Description
1. Pearson’s Coefficient is widely used in the natural sciences as a measure of correlation between two variables.
Given two lists, X = (x1 x2
. . . xn
) and Y = ( y1 y2
. . . yn
), each containing n values, Pearson’s Coefficient
is defined by the expression:
Pn
i=1
(xi − X¯)( yi − Y¯)
qPn
i=1
(xi − X¯)
2
qPn
i=1
( yi − Y¯)
2
,
where X¯ and Y¯ denote the averages of the xi and the yi
, which is to say that
X¯ =
1
n
X
i
xi and Y¯ =
1
n
X
i
yi
.
(a) Write a function that takes a list and returns its sum.
(b) Write a function that takes a list X = (x1 x2
. . . xn
) and returns the average X¯. (Note: You will have
to compute the length of the list in order to compute the average.)
(c) Write a function that maps a list X to the square of its deviation. Thus the list X = (x1 x2
. . . xn
)
should be carried to the list
((x1 − X¯)
2
. . . (xn − X¯)
2
).
You should use the map function. (For example, your function, when evaluated on the list (1 2 3 4 5),
should return (4 1 0 1 4).)
(d) Write a function that returns the standard deviation of a list, q
(
1/n)
Pn
i=1
(xi − X¯)
2. This should be
easy, you already have the functions you need.
(e) Write a new version (called map2) of the map function which operates on two lists. Your function
should accept three parameters, a function f and two lists X and Y , and return a new list composed of the function f applied to corresponding elements of X and Y . In particular, given two lists
(x1 x2
. . . xn
) and ( y1 y2
. . . yn
) (and the function f ), map2 should return
(f (x1
, y1
) . . . f (xn
, yn
)).
(f) Write a function that, given two lists X = (x1 x2
. . . xn
) and Y = ( y1 y2
. . . yn
), computes the
covariance list:
((x1 − X¯)( y1 − Y¯) . . . (Xn − X¯)(Yn − Y¯)).
(g) Write a function that computes Pearson’s Coefficient. It might be useful to observe that an equivalent
way to write Pearson’s coefficient, by dividing the top and bottom by n, is
1/n
Pn
i=1
(xi − X¯)( yi − Y¯)
σ(X)σ(Y )
.
2. Least-Squares Line Fitting. Line fitting refers to the process of finding the “best” fitting line for a set of
points. This is one of the most fundamental tools that natural scientists use to fit mathematical models to
experimental data.
As an example, consider the red points in the scatter plot of Figure 1. They do not lie on a line, but there is
a line that “fits” them very well, shown in black. This line has been chosen—among all possible lines—to
be the one that minimizes the sum of squares of the vertical distances from the points to the line. (This may
1
160 180 200 220 240 260 280
20
40
60
80
100
120
140
Figure 1: An example of a scatter plot and a best-fit line.
seem like an odd thing to minimize, but it turns out that there are several reasons that it is the “right” thing
minimize.)
Since the equation of a line is y = a+bx, this boils down to finding a slope b and x-intercept a for given lists
X and Y corresponding to point data. It turns out that the best fitting line can be found with the following
two equations:
b = r ·
σ(Y )
σ(X)
, and a = Y¯ − bX¯ ,
where σ(X) and σ(Y ) refer to the standard deviations of X and Y and r is Pearson’s Coefficient.
(a) Write a function that takes two lists X and Y (of the same length) and returns a pair (a, b) corresponding to the best fit line for the data.
(b) Write an alternate version of best-fit that returns the best fitting line as a function. Specifically, write
a new function best-fit-fn so that (best-fit-fn pX pY) returns the function which, given a
number x, returns ax + b where a and b are the best fit coefficients.
Use this to the best fit line for the following data:
X = {160, 180, 200, 220, 240, 260, 280} , Y = {126, 103, 82, 75, 78, 40, 20} .
Call the function fitline so that you can graph it in the next problem.
(c) We would like to plot the original data along with the best fit line. To do this we will need to “zip”
lists X and Y into a list of data points. You wrote zip in last week’s homework. Rewrite this function
as zip2 and make use of map2 function (the implementation should be much simpler). The plotting
package requires a list of vectors rather than pairs. Vectors haven’t been covered in class, but it will
suffice to replace the cons keyword with vector in your implementation of zip2.
Now plot the best fit line and data using the following command:
( plot ( list ( axes )
( function fitline 140 300)
( points ( zip2 X Y ))))
This produces the image of Figure 2.
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Figure 2: The result of the Racket plot.
3. Define a SCHEME function, merge, which takes two lists `1 and `2 as arguments. Assuming that each of `1
and `2 are sorted lists of integers (in increasing order, say), merge must return the sorted list containing all
elements of `1 and `2
.
To carry out the merge, observe that since `1 and `2 are already sorted, it is easy to find the smallest element
among all those in `1 and `2
: it is simply the smaller of the first elements of `1 and `2
. Removing this smallest
element from whichever of the two lists it came from, we can recurse on the resulting two lists (which are
still sorted), and place this smallest element at the beginning of the result.
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