Linear classification

Assume that we wish to classify a data vector $yin{R}^2$ into two classes. For this purpose, you design two classifiers, namely linear classifier with least square and logistic regression. The parameters of the classifiers are estimated in a supervised manner using data set given in $\text{'}$hw$2$data.txt$\text{'}$ file $($Fig$.1).$ The data set contains $200$ data samples, $y_n\text{'}$s$,$ and their labels, $z_n\text{'}$s$.$ Each row in the data file is associated with one data sample. First two columns in the data file correspond to the first and second coordinates of data vector, $y_n,$ and the last column corresponds to the label, $z_n.$

Figure $1$: Scatter plot of data.

a$.$ First split the entire data set randomly in training and test sets. Allocate the $\%80$ of the data for training and the rest for test.

b$.$ Consider a linear classifier $g_1(y)=w_1^{T}y$ with the following classification rule:

$z=\{\begin{array}{cc}1,& g_1(y)0\\ 0,& g_1(y)<mn\; id="EditorElement\_18794">0</mn>\end{array}$

Train the classifier with least square $($LS$)$ approach using training data.

c$.$ After training classify the test data and calculate the overall accuracy $($OA$)$ and $F_1$ score metrics.

d$.$ Consider a logistic regression $g(y)=(w_2^{T}y)$ with the following classification rule:

$z=\{\begin{array}{cc}1,& g_2(y)0\mathrm{.}5\\ 0,& g_2(y)<mn\; id="EditorElement\_18326">0</mn><mn\; id="EditorElement\_18327">.</mn><mn\; id="EditorElement\_18328">5</mn>\end{array}$

To train the classifier write your own gradient descent $($GD$)$ algorithm. Choose the initial values as zero. Set the learning rate $=0\mathrm{.}001$ and maximum number of iterations to $100.$

e$.$ After training classify the test data and calculate the overall accuracy $($OA$)$ and $F_1$ score metrics.

f$.$ List the performance metrics of the classifiers in a table as follows:

$\backslash$table$[[,$OA$,F_1$