + total_cases_thousand + hosp_patients_thousand +
people_fully_vaccinated_thousand + positive_rate + total_vaccinations_thousand +
total_boosters_thousand, data = train.dat)
mlr_predictions <- predict(model2, test.dat[-23])
## Prediction Error
mlr_mse <- mean((mlr_predictions - test.dat$new_deaths_thousand)^2)
mlr_mse## [1] 0.1620632.3.6 Forward Selection
require(leaps)
set.seed(101)
regfit.fwd <- regsubsets(new_deaths_thousand~.-date, data = train.dat, nvmax = 60,
method = "forward", really.big = T)## Reordering variables and trying again:fwd.sum <- summary(regfit.fwd)
forward_select=which.min(fwd.sum$bic)
forward_select## [1] 26The forward selection method chooses the following variables (shown below).
coef(regfit.fwd, forward_select)## (Intercept) iso_codeRUS
## -1.274865e+00 -1.264948e+00
## total_tests_per_thousand new_tests_per_thousand
## -1.085664e-04 1.516359e-02
## positive_rate stringency_index
## 2.680906e+00 5.670270e-03
## total_boosters_thousand new_vaccinations_thousand
## -1.713454e-05 -5.083220e-05
## total_deaths_thousand day_of_weekMonday
## 2.291455e-03 -4.388191e-02
## day_of_weekThursday day_of_weekTuesday
## 1.227930e-01 1.293616e-01
## day_of_weekWednesday seasonWinter
## 1.330975e-01 2.167823e-01
## year2021 month2
## -7.177666e-02 -9.431512e-02
## month3 month7
## -3.208461e-02 -1.406207e-01
## month9 month10
## -1.134665e-01 -1.496193e-01
## month12 aged_70_older
## -1.073922e-01 -2.914047e-03
## extreme_poverty cardiovasc_death_rate
## -2.042810e-02 5.654257e-03
## female_smokers hospital_beds_per_thousand
## 2.643003e-02 -7.330337e-02
## month11
## -6.685557e-022.3.7 Backward Selection
set.seed(101)
regfit.bwd <- regsubsets(new_deaths_thousand ~.-date, data = train.dat, nvmax = 60, really.big = T,
method = "backward")## Reordering variables and trying again:bwd.sum <- summary(regfit.bwd)
backward_select <- which.min(bwd.sum$bic)
backward_select## [1] 28Backward selection chooses 26 variables (shown below). Recall that forward selection selected 28 variables.
coef(regfit.bwd, backward_select)## (Intercept) iso_codeCOL
## 4.094570e-01 -1.258029e+00
## iso_codeDEU iso_codeFRA
## -7.865215e-01 -8.891055e-01
## iso_codeGBR iso_codeIND
## -7.712876e-01 -4.114450e-01
## iso_codeITA iso_codeRUS
## -8.897437e-01 -5.280417e-01
## iso_codeTUR iso_codeUSA
## -9.933587e-01 7.344222e-02
## total_tests_per_thousand new_tests_per_thousand
## -7.832633e-05 7.091211e-03
## positive_rate stringency_index
## 3.134842e+00 5.831975e-03
## day_of_weekTuesday day_of_weekWednesday
## 1.161828e-01 1.157296e-01
## seasonWinter year2021
## 5.202780e-02 2.693821e-01
## month2 month3
## -1.802960e-02 -1.195880e-01
## month7 month9
## -8.968866e-02 -3.405191e-02
## month10 month12
## -5.845813e-02 1.539332e-01
## aged_70_older extreme_poverty
## 0.000000e+00 0.000000e+00
## cardiovasc_death_rate female_smokers
## 0.000000e+00 0.000000e+00
## hospital_beds_per_thousand
## 0.000000e+00Comparison of Forward and Backward Selection
par(mfrow = c(1, 2))
plot(fwd.sum$bic,xlab=" Number of Variables ", ylab=" BIC",
type="l", main="Forward Selection: BIC plot")
points (forward_select, fwd.sum$bic[forward_select], col =" red", cex =2, pch =20)
plot(bwd.sum$bic,xlab=" Number of Variables ", ylab=" BIC",
type="l", main="Backward Selection: BIC plot")
points (backward_select, bwd.sum$bic[backward_select], col =" red",cex =2, pch =20)
The above figure reiterates the optimal number of features selected in each method using BIC as a metric.
Forward and Backward Selection MLR Models
Recall that in the list of coefficients, dummy variables for the categorical variables are included, so the actual number of variables used in both models are not 26 (forward selection) nor 28 (backward selection).
forward_mlr <- glm(new_deaths_thousand ~ new_tests_per_thousand + total_boosters_thousand + season + month + aged_70_older + hospital_beds_per_thousand +
iso_code + positive_rate + total_deaths_thousand + year + extreme_poverty + stringency_index + day_of_week + cardiovasc_death_rate +
total_tests_per_thousand + people_vaccinated_thousand + female_smokers, data = train.dat)
backward_mlr <- glm(new_deaths_thousand ~ iso_code + positive_rate + year + month + extreme_poverty + stringency_index + cardiovasc_death_rate +
total_tests_per_thousand + day_of_week + female_smokers + season + aged_70_older + hospital_beds_per_thousand,
data = train.dat)Forward and Backward Selection MLR Test Errors
forward_pred <- predict(forward_mlr, test.dat[-23])
backward_pred <- predict(backward_mlr, test.dat[-23])
sprintf("Forward-Selected Model MSE: %.4f", mean((forward_pred - test.dat$new_deaths_thousand)^2))## [1] "Forward-Selected Model MSE: 0.1940"sprintf("Backward-Selected Model MSE %.4f", mean((backward_pred - test.dat$new_deaths_thousand)^2))## [1] "Backward-Selected Model MSE 0.2029"f_mse <- mean((forward_pred - test.dat$new_deaths_thousand)^2)
b_mse <- mean((backward_pred - test.dat$new_deaths_thousand)^2)2.3.7 Ridge Regression
Recall that ridge regression shrinks the coefficients in the model towards 0, but never to 0. This shrinkage leads to a substantial reduction in the variance of the predictions, with a penalty of slightly increased bias. This can result in a lower MSE when an optimal “λ” is selected. The figure below represents how noise is removed from the model. The number of variables that minimize noise while retaining the predictive power of the model will be chosen based on log lambda on the graph. We believe that this and LASSO may be highly accurate as it shrinks coefficients and may be able to reduce noise better than other models.
require(glmnet)
set.seed(101)
x_train = model.matrix(new_deaths_thousand~.-date, train.dat)[,-1]
y_train <- train.dat$new_deaths_thousand
x_test <- model.matrix(new_deaths_thousand ~.-date, test.dat)[, -1]
y_test <- test.dat$new_deaths_thousand
# train = sample(6720, 5377) # 80% training
# test = (-train)
ridge.model1 <- glmnet(x = x_train,
y = y_train,
alpha = 0, standardize = F)
cv.out <- cv.glmnet(x = x_train,
y = y_train,
alpha = 0, standardize = F)
plot(ridge.model1, xvar = "lambda")
Finding the best Lambda Using Cross-Validation
set.seed(101)
cv.out <- cv.glmnet(x = x_train,
y = y_train,
alpha = 0, standardize = F)
plot(cv.out)
bestlam_ridge <- cv.out$lambda.minAbove we chart the impact of different “λ” parameters and select the best one for our ridge regression model. The Lowest MSE occurs at 8.7548363. Now, the training and testing errors are compared.
ridge_train_preds <- predict(cv.out, x_train, s = "lambda.min")
ridge_test_preds <- predict(ridge.model1, x_test, s = bestlam_ridge)
sprintf("Test MSE: %.4f", mean((ridge_test_preds - y_test)^2))## [1] "Test MSE: 0.2119"sprintf("Train MSE: %.4f", mean((ridge_train_preds - y_train)^2))## [1] "Train MSE: 0.2252"ridge_mse <- mean((ridge_test_preds - y_test)^2)2.3.8 LASSO Regression
Recall that in LASSO, as opposed to ridge, the coefficients are indeed shrunk to 0. As previously mentioned, we believe that this and ridge regression may be highly accurate as it shrinks coefficients and may be able to reduce noise better than other models.
require(glmnet)
set.seed(101)
lasso.model1 <- glmnet(x = x_train,
y = y_train,
alpha = 1, standardize = F)
plot(lasso.model1, xvar = "lambda")
Based on the figure above, it is clear that many of the coefficients shrink to 0, which indicates they are not important to the model. This is explored further below.
set.seed(101)
lasso.cv <- cv.glmnet(x = x_train,
y = y_train,
alpha = 1, standardize = F)
plot(lasso.cv)
best_lasso <- lasso.cv$lambda.minThe ideal lambda value occurs at 1.847081 , as indicated above.
Comparing LASSO training and testing MSE
lasso_train_preds <- predict(lasso.cv, x_train, s = "lambda.min")
lasso_test_preds <- predict(lasso.model1, x_test, s = best_lasso)
sprintf("Test MSE: %.4f", mean((lasso_test_preds - y_test)^2))## [1] "Test MSE: 0.1677"sprintf("Train MSE: %.4f", mean((lasso_train_preds - y_train)^2))## [1] "Train MSE: 0.1832"lasso_mse <- mean((lasso_test_preds - y_test)^2)This is an improvement in both the training and testing errors compared to the results from Ridge regression.
LASSO Regression Most Influential Features
require(broom)
coef(lasso.cv, s = "lambda.min") %>%
tidy() %>%
filter(row != "(Intercept)") %>%
ggplot(aes(value, reorder(row, value))) +
geom_point() +
xlab("Coefficient") +
ylab(NULL)
Notice the top 10 variables that contributed to new deaths.
2.3.8 PCR
Our motivation behind using principal component regression (PCR), is this method helps to avoid multicollinearity. In this data set, it is clear that many of the variables are correlated with each other. We assume this will perform better than that of LASSO or ridge regression using the principal component of the variables with assumptions for the best bias variance trade-off model.
require(pls)
set.seed(101)
pcr_model <- pls::pcr(new_deaths_thousand~., data = no_date, scale = F, validation = "CV")
validationplot(pcr_model, val.type = "MSEP")
validationplot(pcr_model, val.type = "R2")
The first figure above represents MSE for each component derived using principal component analysis (PCA). The second figure displays the \(R^2\) score for the number of components. Observe that dimensionality reduction has occured, the number of components (15) that explain about 80% of the variability is less than the number of predictors in the data set.
Below displays the predicted vs actual deaths based on a trained PCR model using the same training and testing split data sets that were used in all the other models.
PCR with 15 Compoonents
set.seed(101)
pcr_trained <- pcr(new_deaths_thousand~.-date, data = train.dat, scale = F, validation = "CV")
pcr_preds <- predict(pcr_trained, test.dat[-23], ncomp = 15)
pcr_mse <- mean((pcr_preds - test.dat$new_deaths_thousand)^2)3. Results
results_tab <- data.frame(Method = c("Decision Tree", "Bagged Tree",
"Boosted Tree", "Random Forest",
"MLR", "MLR - Forward", "MLR-Backward",
"LASSO", "Ridge", "PCR"),
MSE = c(tree_mse, bag_mse, boost_mse,
rf_mse, mlr_mse, f_mse, b_mse,
lasso_mse, ridge_mse, pcr_mse))
knitr::kable(results_tab)| Method | MSE |
|---|---|
| Decision Tree | 0.1040313 |
| Bagged Tree | 0.0311921 |
| Boosted Tree | 0.0303585 |
| Random Forest | 0.0270787 |
| MLR | 0.1620630 |
| MLR - Forward | 0.1940056 |
| MLR-Backward | 0.2029158 |
| LASSO | 0.1676736 |
| Ridge | 0.2119213 |
| PCR | 0.1245515 |
After each model was tested final results shown above were compared. As previously predicted, MLR models had the highest error rate and were not considered for use. Ridge regression and LASSO showed similar results with a higher MSE. PCR was on the lower end of the models compared, but did not compare to the tree based models. The general decision tree, although not faring well, did prove to have a lower MSE than all other non-tree based models. Boosting and bagging methods were applied to try and reduce error rate further, but Random Forest proved to yield the lower MSE. Bagged tree lagged close behind by a difference in MSE of 0.0006745. Boosting however performed even worse than bagging.
4. Discussion
There were other potential problems we could have analyzed, but this one was selected due to the relevancy for current issues as well as ease of understanding aspects of the data. There was little interpretation needed to understand initial data and additionally little cleaning and manipulation of data due to succinct data maintenance from our source. We were able to utilize most all of the skills we learned this semester in this project. Additional methods such as historical clustering and PCA were attempted, but this lead to computational issues and visualizations so large that they were unable to be interpreted. These would have been supplementary to our analysis, and therefore were unused. Each group member contributed equally to this work. Tasks such as data extraction, coding of methods, and compilation of this document were divided equally among members, and assigned based on each members specialized skills. It was an immensely helpful learning experience to get to show what we have learned this semester in one project.
5. Future Work
The models created in this project are applicable to other results as well from the same dataset. We chose new_deaths as the response variable for our models, but others can be easily interchanged and compared for different results. Note that doing so may yield different results for which model is most accurate. Other alternative responses possible could be new cases, new vaccinations, or other COVID related responses. On the other hand, our methodologies would likely not produce accurate results for something such as population, since the data is geared towards COVID information. Other models, such as (SVM) support vector machines which not been covered, could be applied for different results as well and compared to see if error rate is higher or lower than the current results. Further time and information could also yield usable results with PCA and clustering methods attempted.
6. Appendix
- A. Liaw and M. Wiener (2002). Classification and Regression by randomForest. R News 2(3), 18–22.
- Achim Zeileis and Gabor Grothendieck (2005). zoo: S3 Infrastructure for Regular and Irregular Time Series. Journal of Statistical Software, 14(6), 1-27. doi:10.18637/jss.v014.i06
- Alboukadel Kassambara (2020). ggpubr: ‘ggplot2’ Based Publication Ready Plots. R package version 0.4.0. https://CRAN.R-project.org/package=ggpubr
- Aravind Hebbali (2020). olsrr: Tools for Building OLS Regression Models. R package version 0.5.3. https://CRAN.R-project.org/package=olsrr
- Brandon Greenwell, Bradley Boehmke, Jay Cunningham and GBM Developers (2020). gbm: Generalized Boosted Regression Models. R package version 2.1.8. https://CRAN.R-project.org/package=gbm
- Brian Ripley (2021). tree: Classification and Regression Trees. R package version 1.0-41. https://CRAN.R-project.org/package=tree
- David Robinson, Alex Hayes and Simon Couch (2021). broom: Convert Statistical Objects into Tidy Tibbles. R package version 0.7.9. https://CRAN.R-project.org/package=broom
- Douglas Bates and Martin Maechler (2019). Matrix: Sparse and Dense Matrix Classes and Methods. R package version 1.2-18. https://CRAN.R-project.org/package=Matrix
- H. Wickham. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2016.
- Hadley Wickham (2019). stringr: Simple, Consistent Wrappers for Common String Operations. R package version 1.4.0. https://CRAN.R-project.org/package=stringr
- Hadley Wickham, Romain François, Lionel Henry and Kirill Müller (2021). dplyr: A Grammar of Data Manipulation. R package version 1.0.7. https://CRAN.R-project.org/package=dplyr
- Jarek Tuszynski (2021). caTools: Tools: Moving Window Statistics, GIF, Base64, ROC AUC, etc. R package version 1.18.2. https://CRAN.R-project.org/package=caTools
- Jerome Friedman, Trevor Hastie, Robert Tibshirani (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1-22. URL https://www.jstatsoft.org/v33/i01/.
- Julia Silge, Fanny Chow, Max Kuhn and Hadley Wickham (2021). rsample: General Resampling Infrastructure. R package version 0.1.1. https://CRAN.R-project.org/package=rsample
- Kristian Hovde Liland, Bjørn-Helge Mevik and Ron Wehrens (2021). pls: Partial Least Squares and Principal Component Regression. R package version 2.8-0. https://CRAN.R-project.org/package=pls
- Max Kuhn (2021). caret: Classification and Regression Training. R package version 6.0-88. https://CRAN.R-project.org/package=caret
- Ponce et al. (2021). covid19.analytics: An R Package to Obtain, Analyze and Visualize Data from the Coronavirus Disease Pandemic. Journal of Open Source Software, 6(59), 2995. https://doi.org/10.21105/joss.02995
- R Core Team (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.
- Sarkar, Deepayan (2008) Lattice: Multivariate Data Visualization with R. Springer, New York. ISBN 978-0-387-75968-5
- Thomas Lumley based on Fortran code by Alan Miller (2020). leaps: Regression Subset Selection. R package version 3.1. https://CRAN.R-project.org/package=leaps
6.1 Web App Code
require(ggplot2)
require(ggpubr)
res_dataset <- data.frame(date = test.dat$date, actual_deaths = test.dat$new_deaths_thousand,
decision_tree_pred = yhat,
bagged_tree_pred = yhat.bag,
boosted_tree_pred = yhat.boost,
rf_predicted_cases = rf_predictions,
MLR = mlr_predictions,
Ridge = ridge_test_preds,
LASSO = lasso_test_preds,
ForwardMLR = forward_pred,
BackwardMLR = backward_pred,
country = test.dat$iso_code)
# save(res_dataset, file = "shiny/rf_results.RData")
# res_dataset
res_dataset## date actual_deaths decision_tree_pred bagged_tree_pred
## 1 2020-02-26 0.000 0.1075004 -1.143530e-17
## 21 2020-03-17 0.001 0.1075004 1.019267e-03
## 27 2020-03-23 0.009 0.1075004 1.285657e-02
## 29 2020-03-25 0.013 0.1075004 1.508700e-02
## 43 2020-04-08 0.133 0.1075004 1.494007e-01
## 45 2020-04-10 0.107 0.1075004 1.052728e-01
## 60 2020-04-25 0.353 0.1075004 3.317895e-01
## 65 2020-04-30 0.493 0.1075004 3.280617e-01
## 66 2020-05-01 0.406 0.1075004 3.439939e-01
## 71 2020-05-06 0.650 0.1075004 7.166098e-01
## 80 2020-05-15 0.963 1.4568194 9.405024e-01
## 87 2020-05-22 1.001 1.4568194 1.024589e+00
## 88 2020-05-23 0.965 1.4568194 8.774607e-01
## 103 2020-06-07 0.525 1.4568194 6.456220e-01
## 107 2020-06-11 1.239 1.4568194 1.233912e+00
## 108 2020-06-12 0.909 1.4568194 9.896673e-01
## 110 2020-06-14 0.612 1.4568194 6.461896e-01
## 122 2020-06-26 0.990 1.4568194 1.141935e+00
## 134 2020-07-08 1.223 0.8035981 1.149791e+00
## 135 2020-07-09 1.220 0.8035981 1.241797e+00
## 141 2020-07-15 1.233 0.8035981 1.147353e+00
## 142 2020-07-16 1.322 0.8035981 1.139333e+00
## 151 2020-07-25 1.211 0.8035981 1.089331e+00
## 154 2020-07-28 0.921 0.8035981 1.200933e+00
## 155 2020-07-29 1.595 0.9053733 1.356927e+00
## 157 2020-07-31 1.212 0.8035981 1.160550e+00
## 163 2020-08-06 1.237 0.8035981 1.159327e+00
## 172 2020-08-15 0.709 0.8035981 9.501968e-01
## 174 2020-08-17 0.684 0.8035981 5.723898e-01
## 177 2020-08-20 1.204 0.8035981 1.071275e+00
## 190 2020-09-02 1.184 0.8035981 9.987108e-01
## 192 2020-09-04 0.888 0.8035981 8.978090e-01
## 194 2020-09-06 0.447 0.1075004 3.746623e-01
## 195 2020-09-07 0.310 0.1075004 3.085752e-01
## 196 2020-09-08 0.504 0.1075004 4.205177e-01
## 205 2020-09-17 0.829 0.8035981 9.079322e-01
## 206 2020-09-18 0.858 0.8035981 8.772469e-01
## 211 2020-09-23 0.000 0.1075004 2.759227e-02
## 231 2020-10-13 0.309 0.1075004 2.853351e-01
## 233 2020-10-15 0.713 0.8035981 7.170140e-01
## 236 2020-10-18 0.000 0.1075004 1.483350e-02
## 239 2020-10-21 0.566 0.8035981 5.713683e-01
## 241 2020-10-23 0.571 0.8035981 7.386647e-01
## 245 2020-10-27 0.549 0.8035981 7.329810e-01
## 253 2020-11-04 0.610 0.8035981 3.766181e-01
## 255 2020-11-06 0.909 0.8035981 6.850152e-01
## 267 2020-11-18 0.756 0.8035981 6.969716e-01
## 270 2020-11-21 0.376 0.8035981 7.461439e-01
## 272 2020-11-23 0.302 0.8035981 2.594202e-01
## 273 2020-11-24 0.630 0.8035981 6.056001e-01
## 274 2020-11-25 0.654 0.8035981 6.881910e-01
## 276 2020-11-27 0.514 0.8035981 7.029637e-01
## 277 2020-11-28 0.587 0.8035981 8.739747e-01
## 284 2020-12-05 0.664 0.8035981 7.270517e-01
## 300 2020-12-21 0.527 0.8035981 4.114004e-01
## 301 2020-12-22 0.968 0.8035981 9.600528e-01
## 305 2020-12-26 0.307 0.8035981 4.409463e-01
## 319 2021-01-09 1.171 0.8035981 1.099778e+00
## 320 2021-01-10 0.469 0.8035981 6.006907e-01
## 321 2021-01-11 0.480 0.8035981 4.808391e-01
## 324 2021-01-14 1.131 2.3296739 1.525737e+00
## 343 2021-02-02 1.210 0.8035981 1.239181e+00
## 349 2021-02-08 0.000 0.1075004 1.329723e-02
## 350 2021-02-09 1.986 2.3296739 1.565155e+00
## 356 2021-02-15 0.528 0.8035981 8.580516e-01
## 375 2021-03-06 1.555 2.4070561 1.827249e+00
## 376 2021-03-07 1.086 2.4070561 1.678921e+00
## 378 2021-03-09 1.972 2.4070561 2.038162e+00
## 379 2021-03-10 2.286 2.4070561 2.087018e+00
## 381 2021-03-12 2.216 2.4070561 2.114480e+00
## 395 2021-03-26 3.650 2.4070561 3.083084e+00
## 405 2021-04-05 1.319 0.8035981 1.228952e+00
## 406 2021-04-06 4.195 2.4070561 3.717434e+00
## 409 2021-04-09 3.693 2.4070561 3.611454e+00
## 414 2021-04-14 3.459 2.4070561 3.492130e+00
## 425 2021-04-25 1.305 0.8035981 1.292530e+00
## 433 2021-05-03 0.983 0.8035981 7.937491e-01
## 435 2021-05-05 2.811 2.3296739 2.645239e+00
## 439 2021-05-09 1.024 0.8035981 9.447275e-01
## 445 2021-05-15 2.087 2.3296739 2.092445e+00
## 447 2021-05-17 0.786 0.8035981 9.034015e-01
## 458 2021-05-28 2.371 0.8035981 1.297818e+00
## 463 2021-06-02 2.507 3.6000806 2.762674e+00
## 468 2021-06-07 1.010 0.8035981 9.629899e-01
## 469 2021-06-08 2.378 0.8035981 1.535741e+00
## 484 2021-06-23 2.392 3.6000806 2.652773e+00
## 486 2021-06-25 2.001 2.3296739 2.086825e+00
## 495 2021-07-04 0.830 0.8035981 7.474744e-01
## 499 2021-07-08 1.639 0.8035981 1.537107e+00
## 502 2021-07-11 0.595 0.8035981 6.167046e-01
## 504 2021-07-13 1.605 0.8035981 1.397327e+00
## 505 2021-07-14 1.556 0.8035981 1.457750e+00
## 513 2021-07-22 1.412 0.8035981 1.339016e+00
## 514 2021-07-23 1.324 1.3909884 1.961286e+00
## 517 2021-07-26 0.578 0.8035981 6.426231e-01
## 519 2021-07-28 1.344 0.8035981 1.298936e+00
## 520 2021-07-29 1.318 0.8035981 1.196549e+00
## 521 2021-07-30 0.963 0.8035981 1.156420e+00
## 524 2021-08-02 0.389 0.1075004 4.432611e-01
## 534 2021-08-12 1.148 0.8035981 1.065460e+00
## 543 2021-08-21 0.698 0.8035981 8.821170e-01
## 552 2021-08-30 0.266 0.1075004 3.097336e-01
## 575 2021-09-22 0.876 0.8035981 7.806122e-01
## 582 2021-09-29 0.676 0.8035981 6.372190e-01
## 596 2021-10-13 0.176 0.1075004 2.656678e-01
## 597 2021-10-14 0.525 0.1075004 4.666528e-01
## 599 2021-10-16 0.483 0.1075004 3.539170e-01
## 604 2021-10-21 0.451 0.8035981 4.529890e-01
## 606 2021-10-23 0.318 0.1075004 3.666513e-01
## 608 2021-10-25 0.160 0.1075004 1.649481e-01
## 619 2021-11-05 0.389 0.1075004 3.756171e-01
## 624 2021-11-10 0.280 0.1075004 3.236998e-01
## 628 2021-11-14 0.061 0.1075004 9.131047e-02
## 635 2021-11-21 0.072 0.1075004 9.746640e-02
## 636 2021-11-22 0.123 0.1075004 2.776155e-01
## 637 2021-11-23 0.284 0.1075004 2.639607e-01
## 645 2021-12-01 0.283 0.1075004 2.825900e-01
## 647 2021-12-03 0.221 0.1075004 2.903081e-01
## 651 2021-12-07 0.274 0.1075004 2.584654e-01
## 654 2021-12-10 0.000 0.1075004 1.290711e-01
## 659 2020-03-08 0.000 0.1075004 3.220000e-04
## 670 2020-03-19 0.000 0.1075004 -1.600000e-06
## 672 2020-03-21 0.000 0.1075004 2.863333e-05
## 674 2020-03-23 0.001 0.1075004 1.356600e-03
## 678 2020-03-27 0.000 0.1075004 2.664100e-03
## 679 2020-03-28 0.000 0.1075004 3.201733e-03
## 689 2020-04-07 0.004 0.1075004 9.651567e-03
## 697 2020-04-15 0.004 0.1075004 1.094717e-02
## 705 2020-04-23 0.009 0.1075004 1.214927e-02
## 707 2020-04-25 0.008 0.1075004 1.263613e-02
## 708 2020-04-26 0.011 0.1075004 1.150210e-02
## 721 2020-05-09 0.017 0.1075004 1.581780e-02
## 723 2020-05-11 0.016 0.1075004 1.790467e-02
## 727 2020-05-15 0.021 0.1075004 2.008627e-02
## 729 2020-05-17 0.012 0.1075004 1.893063e-02
## 740 2020-05-28 0.019 0.1075004 3.345960e-02
## 742 2020-05-30 0.037 0.1075004 4.054403e-02
## 746 2020-06-03 0.036 0.1075004 5.074943e-02
## 751 2020-06-08 0.049 0.1075004 5.254847e-02
## 768 2020-06-25 0.163 0.1075004 1.253376e-01
## 774 2020-07-01 0.136 0.1075004 1.637666e-01
## 776 2020-07-03 0.136 0.1075004 1.541165e-01
## 777 2020-07-04 0.165 0.1075004 1.567269e-01
## 789 2020-07-16 0.215 0.1075004 2.689389e-01
## 795 2020-07-22 0.207 0.1075004 2.722396e-01
## 799 2020-07-26 0.256 0.1075004 2.947862e-01
## 802 2020-07-29 0.380 0.1075004 3.122654e-01
## 809 2020-08-05 0.309 0.1075004 3.336206e-01
## 812 2020-08-08 0.290 0.1075004 3.120647e-01
## 814 2020-08-10 0.312 0.1075004 3.195226e-01
## 819 2020-08-15 0.318 0.1075004 3.108111e-01
## 826 2020-08-22 0.400 0.1075004 3.203328e-01
## 831 2020-08-27 0.283 0.1075004 2.927406e-01
## 834 2020-08-30 0.300 0.1075004 2.981201e-01
## 838 2020-09-03 0.273 0.1075004 2.707961e-01
## 846 2020-09-11 0.243 0.1075004 2.303488e-01
## 847 2020-09-12 0.216 0.1075004 2.023215e-01
## 848 2020-09-13 0.190 0.1075004 2.177710e-01
## 851 2020-09-16 0.190 0.1075004 2.611305e-01
## 858 2020-09-23 0.349 0.1075004 1.549166e-01
## 859 2020-09-24 0.000 0.1075004 1.331951e-01
## 862 2020-09-27 0.192 0.1075004 1.583395e-01
## 864 2020-09-29 0.187 0.1075004 1.603368e-01
## 866 2020-10-01 0.198 0.1075004 1.767738e-01
## 868 2020-10-03 0.159 0.1075004 1.692243e-01
## 870 2020-10-05 0.132 0.1075004 1.480973e-01
## 876 2020-10-11 0.174 0.1075004 1.577459e-01
## 891 2020-10-26 0.194 0.1075004 1.694415e-01
## 892 2020-10-27 0.217 0.1075004 1.762970e-01
## 897 2020-11-01 0.201 0.1075004 1.787461e-01
## 911 2020-11-15 0.202 0.1075004 1.745441e-01
## 916 2020-11-20 0.168 0.1075004 1.840082e-01
## 917 2020-11-21 0.175 0.1075004 1.828145e-01
## 918 2020-11-22 0.183 0.1075004 1.818912e-01
## 919 2020-11-23 0.192 