Outline

• cross validation to evaluate machine learning accuracy
  • holdout method
  • k-fold cross validation
  • leave one out cross validation
• cross validation to select tuning parameters
• Other machine learning approaches
  • K nearest neighbour (KNN)
  • Linear discriminant analysis (LDA)
  • Support vector machine (SVM)

Holdout method to evaluate machine learning accuracy

• Pros: easy to implement
• Cons: its evaluation may have a high variance.
  • depend heavily on which data points in the training set and which in the test set

Holdout method example

library(ElemStatLearn)
library(randomForest)

## randomForest 4.6-14

## Type rfNews() to see new features/changes/bug fixes.

prostate0 <- prostate
prostate0$svi <- as.factor(prostate0$svi)
trainLabel <- prostate$train
prostate0$train <- NULL

prostate_train <- subset(prostate0, trainLabel)
prostate_test <- subset(prostate0, !trainLabel)

rfFit <- randomForest(svi ~ ., data=prostate_train)
rfFit

## 
## Call:
##  randomForest(formula = svi ~ ., data = prostate_train) 
##                Type of random forest: classification
##                      Number of trees: 500
## No. of variables tried at each split: 2
## 
##         OOB estimate of  error rate: 14.93%
## Confusion matrix:
##    0 1 class.error
## 0 48 4  0.07692308
## 1  6 9  0.40000000

rfPred <- predict(rfFit, prostate_test)
rfTable <- table(rfPred, truth=prostate_test$svi)
rfTable

##       truth
## rfPred  0  1
##      0 22  4
##      1  2  2

1 - sum(diag(rfTable))/sum(rfTable) ## error rate

## [1] 0.2

k-fold cross validation

• Only split the data into two parts may result in high variance.
• Another commonly used approach is to split the data into \(K\) folds.
• Normally \(K = 5\) or \(K = 10\) are recommended to balance the bias and variance.

Algorithm:

• Split the original dataset \(D\) into \(K\) folds, and obtain \(D_1, \ldots, D_k,\ldots, D_K\) such that \(D_1, \ldots, D_K\) are disjoint; \(|D_k|\) (size of \(D_k\)) is roughly the same for different \(k\) and \(\sum_k |D_k| = |D|\).

• For each iteration \(k = 1, \ldots, K\), use the \(D_k\) as the testing dataset and \(D_{-k} = D - D_k\) as the training dataset. Build the classifier \(f^k\) based on \(D_{-k}\). Then predict \(D_k\) and obtain the prediction error rate \(E_k\)

• Use \(E = \frac{1}{K} \sum_k E_k\) as the final prediction error rate.

k-fold cross validation example

#Create K = 5 equally size folds

K <- 5
folds <- cut(seq(1,nrow(prostate0)),breaks=5,labels=FALSE)

EKs <- numeric(K)

for(k in 1:K){
  atrain <- subset(prostate0, folds != k)
  atest <- subset(prostate0, folds == k)

  krfFit <- randomForest(svi ~ ., data=atrain)
  krfPred <- predict(krfFit, atest)
  krfTable <- table(krfPred, truth=atest$svi)
  
  EKs[k] <- 1 - sum(diag(krfTable))/sum(krfTable)
}

EKs

## [1] 0.0000000 0.0000000 0.0000000 0.3157895 0.3500000

mean(EKs)

## [1] 0.1331579

Leave one out cross validation

For K-fold cross validation

• \(K = n\), leave one out cross validation
  • Each time, only use one sample as the testing sample and the rest of all sample as the training data.
  • Iterate total \(n\) times.

leave one out cross validation example

n <- nrow(prostate0)

EKs <- numeric(n)

for(k in 1:n){
  #cat("leaving the",k,"sample out!","\n")
  atrain <- prostate0[-k,]
  atest <- prostate0[k,]

  krfFit <- randomForest(svi ~ ., data=atrain)
  krfPred <- predict(krfFit, atest)
  krfTable <- table(krfPred, truth=atest$svi)
  
  EKs[k] <- 1 - sum(diag(krfTable))/sum(krfTable)
}

EKs

##  [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## [36] 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0
## [71] 1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0

mean(EKs)

## [1] 0.1443299

Important note for cross validation

• the classifier should only build on the training data.
• the classifier should not use the information of the testing data.

