The Assignment

First, download the CSV file with the data that you$$ll work with $($see the Downloads section$).$

The file you download, streifCourseRecords.csv contains $2$ columns of data:

The year the record was set

The time in number of seconds $($to the $1/100$th of a second$)$

Then you are required to do the following:

Open the streifCourseRecords.csv CSV file and read the data into a matrix

Plot the year $($as the X$-$axis$)$ and time in seconds $($as the Y$-$axis$)$

Title the plot, along with providing labels for the X and Y axes

Use the built$-$in function polyfit function to generate the coefficients of both the $1$st and $2$nd order polynomials that best$-$fit this data

Create your own function that calculates and returns the coefficients of the $1$st order polynomial that best$-$fit this data

You will test your routine against MATLAB$$s built$-$in$$the results generated by the two routines should be the same

Print the equations for the $3$ sets of coefficients that you have produced

Use linspace to generate more data points along the X$-$axis

For all $3$ sets of best$-$fit line coefficients:

Use polyval to generate the Y$-$axis coordinates corresponding to the X coordinates produced in step $7$ above

Plot the increased X coordinates $($from step $7)$ and the Y coordinates $($from step $8.$a above$)$

Add a legend for all $4$ plotted data

Describe what the plots are telling you about this data

Step $5$: Create a Best$-$Fit Coefficients Function

As in past assignments, you are free to implement this function either as:

A subfunction contained in the script M$-$file for your program

As a function implemented in a separate M$-$file

You must implement the least squares formula, which generates the coefficients of a $1$st degree polynomial that best$-$fits a given data set. The formula works by minimizing the squares of the differences between each data point and the line that represents all of the data points.

Recall that a $1$st degree polynomial has the form:

and the slope$-$intercept formula for a straight line is the $1$st degree polynomial:

The $1$st degree least squares formula has two parts:

The formula for calculating the slope:

The formula for calculating the Y$-$intercept:

where:

and

are vectors of values associated with the X and Y axes, respectively

is the length of the

$($or

$)$ vector

All

are

$,$ so

Note that MATLAB supports vectorized versions of all of the required functionality: sum, product, and squaring. Thus, no loops are required anywhere in your implementation.

Your function must $($for full credit$)$:

Not use any loops

Generate and return the slope and Y$-$intercept as a $2-$element vector $($just like ployfit does$).$ The vector returned must have the slope in out$\_$vec$(1)$ and the Y$-$intercept in out$\_$vec$(2).$

Step $6$: Output the Polynomials

Example output $($note that these are not solutions$).$

Output for a $1$st$-$degree polynomial will look like this:

y $=-9\mathrm{.}127$x $+814\mathrm{.}906$

And output for a $2$nd$-$degree polynomial:

y $=0\mathrm{.}913$x$^2+-632\mathrm{.}098$x $+278554\mathrm{.}575$

For clarity, each of these equations should be labeled, either on the same line, or just above. Like either of these:

Linear: y $=-9\mathrm{.}127$x $+814\mathrm{.}906$

Quadratic:

y $=0\mathrm{.}913$x$^2+-632\mathrm{.}098$x $+278554\mathrm{.}575$

Step $10$: Describe What the Plots Indicate

This is not a data analysis course, so do not worry about whether your answer makes sense or not. You will get credit here for the effort, not for the answer.

I would like you to produce a short statement in a word processed document $($using Microsoft$$ Word or Google Docs$)$ describing what you think that the plots are indicating about this data:

Will new course record times be set on the Streif?

If so$,$ when will they be likely to be set?

If not, why not?