CAP 5610 Extra Credit  Danilo Martinez

Bonus credits (5 bonus points added to final score, submit in a separate report) . apply PCA (method A) to classify digit images. . Plot the obtained eigenvectors obtained by PCA (method A + method B)

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#Loading necessary Libraries
library(class)
library(caret)
library(mnist)
library(doParallel)
# Setting up parallel processing.
cl <-makeCluster(detectCores())
registerDoParallel(cl)
#Fetching the data set from the minIST website
mnist <- download_mnist()
#Extracting the the first 60k observations
inTrain =head(mnist, 60000)
#Removing Unnecessary files
rm(mnist)
#Seperating labels in order to prepare for pca analysis
responseY <- as.factor(inTrain[,dim(inTrain)[2]])
predictorX <- as.matrix(inTrain[,1:(dim(inTrain)[2]-1)])
#Performing Principal Component Analysis
pca <- princomp(predictorX)
#Plotting Eigenvalues for all digits
plot(pca,type = "l",main = "Total Digits Eigenvectors")

#Dividing the data by digit and performing pca
for (i in 0:9){
  data <- inTrain[ which(inTrain$Label==i),]
  predictorX <- as.matrix(data[,1:(dim(data)[2]-1)])
#Performing Principal Component Analysis
  pca <- princomp(predictorX)
#Plotting Eigenvalues by specific digit
cat("Plotting Eigenvectors Digit = ", i, "\n", sep = "")
plot(pca,type = "l",main = "Eigenvectors" )
}

## Plotting Eigenvectors Digit = 0

## Plotting Eigenvectors Digit = 1

## Plotting Eigenvectors Digit = 2

## Plotting Eigenvectors Digit = 3

## Plotting Eigenvectors Digit = 4

## Plotting Eigenvectors Digit = 5

## Plotting Eigenvectors Digit = 6

## Plotting Eigenvectors Digit = 7

## Plotting Eigenvectors Digit = 8

## Plotting Eigenvectors Digit = 9

#Dividing data with training having 50k and testing having 10k.
trainIndex = createDataPartition(inTrain$Label, p=0.833245, list=FALSE)
training = inTrain[trainIndex,]
testing = inTrain[-trainIndex,]
#Removing Unnecessary files
rm(inTrain)
#Setting up folds
n=10
#Establishing vector for results
results=rep(0.00,n)
min=1
#Creating for loops to iterate through folds
for(i in 1:n)
  {
  count=nrow(training)
  filter=rep(TRUE,count)
  max=count/n
  up=max+min-1
  res=list()
    for(j in min:up){
      filter[j]=FALSE
    }
#Seperating the training databy label
      for(k in 0:9){
      X=training[filter & training$Label==k,-ncol(training)]
      pca=princomp(X)
      testing=training[!filter,-ncol(training)]
      changeddata=as.matrix(testing)%*%pca$loadings[,1:154]
      rebuilt=(changeddata)%*%t(pca$loadings[,1:154])
      error=(rebuilt-testing)^2
      error=rowSums(error)
      res[[k+1]]=error
      }
#Setting up testing label
    testingy = training[!filter,ncol(training)]
#Initializing accurate predictions to zero
    accurates=0
#Checking accuracy in testing data
    for(l in 1:length(testingy)){
      squarederror=0
      prediction=-1
      for(m in 1:10){
        if(res[[m]][l]<squarederror|squarederror==0){
          prediction=m-1
          squarederror=res[[m]][l]
        }
      }
#Checking if prediciton is accurate
      if(prediction==testingy[i])
        {
        accurates=accurates+1
        }
    }
#Storing results for digit
    results[i]=accurates/sum(!filter)
    min=min+max
}
#Printing and plotting results
print("Average Error Rate")

## [1] "Average Error Rate"

mean(results)

## [1] 0.09644

print("Average Standard Deviation")

## [1] "Average Standard Deviation"

sd(results)

## [1] 0.01397579

print("Displaying the Error Rates per Digit")

## [1] "Displaying the Error Rates per Digit"

plot(results,type = "l",main = "Error Rates per Digit",
xlab = "Digit" )

The eigenvector plots show that the variance held by the principal components is very similar for all digits in the MNIST. They decline similarly and most of the variability is contained by the first 10 eigenvectors.