Choosing tuning parameters

Some machine learning classifiers contain tuning parameters

• CART: pruning criteria.
• K-nearest-neighbors (KNN): K (will be introduced shortly)
• Lasso: regularization penalty \(\lambda\).

The tuning parameter is crucial since it controls the amount of regularization (model complexity).

Tuning parameter selection

Lasso problem

\[\hat{\beta}^{lasso} = \frac{1}{2}\arg\min_{\beta \in \mathbb{R}^P} \|y - X\beta\|_2^2 + \lambda \|\beta\|_1\]

library(lars)

## Loaded lars 1.2

library(ggplot2)

## Registered S3 methods overwritten by 'ggplot2':
##   method         from 
##   [.quosures     rlang
##   c.quosures     rlang
##   print.quosures rlang

## 
## Attaching package: 'ggplot2'

## The following object is masked from 'package:randomForest':
## 
##     margin

options(stringsAsFactors=F)
x <- as.matrix(prostate[,1:8])
y <- prostate[,9]
lassoFit <- lars(x, y) ## lar for least angle regression

lambdas <- seq(0,5,0.5)
coefLassoFit <- coef(lassoFit, s=lambdas, mode="lambda") 
adataFrame <- NULL
for(i in 1:ncol(coefLassoFit)){
  acoef <- colnames(coefLassoFit)[i]
  adataFrame0 <- data.frame(lambda=lambdas, coef = coefLassoFit[,i], variable=acoef)
  adataFrame <- rbind(adataFrame, adataFrame0)
}

ggplot(adataFrame) + aes(x=lambda, y=coef, col=variable) + 
  geom_point() +
  geom_path()

How to select the best tuning parameter

• Use cross validation.
• We should choose the tuning parameter such that the cross validation error is minimized.

Tuning parameter selection algorithm

• Split the original dataset \(D\) into \(K\) folds, and obtain \(D_1, \ldots, D_k,\ldots, D_K\) such that \(D_1, \ldots, D_K\) are disjoint; \(|D_k|\) (size of \(D_k\)) is roughly the same for different \(k\) and \(\sum_k |D_k| = |D|\).

• pre-specify a range of tuning parameters \(\lambda_1, \ldots, \lambda_B\)

• For each iteration \(k = 1, \ldots, K\), use the \(D_k\) as the testing dataset and \(D_{-k} = D - D_k\) as the training dataset. Build the classifier \(f^k_{\lambda_b}\) based on \(D_{-k}\) and tuning parameter \(\lambda_b\), \((1 \le b \le B)\). Then predict \(D_k\) and obtain the
  • mean squared error (MSE) or root mean squared error (rMSE) \(e_k(\lambda_b)\) for continuous outcome.
  • classification error rate for categorical outcome variable.

• Use \(e(\lambda_b) = \frac{1}{K} \sum_k e_k(\lambda_b)\) as the average (MSE, rMSE or classification error rate) for tuning parameter \((\lambda_b)\).

• Choose the tuning parameter \(\hat{\lambda} = \arg \min_{\lambda\in (\lambda_1, \ldots, \lambda_B)} e(\lambda)\)

Choose the tuning parameter \(\hat{\lambda} = \arg \min_{\lambda\in (\lambda_1, \ldots, \lambda_B)} e(\lambda)\)

• Continuous outcome MSE: \[e(\lambda) = \frac{1}{K} \sum_{k=1}^K \sum_{i \in D_k} (y_i - \hat{f}_\lambda^{-k}(x_i))^2\]

• Continuous outcome rMSE: \[e(\lambda) = \frac{1}{K} \sum_{k=1}^K \sqrt{\sum_{i \in D_k} (y_i - \hat{f}_\lambda^{-k}(x_i))^2}\]

• Categorical outcome variable classification error rate: \[e(\lambda) = \frac{1}{K} \sum_{k=1}^K \frac{1}{|D_k|}\sum_{i \in D_k} \mathbb{I}(y_i \ne \hat{f}_\lambda^{-k}(x_i))\]

