## Description

1. In this problem you will code a nearest-means classifier, using Euclidean distance (L2 norm).

You will then use the classifier on data we provide.

Note that you are to write the code for the classifier and error-rate computation yourself; using

a toolbox’s or library’s function or module that implements this (or similar) classifier is not

sufficient. If you use python, this means that you can use built-in functions, numpy, and

matplotlib. Use of other libraries or packages (e.g., pandas or scikit-learn) are not in the spirit

of this homework problem and will result in points deducted.

Your code should take as input a set of data points (training dataset) that are labeled according

to class, and should calculate a representation of the classifier (class means and a way to use

them to classify data points). Once the classifier is “trained” (class means are calculated), your

code should be able to calculate the error rate obtained when classifying training data, and

calculate the error rate obtained when classifying test data. Error rate is defined as the number

of misclassified data points divided by the total number of data points classified, usually

expressed as percentage. In addition, with the aid of a supplied plotting function, you will plot

the training data points, resulting class means, decision boundaries, and decision regions.

You will use your implemented nearest-means classifier on 3 different datasets: two synthetic

datasets, and one real dataset (“wine” from UCI machine learning repository).

To help you, the following files are available for download on the course web site, in “Week

2”, along with this HW assignment:

(i) Dataset files (containing labeled data points).

MATLAB: synthetic1.mat, synthetic2.mat, wine.mat.

For each dataset file, there are in total 4 variables: ‘feature_train’, ‘label_train’,

‘feature_test’ and ‘label_test’. While feature_xxx stores the data points (inputs

) for training (or testing, respectively), label_xxx are the class labels

for these data points ( ). Each row in feature_xxx corresponds to a data

point, and each column corresponds to a feature. Class labels are stored as a column

xn , n = 1,2,!,N

yn , n = 1,2,!,N

p. 2 of 3

vector in label_xxx, where the i

th entry corresponds to the i

th data point in the

feature_xxx file.

Python:

sythetic1_train.csv, synthetic1_test.csv, sythetic2_train.csv, synthetic2_test.csv,

wine_train.csv, wine_test.csv.

Each csv file has one row for each data point, and one column for each feature; except

the last column contains the class labels (each label is an integer from 1 to C, in which C

is the number of classes).

(ii) PlotDecBoundaries.m or PlotDecBoundaries.py function helps you plot the decision

boundary and regions.

To use PlotDecBoundaries.m (or PlotDecBoundaries.py), you only need to pass in 3

variables, which are training data points of all classes, class labels of all training data

points, and sample means for all classes.

For training data points, you need to be consistent with the format of feature_train (or

any of the csv files), where a row corresponds to a data point; it should contain no class

labels. The class labels variable should also follow the format of label_train (or in

python, one column of class labels, same order as rows of “training”). For sample

mean, the i

th row needs to be set as the sample mean of i

th class.

Please answer the following parts. Note that the data is unnormalized; please keep it

unnormalized throughout your work on this problem (that is, do not normalize the data

yourself in this problem).

(a) For each of the two synthetic datasets, there are in total C=2 classes and D=2 features.

For each synthetic dataset: (i) train the classifier, plot the (training-set) data points, the

resulting class means, decision boundaries, and decision regions (using

PlotDecBoundaries()); also run the trained classifier to classify the data points from

their inputs; give the classification error rate on the training set, and separately give the

classification error rate on the test set. The test-set data points should never be used for

training. Turn in the plots and error rates.

Hint: You might want to check your code first, by trying it out with just two or three

training data points in each class, for which you can check the results.

(b) Is there much difference in error rate between the two synthetic datasets? Why or why

not?

Parts (c)-(e) below use the “wine” dataset, which is for classifying the cultivar of the grape

plant a wine was made from, given measured attributes of the wine. This dataset is briefly

described at (in which “attribute” means “feature”):

http://archive.ics.uci.edu/ml/datasets/Wine

(c) For the wine dataset, there are in total C=3 classes (grape cultivars) and D=13 features

(measured attributes of the wine). In this problem you are to use only 2 features for

classification.

p. 3 of 3

Pick the first two features ( alcohol content, and malic acid content), and

repeat the procedure of part (a) for this dataset.

(d) Again for the “wine” dataset, find the 2 features among the 13 that achieve the

minimum classification error on the training set. (We haven’t yet covered how to do

feature selection in class, but will later in the semester. For this problem, try coming up

with your own method – one that you think will give good results – and see how well it

works.

Hint: 13 is not a lot of features, so computation time probably need not be a

consideration.)

Plot the data points and decision boundaries in 2D for your best performing pair of

features, and give its classification error on the training set.

Then give its classification error on the test set. The purpose of the test set is only to

estimate the error of the final classifier on unknown (previously unseen) data. Describe

the method you used to choose the best pair of features.

Turn in a description of your method for choosing the best pair of features; a plot of the

training data, decision boundaries, and decision regions using your chosen pair of

features; state which features you chose; and give both error rates (training and testing).

(e) For the wine dataset, is there much difference in training-set error rate for different pairs

of features? Justify your answer (e.g., by giving the error rate for a few different

example pairs of features; or by giving the standard deviation of error rates over all

possible pairs of features).

Also, answer the same question for the test-set error rate.

x1 = x2 =