Homework_4_LinAlg
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Georgia Institute Of Technology *
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Homework 4
Submission Requirements:
Please submit to Canvas the following files:
1.
A .pdf of this document with the required answer (highlighted yellow) and figures.
2.
Your script Homework_4_LastName.m
(replace LastName
with your last name).
3.
Two functions (2) linearRegression.m
and rootMeanSquaredError.m
4.
The data file Homework_4_Data.mat
Part 1: Linear Regression
For this homework, you are going to write a function in MATLAB called linearRegression
to solve for the coefficients of a line using least-squares regression. You will then use your
function to create a regression model for nonlinear data after linearizing it. Remember that we
are representing out linear fit in the form: ^
y
i
=
β
0
+
β
1
x
i
a.
Hand calculation
First you will do a hand calculation on a smaller dataset to check the output of your
function. i
x
y
1
0
0.9351
2
1
2.1181
3
2
2.9242
4
3
3.8890
5
4
4.9154
6
5
5.9427
What are the values of your matrices below?
X
=
[
¿
]
, X
T
=
[
¿
]
,β
=
[
¿
]
, y
=
[
¿
]
What are the values of your parameters for your linear fit equation?
β
1
=
¿
¿
β
0
=
¿
¿
ME 2016A– Spring 2023
Copyright Georgia Institute of Technology 1
b.
Linear regression MATLAB function
Write a function named linearRegression
which has the independent variable
values, x
, and the dependent variable values, y
, as inputs and the two regression
coefficients, a
1
& a
0
, as the outputs. Calling the function in your MATLAB script would
look like: [ beta1, beta0 ] = linearRegression( x, y );
What are the values of the slope, β
1
, and the intercept, β
0
, of the line, ^
y
=
β
1
x
+
β
0
?
(Hint: Both should be close to a value of 1 and should agree with your hand
calculations):
β
1
=
¿
¿
β
0
=
¿
¿
c.
Plotting the data and model
In your script, plot the dataset of x
and y
as markers in a figure along with your fit line
from your linear regression. Copy and paste your figure below making sure to include
axis labels that are legible, and a legend that denotes the markers as data and has the
equation of the line with the actual coefficient values.
Figure here
d.
Root Mean Squared Error MATLAB function
Write a function named rootMeanSquaredError
which has the dependent data
values, y
, and the predicted model values, y
pred
, as inputs and the model root mean
squared error, RMSE
, as the output. Calling the function in you MATLAB script would
look like: [ RMSE ] = rootMeanSquaredError( y, y_pred );
What is the linear model root mean squared error? (Hint: 0.06
<
RMSE
<
0.08
):
RMSE
=
¿
¿
Part 2: Linearization of Non-Linear Relationships You are now going to use the dataset contained in Homework_45_Data.mat
and determine
which model that can be linearized best fits the data. You are going to call your previously
created functions to help you do this.
ME 2016A– Spring 2023
Copyright Georgia Institute of Technology 2
a.
Exponential Equation
In class we saw that the exponential equation:
y
=
A e
bx
Can be linearized by taking the natural logarithm of both sides so that it can be written
as:
ln
(
y
)
=
ln
(
A
)
+
bx
Using your functions linearRegression
and rootMeanSquaredError
, what
are the two coefficients of the of the exponential model for the data and what is the
RMSE for model? (Hint: 0.4
<
A
<
0.6
, 0.1
<
b
<
0.4
, & 0.01
<
RMSE
<
0.06
)
A
=
¿
¿
b
=
¿
¿
RMSE
=
¿
¿
b.
Power Equation
In class we saw that the power equation:
y
=
A x
b
Can be linearized by taking the base-10 logarithm of both sides so that it can be written
as:
log
(
y
)
=
log
(
A
)
+
b∙
log
(
x
)
Using your functions linearRegression
and rootMeanSquaredError
, what
are the two coefficients of the of the power model for the data and what is the RMSE
for model? (Hint: 0.4
<
A
<
0.6
, 0.2
<
b
<
0.4
, & 0.01
<
RMSE
<
0.06
)
A
=
¿
¿
b
=
¿
¿
RMSE
=
¿
¿
c.
Saturation-Growth-Rate Equation
In class we saw that the Saturation-Growth-Rate equation:
ME 2016A– Spring 2023
Copyright Georgia Institute of Technology 3
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y
=
Ax
x
+
b
Can be linearized by taking the inverse of both sides so that it can be written as:
1
y
=
1
A
+
b
A
1
x
Using your functions linearRegression
and rootMeanSquaredError
, what
are the two coefficients of the of the Saturation-Growth-Rate model for the data and
what is the RMSE for model? (Hint: 0.9
<
A
<
1.1
, 0.9
<
b
<
1.1
, & 0.01
<
RMSE
<
0.06
)
A
=
¿
¿
b
=
¿
¿
RMSE
=
¿
¿
d.
Selecting a Model
Based upon your calculation for the error for each of the models, select the model
which minimizes the error and create a figure in your script which plots the data as
markers and plots a line for the model predicted values from 1
to 5
. Copy and paste
your figure below making sure to include axis labels that are legible, and a legend
denotes the markers as data and has the equation of the line with the actual coefficient
values.
Figure here
Part 3: Polynomial Regression
For this part you are going to derive the equations to determine the coefficients of a polynomial
using least-squares regression
. You want to create a 2
nd
order model so that the y-intercept is
always equal to zero: ^
y
=
β
2
x
2
+
β
1
x
a.
Solving for the coefficients
In MATLAB solve for the coefficients using the dataset from Homework_4_Data.mat
.
(Hint: 0.4
<
a
1
<
0.6
& −
0.07
<
a
2
←
0.05
)
β
1
=
¿
¿
β
2
=
¿
¿
b.
Evaluating the model
ME 2016A– Spring 2023
Copyright Georgia Institute of Technology 4
What is the RMSE for the 2
nd
order model? (Hint: 0.05
<
RMSE
<
0.07
)
RMSE
=
¿
¿
c.
Plotting the model and data
Create a figure in your script which plots the data as markers and plots a line for the
model predicted values from 1
to 5
. Copy and paste your figure below making sure to
include axis labels that are legible, and a legend denotes the markers as data and has
the equation of the line with the actual coefficient values.
Figure here
ME 2016A– Spring 2023
Copyright Georgia Institute of Technology 5
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