6. Process or Product Monitoring and Control
6.5. Tutorials
6.5.5. Principal Components

6.5.5.2.

Numerical Example

Calculation of principal components example

A numerical example may clarify the mechanics of principal component analysis.

Sample data set

Let us analyze the following 3-variate dataset with 10 observations. Each observation consists of 3 measurements on a wafer: thickness, horizontal displacement and vertical displacement.

Compute the correlation matrix

First compute the correlation matrix

Solve for the roots of R

Next solve for the roots of R, using software

	value	proportion
1	1.769	.590
2	.927	.899
3	.304	1.000

Notice that

• Each eigenvalue satisfies |R- I| = 0.
• The sum of the eigenvalues = 3 = p, which is equal to the trace of R (i.e., the sum of the main diagonal elements).
• The determinant of R is the product of the eigenvalues.
• The product is ₁ x ₂ x ₃ = .499.

Compute the first column of the V matrix

Substituting the first eigenvalue of 1.769 and R in the appropriate equation we obtain This is the matrix expression for 3 homogeneous equations with 3 unknowns and yields the first column of V: .64  .69  -.34  (again, a computerized solution is indispensable).

Compute the remaining columns of the V matrix

Repeating this procedure for the other 2 eigenvalues yields the matrix V Notice that if you multiply V by its transpose, the result is an identity matrix, V'V=I.

Compute the L^1/2 matrix

Now form the matrix L^1/2, which is a diagonal matrix whose elements are the square roots of the eigenvalues of R. Then obtain S, the factor structure, using S = V L^1/2 So, for example, .91 is the correlation between variable 2 and the first principal component.

Compute the communality

Next compute the communality, using the first two eigenvalues only

Diagonal elements report how much of the variability is explained

Communality consists of the diagonal elements.

var
1	.8662
2	.8420
3	.9876

This means that the first two principal components "explain" 86.62% of the first variable, 84.20 % of the second variable, and 98.76% of the third.

Compute the coefficient matrix

The coefficient matrix, B, is formed using the reciprocals of the diagonals of L^1/2

B = VL^-1/2 =

Compute the principal factors

Finally, we can compute the factor scores from ZB, where Z is X converted to standard score form. These columns are the principal factors.

Principal factors control chart

These factors can be plotted against the indices, which could be times. If time is used, the resulting plot is an example of a principal factors control chart.