Implement tuning parameter selection for lasso problem

K <- 10
folds <- cut(seq(1,nrow(prostate)),breaks=K,labels=FALSE)
lambdas <- seq(0.5,5,0.5)

rMSEs <- matrix(NA,nrow=length(lambdas),ncol=K)
rownames(rMSEs) <- lambdas
colnames(rMSEs) <- 1:K

for(k in 1:K){
  atrainx <- as.matrix(prostate[!folds==k,1:8])
  atestx <- as.matrix(prostate[folds==k,1:8])
 
  atrainy <- prostate[!folds==k,9]
  atesty <- prostate[folds==k,9]

  klassoFit <- lars(atrainx, atrainy) ## lar for least angle regression

  predictValue <- predict.lars(klassoFit, atestx, s=lambdas, type="fit",  mode="lambda")
  rMSE <- apply(predictValue$fit,2, function(x) sqrt(sum((x - atesty)^2))) ## square root MSE
  
  rMSEs[,k] <- rMSE
}

rMSEdataFrame <- data.frame(lambda=lambdas, rMSE = rowMeans(rMSEs))
ggplot(rMSEdataFrame) + aes(x=lambda, y = rMSE) + geom_point() + geom_path()

Common mistake about tuning parameter selection in machine learning

Combine selecting tuning parameters and evaluating prediction accuracy

1. Split the data into training set and testing set.
2. In training set, use cross validation to determine the best tuning parameter.
3. Use the best tuning parameter and the entire training set to build a classifier.
4. Evaluation:
   • Predict the testing set
   • Report the accuracy.

For Step 1, can we also use cross validation?

Nested Cross validation

1. Use K-folder cross validation (outer) to split the original data into training set and testing set.
2. For \(k^{th}\) fold training set, use cross validation (inner) to determine the best tuning parameter of the \(k^{th}\) fold.
3. Use the best tuning parameter of the \(k^{th}\) fold and the \(k^{th}\) fold training set to build a classifier.
4. Predict the \(k^{th}\) fold testing set and report the accuracy.
5. Evaluation:
   • Report the average accuracy from outer K-folder cross validation..

Other machine learning approaches

• K nearest neighbour (KNN)
• Linear discriminant analysis (LDA)
• Support vector machine (SVM)

KNN motivating example

library(MASS)
set.seed(32611)
N<-100
d1<-mvrnorm(N, c(1,-1), matrix(c(2, 1, 1, 2), 2, 2))
d2<-mvrnorm(N, c(-1,1), matrix(c(2, 1, 1, 2), 2, 2))
atest <- rbind(c(1,1),c(2,-2),c(-2,2))
d <- rbind(d1, d2,atest)
label <- c(rep("A", N), rep("B", N),rep("test",3))
text <- label
text[label!="test"] <- ""
data_train <- data.frame(label=label, x=d[,1], y=d[,2], text=text)
names(data_train)<-c("label", "x", "y", "text")

ggplot(data_train, aes(x=x,y=y,col=label, label = text)) + 
  geom_point() + geom_text(vjust = -1,size=5)

KNN algorithm

• pre-specify \(K\) (e.g., \(K = 5\)).
• for each testing sample, find the top \(K\) training samples that are closest to the testing sample.
• The label of the testing sample will depend on the majority vote of the training labels for categorical outcome.

\[\hat{f}(x) = mode_{i \in N_k(x)} y_i\]

• KNN regression

\[\hat{f}(x) = \frac{1}{K} \sum_{i \in N_k(x)} y_i\]

• KNN missing value imputation

KNN example

library(ElemStatLearn)
library(class)

prostate0 <- prostate
prostate0$svi <- as.factor(prostate0$svi)
trainLabel <- prostate$train

## here 5 is the index of svi in the prostate data
adata_train <- prostate0[trainLabel, -5]
adata_test <- prostate0[!trainLabel, -5]

alabel_train <- prostate0[trainLabel, 5]
alabel_test <- prostate0[!trainLabel, 5]

knnFit <- knn(adata_train, adata_test, alabel_train, k=5)

knnTable <- table(knnFit, truth=alabel_test)
knnTable

##       truth
## knnFit  0  1
##      0 22  5
##      1  2  1

1 - sum(diag(knnTable))/sum(knnTable)

## [1] 0.2333333

KNN Exercise (Will be in HW)

• use cross validation to determine the optimum \(K\) for KNN (with prostate cancer data).

