Assignment 4

Due: Friday, ***, by 5:00pm
You can turn in your assignment until 5:00pm on 3/**/18. You should
hand in both a hardcopy and electronic copy of your solutions. Your hardcopy
submission should include printouts of four code files:
spin.m, lineFit.m, poleVault.m
, and virus.m
.
To save paper, you can cut and paste all of your code files into one file, but your
electronic submission should contain the three separate files.
Your electronic submission is described in the
section Uploading your saved work.
Reading
The following new material in the fifth edition of the text is especially useful to review for this assignment: pages 187188, 192196, 200202, 221231 (fourth edition: pages 190194, 198200, 219229. You should also review notes and examples from Lectures #8, 9 and 11, and Lab #6.Getting Started: Download assign4_programs
Use Cyberduck to connect to the CS server and download a copy of the
assign4_exercises
folder onto your Desktop. This folder contains one file named rotate.m
for
the exercise in this assignment.
Uploading your completed work
When you have completed all of the work for this assignment, your
assign4_exercises
folder should contain one additional file:

spin.m
assign4_problems
folder should contain three additional
code files:

lineFit.m

poleVault.m

virus.m
cs112/drop/assign4
folder. Drag your assign4_exercises
and
assign4_problems
folders to this drop folder. More details about this process
can be found on the webpage on Managing Assignment Work.
Exercise: Spirograph
In this exercise, you'll write a function called
spin.m
that will spin a set of coordinates around in a
circle. You are provided with a function called rotate.m
(in the
assign4_exercises
folder)
that has three inputs: 1) a vector of x coordinates, 2) a vector of y
coordinates, and 3) the angle in degrees to rotate those coordinates.
First, make sure you understand how rotate
works, because your
function spin
will rely upon rotate
.
Understanding rotate.m
Let's take a concrete example of a square. A square with side length 5 can be drawn using these two vectors:
xsquare = [3 8 8 3 3];
ysquare = [3 3 8 8 3];
plot (xsquare, y square);
The following code (the code to generate the yellow axis is not shown here) produced the green, red and black rotated squares at left:

