Part 1 PCA

Material adapted from UC Busisness Analytics R Programming Guide.

Part 1. Concepts Explanation

Data

USArrests data set is built into R.
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

Each variable should be centered at zero for PCA. Notice that variables are not scaled (each is measured at a different scale)
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.
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 a 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 component will be a linear combination of variables that present the largest variance.

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

# Calculate eigenvalues and eigenvectors
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 $:

# Extract the loadings
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 calculate PC for each state.

# Calculate Principal Components scores
PC1 <- as.matrix(scaled_df) %*% phi[,1]
PC2 <- as.matrix(scaled_df) %*% phi[,2]

# Create data frame with Principal Components scores
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

Plotting

We should also plot Proportion of Variance Explained (PVE) which would give us insights on how many components 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.

Part 2. PCA Built-In Analysis

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 # change to positive direction; x is returned values

For visualization we will use a bibplot. Notice PC1 (x-axis) is represented by three vectors (serious crimes). The further from zero to the right - more positive directions. PC2 (y-axis) is mainly represented by UrbanPop (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 Discriminant Analysis

Adapted from SDS 293 - Machine Learning Jordan Crouser (2016) and “Introduction to Statistical Learning with Applications in R” by Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani.

4.6.3 Linear Discriminant Analysis

Now we will perform LDA on the Smarket data. In R, we can fit a LDA model using the lda() function, which is part of the MASS library. Note: dplyr and MASS have a name clash around the word select(), so we need to do a little magic to make them play nicely.

pacman::p_load(MASS,dplyr,ISLR)
select <- dplyr::select

The syntax for the lda() function is identical to that of lm(), and to that of glm() except for the absence of the family option. As we did with logistic regression and KNN, we’ll fit the model using only the observations before 2005, and then test the model on the data from 2005.

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:

# Logistic model, for comparison
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)

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

4.6.4 Quadratic Discriminant Analysis

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!

An Application to Carseats Data

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!

#convert to factor
Carseats$ShelveLoc <- factor(Carseats$ShelveLoc)
pacman::p_load(caret)

# Split into train and test data
t.idx = createDataPartition(Carseats$ShelveLoc,p=0.7,list=FALSE)
cseat_train = Carseats[t.idx,]
cseat_test = Carseats[-t.idx,]
model_LDA=lda(ShelveLoc~Sales+CompPrice+Income+Advertising+Population+Price+Age+Education+Urban+US,data=cseat_train)
print(model_LDA)
## Call:
## lda(ShelveLoc ~ Sales + CompPrice + Income + Advertising + Population + 
##     Price + Age + Education + Urban + US, data = cseat_train)
## 
## Prior probabilities of groups:
##       Bad      Good    Medium 
## 0.2411348 0.2127660 0.5460993 
## 
## Group means:
##            Sales CompPrice   Income Advertising Population    Price      Age
## Bad     5.606471  123.8529 74.57353    6.514706   279.1765 115.6029 51.10294
## Good   10.332500  126.1167 70.75000    6.583333   270.8167 116.3000 51.96667
## Medium  7.263247  124.3636 67.27922    6.883117   256.0779 115.2338 55.12338
##        Education  UrbanYes     USYes
## Bad     13.77941 0.7794118 0.6617647
## Good    13.70000 0.6666667 0.6833333
## Medium  13.87662 0.7012987 0.6363636
## 
## Coefficients of linear discriminants:
##                       LD1          LD2
## Sales        1.0047593184 -0.020508094
## CompPrice   -0.0900719250 -0.003806911
## Income      -0.0151494835 -0.018545360
## Advertising -0.1177273168  0.098632522
## Population  -0.0003347859 -0.003323486
## Price        0.0941952405 -0.003812730
## Age          0.0468812991  0.033053262
## Education    0.0287507551 -0.016147239
## UrbanYes    -0.2839039507 -0.516285602
## USYes        0.1881758685 -1.052780870
## 
## Proportion of trace:
##    LD1    LD2 
## 0.9859 0.0141
plot(model_LDA)

predictions_LDA = data.frame(predict(model_LDA, cseat_test))
names(predictions_LDA)
## [1] "class"            "posterior.Bad"    "posterior.Good"   "posterior.Medium"
## [5] "x.LD1"            "x.LD2"
predictions_LDA = cbind(cseat_test, predictions_LDA)

predictions_LDA %>%
  count(class, ShelveLoc)
##    class ShelveLoc  n
## 1    Bad       Bad 23
## 2    Bad    Medium  5
## 3   Good      Good 21
## 4   Good    Medium  1
## 5 Medium       Bad  5
## 6 Medium      Good  4
## 7 Medium    Medium 59
predictions_LDA %>%
  summarize(score = mean(class == ShelveLoc))
##       score
## 1 0.8728814