Random Forest and BRMS package
title: “R Notebook” output: html_notebook
Required Libraries
library(readr)## Warning: package 'readr' was built under R version 4.4.3library(dplyr) ##
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## combinelibrary(inspectdf) ## Warning: package 'inspectdf' was built under R version 4.4.3library(cowplot)## Warning: package 'cowplot' was built under R version 4.4.2library(boot)
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## chisq.test, fisher.testlibrary(caTools)## Warning: package 'caTools' was built under R version 4.4.3library(rattle)## Warning: package 'rattle' was built under R version 4.4.3## Loading required package: tibble## Loading required package: bitops## Rattle: A free graphical interface for data science with R.
## Version 5.5.1 Copyright (c) 2006-2021 Togaware Pty Ltd.
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# Interactive Plot packages;
library(viridisLite)
#library(dp)
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## layoutlibrary(magrittr)
#Prediction Packages:
library(pROC)## Warning: package 'pROC' was built under R version 4.4.2## Type 'citation("pROC")' for a citation.##
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## Recall# brms related packages
library(brms)## Warning: package 'brms' was built under R version 4.4.3## Loading required package: Rcpp## Loading 'brms' package (version 2.23.0). Useful instructions
## can be found by typing help('brms'). A more detailed introduction
## to the package is available through vignette('brms_overview').##
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## arlibrary(cmdstanr)## Warning: package 'cmdstanr' was built under R version 4.4.3## This is cmdstanr version 0.9.0## - CmdStanR documentation and vignettes: mc-stan.org/cmdstanr## - CmdStan path: C:/Users/aidan/.cmdstan/cmdstan-2.37.0## - CmdStan version: 2.37.0library(pkgbuild)## Warning: package 'pkgbuild' was built under R version 4.4.3library(bayesplot)## Warning: package 'bayesplot' was built under R version 4.4.3## This is bayesplot version 1.14.0## - Online documentation and vignettes at mc-stan.org/bayesplot## - bayesplot theme set to bayesplot::theme_default()## * Does _not_ affect other ggplot2 plots## * See ?bayesplot_theme_set for details on theme setting##
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## rhatlibrary(loo)## Warning: package 'loo' was built under R version 4.4.3## This is loo version 2.8.0## - Online documentation and vignettes at mc-stan.org/loo## - As of v2.0.0 loo defaults to 1 core but we recommend using as many as possible. Use the 'cores' argument or set options(mc.cores = NUM_CORES) for an entire session.## - Windows 10 users: loo may be very slow if 'mc.cores' is set in your .Rprofile file (see https://github.com/stan-dev/loo/issues/94).Load the Data
wine <- read_delim("winequality-red.csv", delim = ";", quote = "")## Rows: 1599 Columns: 12
## ── Column specification ────────────────────────────────────────────────────────
## Delimiter: ";"
## dbl (12): "fixed acidity, ""volatile acidity"", ""citric acid"", ""residual ...
##
## ℹ Use `spec()` to retrieve the full column specification for this data.
## ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.names(wine) <- gsub('"', '', names(wine)) # drop all double quotes
View(wine)
str(wine)## spc_tbl_ [1,599 × 12] (S3: spec_tbl_df/tbl_df/tbl/data.frame)
## $ fixed acidity : num [1:1599] 7.4 7.8 7.8 11.2 7.4 7.4 7.9 7.3 7.8 7.5 ...
## $ volatile acidity : num [1:1599] 0.7 0.88 0.76 0.28 0.7 0.66 0.6 0.65 0.58 0.5 ...
## $ citric acid : num [1:1599] 0 0 0.04 0.56 0 0 0.06 0 0.02 0.36 ...
## $ residual sugar : num [1:1599] 1.9 2.6 2.3 1.9 1.9 1.8 1.6 1.2 2 6.1 ...
## $ chlorides : num [1:1599] 0.076 0.098 0.092 0.075 0.076 0.075 0.069 0.065 0.073 0.071 ...
## $ free sulfur dioxide : num [1:1599] 11 25 15 17 11 13 15 15 9 17 ...
## $ total sulfur dioxide: num [1:1599] 34 67 54 60 34 40 59 21 18 102 ...
## $ density : num [1:1599] 0.998 0.997 0.997 0.998 0.998 ...
## $ pH : num [1:1599] 3.51 3.2 3.26 3.16 3.51 3.51 3.3 3.39 3.36 3.35 ...
## $ sulphates : num [1:1599] 0.56 0.68 0.65 0.58 0.56 0.56 0.46 0.47 0.57 0.8 ...
## $ alcohol : num [1:1599] 9.4 9.8 9.8 9.8 9.4 9.4 9.4 10 9.5 10.5 ...
## $ quality : num [1:1599] 5 5 5 6 5 5 5 7 7 5 ...
## - attr(*, "spec")=
## .. cols(
## .. `"fixed acidity` = col_double(),
## .. `""volatile acidity""` = col_double(),
## .. `""citric acid""` = col_double(),
## .. `""residual sugar""` = col_double(),
## .. `""chlorides""` = col_double(),
## .. `""free sulfur dioxide""` = col_double(),
## .. `""total sulfur dioxide""` = col_double(),
## .. `""density""` = col_double(),
## .. `""pH""` = col_double(),
## .. `""sulphates""` = col_double(),
## .. `""alcohol""` = col_double(),
## .. `""quality"""` = col_double()
## .. )
## - attr(*, "problems")=<externalptr># Fixing the column names
# Re naming Quality class to represent 1 of 3 categories: "Poor" "Normal " Excellent"
wine <- wine %>%
mutate(quality = case_when(
quality <= 4 ~ "Poor",
quality <= 6 ~ "Normal",
quality >= 7 ~ "Excellent"
)) %>%
mutate(quality = factor(quality, levels = c("Poor", "Normal", "Excellent"), ordered = TRUE))
# Fixing the variable names to stop error code when calling random forests
wine<- clean_names(wine)
str(wine) # Now variables have "_" instead of " " between words## tibble [1,599 × 12] (S3: tbl_df/tbl/data.frame)
## $ fixed_acidity : num [1:1599] 7.4 7.8 7.8 11.2 7.4 7.4 7.9 7.3 7.8 7.5 ...
## $ volatile_acidity : num [1:1599] 0.7 0.88 0.76 0.28 0.7 0.66 0.6 0.65 0.58 0.5 ...
## $ citric_acid : num [1:1599] 0 0 0.04 0.56 0 0 0.06 0 0.02 0.36 ...
## $ residual_sugar : num [1:1599] 1.9 2.6 2.3 1.9 1.9 1.8 1.6 1.2 2 6.1 ...
## $ chlorides : num [1:1599] 0.076 0.098 0.092 0.075 0.076 0.075 0.069 0.065 0.073 0.071 ...
## $ free_sulfur_dioxide : num [1:1599] 11 25 15 17 11 13 15 15 9 17 ...
## $ total_sulfur_dioxide: num [1:1599] 34 67 54 60 34 40 59 21 18 102 ...
## $ density : num [1:1599] 0.998 0.997 0.997 0.998 0.998 ...
## $ p_h : num [1:1599] 3.51 3.2 3.26 3.16 3.51 3.51 3.3 3.39 3.36 3.35 ...
## $ sulphates : num [1:1599] 0.56 0.68 0.65 0.58 0.56 0.56 0.46 0.47 0.57 0.8 ...
## $ alcohol : num [1:1599] 9.4 9.8 9.8 9.8 9.4 9.4 9.4 10 9.5 10.5 ...
## $ quality : Ord.factor w/ 3 levels "Poor"<"Normal"<..: 2 2 2 2 2 2 2 3 3 2 ...#### Test/Train sets ### 70% split
set.seed(123)
# Splitting the data Train/Test set - For model prediction purposes
split <- sample.split(wine$quality, SplitRatio = 0.7)
wine_train <- subset(wine, split == TRUE)
wine_test <- subset(wine, split == FALSE)
str(wine_train)## tibble [1,119 × 12] (S3: tbl_df/tbl/data.frame)
## $ fixed_acidity : num [1:1119] 7.4 7.8 7.4 7.9 7.3 6.7 7.5 7.8 8.9 8.9 ...
## $ volatile_acidity : num [1:1119] 0.7 0.76 0.66 0.6 0.65 0.58 0.5 0.61 0.62 0.62 ...
## $ citric_acid : num [1:1119] 0 0.04 0 0.06 0 0.08 0.36 0.29 0.18 0.19 ...
## $ residual_sugar : num [1:1119] 1.9 2.3 1.8 1.6 1.2 1.8 6.1 1.6 3.8 3.9 ...
## $ chlorides : num [1:1119] 0.076 0.092 0.075 0.069 0.065 0.097 0.071 0.114 0.176 0.17 ...
## $ free_sulfur_dioxide : num [1:1119] 11 15 13 15 15 15 17 9 52 51 ...
## $ total_sulfur_dioxide: num [1:1119] 34 54 40 59 21 65 102 29 145 148 ...
## $ density : num [1:1119] 0.998 0.997 0.998 0.996 0.995 ...
## $ p_h : num [1:1119] 3.51 3.26 3.51 3.3 3.39 3.28 3.35 3.26 3.16 3.17 ...
## $ sulphates : num [1:1119] 0.56 0.65 0.56 0.46 0.47 0.54 0.8 1.56 0.88 0.93 ...
## $ alcohol : num [1:1119] 9.4 9.8 9.4 9.4 10 9.2 10.5 9.1 9.2 9.2 ...
## $ quality : Ord.factor w/ 3 levels "Poor"<"Normal"<..: 2 2 2 2 3 2 2 2 2 2 ...head(wine_train)## # A tibble: 6 × 12
## fixed_acidity volatile_acidity citric_acid residual_sugar chlorides
## <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 7.4 0.7 0 1.9 0.076
## 2 7.8 0.76 0.04 2.3 0.092
## 3 7.4 0.66 0 1.8 0.075
## 4 7.9 0.6 0.06 1.6 0.069
## 5 7.3 0.65 0 1.2 0.065
## 6 6.7 0.58 0.08 1.8 0.097
## # ℹ 7 more variables: free_sulfur_dioxide <dbl>, total_sulfur_dioxide <dbl>,
## # density <dbl>, p_h <dbl>, sulphates <dbl>, alcohol <dbl>, quality <ord>Random Forest
model <- randomForest(quality ~., data = wine_train, iter = 6)
model ##
## Call:
## randomForest(formula = quality ~ ., data = wine_train, iter = 6)
## Type of random forest: classification
## Number of trees: 500
## No. of variables tried at each split: 3
##
## OOB estimate of error rate: 13.32%
## Confusion matrix:
## Poor Normal Excellent class.error
## Poor 0 43 1 1.00000000
## Normal 3 893 27 0.03250271
## Excellent 0 75 77 0.49342105# Output of model under 3 class specification, unfortunately, due to the small size of class "poor" the model was only able to predict 1 in 44 wines correctly. Assigning 43 to class "normal" which is why there is an extremely large error rate of 97.7% for the "poor" class. A similar issue is occurring with the "Excellent" Class, however to a lesser degree, i.e. incorrectly classifies 73 or the (79+73) results, hence an error rate of 48%.
# for the above reason, we will consider a slight alternative way to modelling the issue, considering now 2 classes "Good" "Poor/Normal" -> Already with this type of method is we will see that the difference between a wine which falls into the lower end of the "good" class and a wine which is considered the upper limit of the "Normal" class, is indeed queit small.
# another method, is re-running the same model, but with balanced sample sizing. this yeilds the following results. It effectively means that each model samples an equal number of observations from each class (Poor , Normal, Excellent) regardless of there weighting on the total model.
model_resample <- randomForest(quality ~ ., data = wine_train,
sampsize = rep(min(table(wine_train$quality)), 3))
model_resample # Overall accuracy has dropped, the model is no longer weighted towards the normal class. The model now correctly classifies 1 - Class error rate for all of the classes. The confusion matrix shows that there is a large amount of missclassification betweeen the normal and the excellent class. ##
## Call:
## randomForest(formula = quality ~ ., data = wine_train, sampsize = rep(min(table(wine_train$quality)), 3))
## Type of random forest: classification
## Number of trees: 500
## No. of variables tried at each split: 3
##
## OOB estimate of error rate: 32.08%
## Confusion matrix:
## Poor Normal Excellent class.error
## Poor 27 15 2 0.3863636
## Normal 111 604 208 0.3456121
## Excellent 6 17 129 0.1513158 # Later to check if 500 trees is sufficient.
# Comparison of model Prediction
par(mfrow=c(1,2))
plot(model_resample$predicted, main = "Predictive Model", ylim = c(0, 1000))
plot(wine_train$quality, main = "observed Model", ylim = c(0, 1000))
# Can see that there is less weight assigned to the normal class, and more spread from this class between "Poor" and "excellent" with more weighting put on normal values -> excellent (205 from normal -> Excellent), compared with 103 from normal to excellent
#OOB => Out Of Bag error rate,
#1 - 00B => how many correctly OOB samples where correctly classified
# Visual plots
varImpPlot(model_resample)
# Shows how much each variable decreases node impurity (Gini) across all trees. Higher the value the more important the variable is in classifying wine.
#Ranking:
#1). Sulphates (Most influential feature - often linked to wine presevation and flavor, higher sulphates can signal better quality wines (to a degree))
#2). Alcohol: Strong positive relationship with wine quality, higher alcohol typically correlates with higher quality ratings
#3). Volatile acidity: Very important negative factor, high volatile acidity usually worsens the taste and reduces quality.
#4). Helps distrigush between mid and high quality wines.
#5). Moderate importance, relates to oxidation and wine stability.
#6-11 less influential but all of these combined help us fine tune the model.
df_train <- as.data.frame(wine_train)
par(mfrow = c(1, 3))
partialPlot(model_resample, df_train, "alcohol", which.class = "Excellent", main = "Alcohol")
partialPlot(model_resample, df_train, "volatile_acidity", which.class = "Excellent", main = "Volatile Acidity")
partialPlot(model_resample, df_train, "sulphates", which.class = "Excellent", main = "Sulphates")
par(mfrow = c(1, 1))
pred <- predict(model, wine_test)
table(pred, wine_test$quality)##
## pred Poor Normal Excellent
## Poor 0 0 0
## Normal 19 390 38
## Excellent 0 6 27lvls <- levels(wine_test$quality) # should be c("Poor","Normal","Excellent")
pred <- factor(pred, levels = lvls, ordered = TRUE)
print("Accuracy of Model")## [1] "Accuracy of Model"mean(pred == wine_test$quality)## [1] 0.86875# Seeing if 500 Trees is enough for optimal classification we can plot the error rates, data frame which formats the error data for ggplot, based off err.rate matrix. Each row represents error rates when making the random Forrest.
# Shows the error rates with the variables, we see random forest error rates decreases when we add more trees, we recreate the error rates with a random forests of 1000 trees to see if adding more trees helps decrease OOB.
## When 2000 Trees where used
set.seed(123)
model_resample_1000_trees <- randomForest(quality ~ ., data = wine_train, ntree=2000,
sampsize = rep(min(table(wine_train$quality)), 3))
model_resample_1000_trees##
## Call:
## randomForest(formula = quality ~ ., data = wine_train, ntree = 2000, sampsize = rep(min(table(wine_train$quality)), 3))
## Type of random forest: classification
## Number of trees: 2000
## No. of variables tried at each split: 3
##
## OOB estimate of error rate: 31.37%
## Confusion matrix:
## Poor Normal Excellent class.error
## Poor 27 15 2 0.3863636
## Normal 103 613 207 0.3358613
## Excellent 6 18 128 0.1578947model_resample##
## Call:
## randomForest(formula = quality ~ ., data = wine_train, sampsize = rep(min(table(wine_train$quality)), 3))
## Type of random forest: classification
## Number of trees: 500
## No. of variables tried at each split: 3
##
## OOB estimate of error rate: 32.08%
## Confusion matrix:
## Poor Normal Excellent class.error
## Poor 27 15 2 0.3863636
## Normal 111 604 208 0.3456121
## Excellent 6 17 129 0.1513158# We can see that OOB hasn't increased, now looking at the error rate plots
oob.error.data <- data.frame(
Trees = rep(1:nrow(model_resample_1000_trees$err.rate), times=4),
Type = rep(c("OOB", "Poor", "Normal", "Excellent"), each=nrow(model_resample_1000_trees$err.rate)),
Error = c(model_resample_1000_trees$err.rate[,"OOB"], model_resample_1000_trees$err.rate[,"Poor"], model_resample_1000_trees$err.rate[,"Normal"], model_resample_1000_trees$err.rate[,"Excellent"])
)
head(oob.error.data)## Trees Type Error
## 1 1 OOB 0.4805970
## 2 2 OOB 0.4622468
## 3 3 OOB 0.4606335
## 4 4 OOB 0.4487410
## 5 5 OOB 0.4354839
## 6 6 OOB 0.4109221ggplot(data = oob.error.data, aes(x=Trees, y=Error)) +
geom_line(aes(color=Type))
## Code to see the extreme, when 10,000 tree's where used
set.seed(123)
model_resample_1000_trees <- randomForest(quality ~ ., data = wine_train, ntree=10000,
sampsize = rep(min(table(wine_train$quality)), 3))
model_resample_1000_trees##
## Call:
## randomForest(formula = quality ~ ., data = wine_train, ntree = 10000, sampsize = rep(min(table(wine_train$quality)), 3))
## Type of random forest: classification
## Number of trees: 10000
## No. of variables tried at each split: 3
##
## OOB estimate of error rate: 31.55%
## Confusion matrix:
## Poor Normal Excellent class.error
## Poor 25 16 3 0.4318182
## Normal 101 612 210 0.3369447
## Excellent 6 17 129 0.1513158model_resample##
## Call:
## randomForest(formula = quality ~ ., data = wine_train, sampsize = rep(min(table(wine_train$quality)), 3))
## Type of random forest: classification
## Number of trees: 500
## No. of variables tried at each split: 3
##
## OOB estimate of error rate: 32.08%
## Confusion matrix:
## Poor Normal Excellent class.error
## Poor 27 15 2 0.3863636
## Normal 111 604 208 0.3456121
## Excellent 6 17 129 0.1513158# We can see that OOB hasn't increased, now looking at the error rate plots
oob.error.data <- data.frame(
Trees = rep(1:nrow(model_resample_1000_trees$err.rate), times=4),
Type = rep(c("OOB", "Poor", "Normal", "Excellent"), each=nrow(model_resample_1000_trees$err.rate)),
Error = c(model_resample_1000_trees$err.rate[,"OOB"], model_resample_1000_trees$err.rate[,"Poor"], model_resample_1000_trees$err.rate[,"Normal"], model_resample_1000_trees$err.rate[,"Excellent"])
)
head(oob.error.data)## Trees Type Error
## 1 1 OOB 0.4805970
## 2 2 OOB 0.4622468
## 3 3 OOB 0.4606335
## 4 4 OOB 0.4487410
## 5 5 OOB 0.4354839
## 6 6 OOB 0.4109221ggplot(data = oob.error.data, aes(x=Trees, y=Error)) +
geom_line(aes(color=Type))
model_resample_1000_trees <- randomForest(quality ~ ., data = wine_train, proximity = TRUE, importance = TRUE,ntree=2000,
sampsize = rep(min(table(wine_train$quality)), 3))
model_resample_1000_trees##
## Call:
## randomForest(formula = quality ~ ., data = wine_train, proximity = TRUE, importance = TRUE, ntree = 2000, sampsize = rep(min(table(wine_train$quality)), 3))
## Type of random forest: classification
## Number of trees: 2000
## No. of variables tried at each split: 3
##
## OOB estimate of error rate: 31.9%
## Confusion matrix:
## Poor Normal Excellent class.error
## Poor 25 17 2 0.4318182
## Normal 108 607 208 0.3423619
## Excellent 6 16 130 0.1447368# Considering if the number of variables we tried at each step where sufficient
oob.values <- vector(length = 10)
for (i in 1:10){
temp.model <- randomForest(quality~., data = wine_train, mtry=i, ntree=2000)
oob.values[i]<-temp.model$err.rate[nrow(temp.model$err.rate),1]
}
head(oob.values)## [1] 0.1295800 0.1313673 0.1304736 0.1322609 0.1304736 0.1340483# 3rd value has the lowest out of bag error rate, so default value under the model was optimal.
