Challenging Problem: Multivariable Data Anal$-$ ysis

An environmental scientist has been collecting data for a study on air quality. The scientist is tracking levels of three pollutants: carbon monoxide $($CO$),$ nitrogen dioxide $($NO$),$ and particulate matter $($PM$2\mathrm{.}5).$ These levels are

measured daily for $30$ days. The scientist hypothesizes that the levels of these three pollutants are linearly related, meaning that the level of one pollutant can be predicted using the

levels of the other two pollutants.

However, the data is complicated by the fact that many external variables could be influencing pollutant levels $($e$.$g$..$ wind speed, temperature, hunan activity, etc.$),$ and these factors may not be consistent from day to day.

Here's the data for the first five days:

Day

CO

Level

NO$2$

Level

PM$2\mathrm{.}5$

Level

AA

$1$

$0\mathrm{.}8$

$20$

$12$

$2$

$0\mathrm{.}6$

$18$

$13$

$3$

$0\mathrm{.}7$

$22$

$14$

$4$

$0\mathrm{.}9$

$23$

$15$

$5$

$0\mathrm{.}6$

$21$

$13$

$6$

$0\mathrm{.}7$

$21$

$13$

$7$

$0\mathrm{.}8$

$20$

$13$

$8$

$0\mathrm{.}8$

$22$

$14$

$9$

$0\mathrm{.}8$

$23$

$15$

$10$

$0\mathrm{.}7$

$20$

$12$

a$.$ Represent the data in matrix form.

b$.$ Write down a system of linear equations that relates CO$,$ NO$,$ and PM$2\mathrm{.}5$ levels. Assume the relationship can be modeled as aCO$+$NO$2=$ PM$2\mathrm{.}5,$ where a$,$ and a are constants to be determined.

c$.$ Determine the least squares solution to the system of equations. This will give you the best estimates for a$,$ and as$.$ You can use SageMath for this step.

d$.$ Using the values of of and as obtained in part $($c$),$ predict the PM$2\mathrm{.}5$ levels for days $6$ to $10.$

e$.$ For each day $6$ to $10,$ calculate the prediction error, which is the absolute difference between the predicted PM$2\mathrm{.}5$ level and the actual PM$2\mathrm{.}5$ level.

f$.$ Discuss the limitations of this model and suggest how it might be improved.