Linear discriminant analysis (LDA)

Problem setting:

• \(X \in \mathbb{R}^p\) is a sample with \(p\) features.
• \(Y\) is the class label, with possible value \(1\) and \(2\).
• Suppose we have training dataset:
  • \(X_{train} \in \mathbb{R}^{n \times p}\)
  • \(Y_{train} \in \{1,2\}^n\)
• Goal:
  • build a classifier
  • predict testing data \(X_{test}\)
• Gaussian Assumption:

\[f(X|Y=k) = \frac{1}{(2\pi)^{p/2} |\Sigma_k|^{1/2}} \exp ( - \frac{1}{2} (X - \mu_k)^\top \Sigma_k^{-1} (X - \mu_k))\]

Intuition for Linear discriminant analysis (LDA)

library(MASS)
library(ggplot2)

set.seed(32611)
N<-100
d1<-mvrnorm(N, c(1,-1), matrix(c(2, 1, 1, 2), 2, 2))
d2<-mvrnorm(N, c(-1,1), matrix(c(2, 1, 1, 2), 2, 2))
d <- rbind(d1, d2)
label <- c(rep("1", N), rep("2", N))
data_train <- data.frame(label=label, x1=d[,1], x2=d[,2])
names(data_train)<-c("label", "x1", "x2")

ggplot(data_train) + aes(x=x1,y=x2,col=label) + geom_point() +  stat_ellipse()

• Intuition:
  • We estimate the Gaussian density function for each class, and for a future testing sample, the classifier is “select the class with larger density function value”.

Math behind Linear discriminant analysis (LDA)

• Gaussian density: \[f(X|Y=k) = \frac{1}{(2\pi)^{p/2} |\Sigma_k|^{1/2}} \exp ( - \frac{1}{2} (X - \mu_k)^\top \Sigma_k^{-1} (X - \mu_k))\]

• The Bayes rule:

\(\begin{aligned} C_{Bayes}(X) & = \arg \max_k f(Y=k | X) \\ & = \arg \max_k \frac{f(Y=k)f(X|Y=k)}{f(X)} \\ & = \arg \max_k f(Y=k)f(X|Y=k) \\ & = \arg \max_k \log f(Y=k) + \log f(X|Y=k) \\ & = \arg \max_k \log f(Y=k) - \frac{1}{2} \log |\Sigma_k| - \frac{1}{2} (X - \mu_k)^\top \Sigma_k^{-1} (X - \mu_k) \\ & = \arg \min_k - 2 \log f(Y=k) + \log |\Sigma_k| + (X - \mu_k)^\top \Sigma_k^{-1} (X - \mu_k) \end{aligned}\)

Decision boundary for Linear discriminant analysis (LDA)

The decision boundary for class \(m\) and class \(l\) is: \[f(Y=l|X) = f(Y=m|X)\]

Or equivalently,

\(\begin{aligned} & - 2 \log f(Y=l) + \log |\Sigma_l| + (X - \mu_l)^\top \Sigma_l^{-1} (X - \mu_l) \\ & = - 2 \log f(Y=m) + \log |\Sigma_m| + (X - \mu_m)^\top \Sigma_m^{-1} (X - \mu_m) \end{aligned}\)

Further assume uniform prior \(f(Y=l) = f(Y=m)\), and same covariance structure \(\Sigma_l = \Sigma_m = \Sigma\), the decision hyperlane simplifies as:

\[2 (\mu_l - \mu_m)^\top \Sigma^{-1} X - (\mu_l - \mu_m)^\top \Sigma^{-1} (\mu_l + \mu_m) = 0\]

\[2 (\mu_l - \mu_m)^\top \Sigma^{-1} (X - \frac{\mu_l + \mu_m}{2}) = 0\]

You can see the decision boundary is linear and passes the center between \(\frac{\mu_l + \mu_m}{2}\).