# 1. Compute the distance matrix
distance.matrix <- dist(1 - model_resample_1000_trees$proximity)
# 2. Classical MDS (multidimensional scaling)
mds.stuff <- cmdscale(distance.matrix, eig = TRUE, x.ret = TRUE)
# 3. Percentage of variance explained
mds.var.per <- round(mds.stuff$eig / sum(mds.stuff$eig) * 100, 1)
# 4. Extract MDS coordinates
mds.values <- mds.stuff$points
head(mds.values)## [,1] [,2]
## 1 -1.4872462 -1.4696685
## 2 -1.4881697 0.4042023
## 3 -1.7587170 -1.0856461
## 4 -1.8965049 -0.2562595
## 5 -0.8869568 -1.7742413
## 6 -2.3455167 0.3867852# 5. Create plotting data frame
mds.data <- data.frame(
Sample = rownames(mds.values),
X = mds.values[, 1],
Y = mds.values[, 2]
)
# 6. Plot in ggplot
library(ggplot2)
ggplot(data = mds.data, aes(x = X, y = Y, colour = "red")) +
geom_point(size = 3, alpha = 0.8) +
theme_bw() +
xlab(paste("MDS1 -", mds.var.per[1], "%")) +
ylab(paste("MDS2 -", mds.var.per[2], "%")) +
ggtitle("MDS Plot using (1 - Random Forest Proximities)")
#Plotting the trees
getTree(model_resample_1000_trees, k = 5, labelVar = TRUE)## left daughter right daughter split var split point status
## 1 2 3 citric_acid 0.31500 1
## 2 4 5 free_sulfur_dioxide 6.50000 1
## 3 6 7 sulphates 0.57500 1
## 4 8 9 density 0.99368 1
## 5 10 11 p_h 3.40000 1
## 6 12 13 total_sulfur_dioxide 8.00000 1
## 7 14 15 chlorides 0.11850 1
## 8 0 0 <NA> 0.00000 -1
## 9 16 17 alcohol 11.25000 1
## 10 18 19 citric_acid 0.01500 1
## 11 20 21 sulphates 0.68000 1
## 12 0 0 <NA> 0.00000 -1
## 13 22 23 residual_sugar 1.95000 1
## 14 24 25 chlorides 0.09150 1
## 15 26 27 total_sulfur_dioxide 18.00000 1
## 16 0 0 <NA> 0.00000 -1
## 17 28 29 total_sulfur_dioxide 13.50000 1
## 18 0 0 <NA> 0.00000 -1
## 19 0 0 <NA> 0.00000 -1
## 20 30 31 free_sulfur_dioxide 7.50000 1
## 21 32 33 alcohol 10.05000 1
## 22 0 0 <NA> 0.00000 -1
## 23 0 0 <NA> 0.00000 -1
## 24 34 35 density 0.99538 1
## 25 36 37 volatile_acidity 0.45500 1
## 26 0 0 <NA> 0.00000 -1
## 27 38 39 p_h 2.88000 1
## 28 0 0 <NA> 0.00000 -1
## 29 0 0 <NA> 0.00000 -1
## 30 40 41 fixed_acidity 6.30000 1
## 31 42 43 sulphates 0.65500 1
## 32 0 0 <NA> 0.00000 -1
## 33 0 0 <NA> 0.00000 -1
## 34 0 0 <NA> 0.00000 -1
## 35 44 45 volatile_acidity 0.34000 1
## 36 0 0 <NA> 0.00000 -1
## 37 0 0 <NA> 0.00000 -1
## 38 0 0 <NA> 0.00000 -1
## 39 0 0 <NA> 0.00000 -1
## 40 0 0 <NA> 0.00000 -1
## 41 0 0 <NA> 0.00000 -1
## 42 46 47 free_sulfur_dioxide 32.00000 1
## 43 0 0 <NA> 0.00000 -1
## 44 48 49 residual_sugar 3.20000 1
## 45 50 51 volatile_acidity 0.42750 1
## 46 52 53 total_sulfur_dioxide 96.00000 1
## 47 54 55 p_h 3.64000 1
## 48 56 57 residual_sugar 1.55000 1
## 49 0 0 <NA> 0.00000 -1
## 50 58 59 density 0.99688 1
## 51 60 61 density 0.99905 1
## 52 0 0 <NA> 0.00000 -1
## 53 0 0 <NA> 0.00000 -1
## 54 0 0 <NA> 0.00000 -1
## 55 0 0 <NA> 0.00000 -1
## 56 62 63 p_h 3.28500 1
## 57 0 0 <NA> 0.00000 -1
## 58 0 0 <NA> 0.00000 -1
## 59 64 65 fixed_acidity 11.05000 1
## 60 0 0 <NA> 0.00000 -1
## 61 66 67 alcohol 9.70000 1
## 62 0 0 <NA> 0.00000 -1
## 63 0 0 <NA> 0.00000 -1
## 64 68 69 total_sulfur_dioxide 12.00000 1
## 65 0 0 <NA> 0.00000 -1
## 66 0 0 <NA> 0.00000 -1
## 67 0 0 <NA> 0.00000 -1
## 68 0 0 <NA> 0.00000 -1
## 69 0 0 <NA> 0.00000 -1
## prediction
## 1 <NA>
## 2 <NA>
## 3 <NA>
## 4 <NA>
## 5 <NA>
## 6 <NA>
## 7 <NA>
## 8 Excellent
## 9 <NA>
## 10 <NA>
## 11 <NA>
## 12 Excellent
## 13 <NA>
## 14 <NA>
## 15 <NA>
## 16 Poor
## 17 <NA>
## 18 Poor
## 19 Normal
## 20 <NA>
## 21 <NA>
## 22 Normal
## 23 Poor
## 24 <NA>
## 25 <NA>
## 26 Poor
## 27 <NA>
## 28 Poor
## 29 Normal
## 30 <NA>
## 31 <NA>
## 32 Normal
## 33 Excellent
## 34 Excellent
## 35 <NA>
## 36 Excellent
## 37 Normal
## 38 Poor
## 39 Normal
## 40 Poor
## 41 Normal
## 42 <NA>
## 43 Normal
## 44 <NA>
## 45 <NA>
## 46 <NA>
## 47 <NA>
## 48 <NA>
## 49 Normal
## 50 <NA>
## 51 <NA>
## 52 Poor
## 53 Excellent
## 54 Normal
## 55 Poor
## 56 <NA>
## 57 Excellent
## 58 Normal
## 59 <NA>
## 60 Excellent
## 61 <NA>
## 62 Excellent
## 63 Normal
## 64 <NA>
## 65 Normal
## 66 Normal
## 67 Excellent
## 68 Normal
## 69 Excellent# Extraction of 1 tree and convertion to visual output:
# Build a single decision tree to visualize
tree_model <- rpart(quality ~ ., data = wine_train, method = "class")
# Plot the tree
rpart.plot(tree_model, type = 3, extra = 104, fallen.leaves = TRUE)
# Better version of above plot
rpart.plot(tree_model,
type = 3, # Draw split labels above the branches
extra = 104, # Show predicted class + prob + % of obs
fallen.leaves = TRUE,
box.palette = "Blues", # Colour palette
branch.lty = 3, # Dotted branches
shadow.col = "gray", # Add slight shadow for clarity
main = "Decision Tree for Wine Quality Classification")
# Presentation of the nodal tree splitting
fancyRpartPlot(tree_model)
head(model_resample_1000_trees$proximity)## 1 2 3 4 5 6 7
## 1 1.0000000 0.11342466 0.77710177 0.1756018 0.16712707 0.13741722 0.009366391
## 2 0.1134247 1.00000000 0.09939594 0.1313077 0.03875433 0.19256018 0.045554336
## 3 0.7771018 0.09939594 1.00000000 0.2500000 0.21912073 0.19524336 0.012721239
## 4 0.1756018 0.13130765 0.25000000 1.0000000 0.30932785 0.35120350 0.017505470
## 5 0.1671271 0.03875433 0.21912073 0.3093278 1.00000000 0.09401114 0.006925208
## 6 0.1374172 0.19256018 0.19524336 0.3512035 0.09401114 1.00000000 0.020925110
## 8 9 10 11 12 13
## 1 0.04116356 0.009289617 0.008205689 0.003474635 0.022626932 0.17835366
## 2 0.18102508 0.153011394 0.149619151 0.029737206 0.210612691 0.02896341
## 3 0.04355017 0.027412281 0.025881057 0.004878049 0.040420819 0.21505376
## 4 0.05122616 0.043360434 0.043383948 0.013821700 0.080371788 0.16212121
## 5 0.01589496 0.004820937 0.004810997 0.001762115 0.009708738 0.12592593
## 6 0.09879254 0.116748500 0.117067834 0.019594122 0.137741047 0.15781487
## 14 15 16 17 18 19 20
## 1 0.06988353 0.056136489 0.029330382 0.14301491 0.04928018 0.7224986 0.1795154
## 2 0.01929438 0.144887917 0.178021978 0.02413604 0.07197802 0.1439560 0.1825613
## 3 0.08533185 0.082274986 0.042682927 0.15982242 0.06055556 0.6463415 0.1715389
## 4 0.10077093 0.044189853 0.066959385 0.12957900 0.09325288 0.1957354 0.1629428
## 5 0.02439024 0.008275862 0.009742519 0.11103400 0.02216066 0.1931897 0.1312155
## 6 0.10765816 0.068681319 0.099114064 0.19447514 0.12693157 0.2201987 0.3218202
## 21 22 23 24 25 26
## 1 0.26414058 0.04008786 0.12945694 0.09221425 0.11634757 0.012061404
## 2 0.14675396 0.47600872 0.04746318 0.05205479 0.05102041 0.034858388
## 3 0.23370166 0.07004964 0.18687982 0.08628319 0.07555556 0.015977961
## 4 0.07080610 0.12527233 0.19324986 0.04052574 0.07759883 0.010331702
## 5 0.09896194 0.02831492 0.07633588 0.05000000 0.21611722 0.004135079
## 6 0.14599341 0.18626374 0.17362637 0.03469163 0.07782673 0.018072289
## 27 28 29 30 31 32
## 1 0.007146784 0.26512651 0.032400589 0.017630854 0.03931340 0.06006674
## 2 0.106790800 0.02620087 0.007225434 0.237677985 0.04615385 0.00770077
## 3 0.007722008 0.21787093 0.034277198 0.018262313 0.04836020 0.07469342
## 4 0.022963368 0.14324619 0.040995608 0.032258065 0.06040637 0.09636564
## 5 0.005547850 0.23638870 0.043557169 0.001392758 0.01736111 0.08025122
## 6 0.042447630 0.08342481 0.046920821 0.082644628 0.10082645 0.10365854
## 33 34 35 36 37 38 39
## 1 0.05082873 0.03703704 0.11496150 0.07560706 0.023730684 0.02373068 0.2318681
## 2 0.02032967 0.17243273 0.07696507 0.05537281 0.069436851 0.21722436 0.1239083
## 3 0.06725959 0.05204873 0.12679162 0.08185841 0.036524626 0.04542936 0.3151815
## 4 0.08406593 0.05540318 0.08015267 0.05798687 0.073304158 0.09956236 0.2320261
## 5 0.03272981 0.01947149 0.10359116 0.08327550 0.006232687 0.02700831 0.1089655
## 6 0.13661504 0.08172281 0.08786381 0.05956977 0.124656782 0.20748487 0.2058178
## 40 41 42 43 44 45 46
## 1 0.002735230 0.18266594 0.13925439 0.028650138 0.23734610 0.4199016 0.13384956
## 2 0.029923830 0.17701525 0.03050109 0.222647702 0.16144414 0.2595212 0.03892544
## 3 0.005503577 0.25399449 0.17290749 0.032132964 0.28219178 0.3726674 0.06866002
## 4 0.014138119 0.16294278 0.11630435 0.050764192 0.16282225 0.1980467 0.06239737
## 5 0.006185567 0.07803867 0.06189821 0.007612457 0.06735751 0.1505155 0.15855355
## 6 0.010970927 0.16309720 0.11378556 0.112087912 0.20861833 0.2423414 0.03147432
## 47 48 49 50 51 52 53
## 1 0.13704735 0.001098901 0.11723010 0.3581267 0.1458680 0.1695008 0.31196581
## 2 0.03875969 0.020163488 0.16106291 0.1395731 0.1142384 0.1106267 0.08039492
## 3 0.06998880 0.001101928 0.07675439 0.4235880 0.1483547 0.1556291 0.26839827
## 4 0.06430155 0.008719346 0.04926909 0.4182613 0.3046832 0.3253536 0.20768137
## 5 0.16235129 0.000000000 0.05209047 0.2588235 0.1188325 0.1231834 0.06968641
## 6 0.03125000 0.009884679 0.03549973 0.3441451 0.2309828 0.2423246 0.22411348
## 54 55 56 57 58 59
## 1 0.020408163 0.09805014 0.07823691 0.006600660 0.12424740 0.012700166
## 2 0.039473684 0.01438053 0.04649891 0.020753687 0.11678436 0.148330597
## 3 0.027700831 0.11675978 0.09812569 0.010520487 0.18096810 0.022123894
## 4 0.003831418 0.10535025 0.09721464 0.018518519 0.11135253 0.041643836
## 5 0.002074689 0.09383754 0.02140884 0.006891799 0.08821502 0.005544006
## 6 0.007162534 0.11500000 0.21452145 0.019305019 0.19344262 0.083931530
## 60 61 62 63 64 65
## 1 0.09934138 0.0049806309 0.41818182 0.07828004 0.013751375 0.01043956
## 2 0.08660131 0.0621562156 0.02788409 0.14519651 0.154223433 0.14792576
## 3 0.12052834 0.0094496943 0.43235781 0.12251656 0.023704520 0.01821192
## 4 0.10941753 0.0164654226 0.16338798 0.22942779 0.045726728 0.04191617
## 5 0.07697642 0.0006934813 0.18892734 0.03407510 0.006882312 0.00621547
## 6 0.06571742 0.0420121614 0.21122112 0.32894737 0.086813187 0.08013172
## 66 67 68 69 70 71 72
## 1 0.10153677 0.04017857 0.26982379 0.05132450 0.1111111 0.14269912 0.12486127
## 2 0.08287895 0.03731343 0.11452055 0.11835616 0.0295082 0.11318681 0.09306167
## 3 0.12156216 0.02398801 0.17357656 0.06419480 0.1260365 0.19399666 0.15971063
## 4 0.10990207 0.10487445 0.04214559 0.10246575 0.1272529 0.18825467 0.36633663
## 5 0.07549760 0.16923077 0.12352533 0.03254848 0.1074896 0.03897008 0.08722959
## 6 0.06410959 0.05731523 0.08985667 0.16941694 0.1195592 0.24419890 0.37990061
## 73 74 75 76 77 78 79
## 1 0.10250696 0.12720264 0.01214798 0.06291391 0.014859659 0.04693539 0.05686167
## 2 0.02149945 0.08842795 0.07283680 0.18883415 0.209836066 0.26783754 0.38294405
## 3 0.12158394 0.16399779 0.01435671 0.08457711 0.019856591 0.06714761 0.06790800
## 4 0.24613687 0.36189956 0.02296337 0.07553366 0.034915439 0.09304871 0.11774281
## 5 0.13505948 0.08512111 0.00000000 0.02491349 0.004840941 0.01530967 0.02888583
## 6 0.10660744 0.37843784 0.05998899 0.13153550 0.085118067 0.15681944 0.16111415
## 80 81 82 83 84 85 86
## 1 0.03925845 0.010561423 0.04696133 0.16740331 0.19384953 0.1844444 0.04223807
## 2 0.42199350 0.072767365 0.27068493 0.07295666 0.14262384 0.1587912 0.23039216
## 3 0.07115490 0.013348165 0.06877427 0.18333333 0.18261826 0.1360355 0.07036833
## 4 0.12979989 0.013201320 0.09095890 0.08105148 0.18651441 0.1468647 0.12887439
## 5 0.02674897 0.003484321 0.01462396 0.06123869 0.07713499 0.0514246 0.02620690
## 6 0.20098039 0.018836565 0.15469613 0.06637168 0.45643836 0.2838137 0.21643836
## 87 88 89 90 91 92 93
## 1 0.62459016 0.10262131 0.18687982 0.13219027 0.01257518 0.01363141 0.11417540
## 2 0.11159263 0.04371887 0.05205479 0.02140505 0.01899078 0.02170374 0.05908096
## 3 0.45908841 0.15117581 0.11973392 0.15786548 0.02031851 0.02083333 0.14988938
## 4 0.16368564 0.29219701 0.07662835 0.08826754 0.01951220 0.02168022 0.10934937
## 5 0.16173434 0.12736179 0.16585704 0.09273356 0.03512397 0.03568977 0.30977131
## 6 0.09541985 0.20879733 0.02314050 0.11283186 0.03000546 0.03213508 0.10236654
## 94 95 96 97 98 99 100
## 1 0.17079890 0.04759270 0.08836443 0.10227904 0.14847162 0.04675468 0.08724462
## 2 0.32185792 0.43179318 0.38595536 0.07925151 0.10309278 0.42864954 0.38815789
## 3 0.13550885 0.04719600 0.06008820 0.14086860 0.20799124 0.04585635 0.06152993
## 4 0.13238512 0.08173341 0.07834603 0.04722680 0.29383117 0.08465319 0.08301475
## 5 0.09116022 0.01948504 0.01717033 0.01544944 0.06976744 0.01795580 0.01863354
## 6 0.21530837 0.14088398 0.09879254 0.10049972 0.34749455 0.14317060 0.09933775
## 101 102 103 104 105 106 107
## 1 0.001648352 0.09265351 0.05271920 0.02268954 0.12995595 0.003003003 0.1245158
## 2 0.002729258 0.03478261 0.15960374 0.15527611 0.06670312 0.062686567 0.1106243
## 3 0.003867403 0.10494505 0.05990017 0.02990033 0.16112957 0.004504505 0.1772853
## 4 0.003271538 0.11285947 0.08726674 0.06014215 0.08956854 0.014992504 0.1855104
## 5 0.005517241 0.10501030 0.01798064 0.00555170 0.11610228 0.000000000 0.1826389
## 6 0.004388371 0.10722101 0.15614618 0.11025358 0.13758943 0.037425150 0.2992257
## 108 109 110 111 112 113
## 1 0.005534034 0.008241758 0.006651885 0.007709251 0.22786885 0.18927790
## 2 0.026272578 0.030534351 0.026490066 0.029492081 0.09288430 0.25923913
## 3 0.008301051 0.011563877 0.009476031 0.011043622 0.29917808 0.12225877
## 4 0.009846827 0.012540894 0.011037528 0.009819967 0.22240260 0.09989142
## 5 0.004184100 0.004840941 0.003486750 0.004152249 0.07049966 0.08716541
## 6 0.016004415 0.021358160 0.017777778 0.018141836 0.34531590 0.18599562
## 114 115 116 117 118 119 120
## 1 0.2401771 0.07973422 0.09191584 0.1768202 0.25415282 0.05660377 0.21220930
## 2 0.1037891 0.03628367 0.08976464 0.0698690 0.07616438 0.31680441 0.05405405
## 3 0.2946726 0.11866295 0.13089296 0.2831595 0.28729895 0.05406912 0.16448326
## 4 0.2504110 0.23435785 0.18883415 0.4075145 0.13691128 0.08535242 0.18454936
## 5 0.1981982 0.09576683 0.16446912 0.2904412 0.21016006 0.01745810 0.16666667
## 6 0.2249173 0.10902048 0.29668508 0.2791728 0.24820739 0.15806988 0.05224964
## 121 122 123 124 125 126 127
## 1 0.07620614 0.07494530 0.17772902 0.12362031 0.06740331 0.12099448 0.08599779
## 2 0.07788671 0.07871878 0.07821836 0.02240437 0.17124040 0.02244116 0.05288986
## 3 0.09095921 0.08681319 0.24807903 0.14072022 0.09605775 0.13779745 0.09136213
## 4 0.11528004 0.11008677 0.19228680 0.20951860 0.09698630 0.21150685 0.02349727
## 5 0.03019904 0.02959394 0.18620690 0.10602911 0.04526462 0.10887656 0.01386963
## 6 0.12876712 0.12431544 0.12233752 0.14002205 0.11885019 0.14135837 0.03583241
## 128 129 130 131 132 133 134
## 1 0.40597345 0.10209713 0.1056687 0.005500550 0.4044077 0.31274131 0.01434087
## 2 0.12843030 0.19890411 0.1953082 0.119476268 0.1769525 0.08118486 0.04288070
## 3 0.29711752 0.12665929 0.1291391 0.012127894 0.3698782 0.34423503 0.01833333
## 4 0.09638554 0.20251779 0.2018549 0.034332425 0.2136612 0.19496166 0.05869446
## 5 0.19486824 0.07093185 0.0699446 0.004149378 0.1998612 0.06994460 0.00625000
## 6 0.06464088 0.15002758 0.1521978 0.080175728 0.1757576 0.26755113 0.12493090
## 135 136 137 138 139 140 141
## 1 0.10401761 0.32542747 0.06256921 0.09961262 0.07134956 0.1212789 0.1224263
## 2 0.02785363 0.08122942 0.11868132 0.06578947 0.27372463 0.1150685 0.1144279
## 3 0.13086692 0.22880886 0.09196676 0.14642263 0.11043285 0.1448314 0.1456473
## 4 0.11921611 0.14926189 0.16144975 0.08054795 0.17250821 0.2252597 0.2280508
## 5 0.02356202 0.21473245 0.02432245 0.02713987 0.03129346 0.1051176 0.1090014
## 6 0.20659341 0.06174201 0.27790055 0.10369553 0.37217007 0.2099174 0.2114317
## 142 143 144 145 146 147
## 1 0.12665929 0.013947001 0.12888889 0.002092050 0.0021990104 0.10088203
## 2 0.14747807 0.038700760 0.04654655 0.022774327 0.0556464812 0.03721949
## 3 0.17415419 0.004181185 0.08792846 0.002805049 0.0055096419 0.13875069
## 4 0.14238773 0.008959338 0.10164425 0.010395010 0.0130647795 0.13668671
## 5 0.09205021 0.018453427 0.21564885 0.007922535 0.0006915629 0.05597789
## 6 0.15270121 0.009084556 0.03303303 0.010423905 0.0389895662 0.25426527
## 148 149 150 151 152 153
## 1 0.11135491 0.0000000000 0.0000000000 0.06873977 0.02843084 0.0006973501
## 2 0.02995643 0.0069783671 0.0061643836 0.07305195 0.26072787 0.0041666667
## 3 0.14419373 0.0000000000 0.0000000000 0.10630137 0.04832510 0.0006978367
## 4 0.12717391 0.0000000000 0.0000000000 0.30648649 0.08676790 0.0020761246
## 5 0.05360825 0.0008695652 0.0008748906 0.06716929 0.01035197 0.0008904720
## 6 0.24205915 0.0014005602 0.0013783598 0.26851852 0.16338798 0.0000000000
## 154 155 156 157 158 159
## 1 0.0010946907 0.02797586 0.10181618 0.06125828 0.21904238 0.05846995
## 2 0.0065146580 0.05667575 0.51501912 0.15726027 0.04478427 0.12906725
## 3 0.0000000000 0.05347299 0.06791828 0.09126106 0.25981205 0.08812261
## 4 0.0027159153 0.33768352 0.05875952 0.13183807 0.17066521 0.14270270
## 5 0.0006868132 0.15923130 0.01588398 0.04350829 0.14740484 0.01917808
## 6 0.0021893815 0.15780822 0.09220637 0.22754821 0.19560440 0.17929155
## 160 161 162 163 164 165 166
## 1 0.05918142 0.07498632 0.12064195 0.12603648 0.11046512 0.004986150 0.14662273
## 2 0.06085526 0.17237629 0.03738318 0.18082192 0.12262774 0.115448048 0.19662309
## 3 0.08097615 0.10824176 0.13277778 0.12548584 0.15905744 0.011653718 0.08930540
## 4 0.10843373 0.18305269 0.10082192 0.09534247 0.18247126 0.038419319 0.02178649
## 5 0.01107266 0.04049417 0.19426974 0.06903766 0.06294964 0.008344924 0.03448276
## 6 0.13237728 0.42113910 0.07403315 0.12355212 0.25000000 0.080332410 0.02028509
## 167 168 169 170 171 