**CODE**

% Let's start with a simple example.

data = randnmulti(100,[10 6],[1 .6; .6 1],[3 1]);

figure(999); clf; hold on;

scatter(data(:,1),data(:,2),'r.');

axis square; axis([0 20 0 20]);

set(gca,'XTick',0:1:20,'YTick',0:1:20);

xlabel('X'); ylabel('Y');

title('Data');

% To compute the correlation, let's first standardize each

% variable, i.e. subtract off the mean and divide by the

% standard deviation. This converts each variable into

% z-score units.

dataz = calczscore(data,1);

figure(998); clf; hold on;

scatter(dataz(:,1),dataz(:,2),'r.');

axis square; axis([-5 5 -5 5]);

set(gca,'XTick',-5:1:5,'YTick',-5:1:5);

xlabel('X (z-scored)'); ylabel('Y (z-scored)');

% We would like a single number that represents the relationship

% between X and Y. Points that lie in the upper-right or lower-left

% quadrants support a positive relationship between X and Y (i.e.

% higher on X is associated with higher on Y), whereas points that

% lie in the upper-left or lower-right quadrants support a negative

% relationship between X and Y (i.e. higher on X is associated with

% lower on Y).

set(straightline(0,'h','k-'),'Color',[.6 .6 .6]);

set(straightline(0,'v','k-'),'Color',[.6 .6 .6]);

text(4,4,'+','FontSize',48,'HorizontalAlignment','center');

text(-4,-4,'+','FontSize',48,'HorizontalAlignment','center');

text(4,-4,'-','FontSize',60,'HorizontalAlignment','center');

text(-4,4,'-','FontSize',60,'HorizontalAlignment','center');

% Let's calculate the "average" relationship. To do this, we compute

% the average product of the X and Y variables (in the z-scored space).

% The result is the correlation value.

r = mean(dataz(:,1) .* dataz(:,2));

title(sprintf('r = %.4g',r));

% Now, let's turn to a different (but in my opinion more useful)

% interpretation of correlation. Let's start with a simple example.

data = randnmulti(100,[10 6],[1 .8; .8 1],[3 1]);

% For each set of values, subtract off the mean and scale the values such

% that the the values have unit length (i.e. the sum of the squares of the

% values is 1).

datanorm = unitlength(zeromean(data,1),1);

% The idea is that each set of values represents a vector in a

% 100-dimensional space. After mean-subtraction and unit-length-normalization,

% the vectors lie on the unit sphere. The correlation is simply

% the dot-product between the two vectors. If the two vectors are

% very similar to one another, the dot-product will be high (close to 1);

% if the two vectors are not very similar to one another (think: randomly

% oriented), the dot-product will be low (close to 0); if the two vectors

% are very anti-similar to one another (think: pointing in opposite

% directions), the dot product will be very negative (close to -1).

% Let's visualize this for our example.

% the idea here is to project the 100-dimensional space onto

% two dimensions so that we can actually visualize the data.

% the first dimension is simply the first vector.

% the second dimension is the component of the second vector that

% is orthogonal to the first vector.

dim1 = datanorm(:,1);

dim2 = unitlength(projectionmatrix(datanorm(:,1))*datanorm(:,2));

% compute the coordinates of the two vectors in this reduced space.

c1 = datanorm(:,1)'*[dim1 dim2];

c2 = datanorm(:,2)'*[dim1 dim2];

% make a figure

figure(997); clf; hold on;

axis square; axis([-1.2 1.2 -1.2 1.2]);

h1 = drawarrow([0 0; 0 0],[c1; c2],[],[],[],'LineWidth',4,'Color',[1 .8 .8]);

h2 = drawellipse(0,0,0,1,1,[],[],'k-');

h3 = plot([c2(1) c2(1)],[0 c2(2)],'k--');

h4 = plot([0 c2(1)],[0 0],'r-','LineWidth',4);

xlabel('Dimension 1');

ylabel('Dimension 2');

legend([h1(1) h2 h4],{'Data' 'Unit sphere' 'Projection'});

r = dot(datanorm(:,1),datanorm(:,2));

title(sprintf('r = %.4g',r));

% The vector interpretation lends itself readily to the concept

% of variance explained. Suppose that we are using one vector A

% to predict the other vector B in a linear regression sense. Then,

% the weight applied to vector A is equal to the correlation

% value. (This is because inv(A'*A)*A'*B = inv(1)*A'*B = A'*B = r.)

% Let's visualize this.

figure(996); clf; hold on;

axis square; axis([-1.2 1.2 -1.2 1.2]);

h1 = drawarrow([0 0],c2,[],[],[],'LineWidth',4,'Color',[1 0 0]);

h2 = drawarrow([0 0],r*c1,[],[],[],'LineWidth',4,'Color',[0 1 0]);

h3 = plot([r r],[0 c2(2)],'b-');

h4 = drawellipse(0,0,0,1,1,[],[],'k-');

text(.3,.38,'1');

text(.35,-.1,'r');

xlabel('Dimension 1');

ylabel('Dimension 2');

legend([h1 h2 h3 h4],{'Vector B' 'Vector A (scaled by r)' 'Residuals' 'Unit sphere'});

% So, we ask, how much variance in vector B is explained by vector A?

% The total variance in vector B is 1^2. By the Pythagorean Theorem,

% the total variance in the residuals is 1^2 - r^2. So, the fraction

% of variance that we do not explain is (1^2 - r^2) / 1^2, and

% so the fraction of variance that we do explain is 1 - (1^2 - r^2) / 1^2,

% which simplifies to r^2. (Note that here, in order to keep things simple,

% we have invoked a version of variance that omits the division by the number

% of data points. Under this version of variance, the variance associated

% with a zero-mean vector is simply the sum of the squares of the elements

% of the vector, which is equivalent to the square of the vector's length.

% Thus, there is a nice equivalence between variance and squared distance,

% which we have exploited for our interpretation.)

**FINAL REMARKS**

There is much more to be said about correlation; we have covered here only the basics.

One important caveat to keep in mind is that

**correlation is not sensitive to offset and gain**. This is because the mean of each set of numbers is subtracted off and because the scale of each set of numbers is normalized out. This has several consequences:

- Correlation indicates how well
**deviations relative to the mean**can be predicted. This is natural, since variance (normally defined) is also relative to the mean. - When using correlation as a index of how well one set of numbers predicts another set of numbers, you are
**implicitly allowing scale and offset parameters**in your model. If you want to be completely parameter-free, you should instead use a metric like R^2 (which will be described in a later post).

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