Visualize the decision boundary (LDA)

library(MASS)

alda <- lda(label ~ . ,data=data_train)

avec <- seq(-4,4,0.1)
z <- expand.grid(avec, avec)
z <- data.frame(x1=z[,1],x2=z[,2])
predict_lda <- predict(alda, z)$class

z_lda <- cbind(region=as.factor(predict_lda), z)
ggplot(z_lda) + aes(x=x1,y=x2,col=region) + geom_point(alpha = 0.4)  + geom_point(data = data_train, aes(x=x1,y=x2,col=label)) +
  labs(title = "LDA boundary")

Quadratic discriminant analysis (QDA)

Further assume uniform prior \(f(Y=l) = f(Y=m)\), but different covariance structure \(\Sigma_l\), \(\Sigma_m\), the decision hyperlane simplifies as:

\(\begin{aligned} \log |\Sigma_l| + (X - \mu_l)^\top \Sigma_l^{-1} (X - \mu_l) = \log |\Sigma_m| + (X - \mu_m)^\top \Sigma_m^{-1} (X - \mu_m) \end{aligned}\)

You will see the decision boundary is a quadratic function.

library(MASS)
library(ggplot2)

set.seed(32611)
N<-100
d1<-mvrnorm(N, c(0,2), matrix(c(4, 0, 0, 1), 2, 2))
d2<-mvrnorm(N, c(0,-2), matrix(c(1, 0, 0, 4), 2, 2))
d <- rbind(d1, d2)
label <- c(rep("1", N), rep("2", N))
data_train <- data.frame(label=label, x1=d[,1], x2=d[,2])
names(data_train)<-c("label", "x1", "x2")

ggplot(data_train) + aes(x=x1,y=x2,col=label) + geom_point() +  stat_ellipse()

Visualize the decision boundary (QDA)

library(MASS)
aqda <- qda(label ~ . ,data=data_train)

avec <- seq(-6,6,0.1)
z <- expand.grid(avec, avec)
z <- data.frame(x1=z[,1],x2=z[,2])
predict_qda <- predict(aqda, z)$class

z_qda <- cbind(region=as.factor(predict_qda), z)
ggplot(z_qda) + aes(x=x1,y=x2,col=region) + geom_point(alpha = 0.4)  + geom_point(data = data_train, aes(x=x1,y=x2,col=label)) +
  labs(title = "QDA boundary")

Support vector machine (SVM)

• Motivation: find a boundary to separate data from with different labels

• Linear separable case

library(MASS)
library(ggplot2)

set.seed(32611)
N<-100
d1<-mvrnorm(N, c(3,-3), matrix(c(1, 0, 0, 1), 2, 2))
d2<-mvrnorm(N, c(-3,3), matrix(c(1, 0, 0, 1), 2, 2))
d <- rbind(d1, d2)
label <- c(rep("1", N), rep("-1", N))
data_train <- data.frame(label=label, x1=d[,1], x2=d[,2])
names(data_train)<-c("label", "x1", "x2")

p <- ggplot(data_train) + aes(x=x1,y=x2,col=label) + geom_point()
p + geom_abline(slope=1,intercept=0, col="black") + 
 geom_abline(slope=0.8,intercept=0.5, col="green") + 
 geom_abline(slope=1.4,intercept=-0.4, col = "orange")

SVM margin

p +
geom_abline(slope=1,intercept=max(d1[,2] - d1[,1]), col="black", linetype = "dashed") +
geom_abline(slope=1,intercept=min(d2[,2] - d2[,1]), col="black", linetype = "dashed") +
geom_abline(slope=1,intercept=(max(d1[,2] - d1[,1]) + min(d2[,2] - d2[,1]))/2, col="black")

• Black line (Separation boundary) with intercept included: \[f(x) = w^\top x = w_1 x_1 + w_2 x_2 + c\]

• red dots \[w^\top x \le -1\] Otherwise, we can re-scale \(w\) to make sure this relationship holds

• blue dots \[w^\top x \ge 1\]

• Unify red dots and blue dots: \[y w^\top x \ge 1\]

Formulate optimization problem via SVM margin

• Upper dashed line: \[w^\top x + 1 = 0\]

• lower dashed line: \[w^\top x- 1 = 0\]