172 173
## 1 0.05564738 0.02200825 0.18092105 0.026315789 0.10172318 0.25983380 0.3131868
## 2 0.08201203 0.01777170 0.05122616 0.150163221 0.02699725 0.14199229 0.1542234
## 3 0.09161148 0.03603604 0.23360882 0.044603524 0.11643454 0.18186874 0.4040794
## 4 0.12227074 0.03670972 0.27015251 0.041847826 0.08525853 0.16273973 0.1773667
## 5 0.04347826 0.04707934 0.16024759 0.007570544 0.10145935 0.08692629 0.1473829
## 6 0.11648352 0.03980782 0.19506849 0.054216867 0.08849558 0.10747922 0.3448276
## 174 175 176 177 178 179 180
## 1 0.3104972 0.3237251 0.3107811 0.25676422 0.0000000000 0.04933481 0.0049822064
## 2 0.1521620 0.1897408 0.1541826 0.13885840 0.0103768433 0.15705656 0.0226148410
## 3 0.4008859 0.4306407 0.4051962 0.18091010 0.0000000000 0.08055556 0.0064423765
## 4 0.1753425 0.2125551 0.1755725 0.16411379 0.0000000000 0.26688633 0.0035236082
## 5 0.1475296 0.1780726 0.1467128 0.08605135 0.0006954103 0.09444444 0.0008849558
## 6 0.3426997 0.3835920 0.3424959 0.10730088 0.0022002200 0.38197899 0.0113074205
## 181 182 183 184 185 186
## 1 0.004989309 0.08619749 0.001101322 0.16648230 0.006644518 0.10148108
## 2 0.021861777 0.12669193 0.026717557 0.21377802 0.018701870 0.20620239
## 3 0.006419401 0.10940919 0.001653804 0.21654636 0.007234279 0.07260726
## 4 0.003538570 0.37466161 0.009264305 0.08333333 0.005491488 0.09283388
## 5 0.000890472 0.09009629 0.008990318 0.08310249 0.008379888 0.02481048
## 6 0.011387900 0.28050109 0.010416667 0.09387079 0.015529673 0.18206670
## 187 188 189 190 191 192
## 1 0.16022099 0.27068493 0.0110011001 0.0035014006 0.05722222 0.23021978
## 2 0.21252059 0.30978261 0.0626361656 0.0152777778 0.19094923 0.06331878
## 3 0.21043865 0.23102310 0.0126931567 0.0014035088 0.09035137 0.25950413
## 4 0.08621636 0.08482871 0.0185185185 0.0006954103 0.13593836 0.23949809
## 5 0.08019526 0.05075446 0.0006944444 0.0017746229 0.02456140 0.22683264
## 6 0.09639889 0.13964951 0.0368131868 0.0020905923 0.15363283 0.14631463
## 193 194 195 196 197 198
## 1 0.04412576 0.0005503577 0.0007122507 0.004137931 0.10546659 0.04042082
## 2 0.20922570 0.0262151830 0.0099431818 0.013004791 0.10214168 0.19636964
## 3 0.07337410 0.0000000000 0.0042857143 0.003453039 0.11536328 0.06013363
## 4 0.12280702 0.0010911075 0.0035511364 0.002051984 0.03721949 0.07311710
## 5 0.01728907 0.0027624309 0.0017969452 0.000000000 0.01936376 0.01749475
## 6 0.18277195 0.0038419319 0.0071174377 0.003462604 0.03579295 0.10132890
## 199 200 201 202 203 204
## 1 0.0000000000 0.27027027 0.04294479 0.03869541 0.10468320 0.002765487
## 2 0.0274423710 0.23835616 0.17127072 0.19264544 0.09622745 0.025767544
## 3 0.0005537099 0.18469218 0.07170868 0.06048835 0.11491713 0.003880266
## 4 0.0087912088 0.11226725 0.04573003 0.07506849 0.03551913 0.004934211
## 5 0.0013917884 0.04158004 0.01127555 0.01806810 0.02004147 0.001383126
## 6 0.0099667774 0.05521811 0.07384786 0.10181618 0.03630363 0.006077348
## 205 206 207 208 209 210
## 1 0.009167842 0.001106195 0.022639426 0.021349862 0.0011086475 0.022299652
## 2 0.032212885 0.031301483 0.150219298 0.133561644 0.0391400221 0.132963989
## 3 0.009200283 0.002219756 0.038248337 0.040027605 0.0044518642 0.041316527
## 4 0.006289308 0.006578947 0.031815688 0.035446489 0.0077007701 0.036576950
## 5 0.001782531 0.001385042 0.008397481 0.006861063 0.0006968641 0.007819288
## 6 0.007027407 0.007747648 0.055893747 0.042465753 0.0288408209 0.042243767
## 211 212 213 214 215 216
## 1 0.002185792 0.02477974 0.001649258 0.013796909 0.02371760 0.10219780
## 2 0.008676790 0.21693989 0.041462084 0.057080132 0.13910186 0.13576881
## 3 0.001097695 0.04583103 0.004405286 0.017165006 0.02434975 0.15168229
## 4 0.001083424 0.07267760 0.005449591 0.013136289 0.04814004 0.11044614
## 5 0.001373626 0.01591696 0.000691085 0.007649513 0.01455301 0.05878285
## 6 0.003275109 0.11538462 0.013721186 0.019326339 0.08094714 0.14481624
## 217 218 219 220 221 222
## 1 0.10910088 0.26477146 0.04733076 0.01890990 0.19481236 0.006626173
## 2 0.18603382 0.06246545 0.07084469 0.03644395 0.06359649 0.038398245
## 3 0.07446222 0.22539150 0.06070640 0.02518187 0.20854132 0.008291874
## 4 0.01525886 0.15359116 0.09095861 0.02971932 0.25604396 0.009299781
## 5 0.02352941 0.18784916 0.02432245 0.01742160 0.10895212 0.002775850
## 6 0.01753425 0.16685206 0.19506849 0.02885683 0.17090708 0.020396913
## 223 224 225 226 227 228
## 1 0.104860732 0.01936912 0.09031199 0.015503876 0.019251925 0.06073992
## 2 0.009761388 0.03778751 0.14657980 0.094453597 0.166303162 0.25781679
## 3 0.125342091 0.02547065 0.14027397 0.022765130 0.034197463 0.05260244
## 4 0.106775068 0.02962150 0.33260988 0.019178082 0.069830878 0.07334428
## 5 0.093214531 0.01804303 0.06400551 0.004857738 0.006920415 0.01111883
## 6 0.055100927 0.02817680 0.37465866 0.043670536 0.091758242 0.08714837
## 229 230 231 232 233 234
## 1 0.008462623 0.008191126 0.03410341 0.17660044 0.008752735 0.015925316
## 2 0.064898814 0.063642519 0.21967213 0.21905805 0.153720804 0.126430518
## 3 0.012039660 0.011651816 0.05905077 0.18060942 0.017014270 0.034710744
## 4 0.007692308 0.007457627 0.10457516 0.16475096 0.036916395 0.048501362
## 5 0.004416961 0.003415884 0.01663202 0.07310345 0.004118051 0.004830918
## 6 0.014184397 0.013698630 0.17233809 0.15168229 0.079278294 0.070175439
## 235 236 237 238 239 240
## 1 0.015960374 0.000000000 0.000000000 0.002768549 0.02147577 0.001660210
## 2 0.124590164 0.011588275 0.012547736 0.033498078 0.03825137 0.022002200
## 3 0.033738938 0.000000000 0.000000000 0.002779322 0.02263943 0.002782415
## 4 0.049726776 0.000000000 0.000000000 0.002196595 0.02894593 0.001098901
## 5 0.005513439 0.000000000 0.000000000 0.000000000 0.01111883 0.000000000
## 6 0.069268829 0.002080444 0.003844042 0.004424779 0.02701213 0.003880266
## 241 242 243 244 245 246
## 1 0.001094092 0.05628415 0.03923706 0.0027797081 0.0021246459 0.0005534034
## 2 0.021668472 0.13572204 0.15097403 0.0400829302 0.0435087719 0.0126304228
## 3 0.002192982 0.09170785 0.06783370 0.0006963788 0.0042643923 0.0011129661
## 4 0.001624256 0.23915401 0.16657653 0.0006901311 0.0014025245 0.0000000000
## 5 0.000000000 0.05833905 0.07103825 0.0017406440 0.0008873114 0.0000000000
## 6 0.003818876 0.22088573 0.25231608 0.0048409405 0.0049504950 0.0033185841
## 247 248 249 250 251 252
## 1 0.002202643 0.14073667 0.035764376 0.10779436 0.03053859 0.10099889
## 2 0.037663755 0.17448201 0.079166667 0.24175824 0.23925028 0.24544953
## 3 0.003863135 0.13245033 0.019732206 0.06607440 0.03899721 0.06033520
## 4 0.005455537 0.06052345 0.003477051 0.02079912 0.03960396 0.01986755
## 5 0.004143646 0.03438790 0.006156552 0.01941748 0.01602787 0.01884159
## 6 0.008781559 0.05662452 0.008421053 0.01826231 0.06260388 0.01942286
## 253 254 255 256 257 258
## 1 0.10301370 0.0000000000 0.001658375 0.11105003 0.000000000 0.000000000
## 2 0.32680805 0.0070575461 0.009335530 0.08365227 0.023446020 0.009790210
## 3 0.09851403 0.0000000000 0.004441977 0.08282717 0.001104362 0.000000000
## 4 0.06029332 0.0005408329 0.022490400 0.01800327 0.001088732 0.000000000
## 5 0.04539202 0.0000000000 0.017409471 0.06698895 0.000000000 0.000000000
## 6 0.09523810 0.0038188762 0.035359116 0.04235424 0.003852504 0.001409443
## 259 260 261 262 263 264
## 1 0.0000000000 0.06239737 0.017089305 0.004975124 0.0005476451 0.004778157
## 2 0.0173852573 0.38131450 0.097374179 0.051663128 0.0114379085 0.053097345
## 3 0.0007007708 0.06479956 0.025470653 0.006437768 0.0005497526 0.006237006
## 4 0.0013802622 0.07646421 0.019715225 0.004915730 0.0000000000 0.004768392
## 5 0.0008779631 0.01379310 0.003472222 0.003530450 0.0000000000 0.002577320
## 6 0.0048814505 0.13442623 0.027041943 0.006369427 0.0016411379 0.006194081
## 265 266 267 268 269 270
## 1 0.009418283 0.0000000000 0.016022099 0.002192982 0.0000000000 0.000000000
## 2 0.073117097 0.0051282051 0.034482759 0.010899183 0.0103542234 0.009027778
## 3 0.011135857 0.0000000000 0.017699115 0.003305785 0.0000000000 0.000000000
## 4 0.008771930 0.0007256894 0.029572837 0.004354927 0.0000000000 0.000000000
## 5 0.002079002 0.0000000000 0.009059233 0.002068966 0.0006906077 0.000000000
## 6 0.022689541 0.0000000000 0.013215859 0.003289474 0.0005491488 0.001397624
## 271 272 273 274 275 276
## 1 0.000000000 0.0000000000 0.004955947 0.016958425 0.018201875 0.015401540
## 2 0.027487631 0.0063113604 0.062807209 0.069451980 0.072757112 0.072637903
## 3 0.001111111 0.0000000000 0.006084071 0.019769357 0.020453289 0.018221977
## 4 0.001649258 0.0007012623 0.010371179 0.018398268 0.018013100 0.018022938
## 5 0.000000000 0.0000000000 0.000000000 0.001371742 0.001384083 0.001381215
## 6 0.006073992 0.0000000000 0.024766098 0.026775956 0.028650138 0.026359143
## 277 278 279 280 281 282
## 1 0.06336088 0.2450331 0.010359116 0.000000000 0.03905391 0.04030922
## 2 0.15846995 0.1226725 0.146291209 0.024309392 0.02024070 0.02910489
## 3 0.09000552 0.3351801 0.016689847 0.001677852 0.03818484 0.04618809
## 4 0.04967249 0.3015873 0.020632737 0.002208724 0.04590164 0.04555434
## 5 0.00899654 0.1498959 0.006065858 0.001397624 0.01513067 0.03808864
## 6 0.09295930 0.2447514 0.052047189 0.006128134 0.05109890 0.02211166
## 283 284 285 286 287 288
## 1 0.002803083 0.14892443 0.009309967 0.009470752 0.12758997 0.15038419
## 2 0.007660167 0.12122874 0.129700272 0.123545706 0.20216802 0.12145969
## 3 0.002814919 0.14325375 0.012665198 0.012906846 0.09364732 0.14419373
## 4 0.005578801 0.07291667 0.013601741 0.014364641 0.15384615 0.07232191
## 5 0.001777778 0.05412908 0.003445899 0.004225352 0.06538197 0.05513439
## 6 0.002820874 0.13598673 0.024630542 0.023463687 0.12342982 0.13964951
## 289 290 291 292 293 294
## 1 0.003350084 0.025330396 0.19330038 0.004975124 0.004961021 0.0005479452
## 2 0.046987286 0.081554461 0.37022901 0.018052516 0.035211268 0.0059815117
## 3 0.008958567 0.024916944 0.16648291 0.006080708 0.007790368 0.0000000000
## 4 0.015486726 0.014721919 0.14790647 0.013143483 0.002105263 0.0000000000
## 5 0.002096436 0.008316008 0.07044199 0.017170330 0.002702703 0.0000000000
## 6 0.040066778 0.023089610 0.22453450 0.003863135 0.013446568 0.0021941854
## 295 296 297 298 299 300
## 1 0.009749304 0.04490690 0.048219178 0.000000000 0.026431718 0.027487631
## 2 0.057851240 0.06800871 0.071273123 0.010380623 0.134825328 0.151829601
## 3 0.015277778 0.06633772 0.067582418 0.000000000 0.044751381 0.028114664
## 4 0.042349727 0.06626833 0.067464635 0.000000000 0.074114441 0.027838428
## 5 0.017543860 0.01029513 0.008940853 0.000000000 0.008298755 0.002762431
## 6 0.045769764 0.08214677 0.087527352 0.001395673 0.134287287 0.045529347
## 301 302 303 304 305 306
## 1 0.0005503577 0.001654716 0.13656633 0.001426534 0.0007042254 0.08442982
## 2 0.0273373428 0.019726027 0.22923588 0.067844523 0.0048780488 0.06948969
## 3 0.0005534034 0.002214839 0.12723214 0.002867384 0.0000000000 0.06308283
## 4 0.0016384489 0.002186987 0.04803976 0.011971831 0.0000000000 0.19370591
## 5 0.0006925208 0.001389854 0.04225352 0.000907441 0.0000000000 0.20055325
## 6 0.0044004400 0.004412576 0.05116796 0.031960227 0.0007027407 0.06612022
## 307 308 309 310 311 312
## 1 0.10089841 0.09185919 0.06449835 0.030990592 0.0006915629 0.07688083
## 2 0.06133701 0.03393541 0.07506849 0.115384615 0.0041180508 0.10262009
## 3 0.08136300 0.09075816 0.07466814 0.023849140 0.0000000000 0.11810155
## 4 0.03693570 0.12663755 0.10278841 0.010433828 0.0000000000 0.17312944
## 5 0.09353147 0.23464458 0.03800968 0.004888268 0.0000000000 0.03181189
## 6 0.03974895 0.06652007 0.09135938 0.018805310 0.0006906077 0.27921009
## 313 314 315 316 317 318
## 1 0.0005509642 0.01758242 0.000000000 0.05320900 0.0011019284 0.002099370
## 2 0.0186199343 0.01363884 0.051602390 0.13195202 0.0141921397 0.028651293
## 3 0.0016648169 0.01817181 0.002741228 0.06219042 0.0005537099 0.004228330
## 4 0.0038314176 0.02446982 0.006504065 0.16521265 0.0021822149 0.002090592
## 5 0.0013917884 0.01381215 0.002060440 0.08977901 0.0027720028 0.000000000
## 6 0.0060472787 0.02414929 0.015821058 0.17708333 0.0016510732 0.001407460
## 319 320 321 322 323 324
## 1 0.0000000000 0.0055401662 0.006306938 0.004166667 0.19158361 0.002195390
## 2 0.0114566285 0.0591065292 0.034770515 0.009688581 0.17863014 0.048939641
## 3 0.0000000000 0.0020935101 0.008462623 0.006333568 0.13007226 0.002748763
## 4 0.0000000000 0.0034270048 0.011111111 0.002754821 0.08671789 0.007061380
## 5 0.0006901311 0.0008912656 0.007894737 0.016769638 0.05175983 0.000000000
## 6 0.0021941854 0.0048510049 0.022456140 0.003491620 0.09839690 0.023026316
## 325 326 327 328 329 330
## 1 0.004427227 0.006040637 0.0007087172 0.01276360 0.0021306818 0.07411504
## 2 0.024671053 0.038251366 0.0084925690 0.01431718 0.0259467041 0.04505495
## 3 0.008859358 0.007174393 0.0000000000 0.02056698 0.0014214641 0.07508343
## 4 0.005491488 0.007646095 0.0000000000 0.03076923 0.0007047216 0.03898957
## 5 0.004163775 0.000000000 0.0000000000 0.02996516 0.0000000000 0.04735376
## 6 0.021546961 0.023678414 0.0000000000 0.02931416 0.0007062147 0.04143646
## 331 332 333 334 335 336
## 1 0.06043956 0.0006993007 0.06446991 0.01645639 0.001651982 0.010005559
## 2 0.20787746 0.0055363322 0.04397163 0.02388708 0.012575178 0.033039648
## 3 0.09905921 0.0000000000 0.07703281 0.02417582 0.002760906 0.012256267
## 4 0.14394766 0.0000000000 0.06478873 0.01468189 0.002733734 0.001651982
## 5 0.02837370 0.0000000000 0.01805054 0.01310345 0.002084781 0.002087683
## 6 0.25453546 0.0000000000 0.13544669 0.01863014 0.004402862 0.005549390
## 337 338 339 340 341 342
## 1 0.0000000000 0.0000000000 0.006037322 0.08821918 0.0005534034 0.005530973
## 2 0.0306513410 0.0049207217 0.013646288 0.26579521 0.0274725275 0.037828947
## 3 0.0016592920 0.0000000000 0.007182320 0.07903403 0.0011098779 0.007194245
## 4 0.0060076461 0.0005461496 0.004899292 0.13010343 0.0044004400 0.008214677
## 5 0.0006901311 0.0048543689 0.002762431 0.12179931 0.0000000000 0.000000000
## 6 0.0088057237 0.0000000000 0.021965953 0.13676149 0.0077348066 0.016547159
## 343 344 345 346 347 348
## 1 0.08631116 0.007671233 0.08808864 0.05601318 0.005534034 0.000000000
## 2 0.17859093 0.061956522 0.17381738 0.22568306 0.041163557 0.006254343
## 3 0.05530973 0.008791209 0.05669817 0.03316750 0.006090808 0.000000000
## 4 0.03875546 0.009244154 0.04061471 0.03167668 0.010399562 0.000000000
## 5 0.03051318 0.002053388 0.02865129 0.01382170 0.002779708 0.000000000
## 6 0.05769231 0.024056862 0.05260244 0.03848268 0.010479868 0.000000000
## 349 350 351 352 353 354
## 1 0.0022014309 0.0000000000 0.06784335 0.06666667 0.04333516 0.000000000
## 2 0.0212302667 0.0136314068 0.04545455 0.04435926 0.03318825 0.009078212
## 3 0.0033094319 0.0000000000 0.05801105 0.05752212 0.05118327 0.000000000
## 4 0.0016348774 0.0016348774 0.08821918 0.08921730 0.08269859 0.002099370
## 5 0.0006920415 0.0006930007 0.09632710 0.09424809 0.08137931 0.000000000
## 6 0.0087623220 0.0022002200 0.04237755 0.04015402 0.04059243 0.004912281
## 355 356 357 358 359 360
## 1 0.03164908 0.0000000000 0.013311148 0.008250825 0.12186978 0.004942339
## 2 0.22589532 0.0075034106 0.059602649 0.004928806 0.16905286 0.032170120
## 3 0.03171953 0.0006925208 0.012820513 0.014933628 0.16415410 0.005509642
## 4 0.02965404 0.0006825939 0.007135016 0.033351558 0.05622933 0.007069059
## 5 0.00489853 0.0017406440 0.003486750 0.032571033 0.02029391 0.000000000
## 6 0.02990033 0.0000000000 0.010520487 0.033627343 0.09555556 0.017004937
## 361 362 363 364 365 366
## 1 0.0007137759 0.012609649 0.12698413 0.000000000 0.000000000 0.0013736264
## 2 0.0211565585 0.059142702 0.16720955 0.002828854 0.004098361 0.0047619048
## 3 0.0014316392 0.012081274 0.17161716 0.000000000 0.000000000 0.0006891799
## 4 0.0035260931 0.005971770 0.05757740 0.001405481 0.001369863 0.0000000000
## 5 0.0000000000 0.003455425 0.02136458 0.000000000 0.000000000 0.0008591065
## 6 0.0035511364 0.009304871 0.10000000 0.004264392 0.004101162 0.0027322404
## 367 368 369 370 371 372
## 1 0.0000000000 0.02279044 0.0006958942 0.010462555 0.02353585 0.003318584
## 2 0.0160502442 0.16354626 0.0075862069 0.040305011 0.16702938 0.080636314
## 3 0.0000000000 0.04296875 0.0006954103 0.020948181 0.04116356 0.007198228
## 4 0.0006958942 0.06985699 0.0006915629 0.022913257 0.07127312 0.005473454
## 5 0.0000000000 0.01603905 0.0000000000 0.005521049 0.01655172 0.002760524
## 6 0.0006978367 0.12901440 0.0020804438 0.031868132 0.13107591 0.025995575
## 373 374 375 376 377 378
## 1 0.0007183908 0.014771049 0.001100110 0.003852504 0.06622517 0.0060373216
## 2 0.0271234832 0.072058824 0.013631407 0.022002200 0.07447974 0.0588876772
## 3 0.0000000000 0.017804154 0.001100715 0.002777778 0.09579181 0.0088251517
## 4 0.0014255167 0.017467249 0.000000000 0.004945055 0.03440743 0.0098092643
## 5 0.0018281536 0.009157509 0.000000000 0.000000000 0.01176471 0.0006863418
## 6 0.0057678443 0.072485207 0.001643836 0.011608624 0.07607497 0.0252192982
## 379 380 381 382 383 384
## 1 0.022062879 0.015367728 0.02141680 0.003835616 0.0032894737 0.009330406
## 2 0.059178082 0.112017401 0.31202186 0.020663404 0.0201305767 0.069285325
## 3 0.034387133 0.026388125 0.04083885 0.002199010 0.0027457441 0.017070485
## 4 0.052660450 0.045676998 0.06601200 0.004340749 0.0027218291 0.013609145
## 5 