COM S 573: Home work 5


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1. Suppose you have the following data
Observation X1 X2 Class
1 3 1 +1
2 3 -1 +1
3 6 1 +1
4 6 -1 +1
5 1 0 -1
6 0 1 -1
7 0 -1 -1
8 -1 0 -1
(a) [6 points] Find the equation of the hyperplane (in terms of w) WITHOUT solving a quadratic
programming (QP) problem. Make a sketch of the problem (i.e., plot the data, unique hyperplane and corresponding dashed lines).
(b) [1 point] Calculate the margin.
(c) [2 points] Find the α’s of the SVM for classification (again WITHOUT solving a QP problem).
2. In this problem (based on Problem 7 on p. 371), you will use support vector classification in order
to predict whether a given car gets high or low gas mileage based on the Auto data set. You can
download the data set at It can be helpful
to do the lab starting on p. 359 of the textbook first.
(a) [1 point] Create a class variable that takes on a “1” for cars with gas mileage above the
median, and a “-1” for cars with gas mileage below the median.
(b) [5 points] Fit a support vector classifier with linear kernel to the data with various values of
cost (i.e., the parameter C from class), in order to predict whether a car gets high or low gas
mileage. Report the cross-validation errors associated with different values of this parameter.
Comment on your results.
(c) [5 points] Now repeat (b), this time using SVMs with radial basis and polynomial kernels,
with different values of gamma, degree and cost. Comment on your results.
In class, I said the gaussian kernel had the following form: K(Xi
, Xj ) = exp(−kXi−Xjk
However, in this software package e1071, the radial basis function has the form K(Xi
, Xj ) =
exp(−γkXi − Xjk
). Consequently, γ = 1/h2
(d) [1 point] Which kernel function do you prefer and why?
3. (a) [6 points] Make an R program that solves the LS-SVM for nonlinear regression given the
parameters γ and bandwidth h for the following gaussian kernel (do not implemenent the
cross-validation procedure)
, Xj ) = 1


kXi − Xjk

(b) [3 points] Try your program on the following function (i.e., experiment with some values of γ
and h)
X <- seq(0, 1, length.out = 200) y <- (sin(2*pi*(x-0.5)))^2 + rnorm(200, 0, 0.2) Describe the effect when the two tuning parameters γ and h change. Finally, try the values γ = 9.4365 and h = 0.16006. Does this seems a good fit to you? 2