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\begin{document}

\title{Introduction to Machine Learning\\
Problems:  Logistic Regression}
\author{Prof. Sundeep Rangan}
\date{}

\maketitle

\begin{enumerate}

\item Suggest possible response variables and predictors for the following classification
problems.  For each problem, indicate how many classes there are.  There is no single
correct answer.
\begin{enumerate}[(a)]
\item Given an audio sample, to detect the gender of the voice.
\item A electronic writing pad records motion of a stylus
and it is desired to determine
which letter or number was written.  Assume a segmentation algorithm is already run
which indicates very reliably the beginning and end time of the writing
of each character.
\end{enumerate}

\item Suppose that a logistic regression model for a binary class
label $y=0,1$ is given by
\[
    P(y=1|\xbf) = \frac{1}{1+e^{-z}}, \quad z = \beta_0 + \beta_1x_1 + \beta_2 x_2,
\]
where $\betabf = [1,2,3]\tran$.  Describe the following sets:
\begin{enumerate}[(a)]
\item The set of $\xbf$ such that $P(y=1|\xbf) > P(y=0|\xbf)$.
\item The set of $\xbf$ such that $P(y=1|\xbf) > 0.8$.
\item The set of $x_1$ such that $P(y=1|\xbf) > 0.8$ and $x_2=0.5$.
\end{enumerate}

\item A data scientist is hired by a
political candidate to predict who will donate
money.  The data scientist decides to use two predictors for each possible donor:
\begin{itemize}
\item $x_1$ = the income of the person (in thousands of dollars), and
\item $x_2$ = the number of websites with similar political views as the
candidate the person follow on Facebook.
\end{itemize}
To train the model, the scientist
tries to solicit donations from a randomly selected subset
of people and records who donates or not.  She obtains the following data:

\begin{center}
\begin{tabular}{|l|c|c|c|c|c|} \hline
Income (thousands \$), $x_{i1}$ & 30 & 50 & 70 & 80 & 100 \\ \hline
Num websites, $x_{i2}$          & 0  & 1 & 1 & 2 & 1  \\ \hline
Donate (1=yes or 0=no), $y_i$   & 0  & 1 & 0 & 1 & 1 \\ \hline
\end{tabular}
\end{center}

\begin{enumerate}[(a)]
\item Draw a scatter plot of the data labeling the two classes with different markers.

\item Find a linear classifier that makes at most one error on the training data.
The classifier should be of the form,
\[
    \hat{y}_i = \begin{cases}
        1 & \mbox{if } z_i > 0\\
        0 & \mbox{if } z_i < 0,
        \end{cases}
    \quad
    z_i = \wbf\tran\xbf_i + b.
\]
What is the weight vector $\wbf$ and bias $b$ in your classifier?

\item Now consider a logistic model of the form,
\[
    P(y_i=1|\xbf_i) = \frac{1}{1+e^{-z_i}}, \quad z_i = \wbf\tran\xbf_i +b.
\]
Using $\wbf$ and $b$ from the previous part, which sample $i$ is the
\emph{least} likely (i.e.\ $P(y_i|\xbf_i)$ is the smallest).
If you do the calculations correctly, you should not need a calculator.

\item Now consider a new set of parameters
\[
    \wbf' = \alpha \wbf, \quad b'=\alpha b,
\]
where $\alpha>0$ is a positive scalar.  Would using the new parameters
change the values $\hat{y}$ in part (b)?  Would they change the
likelihoods $P(y_i|\xbf_i)$ in part (c)?  If they do not change, state why.
If they do change, qualitatively describe the change as a function of $\alpha$.

% \item Complete the following python function to generate a vector
% of random labels \pycode{y}, where \pycode{y[i]} uses the data
% record in \pycode{X[i,:]}, weight vector \pycode{w} and bias \pycode{b}.
% \begin{python}
%     import numpy as np
%     def gen_rand(X,w,b):
%       ...
%       return y
% \end{python}

\end{enumerate}

\item Suppose we collect data for a group of students in a machine learning class
with variables $X_1 =$ hours studied, $X_2 =$ undergrad GPA, and $Y =$
receive an A. We fit a logistic regression and produce estimated
coefficient, $\beta_0 = -6, \;\beta_1 = 0.05,\; \beta_2 = 1$.

\begin{enumerate}
    \item  Estimate the probability that a student who studies for 40 h and
has an undergrad GPA of 3.5 gets an A in the class.


\item  How many hours would the student in part (a) need to study to
have a 50 \% chance of getting an A in the class?
\end{enumerate}

\item The loss function for logistic regression for binary classification is the binary cross entropy defined as $$J(\betabf)=\sum_{i=1}^N\ln(1+e^{z_i})-y_iz_i$$ where $z_i = \beta_0+\beta_1x_{1i}+\beta_2x_{2i}$ for two features $x_{1,i}$ and $x_{2,i}$.
\begin{enumerate}
    \item What are the partial derivatives of $z_i$ with respect to $\beta_0$, $\beta_1$, and $\beta_2$.
    \item Compute the partial derivatives of $J(\betabf)$ with respect to $\beta_0$, $\beta_1$, and $\beta_2$. You should use the chain rule of differentiation.
    \item Can you find the close form expressions for the optimal parameters $\hat{\beta}_0$, $\hat{\beta}_1$, and $\hat{\beta}_2$ by putting the derivatives of $J(\betabf)$ to 0? What methods can be used to optimize the loss function $J(\betabf)$?
\end{enumerate}
\end{enumerate}



\end{document}