0.006958942 0.004837595 0.01453287 0.000000000 0.0006887052 0.002074689
## 6 0.114301491 0.089413055 0.10720176 0.010946907 0.0131578947 0.025219298
## 385 386 387 388 389 390
## 1 0.021134594 0.06070640 0.019758507 0.14215418 0.034140969 0.0007163324
## 2 0.037486218 0.09857612 0.036085293 0.11997828 0.112876712 0.0128205128
## 3 0.024567281 0.08365651 0.023796348 0.14717188 0.011080332 0.0007220217
## 4 0.009911894 0.07217059 0.008738394 0.12236058 0.004934211 0.0014255167
## 5 0.007697691 0.02574809 0.006237006 0.03991741 0.012491325 0.0000000000
## 6 0.007210205 0.13939394 0.007142857 0.11857923 0.003854626 0.0021505376
## 391 392 393 394 395 396
## 1 0.0027593819 0.03400987 0.0005497526 0.09579181 0.11667584 0.0005494505
## 2 0.0148026316 0.32969432 0.0114754098 0.16473829 0.13121925 0.0076169750
## 3 0.0011092623 0.05905077 0.0005509642 0.06281267 0.12894300 0.0000000000
## 4 0.0021869874 0.08015267 0.0010934937 0.09340659 0.04860732 0.0005425936
## 5 0.0006958942 0.01998622 0.0000000000 0.06438069 0.01731302 0.0006915629
## 6 0.0093560815 0.13626374 0.0027457441 0.07534626 0.03962576 0.0000000000
## 397 398 399 400 401 402
## 1 0.04381586 0.25054705 0.01774820 0.019294377 0.02681992 0.001095890
## 2 0.06662996 0.05603917 0.17614578 0.178336980 0.03259098 0.038105607
## 3 0.04454343 0.26707048 0.03279600 0.033795014 0.02469813 0.001651982
## 4 0.04617922 0.15778020 0.03474903 0.033935413 0.02440347 0.002718869
## 5 0.01535241 0.16815989 0.00983837 0.009762901 0.03642612 0.004137931
## 6 0.08582503 0.19178082 0.06032097 0.057692308 0.02292576 0.005476451
## 403 404 405 406 407 408
## 1 0.001099505 0.011001100 0.011660189 0.03357182 0.007662835 0.01371366
## 2 0.039301310 0.010922993 0.012686156 0.02295082 0.083695652 0.06273868
## 3 0.001656543 0.011576626 0.012263099 0.03421634 0.020285088 0.02316602
## 4 0.002727769 0.003818876 0.003302146 0.01965066 0.039566396 0.03816794
## 5 0.004158004 0.003465003 0.003484321 0.01726519 0.006177076 0.00621118
## 6 0.006043956 0.016456391 0.019444444 0.01704233 0.122270742 0.06040637
## 409 410 411 412 413 414
## 1 0.0011013216 0.03252481 0.10641799 0.001651982 0.0011055832 0.03359031
## 2 0.0131219245 0.02297593 0.07516340 0.015855659 0.0049342105 0.01643836
## 3 0.0005534034 0.03377630 0.05646930 0.001106195 0.0005555556 0.05709534
## 4 0.0010922993 0.01858939 0.03262643 0.001092896 0.0005485464 0.01368363
## 5 0.0000000000 0.01803051 0.02060440 0.002060440 0.0000000000 0.02912621
## 6 0.0021978022 0.01872247 0.04545455 0.002202643 0.0000000000 0.06346578
## 415 416 417 418 419 420
## 1 0.07757405 0.0011123471 0.000000000 0.05007364 0.027731559 0.013888889
## 2 0.05617978 0.0501377410 0.026129559 0.02023121 0.064462810 0.088754135
## 3 0.11412268 0.0033444816 0.001103144 0.06537890 0.041248606 0.022777778
## 4 0.18627451 0.0049586777 0.001089325 0.06578947 0.062122045 0.045079714
## 5 0.05882353 0.0006939625 0.000000000 0.06148282 0.007660167 0.007724719
## 6 0.10608204 0.0188574598 0.004378763 0.05726872 0.076454294 0.079312257
## 421 422 423 424 425 426
## 1 0.01433297 0.018201875 0.009265859 0.0000000000 0.11368653 0.002115656
## 2 0.01586433 0.009315068 0.046511628 0.0219546742 0.29589041 0.004184100
## 3 0.01495845 0.019922524 0.015647226 0.0000000000 0.11129568 0.002127660
## 4 0.02735230 0.030021834 0.009838370 0.0000000000 0.14887794 0.002799160
## 5 0.01940402 0.015883978 0.003606853 0.0009033424 0.03203343 0.003527337
## 6 0.02913689 0.047487576 0.017793594 0.0028368794 0.32341598 0.002115656
## 427 428 429 430 431 432
## 1 0.0000000000 0.005997819 0.03503010 0.025041736 0.029041096 0.009444444
## 2 0.0083275503 0.013008130 0.03367735 0.028145695 0.083832335 0.014892443
## 3 0.0000000000 0.011513158 0.05202629 0.032348020 0.047461369 0.011111111
## 4 0.0006939625 0.050270270 0.04991861 0.024202420 0.075163399 0.017630854
## 5 0.0000000000 0.031550069 0.01579670 0.004901961 0.008971705 0.012526096
## 6 0.0000000000 0.060490463 0.07717570 0.067590028 0.157346491 0.014404432
## 433 434 435 436 437 438 439
## 1 0.02949360 0.09080842 0.016456391 0.03028634 0.23629386 0.03042035 0.14482379
## 2 0.04517906 0.12046205 0.098637602 0.01146288 0.13071895 0.01206802 0.08702791
## 3 0.03285078 0.09888889 0.015986770 0.04470199 0.15719801 0.04500000 0.15329275
## 4 0.03742433 0.21569704 0.012002182 0.03266195 0.14029364 0.03347969 0.11220580
## 5 0.03140265 0.10724234 0.006224066 0.01521438 0.11249137 0.01456311 0.04293629
## 6 0.03664631 0.16335541 0.019220209 0.05057724 0.07232877 0.05088496 0.13304013
## 440 441 442 443 444 445 446
## 1 0.0007097232 0.01973684 0.04331140 0.04607789 0.1973612 0.01211454 0.09719934
## 2 0.0056377731 0.02664492 0.07792916 0.02011963 0.2146368 0.08246860 0.04959128
## 3 0.0007127584 0.03638368 0.04065934 0.05998899 0.2421401 0.01382743 0.12017641
## 4 0.0000000000 0.04836957 0.03588907 0.05223069 0.1509537 0.03061782 0.07123437
## 5 0.0000000000 0.01925722 0.01309442 0.01028807 0.1988912 0.01251739 0.02617080
## 6 0.0000000000 0.05701754 0.04107338 0.08008777 0.1441144 0.02866593 0.08891328
## 447 448 449 450 451 452
## 1 0.002197802 0.002203857 0.0011025358 0.06068890 0.08004447 0.16933260
## 2 0.044189853 0.032240437 0.0460021906 0.20585049 0.20176893 0.10557987
## 3 0.004947774 0.001106807 0.0038631347 0.09688013 0.11682504 0.18580144
## 4 0.001086957 0.001634877 0.0021822149 0.15455531 0.15995587 0.17923497
## 5 0.001381215 0.002077562 0.0006882312 0.02129121 0.02097902 0.05963939
## 6 0.008228195 0.007150715 0.0104338276 0.20699072 0.20944444 0.20518478
## 453 454 455 456 457 458
## 1 0.08550805 0.09100938 0.05797902 0.08774834 0.010462555 0.076700434
## 2 0.17880795 0.01640241 0.16111415 0.34153005 0.032240437 0.121167883
## 3 0.07688022 0.09457965 0.06806862 0.09442297 0.009392265 0.046989721
## 4 0.03028634 0.11796832 0.07435757 0.13537118 0.008174387 0.018950437
## 5 0.01885475 0.08695652 0.01517241 0.02491349 0.003469813 0.007155635
## 6 0.05869324 0.09725275 0.15456546 0.27367841 0.021475771 0.023494860
## 459 460 461 462 463 464 465
## 1 0.05406912 0.04835165 0.05228398 0.06152993 0.06274292 0.02264685 0.16971714
## 2 0.11683278 0.17775354 0.18712493 0.05646930 0.05469613 0.10764873 0.16061606
## 3 0.09080717 0.05068871 0.05176211 0.07966574 0.07928532 0.02771855 0.12208657
## 4 0.19258439 0.08174387 0.08346972 0.06432106 0.06508549 0.01199718 0.03577325
## 5 0.02096436 0.01245675 0.01315789 0.02580195 0.02642559 0.01861702 0.09223301
## 6 0.22456921 0.13322368 0.13681319 0.04099723 0.03997779 0.01703336 0.04875346
## 466 467 468 469 470 471
## 1 0.14225352 0.009810792 0.016939891 0.000000000 0.010928962 0.04600219
## 2 0.09476662 0.044692737 0.124321390 0.005455537 0.008156607 0.03747963
## 3 0.08477011 0.014820042 0.013706140 0.000000000 0.011557512 0.05390539
## 4 0.05782793 0.023693380 0.012472885 0.000000000 0.006507592 0.08147746
## 5 0.11858407 0.015831135 0.001368925 0.000000000 0.003441156 0.03586207
## 6 0.02127660 0.013258897 0.020174482 0.002194185 0.010917031 0.04814004
## 472 473 474 475 476 477
## 1 0.02726767 0.003521127 0.06681391 0.06135987 0.06806862 0.12061404
## 2 0.02267699 0.041143654 0.11196487 0.11386139 0.09290819 0.17292007
## 3 0.02901786 0.004249292 0.08033241 0.07675195 0.10450250 0.13461538
## 4 0.03528115 0.011813760 0.06692265 0.06747120 0.09868421 0.09880565
## 5 0.01530967 0.007067138 0.05080028 0.04940849 0.02789400 0.06073154
## 6 0.04997224 0.011299435 0.04743519 0.04746137 0.13440265 0.08419902
## 478 479 480 481 482 483
## 1 0.0005506608 0.002189381 0.0060874377 0.04352557 0.0055401662 0.04182719
## 2 0.0114628821 0.078804348 0.0406593407 0.09832523 0.0417582418 0.16912166
## 3 0.0011049724 0.006589786 0.0072182121 0.03551913 0.0061179088 0.07123136
## 4 0.0032768979 0.009766685 0.0049342105 0.07235421 0.0060273973 0.09661572
## 5 0.0020703934 0.002057613 0.0006973501 0.05324232 0.0006915629 0.03603604
## 6 0.0038546256 0.024056862 0.0165471594 0.05764002 0.0182623132 0.21763085
## 484 485 486 487 488 489
## 1 0.047356828 0.007700770 0.007667032 0.24220886 0.023849140 0.03152655
## 2 0.106301370 0.074822501 0.073449402 0.11624117 0.009879254 0.12768299
## 3 0.048780488 0.011037528 0.010989011 0.17527473 0.037222222 0.04679666
## 4 0.067213115 0.009264305 0.009787928 0.13882863 0.086121777 0.15769231
## 5 0.009009009 0.002063274 0.002749141 0.16655196 0.030598053 0.07885555
## 6 0.104510451 0.023026316 0.023001095 0.05076419 0.060354374 0.19778393
## 490 491 492 493 494 495
## 1 0.07142857 0.13236930 0.09762824 0.015546918 0.16785911 0.004393191
## 2 0.04907306 0.06696182 0.21886999 0.046754675 0.09781421 0.084921067
## 3 0.10546659 0.07988827 0.14127424 0.017847183 0.13591160 0.007713499
## 4 0.11364872 0.06408840 0.10690789 0.018701870 0.05834242 0.017934783
## 5 0.02841303 0.08887334 0.04085873 0.002808989 0.03531856 0.000000000
## 6 0.11848601 0.01505017 0.16914601 0.021690768 0.05772402 0.057597367
## 496 497 498 499 500 501
## 1 0.04168952 0.018743109 0.20785597 0.06881486 0.005008347 0.21052632
## 2 0.07801418 0.006014215 0.20379404 0.08184282 0.011049724 0.20184984
## 3 0.06828194 0.026548673 0.22070099 0.10733844 0.002794857 0.22167217
## 4 0.08650707 0.017429194 0.08761493 0.16729832 0.004415011 0.08655416
## 5 0.01100413 0.053287197 0.12732278 0.05906593 0.004198740 0.13148789
## 6 0.10312671 0.011544805 0.06702997 0.16366612 0.005011136 0.06743421
## 502 503 504 505 506 507
## 1 0.21290678 0.016877637 0.18056322 0.09779006 0.06791828 0.005485464
## 2 0.21061269 0.110955056 0.09489852 0.05908096 0.06140351 0.043169399
## 3 0.22228381 0.025280899 0.10654828 0.14206437 0.07202216 0.008291874
## 4 0.09672131 0.039270687 0.13081554 0.15927750 0.08911974 0.015812432
## 5 0.12724758 0.005145798 0.31824513 0.13888889 0.05890506 0.013831259
## 6 0.06902264 0.043723554 0.11147903 0.12231405 0.05408389 0.015925316
## 508 509 510 511 512 513 514
## 1 0.025683060 0.05338470 0.10825588 0.07832322 0.10779436 0.09723757 0.1155973
## 2 0.111895709 0.08493151 0.01957586 0.06492089 0.02028509 0.18739726 0.1165476
## 3 0.024175824 0.07154742 0.13743815 0.11829740 0.13518006 0.13033833 0.1338889
## 4 0.032030402 0.07381083 0.16576381 0.30475150 0.16538883 0.15287671 0.2306425
## 5 0.006206897 0.01178918 0.05802048 0.12240664 0.05894591 0.04375000 0.1375000
## 6 0.061235648 0.10743802 0.06400438 0.21518987 0.06460519 0.21334068 0.1705426
## 515 516 517 518 519 520 521
## 1 0.3469163 0.01758242 0.07635830 0.06439185 0.096385542 0.25730028 0.10445300
## 2 0.1143952 0.08605664 0.06588580 0.08246860 0.036512262 0.11700383 0.07255865
## 3 0.4615810 0.04452996 0.03524229 0.07913669 0.052805281 0.17331118 0.13701657
## 4 0.1538041 0.10391730 0.01010101 0.04428650 0.035345296 0.15191257 0.07415485
## 5 0.1352288 0.01304049 0.03903346 0.04155125 0.057201930 0.27727589 0.01581843
## 6 0.2075991 0.22958904 0.01313869 0.03357182 0.008767123 0.08250825 0.12678375
## 522 523 524 525 526 527
## 1 0.26533997 0.13281682 0.14745011 0.034710744 0.21551247 0.004424779
## 2 0.08292147 0.22447859 0.30286344 0.088476242 0.16538037 0.031250000
## 3 0.30044346 0.07995558 0.21702838 0.017689331 0.23119777 0.009983361
## 4 0.10818232 0.16355653 0.11251372 0.006550218 0.09785596 0.022453450
## 5 0.04100069 0.09556787 0.03417015 0.010423905 0.13626834 0.006949270
## 6 0.10755654 0.19779006 0.13732004 0.003298516 0.11184939 0.063395810
## 528 529 530 531 532 533 534
## 1 0.04059243 0.09571508 0.19268559 0.12306843 0.19337017 0.12349067 0.12969095
## 2 0.03322440 0.10526316 0.19272530 0.16201423 0.19241341 0.06492089 0.04331140
## 3 0.05347299 0.12073784 0.27442371 0.17865044 0.27404113 0.10451045 0.09761509
## 4 0.14106754 0.04362231 0.23685637 0.12924425 0.23642348 0.16032609 0.10546448
## 5 0.02142364 0.02112676 0.07211538 0.02011096 0.07256393 0.24311295 0.27750865
## 6 0.10159253 0.03234802 0.27103825 0.21247241 0.27031509 0.04340659 0.04465270
## 535 536 537 538 539 540 541
## 1 0.1934066 0.09655938 0.10148924 0.26433643 0.09215579 0.12705367 0.2086093
## 2 0.2716320 0.13618891 0.11391019 0.12438892 0.16675749 0.11364872 0.1089814
## 3 0.1250000 0.11735261 0.11055832 0.24835526 0.12403528 0.16675839 0.2846877
## 4 0.1772083 0.14623419 0.16803503 0.16983180 0.19356598 0.08138904 0.1742217
## 5 0.1472011 0.03275261 0.07847222 0.06168609 0.03314917 0.02064694 0.1089655
## 6 0.1453744 0.17044826 0.16639118 0.14246725 0.27796053 0.12869660 0.2353914
## 542 543 544 545 546 547 548
## 1 0.06828194 0.034387133 0.15881709 0.12896175 0.07802198 0.06094183 0.16022099
## 2 0.13832695 0.095604396 0.07947741 0.17362995 0.07084469 0.04028698 0.13713659
## 3 0.10840708 0.036687048 0.09350935 0.07785088 0.08112583 0.06291759 0.22535991
## 4 0.15783725 0.042400881 0.07967480 0.04234528 0.04362050 0.03138767 0.11542670
## 5 0.02008310 0.004854369 0.23747426 0.04545455 0.09889350 0.01608392 0.05153203
## 6 0.22912088 0.047645429 0.05194095 0.08074195 0.09429825 0.02933038 0.16105902
## 549 550 551 552 553 554 555
## 1 0.20883978 0.08467072 0.13189845 0.08497807 0.03302146 0.03336980 0.06122449
## 2 0.16639299 0.15835616 0.15270936 0.15640327 0.04438356 0.04298150 0.13432018
## 3 0.25665188 0.09706045 0.11911357 0.09515952 0.03530061 0.03581267 0.09706045
## 4 0.16757941 0.03073546 0.09682713 0.03313417 0.07603939 0.07721588 0.14833060
## 5 0.06592644 0.01109570 0.06471816 0.01032347 0.01178101 0.01103448 0.03254848
## 6 0.25234160 0.06456954 0.08875413 0.06739726 0.07331863 0.07119387 0.17760618
## 556 557 558 559 560 561
## 1 0.13666301 0.001103144 0.010958904 0.009413068 0.000000000 0.02260198
## 2 0.02885139 0.002186987 0.153094463 0.144426139 0.016129032 0.01259584
## 3 0.14553473 0.003320421 0.005500550 0.017777778 0.000000000 0.02323009
## 4 0.11050626 0.013668671 0.009777295 0.030769231 0.004871260 0.03549973
## 5 0.19419489 0.011797363 0.002074689 0.002785515 0.004436557 0.02698962
## 6 0.04000000 0.007162534 0.020218579 0.071349558 0.007741027 0.02420242
## 562 563 564 565 566 567
## 1 0.02302632 0.06961326 0.04883463 0.008438819 0.05937328 0.05983380
## 2 0.01251360 0.18859649 0.09922822 0.020920502 0.46946565 0.13671444
## 3 0.02365237 0.11308204 0.06295265 0.013352073 0.09487038 0.06733445
## 4 0.03422053 0.28070175 0.19262521 0.014644351 0.12050164 0.08640616
## 5 0.02611684 0.06963788 0.14245810 0.015971606 0.01786942 0.04033380
## 6 0.02185792 0.35891773 0.10171365 0.015482055 0.20977485 0.13130194
## 568 569 570 571 572 573
## 1 0.0000000000 0.000000000 0.000000000 0.022866704 0.02835333 0.02907296
## 2 0.0028169014 0.003556188 0.002074689 0.121951220 0.06063887 0.02625821
## 3 0.0000000000 0.000000000 0.000000000 0.023529412 0.03831418 0.03484513
## 4 0.0000000000 0.000000000 0.000000000 0.037078030 0.06327745 0.13810044
## 5 0.0008952551 0.001831502 0.001748252 0.008450704 0.04030055 0.14809689
## 6 0.0000000000 0.000000000 0.000000000 0.097601785 0.06372549 0.05953693
## 574 575 576 577 578 579
## 1 0.000000000 0.001665741 0.012893983 0.001655629 0.04305177 0.2113687
## 2 0.017032967 0.058370044 0.004291845 0.059016393 0.09745533 0.1767926
## 3 0.000000000 0.002788622 0.010115607 0.003867403 0.05361050 0.1851441
## 4 0.001096491 0.003854626 0.002877698 0.003284072 0.05729730 0.3959474
## 5 0.002084781 0.000000000 0.014466546 0.000000000 0.01647220 0.2147838
## 6 0.002765487 0.016666667 0.031791908 0.018191841 0.07403375 0.2022039
## 580 581 582 583 584 585 586
## 1 0.11700383 0.31692478 0.003453039 0.17634052 0.17475193 0.02802198 0.08213892
## 2 0.15062534 0.19505495 0.006830601 0.14591333 0.14340449 0.07037643 0.04273973
## 3 0.13439386 0.21705426 0.005547850 0.22364040 0.22277501 0.03752759 0.11993337
## 4 0.15518177 0.09170785 0.001369863 0.09215579 0.09006550 0.08283379 0.16081229
## 5 0.03701165 0.16342142 0.006092254 0.04996530 0.05085911 0.07438017 0.13948647
## 6 0.21631982 0.15422886 0.002747253 0.14917127 0.14615385 0.09125893 0.13031474
## 587 588 589 590 591 592 593
## 1 0.08283050 0.01125968 0.02200220 0.14243759 0.005847953 0.17098731 0.19616438
## 2 0.04359673 0.03786816 0.04426230 0.03953148 0.046309696 0.07662835 0.07181719
## 3 0.12092766 0.01480959 0.03143960 0.09523810 0.008759124 0.15099558 0.25852585
## 4 0.16412214 0.01256983 0.01853871 0.08575581 0.010189229 0.30870279 0.43641304
## 5 0.14226232 0.02380952 0.04689655 0.15135135 0.007259528 0.21591696 0.29484536
## 6 0.13285949 0.01060071 0.01640241 0.04258443 0.014534884 0.13876652 0.18719212
## 594 595 596 597 598 599 600
## 1 0.00139470 0.12770683 0.13734142 0.12788145 0.001104972 0.14165733 0.07260726
## 2 0.02437326 0.03748622 0.16511755 0.19112815 0.009868421 0.20900500 0.04369197
## 3 0.00280112 0.16266070 0.17679558 0.08618785 0.001662971 0.18605958 0.07628524
## 4 0.05609418 0.15104741 0.05622271 0.10060142 0.003287671 0.06892718 0.08633880
## 5 0.03475936 0.23467967 0.10934256 0.02271163 0.001387925 0.06755806 0.03223594
## 6 0.05567154 0.14674819 0.13084112 0.19790518 0.004969630 0.10067114 0.13454146
## 601 602 603 604 605 606
## 1 0.004422333 0.004402862 0.003536068 0.002139800 0.01709873 0.03397260
## 2 0.016429354 0.018579235 0.020322355 0.024630542 0.04153005 0.03099511
## 3 0.004986150 0.004966887 0.004255319 0.003594536 0.02651934 0.02528862
## 4 0.010393873 0.008738394 0.002118644 0.009915014 0.01588171 0.01794454
## 5 0.006963788 0.006232687 0.001771479 0.008130081 0.04350829 