0.1075004 1.762781e-01
## 935 2020-12-09 0.150 0.1075004 1.888991e-01
## 936 2020-12-10 0.176 0.1075004 1.791708e-01
## 938 2020-12-12 0.197 0.1075004 1.736397e-01
## 946 2020-12-20 0.207 0.1075004 2.202041e-01
## 947 2020-12-21 0.205 0.1075004 2.175483e-01
## 949 2020-12-23 0.243 0.1075004 2.581864e-01
## 950 2020-12-24 0.280 0.1075004 2.575597e-01
## 961 2021-01-04 0.222 0.1075004 2.667969e-01
## 969 2021-01-12 0.331 0.1075004 3.817396e-01
## 972 2021-01-15 0.377 0.7794346 4.008797e-01
## 975 2021-01-18 0.373 0.1075004 3.625332e-01
## 984 2021-01-27 0.395 0.1075004 3.727989e-01
## 989 2021-02-01 0.289 0.1075004 3.050404e-01
## 992 2021-02-04 0.254 0.1075004 3.083418e-01
## 999 2021-02-11 0.250 0.1075004 2.358912e-01
## 1004 2021-02-16 0.163 0.1075004 1.980114e-01
## 1011 2021-02-23 0.144 0.1075004 1.471249e-01
## 1014 2021-02-26 0.122 0.1075004 1.417631e-01
## 1020 2021-03-04 0.107 0.1075004 1.067149e-01
## 1025 2021-03-09 0.078 0.1075004 1.048418e-01
## 1028 2021-03-12 0.092 0.1075004 1.040723e-01
## 1031 2021-03-15 0.100 0.1075004 9.471083e-02
## 1033 2021-03-17 0.130 0.1075004 1.196421e-01
## 1038 2021-03-22 0.120 0.1075004 1.275122e-01
## 1041 2021-03-25 0.125 0.1075004 1.433908e-01
## 1055 2021-04-08 0.247 0.1075004 2.966141e-01
## 1061 2021-04-14 0.337 0.2831279 3.656677e-01
## 1069 2021-04-22 0.430 0.2831279 4.309404e-01
## 1072 2021-04-25 0.465 0.2831279 4.115833e-01
## 1077 2021-04-30 0.490 0.2831279 4.349528e-01
## 1078 2021-05-01 0.272 0.2831279 4.644064e-01
## 1086 2021-05-09 0.495 0.2831279 4.629048e-01
## 1091 2021-05-14 0.490 0.2831279 4.708786e-01
## 1092 2021-05-15 0.530 0.2831279 4.981503e-01
## 1094 2021-05-17 0.509 0.1075004 4.827246e-01
## 1100 2021-05-23 0.496 0.2831279 5.105458e-01
## 1102 2021-05-25 0.459 0.2831279 5.042518e-01
## 1103 2021-05-26 0.514 0.2831279 5.134704e-01
## 1110 2021-06-02 0.511 0.2831279 5.447739e-01
## 1113 2021-06-05 0.532 0.2831279 5.436618e-01
## 1114 2021-06-06 0.539 0.2831279 5.336816e-01
## 1121 2021-06-13 0.586 0.2831279 5.787325e-01
## 1123 2021-06-15 0.599 0.2831279 5.684595e-01
## 1128 2021-06-20 0.599 0.2831279 5.870690e-01
## 1129 2021-06-21 0.648 0.2831279 6.132949e-01
## 1131 2021-06-23 0.645 0.2831279 6.461671e-01
## 1132 2021-06-24 0.689 0.2831279 6.673816e-01
## 1141 2021-07-03 0.591 0.2831279 5.761695e-01
## 1147 2021-07-09 0.576 0.2831279 5.525581e-01
## 1164 2021-07-26 0.314 0.1075004 3.080973e-01
## 1181 2021-08-12 0.144 0.1075004 1.464301e-01
## 1186 2021-08-17 0.108 0.1075004 1.209776e-01
## 1188 2021-08-19 0.120 0.1075004 1.184989e-01
## 1201 2021-09-01 0.071 0.1075004 6.702683e-02
## 1203 2021-09-03 0.061 0.1075004 6.406310e-02
## 1206 2021-09-06 0.053 0.1075004 4.058560e-02
## 1209 2021-09-09 0.053 0.1075004 5.656873e-02
## 1221 2021-09-21 0.038 0.1075004 3.210697e-02
## 1222 2021-09-22 0.044 0.1075004 3.453903e-02
## 1228 2021-09-28 0.041 0.1075004 3.593383e-02
## 1230 2021-09-30 0.038 0.1075004 3.237953e-02
## 1236 2021-10-06 0.032 0.1075004 3.084003e-02
## 1239 2021-10-09 0.033 0.1075004 3.596790e-02
## 1243 2021-10-13 0.034 0.1075004 3.120880e-02
## 1257 2021-10-27 0.026 0.1075004 3.228880e-02
## 1260 2021-10-30 0.033 0.1075004 3.571503e-02
## 1261 2021-10-31 0.023 0.1075004 3.650503e-02
## 1268 2021-11-07 0.045 0.1075004 3.746970e-02
## 1272 2021-11-11 0.040 0.1075004 4.233687e-02
## 1278 2021-11-17 0.047 0.1075004 3.987097e-02
## 1284 2021-11-23 0.050 0.1075004 4.743817e-02
## 1292 2021-12-01 0.058 0.1075004 4.203527e-02
## 1307 2020-01-27 0.000 0.1075004 -2.775558e-17
## 1311 2020-01-31 0.000 0.1075004 -3.352874e-17
## 1313 2020-02-02 0.000 0.1075004 -3.275158e-17
## 1314 2020-02-03 0.000 0.1075004 -3.397282e-17
## 1320 2020-02-09 0.000 0.1075004 -3.264056e-17
## 1327 2020-02-16 0.000 0.1075004 3.733333e-05
## 1332 2020-02-21 0.000 0.1075004 1.794333e-04
## 1336 2020-02-25 0.000 0.1075004 1.876000e-04
## 1337 2020-02-26 0.001 0.1075004 8.447333e-04
## 1340 2020-02-29 0.000 0.1075004 3.852333e-04
## 1348 2020-03-08 0.003 0.1075004 4.392667e-03
## 1353 2020-03-13 0.031 0.1075004 1.429613e-02
## 1356 2020-03-16 0.058 0.1075004 2.514177e-02
## 1368 2020-03-28 0.318 0.3836605 4.711112e-01
## 1373 2020-04-02 0.609 0.3836605 8.726719e-01
## 1383 2020-04-12 0.561 0.7794346 5.067348e-01
## 1385 2020-04-14 0.743 0.7794346 5.828823e-01
## 1386 2020-04-15 1.438 0.3836605 6.851814e-01
## 1400 2020-04-29 0.426 0.3836605 3.197367e-01
## 1410 2020-05-09 0.080 0.3836605 2.121807e-01
## 1412 2020-05-11 0.263 0.3836605 2.052425e-01
## 1415 2020-05-14 0.351 0.3836605 1.601466e-01
## 1424 2020-05-23 0.086 0.1075004 1.009292e-01
## 1435 2020-06-03 0.081 0.1075004 3.849803e-02
## 1437 2020-06-05 0.045 0.1075004 4.768070e-02
## 1438 2020-06-06 0.031 0.1075004 3.348127e-02
## 1444 2020-06-12 0.028 0.1075004 4.574630e-02
## 1446 2020-06-14 0.010 0.1075004 3.014643e-02
## 1452 2020-06-20 0.018 0.1075004 2.620987e-02
## 1456 2020-06-24 0.013 0.1075004 2.995770e-02
## 1457 2020-06-25 0.019 0.1075004 3.120180e-02
## 1459 2020-06-27 0.001 0.1075004 3.456363e-02
## 1463 2020-07-01 0.019 0.1075004 4.596150e-02
## 1469 2020-07-07 0.011 0.1075004 1.647360e-02
## 1470 2020-07-08 0.031 0.1075004 1.913157e-02
## 1473 2020-07-11 0.003 0.1075004 4.118667e-03
## 1477 2020-07-15 0.088 0.1075004 1.521370e-02
## 1492 2020-07-30 0.016 0.1075004 1.189490e-02
## 1494 2020-08-01 0.000 0.1075004 3.582567e-03
## 1513 2020-08-20 0.011 0.1075004 1.921320e-02
## 1514 2020-08-21 0.024 0.1075004 1.834687e-02
## 1520 2020-08-27 0.035 0.1075004 1.978567e-02
## 1532 2020-09-08 0.038 0.1075004 3.295793e-02
## 1533 2020-09-09 0.032 0.1075004 3.058937e-02
## 1535 2020-09-11 0.082 0.1075004 2.629157e-02
## 1540 2020-09-16 0.048 0.1075004 6.118870e-02
## 1541 2020-09-17 0.047 0.1075004 6.649590e-02
## 1553 2020-09-29 0.073 0.1075004 7.423967e-02
## 1555 2020-10-01 0.057 0.1075004 8.323047e-02
## 1559 2020-10-05 0.071 0.1075004 9.428340e-02
## 1561 2020-10-07 0.080 0.2831279 9.947917e-02
## 1567 2020-10-13 0.128 0.1075004 1.003734e-01
## 1570 2020-10-16 0.179 0.2831279 1.428967e-01
## 1579 2020-10-25 -0.021 0.3836605 2.134536e-01
## 1581 2020-10-27 0.528 0.2831279 3.554559e-01
## 1583 2020-10-29 0.233 0.2831279 2.771181e-01
## 1587 2020-11-02 0.428 0.9053733 8.435181e-01
## 1593 2020-11-08 0.270 0.2831279 3.601582e-01
## 1595 2020-11-10 1.231 0.2831279 6.745993e-01
## 1598 2020-11-13 0.933 0.2831279 6.499101e-01
## 1607 2020-11-22 0.214 0.3836605 3.619335e-01
## 1610 2020-11-25 0.388 0.2831279 4.671368e-01
## 1619 2020-12-04 0.630 0.3836605 5.826760e-01
## 1621 2020-12-06 0.174 0.3836605 1.929876e-01
## 1622 2020-12-07 0.366 0.3836605 3.702991e-01
## 1629 2020-12-14 0.376 0.3836605 3.793095e-01
## 1634 2020-12-19 0.189 0.2831279 2.883299e-01
## 1639 2020-12-24 0.291 0.2831279 3.692896e-01
## 1647 2021-01-01 0.133 0.2831279 4.100030e-01
## 1652 2021-01-06 0.282 0.2831279 2.883595e-01
## 1668 2021-01-22 0.649 0.2831279 6.017869e-01
## 1676 2021-01-30 0.241 0.2831279 2.146078e-01
## 1689 2021-02-12 0.645 0.2831279 5.672022e-01
## 1691 2021-02-14 0.167 0.2831279 1.662540e-01
## 1694 2021-02-17 0.310 0.2831279 3.200135e-01
## 1700 2021-02-23 0.431 0.2831279 5.478428e-01
## 1704 2021-02-27 0.185 0.2831279 1.769355e-01
## 1707 2021-03-02 0.418 0.2831279 4.111476e-01
## 1712 2021-03-07 0.130 0.2831279 1.535037e-01
## 1716 2021-03-11 0.265 0.2831279 3.003620e-01
## 1719 2021-03-14 0.141 0.2831279 1.544636e-01
## 1745 2021-04-09 0.331 0.3836605 3.503965e-01
## 1749 2021-04-13 0.345 0.2831279 3.954019e-01
## 1754 2021-04-18 0.140 0.2831279 1.775911e-01
## 1760 2021-04-24 0.217 0.2831279 1.999315e-01
## 1764 2021-04-28 0.315 0.2831279 2.961765e-01
## 1765 2021-04-29 0.306 0.2831279 2.875741e-01
## 1774 2021-05-08 0.176 0.2831279 1.687529e-01
## 1787 2021-05-21 0.123 0.3836605 1.637136e-01
## 1789 2021-05-23 0.071 0.3836605 9.823547e-02
## 1790 2021-05-24 0.062 0.3836605 1.617899e-01
## 1796 2021-05-30 0.044 0.3836605 9.249287e-02
## 1797 2021-05-31 0.126 0.3836605 9.554700e-02
## 1805 2021-06-08 0.075 0.3836605 9.371877e-02
## 1808 2021-06-11 0.074 0.3836605 8.297953e-02
## 1814 2021-06-17 0.056 0.1075004 6.227043e-02
## 1818 2021-06-21 0.040 0.1075004 5.421477e-02
## 1822 2021-06-25 0.033 0.1075004 4.227603e-02
## 1829 2021-07-02 0.024 0.1075004 3.677123e-02
## 1830 2021-07-03 0.017 0.1075004 2.853007e-02
## 1831 2021-07-04 0.011 0.1075004 2.354410e-02
## 1840 2021-07-13 0.054 0.1075004 3.427917e-02
## 1847 2021-07-20 0.033 0.2831279 3.407403e-02
## 1851 2021-07-24 0.022 0.2831279 3.951410e-02
## 1854 2021-07-27 0.029 0.2831279 4.286750e-02
## 1865 2021-08-07 0.032 0.2831279 5.346550e-02
## 1867 2021-08-09 0.084 0.1075004 9.660930e-02
## 1868 2021-08-10 0.080 0.2831279 7.170497e-02
## 1875 2021-08-17 0.124 0.2831279 1.220783e-01
## 1888 2021-08-30 0.151 0.3836605 1.396664e-01
## 1894 2021-09-05 0.049 0.3836605 6.434543e-02
## 1898 2021-09-09 0.095 0.2831279 1.116123e-01
## 1903 2021-09-14 0.209 0.1075004 1.011055e-01
## 1904 2021-09-15 0.016 0.1075004 9.275480e-02
## 1911 2021-09-22 0.080 0.1075004 8.610793e-02
## 1916 2021-09-27 0.099 0.1075004 7.314377e-02
## 1920 2021-10-01 0.061 0.1075004 5.429057e-02
## 1924 2021-10-05 0.071 0.1075004 5.595757e-02
## 1932 2021-10-13 0.053 0.1075004 4.639697e-02
## 1950 2021-10-31 0.013 0.1075004 1.391670e-02
## 1966 2021-11-16 0.048 0.2831279 6.345133e-02
## 1983 2021-12-03 0.129 0.2831279 1.117933e-01
## 1995 2020-01-29 0.000 0.1075004 -2.409184e-17
## 2000 2020-02-03 0.000 0.1075004 -2.864375e-17
## 2002 2020-02-05 0.000 0.1075004 -3.075318e-17
## 2006 2020-02-09 0.000 0.1075004 -2.897682e-17
## 2010 2020-02-13 0.000 0.1075004 -3.608225e-17
## 2014 2020-02-17 0.000 0.1075004 -3.719247e-17
## 2020 2020-02-23 0.000 0.1075004 -3.563816e-17
## 2024 2020-02-27 0.000 0.1075004 1.450000e-05
## 2027 2020-03-01 0.000 0.1075004 1.460000e-05
## 2037 2020-03-11 0.001 0.1075004 2.796767e-03
## 2044 2020-03-18 0.004 0.1075004 3.565630e-02
## 2051 2020-03-25 0.049 0.1075004 6.289020e-02
## 2063 2020-04-06 0.226 0.3836605 2.303950e-01
## 2068 2020-04-11 -0.031 0.3836605 1.917992e-01
## 2077 2020-04-20 0.276 0.3836605 1.707440e-01
## 2088 2020-05-01 0.113 0.3836605 1.511970e-01
## 2100 2020-05-13 0.123 0.1075004 7.435207e-02
## 2107 2020-05-20 0.063 0.1075004 5.579840e-02
## 2115 2020-05-28 0.042 0.1075004 4.616867e-02
## 2116 2020-05-29 0.034 0.1075004 3.028107e-02
## 2118 2020-05-31 0.010 0.1075004 2.291733e-02
## 2122 2020-06-04 0.033 0.1075004 2.774523e-02
## 2130 2020-06-12 0.011 0.1075004 1.822787e-02
## 2142 2020-06-24 0.014 0.1075004 1.540427e-02
## 2144 2020-06-26 0.025 0.1075004 1.273743e-02
## 2149 2020-07-01 0.005 0.1075004 9.586233e-03
## 2152 2020-07-04 0.010 0.1075004 7.474833e-03
## 2153 2020-07-05 0.003 0.1075004 5.821333e-03
## 2155 2020-07-07 0.010 0.1075004 8.337033e-03
## 2158 2020-07-10 0.006 0.1075004 8.990667e-03
## 2160 2020-07-12 0.001 0.1075004 4.848933e-03
## 2162 2020-07-14 0.004 0.1075004 5.964033e-03
## 2172 2020-07-24 0.010 0.1075004 4.266233e-03
## 2193 2020-08-14 0.013 0.1075004 5.643267e-03
## 2195 2020-08-16 0.000 0.1075004 2.381933e-03
## 2196 2020-08-17 0.001 0.1075004 2.738000e-03
## 2199 2020-08-20 0.014 0.1075004 5.300800e-03
## 2205 2020-08-26 0.004 0.1075004 7.631800e-03
## 2207 2020-08-28 0.000 0.1075004 5.887767e-03
## 2208 2020-08-29 0.009 0.1075004 5.479233e-03
## 2212 2020-09-02 0.015 0.1075004 5.780667e-03
## 2233 2020-09-23 0.018 0.1075004 9.574767e-03
## 2236 2020-09-26 0.008 0.1075004 6.095833e-03
## 2243 2020-10-03 0.013 0.1075004 1.010423e-02
## 2248 2020-10-08 0.012 0.1075004 2.660127e-02
## 2249 2020-10-09 0.005 0.1075004 2.519710e-02
## 2252 2020-10-12 0.014 0.1075004 3.928710e-02
## 2258 2020-10-18 0.013 0.1075004 1.983830e-02
## 2276 2020-11-05 0.161 0.2831279 1.927268e-01
## 2295 2020-11-24 0.372 0.2831279 3.759579e-01
## 2304 2020-12-03 0.438 0.2831279 4.649219e-01
## 2310 2020-12-09 0.458 0.2831279 4.477472e-01
## 2318 2020-12-17 0.754 0.7794346 6.589154e-01
## 2321 2020-12-20 0.229 0.3836605 3.373414e-01
## 2328 2020-12-27 0.351 0.3836605 4.062439e-01
## 2334 2021-01-02 0.330 0.3836605 5.511824e-01
## 2336 2021-01-04 0.957 0.3836605 7.250673e-01
## 2337 2021-01-05 1.009 0.7794346 9.617226e-01
## 2347 2021-01-15 0.498 0.3836605 5.826377e-01
## 2353 2021-01-21 0.866 0.3836605 7.429972e-01
## 2355 2021-01-23 0.160 0.3836605 6.980092e-01
## 2361 2021-01-29 0.403 0.3836605 8.031984e-01
## 2366 2021-02-03 0.784 0.3836605 8.278475e-01
## 2372 2021-02-09 0.815 0.3836605 5.064951e-01
## 2373 2021-02-10 0.666 0.3836605 6.605352e-01
## 2376 2021-02-13 0.219 0.3836605 3.145398e-01
## 2378 2021-02-15 0.181 0.3836605 4.395974e-01
## 2379 2021-02-16 0.541 0.3836605 4.858741e-01
## 2386 2021-02-23 0.422 0.3836605 4.309732e-01
## 2388 2021-02-25 0.173 0.3836605 5.417594e-01
## 2400 2021-03-09 0.298 0.3836605 2.559227e-01
## 2404 2021-03-13 0.021 0.3836605 1.333736e-01
## 2405 2021-03-14 0.094 0.3836605 1.051010e-01
## 2412 2021-03-21 0.049 0.3836605 9.658773e-02
## 2420 2021-03-29 0.180 0.3836605 2.533696e-01
## 2424 2021-04-02 0.117 0.3836605 2.593795e-01
## 2425 2021-04-03 0.070 0.3836605 1.617267e-01
## 2432 2021-04-10 0.107 0.2831279 1.760723e-01
## 2435 2021-04-13 0.341 0.2831279 3.411782e-01
## 2436 2021-04-14 0.290 0.2831279 3.329547e-01
## 2437 2021-04-15 0.245 0.2831279 2.792793e-01
## 2447 2021-04-25 0.061 0.3836605 1.445056e-01
## 2459 2021-05-07 0.234 0.3836605 2.536054e-01
## 2469 2021-05-17 0.220 0.3836605 2.048985e-01
## 2471 2021-05-19 0.237 0.3836605 2.489752e-01
## 2475 2021-05-23 0.044 0.3836605 6.992360e-02
## 2477 2021-05-25 0.272 0.3836605 1.445571e-01
## 2484 2021-06-01 0.180 0.3836605 1.474504e-01
## 2489 2021-06-06 0.021 0.1075004 5.633550e-02
## 2492 2021-06-09 0.095 0.1075004 1.044479e-01
## 2517 2021-07-04 -0.001 0.1075004 7.599600e-03
## 2521 2021-07-08 0.049 0.1075004 3.211460e-02
## 2526 2021-07-13 0.027 0.1075004 3.456617e-02
## 2531 2021-07-18 0.001 0.1075004 4.805767e-03
## 2532 2021-07-19 0.034 0.1075004 2.165170e-02
## 2533 2021-07-20 0.019 0.1075004 2.645313e-02
## 2538 2021-07-25 0.003 0.1075004 6.166100e-03
## 2540 2021-07-27 0.019 0.1075004 2.194453e-02
## 2543 2021-07-30 0.041 0.1075004 9.307867e-03
## 2547 2021-08-03 0.025 0.1075004 1.818830e-02
## 2550 2021-08-06 0.024 0.1075004 1.953957e-02
## 2555 2021-08-11 0.017 0.1075004 2.168030e-02
## 2556 2021-08-12 0.019 0.1075004 2.315590e-02
## 2561 2021-08-17 0.022 0.1075004 2.857117e-02
## 2565 2021-08-21 0.003 0.1075004 2.015577e-02
## 2573 2021-08-29 0.010 0.1075004 3.594600e-02
## 2577 2021-09-02 0.045 0.1075004 3.830093e-02
## 2578 2021-09-03 0.024 0.1075004 4.358973e-02
## 2581 2021-09-06 0.059 0.1075004 5.357743e-02
## 2584 2021-09-09 0.096 0.2831279 1.227643e-01
## 2590 2021-09-15 0.067 0.1075004 6.512460e-02
## 2595 2021-09-20 0.081 0.1075004 7.533200e-02
## 2614 2021-10-09 0.024 0.1075004 3.200167e-02
## 2616 2021-10-11 0.095 0.1075004 7.331123e-02
## 2626 2021-10-21 0.109 0.2831279 8.995930e-02
## 2642 2021-11-06 0.037 0.2831279 1.146104e-01
## 2645 2021-11-09 0.237 0.2831279 1.935641e-01
## 2648 2021-11-12 0.229 0.2831279 1.799088e-01
## 2660 2021-11-24 0.350 0.9053733 4.770642e-01
## 2661 2021-11-25 0.357 0.9053733 5.000040e-01
## 2662 2021-11-26 0.303 0.2831279 3.657203e-01
## 2664 2021-11-28 0.073 0.2831279 1.545201e-01
## 2666 2021-11-30 0.444 0.9053733 5.331574e-01
## 2670 2021-12-04 0.092 0.7794346 7.696072e-01
## 2682 2020-02-02 0.000 0.1075004 3.138000e-03
## 2686 2020-02-06 0.000 0.1075004 -3.574918e-17
## 2695 2020-02-15 0.000 0.1075004 -3.841372e-17
## 2697 2020-02-17 0.000 0.1075004 -3.652634e-17
## 2700 2020-02-20 0.000 0.1075004 -3.574918e-17
## 2712 2020-03-03 0.000 0.1075004 -3.752554e-17
## 2718 2020-03-09 0.000 0.1075004 -2.686740e-17
## 2724 2020-03-15 0.000 0.1075004 3.030333e-04
## 2729 2020-03-20 0.001 0.1075004 1.388100e-03
## 2730 2020-03-21 -0.001 0.1075004 1.310233e-03
## 2732 2020-03-23 0.003 0.1075004 2.713267e-03
## 2733 2020-03-24 0.000 0.1075004 2.744733e-03
## 2735 2020-03-26 0.008 0.1075004 2.395033e-03
## 2741 2020-04-01 0.023 0.1075004 2.081000e-02
## 2743 2020-04-03 0.000 0.1075004 1.647980e-02
## 2749 2020-04-09 0.048 0.1075004 2.751257e-02
## 2765 2020-04-25 0.045 0.1075004 5.791597e-02
## 2766 2020-04-26 0.056 0.1075004 5.369720e-02
## 2770 2020-04-30 0.075 0.1075004 6.554263e-02
## 2775 2020-05-05 0.127 0.1075004 1.112540e-01
## 2776 2020-05-06 0.092 0.1075004 1.216963e-01
## 2777 2020-05-07 0.104 0.1075004 1.051696e-01
## 2786 2020-05-16 0.118 0.1075004 1.336680e-01
## 2797 2020-05-27 0.190 0.1075004 1.870084e-01
## 2800 2020-05-30 0.205 0.1075004 2.534714e-01
## 2803 2020-06-02 0.221 0.1075004 4.906991e-01
## 2806 2020-06-05 0.286 0.1075004 2.893202e-01
## 2808 2020-06-07 0.261 0.1075004 3.562408e-01
## 2815 2020-06-14 0.325 0.1075004 4.216213e-01
## 2819 2020-06-18 0.336 0.1075004 4.807245e-01
## 2821 2020-06-20 0.306 0.1075004 4.451001e-01
## 2825 2020-06-24 0.418 0.7794346 4.879239e-01
## 2826 2020-06-25 0.407 0.7794346 4.967266e-01
## 2827 2020-06-26 0.384 0.7794346 4.838627e-01
## 2843 2020-07-12 0.501 0.7794346 5.234780e-01
## 2852 2020-07-21 0.650 0.7794346 6.519474e-01
## 2856 2020-07-25 0.702 0.7794346 7.393691e-01
## 2859 2020-07-28 0.785 0.7794346 7.979754e-01
## 2870 2020-08-08 0.861 0.7794346 9.162731e-01
## 2874 2020-08-12 0.942 0.9053733 9.168061e-01
## 2876 2020-08-14 0.996 0.7794346 9.452132e-01
## 2880 2020-08-18 1.091 0.7794346 9.622910e-01
## 2882 2020-08-20 0.983 0.9053733 9.864560e-01
## 2895 2020-09-02 1.043 0.9053733 1.088590e+00
## 2896 2020-09-03 1.096 0.9053733 1.106720e+00
## 2899 2020-09-06 1.016 0.9053733 1.071045e+00
## 2902 2020-09-09 1.172 0.9053733 1.153640e+00
## 2913 2020-09-20 1.130 0.9053733 1.073057e+00
## 2916 2020-09-23 1.129 0.9053733 1.132168e+00
## 2918 2020-09-25 1.089 0.9053733 1.148998e+00
## 2921 2020-09-28 0.776 0.9053733 1.060681e+00
## 2931 2020-10-08 0.964 0.9053733 9.629609e-01
## 2932 2020-10-09 0.926 0.9053733 9.470137e-01
## 2935 2020-10-12 0.706 0.2831279 7.716035e-01
## 2942 2020-10-19 0.587 0.2831279 5.821300e-01
## 2948 2020-10-25 0.480 0.2831279 5.125298e-01
## 2954 2020-10-31 0.470 0.2831279 5.478075e-01
## 2959 2020-11-05 0.670 0.2831279 5.647721e-01
## 2966 2020-11-12 0.547 0.2831279 5.283691e-01
## 2968 2020-11-14 0.447 0.2831279 5.044685e-01
## 2969 2020-11-15 0.435 0.2831279 4.572431e-01
## 2972 2020-11-18 0.585 0.2831279 5.360158e-01
## 2980 2020-11-26 0.492 0.2831279 5.175132e-01
## 2982 2020-11-28 0.496 0.2831279 5.004991e-01
## 2989 2020-12-05 0.482 0.2831279 4.641739e-01
## 2990 2020-12-06 0.391 0.2831279 4.147716e-01
## 3000 2020-12-16 0.355 0.2831279 3.468242e-01
## 3003 2020-12-19 0.341 0.2831279 3.491986e-01
## 3018 2021-01-03 0.214 0.2831279 2.158558e-01
## 3023 2021-01-08 0.000 0.1075004 2.071673e-02
## 3027 2021-01-12 0.202 0.1075004 2.094753e-01
## 3047 2021-02-01 0.094 0.1075004 9.681230e-02
## 3056 2021-02-10 0.108 0.1075004 9.712073e-02
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## 6045 2021-12-08 0.163 0.2831279 1.656386e-01
## 6046 2021-12-09 0.148 0.2831279 1.514282e-01
## 6047 2021-12-10 0.120 0.2831279 1.449374e-01
## 6064 2020-02-05 0.000 0.1075004 -3.230749e-17
## 6069 2020-02-10 0.000 0.1075004 -3.474998e-17
## 6074 2020-02-15 0.000 0.1075004 -3.508305e-17
## 6075 2020-02-16 0.000 0.1075004 -3.574918e-17
## 6078 2020-02-19 0.000 0.1075004 -3.519407e-17
## 6079 2020-02-20 0.000 0.1075004 -3.375078e-17
## 6080 2020-02-21 0.000 0.1075004 -2.675637e-17
## 6087 2020-02-28 0.000 0.1075004 -3.619327e-17
## 6097 2020-03-09 0.001 0.1075004 4.123833e-03
## 6098 2020-03-10 0.006 0.1075004 5.760500e-03
## 6102 2020-03-14 0.007 0.1075004 1.492743e-02
## 6105 2020-03-17 0.037 0.1075004 3.386720e-02
## 6109 2020-03-21 0.103 0.1075004 1.305665e-01
## 6117 2020-03-29 0.555 0.1075004 6.466045e-01
## 6118 2020-03-30 0.707 1.4568194 9.431008e-01
## 6125 2020-04-06 1.733 1.4568194 1.641138e+00
## 6130 2020-04-11 2.141 1.4568194 1.885405e+00
## 6131 2020-04-12 1.889 1.4568194 1.838448e+00
## 6134 2020-04-15 2.587 1.4568194 2.191732e+00
## 6139 2020-04-20 2.245 1.4568194 1.768874e+00
## 6140 2020-04-21 2.483 1.4568194 2.317890e+00
## 6148 2020-04-29 2.391 1.4568194 2.001202e+00
## 6159 2020-05-10 1.001 1.4568194 1.215316e+00
## 6160 2020-05-11 1.034 1.4568194 1.248454e+00
## 6176 2020-05-27 1.478 0.8035981 8.769998e-01
## 6177 2020-05-28 1.089 0.8035981 1.014889e+00
## 6179 2020-05-30 0.950 0.8035981 9.524260e-01
## 6181 2020-06-01 0.771 0.8035981 5.877778e-01
## 6187 2020-06-07 0.457 0.8035981 5.519483e-01
## 6188 2020-06-08 0.511 0.8035981 4.880339e-01
## 6199 2020-06-19 0.626 0.8035981 6.788471e-01
## 6209 2020-06-29 0.369 0.8035981 3.970777e-01
## 6211 2020-07-01 0.698 0.8035981 8.429192e-01
## 6212 2020-07-02 0.739 0.8035981 8.418413e-01
## 6215 2020-07-05 0.322 0.8035981 4.319294e-01
## 6220 2020-07-10 0.816 0.9053733 9.537813e-01
## 6223 2020-07-13 0.435 0.8035981 6.379073e-01
## 6226 2020-07-16 0.961 0.9053733 9.963843e-01
## 6230 2020-07-20 0.529 0.8035981 7.274729e-01
## 6232 2020-07-22 1.225 0.9053733 1.064369e+00
## 6234 2020-07-24 1.094 0.9053733 1.055964e+00
## 6236 2020-07-26 0.534 0.8035981 6.535334e-01
## 6248 2020-08-07 1.291 0.8035981 1.136491e+00
## 6250 2020-08-09 0.572 0.8035981 5.363851e-01
## 6255 2020-08-14 1.309 0.8035981 1.036967e+00
## 6258 2020-08-17 0.469 0.8035981 5.362178e-01
## 6263 2020-08-22 0.979 0.8035981 9.568802e-01
## 6270 2020-08-29 0.983 0.8035981 8.968638e-01
## 6271 2020-08-30 0.449 0.8035981 4.947046e-01
## 6283 2020-09-11 1.156 0.8035981 9.003650e-01
## 6285 2020-09-13 0.400 0.8035981 3.843201e-01
## 6288 2020-09-16 0.955 0.8035981 9.946768e-01
## 6297 2020-09-25 0.915 0.8035981 8.861424e-01
## 6300 2020-09-28 0.341 0.8035981 4.088597e-01
## 6304 2020-10-02 0.841 0.8035981 8.549034e-01
## 6306 2020-10-04 0.369 0.8035981 4.439470e-01
## 6310 2020-10-08 0.982 0.8035981 9.290467e-01
## 6318 2020-10-16 0.943 1.3909884 9.264559e-01
## 6329 2020-10-27 1.011 1.3909884 9.680924e-01
## 6342 2020-11-09 0.759 1.3909884 8.382849e-01
## 6344 2020-11-11 1.443 1.3909884 1.241410e+00
## 6366 2020-12-03 2.969 2.4070561 2.756750e+00
## 6368 2020-12-05 2.390 2.4070561 2.283174e+00
## 6369 2020-12-06 1.403 2.4070561 1.650188e+00
## 6375 2020-12-12 2.533 2.4070561 2.550413e+00
## 6382 2020-12-19 2.754 2.4070561 2.524603e+00
## 6393 2020-12-30 3.933 2.4070561 3.358346e+00
## 6395 2021-01-01 2.194 3.6000806 2.608210e+00
## 6397 2021-01-03 1.437 3.6000806 2.492703e+00
## 6399 2021-01-05 3.604 3.6000806 3.826297e+00
## 6403 2021-01-09 3.256 3.6000806 3.371334e+00
## 6411 2021-01-17 1.991 3.6000806 2.712348e+00
## 6412 2021-01-18 1.565 3.6000806 2.793244e+00
## 6413 2021-01-19 2.401 3.6000806 3.911049e+00
## 6418 2021-01-24 1.858 3.6000806 2.659198e+00
## 6419 2021-01-25 1.770 3.6000806 2.874924e+00
## 6421 2021-01-27 4.128 3.6000806 3.817899e+00
## 6422 2021-01-28 3.850 3.6000806 3.693605e+00
## 6427 2021-02-02 3.533 3.6000806 3.080864e+00
## 6436 2021-02-11 3.191 3.6000806 2.806475e+00
## 6440 2021-02-15 0.949 0.8035981 1.177726e+00
## 6443 2021-02-18 2.607 2.3296739 2.346740e+00
## 6452 2021-02-27 1.521 0.8035981 1.616320e+00
## 6454 2021-03-01 1.319 0.8035981 1.058313e+00
## 6455 2021-03-02 1.933 0.8035981 1.401090e+00
## 6456 2021-03-03 2.581 2.4070561 1.868657e+00
## 6458 2021-03-05 1.723 0.8035981 1.547759e+00
## 6461 2021-03-08 0.705 0.8035981 8.615139e-01
## 6462 2021-03-09 1.784 0.8035981 1.356759e+00
## 6473 2021-03-20 0.773 0.8035981 1.018258e+00
## 6479 2021-03-26 1.168 2.4070561 1.515043e+00
## 6480 2021-03-27 0.814 0.8035981 9.401268e-01
## 6484 2021-03-31 1.101 2.4070561 1.366514e+00
## 6494 2021-04-10 0.724 2.4070561 7.349564e-01
## 6499 2021-04-15 0.905 2.4070561 9.961885e-01
## 6504 2021-04-20 0.847 0.8035981 8.240071e-01
## 6512 2021-04-28 0.994 0.8035981 8.016706e-01
## 6519 2021-05-05 0.813 0.8035981 8.182445e-01
## 6521 2021-05-07 0.824 0.8035981 7.026016e-01
## 6525 2021-05-11 0.700 0.8035981 7.633724e-01
## 6527 2021-05-13 0.790 0.8035981 7.088294e-01
## 6529 2021-05-15 0.529 0.8035981 5.617291e-01
## 6532 2021-05-18 0.780 0.8035981 6.842800e-01
## 6534 2021-05-20 0.684 0.8035981 7.314389e-01
## 6536 2021-05-22 0.484 0.8035981 6.100155e-01
## 6538 2021-05-24 0.438 0.8035981 5.304383e-01
## 6547 2021-06-02 0.564 0.8035981 6.118820e-01
## 6549 2021-06-04 0.527 0.8035981 5.983315e-01
## 6553 2021-06-08 0.331 0.3836605 3.493270e-01
## 6560 2021-06-15 0.334 0.3836605 3.025145e-01
## 6561 2021-06-16 0.383 0.3836605 3.282677e-01
## 6569 2021-06-24 0.340 0.3836605 3.131747e-01
## 6575 2021-06-30 0.295 0.8035981 4.649034e-01
## 6582 2021-07-07 0.321 0.8035981 3.873233e-01
## 6585 2021-07-10 0.083 0.3836605 1.501283e-01
## 6591 2021-07-16 0.358 1.3909884 4.936368e-01
## 6596 2021-07-21 0.352 0.8035981 3.826164e-01
## 6608 2021-08-02 0.361 1.3909884 5.809516e-01
## 6615 2021-08-09 0.506 1.3909884 8.416411e-01
## 6616 2021-08-10 0.865 1.3909884 9.749820e-01
## 6617 2021-08-11 0.779 1.3909884 9.628324e-01
## 6620 2021-08-14 0.254 0.8035981 4.753206e-01
## 6623 