CS 112

Assignment 4

Due: Friday, March 13, at 10:00am (notice the change in submission time!)

You can turn in your assignment up until 10:00am on 3/13/15. 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 187-188, 192-196, 200-202, 221-231 (fourth edition: pages 190-194, 198-200, 219-229. You should also review notes and examples from Lectures #8, 9 and 11, and Lab #6.

Getting Started: Download assign4_programs from cs112d

Use Fetch or WinSCP to connect to the CS server using the cs112d account and download a copy of the assign4_exercises folder from the cs112d directory 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 code file for the exercise, spin.m. Your assign4_problems folder should contain three additional code files, lineFit.m, poleVault.m, and virus.m. Use Fetch or WinSCP to connect to your personal account on the CS file server and navigate to your 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 spirograph toy

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);

squares graphic

The following code (the code to generate the yellow axis is not shown here) produced the green, red and black rotated squares at left:


 hold on;
 [xs2 ys2] = rotate(xsquare, ysquare, 45);
 plot (xs2, ys2, 'g-');  % the green diamond
 [xs3 ys3] =  rotate(xs2, ys2, 45);
 plot (xs3, ys3, 'r-');  % the red square
 [xs4 ys4] =  rotate(xs3, ys3, 45);
 plot (xs4, ys4, 'k-');  % the black diamond

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 function spin 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];

  1. For this first simple version, spin takes only two inputs: the x and the y coordinate vectors. This version of spin will always produce 8 sets of rotated coordinates and plot them in the default blue color.

    spin(xsquare, ysquare) spin(xpetal, ypetal)

  2. Now edit your version of spin so that there is a third input, namely, the number of rotations to be plotted. Your edited spin should plot the user-specified 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)

  3. The final version of spin produces a user-specified 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)
This should be the version of 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 (x1,y1), (x2,y2), ... (xn,yn), the best fit line associated with these points has the form

   y = mx + b

where

   slope   m = (n(Σxy) - (Σx)(Σy)) / (n(Σx2) - (Σx)2)

   intercept   b = (Σy - m(Σx)) / n

The Σ means "sum of", so

   Σx = sum of x coordinates = x1 + x2 + ... + xn
   Σy = sum of y coordinates = y1 + y2 + ... + yn
   Σxy = sum of xy products = x1y1 + x2y2 + ... + xnyn
   Σx2 = sum of squares of x coordinates = x12 + x22 + ... + yn2

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(Σx2) - (Σx)2]0.5[n(Σy2) - (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.

Hint: The following function illustrates the use of multiple inputs and outputs to perform simple computations on two input vectors and return the results:
function [sumV diffV prodV divV] = compute(vect1, vect2)
% [sumV diffV prodV divV] = compute(vect1, vect2)
% computes the element-by-element 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):

    
YearHeight  YearHeight
1896130  1960185
1900130  1964200.75
1904137.75  1968212.5
1908146  1972216.5
1912155.5  1976216.5
1920161  1980227.5
1924155.5  1984226.25
1928165.25  1988237.25
1932169.75  1992228.25
1936171.25  1996233
1948169.25  2000232.28
1952179  2004234.25

In the assign4_problems folder, there is a MAT-file 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:

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*(x-1895))

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 two-dimensional 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:

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 built-in 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.

Day 1
00000
00000
00100
00000
00000
   
Day 2
00000
00100
01200
00000
00000
   
Day 3
00100
01200
02310
00000
00000
   
Day 4
00200
12310
03421
01000
00000

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 3-7 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 built-in 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.