library(ISLR2)
library(psych)
library(tidyverse)
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library(vip)
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## vi
library(car)
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## logit
# Q1
data(mtcars)
# (a) Descriptive statistics
summary(mtcars)
## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000
# Q1
data(mtcars)
# (a) Descriptive statistics
summary(mtcars)
## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000
library(ggplot2)
ggplot(data = mtcars, aes(x = mpg, y = disp)) +
geom_point() +
geom_smooth(method = "lm")
## `geom_smooth()` using formula = 'y ~ x'
# (c) Multiple regression on mpg across all predictors
model <- lm(mpg ~ ., data = mtcars)
summary(model)
##
## Call:
## lm(formula = mpg ~ ., data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.4506 -1.6044 -0.1196 1.2193 4.6271
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.30337 18.71788 0.657 0.5181
## cyl -0.11144 1.04502 -0.107 0.9161
## disp 0.01334 0.01786 0.747 0.4635
## hp -0.02148 0.02177 -0.987 0.3350
## drat 0.78711 1.63537 0.481 0.6353
## wt -3.71530 1.89441 -1.961 0.0633 .
## qsec 0.82104 0.73084 1.123 0.2739
## vs 0.31776 2.10451 0.151 0.8814
## am 2.52023 2.05665 1.225 0.2340
## gear 0.65541 1.49326 0.439 0.6652
## carb -0.19942 0.82875 -0.241 0.8122
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.65 on 21 degrees of freedom
## Multiple R-squared: 0.869, Adjusted R-squared: 0.8066
## F-statistic: 13.93 on 10 and 21 DF, p-value: 3.793e-07
# (d) Checking for multicollinearity
vif_values <- car::vif(model)
high_vif_predictors <- names(vif_values[vif_values > 10])
high_vif_predictors
## [1] "cyl" "disp" "wt"
# (e) Rerun regression after excluding disp, and disp and cyl
model_excl_disp <- lm(mpg ~ . - disp, data = mtcars)
summary(model_excl_disp)
##
## Call:
## lm(formula = mpg ~ . - disp, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.7863 -1.4055 -0.2635 1.2029 4.4753
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.55052 18.52585 0.677 0.5052
## cyl 0.09627 0.99715 0.097 0.9240
## hp -0.01295 0.01834 -0.706 0.4876
## drat 0.92864 1.60794 0.578 0.5694
## wt -2.62694 1.19800 -2.193 0.0392 *
## qsec 0.66523 0.69335 0.959 0.3478
## vs 0.16035 2.07277 0.077 0.9390
## am 2.47882 2.03513 1.218 0.2361
## gear 0.74300 1.47360 0.504 0.6191
## carb -0.61686 0.60566 -1.018 0.3195
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.623 on 22 degrees of freedom
## Multiple R-squared: 0.8655, Adjusted R-squared: 0.8105
## F-statistic: 15.73 on 9 and 22 DF, p-value: 1.183e-07
model_excl_disp_cyl <- lm(mpg ~ . - disp - cyl, data = mtcars)
summary(model_excl_disp_cyl)
##
## Call:
## lm(formula = mpg ~ . - disp - cyl, data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.8187 -1.3903 -0.3045 1.2269 4.5183
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.80810 12.88582 1.072 0.2950
## hp -0.01225 0.01649 -0.743 0.4650
## drat 0.88894 1.52061 0.585 0.5645
## wt -2.60968 1.15878 -2.252 0.0342 *
## qsec 0.63983 0.62752 1.020 0.3185
## vs 0.08786 1.88992 0.046 0.9633
## am 2.42418 1.91227 1.268 0.2176
## gear 0.69390 1.35294 0.513 0.6129
## carb -0.61286 0.59109 -1.037 0.3106
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.566 on 23 degrees of freedom
## Multiple R-squared: 0.8655, Adjusted R-squared: 0.8187
## F-statistic: 18.5 on 8 and 23 DF, p-value: 2.627e-08
# Q2
data(Carseats)
# (a) Fit multiple regression model
model_carseats <- lm(Sales ~ Price + Urban + US, data = Carseats)
summary(model_carseats)
##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16
# (b) Interpretation of coefficients
# - Price: For a one-unit increase in Price, Sales decrease by the coefficient value.
# - Urban: If the store is in an urban location, Sales are expected to increase by the coefficient value.
# - US: If the store is in the US, Sales are expected to increase by the coefficient value.
summary(model)$coefficients
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 12.30337416 18.71788443 0.6573058 0.51812440
## cyl -0.11144048 1.04502336 -0.1066392 0.91608738
## disp 0.01333524 0.01785750 0.7467585 0.46348865
## hp -0.02148212 0.02176858 -0.9868407 0.33495531
## drat 0.78711097 1.63537307 0.4813036 0.63527790
## wt -3.71530393 1.89441430 -1.9611887 0.06325215
## qsec 0.82104075 0.73084480 1.1234133 0.27394127
## vs 0.31776281 2.10450861 0.1509915 0.88142347
## am 2.52022689 2.05665055 1.2254035 0.23398971
## gear 0.65541302 1.49325996 0.4389142 0.66520643
## carb -0.19941925 0.82875250 -0.2406258 0.81217871
# (c) Model equation
# Sales = β0 + β1*Price + β2*Urban + β3*US
# (d) Test for significance of predictors
summary(model_carseats)
##
## Call:
## lm(formula = Sales ~ Price + Urban + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9206 -1.6220 -0.0564 1.5786 7.0581
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.043469 0.651012 20.036 < 2e-16 ***
## Price -0.054459 0.005242 -10.389 < 2e-16 ***
## UrbanYes -0.021916 0.271650 -0.081 0.936
## USYes 1.200573 0.259042 4.635 4.86e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.472 on 396 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2335
