Question

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Best fitting line. Matrix.

$y=b_{0}+b_{1}x$

$b_{0}=\bar{y}-b_{1}\bar{x}$

$p=c_{0}+c_{1}w+c_{2}m$

$\vec{y}=\begin{bmatrix} p_{1}\\ p_{2}\\ ... \\ p_{n} \end{bmatrix}$

$\vec{b_{0}}=(X_{0}^{T}X_{0})^{-1}X_{0}^{T}\vec{y}$

$\vec{b_{1}}=(X_{1}^{T}X_{1})^{-1}X_{1}^{T}\vec{y}$

$X^{T}X\vec{b}=X^{T}\vec{y}$

5. Predicting Populatioin The data below records the population of Irvine, CA (in thousands of people) for the years 2010-2016: Year Population 220 2010 229 2011 2012 236 2013 247 2014 256 2015 266 2016 277 Suppose we want to use this data to predict the population in future years. (a) To use the year as a predictor variable, encode 2010 as 1, 2011 as 2, 2012 as 3, etc. We obtain r 1, r2 - 2, T3- 3, and so on. Write down the design matrix X and the parameter vector f that you would use to predict the population of Irvine based only on the year, using a linear model (b) Solve the normal equations to find the best-fitting line, and interpret the mean- ing of the slope and intercept of this line. (c) Assume we now use the year as the predictor variable, e.g., x1 2010, x2 - 2011, etc. How do slope and intercept compare to the values obtained earlier? (d) Use the fitted line to predict the population of Irvine in the year 2030. How accurate do you think your prediction will be? 71 1 I W2 m2 1 wn mn 012

Answer 1

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