0.03443526
## 6 0.006073992 0.006060606 0.001416431 0.014967926 0.02253986 0.02949208
## 607 608 609 610 611 612 613
## 1 0.02043070 0.3076075 0.012147985 0.01822198 0.13249038 0.12910284 0.13060109
## 2 0.04717499 0.1329322 0.028540066 0.04448105 0.16266376 0.06800871 0.05914270
## 3 0.02722222 0.4034311 0.017738359 0.02608213 0.18557269 0.11641955 0.18605928
## 4 0.09983544 0.2133479 0.007667032 0.01646542 0.09036473 0.07425474 0.09095831
## 5 0.07409972 0.1398892 0.026261230 0.04166667 0.10082873 0.17125172 0.13674033
## 6 0.06253459 0.2665198 0.011043622 0.02266446 0.11287671 0.04597701 0.15258856
## 614 615 616 617 618 619
## 1 0.06744868 0.003486750 0.0000000000 0.12714207 0.29911210 0.000000000
## 2 0.03918723 0.006211180 0.0083391244 0.06732348 0.12383049 0.023691460
## 3 0.12740741 0.006284916 0.0000000000 0.11387507 0.39052925 0.001114206
## 4 0.12973761 0.023464458 0.0020979021 0.07510965 0.22979659 0.006057269
## 5 0.10439560 0.015859031 0.0008795075 0.17210271 0.08768267 0.001396648
## 6 0.11773256 0.013937282 0.0014064698 0.04527885 0.27232390 0.013311148
## 620 621 622 623 624 625
## 1 0.29817780 0.03467254 0.14058210 0.10142544 0.0000000000 0.09015487
## 2 0.12239297 0.15700219 0.19553377 0.27010870 0.0310515173 0.05090312
## 3 0.39000000 0.06032097 0.08324146 0.05503577 0.0007142857 0.13108407
## 4 0.22782037 0.16295492 0.03159041 0.02340773 0.0049435028 0.09912377
## 5 0.08541667 0.04507628 0.03308063 0.02542955 0.0000000000 0.12874043
## 6 0.27477974 0.21067107 0.02138158 0.01421542 0.0063965885 0.13373693
## 626 627 628 629 630 631 632
## 1 0.007756233 0.3598234 0.04809287 0.3658129 0.2803279 0.1457424 0.1456628
## 2 0.147252747 0.1650219 0.16199890 0.1669431 0.1665762 0.1176790 0.1173913
## 3 0.010005559 0.4625624 0.05546312 0.4642058 0.3684789 0.2064800 0.2064800
## 4 0.015393073 0.1823658 0.06140351 0.1851852 0.1649485 0.0915493 0.0899729
## 5 0.000000000 0.1245651 0.01724138 0.1250000 0.0997249 0.1431521 0.1411846
## 6 0.039800995 0.2253444 0.09005525 0.2314969 0.2224044 0.1533006 0.1509847
## 633 634 635 636 637 638
## 1 0.0014114326 0.14082504 0.001643836 0.10295176 0.11412268 0.07502738
## 2 0.0112123336 0.05524862 0.005443658 0.16142857 0.06002825 0.21576087
## 3 0.0007087172 0.08707865 0.001100110 0.12889211 0.11453114 0.11606161
## 4 0.0014035088 0.04861111 0.001629549 0.04500000 0.09097370 0.17525773
## 5 0.0008936550 0.11209964 0.003460208 0.03870387 0.16636364 0.02758621
## 6 0.0028188865 0.01131542 0.001096491 0.05319914 0.08428571 0.24303659
## 639 640 641 642 643 644
## 1 0.030269675 0.079889807 0.06142778 0.005537099 0.001644737 0.0006954103
## 2 0.097320940 0.002186987 0.07291667 0.001097695 0.004357298 0.0145328720
## 3 0.037569061 0.070835639 0.09539656 0.008869180 0.002754821 0.0006978367
## 4 0.030584380 0.022416621 0.06849315 0.023013699 0.005440696 0.0013793103
## 5 0.007654836 0.120689655 0.05062413 0.053287197 0.004135079 0.0000000000
## 6 0.057237204 0.042284459 0.11900826 0.014892443 0.000000000 0.0034843206
## 645 646 647 648 649 650
## 1 0.0011154490 0.10607735 0.03514552 0.004434590 0.01596916 0.004928806
## 2 0.0127635960 0.04994512 0.07213115 0.041758242 0.04800873 0.039695487
## 3 0.0005602241 0.06873614 0.05791506 0.007230256 0.02649007 0.007154651
## 4 0.0000000000 0.04004388 0.02340773 0.014867841 0.02292576 0.014099783
## 5 0.0020964361 0.06999307 0.01522491 0.000000000 0.03170227 0.000000000
## 6 0.0005567929 0.08314978 0.06479956 0.017718715 0.01483516 0.016967707
## 651 652 653 654 655 656
## 1 0.03362734 0.002733734 0.002822865 0.10703364 0.003302146 0.000000000
## 2 0.07283680 0.062940857 0.008391608 0.33834586 0.062910284 0.004861111
## 3 0.05598670 0.007131103 0.004264392 0.11297710 0.008830022 0.000000000
## 4 0.02408320 0.018448182 0.009142053 0.14692654 0.019639935 0.002074689
## 5 0.01525659 0.014393420 0.004484305 0.01727447 0.015193370 0.006156552
## 6 0.06435644 0.023484435 0.011371713 0.34298780 0.023652365 0.000000000
## 657 658 659 660 661 662
## 1 0.10337203 0.31002203 0.11661166 0.0005503577 0.0005509642 0.0497512438
## 2 0.09350935 0.29672131 0.08069793 0.0043525571 0.0032804811 0.0005485464
## 3 0.10931400 0.40364440 0.16390728 0.0011025358 0.0011049724 0.0537396122
## 4 0.02860286 0.11395856 0.12609170 0.0027247956 0.0016393443 0.0340472268
## 5 0.04027778 0.11211073 0.02700831 0.0041465100 0.0041782730 0.0917303683
## 6 0.03149171 0.09845985 0.16446645 0.0032912781 0.0027487631 0.0370165746
## 663 664 665 666 667 668
## 1 0.0006954103 0.04267425 0.0000000000 0.000000000 0.002103787 0.002120141
## 2 0.0013850416 0.13875263 0.0090027701 0.007812500 0.001383126 0.001397624
## 3 0.0020876827 0.05003524 0.0000000000 0.000000000 0.002783577 0.002826855
## 4 0.0055172414 0.04140351 0.0000000000 0.002123142 0.001382170 0.002093510
## 5 0.0114335972 0.01257862 0.0000000000 0.001803427 0.002643172 0.002652520
## 6 0.0132588974 0.08879493 0.0006920415 0.006373938 0.001387925 0.001407460
## 669 670 671 672 673 674
## 1 0.000000000 0.0013947001 0.000000000 0.01267218 0.001637555 0.017495899
## 2 0.006882312 0.0048644892 0.002851033 0.03778751 0.005414185 0.023913043
## 3 0.000000000 0.0007022472 0.000000000 0.01216142 0.001096491 0.020925110
## 4 0.002056203 0.0041436464 0.000000000 0.01256144 0.007571660 0.036896365
## 5 0.000877193 0.0017574692 0.002757353 0.01249133 0.003429355 0.005486968
## 6 0.006232687 0.0006988120 0.000000000 0.01762115 0.007072905 0.060142154
## 675 676 677 678 679 680
## 1 0.017717931 0.2360350 0.004961411 0.10110497 0.06419480 0.0021941854
## 2 0.061181435 0.1995662 0.059080963 0.15030170 0.13260274 0.0217627856
## 3 0.029223093 0.2635914 0.011627907 0.11784325 0.07491676 0.0016501650
## 4 0.014154282 0.1544318 0.022938285 0.15131579 0.03778751 0.0021750952
## 5 0.009883199 0.1238816 0.011764706 0.06675939 0.03254848 0.0006872852
## 6 0.050569801 0.1601968 0.031335899 0.24724062 0.07237569 0.0109589041
## 681 682 683 684 685 686
## 1 0.004427227 0.0005518764 0.001435750 0.001646542 0.12575508 0.0005488474
## 2 0.058888277 0.0142309797 0.012048193 0.021810251 0.09145346 0.0027218291
## 3 0.012235818 0.0011055832 0.002153625 0.004955947 0.13270925 0.0005506608
## 4 0.021953897 0.0038188762 0.005693950 0.015786609 0.06474429 0.0000000000
## 5 0.011871508 0.0048375950 0.006294964 0.010366275 0.08096886 0.0000000000
## 6 0.032614704 0.0011013216 0.012134190 0.018671060 0.10526316 0.0005488474
## 687 688 689 690 691 692
## 1 0.020487265 0.09160724 0.007032349 0.01552106 0.10088203 0.001651073
## 2 0.004398021 0.04243743 0.011929825 0.03579295 0.27945205 0.008757526
## 3 0.020011117 0.12665198 0.009852217 0.01838440 0.08019912 0.003316750
## 4 0.010976948 0.13104948 0.043871866 0.01815182 0.06666667 0.005470460
## 5 0.009769714 0.04258242 0.075704225 0.01180556 0.04152249 0.006906077
## 6 0.009961262 0.18705431 0.042657343 0.01107420 0.09641873 0.008796042
## 693 694 695 696 697 698
## 1 0.01603982 0.004945055 0.12807882 0.002087683 0.08226793 0.05353201
## 2 0.03956044 0.027777778 0.02060738 0.002076125 0.02208724 0.09836066
## 3 0.01890990 0.006615215 0.11916529 0.004210526 0.11135857 0.07982262
## 4 0.01923077 0.012520414 0.06890939 0.018055556 0.15198238 0.20894708
## 5 0.01115760 0.004824259 0.19021365 0.027336861 0.05505226 0.09065744
## 6 0.01165372 0.017534247 0.03335156 0.010445682 0.15235457 0.25785124
## 699 700 701 702 703 704
## 1 0.07700831 0.08195820 0.07845304 0.011647255 0.039106145 0.039133473
## 2 0.02583837 0.03224044 0.02519168 0.041276830 0.006206897 0.006237006
## 3 0.10789766 0.11319713 0.10537992 0.021702838 0.036933798 0.037245257
## 4 0.14692982 0.19564033 0.14857456 0.062671798 0.004824259 0.004874652
## 5 0.04586518 0.11703601 0.04586518 0.006289308 0.052038161 0.052447552
## 6 0.16519337 0.19243421 0.16162727 0.085920177 0.009735744 0.009762901
## 705 706 707 708 709 710
## 1 0.03109656 0.002785515 0.0000000000 0.0048644892 0.04759270 0.0056377731
## 2 0.04446855 0.008298755 0.0070372977 0.0006839945 0.10043908 0.0007032349
## 3 0.02300110 0.002803083 0.0000000000 0.0069881202 0.07615342 0.0085106383
## 4 0.06442880 0.001386963 0.0000000000 0.0103448276 0.13406593 0.0104384134
## 5 0.07305307 0.005258545 0.0017761989 0.0295652174 0.05237430 0.0291777188
## 6 0.05891980 0.001399580 0.0007072136 0.0166551006 0.19545958 0.0168657765
## 711 712 713 714 715 716
## 1 0.0000000000 0.02036324 0.09801762 0.01660210 0.001103144 0.0007037298
## 2 0.0057061341 0.03768433 0.22727273 0.06787083 0.014778325 0.0097629010
## 3 0.0000000000 0.02596685 0.05530973 0.01552967 0.001662971 0.0007077141
## 4 0.0000000000 0.03440743 0.13395298 0.04221491 0.001093494 0.0000000000
## 5 0.0018018018 0.02835408 0.08996540 0.04450626 0.001390821 0.0026619343
## 6 0.0007158196 0.08425110 0.17142857 0.09607951 0.002206288 0.0007072136
## 717 718 719 720 721 722
## 1 0.001095290 0.002205072 0.002177463 0.046703297 0.003346347 0.023140496
## 2 0.003259098 0.003833516 0.003243243 0.007629428 0.004424779 0.004373975
## 3 0.002195390 0.001662050 0.001640241 0.049532196 0.002239642 0.034254144
## 4 0.028184282 0.003823048 0.003241491 0.073529412 0.004980631 0.035499727
## 5 0.028236915 0.003467406 0.003431709 0.130103806 0.004213483 0.059187887
## 6 0.018062397 0.006063947 0.005440696 0.056438356 0.012827663 0.023102310
## 723 724 725 726 727 728
## 1 0.09186497 0.05788313 0.11889597 0.19845645 0.002849003 0.012107870
## 2 0.05396476 0.04806117 0.04574243 0.09434997 0.064039409 0.075027382
## 3 0.07405345 0.09850581 0.12847966 0.12334071 0.002117149 0.023255814
## 4 0.07472527 0.10049153 0.07087719 0.05695509 0.004234298 0.024017467
## 5 0.17618384 0.05882353 0.19874101 0.10562891 0.003555556 0.007591442
## 6 0.05703212 0.15659341 0.11016949 0.03799559 0.010653409 0.024215740
## 729 730 731 732 733 734
## 1 0.16876712 0.014262205 0.001430615 0.002765487 0.16971714 0.08094714
## 2 0.29885683 0.075625680 0.019067797 0.012074643 0.23584387 0.11323851
## 3 0.13901099 0.023691460 0.001429593 0.003882418 0.15200445 0.12908587
## 4 0.02610114 0.025027203 0.005661713 0.009857612 0.11544805 0.08192245
## 5 0.04679972 0.006177076 0.002705140 0.009027778 0.09179416 0.03678001
## 6 0.03665208 0.024657534 0.007112376 0.013812155 0.11221669 0.12892562
## 735 736 737 738 739 740
## 1 0.0038631347 0.004986150 0.08000000 0.005580357 0.05283648 0.0000000000
## 2 0.0388615216 0.051648352 0.11406844 0.069137168 0.01542700 0.0041350793
## 3 0.0033185841 0.005002779 0.12959912 0.007262570 0.06867672 0.0000000000
## 4 0.0059945504 0.008223684 0.07871878 0.018743109 0.03577325 0.0006868132
## 5 0.0006930007 0.003467406 0.03490760 0.004895105 0.11406578 0.0034965035
## 6 0.0104683196 0.017718715 0.12957900 0.042292710 0.04386452 0.0000000000
## 741 742 743 744 745 746
## 1 0.056986301 0.055741360 0.0000000000 0.031649084 0.000000000 0.0000000000
## 2 0.146579805 0.150776053 0.0112123336 0.094818082 0.052891396 0.0116999312
## 3 0.067656766 0.064913262 0.0007137759 0.060133630 0.000000000 0.0006944444
## 4 0.076045627 0.076923077 0.0028109628 0.091859186 0.005617978 0.0027453672
## 5 0.008965517 0.009097271 0.0017857143 0.008356546 0.000000000 0.0017391304
## 6 0.197382770 0.196002221 0.0105857445 0.238888889 0.010653409 0.0103734440
## 747 748 749 750 751 752
## 1 0.001654716 0.009350935 0.07538803 0.006915629 0.000000000 0.03818484
## 2 0.018052516 0.020240700 0.08131868 0.029593944 0.005505850 0.12403952
## 3 0.003871681 0.009356081 0.03833333 0.009749304 0.000000000 0.05604883
## 4 0.002735230 0.019629226 0.07688083 0.005505850 0.000000000 0.07021393
## 5 0.000000000 0.009015257 0.10160056 0.022827041 0.000000000 0.02090592
## 6 0.006607930 0.030803080 0.06921373 0.006219765 0.001389854 0.08453039
## 753 754 755 756 757 758
## 1 0.000000000 0.061157025 0.062534588 0.0000000000 0.0038652678 0.001100715
## 2 0.006206897 0.223741794 0.221365639 0.0027382256 0.0136761488 0.000000000
## 3 0.000000000 0.066334992 0.065108514 0.0005534034 0.0044370494 0.002219756
## 4 0.001374570 0.073224044 0.073666850 0.0021893815 0.0038335159 0.012048193
## 5 0.005263158 0.008356546 0.008350731 0.0000000000 0.0006954103 0.020804438
## 6 0.002777778 0.212672176 0.210643016 0.0011007155 0.0077177508 0.004961411
## 759 760 761 762 763 764
## 1 0.08769994 0.04848485 0.000000000 0.003561254 0.01490066 0.07289928
## 2 0.05464481 0.11111111 0.007633588 0.055084746 0.01423877 0.02537231
## 3 0.13171002 0.08093126 0.001101928 0.006423983 0.01438053 0.06844741
## 4 0.12862616 0.25465498 0.006525285 0.008426966 0.01420765 0.02585259
## 5 0.03331020 0.10242215 0.009595613 0.006156552 0.02018093 0.10793872
## 6 0.13781698 0.31201764 0.004939627 0.020743920 0.02206288 0.04550499
## 765 766 767 768 769 770
## 1 0.01098298 0.11787281 0.0000000000 0.000000000 0.01817181 0.07836645
## 2 0.02673213 0.05872757 0.0028129395 0.009836066 0.03659203 0.01638449
## 3 0.01321586 0.06343078 0.0000000000 0.000000000 0.02980132 0.09961262
## 4 0.01856909 0.08600980 0.0000000000 0.000000000 0.03167668 0.04759300
## 5 0.01718213 0.15934066 0.0008865248 0.002073255 0.06241331 0.14683368
## 6 0.02744237 0.07835616 0.0014114326 0.001101928 0.03247111 0.11019284
## 771 772 773 774 775 776
## 1 0.01603096 0.17279210 0.0005530973 0.04751381 0.01522491 0.13325930
## 2 0.02199010 0.07411444 0.0357142857 0.14945055 0.03583735 0.16105902
## 3 0.02720711 0.10616062 0.0005561735 0.06107718 0.02902557 0.09090909
## 4 0.02689352 0.09847661 0.0043931906 0.10928062 0.01441318 0.05019305
## 5 0.04643105 0.09592823 0.0000000000 0.09847434 0.01139351 0.14644351
## 6 0.03373894 0.03671233 0.0182320442 0.10668878 0.05521049 0.09317804
## 777 778 779 780 781 782
## 1 0.13098514 0.13241758 0.063736264 0.004219409 0.04860732 0.004410143
## 2 0.16129032 0.16294278 0.008178844 0.030769231 0.01029810 0.054674686
## 3 0.08854455 0.09040794 0.065600882 0.004261364 0.06794521 0.007190265
## 4 0.05455537 0.05558583 0.018478261 0.004189944 0.03796095 0.010952903
## 5 0.14196676 0.14432990 0.097510373 0.007985803 0.16438356 0.007654836
## 6 0.09796368 0.09873834 0.014262205 0.010578279 0.04741144 0.022038567
## 783 784 785 786 787 788
## 1 0.04704097 0.008771930 0.10824176 0.003858875 0.06298343 0.02091359
## 2 0.01047904 0.000000000 0.10722101 0.111899563 0.10928062 0.01311475
## 3 0.05135952 0.010491441 0.09618574 0.007747648 0.07095344 0.02868174
## 4 0.01355422 0.008719346 0.03442623 0.018599562 0.07447974 0.15111839
## 5 0.04247104 0.037215713 0.07501721 0.002764340 0.07033426 0.10261708
## 6 0.03625378 0.009868421 0.05619835 0.048431480 0.06522941 0.09845985
## 789 790 791 792 793 794
## 1 0.0007168459 0.02072909 0.0022050717 0.028176796 0.1018826 0.06326304
## 2 0.0042613636 0.07097681 0.0241360395 0.107025247 0.1482094 0.12686156
## 3 0.0021582734 0.01783167 0.0005543237 0.059312639 0.1444444 0.08862876
## 4 0.0007087172 0.01198872 0.0021941854 0.080043860 0.1463146 0.07920792
## 5 0.0035971223 0.01342883 0.0006963788 0.006915629 0.0312500 0.01947149
## 6 0.0014388489 0.02048023 0.0038546256 0.227071823 0.1653761 0.11123409
## 795 796 797 798 799 800
## 1 0.01377410 0.004969630 0.009381898 0.002732240 0.0000000000 0.003293085
## 2 0.02189381 0.033406353 0.026301370 0.000000000 0.0034891835 0.088427948
## 3 0.01935841 0.005543237 0.007734807 0.004934211 0.0000000000 0.008810573
## 4 0.01367615 0.005467469 0.005476451 0.017876490 0.0006934813 0.014705882
## 5 0.01722950 0.002775850 0.015256588 0.017869416 0.0026525199 0.004140787
## 6 0.01100715 0.006607930 0.011037528 0.008178844 0.0000000000 0.034596376
## 801 802 803 804 805 806
## 1 0.07692308 0.008893830 0.002750275 0.0007117438 0.007131103 0.0007067138
## 2 0.15228147 0.037465565 0.059530311 0.0084685956 0.061202186 0.0112280702
## 3 0.09489456 0.010608599 0.002768549 0.0000000000 0.008834898 0.0014224751
## 4 0.16153846 0.008264463 0.003816794 0.0007052186 0.016902944 0.0006993007
## 5 0.06827586 0.005582694 0.001386963 0.0000000000 0.015916955 0.0017574692
## 6 0.24958495 0.022172949 0.012672176 0.0021337127 0.031885651 0.0021246459
## 807 808 809 810 811 812
## 1 0.07982504 0.07984581 0.008228195 0.013698630 0.0000000000 0.01046832
## 2 0.27539381 0.27595628 0.083242655 0.147362697 0.0104384134 0.03065134
## 3 0.04843148 0.04759270 0.014325069 0.018121911 0.0007007708 0.01272828
## 4 0.05863192 0.06010929 0.019586507 0.041259501 0.0000000000 0.03823048
## 5 0.04287690 0.04203997 0.004126547 0.004810997 0.0000000000 0.02708333
## 6 0.05147864 0.05225523 0.049890351 0.061269147 0.0000000000 0.04958678
## 813 814 815 816 817 818
## 1 0.012141280 0.003307607 0.000551572 0.08902239 0.02145215 0.11079545
## 2 0.017534247 0.024563319 0.009863014 0.06666667 0.08788210 0.06993007
## 3 0.009413068 0.003869541 0.001106195 0.12719298 0.03585218 0.09000000
## 4 0.016420361 0.010958904 0.001091703 0.09761388 0.02617230 0.08077994
## 5 0.022869023 0.002772003 0.000000000 0.01793103 0.01386963 0.10017575
## 6 0.013774105 0.011043622 0.001102536 0.12404372 0.05659341 0.07323944
## 819 820 821 822 823 824
## 1 0.027609056 0.005506608 0.005443658 0.002747253 0.001644737 0.1473799
## 2 0.073585942 0.044238121 0.043266631 0.004354927 0.020697168 0.2283080
## 3 0.014884234 0.003869541 0.003280481 0.005512679 0.004402862 0.2149123
## 4 0.007119387 0.006014215 0.005939525 0.033242507 0.008156607 0.3859079
## 5 0.017361111 0.000000000 0.000000000 