The function rotate
always takes three inputs and returns
two output vectors.
Note that the examples above rotate a square, yet
rotate
can rotate any set of x and y coordinates.
Writing spin.m using rotate.m
Your task is to write the functionspin
using
rotate
. The spin
function will take in three inputs: 1) a
vector of x coordinates, 2) a vector of y coordinates, and 3) the
number of times to repeat the coordinates in the design.
spin
will create a design in MATLAB's figure window.
The steps below will incrementally build your spin
function. You need only turn in the final version of spin
. In the examples below, the same x and y coordinates are used for the square as above. The flower petal coordinates are as follows:
xpetal = [0 2 8 6 0];
ypetal = [0 4 5 2 0];
 For this first simple version,
spin
takes only two inputs: the x and the y coordinate vectors. This version ofspin
will always produce 8 sets of rotated coordinates and plot them in the default blue color.spin(xsquare, ysquare)
spin(xpetal, ypetal)
 Now edit your version of
spin
so that there is a third input, namely, the number of rotations to be plotted. Your editedspin
should plot the userspecified number of rotations of the x and y coordinates, as in the figures below.spin(xpetal, ypetal, 5)
spin(xsquare, ysquare, 11)
spin(xsquare, ysquare, 20)
spin(xpetal, ypetal, 20)
 The final version of
spin
produces a userspecified number of rotations of the given x and y coordinates in multiple colors. The examples below cycle through the available colors and show 50 rotations of each set of coordinates.
spin(xsquare, ysquare, 50)
spin(xpetal, ypetal, 50)
spin
that you submit: the function takes three inputs
(x coordinates, y coordinates, and the number of rotations to be plotted) and produces one colorful
figure.
Problem 1: Able to leap tall buildings in a single bound!
One true test of any scientific theory is whether or not it can be used to make accurate predictions. Given some data that captures the relationship between two or more variables, we can try to formulate a mathematical model that summarizes this relationship. If the model is valid, it can be used to predict the relationship between the variables in cases not explicitly given in the original data.
In some cases, variables may have a simple linear relationship, such as in
the forearm and hand data that we used to test the existence of the Golden Ratio:
The line drawn through the points is the best fit line for the data, also
referred to as the regression line. MATLAB provides functions for fitting lines
and other curves to data, but you'll instead write your own function to compute the
regression line for a set of data, and tailor the information that is returned.
You will then apply this analysis to data on the achievements
of olympic pole vaulters in the summer olympics, from 1896 to 2004. Finally, you
will analyze the pole vaulting data using the Curve Fitting Toolbox that you explored in
lab.
Computing a regression line
A nice introduction to the computation of regression lines is provided online at this Finite Mathematics & Applied Calculus resource developed by Stefan Waner and Steven Costenoble at Hofstra University.
Given the (x,y) coordinates for a set of n points (x_{1},y_{1}), (x_{2},y_{2}), ... (x_{n},y_{n}), the best fit line associated with these points has the form
y = mx + b
where
slope m = (n(Σxy)  (Σx)(Σy)) / (n(Σx^{2})  (Σx)^{2})
intercept b = (Σy  m(Σx)) / n
The Σ means "sum of", so
Σx = sum of x coordinates = x_{1} +
x_{2} + ... + x_{n}
Σy = sum of y coordinates = y_{1} +
y_{2} + ... + y_{n}
Σxy = sum of xy products = x_{1}y_{1} +
x_{2}y_{2} + ... + x_{n}y_{n}
Σx^{2} = sum of squares of x coordinates =
x_{1}^{2} + x_{2}^{2} + ... +
y_{n}^{2}
When performing linear regression, it is valuable to know how well the line actually
fits the data. Two measures used to assess the quality of fit are the correlation
coefficient and the size of the residuals that capture the difference between
the actual data values and the values predicted by the regression line. The correlation
coefficient, also described in the Waner and Costenoble online chapter, is a number
r
between 1 and 1 calculated as follows:
coefficient r = (n(Σxy)  (Σx)(Σy)) / [n(Σx^{2})  (Σx)^{2}]^{0.5}[n(Σy^{2})  (Σy)^{2}]^{0.5}
A better fit corresponds to a value of r
whose magnitude is closer to 1,
while a worse fit yields a value of r
closer to 0. The residuals are the
discrepancies between the actual data (actual y
values) and those predicted by the
best fit line (the values mx
+
b
). A rough estimate of the average size of the
residuals is given by the RMS error between these two quantities:
average residual RMS = ((Σ(y  (mx + b))^{2}) / n)^{0.5}
Implementing linear regression
Write a function named lineFit
that has two inputs that are vectors containing
the x and y coordinates of a set of points. This function should return four values, all obtained using
the above calculations: the (1) slope m
and (2) intercept b
of the best fit line,
the (3) correlation coefficient and (4) average residual. Test your function with a small number
of points that you create. You can check your results for the best fit line against
those obtained with the MATLAB polyfit
function, which returns a vector
containing the m
and b
values:
>> lineMB = polyfit(xcoords, ycoords, 1)
Note: you do not need to use any loops (for
statements) in your
lineFit
function  all of the calculations can be done by performing
arithmetic operations on the entire vectors of x and y coordinates all at once. This
problem primarily provides practice with writing a function with multiple inputs and
outputs, and more experience with curve fitting.
function [sumV diffV prodV divV] = compute(vect1, vect2) % [sumV diffV prodV divV] = compute(vect1, vect2) % computes the elementbyelement sum, difference, product and division % of the values in two input vectors and returns the four results sumV = vect1 + vect2; diffV = vect1  vect2; prodV = vect1 .* vect2; divV = vect1 ./ vect2;
The future of olympic pole vaulting
From the time the summer olympics began in 1896, until 2004, pole vaulters have been
achieving heights that
have increased in a roughly linear fashion (heights are given in inches):

