%Candace Metoyer
%Statistics 135 Discussion 4-12-07
%Analysis of the Turtle Data

clear;clc;
format short
format compact

load turtledata.txt         %Load the turtle data

X = turtledata;
x1 = X(:,1);
x2 = X(:,2);
x3 = X(:,3);

[n, p] = size(X)

%Compute the mean of each column (a p by 1 vector)
Xbar = (mean(X))'

%Compute the mean across each row (an n by 1 vector)
%Xbar = mean(X,2)

%Obtain the deviation vectors.

d1 = x1 - Xbar(1)*ones(n,1);
d2 = x2 - Xbar(2)*ones(n,1);
d3 = x3 - Xbar(3)*ones(n,1);

%Compute the sample variance matrix, S

s11 = d1'*d1/(n-1);
s22 = d2'*d2/(n-1);
s33 = d3'*d3/(n-1);
s12 = d1'*d2/(n-1);
s13 = d1'*d3/(n-1);
s23 = d2'*d3/(n-1);

S = [s11 s12 s13; s12 s22 s23; s13 s23 s33]

%Compute the generalized sample variance

g_sample_variance = det(S)

%Compute the total sample variance;

t_sample_variance = trace(S)

%Compute the sample correlation matrix, R

r11 = 1;
r22 = 1;
r33 = 1;
r12 = s12/sqrt(s11*s22);
r13 = s13/sqrt(s11*s33);
r23 = s23/sqrt(s22*s33);

R = [r11 r12 r13; r12 r22 r23; r13 r23 r33]

%Find the eigenvalues and eigenvectors of R

[eigVEC, eigVAL] = eig(R)

%Compute histograms for each variable

figure      %opens up a new figure window
rhist(x1)   %rhist.m forms a relative frequency histogram and hist.m uses the original counts.

figure
rhist(x2)

figure
rhist(x3)

%Yikes.  These plots don't look like Prof. Burman's.
%We can adjust the histograms so that the bins
%are centered on specific values.

centers1 = 100:10:180;
centers2 = 80:5:130;
centers3 = 40:5:65;

figure      %opens up a new figure window
rhist(x1,centers1')

figure
rhist(x2,centers2')

figure
rhist(x3,centers3')

%Construct normal probability plots

figure
qqplot(x1)

figure
qqplot(x2)

figure
qqplot(x3)

%Construct chi-square plots
%Use function qqchi2.m
%input is the n x p data matrix

figure
qqchi2(X)
xlabel('Quantiles of the \chi_3^2 distribution')
ylabel('d_j^2 Quantiles')
title('Chi-square plot for the turtle data (female)')