An auto$-$regressive $($AR$)$ model is a time$-$domain model of the

form

yk$+1=$

X

l$1$

i$=0$

iyk$$i

$,(16)$

where l $>0$ is the model memory and i are the model parameters. AR models are routinely used in various

applications that require prediction such as signal processing and economics.

In this problem, you will develop an AR model to predict the new COVID cases. Download the Howard

county COVID data here. You will use the first half of the data to train the AR model and the second half

of the data to validate$/$test the trained model. Specifically, you will use linear regression to find the AR

model parameters $0,$ $1,...,$ l$1$ using the training data.

$1.$ Plot the new cases per day against the day.

$2.$ Note that

yl$+1=$ $0$yl $+$ $1$yl$1++$ l$1$y$1,(17)$

yl$+2=$ $0$yl$+1+$ $1$yl $++$ l$1$y$2,(18)$

$.$

$.$

$.(19)$

yl$+$N $=$ $0$yl$+$N$1+$ $1$yl$+$N$2++$ l$1$yN $,(20)$

where N is the length of the training data. Write these equations in the $$ $=$ Y form.

$3.$ Choose a value of l and use the least$-$square solution to find the values of i that minimize the training

error. Hint: try l $=10.$

$4.$ With the trained model, predict the new cases for the second half of the pandemic, that is$,$yl$+$N$+1,...,$ y$$M$,$

where M is the total length of the data. Plot the true value and the predicted value on the same plot.

$5.$ For each predicted value, compute the relative error, that is$,$

ek$,$rel $=$

yk $$ y$$k

yk

$(21)$

and plot it$.$

$6.$ Repeat the steps above for five different choices of l$.$

$7.$ Comment on the AR model$$s ability to predict the new cases as well as its limitations$.\%$ Import data from text file

MDCOVID$19\_$Cases$\_$Howard$\_$County $=$ readtable$("$C:$\backslash$Users$\backslash$arjin$\backslash$Downloads$\backslash$MDCOVID$19\_$Cases$-$Howard$-$County.csv$",$ "TextType", "string"$)$;

$\%$ Display results

MDCOVID$19\_$Cases$\_$Howard$\_$County

Can you do it in Matlab