options(repos = "https://cran.rstudio.com/")

Part 1 - PCA

library(tidyverse)  
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## ✔ purrr     1.0.2     
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library(gridExtra) 
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install.packages("devtools")
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library(devtools)
## Loading required package: usethis
library(kableExtra)
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## Attaching package: 'kableExtra'
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## The following object is masked from 'package:dplyr':
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##     group_rows
data("USArrests")
dt = head(USArrests, 10)
kbl(dt) %>%
   kable_minimal()
Murder Assault UrbanPop Rape
Alabama 13.2 236 58 21.2
Alaska 10.0 263 48 44.5
Arizona 8.1 294 80 31.0
Arkansas 8.8 190 50 19.5
California 9.0 276 91 40.6
Colorado 7.9 204 78 38.7
Connecticut 3.3 110 77 11.1
Delaware 5.9 238 72 15.8
Florida 15.4 335 80 31.9
Georgia 17.4 211 60 25.8

Normalization

dt = apply(USArrests, 2, var)
kbl(dt) %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
x
Murder 18.97047
Assault 6945.16571
UrbanPop 209.51878
Rape 87.72916

We need to create a scaled data for PCA. Note: PCA is influenced by the magnitude of each variable and the results will also depend on whether the variables have been scaled.

scaled_df <- apply(USArrests, 2, scale)
dt = head(scaled_df)
kbl(dt)%>%
  kable_styling(bootstrap_options = c("striped", "hover"))
Murder Assault UrbanPop Rape
1.2425641 0.7828393 -0.5209066 -0.0034165
0.5078625 1.1068225 -1.2117642 2.4842029
0.0716334 1.4788032 0.9989801 1.0428784
0.2323494 0.2308680 -1.0735927 -0.1849166
0.2782682 1.2628144 1.7589234 2.0678203
0.0257146 0.3988593 0.8608085 1.8649672

Principal Components

We need to generate Covariance matrix. The idea is that not all data dimensions are interesting and PCA will combine only interesting variables in a linear combination (=component). For example, the first principal componnet will be a linear combination of variable that present the largest variance.

Eigenvalues are calculated from Cov matrix and eigenvectors (a set of loadings) explain the proportion of the variability.

arrests.cov <- cov(scaled_df)
arrests.eigen <- eigen(arrests.cov)
str(arrests.eigen)
## List of 2
##  $ values : num [1:4] 2.48 0.99 0.357 0.173
##  $ vectors: num [1:4, 1:4] 0.536 0.583 0.278 0.543 0.418 ...
##  - attr(*, "class")= chr "eigen"

What are the first two sets of loadings (Principal component PC1 and Principal component PC2?). We can extract them from eigenvector - we will have two vectors. To access vectors from our arrest.eigens we need to use $:

phi <- arrests.eigen$vectors[,1:2]
print(phi)
##           [,1]       [,2]
## [1,] 0.5358995  0.4181809
## [2,] 0.5831836  0.1879856
## [3,] 0.2781909 -0.8728062
## [4,] 0.5434321 -0.1673186

By default, eigenvectors in R point into the negative direction. Positive direction leads to more logical interpretation of graphical results. To use the positive-pointing vector, we multiply the default loadings by -1. The set of loadings for the first principal component (PC1) and second principal component (PC2) are shown below:

phi <- -1*phi
row.names(phi) <- c("Murder", "Assault", "UrbanPop", "Rape")
colnames(phi) <- c("PC1", "PC2")
kbl(phi) %>%
  kable_styling(bootstrap_options = c("striped", "hover"))
PC1 PC2
Murder -0.5358995 -0.4181809
Assault -0.5831836 -0.1879856
UrbanPop -0.2781909 0.8728062
Rape -0.5434321 0.1673186

Each principal component vector defines a direction in feature space. Because eigenvectors are orthogonal to every other eigenvector, principal components are uncorrelated with one another.

From the table, we can infer that PC1 corresponds to an overall rate of serious crimes: Murder, Assault, and Rape (largest values). PC2 is affected by UrbanPop more than the other three variables (the level of urbanization of the state).

We can calculaate PC for each state.

PC1 <- as.matrix(scaled_df) %*% phi[,1]
PC2 <- as.matrix(scaled_df) %*% phi[,2]

PC <- data.frame(State = row.names(USArrests), PC1, PC2)
head(PC)
##        State        PC1        PC2
## 1    Alabama -0.9756604 -1.1220012
## 2     Alaska -1.9305379 -1.0624269
## 3    Arizona -1.7454429  0.7384595
## 4   Arkansas  0.1399989 -1.1085423
## 5 California -2.4986128  1.5274267
## 6   Colorado -1.4993407  0.9776297
ggplot(PC, aes(PC1, PC2)) + 
  modelr::geom_ref_line(h = 0) +
  modelr::geom_ref_line(v = 0) +
  geom_text(aes(label = State), size = 3) +
  xlab("First Principal Component") + 
  ylab("Second Principal Component") + 
  ggtitle("First Two Principal Components of USArrests Data")

We should also plot Proportion of Variance Explained (PVE) which would give us insights on how many componnets to select. The rule of Thumb is components should explain at least 80%, and you can discard other components. Below, the first two component explain 87% of the variability: 62 + 25.