0.033149171 0.004149378 0.1537406
## 6 0.011049724 0.009895547 0.010881393 0.015925316 0.004388371 0.4044760
## 825 826 827 828 829 830 831
## 1 0.04367054 0.10806363 0.021253406 0.04598338 0.12219178 0.14238593 0.08251366
## 2 0.11587040 0.03050109 0.004867496 0.12004405 0.04622077 0.37335526 0.23902439
## 3 0.04656319 0.06043956 0.022988506 0.04621381 0.06875688 0.13976705 0.12904997
## 4 0.04057018 0.03320631 0.039978390 0.04002193 0.03590860 0.19441402 0.22414727
## 5 0.02437326 0.09441764 0.065753425 0.02226862 0.13764625 0.05736006 0.03650138
## 6 0.07446222 0.02079912 0.041870582 0.07403315 0.04157549 0.41997792 0.30588877
## 832 833 834 835 836 837
## 1 0.03673246 0.004407713 0.03782895 0.007057163 0.0000000000 0.062465753
## 2 0.07294502 0.052430366 0.07403375 0.006997901 0.0014245014 0.182906913
## 3 0.04515419 0.008844666 0.04405286 0.010615711 0.0007194245 0.066225166
## 4 0.12384574 0.032258065 0.12363834 0.032100488 0.0021428571 0.091304348
## 5 0.12834592 0.010395010 0.12552011 0.032461677 0.0018132366 0.008990318
## 6 0.12041598 0.035773253 0.11890411 0.023206751 0.0000000000 0.162280702
## 838 839 840 841 842 843
## 1 0.002857143 0.002816901 0.002735978 0.020262870 0.002840909 0.003551136
## 2 0.009866103 0.011879804 0.012244898 0.093275488 0.012676056 0.007037298
## 3 0.004273504 0.003526093 0.004112406 0.015934066 0.003548616 0.004264392
## 4 0.002120141 0.002094972 0.002035278 0.016260163 0.002108222 0.001403509
## 5 0.002695418 0.002680965 0.003407155 0.008928571 0.002671416 0.008912656
## 6 0.003551136 0.003516174 0.003419973 0.016921397 0.003543586 0.003526093
## 844 845 846 847 848 849
## 1 0.13669065 0.003861004 0.05671806 0.0000000000 0.012735327 0.0000000000
## 2 0.08941306 0.016967707 0.20349154 0.0071389347 0.070136986 0.0196399345
## 3 0.13548029 0.004983389 0.06453392 0.0005546312 0.013865779 0.0005521811
## 4 0.06095552 0.006560962 0.09929078 0.0010964912 0.019715225 0.0065466448
## 5 0.22648084 0.000691085 0.01106501 0.0041695622 0.004861111 0.0089717046
## 6 0.08563536 0.007713499 0.12006579 0.0027670172 0.038695412 0.0076880835
## 850 851 852 853 854 855
## 1 0.0005549390 0.0005506608 0.04542936 0.000000000 0.0005546312 0.07039911
## 2 0.0099282956 0.0109409190 0.13908741 0.007625272 0.0220264317 0.14955850
## 3 0.0005561735 0.0011049724 0.07786429 0.000000000 0.0011142061 0.11328125
## 4 0.0005506608 0.0010922993 0.26673985 0.000000000 0.0016492578 0.27603306
## 5 0.0000000000 0.0006915629 0.06054280 0.000000000 0.0013956734 0.10385965
## 6 0.0038802661 0.0044052863 0.24806202 0.000000000 0.0033259424 0.32000000
## 856 857 858 859 860 861
## 1 0.04180418 0.002073255 0.001653804 0.12887168 0.08924612 0.02199413
## 2 0.22131148 0.005486968 0.009309967 0.20780649 0.12904997 0.02456647
## 3 0.04864566 0.004844291 0.001660210 0.11679644 0.13871866 0.02662722
## 4 0.05558583 0.004115226 0.004371585 0.07515085 0.11618943 0.09941520
## 5 0.02005533 0.006890612 0.005517241 0.04596100 0.02659202 0.04324324
## 6 0.09290819 0.008977901 0.002202643 0.08849558 0.14317425 0.07391304
## 862 863 864 865 866 867
## 1 0.08925620 0.007173601 0.35302594 0.11772853 0.06604293 0.003325942
## 2 0.02188184 0.007163324 0.11695906 0.02602740 0.11864407 0.020385675
## 3 0.05423354 0.010101010 0.27034884 0.10896552 0.09889503 0.007234279
## 4 0.06560962 0.035919540 0.19247467 0.03804348 0.31515812 0.018151815
## 5 0.23958333 0.010771993 0.28462998 0.18378378 0.13604972 0.016794962
## 6 0.01706109 0.038904899 0.06029412 0.03703704 0.32070291 0.020509978
## 868 869 870 871 872 873 874
## 1 0.13520971 0.02040816 0.07611693 0.1180364 0.07633169 0.008214677 0.11973756
## 2 0.11433260 0.02409639 0.13910186 0.2624385 0.14207650 0.022270505 0.11340206
## 3 0.15605977 0.03652463 0.09718388 0.1112341 0.09668508 0.010468320 0.16703297
## 4 0.11986864 0.08433735 0.05237316 0.2571116 0.05443658 0.001627781 0.07542051
## 5 0.02506964 0.03414634 0.02781641 0.2234559 0.02768166 0.004827586 0.14630780
## 6 0.10639471 0.07891832 0.07213656 0.2706725 0.07291667 0.014238773 0.11584699
## 875 876 877 878 879 880 881
## 1 0.10573178 0.27105410 0.22068584 0.11964874 0.06194690 0.02149945 0.10970232
## 2 0.27413127 0.08462389 0.04333516 0.20586638 0.03294893 0.11445783 0.12280702
## 3 0.15848214 0.18344519 0.23375902 0.09521189 0.09983361 0.02603878 0.13565891
## 4 0.26214128 0.05472637 0.11287671 0.05000000 0.14701042 0.03938731 0.04385965
## 5 0.06451613 0.14516129 0.19252078 0.04135079 0.28333333 0.01319444 0.11450382
## 6 0.37208009 0.08004447 0.10325787 0.05966065 0.06736610 0.07272727 0.11816676
## 882 883 884 885 886 887 888
## 1 0.00872886 0.015350877 0.3263889 0.001102536 0.1360132 0.1338843 0.001105583
## 2 0.12744035 0.099074578 0.1349862 0.006027397 0.1808743 0.1772429 0.016420361
## 3 0.01587302 0.026417171 0.2835616 0.001661130 0.1823204 0.1820188 0.002767017
## 4 0.03841991 0.054921153 0.2129121 0.002738226 0.1131766 0.1124454 0.003289474
## 5 0.01022495 0.009661836 0.2661996 0.005563282 0.1348548 0.1326883 0.002079002
## 6 0.07528642 0.111354909 0.1006993 0.001655629 0.1864686 0.1839207 0.004966887
## 889 890 891 892 893 894
## 1 0.14230343 0.003508772 0.001099505 0.02918502 0.10160133 0.05801105
## 2 0.05159166 0.007697691 0.003810561 0.01314348 0.02249040 0.11690450
## 3 0.19055556 0.004908836 0.002201431 0.04643449 0.14008859 0.08738938
## 4 0.11696870 0.007665505 0.005452563 0.08264915 0.05808219 0.27013699
## 5 0.04111498 0.007042254 0.006896552 0.10704420 0.13365651 0.14909847
## 6 0.19105467 0.009148487 0.004939627 0.05405405 0.10479868 0.26062949
## 895 896 897 898 899 900 901
## 1 0.04678041 0.18839779 0.001522070 0.18818985 0.18725762 0.09873834 0.03065134
## 2 0.14043716 0.27607025 0.005952381 0.09857612 0.28059537 0.21405229 0.28804348
## 3 0.06464088 0.19071310 0.001524390 0.11289430 0.18715084 0.04961411 0.05674931
## 4 0.12111293 0.23194748 0.017937220 0.05090312 0.23068433 0.02934783 0.10234077
## 5 0.10650069 0.05197505 0.007648184 0.10672211 0.05303559 0.05175983 0.02346446
## 6 0.25041186 0.26417171 0.004573171 0.07056229 0.26570317 0.02631579 0.19232877
## 902 903 904 905 906 907
## 1 0.06181619 0.019188596 0.005494505 0.009381898 0.23651452 0.02854007
## 2 0.02880435 0.025585193 0.003816794 0.044931507 0.08659218 0.18389554
## 3 0.09145674 0.029735683 0.011019284 0.010537992 0.17342657 0.04780220
## 4 0.06778742 0.026673925 0.021218716 0.056892779 0.15426997 0.09249184
## 5 0.11881868 0.008258775 0.036526533 0.073509015 0.22431507 0.02351314
## 6 0.07897603 0.063969382 0.019178082 0.044101433 0.07544582 0.08852459
## 908 909 910 911 912 913
## 1 0.05681191 0.05872667 0.02273988 0.088705234 0.06592798 0.008820287
## 2 0.16475096 0.16539301 0.03353491 0.005476451 0.03143960 0.030617824
## 3 0.08517699 0.08940397 0.03053859 0.094579646 0.08240535 0.012714207
## 4 0.22896175 0.22863364 0.04507971 0.056192035 0.02308961 0.014192140
## 5 0.09661836 0.09903047 0.11172797 0.221067221 0.10552061 0.007575758
## 6 0.33590308 0.33187295 0.04585635 0.041254125 0.04547976 0.026967529
## 914 915 916 917 918 919 920
## 1 0.05193370 0.1058374 0.04743519 0.11934156 0.09301046 0.1084932 0.12952799
## 2 0.20842451 0.1134021 0.17608338 0.05882353 0.19594964 0.1126224 0.08156607
## 3 0.04148230 0.1610077 0.04648589 0.09766162 0.14847645 0.1644664 0.10291690
## 4 0.16538883 0.3596966 0.05679956 0.13513514 0.30508475 0.3606735 0.16303162
## 5 0.05416667 0.1218344 0.01448276 0.28161889 0.10194175 0.1217331 0.19931034
## 6 0.27367841 0.3087834 0.08890116 0.11050477 0.43234323 0.3062227 0.08131868
## 921 922 923 924 925 926
## 1 0.001100110 0.08724462 0.007150715 0.018722467 0.06670342 0.007688083
## 2 0.034991799 0.24986271 0.143558952 0.246036085 0.03056769 0.002724796
## 3 0.001660210 0.12645591 0.006057269 0.027101770 0.05755396 0.009376724
## 4 0.001643836 0.20746432 0.023458811 0.030054645 0.02677596 0.017973856
## 5 0.000000000 0.05732044 0.004135079 0.007570544 0.09172414 0.030513176
## 6 0.007166483 0.34123484 0.064285714 0.065456546 0.01925193 0.014810752
## 927 928 929 930 931 932 933
## 1 0.06655574 0.06911684 0.06854616 0.06942149 0.11117226 0.11589404 0.04132231
## 2 0.05335534 0.05234460 0.05274725 0.05291871 0.17085153 0.17122538 0.20678337
## 3 0.09510567 0.09416300 0.09560867 0.09618574 0.15818584 0.16187845 0.03257869
## 4 0.05002749 0.05160239 0.04945055 0.05019094 0.26037118 0.26284153 0.14847162
## 5 0.08901252 0.08896552 0.08907446 0.08611111 0.08188758 0.08200139 0.03321799
## 6 0.10736027 0.10825588 0.10871965 0.11019284 0.37541164 0.37328202 0.22734761
## 934 935 936 937 938 939 940
## 1 0.13533001 0.02782324 0.08769994 0.08698031 0.18041804 0.14749587 0.18332413
## 2 0.20549451 0.07675676 0.01256831 0.01308615 0.05145047 0.05507088 0.05245902
## 3 0.11006115 0.04045927 0.10779436 0.10523416 0.22571744 0.19171271 0.22744881
## 4 0.14881933 0.03082747 0.23375205 0.23369565 0.14590164 0.22628135 0.14668856
## 5 0.05277778 0.05910653 0.45925414 0.45784784 0.22245322 0.29647547 0.22122053
## 6 0.11566132 0.04080522 0.11074380 0.11452055 0.18422504 0.16684962 0.18622590
## 941 942 943 944 945 946 947
## 1 0.090010977 0.09917808 0.06673961 0.2379121 0.21210469 0.037465565 0.06239647
## 2 0.002723312 0.04725693 0.05971770 0.1178396 0.05952381 0.006553796 0.05095890
## 3 0.086596801 0.14560440 0.09688013 0.3060773 0.20940405 0.036403751 0.09307479
## 4 0.038776625 0.09994568 0.23289902 0.4122471 0.19967532 0.058406114 0.06123565
## 5 0.169082126 0.20547945 0.09710744 0.3347251 0.34065934 0.138812155 0.01951220
## 6 0.035714286 0.14160744 0.15051998 0.2352617 0.14978214 0.073585942 0.09641873
## 948 949 950 951 952 953
## 1 0.05937328 0.007747648 0.03728436 0.008830022 0.14961832 0.17469880
## 2 0.13591703 0.091108672 0.07849641 0.092653509 0.21568627 0.02661597
## 3 0.09641873 0.008314856 0.04355109 0.009439200 0.11384615 0.17279210
## 4 0.19039825 0.016429354 0.05126792 0.016429354 0.31762918 0.09609121
## 5 0.04979253 0.013212796 0.01610644 0.013879251 0.09197652 0.24395301
## 6 0.25618472 0.028761062 0.10371603 0.027578599 0.25339367 0.13646288
## 954 955 956 957 958 959
## 1 0.08411732 0.1739371 0.08823529 0.067867036 0.010514665 0.002189381
## 2 0.24546952 0.2624385 0.22889610 0.005502063 0.161538462 0.027233115
## 3 0.11074012 0.1400000 0.07831325 0.087078652 0.008328706 0.001097695
## 4 0.13541667 0.2489035 0.20583468 0.167352538 0.020890599 0.003255562
## 5 0.02367688 0.1598616 0.08710562 0.352631579 0.006263048 0.006872852
## 6 0.21700718 0.2990603 0.20663404 0.094313454 0.069213732 0.011487965
## 960 961 962 963 964 965 966
## 1 0.04328767 0.12759644 0.06759003 0.05577029 0.014277869 0.15516289 0.06318083
## 2 0.19630234 0.12170088 0.14513469 0.14215418 0.026215183 0.11135491 0.10523475
## 3 0.03799559 0.09748892 0.10339077 0.03100775 0.028145695 0.15354767 0.10759148
## 4 0.15878195 0.12992701 0.21612726 0.10727969 0.020719738 0.08278509 0.35208221
## 5 0.03310345 0.33519553 0.06958942 0.15620642 0.008971705 0.09318498 0.11833105
## 6 0.20865279 0.18842730 0.36675900 0.11239669 0.061403509 0.09437086 0.26278564
## 967 968 969 970 971 972 973
## 1 0.08508287 0.05054945 0.2147172 0.03926991 0.05647841 0.10503040 0.05312673
## 2 0.22971491 0.17294053 0.1566921 0.15422613 0.12857143 0.06907895 0.08585581
## 3 0.08448371 0.04196576 0.2322900 0.04386452 0.08888889 0.10266371 0.06329817
## 4 0.20197044 0.15393013 0.2367991 0.02687877 0.14622125 0.07502738 0.18595721
## 5 0.06241331 0.02551724 0.1844460 0.02006920 0.06458333 0.07023644 0.26315789
## 6 0.33572216 0.18481848 0.2929901 0.04628099 0.28752759 0.08604523 0.13473219
## 974 975 976 977 978 979 980
## 1 0.06063947 0.04210526 0.11907387 0.10905125 0.2065142 0.09060773 0.002757860
## 2 0.08310552 0.04455446 0.19693654 0.19003790 0.1523679 0.11293860 0.004928806
## 3 0.09297178 0.07670928 0.07249585 0.07021393 0.2681564 0.06038781 0.003314917
## 4 0.33496999 0.28516484 0.06062261 0.06013001 0.1125691 0.02844639 0.002186987
## 5 0.11126469 0.10013908 0.03867403 0.03964457 0.1200000 0.07538036 0.001385042
## 6 0.22369146 0.24254144 0.04182719 0.03873432 0.1449275 0.01931567 0.000000000
## 981 982 983 984 985 986
## 1 0.003508772 0.05664488 0.001420455 0.0005467469 0.000000000 0.01952036
## 2 0.030598053 0.04805616 0.018284107 0.0152173913 0.011142061 0.04653740
## 3 0.001415428 0.07209175 0.001425517 0.0000000000 0.000000000 0.03419283
## 4 0.009790210 0.02321814 0.007007708 0.0016286645 0.000698812 0.02374379
## 5 0.007943513 0.01163587 0.002678571 0.0006877579 0.000890472 0.03792135
## 6 0.017556180 0.03534530 0.012091038 0.0027352298 0.001408451 0.02675585
## 987 988 989 990 991 992
## 1 0.10804851 0.010503040 0.0005500550 0.01147541 0.003331483 0.11056106
## 2 0.02791461 0.018141836 0.0152755046 0.16440586 0.026937878 0.10103768
## 3 0.06515737 0.014452474 0.0000000000 0.01974767 0.004449388 0.12548369
## 4 0.04316940 0.007127193 0.0016348774 0.05579632 0.002199010 0.07906216
## 5 0.17614424 0.006925208 0.0006901311 0.01033770 0.000000000 0.08115543
## 6 0.03082003 0.007186291 0.0027472527 0.09557619 0.007190265 0.04607789
## 993 994 995 996 997 998
## 1 0.003852504 0.0006910850 0.06765677 0.04832510 0.07024336 0.02908891
## 2 0.025697102 0.0055096419 0.09651036 0.15720524 0.09599561 0.08006536
## 3 0.004977876 0.0000000000 0.08314978 0.08103638 0.08430394 0.04242424
## 4 0.002183406 0.0006868132 0.18770403 0.26608506 0.19123288 0.02830702
## 5 0.000000000 0.0043478261 0.21724138 0.11203320 0.22176022 0.02076125
## 6 0.007142857 0.0000000000 0.15161467 0.27252747 0.15359116 0.04052574
## 999 1000 1001 1002 1003 1004 1005
## 1 0.033755274 0.020441989 0.02091359 0.02467105 0.1253436 0.05494505 0.04292790
## 2 0.111420613 0.074643249 0.07494530 0.02066340 0.1805783 0.07103825 0.09066084
## 3 0.043356643 0.023849140 0.02434975 0.01976936 0.1743929 0.08669244 0.06949807
## 4 0.018181818 0.024109589 0.02679060 0.03369565 0.1891008 0.09700272 0.17993457
## 5 0.005328597 0.005571031 0.00625000 0.02739726 0.1137147 0.09681881 0.05244997
## 6 0.054016620 0.032578686 0.03195592 0.03608529 0.1539307 0.07734504 0.19989017
## 1006 1007 1008 1009 1010 1011
## 1 0.002178649 0.03257790 0.0005534034 0.001646542 0.004975124 0.05500550
## 2 0.004317323 0.05939902 0.0592755214 0.058791508 0.069741900 0.17130387
## 3 0.002735230 0.04240283 0.0044419767 0.004964148 0.012208657 0.06198118
## 4 0.002695418 0.03006993 0.0109649123 0.011976048 0.030153509 0.06205770
## 5 0.018455229 0.03815439 0.0000000000 0.000000000 0.005536332 0.03583735
## 6 0.001086957 0.03354298 0.0413907285 0.041028446 0.071152785 0.06930693
## 1012 1013 1014 1015 1016 1017
## 1 0.0006988120 0.06263858 0.05665567 0.06050605 0.12431544 0.15659341
## 2 0.0213204952 0.25344353 0.17550574 0.25068418 0.22512235 0.25942108
## 3 0.0007007708 0.05951057 0.06412383 0.05761773 0.15786579 0.16777042
## 4 0.0048275862 0.12252747 0.06222707 0.11542670 0.06902174 0.06434024
## 5 0.0077922078 0.02725367 0.03596127 0.02781641 0.07766323 0.05809129
## 6 0.0069541029 0.19911259 0.06820682 0.19470491 0.08643326 0.08831596
## 1018 1019 1020 1021 1022 1023
## 1 0.0006925208 0.0007147963 0.01380262 0.07326408 0.0000000000 0.02077638
## 2 0.0269337017 0.0226950355 0.06694272 0.05591748 0.0158643326 0.04180239
## 3 0.0013917884 0.0007178751 0.02236198 0.11403509 0.0005527916 0.03793293
## 4 0.0048242591 0.0042462845 0.03161512 0.12086721 0.0005458515 0.20358306
## 5 0.0060922541 0.0044762757 0.03836094 0.02957359 0.0020790021 0.15840220
## 6 0.0131578947 0.0071022727 0.03726708 0.16557377 0.0033039648 0.15191257
## 1024 1025 1026 1027 1028 1029 1030
## 1 0.01918860 0.19150110 0.07806487 0.02142857 0.02079002 0.07472527 0.003325942
## 2 0.01307190 0.07881773 0.19455041 0.13793103 0.14523975 0.01147541 0.014293568
## 3 0.02310231 0.23682751 0.11178414 0.03214286 0.03146853 0.04136790 0.002782415
## 4 0.03695652 0.17194323 0.12948857 0.06524823 0.06409373 0.04577657 0.003291278
## 5 0.03880804 0.28196268 0.08425414 0.03451327 0.03174603 0.15379310 0.000694927
## 6 0.01972603 0.19084391 0.13344316 0.06241135 0.06197772 0.02414929 0.001107420
## 1031 1032 1033 1034 1035 1036
## 1 0.0005540166 0.001413428 0.0005527916 0.001420455 0.16740741 0.09156977
## 2 0.0445544554 0.004915730 0.0455543359 0.004219409 0.05934718 0.03524229
## 3 0.0027793218 0.001408451 0.0033351862 0.001428571 0.10432191 0.11722142
## 4 0.0054945055 0.004210526 0.0076796489 0.003521127 0.04733728 0.03235294
## 5 0.0000000000 0.002650177 0.0000000000 0.001795332 0.11371841 0.14673913
## 6 0.0287292818 0.006346968 0.0309392265 0.004996431 0.02980626 0.04499274
## 1037 1038 1039 1040 1041 1042
## 1 0.004415011 0.05240175 0.003333333 0.079594790 0.25742574 0.04203540
## 2 0.020833333 0.01308140 0.004412576 0.073170732 0.06174863 0.05656233
## 3 0.004434590 0.05029586 0.004997224 0.042335766 0.28468767 0.05219323
## 4 0.001094691 0.04184704 0.015401540 0.008547009 0.18439716 0.02083333
## 5 0.002785515 0.03717472 0.027291812 0.055133080 0.24068966 0.08113731
## 6 0.003305785 0.06432749 0.004988914 0.007194245 0.17621145 0.01767956
## 1043 1044 1045 1046 1047 1048
## 1 0.003320421 0.037239869 0.078657866 0.06045480 0.01795580 0.11919081
## 2 0.021978022 0.002176279 0.006560962 0.21734358 0.05349794 0.07654723
## 3 0.006107718 0.035281147 0.083517699 0.09761509 0.02837370 0.06410959
## 4 0.004388371 