Cogsci 109 Assignment 4

Deadline extended

Due Thursday November 10, 11:59pm

1. Monte-Carlo Simulations

One use of computers in computational models and data analysis is to randomly simulate the behavior of a physical or mathematical system that might be hard to compute exactly ( see for much more on this topic)
In the first question, you will write Matlab programs to estimate the value of pi. Keeping in mind that the area of a circle of radius r is pi*r*r and the area of a square that just encloses the circle is 2r*2r=4*r*r, one can estimate the value of pi by uniformly and randomly generating points from a square of width 2r and seeing what fraction fall in the enclosed circle.

consider a circle inside a square (just touching the edges as I drew on the board)

The circle has radius r, so the square as width 2r

Area of a circle = pi * r^2

Area of square = 4*r^2

If you uniformly randomly generate points in the square and see which fall in the circle you have the following relationship (keeping in mind that uniformly means proportion to area)

# in circle
----------
# in square

is approximately equal to

area circle
------------
area square

which is equal to

pi
---
4
Therfore pi can be approx by

4*# in circle
------------
# in square

You might want to make r=.5 and look at the following matlab commands:

rand

a) First write a program that loops 1000000 times, each time drawing two random numbers. Save this program in a file piloop.m
b) Now improve the speed of the program by removing the loop. Save this program in a file pinoloop.m You might want to look at the following matlab commands/operators:

find  .^2

2. Linear Regression

For this question you will use the data files:
xdata
ydata
xdata holds the x-coordinates of a set of data and ydata holds the corresponding y-coordinates. You might want to look at the matlab command :

 load

Put the following code in a script file called regression.m
a) write the code to plot the 2-dimensional data as green stars.
b) write the code to fit the best (in the least squares sense) linear fit to the data.
c) write the code to draw this on the graph of the data as a red line.
d) write the code to fit the best fifth order fit to the data and draw this on the graph of the data as a black curve. Which looks like a better fit to you? Next homework we will see one way to quantify this.

What to Hand In

Hand in your script files piloop.m, pinoloop.m and your script file regression.m, using the turnin program. There is no need to hand in graphs.