2021-08-17 1.028 1.3909884 1.228617e+00
## 6625 2021-08-19 1.781 1.3909884 1.308806e+00
## 6628 2021-08-22 0.170 0.8035981 3.265273e-01
## 6636 2021-08-30 1.764 1.3909884 1.427079e+00
## 6644 2021-09-07 2.134 2.4070561 1.762512e+00
## 6646 2021-09-09 3.284 2.4070561 2.669145e+00
## 6647 2021-09-10 2.430 2.4070561 2.230108e+00
## 6649 2021-09-12 0.313 0.8035981 3.606927e-01
## 6655 2021-09-18 0.915 0.8035981 6.746952e-01
## 6657 2021-09-20 2.356 2.4070561 1.936376e+00
## 6658 2021-09-21 2.355 2.4070561 2.610628e+00
## 6661 2021-09-24 2.484 2.4070561 2.460796e+00
## 6670 2021-10-03 0.477 0.8035981 3.638645e-01
## 6674 2021-10-07 2.477 1.3909884 2.315945e+00
## 6687 2021-10-20 3.200 1.3909884 2.368086e+00
## 6689 2021-10-22 1.893 1.3909884 1.857884e+00
## 6695 2021-10-28 1.840 1.3909884 1.709340e+00
## 6706 2021-11-08 1.263 1.3909884 1.444001e+00
## 6714 2021-11-16 1.318 1.3909884 1.466804e+00
## 6715 2021-11-17 1.657 1.3909884 1.850411e+00
## 6721 2021-11-23 1.316 1.3909884 1.443133e+00
## 6722 2021-11-24 1.682 1.3909884 1.625036e+00
## 6727 2021-11-29 1.969 1.3909884 1.471064e+00
## 6730 2021-12-02 3.801 2.4070561 2.109720e+00
## 6734 2021-12-06 1.384 2.4070561 1.567196e+00
## 6735 2021-12-07 1.609 2.4070561 1.607512e+00
## 6740 2021-12-12 0.167 0.8035981 5.340603e-01
## boosted_tree_pred rf_predicted_cases MLR s1
## 1 0.0775436830 6.025698e-05 0.707908506 0.211395440
## 21 0.0731960916 3.071552e-03 0.713823868 0.219319919
## 27 0.0498493784 1.149607e-02 0.558315541 0.221735228
## 29 0.1212133882 1.384257e-02 0.738001930 0.221716233
## 43 0.1872469795 1.253206e-01 0.882844835 0.224856440
## 45 0.1825257611 1.084392e-01 0.846698400 0.224569643
## 60 0.3280217977 3.361964e-01 0.772568954 0.231162511
## 65 0.5244170067 4.091075e-01 0.874619923 0.235705024
## 66 0.5088933288 3.732970e-01 0.950501107 0.233869469
## 71 0.5917408871 6.704970e-01 0.993357177 0.242654841
## 80 0.7481402098 8.566679e-01 0.958959719 0.255353749
## 87 0.8240470679 1.002575e+00 0.960221037 0.266905419
## 88 0.6784500360 8.481040e-01 0.886047316 0.264122724
## 103 0.6080148069 6.660656e-01 0.656330593 0.290808721
## 107 0.9008684411 1.208249e+00 0.892109695 0.309503748
## 108 0.8144489367 9.748984e-01 0.871019903 0.307188810
## 110 0.6129206342 6.641214e-01 0.660626884 0.300910161
## 122 1.2932196912 1.116383e+00 0.883721706 0.356598933
## 134 1.1925534595 1.146997e+00 0.830112080 0.384797053
## 135 1.1753262923 1.230036e+00 0.817823099 0.387654027
## 141 1.1562663696 1.146711e+00 0.837817568 0.397719806
## 142 1.1449531552 1.137800e+00 0.827990272 0.405931709
## 151 1.0398939106 1.081906e+00 0.741986062 0.439671886
## 154 1.1312675965 1.180538e+00 0.853735212 0.433496580
## 155 1.3522806331 1.231986e+00 0.843957678 0.465971892
## 157 1.1151994835 1.158704e+00 0.811611210 0.457835616
## 163 1.1537354997 1.157511e+00 0.844372596 0.472200854
## 172 0.8474898926 9.698277e-01 0.758947792 0.491894224
## 174 0.5489263588 5.624779e-01 0.691196858 0.474079885
## 177 0.8978076988 1.063726e+00 0.859179052 0.503727283
## 190 0.9365595684 9.974233e-01 0.917423400 0.538680612
## 192 0.9078789066 9.267290e-01 0.886194014 0.552843497
## 194 0.3056794889 3.846297e-01 0.676573890 0.522633488
## 195 0.3199108320 3.551624e-01 0.742212813 0.514014139
## 196 0.4804767841 4.955447e-01 0.923831587 0.524131574
## 205 0.7982409055 9.320606e-01 0.917724081 0.559151152
## 206 0.8972280319 8.739458e-01 0.899399458 0.570844081
## 211 0.2462293274 2.704960e-01 0.934133016 0.539777788
## 231 0.3343223896 3.172850e-01 0.940214235 0.584567654
## 233 0.6079061294 6.950525e-01 0.928419039 0.605526286
## 236 0.0922201412 1.226725e-01 0.697365884 0.582579009
## 239 0.5650106661 5.549870e-01 0.947197938 0.610877250
## 241 0.6415534925 6.289460e-01 0.903629207 0.618626254
## 245 0.6841041708 6.006643e-01 0.938948129 0.616604537
## 253 0.5179149696 3.493172e-01 1.005367261 0.620245372
## 255 0.7208391023 5.810790e-01 0.970240812 0.639049507
## 267 0.6272607316 6.592905e-01 1.012399477 0.652737255
## 270 0.5543482505 6.475636e-01 0.904670137 0.658409458
## 272 0.3505173324 2.642227e-01 0.837855062 0.645326977
## 273 0.6120298083 5.976501e-01 1.017963626 0.661336135
## 274 0.7628315991 6.824305e-01 1.026637280 0.680755914
## 276 0.6381002359 6.635155e-01 0.994074662 0.672595497
## 277 0.7333992664 7.450738e-01 0.923314056 0.701954656
## 284 0.7419100265 7.284893e-01 1.002479570 0.703236468
## 300 0.5647263308 4.193856e-01 0.961716169 0.731708081
## 301 1.0066243800 9.466442e-01 1.141995228 0.763632319
## 305 0.4978873272 4.383909e-01 1.036393611 0.737892702
## 319 1.0509119766 1.063037e+00 1.027475460 0.819914488
## 320 0.6245509845 6.086370e-01 0.891508392 0.791487851
## 321 0.6222064729 5.200174e-01 0.961319983 0.789476352
## 324 1.6585040898 1.322321e+00 1.151749163 0.843250505
## 343 1.2442974372 1.219995e+00 1.122725163 0.893866517
## 349 0.2995309315 2.707985e-01 0.952905649 0.856996200
## 350 1.9786366313 1.513729e+00 1.138876134 0.933319226
## 356 0.9503970238 7.984948e-01 0.971388589 0.909156923
## 375 1.7173493968 1.755063e+00 0.970629736 0.999197820
## 376 1.7583806684 1.682024e+00 0.838886576 1.011672468
## 378 2.1409873230 1.993370e+00 1.086775179 1.013265477
## 379 2.0675109014 2.043895e+00 1.088949503 1.026682097
## 381 2.3613663621 2.096191e+00 1.059449708 1.040536687
## 395 3.1969137664 3.109370e+00 1.148901607 1.106928841
## 405 1.7257856277 1.373039e+00 1.103400855 1.082074330
## 406 3.5290963944 3.728564e+00 1.283449969 1.147071103
## 409 3.5152286321 3.620257e+00 1.250079753 1.161655869
## 414 3.2845562270 3.467106e+00 1.286370653 1.156275511
## 425 1.5733946350 1.297392e+00 1.011426047 1.123429801
## 433 1.1344438123 9.800689e-01 1.161590382 1.140202298
## 435 2.6899052291 2.663815e+00 1.341163456 1.192510865
## 439 1.0787255306 1.046884e+00 1.098712061 1.154890182
## 445 1.9272345753 2.107550e+00 1.248093495 1.203842296
## 447 0.8750015599 9.163199e-01 1.179341410 1.180163593
## 458 1.2592275815 1.409396e+00 1.343920187 1.232863216
## 463 2.7313016974 2.609852e+00 1.293932635 1.294957624
## 468 0.9206684030 9.624977e-01 1.114674482 1.243255775
## 469 1.4103331061 1.599761e+00 1.294087106 1.269228863
## 484 2.6925464434 2.464011e+00 1.342166733 1.370927767
## 486 2.2265327839 2.137106e+00 1.312736304 1.342261268
## 495 0.8590242514 8.506783e-01 0.999737838 1.216253739
## 499 1.3304901184 1.534217e+00 1.222023506 1.188175906
## 502 0.7545753314 7.319804e-01 0.988196135 1.095610470
## 504 1.3268939529 1.412452e+00 1.232377234 1.112612257
## 505 1.5113212864 1.443960e+00 1.240106892 1.134013339
## 513 1.2290621372 1.315806e+00 1.205392573 1.031483923
## 514 1.9895244163 1.984594e+00 1.193729736 1.104747933
## 517 0.7867922096 6.170364e-01 1.038598510 0.984737927
## 519 1.2802911108 1.273114e+00 1.222594070 1.031417382
## 520 1.2567800791 1.176076e+00 1.195568392 0.993976613
## 521 1.1875781065 1.132299e+00 1.180830057 1.016273529
## 524 0.6601456186 4.864867e-01 1.038462887 0.973802104
## 534 1.0574251711 1.033255e+00 1.165108913 0.962433490
## 543 0.7973053719 8.451538e-01 1.044288702 0.897031051
## 552 0.3701619789 3.451131e-01 0.956376519 0.880930760
## 575 0.8910587637 6.881509e-01 1.041659622 0.821962753
## 582 0.6286905720 5.956841e-01 0.988066741 0.750325815
## 596 0.3878670770 2.867715e-01 0.909895134 0.678929016
## 597 0.4085626080 4.557144e-01 0.874070122 0.679921995
## 599 0.3516907062 3.507065e-01 0.771867837 0.667637854
## 604 0.5032482717 3.929322e-01 0.817194587 0.632504657
## 606 0.2761762272 3.413212e-01 0.709891874 0.611396932
## 608 0.2240471849 2.247490e-01 0.621866379 0.624548874
## 619 0.3107542010 3.525055e-01 0.747198091 0.560874558
## 624 0.3345504509 2.832185e-01 0.752395265 0.520315855
## 628 0.0841539787 9.796905e-02 0.482452775 0.472288278
## 635 0.0768024782 1.213901e-01 0.467130194 0.442981468
## 636 0.1457958092 1.556384e-01 0.516273136 0.418020213
## 637 0.3217767957 2.550288e-01 0.699520786 0.445929636
## 645 0.3117408488 2.626875e-01 0.719176017 0.383252590
## 647 0.2196337211 2.749540e-01 0.676662257 0.401678462
## 651 0.3118967112 2.320492e-01 0.692950670 0.370219224
## 654 0.1639832282 1.522636e-01 0.638117501 0.332062496
## 659 -0.0521763923 1.992060e-03 -0.660489016 0.172898036
## 670 -0.0139948716 1.311907e-04 -0.362191998 0.178706129
## 672 -0.0346725265 1.789266e-03 -0.458458066 0.178740283
## 674 -0.0357505791 9.722807e-04 -0.522834391 0.179165154
## 678 0.0289286861 2.881707e-03 -0.321773040 0.184288663
## 679 0.0079701149 3.286716e-03 -0.396220712 0.184312279
## 689 0.0411534086 8.575673e-03 -0.146389207 0.184523009
## 697 0.0429519532 1.140026e-02 -0.151820125 0.184001495
## 705 0.0338253332 1.156965e-02 -0.166162657 0.184184048
## 707 0.0131476784 1.260067e-02 -0.257744010 0.184689603
## 708 -0.0058376168 1.214664e-02 -0.394413795 0.184682709
## 721 0.0083533370 1.628964e-02 -0.159109328 0.185242395
## 723 0.0072752844 1.771172e-02 -0.228108381 0.185418421
## 727 0.0293119082 1.927792e-02 -0.084539704 0.185651031
## 729 -0.0106319582 1.903226e-02 -0.295617882 0.185768073
## 740 0.0311651735 3.406318e-02 -0.062351271 0.187029720
## 742 0.0104875186 4.235415e-02 -0.158524803 0.187491963
## 746 0.0574754763 5.256309e-02 -0.142343093 0.187220361
## 751 0.0265931489 5.734184e-02 -0.322002004 0.187600772
## 768 0.1782938662 1.290727e-01 0.509711291 0.191854873
## 774 0.2293628464 1.588397e-01 0.508121437 0.193825967
## 776 0.1807174882 1.544956e-01 0.469474318 0.193576201
## 777 0.1797078756 1.531112e-01 0.366842037 0.194277255
## 789 0.2746570254 2.400244e-01 0.470609941 0.201883401
## 795 0.2697418358 2.734196e-01 0.579893861 0.204152377
## 799 0.2393092749 2.919682e-01 0.381510267 0.206860850
## 802 0.2924445665 3.079771e-01 0.657985192 0.209068444
## 809 0.3226912610 3.295152e-01 0.636364498 0.215412088
## 812 0.3136257801 3.131355e-01 0.591343356 0.216365371
## 814 0.2934232729 3.216973e-01 0.480358709 0.218132767
## 819 0.3232159800 3.087158e-01 0.543224351 0.223185364
## 826 0.3155643679 3.003259e-01 0.533926501 0.227451353
## 831 0.3354616133 2.913234e-01 0.605790371 0.229437919
## 834 0.2393241420 3.010701e-01 0.333329826 0.229476048
## 838 0.2425662690 2.726770e-01 0.508244920 0.229535725
## 846 0.2103194596 2.329310e-01 0.445924320 0.232540910
## 847 0.1777936516 2.040302e-01 0.357041115 0.232453154
## 848 0.1588083564 2.219676e-01 0.209177162 0.233352652
## 851 0.2268368345 2.353862e-01 0.454470152 0.235005539
## 858 0.2115713376 1.544107e-01 0.416595873 0.236891808
## 859 0.2104108536 1.554082e-01 0.402256938 0.237085823
## 862 0.1588083564 1.742066e-01 0.144110788 0.238903397
## 864 0.2167624462 1.518523e-01 0.374186635 0.238458998
## 866 0.2012699449 1.615651e-01 0.342792033 0.238958449
## 868 0.1686527430 1.749037e-01 0.241571470 0.240593555
## 870 0.1566471312 1.544377e-01 0.178068939 0.241885774
## 876 0.1519353572 1.586040e-01 0.111691006 0.246170227
## 891 0.1521063846 1.694332e-01 0.148468903 0.252604609
## 892 0.1658110148 1.782279e-01 0.331299258 0.252152266
## 897 0.1262827763 1.770770e-01 0.120314548 0.256005129
## 911 0.0854186494 1.721878e-01 0.110925552 0.260879983
## 916 0.1614718549 1.895374e-01 0.339254332 0.263685502
## 917 0.1026670583 1.875446e-01 0.245491237 0.263228894
## 918 0.0989472599 1.855719e-01 0.107249732 0.264528992
## 919 0.1015890056 1.847467e-01 0.172684777 0.263645861
## 935 0.1941994708 1.846927e-01 0.449732607 0.271717668
## 936 0.2075598630 1.818695e-01 0.431607928 0.272385424
## 938 0.1758353586 1.779589e-01 0.349994329 0.273721118
## 946 0.2124398659 2.298595e-01 0.274274300 0.281803669
## 947 0.2012601437 2.301759e-01 0.342604606 0.282161228
## 949 0.2580376652 2.544635e-01 0.542808131 0.286898511
## 950 0.3037055286 2.564418e-01 0.522798933 0.288441283
## 961 0.2160176783 2.685057e-01 0.505188902 0.291797684
## 969 0.3809841365 3.375218e-01 0.728614550 0.306499473
## 972 0.4735506736 3.864140e-01 0.701366308 0.316212916
## 975 0.3164889305 3.707770e-01 0.517575875 0.313552912
## 984 0.3577544528 3.807031e-01 0.526964481 0.320863301
## 989 0.2657669464 3.018619e-01 0.168251186 0.319738134
## 992 0.2858005834 3.078868e-01 0.287512626 0.321703273
## 999 0.2331028145 2.264031e-01 0.199691659 0.320981578
## 1004 0.2470482327 1.936355e-01 0.184514505 0.320155755
## 1011 0.2445794988 1.465491e-01 0.153330774 0.325195716
## 1014 0.1994525594 1.386179e-01 0.101229162 0.328049111
## 1020 0.1438683766 1.091734e-01 0.037071522 0.338080776
## 1025 0.1563757205 9.999983e-02 0.058963264 0.354343393
## 1028 0.1749179371 1.025380e-01 0.047946897 0.376546141
## 1031 0.0929543052 1.060362e-01 -0.091083590 0.324041964
## 1033 0.1841359511 1.246020e-01 0.108920082 0.325525269
## 1038 0.1518104932 1.277928e-01 0.008997843 0.326643055
## 1041 0.1851334375 1.364875e-01 0.213437932 0.330025746
## 1055 0.3651693942 2.870342e-01 0.634021734 0.340063562
## 1061 0.4220016703 3.518847e-01 0.613769498 0.347080716
## 1069 0.4882739723 4.324587e-01 0.590455844 0.355724030
## 1072 0.3870559987 4.344667e-01 0.350085225 0.355519456
## 1077 0.4429502408 4.526920e-01 0.548306121 0.361569278
## 1078 0.4182474973 4.763771e-01 0.575219275 0.361730693
## 1086 0.4042372838 4.719820e-01 0.476847883 0.378028025
## 1091 0.4183931495 4.793197e-01 0.707691903 0.381549352
## 1092 0.4424836976 5.006828e-01 0.623440713 0.365623559
## 1094 0.3083451772 4.835044e-01 0.548023686 0.363087995
## 1100 0.4499492451 5.058960e-01 0.550139867 0.375429710
## 1102 0.4978046902 4.983031e-01 0.808254756 0.379117964
## 1103 0.4989198581 5.061970e-01 0.824696700 0.383090568
## 1110 0.4956832973 5.443766e-01 0.787630996 0.394755254
## 1113 0.5022977823 5.459967e-01 0.704879868 0.398933906
## 1114 0.4353581800 5.285421e-01 0.572439943 0.393326281
## 1121 0.4838030941 5.849505e-01 0.592308456 0.404470143
## 1123 0.5480385094 5.691912e-01 0.857623487 0.433032025
## 1128 0.5401834782 5.992461e-01 0.628560422 0.412664237
## 1129 0.4705329051 6.107870e-01 0.690303435 0.411196896
## 1131 0.5704449277 6.519347e-01 0.847914741 0.419148023
## 1132 0.6499469398 6.692569e-01 0.835621775 0.422413148
## 1141 0.4632156861 5.801466e-01 0.528281005 0.406293552
## 1147 0.4637216206 5.538088e-01 0.597424325 0.471655167
## 1164 0.1871984345 3.086202e-01 0.127651335 0.242774512
## 1181 0.1365579106 1.466644e-01 0.028206530 0.203322597
## 1186 0.1347436980 1.152245e-01 0.010391570 0.193717013
## 1188 0.1204259695 1.111728e-01 -0.019007195 0.196961371
## 1201 0.0832580836 6.690770e-02 0.005610181 0.476301705
## 1203 0.0744123800 6.813620e-02 -0.112544570 0.182841516
## 1206 0.0524242696 4.636287e-02 -0.183775725 0.475719148
## 1209 0.0678636549 5.283030e-02 -0.112422215 0.172133395
## 1221 0.0810183916 3.357322e-02 -0.025718613 0.476428325
## 1222 0.0761995934 3.536752e-02 -0.113348687 0.165935703
## 1228 0.0810183916 3.671792e-02 -0.037346009 0.476927174
## 1230 0.0670729734 3.327626e-02 -0.047529321 0.476725046
## 1236 0.0675916158 3.333037e-02 -0.126306532 0.124736099
## 1239 0.0377873410 3.568158e-02 -0.249774223 0.113520150
## 1243 0.0675916158 3.368987e-02 -0.146025041 0.097978395
## 1257 0.0746501061 3.413443e-02 -0.132283651 0.055573834
## 1260 0.0385780225 3.468752e-02 -0.248155909 0.043228071
## 1261 0.0262328464 3.511241e-02 -0.382767115 0.044727069
## 1268 0.0360398345 3.659425e-02 -0.309491955 0.030856856
## 1272 0.0753304742 4.065260e-02 -0.068108168 0.023060995
## 1278 0.0874282671 4.078040e-02 -0.038418500 0.011703162
## 1284 0.0962391276 4.633520e-02 -0.030638683 -0.003749393
## 1292 0.0858644993 5.450907e-02 -0.072762487 -0.029574180
## 1307 -0.0586505696 1.740769e-04 -0.240641702 0.181826605
## 1311 -0.0370791732 1.664769e-04 -0.097157857 0.181828107
## 1313 -0.0944675630 1.876923e-06 -0.382451109 0.181825241
## 1314 -0.0510495077 7.435897e-07 -0.314808961 0.181825951
## 1320 -0.0944675630 1.876923e-06 -0.382450883 0.181825561
## 1327 -0.0944675630 4.835366e-05 -0.382451330 0.181825813
## 1332 -0.0294781113 1.201203e-04 -0.171325426 0.181827897
## 1336 -0.0116318390 1.891719e-04 -0.121908053 0.183071855
## 1337 -0.0299135373 1.675560e-04 -0.121142314 0.183074063
## 1340 -0.0546189421 5.009182e-04 -0.196775073 0.186216827
## 1348 -0.1053153972 4.700483e-03 -0.399081787 0.187788461
## 1353 -0.0381917639 1.221100e-02 -0.187869646 0.189097988
## 1356 -0.0612445897 1.836610e-02 -0.309475148 0.191945677
## 1368 0.3561311480 4.079748e-01 -0.023528808 0.203853599
## 1373 0.7181384814 8.314822e-01 0.292299223 0.204676221
## 1383 0.8161010224 6.410167e-01 0.089954233 0.255821562
## 1385 0.9959461062 7.607985e-01 0.336478160 0.226296691
## 1386 0.7807751861 6.878734e-01 0.328940656 0.213370909
## 1400 0.5163266469 3.216980e-01 0.260359522 0.211805830
## 1410 0.2644977955 1.928094e-01 0.201119546 0.214133401
## 1412 0.1835031776 2.183671e-01 0.109822457 0.212328699
## 1415 0.1860275178 1.479426e-01 0.261119082 0.212644556
## 1424 0.1533279803 7.215745e-02 0.189420694 0.212531005
## 1435 0.0612657457 7.224170e-02 0.161686882 0.208078419
## 1437 0.0617011717 5.263075e-02 0.119858242 0.211855047
## 1438 0.0365603410 3.773225e-02 0.042038654 0.211856278
## 1444 0.0617011717 4.724908e-02 0.105605705 0.212111608
## 1446 -0.0032882800 2.326375e-02 -0.109163551 0.211752470
## 1452 0.0365603410 2.382046e-02 0.023564703 0.212071920
## 1456 0.0380045424 2.924816e-02 0.097959145 0.208746720
## 1457 0.0288779224 2.632376e-02 0.079562844 0.208731919
## 1459 0.0132991377 1.992546e-02 -0.022709702 0.208971102
## 1463 0.0192321766 2.872863e-02 -0.025271128 0.209542961
## 1469 0.0375138749 1.866469e-02 -0.032350871 0.209137886
## 1470 0.0192321766 2.075409e-02 -0.033099832 0.209499077
## 1473 -0.0036194884 6.106748e-03 -0.159722989 0.208196921
## 1477 0.0207136060 1.486846e-02 -0.044007614 0.208792090
## 1492 0.0076408024 1.208827e-02 -0.056582869 0.210076395
## 1494 -0.0022467072 3.475433e-03 -0.153343042 0.208893744
## 1513 0.0428792794 1.691160e-02 0.006775503 0.216270834
## 1514 0.0524413254 1.671843e-02 -0.014174649 0.216734858
## 1520 0.1132016478 5.212546e-02 0.021246046 0.224206846
## 1532 0.0895801801 2.919508e-02 0.105552543 0.225057195
## 1533 0.0861916683 3.020606e-02 0.109977022 0.227359811
## 1535 0.1129693114 3.024939e-02 0.074631175 0.229502689
## 1540 0.1125338854 6.320644e-02 0.123837908 0.233008164
## 1541 0.1034072654 6.962212e-02 0.115426356 0.233647807
## 1553 0.1080765228 7.646979e-02 0.175624063 0.240590645
## 1555 0.1172694423 8.391419e-02 0.171925674 0.247084210
## 1559 0.0216660513 9.657284e-02 0.051236481 0.243324637
## 1561 0.1839392322 9.384425e-02 0.259866791 0.257778568
## 1567 0.1588098861 1.064836e-01 0.311346296 0.258902417
## 1570 0.2062295918 1.314704e-01 0.301499401 0.275366735
## 1579 0.0397910826 2.106382e-01 0.245820066 0.276132167
## 1581 0.2749762876 3.950760e-01 0.566606565 0.309588816
## 1583 0.2524401427 3.149450e-01 0.576889901 0.326763018
## 1587 0.5425847012 6.418776e-01 0.557541972 0.402689307
## 1593 0.3585094608 3.372542e-01 0.471939895 0.358065380
## 1595 0.5750700832 7.467646e-01 0.704082085 0.346491910
## 1598 0.5298066890 7.174930e-01 0.635419466 0.354344494
## 1607 0.3873843542 3.807639e-01 0.315749752 0.358729514
## 1610 0.5183131976 4.811290e-01 0.526112171 0.364287453
## 1619 0.4933013516 5.764481e-01 0.499766286 0.365420975
## 1621 0.2919182697 2.252675e-01 0.275802615 0.366474346
## 1622 0.2823264999 3.889719e-01 0.360678390 0.359728935
## 1629 0.2823264999 3.697548e-01 0.349339414 0.365331343
## 1634 0.2241345984 2.879431e-01 0.348195458 0.382488764
## 1639 0.3375286516 3.281312e-01 0.409577496 0.391131652
## 1647 0.3909087552 3.610003e-01 0.397955597 0.396526344
## 1652 0.4368741934 3.124786e-01 0.484128233 0.406802685
## 1668 0.4353885372 5.946856e-01 0.495619381 0.426286201
## 1676 0.3390322057 2.375366e-01 0.416018611 0.435463094
## 1689 0.4297454253 5.549908e-01 0.402251487 0.447553371
## 1691 0.1874160453 1.845692e-01 0.166879341 0.443261242
## 1694 0.4466457169 3.438000e-01 0.434199140 0.455891788
## 1700 0.4986429711 4.809876e-01 0.461832263 0.456743411
## 1704 0.3308123729 1.955278e-01 0.366201704 0.465942626
## 1707 0.3538563098 3.774665e-01 0.388171184 0.467607435
## 1712 0.1679158787 1.731175e-01 0.145949618 0.472940384
## 1716 0.2667661338 3.006956e-01 0.401407413 0.484816463
## 1719 0.2092753485 1.637255e-01 0.169895734 0.483702397
## 1745 0.2108069270 3.203732e-01 0.684391122 0.511719521
## 1749 0.5512060774 3.735825e-01 0.748393605 0.554793523
## 1754 0.2349876068 1.854502e-01 0.488323536 0.545806784
## 1760 0.3101902290 2.056162e-01 0.639424263 0.558375222
## 1764 0.3691628125 2.968856e-01 0.722812786 0.561596912
## 1765 0.3200562740 2.948606e-01 0.695411897 0.557408134
## 1774 0.2420803949 1.602674e-01 0.550365511 0.540219868
## 1787 0.2155368915 1.620787e-01 0.525696989 0.503785027
## 1789 0.0802596520 1.028736e-01 0.282771283 0.490826897
## 1790 0.0685168892 1.562415e-01 0.347257249 0.485669686
## 1796 0.0598259609 9.279870e-02 0.245069264 0.479150114
## 1797 0.0849095640 1.246238e-01 0.331836150 0.477508668
## 1805 0.0880029377 7.842717e-02 0.345980853 0.467497829
## 1808 0.0701566654 7.374667e-02 0.287439960 0.458341213
## 1814 0.0186540524 5.810570e-02 0.269277626 0.443173861
## 1818 0.0078023345 5.119017e-02 0.071955133 0.431133873
## 1822 0.0296975278 4.515383e-02 0.192559396 0.424481768
## 1829 0.0188114322 3.443810e-02 0.056308700 0.408649058
## 1830 -0.0068262511 2.636750e-02 -0.029674422 0.402282890
## 1831 -0.0483560449 2.053757e-02 -0.179768674 0.395072540
## 1840 0.0869972726 2.942978e-02 0.074280157 0.390699964
## 1847 0.2451950523 3.907133e-02 0.122793080 0.386318369
## 1851 0.1629292755 3.548757e-02 0.028011250 0.380059410
## 1854 0.2318387468 4.287793e-02 0.140102617 0.382533714
## 1865 0.1643020567 5.429560e-02 0.057835529 0.366098016
## 1867 -0.0077258640 8.775693e-02 0.033745327 0.353177227
## 1868 0.2548858965 7.402713e-02 0.203247573 0.374348289
## 1875 0.2347646513 1.207692e-01 0.199989136 0.370378024
## 1888 0.0145051951 1.388847e-01 -0.001738991 0.347224354
## 1894 0.0131525085 6.877270e-02 -0.102134299 0.338994546
## 1898 0.1259259336 1.064973e-01 0.146123084 0.347590705
## 1903 0.1438441874 8.668513e-02 0.134139528 0.341461732
## 1904 0.0114437246 8.818563e-02 0.125614311 0.327586230
## 1911 0.0463702159 8.070617e-02 0.098016631 0.328740465
## 1916 -0.0203587381 6.660983e-02 -0.083861091 0.319128600
## 1920 0.0105791023 6.089653e-02 0.051107928 0.320901876
## 1924 0.0625655885 5.795103e-02 0.063189261 0.318735091
## 1932 -0.0070604226 4.702917e-02 0.053151842 0.309715839
## 1950 -0.0511293263 1.571437e-02 -0.194823383 0.299330593
## 1966 0.2206550028 5.769677e-02 0.192310690 0.304303747
## 1983 0.1940035210 1.159458e-01 0.321974169 0.315139584
## 1995 -0.0231174250 6.980000e-04 0.152529376 0.164257327
## 2000 -0.0463986905 -4.074519e-17 -0.101307814 0.164256376
## 2002 -0.0155163631 -4.241052e-17 0.078362344 0.164256993
## 2006 -0.0731213542 -4.141132e-17 -0.168949872 0.164254858
## 2010 -0.0246429830 -4.285461e-17 0.073321065 0.165083763
## 2014 -0.0463986905 -4.285461e-17 -0.091981947 0.165081423
## 2020 -0.0731213542 -4.274359e-17 -0.159624095 0.165080713
## 2024 -0.0246429830 2.560000e-05 0.082664912 0.165931606
## 2027 -0.0839691884 4.426667e-05 -0.217508791 0.167204598
## 2037 -0.0263641973 3.652371e-03 0.043106552 0.168868340
## 2044 -0.0028511486 2.615607e-02 0.076231216 0.174884117
## 2051 0.1041116335 6.949282e-02 0.134709597 0.181377994
## 2063 0.0570804039 2.154788e-01 0.133346882 0.184934723
## 2068 0.0784256327 2.021579e-01 0.204973169 0.186007339
## 2077 0.1929609064 1.881544e-01 0.140989488 0.187028176
## 2088 0.1821055289 1.451107e-01 0.366762704 0.187684619
## 2100 0.0969311338 6.652142e-02 0.366775611 0.186521776
## 2107 0.0969311338 5.870669e-02 0.351560748 0.185936874
## 2115 0.0878045138 4.527318e-02 0.331124529 0.186205493
## 2116 0.0880854301 3.316432e-02 0.309300892 0.186307780
## 2118 0.0382344956 2.648758e-02 0.126873795 0.181444461
## 2122 0.0448444100 2.481378e-02 0.233101687 0.186073956
## 2130 0.0451253264 1.775420e-02 0.208357287 0.186444533
## 2142 0.0539710300 1.357112e-02 0.248970384 0.187248432
## 2144 0.0451253264 1.167923e-02 0.212941728 0.187585195
## 2149 0.0355709745 1.002311e-02 0.138706835 0.187531100
## 2152 0.0057666997 6.739906e-03 0.028034311 0.187564153
## 2153 -0.0288068261 3.749038e-03 -0.094904865 0.180967624
## 2155 0.0403897727 9.427054e-03 0.123362799 0.186268090
## 2158 0.0267252709 8.057681e-03 0.087643483 0.186384034
## 2160 -0.0305355006 5.047802e-03 -0.109986206 0.179249437
## 2162 0.0403897727 6.115923e-03 0.122696732 0.186407809
## 2172 0.0267252709 4.537743e-03 0.087058561 0.187179800
## 2193 0.0302322337 4.868501e-03 0.094905807 0.189354911
## 2195 -0.0388683334 1.883857e-03 -0.092509374 0.178066557
## 2196 0.0081956098 2.719376e-03 -0.048531282 0.189711525
## 2199 0.0299513173 4.897550e-03 0.117028004 0.189912194
## 2205 0.0390779373 6.860639e-03 0.136322980 0.190674587
## 2207 