• Margin:

\[\frac{w^\top x + 1 - (w^\top x - 1)}{2\|w\|_2} = \frac{2}{2\|w\|_2} = \frac{1}{\|w\|_2}\]

• Optimization problem:

\(\max_{w} \frac{1}{\|w\|_2}\) subject to \(y_i (w^\top x_i ) \ge 1\)

SVM: large Margin Classifier

• \(\frac{1}{\|w\|_2}\) is not easy to maximize, but \(\|w\|_2^2\) is easy to minimize:

• SVM (linear separable case):

\(\min_{w} \frac{1}{2} \|w\|_2^2\) subject to \(y_i (w^\top x_i ) \ge 1\) This is the SVM primal objective function.

• Lagrange function:

\[L(w, \alpha) = \frac{1}{2} \|w\|_2^2 - \sum_i \alpha_i [y_i (w^\top x_i ) - 1]\] By KKT condition, \(\frac{\partial L}{\partial w}\) need to be 0 at the optimun solution.

\[\frac{\partial L}{\partial w_j} = w_j - \sum_i \alpha_i y_i x_{ij} = 0\] \[w_j = \sum_i \alpha_i y_i x_{ij}\]

Plugging terms back into Lagrange function \(L\): \[\min_\alpha \frac{1}{2} \sum_{i,i'} \alpha_i \alpha_{i'} y_i y_{i'} x_i^\top x_{i'}\] Subject to \(\alpha_i \ge 0\)

This is the SVM dual objective function.

Geometric interpretation of SVM

p +
geom_abline(slope=1,intercept=max(d1[,2] - d1[,1]), col="black", linetype = "dashed") +
geom_abline(slope=1,intercept=min(d2[,2] - d2[,1]), col="black", linetype = "dashed") +
geom_abline(slope=1,intercept=(max(d1[,2] - d1[,1]) + min(d2[,2] - d2[,1]))/2, col="black")

From KKT condition:

\[\alpha_i [y_i w^\top x_i - 1] = 0\]

• When \(\alpha_i = 0\), no restriction on \(y_i w^\top x_i - 1\), data \(i\) is beyond the boundary.
• When \(\alpha_i > 0\), \(y_i w^\top x_i = 1\), data \(i\) is on the boundary, which are called support vectors.

Use svm function in R

library(e1071)
library(ggplot2)

set.seed(32611)
N<-100
d1<-mvrnorm(N, c(3,-3), matrix(c(1, 0, 0, 1), 2, 2))
d2<-mvrnorm(N, c(-3,3), matrix(c(1, 0, 0, 1), 2, 2))
d <- rbind(d1, d2)
label <- c(rep("1", N), rep("-1", N))
data_train <- data.frame(label=factor(label), x1=d[,1], x2=d[,2])
names(data_train)<-c("label", "x1", "x2")

asvm <- svm(label ~ x1 + x2 ,data=data_train, kernel = "linear")

avec <- seq(-6,6,0.1)
z <- expand.grid(avec, avec)
z <- data.frame(x1=z[,1],x2=z[,2])
predict_svm <- predict(asvm, z)

z_svm <- cbind(region=as.factor(predict_svm), z)
ggplot(z_svm) + aes(x=x1,y=x2,col=region) + geom_point(alpha = 0.4)  + geom_point(data = data_train, aes(x=x1,y=x2,col=label)) +
  labs(title = "SVM boundary")

Property of SVM

\[w_j = \sum_i \alpha_i y_i x_{ij}\]

• Weight vector \(w\) is weighted linear combination of samples.
• Since \(\alpha_i \ne 0\) only if the samples are on the margin, so only the margin samples contribute to \(w\).
• This is quadratic programming so optimization is not too hard.

\[\min_\alpha \frac{1}{2} \sum_{i,i'} \alpha_i \alpha_{i'} y_i y_{i'} x_i^\top x_{i'} \] Subject to \(\alpha_i \ge 0\)

• In the dual formulation, \(x\) plays a part only through \(x_i^\top x_j\), which can be further extended as kernel: \(k(x_i, x_j)\)
  • Linear kernel: \(k(x_i, x_j) = x_i^\top x_j\)
  • Gaussian kernel: \(k(x_i, x_j) = \exp(-\gamma \|x_i - x_j\|_2^2)\)
  • polynomial kernel
  • sigmoid kernel

