Math 232 – Computing Assignment 4 SOLVED

Math 232 – Computing Assignment 4

Programming Preamble:
Matlab: x=[1 1 1]’ produces a column vector. The ’ indicates transpose.
Matlab: help plot gives you all the information you need to make your plots look good - labelling
the axes, putting on a title etc.
Computing Assignment
Required submission: 1 page PDF report and Matlab or Python code (.m or .py respectively,
exported as a .pdf) uploaded to Canvas.
Before you begin, consult the ’Least Squares preamble’ file posted
on Canvas
Machine Learning
This assignment will give you a glimpse into one of the main areas of Machine Learning (Data
Science); that of regression and classification.
We study two modelling examples that use historical data on the behaviour of a specified group of
people to make predictions on their future behaviour. In particular, we study student success in
Math 232 based on their study habits (number of hours spent studying Math 232) and their grades
in previous mathematics courses. (Note: all the data here is purely hypothetical.)
1. Our first approach to this problem will use (hypothetical) historical data on the number of
hours per week students spent studying math 232 in a previous semester, and their subsequent
grade in the course. What we want to obtain from this data is a criteria for “success” in math
232 (i.e., grade > 50%) based solely on how much time a student spends studying the course
materials per week.
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Data for this assignment is contained in the excel file CA4data.xslx that is posted on Canvas.
This (hypothetical) data set contains the number of hours per week students spent studying
math 232 during a past semester, their grade in a previously taken math course (math 152,
taken before they took math 232), and their final grade in math 232 in that semester. In Q1
below, we will use only the studying hours, and Q2 below will use both the studying hours
and the grades in math 152 to build a model from which we can make predictions about
student ‘success’ in math 232.
Read in the data file CA4data.xslx that is available for download on Canvas. Open the file
in excel and remove the first line (which labels the columns). Use these commands to create
(column) vectors with data taken from the columns in the file;
>> data = ’CA4data.xlsx’;
>> D=xlsread(data);
>> h=D(:,1);
>> k=D(:,2);
>> g=D(:,3)
h,k,g are column vectors with the data #hours studied per week, grade in math 152, and
the grade received in math 232, respectively.
The data set for the first question is D1 = {(hi
, gi), i = 1, . . . , 45}. We look for a linear
model of the form g = a+bh. That is, we look for the best fitting (least squares) line through
the data points D1 in R
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.
1a. Let the function gg=a+bx (Matlab notation) be your best fit linear model through the
data (here you have to first compute what those parameters a, b are using our theory of least
squares approximation). The Matlab command
>> x=0:0.2:12;
>> gg=a+b*x;
plot(x,gg)
will plot this line.
Plot your best fit linear line along with the data set using these commands;
>> plot(h,g,’k.’)
>> hold on
>> plot(x,gg)
>> hold on
>> axis([0 15 20 100])
>> hold on
>> xlabel(’hours per week’); ylabel(’grade’)
From your linear model of hours spent studying vs final grade in math 232, what do you
predict is the “success” point; that is, the (minimum) number of hours a student should
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spend per week studying math 232 in order to pass the course? (this will be a point ho on the
h−line (‘hours per week’) where h ≥ ho indicates a grade of at least 50% will be obtained).
1b. Now find the best fit quadratic model to the data ; g = a + b1h + b2h
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. That is, the least
squares quadratic that fits the data. Plot this quadratic along with the data points. From
this quadratic find the predicted success rate. (Notice here that although we are fitting a
nonlinear function to the data set, the least squares algorithm is still a linear problem!)
2. In this second approach we include another ’diagnostic’ from the group of students; that of
their previously taken math 152 grade. So now we use both the number of hours they spent
studying math 232 and their grade in math 152 to make a prediction on student success in
math 232.
Thus, our data set is D2 = {(hi
, ki
, gi), i = 1, . . . , 45}. You can take a look at this 3-dimension
scatter plot with the commands;
>> plot3(h,k,g,’k.’)
>> grid on
(Note that you can drag the mouse on the figure to rotate it and obtain different perspectives).
Our linear model for this data set is g = a + bh + ck. That is, we look for the best fitting
plane through the data points in R
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.
Compute (using the algorithm discussed in class) the least squares plane that fits the data
points. You can plot the plane with the data points using these commands;
>> plot3(h,k,g,’k.’)
>> axis([0 15 20 100 20 100])
>> grid on
>> hold on
>> [X,Y]=meshgrid(x,y);
>> ggg=a+b*X+c*Y;
>> mesh(ggg)
where a+b*X+c*Y is your least squares fitting plane to the data.
From this determine the ‘success line’ in the h, k-plane; one side of this line predicts the
student will receive a grade of ≥ 50%. Plot this line in the h, k-plane indicating the ‘success’
side of it. (Hint: Recall that the intersection of two planes is a line (in 3 dimensions); then
project this line down into the h, k-plane.)
(Remark: In a similar way as one fitted the least squares quadratic curve through the 2
dimensional scatter plot as in Q1, one could here find the least squares quadratic surface
g = ao + a1h + a2k + a3hk + a4h
2 + a5k
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through the 3 dimensional scatter plot. Again, it is
a linear problem determining the parameters a1, . . . , a5. We will NOT be doing that as part
of this assignment.)
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