0.0302322337 5.189323e-03 0.100460018 0.190993427
## 2208 0.0071394809 5.057655e-03 0.026139560 0.190248662
## 2212 0.0435262198 5.860075e-03 0.163404736 0.191190791
## 2233 0.0435262198 1.116430e-02 0.148974775 0.191917362
## 2236 0.0137219450 7.967417e-03 0.039002267 0.192026706
## 2243 -0.0051059990 1.128713e-02 0.037290240 0.193317897
## 2248 0.0501743503 2.328995e-02 0.136015717 0.197297866
## 2249 0.0504552666 2.399828e-02 0.114872465 0.197355480
## 2252 0.0164790957 3.255622e-02 -0.026449944 0.200012089
## 2258 -0.0280176643 2.489560e-02 0.019367826 0.180221524
## 2276 0.2371480006 1.931212e-01 0.274531738 0.236254260
## 2295 0.2595942960 3.286568e-01 0.324148660 0.252400242
## 2304 0.4124770958 4.579665e-01 0.393434460 0.271994301
## 2310 0.4706974082 4.756557e-01 0.417661236 0.281053871
## 2318 0.7580967541 6.345687e-01 0.446899353 0.301650180
## 2321 0.3613880070 3.316776e-01 0.524232509 0.248490637
## 2328 0.4221224888 3.774538e-01 0.587304681 0.262820559
## 2334 0.5989381283 5.577369e-01 0.351842699 0.306422686
## 2336 0.5978600757 6.748413e-01 0.283715390 0.309150238
## 2337 0.8465085208 1.008736e+00 0.466197034 0.316605189
## 2347 0.6812500437 6.917197e-01 0.421354987 0.321763297
## 2353 0.7222435167 8.027422e-01 0.438021537 0.326815462
## 2355 0.6607609646 7.698728e-01 0.339516323 0.331839033
## 2361 0.7706903278 8.693829e-01 0.407820966 0.336339738
## 2366 0.6614207414 8.431849e-01 0.364999593 0.338978497
## 2372 0.5652129339 5.429140e-01 0.355262400 0.333868424
## 2373 0.5568706511 6.517148e-01 0.353455690 0.337846072
## 2376 0.3590269166 3.268214e-01 0.237076029 0.335034086
## 2378 0.3579488639 4.643129e-01 0.166829745 0.334302729
## 2379 0.4826207816 5.288248e-01 0.343715020 0.335307667
## 2386 0.4453852662 4.154189e-01 0.333499602 0.337029364
## 2388 0.4763246827 4.561296e-01 0.317250580 0.342937472
## 2400 0.2229111236 2.730793e-01 0.241809901 0.340517185
## 2404 0.1781619165 1.532024e-01 0.131899773 0.344508556
## 2405 0.0579596902 1.074371e-01 0.164981042 0.292763039
## 2412 0.0721098354 9.949070e-02 0.200906069 0.294456516
## 2420 0.1259231115 2.135965e-01 0.080368746 0.350484355
## 2424 0.2960303328 2.223580e-01 0.371754234 0.357045539
## 2425 0.1511059857 1.301982e-01 0.298558315 0.345306104
## 2432 0.2997768606 1.406473e-01 0.315657769 0.363544764
## 2435 0.4318007237 3.145479e-01 0.430296073 0.377996700
## 2436 0.4020493521 3.015132e-01 0.432366299 0.384078905
## 2437 0.3631698994 2.721541e-01 0.419446422 0.376855435
## 2447 0.2341030643 1.334160e-01 0.529619568 0.305747849
## 2459 0.3510175107 2.386673e-01 0.511452948 0.367437061
## 2469 0.1666659265 2.021939e-01 0.349263188 0.338931834
## 2471 0.2545246781 2.289111e-01 0.518361270 0.350341017
## 2475 0.0959268180 7.486589e-02 0.393035038 0.255958310
## 2477 0.1193918275 1.556455e-01 0.496424101 0.318742949
## 2484 0.1076691266 1.447170e-01 0.353819648 0.280346925
## 2489 -0.0031708154 5.683423e-02 0.141498867 0.170220836
## 2492 0.0977414371 1.064081e-01 0.310131130 0.232847094
## 2517 -0.0226359860 9.667533e-03 -0.153021645 0.010045655
## 2521 0.0507222245 3.106548e-02 0.071267185 0.074280707
## 2526 0.0489307690 3.345958e-02 0.071463135 0.058873927
## 2531 -0.0226359860 7.443233e-03 -0.174809587 -0.037894923
## 2532 0.0132296433 2.440846e-02 -0.124843886 0.032070382
## 2533 0.0492545659 2.892997e-02 0.050738377 0.034139579
## 2538 -0.0226359860 9.095000e-03 -0.172375155 -0.057532794
## 2540 0.0492545659 2.923233e-02 0.034209787 0.016459509
## 2543 0.0274649890 7.475067e-03 -0.011089553 0.006724989
## 2547 0.0411125886 1.825065e-02 0.009462693 -0.001765115
## 2550 0.0503083832 1.778228e-02 -0.033874490 -0.009397429
## 2555 0.0682196992 2.099248e-02 -0.004421729 -0.013871735
## 2556 0.0590930792 1.875902e-02 -0.020696047 -0.017949446
## 2561 0.0730384974 2.637188e-02 -0.010763889 -0.028787663
## 2565 0.0264758772 1.797473e-02 -0.127289818 -0.039532364
## 2573 -0.0022393080 2.012550e-02 -0.054317206 -0.122236066
## 2577 0.1279552043 3.876590e-02 0.006450570 -0.043361154
## 2578 0.0292195836 3.451457e-02 -0.015274036 -0.055723746
## 2581 0.0298461072 4.723890e-02 -0.156872154 -0.055371574
## 2584 0.2216900166 8.308815e-02 0.008982126 -0.040054376
## 2590 0.1370818243 6.526910e-02 0.013768326 -0.056948056
## 2595 0.0298461072 7.095424e-02 -0.167927023 -0.071952377
## 2614 0.0135776451 2.816686e-02 -0.129954128 -0.089667056
## 2616 0.0124995925 7.595864e-02 -0.214397089 -0.092063668
## 2626 0.1480579243 8.704103e-02 -0.038754206 -0.080951805
## 2642 0.1742786092 9.922927e-02 -0.054536187 -0.077961712
## 2645 0.2630281511 1.942139e-01 0.060151258 -0.056442788
## 2648 0.2338445837 1.816074e-01 0.030454615 -0.054315063
## 2660 0.5262876720 4.151969e-01 0.132196348 -0.020166510
## 2661 0.5258972384 4.073428e-01 0.117970054 -0.029164390
## 2662 0.4195241467 3.485171e-01 0.100658179 -0.042910147
## 2664 0.3825647425 1.343052e-01 0.446552461 -0.187769907
## 2666 0.5119316938 4.302705e-01 0.135595879 -0.051374055
## 2670 0.4545659329 3.413678e-01 0.088045232 -0.103283431
## 2682 -0.0584174468 -3.830269e-17 -0.045270587 0.241047220
## 2686 -0.0187544967 -4.329870e-17 0.187674619 0.241049398
## 2695 -0.0394321515 5.000000e-07 0.091404565 0.241047996
## 2697 -0.0405102042 -4.329870e-17 0.022371607 0.241047059
## 2700 -0.0187544967 -4.329870e-17 0.187674619 0.241049398
## 2712 -0.0146327462 3.675144e-03 0.120046528 0.241049783
## 2718 -0.0557354620 1.291053e-05 -0.030861998 0.243534835
## 2724 -0.0736427046 3.380439e-04 -0.078271104 0.245338858
## 2729 -0.0160243236 6.714849e-03 0.244811241 0.250142916
## 2730 0.0023063156 8.398381e-03 0.203471615 0.251009533
## 2732 0.0460868186 4.707400e-03 0.174472830 0.252552379
## 2733 0.0887588478 4.774100e-03 0.371471987 0.252495587
## 2735 0.0669288659 3.328462e-03 0.372520217 0.254541955
## 2741 0.1020728022 2.225344e-02 0.606918321 0.255129659
## 2743 0.0787980797 1.244505e-02 0.489900957 0.254623428
## 2749 0.0785171633 2.775224e-02 0.519456492 0.255648631
## 2765 0.0692302394 5.746380e-02 0.439396599 0.257314110
## 2766 0.0502449442 5.995780e-02 0.288928527 0.257286206
## 2770 0.0806513449 7.353375e-02 0.493518512 0.257954765
## 2775 0.1120328198 1.188567e-01 0.588171196 0.257910523
## 2776 0.1176556205 1.220566e-01 0.593012876 0.258735303
## 2777 0.1085290006 1.058429e-01 0.582692686 0.258755923
## 2786 0.1432153334 1.357612e-01 0.501094145 0.262484746
## 2797 0.2627390709 1.926122e-01 0.663510576 0.269309820
## 2800 0.2937802919 2.416214e-01 0.558749093 0.271899844
## 2803 0.3273151792 3.614540e-01 0.589592369 0.274898642
## 2806 0.3163985472 2.831270e-01 0.561907034 0.277485363
## 2808 0.2645151336 3.423809e-01 0.358532284 0.280234522
## 2815 0.3249194323 4.520711e-01 0.370693176 0.285990989
## 2819 0.4177522018 4.702874e-01 0.617355608 0.291554012
## 2821 0.4091380525 4.303384e-01 0.523865562 0.295268664
## 2825 0.5171191761 4.349605e-01 0.645077602 0.300955558
## 2826 0.5221600595 4.361471e-01 0.631549563 0.302473582
## 2827 0.5493488297 4.340022e-01 0.612098802 0.304891314
## 2843 0.5540889579 5.443936e-01 0.350166468 0.339170186
## 2852 0.7576334154 6.845870e-01 0.641140558 0.368579221
## 2856 0.7239086102 7.620388e-01 0.581866816 0.391580456
## 2859 0.8048306475 7.832477e-01 0.661095224 0.404010279
## 2870 0.9009986777 9.213259e-01 0.534275799 0.456745261
## 2874 0.9909346434 9.515469e-01 0.641756474 0.475262940
## 2876 0.9096193403 9.525692e-01 0.592925144 0.481670621
## 2880 0.9736469497 9.440127e-01 0.604067250 0.497458018
## 2882 0.9337148681 9.680913e-01 0.593523061 0.510618714
## 2895 1.1982339811 1.083495e+00 0.684613920 0.586786650
## 2896 1.1067912221 1.098987e+00 0.668278119 0.591735785
## 2899 1.0232001573 1.088964e+00 0.445572634 0.616209818
## 2902 1.2273050737 1.154958e+00 0.711067411 0.637971547
## 2913 0.9014720906 1.104277e+00 0.516573338 0.696687649
## 2916 1.1410239334 1.118846e+00 0.778187050 0.712085962
## 2918 1.1060669897 1.137125e+00 0.725631580 0.722216294
## 2921 0.9823848721 1.035052e+00 0.577557133 0.724171422
## 2931 0.9236894771 9.406503e-01 0.719303803 0.772361767
## 2932 0.9067798784 9.261613e-01 0.696278611 0.779738984
## 2935 0.6219266022 7.858505e-01 0.555062726 0.775768083
## 2942 0.4443735842 6.062248e-01 0.544592942 0.793843968
## 2948 0.4105908764 5.270816e-01 0.447185555 0.812547151
## 2954 0.4498633480 5.515451e-01 0.589120507 0.831532644
## 2959 0.4837818249 5.711840e-01 0.751663499 0.846910014
## 2966 0.4813060772 5.296067e-01 0.760274340 0.864790189
## 2968 0.4384103109 5.024680e-01 0.666835638 0.866784984
## 2969 0.3239614199 4.633036e-01 0.531212063 0.858894269
## 2972 0.4948302613 5.364595e-01 0.782684301 0.880315869
## 2980 0.4769482158 5.112407e-01 0.795446597 0.901776325
## 2982 0.4355964249 4.938839e-01 0.697513892 0.905936772
## 2989 0.4516610363 4.790277e-01 0.751302507 0.916742561
## 2990 0.3993909772 4.213574e-01 0.615724476 0.916017743
## 3000 0.3431438035 3.484976e-01 0.859329136 0.925599486
## 3003 0.2913574957 3.500449e-01 0.739858361 0.932791404
## 3018 0.1816304589 2.241982e-01 0.575769894 0.942936446
## 3023 0.0379969564 1.179801e-01 0.783331029 0.932055467
## 3027 0.2773631300 2.011858e-01 0.828283945 0.952189311
## 3047 0.0545017494 1.045274e-01 0.575070803 0.958124007
## 3056 0.1527528058 9.559870e-02 0.757816885 0.967561828
## 3070 0.2191092115 1.132510e-01 0.769686686 0.970664336
## 3071 0.2081976182 1.165445e-01 0.754827811 0.974319053
## 3075 0.0542794023 1.154731e-01 0.509000279 0.966343426
## 3076 0.1635002961 1.235631e-01 0.688691319 0.972498726
## 3083 0.2469652610 1.289141e-01 0.691037276 0.974997139
## 3084 0.2742148211 1.294655e-01 0.697560192 0.977253907
## 3090 0.3380561818 1.766750e-01 0.720909410 0.988740478
## 3097 0.3737140522 2.564927e-01 0.766632367 0.997309186
## 3098 0.4032544203 3.043794e-01 0.777300285 1.001209979
## 3107 0.6572563165 6.349711e-01 0.983470932 1.074051177
## 3110 0.4604168613 6.811396e-01 0.924776528 1.077270963
## 3112 0.9428743209 7.551292e-01 1.144222705 1.089195014
## 3123 1.8791383370 1.525972e+00 1.171718300 1.255988357
## 3124 2.0480574438 1.938599e+00 1.271362915 1.292845973
## 3128 2.5491231804 2.490719e+00 1.525620010 1.654372924
## 3130 2.7044438542 2.717915e+00 1.372919417 1.414185147
## 3137 3.5392536893 3.553739e+00 1.591552432 1.570585566
## 3138 3.7234052953 3.617405e+00 1.670663887 1.604514212
## 3140 3.9221595691 3.854796e+00 1.893897867 1.703505896
## 3146 4.2100307701 4.011872e+00 1.944281221 1.764684135
## 3164 4.1246171976 3.832545e+00 1.650258370 1.848233078
## 3172 2.8489062580 2.959292e+00 1.318960667 1.785224963
## 3174 3.9728161479 4.029231e+00 1.559255086 1.793115283
## 3175 3.6908224560 4.310664e+00 1.565932702 1.793613994
## 3186 1.3687024759 1.427915e+00 1.340773443 1.738366753
## 3187 1.3044161121 1.338834e+00 1.410291337 1.827050589
## 3191 1.0984926061 1.283458e+00 1.561939938 1.750688800
## 3193 1.1650099743 1.035206e+00 1.347706971 1.661860759
## 3196 0.8565440868 9.768031e-01 1.583156087 1.618285861
## 3202 0.8505166076 8.432104e-01 1.446753286 1.595395194
## 3211 1.3080377215 6.356909e-01 1.382586969 1.530676500
## 3215 1.0468841857 1.750212e+00 1.187931737 1.408358933
## 3223 1.0429249788 5.744516e-01 1.321925331 1.440085675
## 3227 0.6643353676 7.139344e-01 1.436949597 2.362342493
## 3228 0.4761648994 4.757351e-01 1.063791238 1.302668096
## 3243 0.4076370408 3.749861e-01 1.088508079 1.146675558
## 3245 0.4413922816 4.898736e-01 1.263252415 1.248725371
## 3246 0.6003054014 4.893896e-01 1.248642707 1.217251680
## 3247 0.3847211363 4.963285e-01 1.222980656 1.245728677
## 3252 0.4391266768 5.352153e-01 1.235324138 1.259030183
## 3255 0.4174307298 4.818190e-01 1.112455369 1.223081171
## 3268 0.6458726662 3.034243e-01 1.112989433 0.863804385
## 3275 0.3318638783 3.157948e-01 1.016579311 0.797776441
## 3286 0.3860689199 2.761894e-01 0.846812629 0.511284089
## 3288 0.2450943017 2.901169e-01 0.799042556 0.530542549
## 3292 0.3006384152 2.169603e-01 0.567666562 0.423113783
## 3299 0.0254886481 2.014371e-01 0.438097954 0.279790318
## 3311 0.5234870729 5.077547e-01 0.325126132 0.120540221
## 3313 0.3747986189 3.843096e-01 0.207798313 0.049581411
## 3314 0.4787744495 5.031626e-01 0.362425568 0.127637603
## 3324 0.3055623608 3.961598e-01 0.210389839 -0.133315927
## 3327 0.2227065833 3.347847e-01 0.021886396 -0.077962678
## 3328 0.3304018019 3.499451e-01 0.173427855 -0.113401590
## 3338 0.4258646970 4.319265e-01 -0.151871458 -0.294291429
## 3340 0.4221261433 2.650713e-01 -0.424291434 -0.350527268
## 3345 0.3389646037 4.222494e-01 -0.388331116 -0.328620416
## 3348 0.2549987849 2.662521e-01 -0.610222683 -0.525517718
## 3350 0.5131755399 4.045721e-01 -0.433022110 -0.513404395
## 3364 -0.0486429698 4.443838e-06 -0.273655185 0.155627683
## 3365 -0.0307357272 1.277171e-06 -0.206013037 0.155628392
## 3366 0.0103669886 7.771709e-07 -0.027108929 0.155630987
## 3369 -0.0086991033 1.267717e-05 -0.062529236 0.155630766
## 3370 -0.0296576745 6.777171e-06 -0.136980033 0.155629394
## 3376 -0.0086991033 6.677171e-06 -0.062529236 0.155629831
## 3386 -0.0322171566 3.412886e-03 -0.129452906 0.162414854
## 3387 0.0073162459 4.151052e-03 0.050035571 0.162439114
## 3388 -0.0029041425 4.776698e-03 0.050658249 0.162483865
## 3390 -0.0117498461 7.480379e-03 0.016329764 0.162633838
## 3391 -0.0327084173 1.002778e-02 -0.056897769 0.162667179
## 3392 -0.0625415467 1.593759e-02 -0.272892812 0.163042911
## 3393 -0.0110306720 3.798100e-02 0.073945670 0.163702946
## 3396 0.1094816406 4.332004e-02 0.432847048 0.165092377
## 3408 0.3561685681 4.365610e-01 0.793997525 0.172713568
## 3415 0.7890259849 7.273686e-01 0.903215045 0.179224275
## 3431 0.6442244474 6.133377e-01 0.643511947 0.185708010
## 3432 0.6490306079 6.125703e-01 0.597183058 0.185713780
## 3447 0.3207075380 3.772973e-01 0.329589446 0.187208606
## 3451 0.3166504688 3.630189e-01 0.400895291 0.187062775
## 3467 0.1700764886 1.895966e-01 0.297491721 0.182410747
## 3469 0.1403587678 1.554054e-01 0.085480718 0.182748260
## 3478 0.1250396351 1.016708e-01 0.280165748 0.180222243
## 3480 0.1056926268 1.086014e-01 0.263423450 0.180231616
## 3481 0.1059735431 1.036162e-01 0.238821995 0.180062619
## 3489 0.0420548682 7.210552e-02 0.048259230 0.179168653
## 3496 0.0363737057 5.010745e-02 0.036098614 0.178716174
## 3497 0.0173574539 4.531765e-02 -0.100240452 0.178633275
## 3506 0.0816764524 2.060382e-02 0.130568847 0.177727909
## 3511 0.0201907463 1.746515e-02 -0.114710275 0.177415398
## 3512 0.0380979889 1.981972e-02 -0.048872985 0.177335029
## 3519 0.0158216222 1.502780e-02 -0.163110506 0.177038324
## 3538 0.0065393617 7.641070e-03 -0.091631528 0.176138817
## 3548 0.0535538806 9.188873e-03 0.017858165 0.175162876
## 3552 0.0065224594 6.279992e-03 -0.087931172 0.175097893
## 3553 -0.0054560778 5.928373e-03 -0.223439573 0.175179335
## 3554 0.0124511648 6.344640e-03 -0.155931061 0.174970481
## 3559 0.0065224594 5.728726e-03 -0.076209030 0.175169706
## 3560 -0.0054560778 5.807640e-03 -0.213628543 0.175007007
## 3564 0.0248037381 6.221106e-03 0.027953715 0.175730452
## 3570 0.0299841744 7.248540e-03 0.054993751 0.176051159
## 3575 0.0030480306 6.832403e-03 -0.123207497 0.175559066
## 3578 0.0253058370 9.795133e-03 0.071672408 0.175818717
## 3590 0.0509954052 1.263592e-02 0.093065889 0.175149228
## 3605 0.0372618349 2.157451e-02 0.100203003 0.174953577
## 3611 0.0286542792 2.614732e-02 0.115108796 0.176087611
## 3615 0.0120371941 3.042262e-02 0.043267160 0.179558664
## 3619 0.0636492144 6.111523e-02 0.199459694 0.182388328
## 3624 0.0649214689 8.113543e-02 0.086778119 0.186875205
## 3628 0.2838823472 1.745234e-01 0.334315338 0.201043746
## 3629 0.2556627274 1.692377e-01 0.285289348 0.202651944
## 3631 0.2283932207 2.043210e-01 0.263120764 0.202946067
## 3632 0.3942524289 2.665504e-01 0.473225387 0.208962137
## 3633 0.3904316434 2.797137e-01 0.501812998 0.213281578
## 3635 0.4838640544 3.061705e-01 0.517265479 0.222473901
## 3636 0.4463433155 2.696036e-01 0.466574410 0.225032729
## 3639 0.5253173946 4.084565e-01 0.690888375 0.226812019
## 3640 0.5255098726 3.674676e-01 0.708825813 0.230835557
## 3641 0.4936019123 3.843758e-01 0.713919308 0.236678659
## 3644 0.4365614870 4.334803e-01 0.534201527 0.242286787
## 3669 0.7274336591 7.171210e-01 0.765240531 0.276703744
## 3672 0.4940151433 5.850606e-01 0.506055811 0.275810579
## 3680 0.4585789873 4.987440e-01 0.537374500 0.276285466
## 3681 0.7011546449 6.320451e-01 0.712325704 0.279720956
## 3682 0.8020167848 5.667571e-01 0.737823324 0.318498166
## 3685 0.4812609659 5.191704e-01 0.640932817 0.284851681
## 3692 0.3897636725 4.365531e-01 0.569878728 0.284887030
## 3699 0.4602455391 4.397743e-01 0.602833771 0.292763597
## 3701 0.4027438227 4.185009e-01 0.540656760 0.293754719
## 3705 0.4508739189 5.140607e-01 0.707928141 0.303638468
## 3706 0.4505584177 4.473780e-01 0.619061364 0.306713960
## 3707 0.3643930224 4.108769e-01 0.481199912 0.305874838
## 3708 0.3447535479 4.194389e-01 0.544187646 0.300960367
## 3710 0.4785180889 4.977687e-01 0.711169313 0.305607516
## 3713 0.3647056326 4.436097e-01 0.513470965 0.307012652
## 3714 0.3017389896 3.838738e-01 0.346387537 0.303402928
## 3717 0.3866433115 4.955428e-01 0.539452823 0.305371723
## 3721 0.3061846105 3.677117e-01 0.252957717 0.305186129
## 3727 0.3133621121 4.072769e-01 0.384394638 0.308339657
## 3729 0.2378864154 3.538147e-01 0.228616896 0.304040847
## 3737 0.3604292436 3.940482e-01 0.408746732 0.307516787
## 3738 0.3592482148 3.663673e-01 0.407972290 0.310043195
## 3742 0.2756760179 2.609040e-01 0.147032639 0.308885962
## 3745 0.3382733561 3.373088e-01 0.401582093 0.311960541
## 3747 0.3452817897 3.138967e-01 0.364776391 0.316147208
## 3765 0.4833452808 3.720725e-01 0.426531944 0.331641744
## 3768 0.4695891759 3.662018e-01 0.410645147 0.340651807
## 3771 0.3878425068 3.543716e-01 0.272335381 0.331211220
## 3772 0.5550504087 3.970266e-01 0.469389921 0.335894313
## 3773 0.5058348117 3.833747e-01 0.473926530 0.339515379
## 3774 0.4781043082 3.876232e-01 0.460471510 0.341937028
## 3779 0.5631441855 4.517008e-01 0.497012016 0.340838937
## 3780 0.5370769833 4.333699e-01 0.499322748 0.344098616
## 3782 0.5385003056 4.611717e-01 0.462034066 0.347940681
## 3787 0.5116179477 4.767694e-01 0.493855106 0.350140682
## 3788 0.5550725610 4.832552e-01 0.615659205 0.348428868
## 3792 0.4157342506 4.289851e-01 0.437790751 0.337389139
## 3804 0.3312968127 3.698072e-01 0.442084866 0.342719490
## 3805 0.2882497408 3.118016e-01 0.293153380 0.337745944
## 3807 0.4188345189 4.053562e-01 0.538688709 0.338235384
## 3814 0.3440820744 3.396832e-01 0.496122870 0.331333149
## 3816 0.2792644637 3.166130e-01 0.476225512 0.335138355
## 3822 0.3190111804 2.643787e-01 0.549355748 0.319024283
## 3825 0.2544242788 2.098741e-01 0.416606574 0.312489699
## 3836 0.1625892099 1.750844e-01 0.440304706 0.283215280
## 3838 0.1537435063 1.561411e-01 0.385424528 0.278508024
## 3839 0.1208453880 1.386862e-01 0.304821394 0.272635131
## 3841 0.0730595095 1.133251e-01 0.209226910 0.261451497
## 3842 0.1377669525 1.296711e-01 0.392013857 0.258731461
## 3848 0.0764547853 8.503723e-02 0.174593135 0.236314057
## 3856 0.0820134486 7.303873e-02 0.237070516 0.199261910
## 3860 0.0419887854 4.946983e-02 0.111527069 0.184312154
## 3862 0.0459228704 4.339073e-02 0.027795309 0.179436664
## 3863 0.0820134486 4.988257e-02 0.212397516 0.177637508
## 3865 0.0505651431 3.999380e-02 0.187648228 0.174208173
## 3877 0.0627134692 3.264003e-02 0.156775698 0.146751281
## 3878 0.0524930808 2.864160e-02 0.153238778 0.144720533
## 3879 0.0288442109 2.014803e-02 -0.010098859 0.138971203
## 3880 0.0291251273 2.207173e-02 -0.034216586 0.136292050
## 3881 0.0081665561 1.816633e-02 -0.109647846 0.133631825
## 3894 0.0261922508 1.810423e-02 -0.013456892 0.110834926
## 3897 0.0269562087 1.763652e-02 -0.164812667 0.103358866
## 3898 0.0754473663 2.271273e-02 0.023720387 0.102268326
## 3902 0.0567583784 1.611847e-02 -0.092766217 0.091730518
## 3904 0.0318719139 1.855220e-02 -0.172559761 0.087588672
## 3906 0.0985022003 2.217743e-02 0.017086513 0.084752730
## 3907 0.0893755803 2.239017e-02 0.001302808 0.083104475
## 3911 0.0245262846 2.114960e-02 -0.195987494 0.070082863
## 3915 0.0611199265 2.585630e-02 -0.027286442 0.066807833
## 3916 0.0282218082 2.367923e-02 -0.099496170 0.063964801
## 3920 0.0699656301 3.183097e-02 0.005174109 0.059596472
## 3927 0.0699656301 4.837167e-02 0.015333851 0.053126910
## 3929 0.0611199265 4.853753e-02 -0.020772697 0.051196011
## 3933 0.0801860185 5.304623e-02 0.017866589 0.048371006
## 3936 0.0760131130 5.363467e-02 -0.022500426 0.047384099
## 3937 0.0282218082 4.686610e-02 -0.098811360 0.044429862
## 3938 0.0092365130 4.100220e-02 -0.242732674 0.041764424
## 3942 0.0652872927 6.398227e-02 0.015953148 0.041565451
## 3944 0.0326700907 5.666603e-02 -0.084367965 0.037599199
## 3949 0.0652872927 6.568203e-02 -0.004364583 0.033377737
## 3952 0.0136847955 4.378820e-02 -0.245999883 0.027201942
## 3967 -0.0013797012 4.430690e-02 -0.224525896 0.011965777
## 3980 -0.0367252045 3.409177e-02 -0.297212509 -0.001693378
## 3989 0.0110104348 4.177260e-02 -0.043761072 -0.012021446
## 3990 0.0418010473 3.988823e-02 -0.056755522 -0.012797616
## 3995 -0.0238542113 3.681913e-02 -0.249015864 -0.020759195
## 3996 0.0676631873 4.546963e-02 -0.040640850 -0.021296009
## 4004 0.0705251975 5.065527e-02 0.026885407 -0.031853155
## 4009 0.0375879587 5.020203e-02 -0.184408767 -0.040864109
## 4014 0.0500828686 5.250260e-02 -0.092205622 -0.048735144
## 4016 0.0375879587 5.881980e-02 -0.180460562 -0.053831229
## 4020 0.0990251350 6.473833e-02 -0.016172881 -0.058205806
## 4024 0.1120970618 7.654083e-02 0.033167179 -0.066806472
## 4036 0.0639171470 7.563867e-02 -0.163426251 -0.109279785
## 4037 0.0305823714 9.617731e-02 -0.148096891 -0.115321408
## 4038 0.1701614375 1.047403e-01 0.057626080 -0.114843431
## 4045 0.0256717498 -4.274359e-17 0.151189998 0.245168380
## 4055 0.0602948228 -4.318768e-17 0.261061147 0.245170037
## 4060 -0.0065167982 -4.352074e-17 0.014514891 0.245166733
## 4070 0.0554760246 -4.363176e-17 0.261827197 0.245169931
## 4074 -0.0071446659 -4.085621e-17 -0.066714386 0.245166735
## 4080 0.0158308954 1.287900e-02 0.084083177 0.246375872
## 4083 0.0596669551 8.058000e-03 0.210160896 0.247822165
## 4086 0.0460024533 2.222233e-03 0.176998746 0.247821291
## 4095 -0.0178390820 1.101152e-03 0.039622270 0.253217681
## 4098 0.0441537408 6.691738e-04 0.294358248 0.253316637
## 4106 0.0832536272 1.000645e-02 0.461341024 0.257509483
## 4109 0.0303874244 1.090334e-02 0.233603414 0.257412908
## 4110 0.0482946670 1.271798e-02 0.301418844 0.257453841
## 4112 0.0923802471 1.600151e-02 0.488187216 0.257692022
## 4115 0.0701005158 2.241455e-02 0.385028287 0.258310203
## 4118 0.1047235888 2.609746e-02 0.515448467 0.259697938
## 4122 0.1114535291 5.135744e-02 0.441380390 0.262351274
## 4123 0.0862717392 5.562045e-02 0.317238675 0.263876891
## 4131 0.0971722238 6.867858e-02 0.389985078 0.266273928
## 4147 0.2633881788 1.029241e-01 0.706679976 0.277193302
## 4150 0.2139385628 1.153419e-01 0.584197434 0.277949483
## 4154 0.2130507532 1.337247e-01 0.668408715 0.279418587
## 4160 0.2213359893 1.568184e-01 0.651500184 0.282209306
## 4168 0.2755700882 1.827979e-01 0.542683961 0.284615635
## 4169 0.2846362253 1.839354e-01 0.528665072 0.285295983
## 4170 0.2642486369 1.914215e-01 0.506948533 0.285582311
## 4174 0.2302834565 1.699736e-01 0.530746109 0.286321584
## 4184 0.1844541917 1.800568e-01 0.494944763 0.289561514
## 4189 0.1784067088 1.642723e-01 0.524542767 0.290483737
## 4193 0.1153065470 1.275592e-01 0.274692716 0.291264091
## 4197 0.1992597579 1.593568e-01 0.394597786 0.292392440
## 4204 0.2000041887 1.664883e-01 0.398659039 0.294157587
## 4209 0.1899049284 1.564360e-01 0.403628920 0.294785110
## 4216 0.1899049284 1.450668e-01 0.397007633 0.295774271
## 4224 0.1862635625 1.466756e-01 0.397162790 0.297253298
## 4225 0.1771369426 1.421562e-01 0.382789181 0.297480642
## 4230 0.1917107113 1.104394e-01 0.396441066 0.298199204
## 4231 0.1868919130 1.184548e-01 0.398510172 0.298450490
## 4234 0.1458925218 1.145245e-01 0.288055893 0.298966786
## 4235 0.1129595431 8.298625e-02 0.149131113 0.299133727
## 4236 0.1369471510 7.437922e-02 0.214338026 0.299296649
## 4239 0.1785097238 1.164719e-01 0.382800822 0.299771234
## 4241 0.1458925218 1.125307e-01 0.285512925 0.300093710
## 4247 0.2237081662 1.401751e-01 0.343941612 0.319889560
## 4249 0.1129595431 7.725765e-02 0.130951963 0.299953891
## 4252 0.1876363438 1.167156e-01 0.379379152 0.300329535
## 4255 0.1458925218 1.103900e-01 0.268314115 0.300898279
## 4258 0.1874204567 1.131111e-01 0.373398880 0.298720059
## 4263 0.1086692886 7.616226e-02 0.128867829 0.299883347
## 4266 0.1833460892 1.162928e-01 0.380370521 0.300784238
## 4269 0.1416022673 1.124921e-01 0.272323911 0.301477347
## 4272 0.1881648874 1.257713e-01 0.382433297 0.302122347
## 4275 0.1745003856 1.356473e-01 0.349361370 0.303021673
