**al3xandr3**, and kindly contributed to R-bloggers)

Exercise 5.2 Improves the Logistic Regression implementation done in Exercise 4 by adding a regularization parameter that reduces the problem of over-fitting. We will be using Newton's Method.

## Data

Here's the data we want to fit.

# linear regression # load the data mydata = read.csv("http://spreadsheets.google.com/pub?key=0AnypY27pPCJydHZPN2pFbkZGd1RKeU81OFY3ZHJldWc&output=csv", header = TRUE) # plot the data plot(mydata$u[mydata$y == 0], mydata$v[mydata$y == 0],, xlab="u", ylab="v") points(mydata$u[mydata$y == 1], mydata$v[mydata$y == 1], col="blue", pch=3) legend("topright", c("y=0","y=1"), pch=c(1, 3), col=c("black", "blue"), bty="n")

The idea is to create a mathematical model, that separates the circles from the crosses in the plot above. And that will then allow to predict, for a new u and v value, the probability of it being a cross.

## Theory

Hypothesis is:

[ h_theta(x) = g(theta^T x) = frac{1}{ 1 + e ^{- theta^T x} } ]The idea of the regularization is to loosen up the tight fit, and obtain a more generalized fit. For that we define the cost function, with an added generalization parameter that blunts the fit, like so:

[ J(theta) = frac{1}{m} sum_{i=1}^m [(-y)log(h_theta(x)) - (1 - y) log(1- h_theta(x))] + frac{lambda}{2m} sum_{i=1}^n theta^2] ] (lambda) is called the regularization parameter.The iterative (theta) updates using Newton's method is defined as:

[ theta^{(t+1)} = theta^{(t)} - H^{-1} nabla_{theta}J ]And the gradient and Hessian are defined like so(in vectorized versions):

[ nabla_{theta}J = frac{1}{m} sum_{i=1}^m (h_theta(x) - y) x + frac{lambda}{m} theta ] [ H = frac{1}{m} sum_{i=1}^m [h_theta(x) (1 - h_theta(x)) x^T x] + frac{lambda}{m} begin{bmatrix} 0 & & & \ & 1 & & \ & & ... & \ & & & 1 end{bmatrix} ]## Implementation

Lest first define the functions above, with the added generalization parameter:

# sigmoid function g = function (z) { return (1 / (1 + exp(-z) )) } # plot(g(c(1,2,3,4,5,6)), type="l") # build hight order feature vector # for 2 features, for a given degree hi.features = function (f1,f2,deg) { n = ncol(f1) ma = matrix(rep(1,length(f1))) for (i in 1:deg) { for (j in 0:i) ma = cbind(ma, f1^(i-j) * f2^j) } return(ma) } # hi.features(c(1,2), c(3,4),2) # creates: 1 u v u^2 uv v^2 ... h = function (x,th) { return(g(x %*% th)) } # h(x,th) # derivative of J grad = function (x,y,th,m,la) { G = (la/m * th) G[1,] = 0 return((1/m * t(x) %*% (h(x,th) - y)) + G) } # grad(x,y,th,m,la) H = function (x,y,th,m,la) { n = length(th) L = la/m * diag(n) L[1,] = 0 return((1/m * t(x) %*% x * diag(h(x,th)) * diag(1 - h(x,th))) + L) } # H(x,y,th,m,la) # cost function J = function (x,y,th,m,la) { pt = th pt[1] = 0 A = (la/(2*m))* t(pt) %*% pt return((1/m * sum(-y * log(h(x,th)) - (1 - y) * log(1 - h(x,th)))) + A) } # J(x,y,th,m,la)

Now we can make it run, first for (lambda=1)

# setup variables m = length(mydata$u) # samples x = hi.features(mydata$u, mydata$v,6) n = ncol(x) # features y = matrix(mydata$y, ncol=1) # lambda = 1 # use the cost function to check is works th1 = matrix(0,n) la = 1 jiter = array(0,c(15,1)) for (i in 1:15) { jiter[i] = J(x,y,th1,m,la) th1 = th1 - solve(H(x,y,th1,m,la)) %*% grad(x,y,th1,m,la) }

Validate that is converging properly, by plotting the Cost(J) function against the number of iterations.

# check that is converging correctly plot(jiter, xlab="iterations", ylab="cost J")

Converging well and fast, as is typical from Newton's method.

And now we make it iterate for (lambda=0) and (lambda=10) for comparing fits later:

# lambda = 0 th0 = matrix(0,n) la = 0 for (i in 1:15) { th0 = th0 - solve(H(x,y,th0,m,la)) %*% grad(x,y,th0,m,la) } # lambda = 10 th10 = matrix(0,n) la = 10 for (i in 1:15) { th10 = th10 - solve(H(x,y,th10,m,la)) %*% grad(x,y,th10,m,la) }

Finally calculate the Decision Boundary line and visualize it:

# calculate the decision boundary line # create many points u = seq(-1, 1.2, len=200); v = seq(-1, 1.2, len=200); z0 = matrix(0, length(u), length(v)) z1 = matrix(0, length(u), length(v)) z10 = matrix(0, length(u), length(v)) for (i in 1:length(u)) { for (j in 1:length(v)) { z0[j,i] = hi.features(u[i],v[j],6) %*% th0 z1[j,i] = hi.features(u[i],v[j],6) %*% th1 z10[j,i] = hi.features(u[i],v[j],6) %*% th10 } } # plots contour(u,v,z0,nlev = 0, xlab="u", ylab="v", nlevels=0, col="black",lty=2) contour(u,v,z1,nlev = 0, xlab="u", ylab="v", nlevels=0, col="red",lty=2, add=TRUE) contour(u,v,z10,nlev = 0, xlab="u", ylab="v", nlevels=0, col="green3",lty=2, add=TRUE) points(mydata$u[mydata$y == 0], mydata$v[mydata$y == 0]) points(mydata$u[mydata$y == 1], mydata$v[mydata$y == 1], col="blue", pch=3) legend("topright", c(expression(lambda==0), expression(lambda==1),expression(lambda==10)), lty=1, col=c("black", "red","green3"),bty="n" )

See that the black line ((lambda=0)) is the more tightly fit to the crosses, and as we increase the lambda values it becomes more loose(and more generalized).

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