PVE <- arrests.eigen$values / sum(arrests.eigen$values)
round(PVE, 2)
## [1] 0.62 0.25 0.09 0.04

Cumulative PVE helps to visualize cumulative values to determine how many PCs to choose. These Viz are called scree plots.

PVEplot <- qplot(c(1:4), PVE) + 
  geom_line() + 
  xlab("Principal Component") + 
  ylab("PVE") +
  ggtitle("Scree Plot") +
  ylim(0, 1)
## Warning: `qplot()` was deprecated in ggplot2 3.4.0.
## This warning is displayed once every 8 hours.
## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
## generated.
cumPVE <- qplot(c(1:4), cumsum(PVE)) + 
  geom_line() + 
  xlab("Principal Component") + 
  ylab(NULL) + 
  ggtitle("Cumulative Scree Plot") +
  ylim(0,1)

grid.arrange(PVEplot, cumPVE, ncol = 2)

For the actual analysis, we will use prcomp function and scale our data by setting scale parameter as TRUE.

pca_result <- prcomp(USArrests, scale = TRUE)
pca_result$rotation <- -pca_result$rotation
pca_result$x <- - pca_result$x

For visualization we will use a bibplot. Notice PC1 (x-axis) is represented by three vectors (serious crimes). The further from zero to the righ - more positive directions. PC2 (y-axis) is mainly represented by UrbaanPop (positive direction), the rest of variables are clustered around zero (look where the arrows are in vertical space).

biplot(pca_result, scale = 0)

Part 2 - Discriminint Analysis

library(MASS)
## 
## Attaching package: 'MASS'
## The following object is masked from 'package:dplyr':
## 
##     select
library(dplyr)
library(ISLR)
select <- dplyr::select
train = Smarket %>%
  filter(Year < 2005)

test = Smarket %>%
  filter(Year >= 2005)

model_LDA=lda(Direction~Lag1+Lag2,data=train)
print(model_LDA)
## Call:
## lda(Direction ~ Lag1 + Lag2, data = train)
## 
## Prior probabilities of groups:
##     Down       Up 
## 0.491984 0.508016 
## 
## Group means:
##             Lag1        Lag2
## Down  0.04279022  0.03389409
## Up   -0.03954635 -0.03132544
## 
## Coefficients of linear discriminants:
##             LD1
## Lag1 -0.6420190
## Lag2 -0.5135293

The LDA output indicates prior probabilities of \({\hat{\pi}}_1 = 0.492\) and \({\hat{\pi}}_2 = 0.508\); in other words, 49.2% of the training observations correspond to days during which the market went down.

The function also provides the group means; these are the average of each predictor within each class, and are used by LDA as estimates of \(\mu_k\). These suggest that there is a tendency for the previous 2 days’ returns to be negative on days when the market increases, and a tendency for the previous days’ returns to be positive on days when the market declines.

The coefficients of linear discriminants output provides the linear combination of Lag1 and Lag2 that are used to form the LDA decision rule.

If \(−0.642\times{\tt Lag1}−0.514\times{\tt Lag2}\) is large, then the LDA classifier will predict a market increase, and if it is small, then the LDA classifier will predict a market decline.

We can use the plot() function to produce plots of the linear discriminants, obtained by computing \(−0.642\times{\tt Lag1}−0.514\times{\tt Lag2}\) for each of the training observations.

plot(model_LDA)

The predict() function returns a list with three elements. The first element, class, contains LDA’s predictions about the movement of the market. The second element, posterior, is a matrix whose \(k^{th}\) column contains the posterior probability that the corresponding observation belongs to the \(k^{th}\) class. Finally, x contains the linear discriminants, described earlier.

predictions_LDA = data.frame(predict(model_LDA, test))
names(predictions_LDA)
## [1] "class"          "posterior.Down" "posterior.Up"   "LD1"

Let’s check out the confusion matrix to see how this model is doing. We’ll want to compare the predicted class (which we can find in the class column of the predictions_LDA data frame) to the true class.

predictions_LDA = cbind(test, predictions_LDA)

predictions_LDA %>%
  count(class, Direction)
##   class Direction   n
## 1  Down      Down  35
## 2  Down        Up  35
## 3    Up      Down  76
## 4    Up        Up 106
predictions_LDA %>%
  summarize(score = mean(class == Direction))
##       score
## 1 0.5595238