## 4278 0.1265765312 8.012621e-02 0.204398213 0.303972864
## 4283 0.1416022673 1.324294e-01 0.285331819 0.306707738
## 4288 0.2082036148 1.630586e-01 0.387337684 0.309781152
## 4289 0.2782628448 2.060241e-01 0.379185714 0.339603164
## 4299 0.2205887944 1.671337e-01 0.246500814 0.320746672
## 4303 0.3026109299 2.652752e-01 0.407593272 0.324399241
## 4307 0.3944205704 2.823829e-01 0.441258376 0.328483037
## 4317 0.4010366025 3.454588e-01 0.416491327 0.338261312
## 4320 0.3761562006 2.734633e-01 0.330983511 0.341004114
## 4325 0.4336355797 3.542346e-01 0.419381621 0.347905403
## 4332 0.4502199509 3.968464e-01 0.428319555 0.357686416
## 4336 0.4945075972 4.313941e-01 0.547673612 0.361153374
## 4337 0.4813557235 4.434929e-01 0.537098693 0.364827538
## 4349 0.5451745644 5.070696e-01 0.655223463 0.384307017
## 4352 0.5215524951 5.184231e-01 0.631236728 0.389685861
## 4356 0.5604578979 5.239938e-01 0.673739695 0.394846381
## 4360 0.5120247363 5.565066e-01 0.571465199 0.402847979
## 4363 0.6108264564 5.301605e-01 0.665525890 0.406133358
## 4365 0.5558515397 5.750298e-01 0.670088386 0.410660989
## 4369 0.5109279055 4.910720e-01 0.510274387 0.418287000
## 4370 0.5896080285 5.458720e-01 0.693639971 0.419340795
## 4378 0.5646451044 5.494638e-01 0.711003670 0.430215016
## 4401 0.5702212318 5.505859e-01 0.649189236 0.457661359
## 4404 0.5090921436 4.752487e-01 0.496446022 0.459113775
## 4409 0.5302126278 5.252152e-01 0.554970655 0.463543613
## 4411 0.4689000386 4.620474e-01 0.405287726 0.464245101
## 4423 0.3520663078 4.641742e-01 0.453640973 0.472181782
## 4424 0.3005698891 4.248922e-01 0.327772447 0.472351187
## 4428 0.3645849575 4.529848e-01 0.557887816 0.474665006
## 4429 0.3648658739 4.454562e-01 0.534947823 0.475378842
## 4435 0.3667129342 4.229326e-01 0.562594910 0.477480935
## 4437 0.3340494814 4.213642e-01 0.463508527 0.478938593
## 4442 0.3282817028 4.163423e-01 0.453889442 0.478815132
## 4443 0.2994756543 4.133953e-01 0.422137349 0.472878312
## 4444 0.2665775360 4.096815e-01 0.347856911 0.472529279
## 4454 0.3473862631 4.046052e-01 0.475127019 0.554280884
## 4461 0.3210440461 3.854731e-01 0.475241104 0.558062161
## 4466 0.2073023403 3.535494e-01 0.198168555 0.490330267
## 4471 0.3516003420 3.798185e-01 0.524859763 0.464202685
## 4472 0.3187022237 3.808707e-01 0.450180678 0.466706697
## 4473 0.2857692450 3.344001e-01 0.310777352 0.463218035
## 4474 0.3097568528 3.158857e-01 0.385692921 0.464411767
## 4475 0.3645204131 3.462531e-01 0.567397171 0.467245378
## 4483 0.4319382519 3.831733e-01 0.588454861 0.540674217
## 4486 0.4165366470 3.896630e-01 0.481222715 0.538836168
## 4489 0.3707955509 3.576498e-01 0.553725167 0.463265530
## 4495 0.3160319906 3.354817e-01 0.372863235 0.458668082
## 4499 0.4230925483 3.826741e-01 0.540857836 0.535836806
## 4522 0.2899251638 3.604554e-01 0.379969923 0.444550777
## 4529 0.3284635117 3.683008e-01 0.375162417 0.443635292
## 4531 0.4022293758 3.751209e-01 0.519675642 0.456149502
## 4542 0.3690168467 3.823677e-01 0.423890352 0.448579350
## 4545 0.4273364280 3.852962e-01 0.562328427 0.444381668
## 4552 0.5289604906 4.838396e-01 0.572676239 0.458736187
## 4557 0.5202072450 5.651220e-01 0.343412138 0.453882773
## 4558 0.5419179631 5.645228e-01 0.400930141 0.454556240
## 4559 0.6119814015 6.272500e-01 0.575051514 0.456667486
## 4564 0.5555034509 6.632560e-01 0.228723347 0.455950071
## 4569 0.6646046740 7.403479e-01 0.420335778 0.465832124
## 4577 0.5844600551 7.481795e-01 0.341279523 0.443536636
## 4586 0.5892577529 7.602116e-01 0.250169054 0.441487140
## 4590 0.7342939664 7.746678e-01 0.426174077 0.548773236
## 4592 0.6305537275 7.691481e-01 0.215406191 0.535619058
## 4607 0.5764538651 7.606896e-01 0.224877415 0.420786372
## 4608 0.7101417679 7.658632e-01 0.446980873 0.526203751
## 4612 0.6495963265 7.713587e-01 0.326833604 0.530432498
## 4628 0.6841124678 7.572855e-01 0.278814127 0.516490655
## 4631 0.7324513600 7.748246e-01 0.443937948 0.523979494
## 4634 0.7127834219 7.623505e-01 0.221404820 0.520248844
## 4645 0.7551723912 7.434416e-01 0.620586405 0.660532861
## 4646 0.8016117804 7.990349e-01 0.411487104 0.469894961
## 4648 0.7357318041 7.949988e-01 0.202628909 0.474833042
## 4651 0.8233196256 8.337412e-01 0.396762196 0.362490614
## 4653 0.8468144930 8.645549e-01 0.406689155 0.478636995
## 4656 0.7315023358 8.657596e-01 0.215154355 0.356367032
## 4657 0.9205158045 8.889071e-01 0.445398536 0.473468655
## 4666 0.8764070570 9.729792e-01 0.385446453 0.345731720
## 4674 1.0002032020 1.035013e+00 0.819073295 4.028003509
## 4675 0.9201910083 1.036587e+00 0.747014629 4.056201848
## 4678 1.0427676480 1.045112e+00 0.740645384 0.851208764
## 4681 1.1362898907 1.102551e+00 0.447174043 0.550427892
## 4682 1.0720145707 1.102601e+00 0.376031595 0.552164894
## 4690 1.1400742577 1.158869e+00 0.319359961 0.507656723
## 4696 1.1706844644 1.199260e+00 0.381093845 0.395322446
## 4698 0.9960745162 1.202853e+00 0.300417616 0.388815827
## 4709 1.1830626325 1.208074e+00 0.473854946 0.496859750
## 4714 1.2846104790 1.179362e+00 0.568384273 0.483606584
## 4725 0.9107994939 1.124267e+00 0.128176443 0.311017899
## 4726 -0.0200174456 2.479651e-04 -0.171632269 0.183507656
## 4727 -0.0291440656 1.337267e-02 -0.185999415 0.183506572
## 4729 -0.0498217204 1.269900e-02 -0.282269288 0.183505425
## 4732 -0.0166800767 4.883095e-04 -0.122609329 0.187950419
## 4735 -0.0303445786 1.923000e-03 -0.148691341 0.188925454
## 4743 0.0016325128 2.953451e-02 -0.192924137 0.193438639
## 4749 0.0617555300 8.173315e-02 0.480895119 0.195289448
## 4753 0.1281255247 8.704198e-02 0.504473279 0.197141714
## 4755 0.1221802336 9.357070e-02 0.498348987 0.197797207
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## 4759 0.0884849790 1.038586e-01 0.300839779 0.198964825
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## 4765 0.0600813196 1.171800e-01 0.138874609 0.200652442
## 4771 0.0466096328 1.033317e-01 0.218514990 0.200597081
## 4788 0.0824168492 5.493136e-02 0.303895317 0.201310571
## 4789 0.0775980509 5.225201e-02 0.295439067 0.201331223
## 4790 0.0684714310 4.909401e-02 0.274952082 0.201406796
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## 4807 0.0551798275 3.040747e-02 0.011586551 0.201446034
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## 4825 0.0315794449 2.119801e-02 0.120584982 0.200638781
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## 4853 0.0140412699 1.777642e-02 -0.015166012 0.201143891
## 4854 0.0143221863 1.755350e-02 -0.036181230 0.201156940
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## 4880 0.0241683608 1.837249e-02 -0.028268432 0.199577584
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## 4890 -0.0035017324 2.037413e-02 -0.149375902 0.199709441
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## 4929 0.0196842003 6.219082e-02 0.021180962 0.203160139
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## 4986 0.2516884438 1.833079e-01 0.208327377 0.240076646
## 4990 0.1992509164 1.859930e-01 0.305491456 0.243658986
## 4992 0.3492994286 2.001969e-01 0.648009829 0.247985348
## 5002 0.2409510854 2.317987e-01 0.426724753 0.314060578
## 5003 0.2050764887 2.348847e-01 0.281636858 0.313560663
## 5013 0.2462254116 2.517917e-01 0.489003474 0.320944620
## 5024 0.1838692881 1.992294e-01 0.146571363 0.323130799
## 5031 0.0729555516 1.789614e-01 0.089573653 0.324603018
## 5032 0.0916072249 1.802839e-01 0.150753134 0.326118742
## 5034 0.1344290995 1.757030e-01 0.311967413 0.326440641
## 5036 0.2152076242 1.745928e-01 0.280008836 0.378120111
## 5042 0.1721749231 1.520491e-01 0.267918152 0.408455691
## 5045 0.1205724260 1.366972e-01 0.022498230 0.416076261
## 5057 0.1095034235 1.122835e-01 0.196167604 0.516338440
## 5061 0.1238344680 1.010966e-01 0.245053869 0.531285057
## 5066 0.0486242852 9.649078e-02 0.008115945 0.329144469
## 5067 0.1021344473 9.204803e-02 0.085036774 0.335562218
## 5072 0.0841227713 8.231087e-02 0.161551022 0.328513513
## 5074 0.0830447187 8.163764e-02 0.094534607 0.330682930
## 5075 0.1770832857 7.983990e-02 0.276170750 0.334868239
## 5079 0.1104649884 7.616727e-02 0.182899069 0.328920169
## 5082 0.1112444025 7.134850e-02 0.212276551 0.331914037
## 5083 0.0924427830 6.931790e-02 0.215693165 0.330406777
## 5084 0.0833161631 6.898797e-02 0.203516562 0.330073774
## 5088 0.0496209085 6.853997e-02 0.058816824 0.329903026
## 5089 0.1351987061 7.018253e-02 0.254386082 0.334372249
## 5091 0.1072704667 7.000553e-02 0.254022357 0.333792979
## 5101 0.1226838424 1.008156e-01 0.061627353 0.337502866
## 5104 0.2335041283 1.429852e-01 0.328186218 0.354371048
## 5105 0.2203522546 1.488898e-01 0.321627791 0.351607171
## 5108 0.1528895259 1.639894e-01 0.150612925 0.390279449
## 5113 0.3519416072 1.898779e-01 0.545366558 0.373006731
## 5123 0.2936925532 2.531315e-01 0.497089892 0.409889458
## 5128 0.4000562641 3.115757e-01 0.630232464 0.432806406
## 5130 0.3727241913 3.274213e-01 0.566958415 0.431692475
## 5135 0.3564687508 3.392246e-01 0.600165079 0.433419848
## 5137 0.2999740576 3.438883e-01 0.512732242 0.436089002
## 5140 0.3699234127 3.421860e-01 0.622656486 0.443089079
## 5141 0.3803523702 3.400119e-01 0.582296118 0.434920069
## 5142 0.3782488416 3.353030e-01 0.596337141 0.430119171
## 5154 0.2258036079 2.393563e-01 0.494385016 0.418425292
## 5155 0.2260845243 2.377048e-01 0.462335957 0.420543867
## 5162 0.1535176787 2.132819e-01 0.399536496 0.424573242
## 5171 0.0985174569 1.380112e-01 0.144345922 0.411223655
## 5178 0.0109996786 9.051983e-02 0.009245767 0.407328019
## 5185 0.0109996786 7.534589e-02 -0.001421552 0.407684312
## 5194 0.0765475940 5.902153e-02 0.247697682 0.387070504
## 5203 0.0367322525 4.755789e-02 0.100820548 0.369122362
## 5204 0.0762462128 4.626626e-02 0.074328728 0.369136993
## 5206 -0.0148702447 4.288506e-02 -0.144056885 0.358770158
## 5208 0.0664145444 4.880089e-02 0.093919236 0.350351472
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## 5216 0.0672344122 4.662376e-02 0.076285679 0.324751654
## 5219 0.0210899629 4.959553e-02 -0.036567014 0.308820745
## 5223 0.0600107409 5.723167e-02 0.084392555 0.304516669
## 5226 0.0677875749 6.027083e-02 -0.006830210 0.303457487
## 5230 0.2388275831 8.097110e-02 0.145979860 0.308850147
## 5232 0.2345412822 8.667550e-02 0.128877345 0.297593103
## 5239 0.2166202495 1.257294e-01 0.173492256 0.286965529
## 5240 0.2064047572 1.469160e-01 0.103441705 0.286233938
## 5253 0.2558700072 2.153261e-01 0.028443743 0.241854800
## 5256 0.2220193013 2.189857e-01 -0.117451839 0.237174168
## 5261 0.1429061011 2.449236e-01 -0.064081319 0.215801179
## 5264 0.2505803058 2.486804e-01 0.043110823 0.226401703
## 5265 0.2245153490 2.679753e-01 0.073848021 0.223497421
## 5274 0.2128560909 2.591681e-01 0.093524267 0.214070079
## 5290 0.1787621721 2.044611e-01 -0.117777465 0.196963882
## 5291 0.1966694146 2.236798e-01 -0.052096988 0.202609719
## 5292 0.2751822101 2.269825e-01 0.115871938 0.202045779
## 5294 0.2472539707 2.137496e-01 0.102754920 0.192989246
## 5296 0.1549991904 2.134138e-01 0.013016191 0.197402024
## 5300 0.2315679672 2.181424e-01 0.126791313 0.201519594
## 5301 0.2184160935 2.063648e-01 0.112779703 0.200668919
## 5306 0.2401607148 2.173727e-01 0.134880340 0.206344752
## 5318 0.1305979704 2.033088e-01 -0.128953060 0.201868068
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## 5325 0.1053715200 2.072015e-01 -0.187614177 0.200671277
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## 5329 0.1693811064 2.153119e-01 0.113080332 0.209948390
## 5331 0.1588237857 2.021887e-01 0.023959098 0.207354243
## 5348 0.2100228962 1.980535e-01 0.112368048 0.216505486
## 5357 0.1709950419 2.016365e-01 0.177033516 0.215989198
## 5360 0.1289588641 2.016449e-01 -0.064343284 0.213458787
## 5367 0.0872405365 1.886508e-01 -0.080439480 0.213526272
## 5371 -0.0359364492 2.433333e-05 -0.158026706 0.141509094
## 5379 0.0166844722 -4.274359e-17 0.020877718 0.141507457
## 5382 -0.0006965726 1.200000e-06 -0.014542498 0.141506429
## 5383 -0.0216551438 1.600000e-06 -0.088993295 0.141505057
## 5395 -0.0009774890 7.266667e-06 0.007277936 0.141514668
## 5399 -0.0467842834 1.784155e-04 -0.239250277 0.141548950
## 5408 -0.0026987032 6.822831e-03 -0.059530712 0.141975297
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## 5416 -0.0118253232 3.318556e-02 -0.039510319 0.145984802
## 5420 0.0202090625 1.180888e-01 -0.126059977 0.154581977
## 5422 0.1349638443 1.636538e-01 0.056009944 0.155525219
## 5426 0.0415681067 2.725397e-01 -0.149254151 0.157962674
## 5436 1.0099124248 9.759008e-01 1.309479226 0.165701237
## 5441 0.8361346055 7.403597e-01 1.055135175 0.166956987
## 5445 0.9014228606 9.114657e-01 1.148329308 0.169332997
## 5449 0.8497376747 8.199761e-01 1.049203112 0.170791696
## 5452 0.7676228609 7.296897e-01 0.947539432 0.172087912
## 5470 0.4026170240 3.664671e-01 0.662148618 0.174940818
## 5472 0.2332912278 3.370174e-01 0.611847775 0.173599239
## 5474 0.2126135730 2.729294e-01 0.492575958 0.173339507
## 5475 0.1490221102 2.801838e-01 0.350032296 0.173198740
## 5479 0.2332912278 3.087265e-01 0.536980967 0.174512427
## 5480 0.2335721442 3.089089e-01 0.508787280 0.174154609
## 5484 0.2631354047 2.850699e-01 0.496235813 0.174133377
## 5486 0.2251286690 2.685122e-01 0.457470780 0.174365421
## 5489 0.0804325141 1.813712e-01 0.200352702 0.173572016
## 5492 0.0951441094 1.794556e-01 0.328203124 0.173742489
## 5498 0.2369572089 1.222265e-01 0.298007866 0.174507268
## 5500 0.2010556808 1.111480e-01 0.273765524 0.174362492
## 5504 0.0705302805 6.952438e-02 0.101290218 0.174172541
## 5505 0.1712957350 9.442025e-02 0.279567476 0.174259241
## 5508 0.1371730718 6.842573e-02 0.237016771 0.174175390
## 5513 0.1460187754 8.109810e-02 0.260587608 0.173976842
## 5516 0.1162145006 5.297727e-02 0.135543844 0.173748877
## 5520 0.1206119620 4.288924e-02 0.118967585 0.173081159
## 5529 0.0330793732 3.164439e-02 0.066848655 0.172123424
## 5539 -0.0021605034 1.579161e-02 -0.092892471 0.171308441
## 5540 0.0611216712 2.027238e-02 0.085135544 0.171045987
## 5542 0.0274625224 2.011658e-02 0.072117024 0.171098407
## 5546 -0.0095703403 1.587351e-02 -0.097046210 0.170600017
## 5551 0.0179216628 1.013246e-02 -0.011413499 0.170863275
## 5555 0.0412405020 1.619420e-02 0.099675896 0.170510326
## 5565 -0.0089578023 1.095446e-02 -0.014133207 0.168747842
## 5573 -0.0411463502 6.088024e-03 -0.154568982 0.167622031
## 5577 0.0138540342 1.091379e-02 0.081135431 0.167561270
## 5578 0.0141349505 8.951810e-03 0.059467420 0.167198261
## 5579 -0.0089578023 1.008542e-02 -0.015289619 0.166882189
## 5588 -0.0096395361 1.327788e-02 -0.047017558 0.167498227
## 5592 0.0418765312 2.019913e-02 0.107837335 0.167707396
## 5593 0.0089784129 1.882100e-02 0.033255870 0.167618109
## 5594 -0.0232101351 1.482933e-02 -0.104196085 0.167409212
## 5597 0.0624646328 2.466390e-02 0.149233481 0.168066130
## 5602 0.0064395055 2.426910e-02 -0.023102249 0.168358130
## 5631 0.2748643614 1.525178e-01 0.347105403 0.199772310
## 5638 0.4611780954 2.219843e-01 0.418195814 0.210714988
## 5642 0.3651994868 2.489419e-01 0.328846525 0.214730381
## 5643 0.2522810958 2.536286e-01 0.260445225 0.217177643
## 5649 0.3793382373 3.732302e-01 0.404338722 0.225640256
## 5650 0.2786658719 3.170485e-01 0.260991872 0.222560616
## 5653 0.4138644545 4.544032e-01 0.495577059 0.227173148
## 5654 0.4440348733 4.162131e-01 0.492328913 0.238828391
## 5656 0.3272226701 3.917532e-01 0.402515757 0.235411779
## 5657 0.2846610993 3.235471e-01 0.271859600 0.235003879
## 5660 0.3732615977 5.087884e-01 0.528442934 0.233214894
## 5661 0.3537753485 4.453248e-01 0.502181087 0.237394200
## 5668 0.3614398151 4.875205e-01 0.456991680 0.238420551
## 5673 0.3553572412 5.274866e-01 0.512813226 0.237711045
## 5681 0.3159176519 5.073047e-01 0.507669072 0.246075391
## 5692 0.3573640839 3.697317e-01 0.384117284 0.279703166
## 5701 0.7183653290 6.999082e-01 0.756794637 0.315358828
## 5709 1.0664616636 9.344607e-01 0.879881833 0.351803687
## 5711 1.3014428610 1.127809e+00 0.840560765 0.363957936
## 5725 1.3034791607 1.332060e+00 0.765850002 0.369416690
## 5726 1.0800911883 1.093027e+00 0.674041752 0.364260327
## 5733 0.9025906880 9.013794e-01 0.584467802 0.359560357
## 5754 0.4169260318 4.178383e-01 0.363268017 0.331996676
## 5756 0.3743060724 3.719472e-01 0.297488007 0.328338980
## 5766 0.2133407762 2.908402e-01 0.328959475 0.317841122
## 5786 0.1242859265 9.133403e-02 0.309814303 0.274751420
## 5787 0.1151593065 8.317430e-02 0.259245248 0.272176821
## 5794 0.1220343985 5.729836e-02 0.344739967 0.252748181
## 5803 0.0584560208 3.334347e-02 0.185606956 0.232257079
## 5812 0.0209468295 2.257100e-02 0.109533323 0.203909312
## 5814 0.0541116603 2.765993e-02 0.301960353 0.200790221
## 5816 0.0512808779 2.265393e-02 0.216333870 0.196444912
## 5817 0.0417059811 2.142977e-02 0.123853369 0.194845419
## 5822 -0.0056064091 1.998367e-02 0.239021916 0.178638313
## 5830 0.0047956194 1.383610e-02 0.267708811 0.158093076
## 5833 -0.0236493109 6.604000e-03 0.132842844 0.145768455
## 5834 0.0217832789 1.188063e-02 0.293627857 0.146287207
## 5835 0.0072330165 1.328193e-02 0.319958006 0.145705119
## 5840 -0.0236493109 4.892881e-03 0.120752740 0.127607704
## 5853 -0.0270505417 6.054681e-03 -0.003306090 0.092007561
## 5857 0.0018651100 1.157359e-02 0.142870796 0.090076142
## 5863 0.0109917300 1.050832e-02 0.145027129 0.074985414
## 5866 -0.0060547370 1.337155e-02 -0.003679306 0.070588597
## 5871 0.0431005136 1.301439e-02 0.125560403 0.062071419
## 5874 -0.0085019836 1.012795e-02 -0.102840667 0.051918014
## 5876 0.1038322608 1.918175e-02 0.125776189 0.057162543
## 5878 0.1144390106 2.175343e-02 0.132087325 0.056141265
## 5880 0.1019414260 2.746853e-02 0.010604270 0.056029154
## 5883 0.1925919866 2.592297e-02 0.146767263 0.052654921
## 5888 0.0814527578 1.619050e-02 -0.167345878 0.051357152
## 5893 0.1456002035 3.678717e-02 0.036705910 0.063453998
## 5895 0.0822279789 2.580727e-02 -0.153591142 0.057109192
## 5902 0.0938588940 4.281133e-02 -0.109445943 0.078251472
## 5904 0.1980381427 7.603590e-02 0.124610647 0.078941469
## 5906 0.1527906907 7.788380e-02 0.128721788 0.074543544
## 5910 0.1217835318 5.488163e-02 -0.066210604 0.060293756
## 5914 0.1480980981 9.949833e-02 0.059367212 0.064793151
## 5915 0.1469416730 9.206527e-02 -0.027470933 0.062234590
## 5919 0.1583165829 1.086819e-01 0.098400268 0.064485993
## 5926 0.1583165829 1.158457e-01 0.104804518 0.066059419
## 5927 0.1416578768 1.033995e-01 0.091820079 0.069983139
## 5929 0.1262545730 9.880787e-02 -0.016423290 0.067229361
## 5933 0.1548097505 1.279462e-01 0.114382001 0.072150048
## 5937 0.0970895695 6.247007e-02 -0.129331175 0.071539543
## 5939 0.1866776901 1.461572e-01 0.125422574 0.071938638
## 5946 0.1651775841 1.570314e-01 0.130348997 0.076836029
## 5955 0.1498478001 1.707727e-01 0.151241388 0.086526166
## 5958 0.0984272706 8.186357e-02 -0.102753953 0.078043458
## 5966 0.1128276807 7.608117e-02 -0.042034428 0.088395279
## 5969 0.1568959582 1.614774e-01 0.134657637 0.090519414
## 5985 0.0883058133 1.235808e-01 0.013866791 0.085115322
## 5993 0.0728946571 6.210023e-02 -0.085369985 0.076964182
## 5997 0.1348701771 1.781763e-01 0.173174013 0.086694488
## 5999 0.1000652304 1.640835e-01 0.056508840 0.071254057
## 6003 0.1394259997 2.034188e-01 0.184213182 0.063010442
## 6004 0.1157868560 1.981965e-01 0.166366557 0.056640818
## 6007 0.0421714350 8.432317e-02 -0.069616721 0.046453077
## 6011 0.1342874448 1.992225e-01 0.205501829 0.039626443
## 6014 0.0659505588 8.228550e-02 -0.050076376 0.021379759
## 6021 0.0597669407 7.410457e-02 -0.057676101 0.004963644
## 6027 0.1096001212 1.511201e-01 0.060890473 -0.004271256
## 6028 0.0742268820 6.132123e-02 -0.056235816 -0.011433495
## 6031 0.1437471607 1.610182e-01 0.190306244 -0.013270259
## 6033 0.1372137132 1.557875e-01 0.137528323 -0.012680069
## 6040 0.0962939392 1.458084e-01 0.191015957 -0.032597127
## 6045 0.1167632088 1.597603e-01 0.155725548 -0.054237276
## 6046 0.1028206535 1.474660e-01 0.139917969 -0.057591038
## 6047 0.0868676055 1.471663e-01 0.056485746 -0.055246003
## 6064 0.0697735579 5.501702e-05 0.192211796 0.274305839
## 6069 0.0205313332 5.501702e-05 0.012541684 0.274303415
## 6074 0.0348126386 6.225035e-05 0.081574732 0.274304480
## 6075 -0.0131403650 5.501702e-05 -0.055100374 0.274302833
## 6078 0.0697735579 5.501702e-05 0.192211932 0.274306031
## 6079 0.0554902935 5.501702e-05 0.177844787 0.274305882
## 6080 0.0557712098 1.310170e-04 0.156025620 0.274306916
## 6087 0.0557712098 5.501702e-05 0.156025665 0.274305108
## 6097 0.0340811237 4.889171e-03 0.108163816 0.276618014
## 6098 0.0864570995 5.758162e-03 0.290763709 0.276738603
## 6102 0.0448997289 1.512985e-02 0.209883532 0.279683005
## 6105 0.0925002136 4.557487e-02 0.329397062 0.283804102
## 6109 0.2690058879 1.352667e-01 0.304281468 0.291559953
## 6117 0.4646248024 5.857234e-01 0.368367296 0.308899712
## 6118 0.8308146186 8.850647e-01 0.453796041 0.316054750
## 6125 1.6831781808 1.741279e+00 0.777159817 0.340645490
## 6130 1.7765601591 2.002520e+00 0.799216563 0.350436840
## 6131 1.6078246390 1.878928e+00 0.667209180 0.350226193
## 6134 1.7855788868 2.121048e+00 0.904997961 0.355715969
## 6139 1.5271402580 1.924446e+00 0.681595237 0.370368498
## 6140 1.8625454838 2.200603e+00 0.833205707 0.369169196
## 6148 1.6275653856 2.009814e+00 0.715854521 0.386256152
## 6159 1.2145769625 1.118829e+00 0.448606624 0.400696461
## 6160 1.2436892581 1.135557e+00 0.505591647 0.401918262
## 6176 1.1492466845 8.606701e-01 0.631410836 0.426673544
## 6177 1.0800446220 1.025893e+00 0.609855119 0.430215980
## 6179 1.1692649232 9.621309e-01 0.512760850 0.435741887
## 6181 0.7211400165 5.888245e-01 0.341358494 0.431453869
## 6187 0.7087726679 5.733488e-01 0.267772139 0.441062782
## 6188 0.6832164157 5.408166e-01 0.335573689 0.440311540
## 6199 0.7019146547 6.578306e-01 0.487022911 0.470377159
## 6209 0.7049123257 4.135668e-01 0.401657131 0.500901501
## 6211 0.7999763408 8.104775e-01 0.483739612 0.518223441
## 6212 0.8188985140 7.859481e-01 0.477068931 0.526211452
## 6215 0.5999880436 4.440835e-01 0.261747852 0.530122898
## 6220 0.8863613617 9.151848e-01 0.491378954 0.564460230
## 6223 0.7127809149 6.612609e-01 0.353894544 0.566056318
## 6226 0.9673227580 9.749785e-01 0.908230074 0.602154865
## 6230 0.7575463174 7.830791e-01 0.790021200 0.603529449
## 6232 1.1451818806 1.105602e+00 1.020961923 0.622336797
## 6234 0.9215767953 1.061028e+00 1.010749298 0.634923180
## 6236 0.7103978458 6.980343e-01 0.759777121 0.625305650
## 6248 1.0164332278 1.147158e+00 1.015726163 0.674630533
## 6250 0.6482089724 5.857838e-01 0.781875362 0.668809448
## 6255 0.9384554098 1.065523e+00 0.982126922 0.701319517
## 6258 0.6115827025 5.915663e-01 0.818958490 0.681539211
## 6263 0.7941999309 9.228057e-01 0.830904379 0.703336285
## 6270 0.8713433708 8.603979e-01 0.796120890 0.720883403
## 6271 0.6214952137 5.116354e-01 0.653761196 0.714304292
## 6283 0.9294540744 9.225184e-01 0.860446032 0.754196078
## 6285 0.5564088127 4.088013e-01 0.626359133 0.745427650
## 6288 0.9684169315 9.885610e-01 0.877548850 0.756377604
## 6297 0.9015798628 8.786850e-01 0.857234196 0.789999800
## 6300 0.6595987777 4.168805e-01 0.712408395 0.782116069
## 6304 0.8562840321 8.619674e-01 0.870080640 0.814616642
## 6306 0.4979788947 4.356384e-01 0.659865431 0.800726988
## 6310 0.9882483446 9.067463e-01 0.932678305 0.835580173
## 6318 1.0908839150 9.276863e-01 0.988822619 0.873478365
## 6329 1.1811411192 9.885223e-01 1.139597072 0.927683163
## 6342 0.6735827483 8.551527e-01 1.286459808 1.056235378
## 6344 1.4990107049 1.234503e+00 1.528820132 1.104526058
## 6366 2.9843231566 2.706687e+00 2.122497292 1.420216826
## 6368 2.2134709796 2.311138e+00 2.035940097 1.447178170
## 6369 1.3630432754 1.660481e+00 1.918134543 1.418811483
## 6375 2.4229272130 2.638220e+00 2.184526722 1.545549304
## 6382 2.3821109645 2.639083e+00 2.305844453 1.630087738
## 6393 3.2117131606 3.435828e+00 2.608436815 1.795375911
## 6395 2.7552317381 2.492026e+00 2.555707018 1.753726072
## 6397 2.3163745142 2.295801e+00 2.419033511 1.822505999
## 6399 4.1470497834 3.704500e+00 2.722324373 1.888209885
## 6403 3.1388951478 3.290213e+00 2.604779744 1.975832682
## 6411 2.2096519085 2.567049e+00 2.460483085 1.999455037
## 6412 2.4706208354 2.754363e+00 2.523296275 1.985977000
## 6413 3.8732239078 3.899998e+00 2.697321101 2.025872033
## 6418 2.3129150831 2.462021e+00 2.317829727 2.031559579
## 6419 2.6504616288 2.738082e+00 2.387382495 2.062604140
## 6421 4.0294832842 3.775482e+00 2.542497418 2.101083431
## 6422 3.9520700502 3.665784e+00 2.503741185 2.121982530
## 6427 3.1429269862 3.103596e+00 2.347690233 2.104379262
## 6436 2.9799480844 2.781635e+00 2.100861198 2.153821477
## 6440 1.0719613253 1.237288e+00 1.848770941 2.094465260
## 6443 2.5142360149 2.311881e+00 1.951115065 2.130997758
## 6452 1.5375589298 1.613672e+00 1.666190453 2.148543087
## 6454 0.8964821899 1.095614e+00 1.498357000 2.143606320
## 6455 