YGC

Exercise 4 required implementing Logistic Regression using Newton's Method.

The dataset in use is 80 students and their grades of 2 exams, 40 students were admitted to college and the other 40 students were not. We need to implement a binary classification model to estimates college admission based on the student's scores on these two exams.

plot the data

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x <- read.table("ex4x.dat",header=F, stringsAsFactors=F)
x <- cbind(rep(1, nrow(x)), x)
colnames(x) <- c("X0", "Exam1", "Exam2")
x <- as.matrix(x)
 
y <- read.table("ex4y.dat",header=F, stringsAsFactors=F)
y <- y[,1]
 
## plotting data
d <- data.frame">data.frame(x,
                y = factor(y,
                levels=c(0,1),
                labels=c("Not admitted","Admitted" )
                )
                )
 
require(ggplot2)
p <- ggplot(d, aes(x=Exam1, y=Exam2)) +
    geom_point(aes(shape=y, colour=y)) +
    xlab("Exam 1 score") +
    ylab("Exam 2 score")

Logistic Regression

We first need to define our Hypothesis Function that return values between[0,1]，suitable for binary classification.

h_\theta(x) = g(\theta^T x) = \frac{1}{ 1 + e ^{- \theta^T x} }$

function g is the sigmoid function, and function h return the probability of y=1：

h_\theta(x) = P (y=1 | x; \theta) $

What we need is to compute \theta $，to find out the proper Hypothesis Function.

Similar to the linear regression，We defined the cost function, which estimate the error of hypothesis function fitting the sample data, at a given \theta$.

The cost function was defined as:
J(\theta) = \frac{1}{m} \sum_{i=1}^m ((-y)log(h_\theta(x)) - (1 - y) log(1- h_\theta(x)) )$

To determine the most suitable hypothesis function, we need to find the \theta$ value which minimize the value of J(\theta)$. This can be achieved by the Newton's method, by finding the root of the derivative function of the cost function.

And the \theta$ can be updated by:
\theta^{(t+1)} = \theta^{(t)} - H^{-1} \nabla_{\theta}J$

the gradient and Hessian are defined as:
\nabla_{\theta}J = \frac{1}{m} \sum_{i=1}^m (h_\theta(x) - y) x$
H = \frac{1}{m} \sum_{i=1}^m [h_\theta(x) (1 - h_\theta(x)) x^T x]$

The above equations were implemented using R:

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### Newton's Method
## sigmoid function
g <- function(z) {
    1/(1+exp(-z))
}
 
## hypothesis function
h <- function(theta, x) {
    g(x %*% theta)
}
 
## cost function
J <- function(theta, x, y) {
    m <- length(y)
    s <- sapply(1:m, function(i)
                y[i]*log(h(theta,x[i,])) + (1-y[i])*log(1-h(theta,x[i,]))
                )
    j <- -1/m * sum(s)
    return(j)
}
 
 
## gradient
grad <- function(theta, x, y) {
    m <- length(y)
    g <- 1/m * t(x) %*% (h(theta,x)-y)
    return(g)
}
 
## Hessian
Hessian <- function(theta, x) {
    m <- nrow(x)
    H <- 1/m * t(x) %*% x * diag(h(theta,x)) * diag(1-h(theta,x))
    return(H)
}

The first question need to determine how many iteration until convergence.

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theta <- rep(0, ncol(x))
j <- rep(0,10)
for (i in 1:10) {
    theta <- theta - solve(Hessian(theta,x)) %*% grad(theta,x,y)
    j[i] <- J(theta,x,y)
}
 
ggplot()+
    aes(x=1:10,y=j)+
    geom_point(colour="red")+
    geom_path()+xlab("Iteration")+
    ylab("Cost J")

As illustrated in the above figure, Newton's method converge very fast, only 4-5 iterations was needed.

The second question:What is the probability that a student with a score of 20 on Exam 1 and a score of 80 on Exam 2 will not be admitted?

> (1 - g(c(1, 20, 80) %*% theta))* 100
         [,1]
[1,] 64.24722

In our model, we predicted that the probability of the student will not admitted is 64%.

At last, we calculate our classification model based on \theta^T x = 0$, and visualize the fit as below:

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x1 <- c(min(x[,2]),max(x[,2]))
x2 <- -1/theta[3,] * (theta[2,]*x1+theta[1,])
a <- (x2[2]-x2[1])/(x1[2]-x1[1])
b <- x2[2]-a*x1[2]
p+geom_abline(slope=a, intercept=b)

Fantastic.

References:
Machine Learning Course
Exercise 4

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