In the assign4_problems
folder, there is a MATfile
named poleVaultData.mat
that contains two variables years
and heights
that store the above data.
Write a script file named poleVault.m
that performs the following actions:
 loads the
poleVaultData.mat
file  plots the data (height vs. year) using the
scatter
function to create a scatter plot:scatter(xcoords, ycoords)
Check the MATLAB help pages for properties that can be used to change the appearance of the dots, and incorporate some of these properties into your scatter plot.
 calculates the best fit line using your
lineFit
function  draws the bestfit line superimposed on the scatter plot, using the
plot
function (remember that you only need two points to draw a line!)  assuming that this is an accurate model for predicting the future of pole vaulting, predict the year in which pole vaulters will be able to leap tall buildings in a single bound  in this case, Green Hall, which reaches 182 feet from the ground to the highest finial
 prints the predicted year in which a pole vaulter will leap over Green Hall  the floating
point value for this year can be converted to an integer using the
uint16
function:>> uint16(5626.7864)
ans =
5627
In a comment at the end of the poleVault.m
script, write the predicted
year that is printed by your script, and also comment on the reasonableness of the model.
Using the curve fitting tool
After running your poleVault.m
script, the two variables years
and
heights
will be stored in your Workspace. Open the curve fitting tool with the
cftool
function and create a data set with years
as the X data and
heights
as the Y data. A linear polynomial fit to this data should yield a line
similar to what you obtained with your linefit
function. A better model of
pole vaulting performance, though, would use a function that reaches a plateau as
the year increases. One example is a logarithmic function. To obtain a fit to a log function,
select Custom Equation
for the type of fit.
A default exponential equation will appear in the
equation box. Replace this expression with the following general logarithmic expression:
a * log(b*(x1895))
The logarithmic function does not
fit the past data as tightly, but probably has better predictive capability for the future.
Use both the logarithmic curve fit and a linear curve fit to predict pole vaulting heights
for the year 3000. Record this information in your poleVault.m
script file
and comment on which fit appears to yield a more reasonable prediction.
Problem 2: Gesundheit! The Spread of Disease
Imagine that the Wellesley College community has been quarantined due to a sudden outbreak of an annoying stomach virus. How quickly can this virus spread through the population, and will it eventually die out? This depends on factors such as the ease with which the virus is passed from one individual to another, the time it takes to recover from the virus, and the time during which an individual remains immune to the virus after recovering. One of the simplest models of the spread of disease was developed by W. O. Kermack and A. G. McKendrick in 1927 and is known as the SIR model. Its name is derived from the three populations it considers: Susceptibles (S) have no immunity from the disease, Infecteds (I) have the disease and can spread it to others, and Recovereds (R) have recovered from the disease and are immune to it.
In this problem, you will model the spread of a virus over time, through a population that is represented on a twodimensional grid. Suppose that each cell on a 100x100 grid is an individual in a group of 10,000 people. Each individual can be susceptible, infectious or immune. Assume that the infection lasts two days and immunity lasts 5 days before the individual becomes susceptible again. The state of this virus in the population can be stored in a 100x100 matrix that contains values from 0 to 7 representing the following conditions:
 0: susceptible individual
 1,2: infectious individual in the first or second day of the infection
 3,4,5,6,7: immune individual in the first, second, third, fourth or fifth day after recovery
The virus.m
code file in the assign5_problems
folder contains
partial code for a function named virus
that simulates the spread of
the virus through the population, over a period of days.
The virus
function has two inputs: the number of days to run the simulation
and the probability that a susceptible individual will become infected if one of their
neighbors is infected. The 100x100 matrix grid1
is filled with the initial
state of the simulation on day 1, in which all of the individuals are susceptible, except
for a single infected individual in the center of the grid. The function
displayGrid
, also provided in the assign5_problems
folder,
displays the current grid as a color image. Susceptible individuals are shown in blue,
infected individuals are bright red on the first day and dark red on the second, and
recovered individuals are shown in shades of green from bright (value 3, first day of
recovery) to dark (value 7, fifth day after recovery). The code in the
virus
function displays the initial grid on
day 1.
Your task is to complete the virus
function to simulate the spread of the
virus, starting with day 2 and continuing over the input number of days. A second 100x100 matrix
named grid2
is created in the initial code, to assist with the simulation. For
each new day, assume that the current state of the virus is stored in grid1
.
Loop through all of the locations of grid1
(individuals in the population),
compute the new value for each individual and store it in the same location of grid2
.
For example, if the value in grid1
at a particular location is 0 (susceptible