1.4943305282 1.466068e+00 1.666719925 2.155099410
## 6456 1.9449592629 1.887076e+00 1.653614758 2.173368766
## 6458 1.6401251181 1.573201e+00 1.573905199 2.165873953
## 6461 0.8645982462 8.860703e-01 1.392310525 2.088025755
## 6462 1.2749880269 1.363130e+00 1.568592191 2.092973682
## 6473 0.9017232416 9.634476e-01 1.369539477 1.932487998
## 6479 1.3208715926 1.310564e+00 1.440338818 1.936662923
## 6480 0.7955834327 8.747988e-01 1.352741448 1.887286754
## 6484 1.1744571557 1.191981e+00 1.462226181 1.886391027
## 6494 0.9939771955 7.792592e-01 1.433213723 1.561953855
## 6499 0.9141378388 9.601354e-01 1.517186736 1.526899803
## 6504 1.1856946399 8.333628e-01 1.504943617 1.496541034
## 6512 0.6815950527 8.238202e-01 1.396547169 1.456795694
## 6519 0.8729183010 8.253345e-01 1.407282942 1.381352476
## 6521 0.7439945380 7.228350e-01 1.329507678 1.362659411
## 6525 0.7335044472 7.433636e-01 1.307288022 1.314472178
## 6527 0.7571361410 7.351719e-01 1.259975268 1.305823437
## 6529 0.4733826225 5.539965e-01 1.125156498 1.271724761
## 6532 0.6957477257 6.889753e-01 1.212583400 1.251926527
## 6534 0.5962598115 7.032920e-01 1.170844168 1.233486327
## 6536 0.4979383985 5.869114e-01 1.039928334 1.197302791
## 6538 0.4839975120 4.601818e-01 0.952742189 1.187688860
## 6547 0.5453486151 6.246859e-01 0.958969565 1.129671628
## 6549 0.4403755880 5.739549e-01 0.889231381 1.121661084
## 6553 0.3849263423 4.178232e-01 0.889994985 1.094587188
## 6560 0.3396094033 3.432878e-01 0.864523801 1.056573893
## 6561 0.3801331195 3.592996e-01 0.855795107 1.052257504
## 6569 0.3585254615 3.461258e-01 0.818238397 1.019544204
## 6575 0.5074959708 3.669593e-01 0.832542277 1.005351538
## 6582 0.3753777875 3.392410e-01 0.757512123 0.998384735
## 6585 0.0661876325 1.278473e-01 0.647340701 0.974588786
## 6591 0.6592708661 4.877977e-01 0.817413788 1.041162339
## 6596 0.3664890453 3.820194e-01 0.940078630 1.015567270
## 6608 0.4439045648 5.688425e-01 1.120570045 1.108582970
## 6615 0.6559775624 8.782288e-01 1.344365209 1.196243337
## 6616 1.0031621682 9.648316e-01 1.565088205 1.147005619
## 6617 1.1039767421 1.003801e+00 1.610628515 1.185548645
## 6620 0.3049360040 4.689807e-01 1.543172282 1.097984133
## 6623 1.3638494947 1.190165e+00 1.754211164 1.215499738
## 6625 1.4835213058 1.310730e+00 1.770744342 1.229034411
## 6628 0.0606806664 3.417400e-01 1.573320556 1.115999878
## 6636 1.0748192072 1.496151e+00 1.719594659 1.384086146
## 6644 2.4486166834 1.791317e+00 1.959250889 1.437373206
## 6646 2.4169029671 2.449731e+00 1.940095652 1.350271823
## 6647 2.2670813754 2.192100e+00 1.907721500 1.426636403
## 6649 0.2383499598 3.677030e-01 1.655967073 1.238751499
## 6655 1.1080691847 7.024303e-01 1.748844983 1.300679635
## 6657 2.1727504211 2.102587e+00 1.664328675 1.439514868
## 6658 2.7126095264 2.612314e+00 1.835857817 1.374377002
## 6661 2.6572845239 2.511505e+00 1.733425314 1.446496426
## 6670 0.3817615016 3.901486e-01 1.352851141 1.302743426
## 6674 2.3290350582 2.282743e+00 1.541885127 1.369583046
## 6687 2.2884161830 2.172168e+00 1.425975788 1.347759730
## 6689 2.0089801692 1.836482e+00 1.355786460 1.354022449
## 6695 1.6215614160 1.642861e+00 1.309837740 1.298826450
## 6706 1.1855797776 1.520004e+00 1.064942975 1.254228826
## 6714 1.3984798037 1.457860e+00 1.265974853 1.186004313
## 6715 1.5976022908 1.777616e+00 1.269359462 1.202397436
## 6721 1.4748673620 1.427157e+00 1.286466721 1.144494313
## 6722 1.4085160999 1.592859e+00 1.289389369 1.150399958
## 6727 1.4425947006 1.463368e+00 1.154372519 1.224813174
## 6730 2.0310810782 1.876187e+00 1.313573815 1.146533211
## 6734 1.2229432691 1.519843e+00 1.162382832 1.160922699
## 6735 1.5545520624 1.599474e+00 1.357605404 1.088744596
## 6740 0.4730991223 4.322305e-01 0.954091633 1.159339648
## s1.1 ForwardMLR BackwardMLR country
## 1 0.183506235 0.4294926639 0.6020000258 BRA
## 21 0.184011114 0.5365342646 0.7534507700 BRA
## 27 0.185144887 0.4274863777 0.6594021186 BRA
## 29 0.184893034 0.6032934950 0.8331675441 BRA
## 43 0.193088090 0.7246007426 0.9551973553 BRA
## 45 0.190894144 0.6859789042 0.9201912877 BRA
## 60 0.208308863 0.6157943916 0.8377070776 BRA
## 65 0.218808735 0.7281938853 0.9533984581 BRA
## 66 0.209110969 0.7844997833 1.0339965618 BRA
## 71 0.235854859 0.8434784403 1.0899868018 BRA
## 80 0.265350157 0.8146984081 1.0549370570 BRA
## 87 0.286486074 0.8218395604 1.0548750420 BRA
## 88 0.270014979 0.7488165898 0.9723907763 BRA
## 103 0.299755905 0.4927998133 0.7109913542 BRA
## 107 0.352739297 0.7338930009 0.9485488224 BRA
## 108 0.336262825 0.7141323781 0.9310050822 BRA
## 110 0.301743662 0.5019815327 0.7105165728 BRA
## 122 0.445118362 0.7330372351 0.9301852729 BRA
## 134 0.460428706 0.7088933655 0.9019441756 BRA
## 135 0.456746447 0.6938476769 0.8855952285 BRA
## 141 0.455762237 0.7190310818 0.9017397373 BRA
## 142 0.480508962 0.7087153909 0.8842361223 BRA
## 151 0.527862304 0.6225826846 0.7855672781 BRA
## 154 0.488382781 0.7343051023 0.9005425755 BRA
## 155 0.609576073 0.7027629510 0.8557683479 BRA
## 157 0.548202469 0.6670824379 0.8207622804 BRA
## 163 0.562003948 0.7276634567 0.8734919050 BRA
## 172 0.541827297 0.6340035619 0.7598833829 BRA
## 174 0.454228545 0.5733903529 0.7036082351 BRA
## 177 0.562999414 0.7335115805 0.8570599220 BRA
## 190 0.597498348 0.7948278173 0.9000033953 BRA
## 192 0.620960159 0.7605627628 0.8680962990 BRA
## 194 0.479054861 0.5483506842 0.6480825710 BRA
## 195 0.456862561 0.6221609687 0.7258908304 BRA
## 196 0.479478020 0.7994621791 0.9022387853 BRA
## 205 0.579628181 0.7915765487 0.8819612057 BRA
## 206 0.602002420 0.7749560956 0.8680962990 BRA
## 211 0.446135956 0.8043595116 0.8873203740 BRA
## 231 0.517450223 0.7824809447 0.8379857570 BRA
## 233 0.594854360 0.7694024282 0.8213870110 BRA
## 236 0.482344518 0.5362520777 0.5838295427 BRA
## 239 0.587232490 0.7914413382 0.8388493382 BRA
## 241 0.611489256 0.7284104916 0.7722792856 BRA
## 245 0.607595328 0.7615127225 0.8015562627 BRA
## 253 0.591470954 0.8479027523 0.8594069894 BRA
## 255 0.663417694 0.8095726004 0.8243583560 BRA
## 267 0.652487401 0.8402803029 0.8381841820 BRA
## 270 0.652576151 0.7269328339 0.7206493936 BRA
## 272 0.587964432 0.6668181671 0.6643467391 BRA
## 273 0.650099212 0.8408700476 0.8372286897 BRA
## 274 0.720922273 0.8613153248 0.8590790768 BRA
## 276 0.669108356 0.8258314578 0.8240162178 BRA
## 277 0.753848803 0.7541735663 0.7475044510 BRA
## 284 0.724064873 0.9433504560 0.9767591298 BRA
## 300 0.689062715 0.9017339207 0.9253896119 BRA
## 301 0.814582753 1.0748726565 1.0982771194 BRA
## 305 0.668991012 0.9658039054 0.9815907418 BRA
## 319 0.885606413 1.0186013563 1.0853470856 BRA
## 320 0.755524716 0.8790684197 0.9478091769 BRA
## 321 0.740864429 0.9590127260 1.0290388187 BRA
## 324 0.921841533 1.1719081142 1.2484165409 BRA
## 343 0.917480310 1.1420777402 1.2276268159 BRA
## 349 0.710291712 0.9736017855 1.0547203594 BRA
## 350 1.018048264 1.1570897906 1.2275644675 BRA
## 356 0.857590914 1.0013884122 1.0755671870 BRA
## 375 1.056790389 0.9328839088 0.9495631083 BRA
## 376 1.105073613 0.7984170678 0.8120041389 BRA
## 378 1.070255493 1.0506814996 1.0661206769 BRA
## 379 1.110833718 1.0560831760 1.0669653647 BRA
## 381 1.141575699 1.0282621908 1.0318751101 BRA
## 395 1.183854269 1.1811671427 1.0518124957 BRA
## 405 0.983105899 1.0770589768 1.0191493642 BRA
## 406 1.223181936 1.2553741901 1.1920330374 BRA
## 409 1.258207409 1.2333119039 1.1578069198 BRA
## 414 1.187460437 1.2863880166 1.1927493052 BRA
## 425 1.036506027 1.0216325770 0.8797943065 BRA
## 433 1.011617334 1.1830383442 1.0372943500 BRA
## 435 1.213364682 1.3655138147 1.2110318243 BRA
## 439 1.082221648 1.1207532502 0.9559437900 BRA
## 445 1.210631728 1.2900289388 1.1144863761 BRA
## 447 1.063882992 1.2281510251 1.0581998922 BRA
## 458 1.167912122 1.4156076670 1.2278776010 BRA
## 463 1.366943679 1.3448510440 1.1598280622 BRA
## 468 1.139708785 1.1509539776 0.9544520075 BRA
## 469 1.205502952 1.3242709107 1.1273424602 BRA
## 484 1.497134157 1.3493767550 1.1274439683 BRA
## 486 1.354834146 1.3171111386 1.0923888889 BRA
## 495 1.105697076 1.0144043304 0.7881181508 BRA
## 499 1.157521094 1.2308887558 1.0045885330 BRA
## 502 0.995078384 0.9973860886 0.7669853870 BRA
## 504 1.074642502 1.2497714627 1.0211103715 BRA
## 505 1.139484022 1.2665981069 1.0361321522 BRA
## 513 1.025381360 1.2415250779 1.0044158801 BRA
## 514 1.287860105 1.2323072724 1.0011260847 BRA
## 517 0.895003131 1.0814145244 0.8480755507 BRA
## 519 1.028327747 1.2736373013 1.0361321522 BRA
## 520 0.977786883 1.2415736112 1.0043732031 BRA
## 521 0.992191615 1.2356408559 1.0011260847 BRA
## 524 0.882807922 1.1330615371 0.8997540320 BRA
## 534 0.936052981 1.2511548574 1.0100018990 BRA
## 543 0.859924688 1.1538711285 0.9098196335 BRA
## 552 0.787192327 1.1056531139 0.8682468166 BRA
## 575 0.813882636 1.3179542691 1.0835229406 BRA
## 582 0.684679954 1.2976090813 1.0685856083 BRA
## 596 0.598677404 1.2625333055 1.0560567150 BRA
## 597 0.592260336 1.2241639972 1.0218317255 BRA
## 599 0.590323883 1.1396136410 0.9385664373 BRA
## 604 0.579433488 1.1744284514 0.9860255925 BRA
## 606 0.548249373 1.0737157647 0.8859976420 BRA
## 608 0.511060613 1.0030503492 0.8297224943 BRA
## 619 0.476895868 1.1627321020 1.0051296478 BRA
## 624 0.442917185 1.1851119134 1.0401357153 BRA
## 628 0.388478636 0.9214316369 0.7851159198 BRA
## 635 0.355268681 0.9662158636 0.8692487002 BRA
## 636 0.311404007 1.0205915795 0.9335767573 BRA
## 637 0.363462304 1.2072371773 1.1234049145 BRA
## 645 0.299552724 1.3554348835 1.3432293751 BRA
## 647 0.308179157 1.3242384138 1.3252009663 BRA
## 651 0.284577871 1.3441965652 1.3593434526 BRA
## 654 0.218245247 1.2908119764 1.3252009663 BRA
## 659 0.187698057 -0.9382422213 -1.0307132297 COL
## 670 0.187728506 -0.5245413840 -0.5723781751 COL
## 672 0.187863293 -0.6199537760 -0.6724061255 COL
## 674 0.187900365 -0.6696476589 -0.7129560505 COL
## 678 0.187923461 -0.3752677753 -0.3797035039 COL
## 679 0.188012559 -0.4496481871 -0.4621877141 COL
## 689 0.188614830 -0.2315915874 -0.2393131990 COL
## 697 0.188386514 -0.2456948063 -0.2594544280 COL
## 705 0.188790127 -0.2635000540 -0.2769167553 COL
## 707 0.189050429 -0.3461023339 -0.3611627132 COL
## 708 0.188966671 -0.4855242466 -0.4986922310 COL
## 721 0.190098655 -0.2675135724 -0.2630826618 COL
## 723 0.190592414 -0.3299030454 -0.3193578096 COL
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## 1587 0.688662260 0.5125652168 0.4458718312 FRA
## 1593 0.430318136 0.3450736110 0.3164883765 FRA
## 1595 0.368167400 0.5825977918 0.5193069004 FRA
## 1598 0.379664201 0.4957908111 0.4370353294 FRA
## 1607 0.347539833 0.1592747677 0.1207634321 FRA
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## 1619 0.348470222 0.4929406696 0.4664487710 FRA
## 1621 0.347554608 0.2573952230 0.2464350430 FRA
## 1622 0.317184772 0.3648566493 0.3308980186 FRA
## 1629 0.321144733 0.3674721463 0.3212722017 FRA
## 1634 0.383510307 0.3278477976 0.2599298531 FRA
## 1639 0.404759746 0.3940359758 0.3086201132 FRA
## 1647 0.401634836 0.4435025215 0.4849909456 FRA
## 1652 0.429041756 0.5495105524 0.5520830696 FRA
## 1668 0.437978644 0.5483844829 0.5395372416 FRA
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## 1689 0.447426130 0.4486284892 0.4560166623 FRA
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## 1694 0.468613038 0.4822459840 0.4910227299 FRA
## 1700 0.453763243 0.5115048371 0.5190365995 FRA
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## 1707 0.471433449 0.3856945411 0.3775969298 FRA
## 1712 0.472290527 0.1497897908 0.1812323285 FRA
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## 1749 0.595255704 0.6522083527 0.6449272945 FRA
## 1754 0.562008011 0.3697748893 0.3971882915 FRA
## 1760 0.580488389 0.5332886817 0.5379264150 FRA
## 1764 0.577218279 0.6224880111 0.6201220306 FRA
## 1765 0.556557777 0.5967796077 0.5866166751 FRA
## 1774 0.530127651 0.4068828513 0.4349652231 FRA
## 1787 0.469967130 0.4365249184 0.4231687646 FRA
## 1789 0.456791678 0.1726397470 0.1999464309 FRA
## 1790 0.426085456 0.2481219939 0.2747835897 FRA
## 1796 0.445165131 0.1579132643 0.1871120083 FRA
## 1797 0.410432402 0.2750719510 0.2683663784 FRA
## 1805 0.420024716 0.2785805757 0.2997734752 FRA
## 1808 0.404907396 0.2346811830 0.2527965663 FRA
## 1814 0.387296274 0.2341680953 0.2575058840 FRA
## 1818 0.374427983 0.0345731334 0.0527191843 FRA
## 1822 0.371340205 0.1718648691 0.1882699366 FRA
## 1829 0.360114988 0.0757609071 0.0995723721 FRA
## 1830 0.359325107 -0.0137651423 0.0170881619 FRA
## 1831 0.356648035 -0.1764529241 -0.1204413559 FRA
## 1840 0.357716490 0.1249308947 0.1456726928 FRA
## 1847 0.391059164 0.1949660073 0.2162620170 FRA
## 1851 0.415174804 0.1127216840 0.1124697431 FRA
## 1854 0.418628001 0.2214493741 0.2323050453 FRA
## 1865 0.408165975 0.1543245953 0.1562742597 FRA
## 1867 0.332389303 0.1958738572 0.1850084804 FRA
## 1868 0.420149369 0.3512509074 0.3547017191 FRA
## 1875 0.404572440 0.3553007495 0.3664168735 FRA
## 1888 0.329421851 0.1600498667 0.1710547897 FRA
## 1894 0.336727708 0.0230427400 0.1091119142 FRA
## 1898 0.365660208 0.3017642310 0.3370435655 FRA
## 1903 0.348328445 0.2951050386 0.3408078890 FRA
## 1904 0.289876984 0.2878578576 0.3384628646 FRA
## 1911 0.312888192 0.2697178682 0.3288370476 FRA
## 1916 0.287183756 0.1023789442 0.1518630165 FRA
## 1920 0.298945291 0.2286848700 0.2599475433 FRA
## 1924 0.300372332 0.2309810248 0.2908814240 FRA
## 1932 0.277052686 0.2264432216 0.2949536108 FRA
## 1950 0.282914761 -0.0456449394 0.0752284774 FRA
## 1966 0.322708615 0.3700376115 0.4544747858 FRA
## 1983 0.410298976 0.5623097083 0.6921443409 FRA
## 1995 0.163685813 -0.0200572358 -0.1682588695 DEU
## 2000 0.163694381 -0.2499568510 -0.3628281907 DEU
## 2002 0.163686271 -0.0741846995 -0.1890627652 DEU
## 2006 0.163690440 -0.3269571221 -0.4440825607 DEU
## 2010 0.163686500 -0.0665835478 -0.1750178771 DEU
## 2014 0.163686500 -0.2244329664 -0.3313209754 DEU
## 2020 0.163686500 -0.3014332375 -0.4125753454 DEU
## 2024 0.163765257 -0.0410136743 -0.1434538920 DEU
## 2027 0.163899815 -0.3682832861 -0.4783704277 DEU
## 2037 0.165623432 -0.0793133395 -0.1786728332 DEU
## 2044 0.176838892 0.0101242056 -0.0683124249 DEU
## 2051 0.183406116 0.1232301220 0.0710005596 DEU
## 2063 0.182780851 0.0571607278 0.0034829527 DEU
## 2068 0.181928475 0.1208928885 0.0597581005 DEU
## 2077 0.179724999 0.0614293558 0.0034829527 DEU
## 2088 0.177401576 0.2791600838 0.2403223620 DEU
## 2100 0.177403070 0.2622020327 0.2043662329 DEU
## 2107 0.176712618 0.2413049476 0.1780818352 DEU
## 2115 0.176832320 0.2238381691 0.1606195079 DEU
## 2116 0.177092806 0.2028537424 0.1430757677 DEU
## 2118 0.170486183 0.0214834115 -0.0444887081 DEU
## 2122 0.175660926 0.1041675501 0.0576280283 DEU
## 2130 0.176564539 0.0833425676 0.0400842880 DEU
## 2142 0.176303554 0.1395619945 0.0961519355 DEU
## 2144 0.177481204 0.1006590308 0.0611458680 DEU
## 2149 0.176817019 0.0430161035 0.0121367616 DEU
## 2152 0.176660545 -0.0702840555 -0.1053535161 DEU
## 2153 0.169536169 -0.1948273395 -0.2278285255 DEU
## 2155 0.176162426 0.0019136278 -0.0360727972 DEU
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## 2160 0.168658613 -0.2334726680 -0.2755146409 DEU
## 2162 0.176271129 0.0019779649 -0.0360727972 DEU
## 2172 0.178456219 -0.0341185907 -0.0702152836 DEU
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## 2295 0.285158596 0.1596043247 0.0967413453 DEU
## 2304 0.324640219 0.3383973100 0.3160811373 DEU
## 2310 0.337167436 0.3596250127 0.3335434646 DEU
## 2318 0.369226656 0.4162455071 0.4002139177 DEU
## 2321 0.240088508 0.5231243316 0.5026582104 DEU
## 2328 0.271260625 0.5824794821 0.5564823319 DEU
## 2334 0.309639450 0.3814453266 0.4203026101 DEU
## 2336 0.316057852 0.3207028715 0.3640274623 DEU
## 2337 0.341503647 0.5078112953 0.5527112991 DEU
## 2347 0.313146954 0.4747996489 0.5080096379 DEU
## 2353 0.315688854 0.5026244979 0.5255533782 DEU
## 2355 0.333712318 0.4084809459 0.4255254277 DEU
## 2361 0.338019157 0.4887159239 0.5080096379 DEU
## 2366 0.337835211 0.4781383796 0.5222118098 DEU
## 2372 0.311499295 0.4794902713 0.5213482286 DEU
## 2373 0.325946643 0.4831317678 0.5222118098 DEU
## 2376 0.315151612 0.3713896296 0.4047215321 DEU
## 2378 0.309835995 0.3092431872 0.3484463843 DEU
## 2379 0.312958083 0.4828825166 0.5213482286 DEU
## 2386 0.313457292 0.4777821395 0.5108458235 DEU
## 2388 0.335362185 0.4631535444 0.4942470774 DEU
## 2400 0.319259306 0.3328425267 0.3451927379 DEU
## 2404 0.335207554 0.2224469502 0.2285660415 DEU
## 2405 0.278025177 0.2653826196 0.2744822039 DEU
## 2412 0.283847851 0.2909071643 0.2962935370 DEU
## 2420 0.339460311 0.1477493984 0.1565089011 DEU
## 2424 0.365222320 0.3921486565 0.4015160777 DEU
## 2425 0.314886577 0.3176732554 0.3190318675 DEU
## 2432 0.378069426 0.3165885459 0.3190318675 DEU
## 2435 0.422349579 0.4269329043 0.4356585640 DEU
## 2436 0.430240857 0.4293409527 0.4365221452 DEU
## 2437 0.406168407 0.4110575814 0.4190598180 DEU
## 2447 0.310854108 0.5344878126 0.5425958959 DEU
## 2459 0.365726533 0.4620362351 0.4995961290 DEU
## 2469 0.316037695 0.3227033323 0.3608367710 DEU
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## 5513 0.183824425 0.2279283887 0.2389606371 GBR
## 5516 0.183160116 0.1018757471 0.1088344900 GBR
## 5520 0.180702138 0.1014005246 0.1238790074 GBR
## 5529 0.182241389 0.0324216193 0.0506614519 GBR
## 5539 0.181561688 -0.1177234868 -0.1052972189 GBR
## 5540 0.180855610 0.0561614306 0.0642968846 GBR
## 5542 0.181937585 0.0432583835 0.0484441679 GBR
## 5546 0.181131546 -0.1202282923 -0.1075939669 GBR
## 5551 0.180752494 0.0072987631 0.0180194138 GBR
## 5555 0.180783526 0.1228030886 0.1350683070 GBR
## 5565 0.179865266 -0.0027852745 -0.0012348384 GBR
## 5573 0.178645614 -0.1472926708 -0.1417596671 GBR
## 5577 0.180053897 0.0913252753 0.0975021942 GBR
## 5578 0.178913138 0.0705088618 0.0807836135 GBR
## 5579 0.177994352 -0.0044254134 -0.0024812958 GBR
## 5588 0.184424218 -0.0345279701 -0.0303607797 GBR
## 5592 0.186360964 0.1162957823 0.1154011742 GBR
## 5593 0.186108794 0.0418638695 0.0349828333 GBR
## 5594 0.185346617 -0.0993151287 -0.1033621143 GBR
## 5597 0.187735179 0.1626771056 0.1593226281 GBR
## 5602 0.188916612 -0.0116659195 -0.0073470626 GBR
## 5631 0.269258174 0.3069436031 0.2923796273 GBR
## 5638 0.281863026 0.3866579906 0.3802388877 GBR
## 5642 0.281646785 0.2882793564 0.2719708078 GBR
## 5643 0.288097195 0.2348611411 0.2041438188 GBR
## 5649 0.300634560 0.3588818594 0.3129451376 GBR
## 5650 0.283625157 0.2074154953 0.1681398016 GBR
## 5653 0.296005736 0.4222516868 0.3653047004 GBR
## 5654 0.340090376 0.4153873748 0.3571724520 GBR
## 5656 0.315349903 0.3203924652 0.2601078865 GBR
## 5657 0.308733720 0.1802321079 0.1294233605 GBR
## 5660 0.289521285 0.4422693016 0.3831261606 GBR
## 5661 0.303705384 0.4116673025 0.3499696221 GBR
## 5668 0.286116955 0.3699197536 0.3075884131 GBR
## 5673 0.271912355 0.5278446221 0.5115347021 GBR
## 5681 0.288454448 0.5293446278 0.4981654222 GBR
## 5692 0.377940407 0.4153018688 0.3804738983 GBR
## 5701 0.462788033 0.7628432524 0.7387299383 GBR
## 5709 0.521731813 0.9490752436 0.9714442126 GBR
## 5711 0.549867723 0.8939405466 0.9046200076 GBR
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## 5726 0.428601054 0.6512983623 0.6582485099 GBR
## 5733 0.389013094 0.5746268962 0.5785758961 GBR
## 5754 0.327790318 0.4537278374 0.4849510352 GBR
## 5756 0.327457099 0.4034281557 0.4078737856 GBR
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## 5786 0.257366499 0.4587674693 0.3157995925 GBR
## 5787 0.257400907 0.3802649929 0.2976265432 GBR
## 5794 0.234012773 0.4047719215 0.3515274368 GBR
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## 6046 0.019590647 -0.0840527010 0.0999947790 GBR
## 6047 0.048200523 -0.1280438888 0.0818097779 GBR
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## 6069 0.151657288 0.2327834147 0.4957926062 USA
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## 6674 1.418265737 1.3942807820 1.2235950146 USA
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## 6695 1.246649568 1.2237420818 1.1491566048 USA
## 6706 1.302738583 1.0043290521 1.0134748767 USA
## 6714 1.144949727 1.1444073344 1.2123348394 USA
## 6715 1.233009776 1.1410620966 1.2157640879 USA
## 6721 1.107088799 1.0836207274 1.2318350648 USA
## 6722 1.178186902 1.0851703740 1.2388434026 USA
## 6727 1.509844254 0.9076477798 1.1032106920 USA
## 6730 1.240927135 0.9882027952 1.1890491389 USA
## 6734 1.392125607 0.7815772533 1.0421715975 USA
## 6735 1.097468994 0.9571343561 1.2183337318 USA
## 6740 0.965116475 0.5077167094 0.8232138482 USAlibrary(shiny)
library(shinydashboard)
library(leaflet)
library(plotly)
load("external1.RData")
load("rf_results.RData")
ui <- dashboardPage(skin = "green",
dashboardHeader(title = "COVID-19 Metrics by Country",
titleWidth = 300),
dashboardSidebar(width = 300,
sidebarMenu(
menuItem("Map", tabName = "map", icon = icon("map")
),
selectInput(inputId = "vars",
label = "Select the Metric (for Map)",
choices = c(colnames(map_ds1)[2:5], colnames(map_ds1)[8]),
selected = colnames(map_ds1)[2]),
selectInput(inputId = "selected_country",
label = "Select the Country (for Time Series)",
choices = c(levels(res_dataset$country)),
selected = "USA"),
menuItem("Time-Series (Trees)", tabName = "ts-rf", icon = icon("chart-line")),
menuItem("Time-Series (Linear Methods)", tabName = "ts-lin", icon = icon("chart-line")),
menuItem("Time-Series", tabName = "ts-final", icon = icon("chart-line"))
)),
dashboardBody(
tabItems(
tabItem(tabName = "map",
fluidRow(
column(8, leafletOutput("map1", height = 800)),
column(4, dataTableOutput("table1"))
)
),
tabItem(tabName = "ts-rf",
fluidRow(
box(plotlyOutput("time_series_trees"))
)),
tabItem(tabName = "ts-lin",
fluidRow(
box(plotlyOutput("time_series_linear"))
)),
tabItem(tabName = "ts-final",
fluidRow(
box(plotlyOutput("time_series_final"))
))
))
)library(shinydashboard)
library(shiny)
library(leaflet)
library(dplyr)
library(stats)
library(sf)
library(lubridate)
library(plotly)
load("external1.RData")
load("rf_results.RData")
server <- function(input, output) {
time_series <- reactive({
filter(res_dataset, country == input$selected_country)
})
reactive_ds <- reactive({
select(map_ds1, geometry, "selection" = input$vars)
})
bw <- reactive({2 * IQR(reactive_ds()$selection) / length(reactive_ds()$selection)^(1/3)})
pal1 <- reactive({colorBin("viridis",
domain = seq(from = min(range(reactive_ds()$selection)),
to = max(range(reactive_ds()$selection)), by = bw()))})
pop <- reactive({paste("Country: ", map_ds1$name, "<br/>",
"Value: ", round(reactive_ds()$selection, 3))})
output$map1 <- renderLeaflet({
leaflet(reactive_ds()) %>%
addProviderTiles(providers$CartoDB) %>%
addPolygons(data = reactive_ds()$geometry , fillColor = pal1()(reactive_ds()$selection),
color = "black", popup = pop()) %>%
addLegend(pal = pal1(), values = reactive_ds()$selection)
})
output$table1 <- renderDataTable({
reactive_ds() %>% select(-geometry) %>%
cbind("Country" = map_ds1$name)
})
output$time_series_trees <- renderPlotly({
p <- time_series() %>% plot_ly(x = ~ date, y = ~rf_predicted_cases, name = "Random Forest", type = "scatter", mode = "lines") %>%
add_trace(y = ~actual_deaths, name = "Actual Deaths", mode = "lines+markers") %>%
add_trace(y = ~decision_tree_pred, name = "Decision Tree") %>%
add_trace(y = ~bagged_tree_pred, name = "Bagged Tree") %>%
add_trace(y = ~boosted_tree_pred, name = "Boosted Tree") %>%
layout(title = paste0("Actual vs. Predicted Daily Deaths (thousands) for", " ", input$selected_country),
yaxis = list(title = "Deaths"))
return(p)
})
output$time_series_linear <- renderPlotly({
p.all <- time_series() %>% plot_ly(x = ~ date, y = ~MLR, name = "MLR (Reduced by p-values)", type = "scatter", mode = "lines") %>%
add_trace(y = ~actual_deaths, name = "Actual Deaths", mode = "lines+markers") %>%
add_trace(y = ~ForwardMLR, "MLR (Forward Selection)") %>%
add_trace(y = ~BackwardMLR, "MLR (Backward Selection)") %>%
add_trace(y = ~s1, name = "Ridge") %>%
add_trace(y = ~s1.1, name = "LASSO") %>%
layout(title = paste0("Actual vs. Predicted Daily Deaths (thousands) for", " ", input$selected_country),
yaxis = list(title = "Deaths"))
return(p.all)
})
output$time_series_final <- renderPlotly({
p.final <- time_series() %>% plot_ly(x = ~date, y = ~new_deaths_thousand.15.comps, name = "PCR",
type = "scatter", mode = "lines") %>%
add_trace(y = ~actual_deaths, name = "Actual Deaths", mode = "lines+markers") %>%
add_trace(y = ~MLR, name = "MLR (Reduced by p-values") %>%
add_trace(y = ~s1.1, name = "LASSO") %>%
layout(title = paste0("Actual vs. Predicted Daily Deaths (thousands) for", " ", input$selected_country),
yaxis = list(title = "Deaths"))
return(p.final)
})
}---
title: "MATH 624 Group 2 Final Project"
author: "Collin Hoskins, Meiren Jiang, Cordell Jones, Samuel Harold"
date: "12/13/2021"
output:
   openintro::lab_report

