Project 8 Matrices Solution

The objective of this assignment is to write a Matrix class
for 2D matrices of arbitrary size, and to use it for a)
calculation of correlations matrices for the Iris database,
and b) rotating geometrical objects (2D points) with
rotation matrices.

Matrix class design.

You will want several constructors, including a constructor
that takes dimensions P and Q and allocates space on the
heap for P*Q floats, and a copy constructor.
You will want to implement setter and getter functions
that look like this

float Matrix::get(int row,int col);
void Matrix::set(int row,int col,float val);

You will want to implement a print() function. You will need to be
able to generate the transpose of a matrix, where the rows and columns
are swapped. Finally, our calculations will rely on matrix
multiplication, so you will want to implement a prod() function.


Matrix Multiplication.

Two matrices, A and B, can be multiplied if the number of columns of A
matches the number of rows of B. If the dimensions of A are (rA x cA)
and the dimensions of B are (rB x cB), the the dimenions of the
product will be (rA x cB). The pseudocode for matrix multiplication
is this (using 1-based indexing):

for i = 1..rA
for j = 1..cB
C[i,j] = 0
for k = 1..cA
C[i,j] += A[i,k]*B[k,j]

(Of course, your C++ program will probably want to use 0-based indexing.)


Application 1: Data Science (correlation matrix of Iris attributes)

Read in the Iris database. Select only the "iris-setosa" records.
Extract the 4 floats. Use them to initialize an Nx4 matrix. Then
"center" the data (for the values in each column, subtract the mean of
the column).

If X is a centered (M x N) data matrix (with M instances as rows and
N attributes as columns), then the covariance matrix is defined

cov(X) = X^T * X

or in other words, the transpose of the matrix times itself. This
produces a symmetric NxN matrix that looks like this:

S(1,1) S(1,2) S(1,3) ...
S(2,1) S(2,2)
S(3,1) ...
... S(N,N)

Where S(i,j) is the covariance between attributes i and j, and
the variances of each attribute are on the diagonals, S(i,i).
Finally, the correlation matrix consists of entries cor(i,j)
where:

cor(i,j) = S(i,j)/sqrt(S(i,i)*S(j,j))

In other words, each entry in the covariance matrix is
normalized by dividing by variances on the diagonal.
The correlation matrix should have 1's on the diagonal.

Here is my output for the iris-setosa subset:

covariance matrix:
6.088 4.915 0.791 0.517
4.915 7.114 0.572 0.560
0.791 0.572 1.475 0.279
0.517 0.560 0.279 0.563

correlation matrix:
1.000 0.747 0.264 0.279
0.747 1.000 0.177 0.280
0.264 0.177 1.000 0.306
0.279 0.280 0.306 1.000


See the bottom of this web page for a visual depiction of
the correlations among the attributes for different
subsets of the Iris database:

http://rstudio-pubs-static.s3.amazonaws.com/376329_7bd791576b7240d2b8e6d251d6929aab.html



Application 2: Rotating Geometrical Objects (2D Points)

Operations on 2D points can be doing using matrices.
Derive subclasses for Point2D and Rot2D.
A Point2D is a 1x2 matrix that can be initialized with
2 floats, x and y (e.g. coordinates). A Rot2D is a
2x2 rotation matrix. A matrix representing a rotation
by theta (in radians) looks like this:

cos(theta) -sin(theta)

sin(theta) cos(theta)

To rotate a point, take the product of the vector
(as a 1d matrix) with the rotation matrix.

Here are some test cases.
Let A B C and D be matrices corresponding to
rotations of 30, 60, 90, and -60 degrees.
Then A*B should equal C and B*D should equal
the identity matrix:

A
0.866 -0.500
0.500 0.866

B
0.500 -0.866
0.866 0.500

C
-0.000 -1.000
1.000 -0.000

D
0.500 0.866
-0.866 0.500

A*B
-0.000 -1.000
1.000 -0.000

B*D
1.000 0.000
0.000 1.000

Use this to calculate the coordinates of one of the
hands of a clock on each hour. Define the hand
by 3 points:

(0,5) A
/|\
/ | \
/ | \
/ | \
/ | \
(-1,0)=|=(1,0) B C

The coordinates at each hour can be computed
by rotating by 30 degrees. Here is example output:

----------------
hour=1
2.500 4.330
-0.866 0.500
0.866 -0.500
----------------
hour=2
4.330 2.500
-0.500 0.866
0.500 -0.866
----------------
hour=3
5.000 0.000
0.000 1.000
0.000 -1.000
...
----------------
hour 12
0.000 5.000
-1.000 0.000
1.000 0.000


What to turn-in:
----------------

As usual, check your code into Github. You should make SEPARATE
SOURCE FILES for you classes and main programs. You should have a
matrix.hpp file with the class interface(s), and matrix.cpp with the
implementations of the class methods. Put this all in a distinct
directory for Project8. Don't forget check-in your makefile.

For the first application, call the executable 'iriscorr', which takes
2 command-line arguments: name of iris data file (I can run your
program on my copy of the file, so you don't have to check it in), and
the name of the species to select records. For example:

iriscorr iris.data Iris-virginica

This will print out the correlation matrix for the 4 attributes
for the instances of this species (lines containing Iris-virginica).

For the second program, call the executable 'HourlyRotation', which takes
a file name on the command-line containing a list of coordinates for
an arbitrary number of 2D Points. The program will print out the rotation
of the coordinates for each hour, as shown above.

HourlyRotation clockhand.txt

Here are the contents of the test file:

clockhand.txt:
0.000 5.000
-1.000 0.000
1.000 0.000