prob_simp_lin_reg.tex

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\begin{document}

\title{Introduction to Machine Learning\\
Unit 2:  Simple Linear Regression Problems}
\author{Prof. Sundeep Rangan}
\date{}

\maketitle

\begin{enumerate}
\item A university admissions office wants to predict the success of students based on
their application material.  They have access to past student records to learn
a good algorithm.
\begin{enumerate}[(a)]
\item To formulate this as a supervised learning problem,
identify a possible target variable.  This should be some variable that measures success
in a meaningful way and can be easily collected (in an automated manner) by the
university. There is no one correct answer to this problem.
\item Is the target variable continuous or discrete-valued?
\item State at least one possible variable that can act as the predictor for the target
variable you chose in part (a).
\item Before looking at the data, would a linear model for the data be reasonable?
If so, what sign do you expect the slope to be?
\end{enumerate}

\item Suppose that we are given data samples $(x_i,y_i)$:
\begin{center}
\begin{tabular}[h]{|c|c|c|c|c|c|} \hline
$x_i$ & 0 & 1 & 2 & 3 & 4 \\ \hline
$y_i$ & 0 & 2 & 3 & 8 & 17 \\ \hline
\end{tabular}
\end{center}
\begin{enumerate}[(a)]
\item What are the sample means, $\bar{x}$ and $\bar{y}$?
\item What are the sample variances and co-variances $s_x^2$, $s_y^2$ and $s_{xy}$?
\item What are the least squares parameters for the regression line
\[
    y = \beta_0 + \beta_1x + \epsilon.
\]
\item Using the linear model, what is the predicted value at $x=2.5$?
\end{enumerate}

\item A medical researcher wants to model, $z(t)$, the concentration
of some chemical in the blood
over time. She believes the concentration should decay exponentially in that
\beq \label{eq:zexp}
    z(t) \approx z_0 e^{-\alpha t},
\eeq
for some parameters $z_0$ and $\alpha$.
To confirm this model, and to estimate the parameters $z_0,\alpha$,
she collects a large number of time-stamped samples $(t_i,z(t_i))$, $i=1,\ldots,N$.
Unfortunately, the model \eqref{eq:zexp} is non linear, so she can't directly apply
the linear regression formula.
\begin{enumerate}[(a)]
\item  Taking logarithms, show that we can rewrite the model in a form where the parameters
$z_0$ and $\alpha$ appear linearly.

\item Using the transform in part (a), write the least-squares solution for
the best estimates of the parameters $z_0$ and $\alpha$ from the data.

\item Write a few lines of python code that you would compute these estimates
from vectors of samples \pycode{t} and \pycode{z}.
\end{enumerate}

\item Consider a linear model of the form,
\[
    y \approx \beta x,
\]
which is a linear model, but with the intercept forced to zero.  This occurs
in applications where we want to force the predicted value $\hat{y}=0$ when
$x=0$.  For example, if we are modeling $y=$ output power of a motor
vs.\ $x=$ the input power, we would expect $x=0 \Rightarrow y=0$.
\begin{enumerate}[(a)]
\item Given data $(x_i,y_i)$,
write a cost function representing the residual sum of squares (RSS) between
$y_i$ and the predicted value $\hat{y}_i$ as a function of $\beta$.
\item Taking the derivative with respect to $\beta$, find the $\beta$ that minimizes
the RSS.
\end{enumerate}
\end{enumerate}

\end{document}