---


<style type="text/css">
.main-container {
  max-width: 1800px;
  margin-left: auto;
  margin-right: auto;
}
</style>



```{r setup, include = F}

# setwd("C:/Users/Collin/Desktop/Academic/Courses/MATH 624/Project 2")

```

# Abstract
For this analysis, we developed a statistical model for the global prediction of new COVID deaths for the top 10 countries based on the highest amount of deaths. A variety of supervised and unsupervised learning methods were used. 

# 1. Intro
The COVID-19 virus has caused a global pandemic which has detrimentally affected the world causing loss of population and extreme use of various resources. Medical facilities resources have been exacerbated, labor forces have dwindled, and businesses have needed to adapt to wildly changing markets. The purpose of this project was to develop a statistical model for the global prediction of new COVID deaths for the top ten leading countries for benefit of global preparations and accommodations of physical, medical, and populous resources. Model evaluation was based on MSE comparison between models for prediction accuracy. Work for this group was divided equally among colleagues. Work was divided into four parts, data discovery and visualization, tree based methods, linear methods, and unsupervised methods. A number of models were tested and compared with the mentioned methods before deciding on the final working model. Our initial presumption is that tree methods will likely be the most accurate for this analysis. We are testing multiple models to see if other models may be more accurate than our initial thought. This may also prove to be incorrect, as a more complex model does not always mean more accurate predictions. There also is potential for certain models to over-fit due to variables tha directly impact new deaths included in models.

# 2. Method
The methodology in this analysis consisted of diagnosing a world issue, exploring data based around the issue, building various models to compare accuracy, and finally analyze model results for insights.


## 2.1 Data
Data was collected from one source using the code below to extract a csv file at the link "https://raw.githubusercontent.com/owid/covid-19-data/master/public/data/owid-covid-data.csv". This resource can be found through a browser at with the link "https://github.com/owid/covid-19-data". The input data was filtered to the top 10 leading countries in deaths. Variables that were not scaled to thousands were appropriately changed to this scaling and variables that were scaled in another fashion which had a thousand counterpart were removed to reduce duplicated variables and model noise. Some other variables with unneeded information, such as excess_mortality_rate, were also removed for noise reduction.


#### Reading the initial data set and filtering

```{r, echo = T, message = F, warning = F}

require(dplyr)

dataset <- read.csv("https://raw.githubusercontent.com/owid/covid-19-data/master/public/data/owid-covid-data.csv", stringsAsFactors = T) %>%   select(-c(continent, location, weekly_icu_admissions, weekly_icu_admissions_per_million, weekly_hosp_admissions,
            weekly_hosp_admissions_per_million, hosp_patients_per_million, total_boosters_per_hundred, continent,
            location, weekly_icu_admissions, weekly_icu_admissions_per_million, weekly_hosp_admissions,
            weekly_hosp_admissions_per_million, icu_patients_per_million, total_deaths_per_million,
            new_deaths_per_million, new_tests, excess_mortality_cumulative, total_vaccinations_per_hundred,
            people_fully_vaccinated_per_hundred, new_vaccinations_smoothed_per_million, total_cases_per_million,
            new_people_vaccinated_smoothed_per_hundred, excess_mortality_cumulative_absolute, new_cases_per_million,
            excess_mortality_cumulative_per_million, excess_mortality_cumulative, excess_mortality, total_tests,
            people_vaccinated_per_hundred, new_deaths_smoothed_per_million, new_cases_smoothed_per_million,
            new_tests_smoothed, new_deaths_smoothed, reproduction_rate,
            new_tests_smoothed_per_thousand, tests_per_case, new_vaccinations_smoothed, tests_units,
            new_people_vaccinated_smoothed, new_cases_smoothed)) %>%
  filter(iso_code %in% c("USA", "IND", "BRA", "RUS", "GBR", "FRA", "TUR", "ITA", "COL", "DEU")) 


## Creating "by thousands" variable
dataset$total_cases_thousand = dataset$total_cases / 1000
dataset$new_cases_thousand = dataset$new_cases / 1000
dataset$new_deaths_thousand = dataset$new_deaths / 1000
dataset$icu_patients_thousand = dataset$icu_patients / 1000
dataset$hosp_patients_thousand = dataset$hosp_patients / 1000
dataset$total_vaccinations_thousand = dataset$total_vaccinations / 1000
dataset$people_vaccinated_thousand = dataset$people_vaccinated / 1000
dataset$people_fully_vaccinated_thousand = dataset$people_fully_vaccinated / 1000
dataset$total_boosters_thousand = dataset$total_boosters / 1000
dataset$new_vaccinations_thousand = dataset$new_vaccinations / 1000
dataset$population_thousand = dataset$population / 1000
dataset$total_deaths_thousand = dataset$total_deaths / 1000

# Remove old scales
dataset = subset(dataset, select = -c(total_cases, new_cases, new_deaths, icu_patients, hosp_patients,
                                       total_vaccinations, people_vaccinated, people_fully_vaccinated,
                                       total_boosters, new_vaccinations, population, total_deaths))  




```


#### Dates


```{r date, echo = T, message = F, warning = F}

dataset$date <- as.Date(dataset$date)

## Creating day of week variable and 

## Creating season variable (using astronimical start dates)



dataset <- mutate(dataset, day_of_week = as.factor(weekdays(date)),
                  season = as.factor(case_when(months(date) %in% c("March", "April", "May") ~ "Spring",
                                     months(date) %in% c("June", "July", "August") ~ "Summer",
                                     months(date) %in% c("September", "October", "November") ~ "Fall",
                                     months(date) %in% c("December", "January", "February") ~ "Winter")))

dataset$year <- factor(lubridate::year(dataset$date))

dataset$month <- factor(lubridate::month(dataset$date))



dataset <- dataset %>% replace(is.na(.), 0)

dataset$iso_code <- factor(dataset$iso_code)



no_date <- select(dataset, -date)

require(caTools)
require(rsample)

## Splitting the data

set.seed(101)


split <- rsample::initial_split(dataset, prop = 0.8, strata = "new_deaths_thousand")

train.dat <- rsample::training(split)
test.dat <- rsample::testing(split)


```


## 2.2 Data Visualization

We created an interactive geographical shiny app for exploratory information related to average daily cases, average new vaccinations, average daily deaths, population density, and HDI. Additional time series charts are available here as well for our late used methods. This shiny app is available to view via internet browser at "https://choskins.shinyapps.io/shiny/".