Use svm function with Gaussian kernel in R

library(e1071)
library(ggplot2)

N<-100
set.seed(32611)
theta1 <- runif(N,0,2*pi)
r1 <- runif(N,2,3)

theta2 <- runif(N,0,2*pi)
r2 <- runif(N,4,5)

d1 <- cbind(r1 * cos(theta1), r1 * sin(theta1))
d2 <- cbind(r2 * cos(theta2), r2 * sin(theta2))

d <- rbind(d1, d2)
label <- c(rep("1", N), rep("-1", N))
data_train <- data.frame(label=factor(label), x=d[,1], y=d[,2])
names(data_train)<-c("label", "x", "y")

ggplot(data_train, aes(x=x,y=y,col=label))+ geom_point() 

Use svm function with Gaussian kernel in R

asvm <- svm(label ~ . ,data=data_train, kernel = "radial")

avec <- seq(-6,6,0.1)
z <- expand.grid(avec, avec)
z <- data.frame(x=z[,1],y=z[,2])
predict_svm <- predict(asvm, z)

z_svm <- cbind(region=as.factor(predict_svm), z)
ggplot(z_svm) + aes(x=x,y=y,col=region) + geom_point(alpha = 0.4)  + geom_point(data = data_train, aes(x=x,y=y,col=label)) +
  labs(title = "SVM boundary")

Interpretation for Kernel SVM:

• The data is not linearly separable in two dimensional space
• Kernel method can create new features, thus map the two dimensional space to higher dimensional space

\[\exp(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots\]

• The data are linearly separable in this higher dimensional space

SVM when linear separator is impossible

library(MASS)
library(ggplot2)

set.seed(32611)
N<-100
d1<-mvrnorm(N, c(2,-2), matrix(c(2, 1, 1, 2), 2, 2))
d2<-mvrnorm(N, c(-2,2), matrix(c(2, 1, 1, 2), 2, 2))
d <- rbind(d1, d2)
label <- c(rep("1", N), rep("-1", N))
data_train <- data.frame(label=factor(label), x=d[,1], y=d[,2])
names(data_train)<-c("label", "x", "y")

p <- ggplot(data_train) + aes(x=x,y=y,col=label) + geom_point()
p + geom_abline(slope=1,intercept=(max(d1[,2] - d1[,1]) + min(d2[,2] - d2[,1]))/2, col="black")

SVM for linearly no-separable case

\[\min_{w} \frac{1}{2} \|w\|_2^2 + C\|\zeta\|_2^2\] subject to \(y_i (w^\top x_i ) \ge 1 - \zeta_i\) and \(\zeta_i \ge 0\)

• Similar we can write down the Lagrange function and use KKT condition to get the dual problem.

Use svm function in R with non-linear separable case

library(e1071)
library(ggplot2)

asvm <- svm(label ~ . ,data=data_train, kernel = "linear")

avec <- seq(-6,6,0.1)
z <- expand.grid(avec, avec)
z <- data.frame(x=z[,1],y=z[,2])
predict_svm <- predict(asvm, z)

z_svm <- cbind(region=as.factor(predict_svm), z)
ggplot(z_svm) + aes(x=x,y=y,col=region) + geom_point(alpha = 0.4)  + geom_point(data = data_train, aes(x=x,y=y,col=label)) +
  labs(title = "SVM boundary")

Summary for classification functions in R

Deep learning

• Have better accuracy compared to other exisiting machine learning method
• Free online deep learning course
  • https://cds.nyu.edu/deep-learning/
  • Instructor: Yann LeCun
    • 2028 Tuning Awardee for his contribution in deep learning

Resources

• https://www.openml.org/a/estimation-procedures/1
• https://www.cs.cmu.edu/~schneide/tut5/node42.html
• http://www.stat.cmu.edu/~ryantibs/datamining/lectures/18-val1.pdf
• http://www.stat.cmu.edu/~ryantibs/datamining/lectures/19-val2.pdf
• Element of Statistical learning, Chapter 7