The LDA predictions are identical to the ones from our logistic model:

model_logistic = glm(Direction~Lag1+Lag2, data=train ,family=binomial)

logistic_probs = data.frame(probs = predict(model_logistic, test, type="response"))

predictions_logistic = logistic_probs %>%
  mutate(class = ifelse(probs>.5, "Up", "Down"))

predictions_logistic = cbind(test, predictions_logistic)

accuracy_summary = predictions_logistic %>%
  summarize(score = mean(class == Direction))

print(accuracy_summary)
##       score
## 1 0.5595238

We will now fit a QDA model to the Smarket data. QDA is implemented in R using the qda() function, which is also part of the MASS library. The syntax is identical to that of lda().

model_QDA = qda(Direction~Lag1+Lag2, data=train)
model_QDA
## Call:
## qda(Direction ~ Lag1 + Lag2, data = train)
## 
## Prior probabilities of groups:
##     Down       Up 
## 0.491984 0.508016 
## 
## Group means:
##             Lag1        Lag2
## Down  0.04279022  0.03389409
## Up   -0.03954635 -0.03132544

The output contains the group means. But it does not contain the coefficients of the linear discriminants, because the QDA classifier involves a quadratic, rather than a linear, function of the predictors. The predict() function works in exactly the same fashion as for LDA.

predictions_QDA = data.frame(predict(model_QDA, test))

predictions_QDA = cbind(test, predictions_QDA)

predictions_QDA %>%
  count(class, Direction)
##   class Direction   n
## 1  Down      Down  30
## 2  Down        Up  20
## 3    Up      Down  81
## 4    Up        Up 121
predictions_QDA %>%
  summarize(score = mean(class == Direction))
##       score
## 1 0.5992063

Interestingly, the QDA predictions are accurate almost 60% of the time, even though the 2005 data was not used to fit the model. This level of accuracy is quite impressive for stock market data, which is known to be quite hard to model accurately.

This suggests that the quadratic form assumed by QDA may capture the true relationship more accurately than the linear forms assumed by LDA and logistic regression. However, we recommend evaluating this method’s performance on a larger test set before betting that this approach will consistently beat the market!

Let’s see how the LDA/QDA approach performs on the Carseats data set, which is part of the ISLR library.

Recall: this is a simulated data set containing sales of child car seats at 400 different stores.

summary(Carseats)
##      Sales          CompPrice       Income        Advertising    
##  Min.   : 0.000   Min.   : 77   Min.   : 21.00   Min.   : 0.000  
##  1st Qu.: 5.390   1st Qu.:115   1st Qu.: 42.75   1st Qu.: 0.000  
##  Median : 7.490   Median :125   Median : 69.00   Median : 5.000  
##  Mean   : 7.496   Mean   :125   Mean   : 68.66   Mean   : 6.635  
##  3rd Qu.: 9.320   3rd Qu.:135   3rd Qu.: 91.00   3rd Qu.:12.000  
##  Max.   :16.270   Max.   :175   Max.   :120.00   Max.   :29.000  
##    Population        Price        ShelveLoc        Age          Education   
##  Min.   : 10.0   Min.   : 24.0   Bad   : 96   Min.   :25.00   Min.   :10.0  
##  1st Qu.:139.0   1st Qu.:100.0   Good  : 85   1st Qu.:39.75   1st Qu.:12.0  
##  Median :272.0   Median :117.0   Medium:219   Median :54.50   Median :14.0  
##  Mean   :264.8   Mean   :115.8                Mean   :53.32   Mean   :13.9  
##  3rd Qu.:398.5   3rd Qu.:131.0                3rd Qu.:66.00   3rd Qu.:16.0  
##  Max.   :509.0   Max.   :191.0                Max.   :80.00   Max.   :18.0  
##  Urban       US     
##  No :118   No :142  
##  Yes:282   Yes:258  
##                     
##                     
##                     
##

See if you can build a model that predicts ShelveLoc, the shelf location (Bad, Good, or Medium) of the product at each store. Don’t forget to hold out some of the data for testing!

train_index <- sample(1:nrow(Carseats), 0.7 * nrow(Carseats))
train_data <- Carseats[train_index, ]
test_data <- Carseats[-train_index, ]

lda_model <- lda(ShelveLoc ~ ., data = train_data)

lda_predictions <- predict(lda_model, newdata = test_data)

confusion_matrix <- table(lda_predictions$class, test_data$ShelveLoc)
print(confusion_matrix)
##         
##          Bad Good Medium
##   Bad     23    0      2
##   Good     0   21      1
##   Medium   8    9     56
accuracy <- sum(diag(confusion_matrix)) / sum(confusion_matrix)
cat("Accuracy of LDA model:", accuracy, "\n")
## Accuracy of LDA model: 0.8333333