Linear regression!

In this lab we will start by learning to use the function provided by matlab to find the root. In this homework, we are going to use linear regression to fit a line to a set of data.

$m=\frac{n{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i{y}_i-{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i{\displaystyle \underset{i=1}{\overset{n}{}}}{y}_i}{n{\displaystyle \underset{i=1}{\overset{n}{}}}(x_i{)}^2-({\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i{)}^2}$

$b=\frac{{\displaystyle \underset{i=1}{\overset{n}{}}}{y}_i-m{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i}{n}$

Task:

First, you should download the data file from Canvas. You should save the file to the same folder on your computer as your code. The name of the file should be ME$208\_$week$6\_$hwkdata.txt $($if Canvas changes it$,$ you should change it back$).$

We will create a main script and a function. I recommend putting the function in the separate file in the same folder on your computer.

The function will calculate the linear regressions formulas for $m$ and $b.$ Start your function passing in $x$ and $y($two arrays of data$).$ You should pass out $m$ and $b.$ The first line of your function should be: function $[m,b]=$ mylinregression $(x,y)$

The first part of your function should be to create the $4$ sums:

${\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i$

${\displaystyle \underset{i=1}{\overset{n}{}}}{y}_i$

${\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i{y}_i$

${\displaystyle \underset{i=1}{\overset{n}{}}}(x_i{)}^2$

Start with $4$ variables, sumx, sumy, sumxy, sumxx and set them to zero. Use a for loop that goes from $i=1$ to $i=n$ to add to the sums.

$5.$ The last part of the function should be to put together $m$ and $b$ using the completed sums and the formula above.

$6.$ Once you have completed the function, you should work on the main script. Start by loading the data from the file and assigning the second column to a variable exam$1$ and the third to exam$2$: load ME$208\_$week$6\_$hwkdata.txt

exam$1=$ ME$208\_$week$6\_$hwkdata$($:$,2)$;

exam$2=$ ME$208\_$week$6\_$hwkdata$($:$,3)$;

$7.$ Send your data to your function to get $m$ and $b$:$[m,b]=$ mylinregression$($exam$1,$exam$2)$;

$8.$ We want to compare your function with Matlab's built in function polyfit to see if it is giving the right answer. If you send the data to polyfit and fit it to a first order nolvnomial the coefficients of that polynomial should be equal to m and b: p$=$polyfit$($exam$1,$exam$2,1)$;

$9.$ Using your new equations fitting a line to your data, create some new y data

yfit$1=$p$(1)*$exam$1+$p$(2)$

yfit$2=$m$*$exam$1+$b;

$10.$ Plot your data using $*$ symbol $($exam$1$ vs exam$2)$ and this fit data $($yfit$1$ and yfit$2)$ using a red line to create two subplots $($one for the fit from your function and one for the fit from polyfit$).$

$11.$ Display your coefficients to the command window:

disp$([\text{'}$The slope $($m$)$ was calculated as 'num$2$str$($m$)\text{'}$ compared to 'num$2$str$($p$(1))$ 'using the matlab function'$])$

disp$([\text{'}$The intercept $($b$)$ was calculated as 'num$2$str$($b$)\text{'}$ compared to 'num$2$str$($p$(2))$ 'using the matlab function'$])$