individual) and one of the neighbors (above, below, left or right) is infected (i.e. has the
value 1 or 2), then the input probability probInfect
should be used to
determine whether the value stored at this location in grid2
should be a 0
(individual remains susceptible) or 1 (individual becomes infected). For each of the other
values stored in grid1
, from 1 to 7, think about what value should be stored in
grid2
reflecting the state of the individual on the next day. (Remember that
immunity from the virus lasts for only 5 days after recovery, before the individual
becomes susceptible again.) Display the new grid with the displayGrid
function.
After displaying the new grid, add a short pause to your code so that the new grid stays visible
for a short amount of time. In this way, the spreading virus will appear as an animation. The
builtin MATLAB function pause()
has a single input that is the number of seconds
to pause, which can be a fraction. For example, the statement pause(0.1)
will cause
MATLAB to pause for one tenth of a second before continuing the execution of your code.
After each day of the simulation is complete, copy the contents of
grid2
into grid1
in preparation for the next day of the simulation.
To simplify the task of checking whether a neighbor is infected, assume that there is a border
of individuals around the grid (rows 1 and 100, and columns 1 and 100) that remain susceptible
(value 0) throughout the simulation. When looping through grid1
, you only need
to consider individuals in the rows and columns 2 through 99. Finally, the rand
function can be used to determine whether a susceptible individual with an infected
neighbor, becomes infected themselves. The function call rand(1)
returns a
single random number between 0.0 and 1.0. Suppose we want to simulate the flipping
of a coin that is biased towards heads so that on average, 60% of the flips come up heads.
The following loop simulates 100 flips of this coin using a probability of 0.6:
for flips = 1:100
if (rand(1) <= 0.6)
disp('heads');
else
disp('tails');
end
end
An analogous strategy can be used to determine whether a susceptible individual becomes infected.
The following four pictures show the state of the virus for the first four days of a sample
simulation. Results are shown for a 5x5
section around the center of the
grid. On day 1, only the individual at the center is infected. The simulation used a
probability of infection of 0.5, and on day 2, two neighbors of the center individual
become infected. On day 3, three more individuals become infected, and on day 4, an
additional four individuals become infected. Note that once an individual is infected,
the value at that location increments as each day passes.
0  0  0  0  0 
0  0  0  0  0 
0  0  1  0  0 
0  0  0  0  0 
0  0  0  0  0 
0  0  0  0  0 
0  0  1  0  0 
0  1  2  0  0 
0  0  0  0  0 
0  0  0  0  0 
0  0  1  0  0 
0  1  2  0  0 
0  2  3  1  0 
0  0  0  0  0 
0  0  0  0  0 
0  0  2  0  0 
1  2  3  1  0 
0  3  4  2  1 
0  1  0  0  0 
0  0  0  0  0 
The following picture shows an example of the state of the virus for the entire grid, after 60 days:
>> virus(60, 0.5);
Run simulations for a few different values of the probablity (for example, 0.25, 0.5,
0.75, 1.0) and add comments to the virus.m
code file about what you observe.
You now have a colorful and dynamic visualization of a spreading virus. Suppose you then want
to quantify the number of individuals who are either susceptible, infected, or immune to the
virus on each day. Add code to your virus()
function to create three vectors to store
the number of susceptible individuals (value 0 in the grid), infected individuals (values 1 or 2
in the grid) or immune individuals (values 37 in the grid). As your simulation calculates the
state of the virus on each day, count the number of individuals in each category (you should not
need a loop for this calculation!) and store it in the appropriate vector. You can assume that
there is a total of 10,000 individuals in the population (one for each grid location). At the end
of the virus()
function, plot the three populations on a single graph, with different
colors (an example is shown below).
With only one infected individual on the first day, these curves are not very interesting. The
final step in the construction of your program is to add the ability to specify an arbitrary
number of individuals who are infected on the first day of the simulation, and place them at
randomly selected locations of the grid. First, add a third input to the virus()
function that is used to specify the number of individuals who are infected on the first day.
Make this an optional input. If the user does not specify a value for this input, your
function should place a single infected individual at the center of grid1
at the
outset (the code currently does this). However, if the user specifies a value for this input,
place this number of 1's in grid1
at the outset. The row and column number for each
1 (infected individual) should be a random integer in the range between 2 and 99 (omit the outer
rows and columns of the grid). The builtin function randi(imax)
returns a random
integer in the range from 1 to imax
, and can be helpful here. (Note that the
randi()
function is a relatively new addition to MATLAB and is not available in
older versions.)
The following figure is a sample plot generated by the following function call:
>> virus(100, 0.5, 10);
Run your simulation for 100 days, using a couple different values of the probInfect
input, combined with a couple different values for the number of individuals who are initially
infected. Add comments to your code describing how the graphs of the susceptible, infected,
and immune individuals changed with different input parameters.