0.015786609 0.054068815 0.18465753 0.02804378 0.02222222
## 5 0.020905923 0.084195997 0.197779320 0.02855153 0.08836395 0.12396694
## 6 0.006640841 0.016456391 0.037953795 0.32577434 0.02349689 0.01693989
## 1049 1050 1051 1052 1053 1054 1055
## 1 0.16095133 0.12017641 0.06873614 0.1678591 0.004953220 0.017117615 0.20498615
## 2 0.06622879 0.07326408 0.22124381 0.1776927 0.011494253 0.003831418 0.13755158
## 3 0.09151414 0.06467662 0.05552471 0.2341248 0.006087438 0.013858093 0.13988920
## 4 0.03501094 0.02185792 0.12795167 0.3549618 0.002738226 0.013129103 0.03611111
## 5 0.10607118 0.12655602 0.01873699 0.1512431 0.002766252 0.016701461 0.09041591
## 6 0.02976847 0.01596916 0.18995030 0.3880351 0.007705008 0.018753447 0.06077348
## 1056 1057 1058 1059 1060 1061
## 1 0.09156095 0.016930639 0.006060606 0.0005491488 0.13861386 0.05423354
## 2 0.11561644 0.003252033 0.013668671 0.0092643052 0.18951393 0.08786381
## 3 0.04878049 0.013721186 0.007182320 0.0016556291 0.08890116 0.08555556
## 4 0.01750547 0.013001083 0.005461496 0.0032644178 0.03431373 0.09989023
## 5 0.05601660 0.016438356 0.003460208 0.0055286800 0.09695291 0.08027682
## 6 0.01709873 0.019115238 0.004947774 0.0060373216 0.03626374 0.20741150
## 1062 1063 1064 1065 1066 1067
## 1 0.079670330 0.083516484 0.012568306 0.02535832 0.23861766 0.02640264
## 2 0.197052402 0.196399345 0.039587852 0.04266958 0.16539717 0.07962658
## 3 0.074421169 0.076033058 0.019758507 0.02490315 0.17721519 0.03722222
## 4 0.029989095 0.029427793 0.026044493 0.02240437 0.06844106 0.02905702
## 5 0.006194081 0.006185567 0.005479452 0.02006920 0.11395028 0.05551700
## 6 0.064250412 0.065384615 0.027277687 0.04068169 0.14731654 0.04022039
## 1068 1069 1070 1071 1072 1073
## 1 0.01368363 0.07241570 0.05365044 0.03984505 0.007700770 0.008282717
## 2 0.04073873 0.12438356 0.20581778 0.14772103 0.087834152 0.022478070
## 3 0.02145215 0.09756098 0.08398220 0.05336298 0.012134584 0.009983361
## 4 0.02497286 0.11439518 0.12287438 0.09682713 0.023433243 0.006020799
## 5 0.00550585 0.09993060 0.06838800 0.10138889 0.004143646 0.036136206
## 6 0.02408320 0.13072256 0.13709232 0.10325787 0.036732456 0.009933775
## 1074 1075 1076 1077 1078 1079 1080
## 1 0.13002755 0.05528188 0.1613611 0.04908991 0.08185841 0.06027397 0.06810631
## 2 0.20218579 0.05655247 0.1456628 0.09409190 0.07260726 0.06188925 0.02744237
## 3 0.09652510 0.09356081 0.2247934 0.07641196 0.10394664 0.07569940 0.05862832
## 4 0.03058438 0.09635275 0.1724701 0.03663204 0.06425041 0.02711497 0.02631579
## 5 0.04830918 0.01793103 0.1085003 0.03331020 0.15566366 0.08475735 0.02298050
## 6 0.05375754 0.12972085 0.2404162 0.07170436 0.08637874 0.06287589 0.03195592
## 1081 1082 1083 1084 1085 1086 1087
## 1 0.009232955 0.1312910 0.04429679 0.064906491 0.004834254 0.4373626 0.4446903
## 2 0.032348805 0.1225490 0.08044077 0.002183406 0.032458564 0.1185145 0.1206140
## 3 0.014184397 0.1112335 0.06559200 0.072375691 0.006302521 0.3789300 0.4174058
## 4 0.018374558 0.1147363 0.07734504 0.086743044 0.008971705 0.1259542 0.1233553
## 5 0.007174888 0.1771978 0.15389972 0.211432507 0.004382121 0.2048276 0.1862404
## 6 0.011355571 0.1188390 0.08241150 0.043406593 0.020905923 0.1482702 0.1614151
## 1088 1089 1090 1091 1092 1093 1094
## 1 0.38240177 0.03394625 0.33680746 0.15769231 0.04856512 0.0550055 0.05384615
## 2 0.22185612 0.09205903 0.11444142 0.12377317 0.10290093 0.1687602 0.16402406
## 3 0.31722222 0.05516266 0.26074972 0.11852260 0.07313019 0.0839779 0.08678828
## 4 0.08388158 0.04044630 0.09961894 0.05059848 0.07864555 0.2339149 0.23413567
## 5 0.12301313 0.10717449 0.19944979 0.12733564 0.01044568 0.0615917 0.06154910
## 6 0.10657096 0.03745583 0.08776742 0.04772353 0.21014892 0.3204846 0.32050577
## 1095 1096 1097 1098 1099 1100 1101
## 1 0.2498621 0.2455752 0.0000000000 0.2426068 0.13758943 0.0011068069 0.01387347
## 2 0.1336254 0.1353425 0.0092744135 0.1345316 0.09896118 0.0027397260 0.03636364
## 3 0.1772853 0.1736959 0.0005497526 0.1685950 0.20121614 0.0005546312 0.02060134
## 4 0.1648412 0.1689523 0.0021810251 0.1628540 0.20512821 0.0000000000 0.01541002
## 5 0.2315716 0.2275574 0.0055210490 0.2272099 0.07809261 0.0000000000 0.04517026
## 6 0.2005510 0.2037548 0.0054824561 0.2033991 0.30473047 0.0038695412 0.01828255
## 1102 1103 1104 1105 1106 1107
## 1 0.01894150 0.015907844 0.05901820 0.04670330 0.004402862 0.04452996
## 2 0.06329114 0.008169935 0.11822660 0.02128821 0.008192245 0.01909438
## 3 0.03242035 0.024766098 0.05794702 0.06504961 0.006070640 0.06229327
## 4 0.01822198 0.042414356 0.02128821 0.04196185 0.014161220 0.04084967
## 5 0.02712100 0.029532967 0.06167706 0.11019284 0.028178694 0.10880111
## 6 0.03722222 0.047697368 0.05112699 0.05159166 0.010439560 0.05219780
## 1108 1109 1110 1111 1112 1113
## 1 0.000000000 0.001108647 0.0000000000 0.03577325 0.002759382 0.11135612
## 2 0.009161381 0.012644310 0.0081922447 0.06075534 0.049288061 0.13005464
## 3 0.000000000 0.001667593 0.0005521811 0.05300939 0.006084071 0.11332228
## 4 0.001409443 0.004405286 0.0038188762 0.02894593 0.012575178 0.15447598
## 5 0.004456328 0.004861111 0.0048476454 0.09342561 0.014553015 0.03094911
## 6 0.004255319 0.002771619 0.0027517887 0.03951701 0.024820739 0.24573942
## 1114 1115 1116 1117 1118 1119
## 1 0.01438849 0.13611416 0.10049696 0.009955752 0.04367054 0.002748763
## 2 0.03903244 0.05223069 0.16712329 0.034577387 0.11068493 0.016339869
## 3 0.01778766 0.10462555 0.14024390 0.018333333 0.07329262 0.004415011
## 4 0.01701427 0.02773246 0.10224166 0.016492578 0.03620406 0.005997819
## 5 0.01394700 0.14511692 0.04854369 0.018093250 0.03331020 0.008264463
## 6 0.02542841 0.08826754 0.10044150 0.023796348 0.05074462 0.011525796wine.mds <- cmdscale(1 - model_resample_1000_trees$proximity, eig=TRUE)
op <- par(pty="s")
pairs(cbind(wine_train[c("alcohol", "sulphates", "volatile_acidity", "total_sulfur_dioxide", "chlorides")
], wine.mds$points), cex=0.6, gap=0,
col=c("red", "green", "blue")[as.numeric(wine_train$quality)],
main="Wine Data: Predictors and MDS of Proximity Based on RandomForest")
# plot of the most important predictor variables of wine quality and then red, green, blue to represent the different qualitlies "Poor", "Normal", "Excellent"
#Explanatory Plot
ggpairs(data.frame(
alcohol = wine_train$alcohol,
sulphates = wine_train$sulphates,
volatile_acidity = wine_train$volatile_acidity,
MDS1 = wine.mds$points[,1],
MDS2 = wine.mds$points[,2],
quality = wine_train$quality
), mapping = aes(color = quality))## `stat_bin()` using `bins = 30`. Pick better value `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value `binwidth`.
## `stat_bin()` using `bins = 30`. Pick better value `binwidth`.
# Interactive graph:
p <- plot_ly(
data = wine_train,
x = ~alcohol,
y = ~volatile_acidity,
type = "scatter",
mode = "markers", # <-- fix
color = ~quality, # discrete if factor/ordered
colors = viridis(nlevels(wine_train$quality)),
size = ~sulphates, # bubble size mapping
sizes = c(5, 25), # pixel range for bubbles
text = ~paste(
"Quality:", quality,
"<br>Alcohol:", alcohol,
"<br>Volatile acidity:", volatile_acidity,
"<br>Sulphates:", sulphates
),
hoverinfo = "text"
) %>%
layout(
xaxis = list(title = "Alcohol"),
yaxis = list(title = "Volatile acidity"),
legend = list(title = list(text = "Quality"))
)
p## Warning: `line.width` does not currently support multiple values.
## Warning: `line.width` does not currently support multiple values.
## Warning: `line.width` does not currently support multiple values.varImpPlot(model_resample_1000_trees)
ggplot(transform(wine_train, MDS1 = wine.mds$points[,1], MDS2 = wine.mds$points[,2]),
aes(MDS1, MDS2)) +
geom_hex(bins = 40) + # or: geom_bin2d()
facet_wrap(~ quality, nrow = 1) + # show each class separately
theme_bw() + labs(title = "Class-wise density (hex bins)")
mds_df <- data.frame(
MDS1 = wine.mds$points[, 1],
MDS2 = wine.mds$points[, 2],
quality = wine_train$quality, # your class label
row_id = seq_len(nrow(wine_train))# optional: ID for tooltips
)
# Visualize overlap between classes
ggplot(mds_df, aes(MDS1, MDS2, color = quality)) +
geom_point(alpha = 0.6) +
theme_minimal() +
labs(title = "Random Forest proximity MDS by wine quality")
# Cluster tightness per class
aggregate(cbind(MDS1, MDS2) ~ quality, mds_df, function(x) sd(x))## quality MDS1 MDS2
## 1 Poor 0.09883704 0.1826716
## 2 Normal 0.18552602 0.1543027
## 3 Excellent 0.15503251 0.1127544Predictive Power of the Model
## Prepare X (predictors) and y (target) for TEST/VALIDATION of the Random Forrest
wine_test$quality <- factor(wine_test$quality,
levels = levels(wine_train$quality),
ordered = is.ordered(wine_train$quality))
validate_x <- subset(wine_test, select = setdiff(names(wine_test), "quality"))
validate_y <- wine_test$quality
## 2) Get predictions: class labels and class probabilities
# Use your balanced model; change the object name if needed
rf_fit <- model_resample_1000_trees
# class predictions
pred_y <- predict(rf_fit, newdata = validate_x, type = "class")
# probability matrix (cols: "Poor","Normal","Excellent")
probs <- predict(rf_fit, newdata = validate_x, type = "prob")
## 3) Confusion matrix + overall metrics
cm <- confusionMatrix(pred_y, validate_y) # multiclass by default
cm## Confusion Matrix and Statistics
##
## Reference
## Prediction Poor Normal Excellent
## Poor 13 51 3
## Normal 6 263 8
## Excellent 0 82 54
##
## Overall Statistics
##
## Accuracy : 0.6875
## 95% CI : (0.6439, 0.7287)
## No Information Rate : 0.825
## P-Value [Acc > NIR] : 1
##
## Kappa : 0.349
##
## Mcnemar's Test P-Value : <2e-16
##
## Statistics by Class:
##
## Class: Poor Class: Normal Class: Excellent
## Sensitivity 0.68421 0.6641 0.8308
## Specificity 0.88286 0.8333 0.8024
## Pos Pred Value 0.19403 0.9495 0.3971
## Neg Pred Value 0.98547 0.3448 0.9680
## Prevalence 0.03958 0.8250 0.1354
## Detection Rate 0.02708 0.5479 0.1125
## Detection Prevalence 0.13958 0.5771 0.2833
## Balanced Accuracy 0.78354 0.7487 0.8166cm$byclass## NULL## 4) Top-2 accuracy, useful with 3 classes ---
top2_correct <- mean(apply(probs, 1, function(p) {
ord <- order(p, decreasing = TRUE)
levels(validate_y)[ord[1]] == as.character(validate_y)[which.max(p)] ||
levels(validate_y)[ord[2]] == as.character(validate_y)[which.max(p)]
}))
top2_correct## [1] 0.9833333## 5) One-vs-rest ROC AUC per class
auc_per_class <- sapply(levels(validate_y), function(cl){
y_bin <- factor(ifelse(validate_y == cl, cl, paste0("Not_", cl)),
levels = c(paste0("Not_", cl), cl))
auc(roc(y_bin, probs[, cl], levels = levels(y_bin), direction = "<"))
})
auc_per_class## Poor Normal Excellent
## 0.8526658 0.8354227 0.8810195# Confusion Matrix Graphic
cm_table <- as.table(cm$table)
cm_df <- melt(cm_table)
colnames(cm_df) <- c("True", "Predicted", "Count")
ggplot(cm_df, aes(x = Predicted, y = True, fill = Count)) +
geom_tile(color = "white") +
geom_text(aes(label = Count), color = "black", size = 4) +
scale_fill_viridis_c() +
theme_minimal() +
labs(title = "Confusion Matrix — Random Forest Wine Quality",
x = "Predicted Class", y = "True Class")
# Creation of ROC plots:
roc_poor <- roc(validate_y == "Poor", probs[,"Poor"])## Setting levels: control = FALSE, case = TRUE## Setting direction: controls < casesroc_norm <- roc(validate_y == "Normal", probs[,"Normal"])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesroc_ex <- roc(validate_y == "Excellent", probs[,"Excellent"])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesplot.roc(roc_poor, col="purple", print.auc=TRUE)
plot.roc(roc_norm, add=TRUE, col="blue", print.auc=TRUE, print.auc.y=0.4)
plot.roc(roc_ex, add=TRUE, col="gold", print.auc=TRUE, print.auc.y=0.3)
legend("bottomright", c("Poor vs rest","Normal vs rest","Excellent vs rest"),
col=c("purple","blue","gold"), lwd=2)
# varIMplot to see which variables help the model distigusih what type of wine it is
varImpPlot(model_resample_1000_trees, main="Variable Importance — Random Forest")
par(mfrow=c(1,3))
# PPC plots: to show relationship between probability weightings to classes and signficant variables
partialPlot(model_resample_1000_trees, as.data.frame(wine_train), "alcohol", which.class="Excellent")
partialPlot(model_resample_1000_trees, as.data.frame(wine_train), "volatile_acidity", which.class="Excellent")
partialPlot(model_resample_1000_trees, as.data.frame(wine_train), "sulphates", which.class="Excellent")
par(mfrow=c(1,1))Bayesian Ordinal Regression Model
# Problem arising due to corput ggplot code:
#search() # should show only base packages
# Remove broken installs:
#lib <- .libPaths()[1]
#unlink(file.path(lib, "ggplot2"), recursive = TRUE, force = TRUE)
#unlink(file.path(lib, "00LOCK*"), recursive = TRUE, force = TRUE)
# Re-install package cleanly
#options(repos = c(CRAN = "https://cloud.r-project.org"))
#options(pkgType = "binary")
#install.packages("ggplot2") # should NOT complain it's in use
#library(ggplot2) # quick sanity check
# Need To install Rtools First
#install.packages("cmdstanr",
# repos = c("https://mc-stan.org/r-packages/", getOption("repos"))
#)
#check_cmdstan_toolchain(fix = TRUE) # should say toolchain is good
#install_cmdstan() # first-time download & build (~few minutes)
## Check ## After Resatall: ##
#has_build_tools(debug = TRUE)
Sys.which(c("make","g++")) # if in form "C:\\rtools44\\usr\\bin\\make.exe" "C:\\rtools44\\X86_64~1.POS\\bin\\G__~1.EXE" under Make and g++ repspectively then good to go. ## make
## "C:\\rtools44\\usr\\bin\\make.exe"
## g++
## "C:\\rtools44\\X86_64~1.POS\\bin\\G__~1.EXE"#Installing Windows friendly Binary from Stans R-Universe
# Check that install was all good:
check_cmdstan_toolchain(fix = TRUE)## The C++ toolchain required for CmdStan is setup properly!# one-time install of CmdStan (downloads & builds)
#cmdstanr::install_cmdstan() # installs to: %USERPROFILE%/.cmdstanr/cmdstan-<ver>
#cmdstanr::cmdstan_path() # verify path
## Running Bayesian Ordinal Model ##
# make sure the outcome is ordered *and* levels are in order
wine_train$quality <- ordered(wine_train$quality,
levels = c("Poor","Normal","Excellent"))
pri <- c(
set_prior("normal(0,1)", class = "b"), # slopes
set_prior("student_t(3,0,2.5)", class = "Intercept") # cutpoints
)
fit_ord <- brm(
quality ~ .,
data = wine_train,
family = cumulative("logit"),
prior = pri,
chains = 4, cores = 4, iter = 2000, seed = 123,
backend = "cmdstanr" # <— uses your CmdStan at ~/.cmdstan
)## Start sampling## Running MCMC with 4 parallel chains...
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##
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## Total execution time: 33.4 seconds.## Loading required package: rstan## Warning: package 'rstan' was built under R version 4.4.3## Loading required package: StanHeaders## Warning: package 'StanHeaders' was built under R version 4.4.3##
## rstan version 2.32.7 (Stan version 2.32.2)## For execution on a local, multicore CPU with excess RAM we recommend calling
## options(mc.cores = parallel::detectCores()).
## To avoid recompilation of unchanged Stan programs, we recommend calling
## rstan_options(auto_write = TRUE)
## For within-chain threading using `reduce_sum()` or `map_rect()` Stan functions,
## change `threads_per_chain` option:
## rstan_options(threads_per_chain = 1)## Do not specify '-march=native' in 'LOCAL_CPPFLAGS' or a Makevars file##
## Attaching package: 'rstan'## The following object is masked from 'package:magrittr':
##
## extract# Assessing models fit:
summary(fit_ord) # coefficients + cutpoints + Rhat/ESS## Family: cumulative
## Links: mu = logit
## Formula: quality ~ fixed_acidity + volatile_acidity + citric_acid + residual_sugar + chlorides + free_sulfur_dioxide + total_sulfur_dioxide + density + p_h + sulphates + alcohol
## Data: wine_train (Number of observations: 1119)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept[1] 2.97 2.73 -2.24 8.34 1.00 5469
## Intercept[2] 9.34 2.76 3.96 14.83 1.00 5266
## fixed_acidity 0.11 0.07 -0.03 0.24 1.00 3985
## volatile_acidity -2.74 0.51 -3.73 -1.73 1.00 4755
## citric_acid 0.38 0.57 -0.77 1.46 1.00 4235
## residual_sugar -0.01 0.06 -0.13 0.11 1.00 5914
## chlorides -1.53 0.87 -3.21 0.18 1.00 6568
## free_sulfur_dioxide 0.02 0.01 -0.01 0.04 1.00 4957
## total_sulfur_dioxide -0.01 0.00 -0.01 0.00 1.00 4629
## density -0.01 0.99 -1.97 1.90 1.00 6828
## p_h -0.82 0.64 -2.08 0.45 1.00 5061
## sulphates 2.20 0.46 1.30 3.08 1.00 5544
## alcohol 0.84 0.09 0.67 1.03 1.00 4307
## Tail_ESS
## Intercept[1] 3197
## Intercept[2] 2967
## fixed_acidity 3416
## volatile_acidity 3085
## citric_acid 3305
## residual_sugar 3189
## chlorides 3082
## free_sulfur_dioxide 3169
## total_sulfur_dioxide 3054
## density 2794
## p_h 2872
## sulphates 3193
## alcohol 3036
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## disc 1.00 0.00 1.00 1.00 NA NA NA
##
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).bayes_R2(fit_ord)## Warning: Predictions are treated as continuous variables in 'bayes_R2' which is
## likely invalid for ordinal families.## Estimate Est.Error Q2.5 Q97.5
## R2 0.2130358 0.02284927 0.1676189 0.2561668pp_check(fit_ord, type = "bars") # overall class counts## Using 10 posterior draws for ppc type 'bars' by default.
pp_check(fit_ord, type = "bars_grouped", group = "quality")## Using 10 posterior draws for ppc type 'bars_grouped' by default.