The figure below depicts the training and testing split of the response variable (new daily deaths by thousands) for the time series.

```{r viz1, echo = T, message = F, warning = F}

require(ggplot2)
require(ggpubr)

ggplot(data = train.dat, aes(x = date, y = new_deaths_thousand)) + geom_point(color = "blue") +
  geom_point(data = test.dat, aes(x = date, y = new_deaths_thousand), color = "orange") + theme_classic()


```

There is no clear trend, other than the potential of month in the number of COVID-19 deaths. The blue points in the figure above indicate the observations in the training set, and the orange points compose the observations in the testing set. The distribution with time is fairly consistent. 

```{r viz2, echo = T, message = F, warning = F}



A <- ggplot(dataset) + geom_boxplot(aes(x = iso_code, y = new_deaths_thousand, fill = iso_code)) + theme_classic()

B <- ggplot(dataset) + geom_boxplot(aes(x = season, y = new_deaths_thousand, fill = season)) + theme_classic()

C <- ggplot(dataset) + geom_point(aes(x = new_vaccinations_thousand, y = new_deaths_thousand)) + theme_classic() + 
  facet_wrap(~season)
D <- ggplot(dataset) + geom_histogram(aes(x = new_deaths_thousand, fill = month)) + facet_wrap(~month) 

A

```

The new daily deaths vary significantly depending on the country. Notice that the United States (USA) has the highest median new daily deaths, but India (IND) has the highest single value of new daily deaths.

```{r,  echo = T, message = F, warning = F}
B

```

As expected the Winter and Spring seasons have the highest median values for new daily deaths. However, the Summer season has the highest single value of new daily deaths, which occurred, which occurred in late June.

```{r,  echo = T, message = F, warning = F}
C

```

The relationship between new daily vaccinations and new daily deaths is interesting to observe. There are bundles of observations near 0 vaccinations in which there were many deaths, which is due to the lag between vaccine availability and the pandemic. The Spring season has a very unique behavior, as all observations are bundled closely together. 

```{r,  echo = T, message = F, warning = F}
D
```

Each month has a similar distribution of new daily deaths. Each month has the highest counts of new daily deaths near 0, which is likely due to our data preparation, in which we scaled the new daily deaths. 

## 2.3 Models
Multiple models were used for this analysis. They included forward and backward subset selection, multinomial linear models, random forest, ridge and lasso regression, bagging, and generalized boosted regression models.


### 2.3.1 Decision Tree
To demonstrate a comparative analysis with regression tree, bagging, random forests, and boosting we constructed
a regression tree for new_deaths_thousand on the rest of the predictors and calculated the test prediction error. 

```{r dec.Tree, echo = T, message = F, warning = F}

require(tree)

tree.deaths = tree(new_deaths_thousand ~.-date, data = train.dat)
cv.deaths = cv.tree(tree.deaths)
plot(cv.deaths$size, cv.deaths$dev, type = 'b')
prune.deaths = prune.tree(tree.deaths, best = 11)

## prediction error
yhat = predict(prune.deaths, newdata = test.dat)
tree_mse <- mean((yhat - test.dat$new_deaths_thousand)^2)

tree_mse

```

### 2.3.2 Bagged Tree
The main difference between random forest (RF) a and boostrapped aggregated decision tree (Bagged Tree) is that in Bagged Trees, the number of variables selected at each node is not restricted. In RF, the number of variables selected is usually decided using the length of the predictors divided by 3 (for numeric responses). We assume that this model will predict accurately, but has possibility to be inaccurate comparatively due to complexity and over-fitting.

```{r baggedTree, message=FALSE, warning=FALSE}

require(randomForest)

# bag.deaths

bag.deaths = randomForest(new_deaths_thousand ~.-date, data = train.dat, mtry = 34, importance = TRUE)


## prediction error

yhat.bag = predict(bag.deaths, newdata = test.dat)
bag_mse <- mean((yhat.bag - test.dat$new_deaths_thousand)^2)

bag_mse

```

### 2.3.3 Bagged Tree
We assume that this model will predict accurately as well, but has possibility to be inaccurate comparatively for similar reasons.
```{r boost, echo = T, message = F, warning = F}

library(gbm)

boost.deaths = gbm(new_deaths_thousand ~.-date, data = train.dat, distribution = "gaussian", n.trees = 500,
                   interaction.depth = 3)

summary(boost.deaths)

## prediction error

yhat.boost = predict(boost.deaths, newdata = test.dat, n.trees = 500)
boost_mse <- mean((yhat.boost - test.dat$new_deaths_thousand)^2)

boost_mse

```

### 2.3.4 Random Forest
We assume that this model will predict accurately, but has possibility to be inaccurate comparatively for similar reasons. Our hopes is for this to be more accurate than bagging or boosting due to its composition consisting of aspects of both bagging and boosting. Once again this may actually decrease accuracy due to complexity. 
```{r rf, echo = T, message = F, warning = F}



rf <- randomForest(new_deaths_thousand ~.-date, data = train.dat, mtry = (35/3), importance = T)



rf_predictions <- predict(rf, test.dat[-23])


## prediction error

rf_mse <- mean((rf_predictions - test.dat$new_deaths_thousand)^2)

rf_mse

```

### 2.3.5 Multiple Linear Regression (MLR)
Multiple linear regression was used in attempt to find meaningful results from a less complex model. As a simple and easy to use model it was favorable for quick and easy to interpreted results. A minimum error of 0.164 was achieved through this model type. We continued after to test other models to see if we could produce more accurate results with a lower error.
We anticipate this method may possibly have the highest error rate, since the data we are analyzing are composed from time-series data in which the trends with time have been removed. We do not assume this will be as accurate as the more complex models due to the large amount of variables in each model.

```{r mlr1, echo = T, message = F, warning = F}

model1 = glm(new_deaths_thousand ~ ., data=train.dat)
results <- summary(model1)

## Extracting variables with low p-values from the generalized multiple linear regression

pvals <- data.frame(results$coefficients)
pvals <- filter(pvals, pvals$Pr...t.. < 0.05)
print(rownames(pvals))


```

#### A MLR model is built based on the coefficients from the model that have low p-values. 

```{r mlr2, echo = T, message = F, warning = F}

model2 <- glm(new_deaths_thousand ~ iso_code + total_tests_per_thousand + stringency_index +
                icu_patients_thousand + total_deaths_thousand + day_of_week +
                month + new_tests_per_thousand + total_cases_thousand + hosp_patients_thousand +
                people_fully_vaccinated_thousand + positive_rate + total_vaccinations_thousand +
                total_boosters_thousand, data = train.dat)

mlr_predictions <- predict(model2, test.dat[-23])

## Prediction Error

mlr_mse <- mean((mlr_predictions - test.dat$new_deaths_thousand)^2)

mlr_mse

```

### 2.3.6 Forward Selection

```{r forward, echo = T, message = F, warning = F}
require(leaps)

set.seed(101)
regfit.fwd <- regsubsets(new_deaths_thousand~.-date, data = train.dat, nvmax = 60,
                         method = "forward", really.big = T)

fwd.sum <- summary(regfit.fwd)
forward_select=which.min(fwd.sum$bic)
forward_select

```

The forward selection method chooses the following variables (shown below).

```{r forward2, echo = T, message = F, warning = F}

coef(regfit.fwd, forward_select)


```

### 2.3.7 Backward Selection


```{r backward, echo = T, message = F, warning = F}

set.seed(101)
regfit.bwd <- regsubsets(new_deaths_thousand ~.-date, data = train.dat, nvmax = 60, really.big = T,
                         method = "backward")

bwd.sum <- summary(regfit.bwd)

backward_select <- which.min(bwd.sum$bic)
backward_select

```

Backward selection chooses 26 variables (shown below). Recall that forward selection selected 28 variables. 

```{r backward2, echo = T, message = F, warning = F}

coef(regfit.bwd, backward_select)


```

#### Comparison of Forward and Backward Selection

```{r linear_comp, echo = T, message = F, warning = F}

par(mfrow = c(1, 2))

plot(fwd.sum$bic,xlab=" Number of Variables ", ylab=" BIC",
     type="l", main="Forward Selection: BIC plot")
points (forward_select, fwd.sum$bic[forward_select], col =" red", cex =2, pch =20)
plot(bwd.sum$bic,xlab=" Number of Variables ", ylab=" BIC",
     type="l", main="Backward Selection: BIC plot")
points (backward_select, bwd.sum$bic[backward_select], col =" red",cex =2, pch =20)

```

The above figure reiterates the optimal number of features selected in each method using BIC as a metric. 

#### Forward and Backward Selection MLR Models


Recall that in the list of coefficients, dummy variables for the categorical variables are included, so the actual number of variables used in both models are not 26 (forward selection) nor 28 (backward selection).

```{r fb, echo = T, message = F, warning = F}

forward_mlr <- glm(new_deaths_thousand ~ new_tests_per_thousand + total_boosters_thousand + season + month + aged_70_older + hospital_beds_per_thousand +
  iso_code + positive_rate + total_deaths_thousand + year + extreme_poverty + stringency_index + day_of_week + cardiovasc_death_rate +
  total_tests_per_thousand + people_vaccinated_thousand + female_smokers, data = train.dat)

backward_mlr <- glm(new_deaths_thousand ~ iso_code + positive_rate + year + month + extreme_poverty + stringency_index + cardiovasc_death_rate +
  total_tests_per_thousand + day_of_week + female_smokers + season + aged_70_older + hospital_beds_per_thousand,
  data = train.dat)

```

#### Forward and Backward Selection MLR Test Errors

```{r fbmse, echo = T, message = F, warning = F}

forward_pred <- predict(forward_mlr, test.dat[-23])
backward_pred <- predict(backward_mlr, test.dat[-23])

sprintf("Forward-Selected Model MSE: %.4f", mean((forward_pred - test.dat$new_deaths_thousand)^2))
sprintf("Backward-Selected Model MSE %.4f", mean((backward_pred - test.dat$new_deaths_thousand)^2))

f_mse <- mean((forward_pred - test.dat$new_deaths_thousand)^2)
b_mse <- mean((backward_pred - test.dat$new_deaths_thousand)^2)


```


### 2.3.7 Ridge Regression
Recall that ridge regression shrinks the coefficients in the model towards 0, but never to 0. This shrinkage leads to a substantial reduction in the variance of the predictions, with a penalty of slightly increased bias. This can result in a lower MSE when an optimal "λ" is selected. The figure below represents how noise is removed from the model. The number of variables that minimize noise while retaining the predictive power of the model will be chosen based on *log lambda* on the graph. We believe that this and LASSO may be highly accurate as it shrinks coefficients and may be able to reduce noise better than other models.

```{r ridge, echo = T, message = F, warning = F}

require(glmnet)

set.seed(101)



x_train = model.matrix(new_deaths_thousand~.-date, train.dat)[,-1]

y_train <- train.dat$new_deaths_thousand

x_test <- model.matrix(new_deaths_thousand ~.-date, test.dat)[, -1]
y_test <- test.dat$new_deaths_thousand

# train = sample(6720, 5377) # 80% training
# test = (-train)


ridge.model1 <- glmnet(x = x_train,
                       y = y_train,
                       alpha = 0, standardize = F)

cv.out <- cv.glmnet(x = x_train,
                       y = y_train,
                       alpha = 0, standardize = F)


plot(ridge.model1, xvar = "lambda")


```

#### Finding the best Lambda Using Cross-Validation


```{r ridge1, echo = T, message = F, warning = F}

set.seed(101)

cv.out <- cv.glmnet(x = x_train,
                       y = y_train,
                       alpha = 0, standardize = F)

plot(cv.out)



bestlam_ridge <- cv.out$lambda.min


```

Above we chart the impact of different "λ" parameters and select the best one for our ridge regression model. The Lowest MSE occurs at `r log(bestlam_ridge)`. Now, the training and testing errors are compared.

```{r ridge_res, echo = T, message = F, warning = F}
ridge_train_preds <- predict(cv.out, x_train, s = "lambda.min")

ridge_test_preds <- predict(ridge.model1, x_test, s = bestlam_ridge)

sprintf("Test MSE: %.4f", mean((ridge_test_preds - y_test)^2))
sprintf("Train MSE: %.4f", mean((ridge_train_preds - y_train)^2))

ridge_mse <- mean((ridge_test_preds - y_test)^2)

```

### 2.3.8 LASSO Regression

Recall that in LASSO, as opposed to ridge, the coefficients are indeed shrunk to 0. As previously mentioned, we believe that this and ridge regression may be highly accurate as it shrinks coefficients and may be able to reduce noise better than other models.

```{r LASSO, echo = T, message = F, warning = F}

require(glmnet)


set.seed(101)

lasso.model1 <- glmnet(x = x_train,
                       y = y_train,
                       alpha = 1, standardize = F)



plot(lasso.model1, xvar = "lambda")



```
Based on the figure above, it is clear that many of the coefficients shrink to 0, which indicates they are not important to the model. This is explored further below.  


```{r lasso2, echo = T, message = F, warning = F}

set.seed(101)

lasso.cv <- cv.glmnet(x = x_train,
                       y = y_train,
                       alpha = 1, standardize = F)

plot(lasso.cv)


best_lasso <- lasso.cv$lambda.min

```

The ideal lambda value occurs at `r log(lasso.cv$lambda.min)` , as indicated above. 

#### Comparing LASSO training and testing MSE

```{r lassoMSE, echo = T, message = F, warning = F}

lasso_train_preds <- predict(lasso.cv, x_train, s = "lambda.min")

lasso_test_preds <- predict(lasso.model1, x_test, s = best_lasso)

sprintf("Test MSE: %.4f", mean((lasso_test_preds - y_test)^2))
sprintf("Train MSE: %.4f", mean((lasso_train_preds - y_train)^2))

lasso_mse <- mean((lasso_test_preds - y_test)^2)

```

This is an improvement in both the training and testing errors compared to the results from Ridge regression. 

#### LASSO Regression Most Influential Features

```{r lasso_features, echo = T, message = F, warning = F}
require(broom)

coef(lasso.cv, s = "lambda.min") %>%
  tidy() %>%
  filter(row != "(Intercept)") %>%
  ggplot(aes(value, reorder(row, value))) +
  geom_point() +
  xlab("Coefficient") +
  ylab(NULL)




```

Notice the top 10 variables that contributed to new deaths. 

### 2.3.8 PCR

Our motivation behind using principal component regression (PCR), is this method helps to avoid multicollinearity. In this data set, it is clear that many of the variables are correlated with each other. We assume this will perform better than that of LASSO or ridge regression using the principal component of the variables with assumptions for the best bias variance trade-off model. 

```{r pcr, echo = T, message = F, warning = F}
require(pls)


set.seed(101)

pcr_model <- pls::pcr(new_deaths_thousand~., data = no_date, scale = F, validation = "CV")

validationplot(pcr_model, val.type = "MSEP")
validationplot(pcr_model, val.type = "R2")

```

The first figure above represents MSE for each component derived using principal component analysis (PCA). The second figure displays the $R^2$ score for the number of components. Observe that dimensionality reduction has occured, the number of components (15) that explain about 80% of the variability is less than the number of predictors in the data set.

Below displays the predicted vs actual deaths based on a trained PCR model using the same training and testing split data sets that were used in all the other models.

#### PCR with 15 Compoonents

```{r pcr1, echo = T, message = F, warning = F}

set.seed(101)

pcr_trained <- pcr(new_deaths_thousand~.-date, data = train.dat, scale = F, validation = "CV")

pcr_preds <- predict(pcr_trained, test.dat[-23], ncomp = 15)

pcr_mse <- mean((pcr_preds - test.dat$new_deaths_thousand)^2)

```
# 3. Results

```{r results_tab, echo = T, message = F, warning = F}

results_tab <- data.frame(Method = c("Decision Tree", "Bagged Tree", 
                                     "Boosted Tree", "Random Forest",
                                     "MLR", "MLR - Forward", "MLR-Backward",
                                     "LASSO", "Ridge", "PCR"),
                          MSE = c(tree_mse, bag_mse, boost_mse,
                                  rf_mse, mlr_mse, f_mse, b_mse,
                                  lasso_mse, ridge_mse, pcr_mse)) 


knitr::kable(results_tab)

```

After each model was tested final results shown above were compared. As previously predicted, MLR models had the highest error rate and were not considered for use. Ridge regression and LASSO showed similar results with a higher MSE. PCR was on the lower end of the models compared, but did not compare to the tree based models. The general decision tree, although not faring well, did prove to have a lower MSE than all other non-tree based models. Boosting and bagging methods were applied to try and reduce error rate further, but Random Forest proved to yield the lower MSE. Bagged tree lagged close behind by a difference in MSE of 0.0006745. Boosting however performed even worse than bagging. 

# 4. Discussion
There were other potential problems we could have analyzed, but this one was selected due to the relevancy for current issues as well as ease of understanding aspects of the data. There was little interpretation needed to understand initial data and additionally little cleaning and manipulation of data due to succinct data maintenance from our source. We were able to utilize most all of the skills we learned this semester in this project. Additional methods such as historical clustering and PCA were attempted, but this lead to computational issues and visualizations so large that they were unable to be interpreted. These would have been supplementary to our analysis, and therefore were unused. Each group member contributed equally to this work. Tasks such as data extraction, coding of methods, and compilation of this document were divided equally among members, and assigned based on each members specialized skills. It was an immensely helpful learning experience to get to show what we have learned this semester in one project.

# 5. Future Work
The models created in this project are applicable to other results as well from the same dataset. We chose new_deaths as the response variable for our models, but others can be easily interchanged and compared for different results. Note that doing so may yield different results for which model is most accurate. Other alternative responses possible could be new cases, new vaccinations, or other COVID related responses. On the other hand, our methodologies would likely not produce accurate results for something such as population, since the data is geared towards COVID information. Other models, such as (SVM) support vector machines which not been covered, could be applied for different results as well and compared to see if error rate is higher or lower than the current results. Further time and information could also yield usable results with PCA and clustering methods attempted.

# 6. Appendix
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2. Achim Zeileis and Gabor Grothendieck (2005). zoo: S3 Infrastructure for Regular and Irregular Time Series. Journal of Statistical Software, 14(6), 1-27. doi:10.18637/jss.v014.i06
3. Alboukadel Kassambara (2020). ggpubr: 'ggplot2' Based Publication Ready Plots. R package version 0.4.0. https://CRAN.R-project.org/package=ggpubr
4. Aravind Hebbali (2020). olsrr: Tools for Building OLS Regression Models. R package version 0.5.3. https://CRAN.R-project.org/package=olsrr
5. Brandon Greenwell, Bradley Boehmke, Jay Cunningham and GBM Developers (2020). gbm: Generalized Boosted Regression Models. R package version 2.1.8. https://CRAN.R-project.org/package=gbm
6. Brian Ripley (2021). tree: Classification and Regression Trees. R package version 1.0-41. https://CRAN.R-project.org/package=tree
7. David Robinson, Alex Hayes and Simon Couch (2021). broom: Convert Statistical Objects into Tidy Tibbles. R package version 0.7.9. https://CRAN.R-project.org/package=broom
8. Douglas Bates and Martin Maechler (2019). Matrix: Sparse and Dense Matrix Classes and Methods. R package version 1.2-18. https://CRAN.R-project.org/package=Matrix
9. H. Wickham. ggplot2: Elegant Graphics for Data Analysis. Springer-Verlag New York, 2016.
10. Hadley Wickham (2019). stringr: Simple, Consistent Wrappers for Common String Operations. R package version 1.4.0. https://CRAN.R-project.org/package=stringr
11. Hadley Wickham, Romain François, Lionel Henry and Kirill Müller (2021). dplyr: A Grammar of Data Manipulation. R package version 1.0.7. https://CRAN.R-project.org/package=dplyr
12. Jarek Tuszynski (2021). caTools: Tools: Moving Window Statistics, GIF, Base64, ROC AUC, etc. R package version 1.18.2. https://CRAN.R-project.org/package=caTools
13. Jerome Friedman, Trevor Hastie, Robert Tibshirani (2010). Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, 33(1), 1-22. URL https://www.jstatsoft.org/v33/i01/.
14. Julia Silge, Fanny Chow, Max Kuhn and Hadley Wickham (2021). rsample: General Resampling Infrastructure. R package version 0.1.1. https://CRAN.R-project.org/package=rsample
15. Kristian Hovde Liland, Bjørn-Helge Mevik and Ron Wehrens (2021). pls: Partial Least Squares and Principal Component Regression. R package version 2.8-0. https://CRAN.R-project.org/package=pls
16. Max Kuhn (2021). caret: Classification and Regression Training. R package version 6.0-88. https://CRAN.R-project.org/package=caret
17. Ponce et al. (2021). covid19.analytics: An R Package to Obtain, Analyze and Visualize Data from the Coronavirus Disease Pandemic. Journal of Open Source Software, 6(59), 2995. https://doi.org/10.21105/joss.02995
18. R Core Team (2020). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.
19. Sarkar, Deepayan (2008) Lattice: Multivariate Data Visualization with R. Springer, New York. ISBN 978-0-387-75968-5
20. Thomas Lumley based on Fortran code by Alan Miller (2020). leaps: Regression Subset Selection. R package version 3.1. https://CRAN.R-project.org/package=leaps
21. 

## 6.1 Web App Code

```{r ts-res, echo = T, message = F, warning = F, include = T}

require(ggplot2)
require(ggpubr)


res_dataset <- data.frame(date = test.dat$date, actual_deaths = test.dat$new_deaths_thousand, 
                          decision_tree_pred = yhat,
                          bagged_tree_pred = yhat.bag,
                          boosted_tree_pred = yhat.boost,
                          rf_predicted_cases = rf_predictions,
                          MLR = mlr_predictions,
                          Ridge = ridge_test_preds,
                          LASSO = lasso_test_preds,
                          ForwardMLR = forward_pred,
                          BackwardMLR = backward_pred,
                          country = test.dat$iso_code)


# save(res_dataset, file = "shiny/rf_results.RData")
# res_dataset
res_dataset

```


```{r ui, eval = F}

library(shiny)
library(shinydashboard)
library(leaflet)
library(plotly)

load("external1.RData")
load("rf_results.RData")

ui <- dashboardPage(skin = "green",
                    dashboardHeader(title = "COVID-19 Metrics by Country",
                                    titleWidth = 300),
                    dashboardSidebar(width = 300,
                                     sidebarMenu(
                                       menuItem("Map", tabName = "map", icon = icon("map")
                                                ),
                                       
                                       selectInput(inputId = "vars",
                                                   label = "Select the Metric (for Map)",
                                                   choices = c(colnames(map_ds1)[2:5], colnames(map_ds1)[8]),
                                                   selected =  colnames(map_ds1)[2]),
                                       selectInput(inputId = "selected_country",
                                                   label = "Select the Country (for Time Series)",
                                                   choices = c(levels(res_dataset$country)),
                                                   selected = "USA"),
                                       
                                      menuItem("Time-Series (Trees)", tabName = "ts-rf", icon = icon("chart-line")),
          
                                       menuItem("Time-Series (Linear Methods)", tabName = "ts-lin", icon = icon("chart-line")),
                                       menuItem("Time-Series", tabName = "ts-final", icon = icon("chart-line"))
                                       
                                     )),
                    
                    dashboardBody(
                      
                      tabItems(
                        
                        tabItem(tabName = "map",
                                
                      
                      fluidRow(
                                  column(8, leafletOutput("map1", height = 800)),
                                  column(4, dataTableOutput("table1"))
                      )
                      ),
                      
                    tabItem(tabName = "ts-rf",
                            fluidRow(
                              box(plotlyOutput("time_series_trees"))
                            )),
                    tabItem(tabName = "ts-lin",
                            fluidRow(
                              box(plotlyOutput("time_series_linear"))
                            )),
                    tabItem(tabName = "ts-final",
                            fluidRow(
                              box(plotlyOutput("time_series_final"))
                            ))
                      ))         
                      )
                    




```



```{r server, eval = F}

library(shinydashboard)
library(shiny)
library(leaflet)
library(dplyr)
library(stats)
library(sf)
library(lubridate)
library(plotly)

load("external1.RData")
load("rf_results.RData")



server <- function(input, output) {

  time_series <- reactive({
    filter(res_dataset, country == input$selected_country)
    
  })
  
  reactive_ds <- reactive({
    select(map_ds1, geometry, "selection" = input$vars)
  })
  
  bw <- reactive({2 * IQR(reactive_ds()$selection) / length(reactive_ds()$selection)^(1/3)})
  

  pal1 <- reactive({colorBin("viridis",
                      domain = seq(from = min(range(reactive_ds()$selection)),
                                   to = max(range(reactive_ds()$selection)), by = bw()))})

  pop <- reactive({paste("Country: ", map_ds1$name, "<br/>",
                         "Value: ", round(reactive_ds()$selection, 3))})
              
  
  
  output$map1 <- renderLeaflet({
    
    
    leaflet(reactive_ds()) %>%
      addProviderTiles(providers$CartoDB) %>%
      addPolygons(data = reactive_ds()$geometry , fillColor = pal1()(reactive_ds()$selection),
                  color = "black", popup = pop()) %>%
     addLegend(pal = pal1(), values = reactive_ds()$selection)
  })
  
  
  output$table1 <- renderDataTable({
    reactive_ds() %>% select(-geometry) %>%
      cbind("Country" = map_ds1$name)
  })
  
  
  output$time_series_trees <- renderPlotly({
   
   p <- time_series() %>% plot_ly(x = ~ date, y = ~rf_predicted_cases, name = "Random Forest", type = "scatter", mode = "lines") %>%
     add_trace(y = ~actual_deaths, name = "Actual Deaths", mode = "lines+markers") %>%
     add_trace(y = ~decision_tree_pred, name = "Decision Tree") %>%
     add_trace(y = ~bagged_tree_pred, name = "Bagged Tree") %>%
     add_trace(y = ~boosted_tree_pred, name = "Boosted Tree") %>%
     layout(title = paste0("Actual vs. Predicted Daily Deaths (thousands) for", " ", input$selected_country),
            yaxis = list(title = "Deaths"))
   
   return(p)
   
   
     
  })
  
  output$time_series_linear <- renderPlotly({
  
  p.all <- time_series() %>% plot_ly(x = ~ date, y = ~MLR, name = "MLR (Reduced by p-values)", type = "scatter", mode = "lines") %>%
    add_trace(y = ~actual_deaths, name = "Actual Deaths", mode = "lines+markers") %>%
    add_trace(y = ~ForwardMLR, "MLR (Forward Selection)") %>%
    add_trace(y = ~BackwardMLR, "MLR (Backward Selection)") %>%
    add_trace(y = ~s1, name = "Ridge") %>%
    add_trace(y = ~s1.1, name = "LASSO") %>%
    layout(title = paste0("Actual vs. Predicted Daily Deaths (thousands) for", " ", input$selected_country),
           yaxis = list(title = "Deaths"))
  
  return(p.all)
  })
  
  
  output$time_series_final <- renderPlotly({
  
  p.final <- time_series() %>% plot_ly(x = ~date, y = ~new_deaths_thousand.15.comps, name = "PCR",
                                       type = "scatter", mode = "lines") %>%
    add_trace(y = ~actual_deaths, name = "Actual Deaths", mode = "lines+markers") %>%
    add_trace(y = ~MLR, name = "MLR (Reduced by p-values") %>%
    add_trace(y = ~s1.1, name = "LASSO")  %>%
    layout(title = paste0("Actual vs. Predicted Daily Deaths (thousands) for", " ", input$selected_country),
           yaxis = list(title = "Deaths"))
  
  return(p.final)
    
  })
  
}



```