## F-statistic: 41.52 on 3 and 396 DF, p-value: < 2.2e-16
summary(model)$coefficients[, "Pr(>|t|)"] < 0.05
## (Intercept) cyl disp hp drat wt
## FALSE FALSE FALSE FALSE FALSE FALSE
## qsec vs am gear carb
## FALSE FALSE FALSE FALSE FALSE
# We can't reject the null hypothesis.
# (e) Fit smaller model with significant predictors
model_carseats_smaller <- lm(Sales ~ Price + US, data = Carseats)
summary(model_carseats_smaller)
##
## Call:
## lm(formula = Sales ~ Price + US, data = Carseats)
##
## Residuals:
## Min 1Q Median 3Q Max
## -6.9269 -1.6286 -0.0574 1.5766 7.0515
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 13.03079 0.63098 20.652 < 2e-16 ***
## Price -0.05448 0.00523 -10.416 < 2e-16 ***
## USYes 1.19964 0.25846 4.641 4.71e-06 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2.469 on 397 degrees of freedom
## Multiple R-squared: 0.2393, Adjusted R-squared: 0.2354
## F-statistic: 62.43 on 2 and 397 DF, p-value: < 2.2e-16
smaller_model <- lm(Sales ~ Price + US, data = Carseats)
# (f) Model fit comparison
# Compare R-squared values from both models
summary(model)$adj.r.squared
## [1] 0.8066423
summary(smaller_model)$adj.r.squared
## [1] 0.2354305
# (g) Confidence intervals
confint(model_carseats_smaller)
## 2.5 % 97.5 %
## (Intercept) 11.79032020 14.27126531
## Price -0.06475984 -0.04419543
## USYes 0.69151957 1.70776632
# (h) Outliers and leverage
# Use diagnostic plots to identify outliers and leverage points
plot(model_carseats_smaller, which = c(4, 6))
plot(smaller_model)
#Q3
library(quantmod)
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library(TTR)
library(readr)
library(caret)
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# Download S&P 500 data from Yahoo Finance
getSymbols("^GSPC", src = "yahoo", from = as.Date('2019-01-01'), to = as.Date('2023-12-31'))
## [1] "GSPC"
head(GSPC)
## GSPC.Open GSPC.High GSPC.Low GSPC.Close GSPC.Volume GSPC.Adjusted
## 2019-01-02 2476.96 2519.49 2467.47 2510.03 3733160000 2510.03
## 2019-01-03 2491.92 2493.14 2443.96 2447.89 3858830000 2447.89
## 2019-01-04 2474.33 2538.07 2474.33 2531.94 4234140000 2531.94
## 2019-01-07 2535.61 2566.16 2524.56 2549.69 4133120000 2549.69
## 2019-01-08 2568.11 2579.82 2547.56 2574.41 4120060000 2574.41
## 2019-01-09 2580.00 2595.32 2568.89 2584.96 4088740000 2584.96
d_ex1 <- as.data.frame(GSPC)
d_ex1 <- na.omit(d_ex1)
d_ex1$Direction <- NA # Create an empty Direction column first
for (i in 2:nrow(d_ex1)) {
if (Cl(d_ex1)[i] > Cl(d_ex1)[i - 1]) {
d_ex1$Direction[i] <- "Up"
} else {
d_ex1$Direction[i] <- "Down"
}
}
d_ex1 <- na.omit(d_ex1)
idx1 <- c(1:round(nrow(d_ex1) * 0.7))
d_train1 <- d_ex1[idx1, ]
d_test1 <- d_ex1[-idx1, ]
set.seed(999)
cntrl1 <- trainControl(method = "timeslice", initialWindow = 250, horizon = 30,
fixedWindow = TRUE)
prep1 <- c("center", "scale")
logit_ex1 <- train(Direction ~ ., data = d_train1, method = "glm", family = "binomial",
trControl = cntrl1, preProcess = prep1)
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## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred
logit_ex1
## Generalized Linear Model
##
## 880 samples
## 6 predictor
## 2 classes: 'Down', 'Up'
##
## Pre-processing: centered (6), scaled (6)
## Resampling: Rolling Forecasting Origin Resampling (30 held-out with a fixed window)
## Summary of sample sizes: 250, 250, 250, 250, 250, 250, ...
## Resampling results:
##
## Accuracy Kappa
## 0.8719911 0.7369671
summary(logit_ex1$finalModel)
##
## Call:
## NULL
##
## Coefficients: (1 not defined because of singularities)
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.02660 0.15546 0.171 0.864
## GSPC.Open -114.97118 10.21369 -11.257 < 2e-16 ***
## GSPC.High 9.14175 8.56720 1.067 0.286
## GSPC.Low 27.71746 6.54280 4.236 2.27e-05 ***
## GSPC.Close 77.95260 8.23097 9.471 < 2e-16 ***
## GSPC.Volume 0.08424 0.16609 0.507 0.612
## GSPC.Adjusted NA NA NA NA
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 1208.09 on 879 degrees of freedom
## Residual deviance: 538.62 on 874 degrees of freedom
## AIC: 550.62
##
## Number of Fisher Scoring iterations: 7
vip(logit_ex1, geom = "point") + theme_minimal()