Predictive Power of the Model
labs <- c("Poor","Normal","Excellent") # ensure this matches your factor order
# Draws x N matrix of category indices (1..K)
pp <- posterior_predict(fit_ord, newdata = wine_test)
# N x K matrix of posterior mean probs
probs <- t(apply(pp, 2, function(v) prop.table(tabulate(v, nbins = length(labs)))))
# Hard labels via argmax
pred_lab <- factor(labs[max.col(probs, ties.method = "first")], levels = labs, ordered = TRUE)
# Confusion + accuracy
table(Truth = wine_test$quality, Pred = pred_lab)## Pred
## Truth Poor Normal Excellent
## Poor 0 19 0
## Normal 0 384 12
## Excellent 0 52 13mean(pred_lab == wine_test$quality) # Accuarcy ## [1] 0.8270833pred <- factor(pred_lab, levels = levels(wine_test$quality))
ref <- factor(wine_test$quality, levels = levels(wine_test$quality))
confusionMatrix(pred, ref)## Confusion Matrix and Statistics
##
## Reference
## Prediction Poor Normal Excellent
## Poor 0 0 0
## Normal 19 384 52
## Excellent 0 12 13
##
## Overall Statistics
##
## Accuracy : 0.8271
## 95% CI : (0.7902, 0.8599)
## No Information Rate : 0.825
## P-Value [Acc > NIR] : 0.4812
##
## Kappa : 0.1802
##
## Mcnemar's Test P-Value : NA
##
## Statistics by Class:
##
## Class: Poor Class: Normal Class: Excellent
## Sensitivity 0.00000 0.9697 0.20000
## Specificity 1.00000 0.1548 0.97108
## Pos Pred Value NaN 0.8440 0.52000
## Neg Pred Value 0.96042 0.5200 0.88571
## Prevalence 0.03958 0.8250 0.13542
## Detection Rate 0.00000 0.8000 0.02708
## Detection Prevalence 0.00000 0.9479 0.05208
## Balanced Accuracy 0.50000 0.5622 0.58554# ROC for the three classes
roc_poor <- roc(wine_test$quality == "Poor", probs[, 1])## Setting levels: control = FALSE, case = TRUE## Setting direction: controls < casesroc_norm <- roc(wine_test$quality == "Normal", probs[, 2])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesroc_ex <- roc(wine_test$quality == "Excellent", probs[, 3])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesauc(roc_poor); auc(roc_norm); auc(roc_ex)## Area under the curve: 0.7883## Area under the curve: 0.7739## Area under the curve: 0.8579plot.roc(roc_norm, col="blue", main="One-vs-All ROC Curves")
lines.roc(roc_poor, col="red")
lines.roc(roc_ex, col="green")
legend("bottomright", legend=c("Poor","Normal","Excellent"),
col=c("red","blue","green"), lwd=2)
# Metrics for model performance:
# MCM Graph:
posterior_summary(fit_ord, pars = "^b_Intercept")## Warning: Argument 'pars' is deprecated. Please use 'variable' instead.## Estimate Est.Error Q2.5 Q97.5
## b_Intercept[1] 2.970277 2.728673 -2.241332 8.341163
## b_Intercept[2] 9.342572 2.764952 3.957856 14.831309# 95% inner band, 99% outer band (avoids the prob/prob_outer warning)
mcmc_intervals(
fit_ord,
regex_pars = "^b_Intercept",
prob = 0.95,
prob_outer = 0.99
)
## After a quick review of the confusion matrix, we can see that the oridinal bayes model is unable to correctly model and of the poor wines in the test set. We consider an application which considers a Class-weighted likelihood, to try midgate the dominance of the normal set over the poor and excellent wines. Smoothed Bayesian Ordinal regression Model
##Relaxing proportional odds, another method to try prevent normal class dominating the others:
fit_cs <- brm(
quality ~ cs(volatile_acidity) + alcohol + sulphates + citric_acid + p_h,
data = wine_train, family = cumulative("logit"),
prior = pri, chains = 4, iter = 2000, backend = "cmdstanr"
)## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## Start sampling## Running MCMC with 4 sequential chains...
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##
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## Mean chain execution time: 45.9 seconds.
## Total execution time: 184.0 seconds.## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.# Assessing models fit:
model_b<-summary(fit_cs) ## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.summary(fit_cs) ## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## Family: cumulative
## Links:## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## mu = logit
## Formula: quality ~ cs(volatile_acidity) + alcohol + sulphates + citric_acid + p_h
## Data: wine_train (Number of observations: 1119)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept[1] 1.95 2.01 -1.95 5.90 1.00 3841 2811
## Intercept[2] 7.80 2.03 3.88 11.82 1.00 3798 2685
## alcohol 0.86 0.09 0.69 1.03 1.00 4054 3048
## sulphates 2.05 0.45 1.15 2.90 1.00 4791 2490
## citric_acid 0.84 0.50 -0.13 1.83 1.00 3342 2916
## p_h -1.17 0.57 -2.26 -0.04 1.00 3459 3162
## volatile_acidity[1] -1.86 0.67 -3.18 -0.57 1.00 2195 2648
## volatile_acidity[2] -2.52 0.58 -3.68 -1.40 1.00 3523 2660
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## disc 1.00 0.00 1.00 1.00 NA NA NA
##
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).bayes_R2(fit_cs)## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## Warning: Predictions are treated as continuous variables in 'bayes_R2' which is
## likely invalid for ordinal families.## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## Estimate Est.Error Q2.5 Q97.5
## R2 0.201218 0.02247582 0.1559278 0.2444526pp_check(fit_cs, type = "bars") ## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## Using 10 posterior draws for ppc type 'bars' by default.## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
pp_check(fit_cs, type = "bars_grouped", group = "quality")## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## Using 10 posterior draws for ppc type 'bars_grouped' by default.## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
Predictive Power of the Model
labs <- c("Poor","Normal","Excellent") # ensure this matches your factor order
# Draws x N matrix of category indices (1..K)
pp <- posterior_predict(fit_cs, newdata = wine_test) ## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.# N x K matrix of posterior mean probs
probs <- t(apply(pp, 2, function(v) prop.table(tabulate(v, nbins = length(labs)))))
# Hard labels via argmax
pred_lab <- factor(labs[max.col(probs, ties.method = "first")], levels = labs, ordered = TRUE)
# Confusion + accuracy
table(Truth = wine_test$quality, Pred = pred_lab)## Pred
## Truth Poor Normal Excellent
## Poor 0 19 0
## Normal 0 388 8
## Excellent 0 52 13mean(pred_lab == wine_test$quality)## [1] 0.8354167pred <- factor(pred_lab, levels = levels(wine_test$quality))
ref <- factor(wine_test$quality, levels = levels(wine_test$quality))
confusionMatrix(pred, ref)## Confusion Matrix and Statistics
##
## Reference
## Prediction Poor Normal Excellent
## Poor 0 0 0
## Normal 19 388 52
## Excellent 0 8 13
##
## Overall Statistics
##
## Accuracy : 0.8354
## 95% CI : (0.7992, 0.8675)
## No Information Rate : 0.825
## P-Value [Acc > NIR] : 0.2976
##
## Kappa : 0.1978
##
## Mcnemar's Test P-Value : NA
##
## Statistics by Class:
##
## Class: Poor Class: Normal Class: Excellent
## Sensitivity 0.00000 0.9798 0.20000
## Specificity 1.00000 0.1548 0.98072
## Pos Pred Value NaN 0.8453 0.61905
## Neg Pred Value 0.96042 0.6190 0.88671
## Prevalence 0.03958 0.8250 0.13542
## Detection Rate 0.00000 0.8083 0.02708
## Detection Prevalence 0.00000 0.9563 0.04375
## Balanced Accuracy 0.50000 0.5673 0.59036# ROC for the three classes
roc_poor <- roc(wine_test$quality == "Poor", probs[, 1])## Setting levels: control = FALSE, case = TRUE## Setting direction: controls < casesroc_norm <- roc(wine_test$quality == "Normal", probs[, 2])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesroc_ex <- roc(wine_test$quality == "Excellent", probs[, 3])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesauc(roc_poor); auc(roc_norm); auc(roc_ex)## Area under the curve: 0.7884## Area under the curve: 0.751## Area under the curve: 0.8575plot.roc(roc_norm, col="blue", main="One-vs-All ROC Curves")
lines.roc(roc_poor, col="red")
lines.roc(roc_ex, col="green")
legend("bottomright", legend=c("Poor","Normal","Excellent"),
col=c("red","blue","green"), lwd=2)
# Comparing Models
loo_compare(loo(fit_cs), loo(fit_ord))## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.
## Warning: Category specific effects for this family should be considered
## experimental and may have convergence issues.## elpd_diff se_diff
## fit_ord 0.0 0.0
## fit_cs -3.8 2.7# Since no increase in model predictive power under the the smoothed model, we will use the simplier model.
# Output of the first model confusion Matrix:
# Confusion Matrix Graphic
# Draws x N matrix of category indices (1..K)
pp <- posterior_predict(fit_ord, newdata = wine_test)
# N x K matrix of posterior mean probs
probs <- t(apply(pp, 2, function(v) prop.table(tabulate(v, nbins = length(labs)))))
# Hard labels via argmax
pred_lab <- factor(labs[max.col(probs, ties.method = "first")], levels = labs, ordered = TRUE)
# Confusion + accuracy
table(Truth = wine_test$quality, Pred = pred_lab)## Pred
## Truth Poor Normal Excellent
## Poor 0 19 0
## Normal 0 384 12
## Excellent 0 52 13pred <- factor(pred_lab, levels = levels(wine_test$quality))
ref <- factor(wine_test$quality, levels = levels(wine_test$quality))
cm<- confusionMatrix(pred, ref)
cm_table <- as.table(cm$table)
cm_df <- melt(cm_table)
colnames(cm_df) <- c("True", "Predicted", "Count")
ggplot(cm_df, aes(x = Predicted, y = True, fill = Count)) +
geom_tile(color = "white") +
geom_text(aes(label = Count), color = "black", size = 4) +
scale_fill_viridis_c() +
theme_minimal() +
labs(title = "Confusion Matrix — Bayesian Ordinal Regression",
x = "Predicted Class", y = "True Class")
Prior Sensitivity Analysis
# Baseline (used in above fit_ord model)
pri_base <- c(
set_prior("normal(0,1)", class="b"),
set_prior("student_t(3,0,2.5)", class="Intercept")
)
# Smaller variation from baseline prior
pri_tight <- c(
set_prior("normal(0,0.5)", class="b"),
set_prior("student_t(3,0,2.5)", class="Intercept")
)
# more variation from baseline prior
pri_wide <- c(
set_prior("normal(0,2)", class="b"),
set_prior("student_t(3,0,5)", class="Intercept")
)
# Sparsity-favoring (shrink weak predictors)
pri_hs <- c(
set_prior("horseshoe(2)", class="b"),
set_prior("student_t(3,0,2.5)", class="Intercept")
)
# To Improve efficiency (not having to write out same code for)
fit_base <- fit_ord; fit_base## Family: cumulative
## Links: mu = logit
## Formula: quality ~ fixed_acidity + volatile_acidity + citric_acid + residual_sugar + chlorides + free_sulfur_dioxide + total_sulfur_dioxide + density + p_h + sulphates + alcohol
## Data: wine_train (Number of observations: 1119)
## Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
## total post-warmup draws = 4000
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## Intercept[1] 2.97 2.73 -2.24 8.34 1.00 5469
## Intercept[2] 9.34 2.76 3.96 14.83 1.00 5266
## fixed_acidity 0.11 0.07 -0.03 0.24 1.00 3985
## volatile_acidity -2.74 0.51 -3.73 -1.73 1.00 4755
## citric_acid 0.38 0.57 -0.77 1.46 1.00 4235
## residual_sugar -0.01 0.06 -0.13 0.11 1.00 5914
## chlorides -1.53 0.87 -3.21 0.18 1.00 6568
## free_sulfur_dioxide 0.02 0.01 -0.01 0.04 1.00 4957
## total_sulfur_dioxide -0.01 0.00 -0.01 0.00 1.00 4629
## density -0.01 0.99 -1.97 1.90 1.00 6828
## p_h -0.82 0.64 -2.08 0.45 1.00 5061
## sulphates 2.20 0.46 1.30 3.08 1.00 5544
## alcohol 0.84 0.09 0.67 1.03 1.00 4307
## Tail_ESS
## Intercept[1] 3197
## Intercept[2] 2967
## fixed_acidity 3416
## volatile_acidity 3085
## citric_acid 3305
## residual_sugar 3189
## chlorides 3082
## free_sulfur_dioxide 3169
## total_sulfur_dioxide 3054
## density 2794
## p_h 2872
## sulphates 3193
## alcohol 3036
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## disc 1.00 0.00 1.00 1.00 NA NA NA
##
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).# Smaller-variance prior (refit happens because prior changed)
fit_tight <- update(
fit_base,
prior = pri_tight,
seed = 124,
chains = 4,
iter = 4000,
refresh = 0,
recompile = TRUE,
init = "random"
)## Start sampling## Running MCMC with 4 sequential chains...
##
## Chain 1 finished in 27.2 seconds.
## Chain 2 finished in 30.1 seconds.
## Chain 3 finished in 30.4 seconds.
## Chain 4 finished in 27.4 seconds.
##
## All 4 chains finished successfully.
## Mean chain execution time: 28.8 seconds.
## Total execution time: 115.6 seconds.# Wider variance Prior
fit_wide <- update(
fit_base,
prior = pri_wide,
seed = 125,
chains = 4,
iter = 4000,
refresh = 0,
recompile = TRUE,
inits = "random"
)## Warning: Argument 'inits' is deprecated. Please use argument 'init' instead.## Start sampling## Running MCMC with 4 sequential chains...
##
## Chain 1 finished in 38.1 seconds.
## Chain 2 finished in 41.7 seconds.
## Chain 3 finished in 35.6 seconds.
## Chain 4 finished in 38.8 seconds.
##
## All 4 chains finished successfully.
## Mean chain execution time: 38.5 seconds.
## Total execution time: 154.8 seconds.# Horse Shoe Prior
fit_hs <- update(
fit_base,
prior = pri_hs,
seed = 126,
chains = 4,
iter = 4000,
refresh = 0,
recompile = TRUE,
inits = "random"
)## Warning: Argument 'inits' is deprecated. Please use argument 'init' instead.## Start sampling## Running MCMC with 4 sequential chains...
##
## Chain 1 finished in 178.8 seconds.
## Chain 2 finished in 147.9 seconds.
## Chain 3 finished in 322.2 seconds.
## Chain 4 finished in 210.7 seconds.
##
## All 4 chains finished successfully.
## Mean chain execution time: 214.9 seconds.
## Total execution time: 860.2 seconds.## Warning: 2433 of 8000 (30.0%) transitions ended with a divergence.
## See https://mc-stan.org/misc/warnings for details.## Warning: 1 of 8000 (0.0%) transitions hit the maximum treedepth limit of 10.
## See https://mc-stan.org/misc/warnings for details.## Sanity Check fromm above:
# Do the Priors actually differ?
prior_summary(fit_base); prior_summary(fit_tight)## prior class coef group resp dpar nlpar lb ub
## normal(0,1) b
## normal(0,1) b alcohol
## normal(0,1) b chlorides
## normal(0,1) b citric_acid
## normal(0,1) b density
## normal(0,1) b fixed_acidity
## normal(0,1) b free_sulfur_dioxide
## normal(0,1) b p_h
## normal(0,1) b residual_sugar
## normal(0,1) b sulphates
## normal(0,1) b total_sulfur_dioxide
## normal(0,1) b volatile_acidity
## student_t(3,0,2.5) Intercept
## student_t(3,0,2.5) Intercept 1
## student_t(3,0,2.5) Intercept 2
## tag source
## user
## (vectorized)
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## user
## (vectorized)
## (vectorized)## prior class coef group resp dpar nlpar lb ub
## normal(0,0.5) b
## normal(0,0.5) b alcohol
## normal(0,0.5) b chlorides
## normal(0,0.5) b citric_acid
## normal(0,0.5) b density
## normal(0,0.5) b fixed_acidity
## normal(0,0.5) b free_sulfur_dioxide
## normal(0,0.5) b p_h
## normal(0,0.5) b residual_sugar
## normal(0,0.5) b sulphates
## normal(0,0.5) b total_sulfur_dioxide
## normal(0,0.5) b volatile_acidity
## student_t(3,0,2.5) Intercept
## student_t(3,0,2.5) Intercept 1
## student_t(3,0,2.5) Intercept 2
## tag source
## user
## (vectorized)
## (vectorized)
## (vectorized)
## (vectorized)
## (vectorized)
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## (vectorized)
## user
## (vectorized)
## (vectorized)# Posteriors changed, will be False if they've changed.
all.equal(
posterior_summary(fit_base, pars="^b_"),
posterior_summary(fit_tight, pars="^b_")
) ## Warning: Argument 'pars' is deprecated. Please use 'variable' instead.## Warning: Argument 'pars' is deprecated. Please use 'variable' instead.## [1] "Mean relative difference: 0.3166304"loo_base <- loo(fit_base, reloo = TRUE)## No problematic observations found. Returning the original 'loo' object.loo_tight <- loo(fit_tight, reloo = TRUE)## No problematic observations found. Returning the original 'loo' object.loo_hs <- loo(fit_hs, reloo = TRUE)## No problematic observations found. Returning the original 'loo' object.loo_wide <- loo(fit_wide, reloo = TRUE)## No problematic observations found. Returning the original 'loo' object.loo_compare(loo_base, loo_tight, loo_hs, loo_wide)## elpd_diff se_diff
## fit_hs 0.0 0.0
## fit_wide -0.2 0.7
## fit_base -3.9 1.9
## fit_tight -12.3 4.3# Horse shoe is the best model
# Inspect raw elpd_loo estimates (avoid rounding hiding tiny diffs)
c(base = loo_base$estimates[1,1],
tight = loo_tight$estimates[1,1],
hs = loo_hs$estimates[1, 1],
wide = loo_wide$estimates[1, 1])## base tight hs wide
## -499.9911 -508.4596 -496.1311 -496.3388# Horse shoe achieved the best predictive performance, followed closley by the wider standard normal prior. Predictive Power, Horse Prior
pp_check(fit_hs, type = "bars") ## Using 10 posterior draws for ppc type 'bars' by default.
pp_check(fit_hs, type = "bars_grouped", group = "quality")## Using 10 posterior draws for ppc type 'bars_grouped' by default.
labs <- c("Poor","Normal","Excellent") # ensure this matches your factor order
# Draws x N matrix of category indices (1..K)
pp <- posterior_predict(fit_hs, newdata = wine_test)
# N x K matrix of posterior mean probs
probs <- t(apply(pp, 2, function(v) prop.table(tabulate(v, nbins = length(labs)))))
# Hard labels via argmax
pred_lab <- factor(labs[max.col(probs, ties.method = "first")], levels = labs, ordered = TRUE)
# Confusion + accuracy
table(Truth = wine_test$quality, Pred = pred_lab)## Pred
## Truth Poor Normal Excellent
## Poor 1 18 0
## Normal 0 383 13
## Excellent 0 48 17mean(pred_lab == wine_test$quality)## [1] 0.8354167pred <- factor(pred_lab, levels = levels(wine_test$quality))
ref <- factor(wine_test$quality, levels = levels(wine_test$quality))
confusionMatrix(pred, ref)## Confusion Matrix and Statistics
##
## Reference
## Prediction Poor Normal Excellent
## Poor 1 0 0
## Normal 18 383 48
## Excellent 0 13 17
##
## Overall Statistics
##
## Accuracy : 0.8354
## 95% CI : (0.7992, 0.8675)
## No Information Rate : 0.825
## P-Value [Acc > NIR] : 0.2976
##
## Kappa : 0.251
##
## Mcnemar's Test P-Value : NA
##
## Statistics by Class:
##
## Class: Poor Class: Normal Class: Excellent
## Sensitivity 0.052632 0.9672 0.26154
## Specificity 1.000000 0.2143 0.96867
## Pos Pred Value 1.000000 0.8530 0.56667
## Neg Pred Value 0.962422 0.5806 0.89333
## Prevalence 0.039583 0.8250 0.13542
## Detection Rate 0.002083 0.7979 0.03542
## Detection Prevalence 0.002083 0.9354 0.06250
## Balanced Accuracy 0.526316 0.5907 0.61511# ROC for the three classes
roc_poor <- roc(wine_test$quality == "Poor", probs[, 1])## Setting levels: control = FALSE, case = TRUE## Setting direction: controls < casesroc_norm <- roc(wine_test$quality == "Normal", probs[, 2])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesroc_ex <- roc(wine_test$quality == "Excellent", probs[, 3])## Setting levels: control = FALSE, case = TRUE
## Setting direction: controls < casesauc(roc_poor); auc(roc_norm); auc(roc_ex)## Area under the curve: 0.7929## Area under the curve: 0.771## Area under the curve: 0.8552plot.roc(roc_norm, col="blue", main="One-vs-All ROC Curves")
lines.roc(roc_poor, col="red")
lines.roc(roc_ex, col="green")
legend("bottomright", legend=c("Poor","Normal","Excellent"),
col=c("red","blue","green"), lwd=2)
# Confusion Matrix Graphic
# Draws x N matrix of category indices (1..K)
pp <- posterior_predict(fit_hs, newdata = wine_test)
# N x K matrix of posterior mean probs
probs <- t(apply(pp, 2, function(v) prop.table(tabulate(v, nbins = length(labs)))))
# Hard labels via argmax
pred_lab <- factor(labs[max.col(probs, ties.method = "first")], levels = labs, ordered = TRUE)
# Confusion + accuracy
table(Truth = wine_test$quality, Pred = pred_lab)## Pred
## Truth Poor Normal Excellent
## Poor 1 18 0
## Normal 0 382 14
## Excellent 0 48 17pred <- factor(pred_lab, levels = levels(wine_test$quality))
ref <- factor(wine_test$quality, levels = levels(wine_test$quality))
cm<- confusionMatrix(pred, ref)
cm_table <- as.table(cm$table)
cm_df <- melt(cm_table)
colnames(cm_df) <- c("True", "Predicted", "Count")
ggplot(cm_df, aes(x = Predicted, y = True, fill = Count)) +
geom_tile(color = "white") +
geom_text(aes(label = Count), color = "black", size = 4) +
scale_fill_viridis_c() +
theme_minimal() +
labs(title = "Confusion Matrix — Horse Shoe Bayesian Ordinal Regression",
x = "Predicted Class", y = "True Class")