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6.
Process or Product Monitoring and Control
6.5. Tutorials
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| Dimension reduction tool | A Multivariate Analysis problem could start out with a substantial set of variables with a large inter-correlation structure. One of the tools used to reduce such a complicated situation the well known method of Principal Component Analysis. | ||
| Principal factors | The technique of principal component analysis enables us to create and use a reduced set of variables, which are called principal factors. A reduced set is much easier to analyze and interpret. To study a data set that results in the estimation of roughly 500 parameters may be difficult, but if we could reduce these to 5 it would certainly make our day. We will show in what follows how to achieve this reduction. | ||
| Inverse transformaion not possible | While these principal factors represent or replace one or more of the original variables, it should be noted that they are not just a one to one transformation, so inverse transformations are not possible. | ||
| Original data matrix | To shed a light on the structure of principal components analysis, let us consider a multivariate sample variable, X. It is a matrix with n rows and p columns. The p elements of each row are scores or measurements on a subject, such as height, weight and age. | ||
| Linear function that maximizes variance | Next, standardize the X matrix so that each column mean is 0 and each column variance is 1. Call this matrix Z. Each row of Z is a vector variable, z , consisting of p elements. There are n such vector variables. The main idea behind principal component analysis is to derive a linear function y, consisting of n elements, for each of the vector variables z. This linear function possesses an extremely important property, namely, its variance is maximized. The number of y vectors is p, each y vector consists of n elements. | ||
| Linear function is component of z | The linear function y is referred to as a component of z. To illustrate the computation of a single element for the jth y vector, consider the product y = z v' , v' is a column vector of V, where V is a p x p coefficient matrix that carries the p-element variable z into the derived n-element variable y. V is known as the eigen vector matrix. The dimension of z is 1 x p, the dimension of v' is p x 1. The scalar algebra for the component score for the ith individual of yj, j = 1, ...p is: yji = v'1z1i + v'2z2i + ... + v'pzpi. This becomes in matrix notation for all of the y: Y = ZV. | ||
| Mean and dispersion matrix of y |
The mean of y is my =
V'mz = 0, because
mz = 0.
The dispersion matrix of y is
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| R is correlation matrix | Now, it can be shown that the dispersion matrix Dz of a standardized variable is a correlation matrix. Thus R is the correlation matrix for z. | ||
| Number of parameters to estimate increases rapidly as p increases |
At this juncture you may be tempted to say: "so what?". To answer
this let us look at the intercorrelations among the elements of a
vector variable. The number of parameters to be estimated for a
p-element variable is
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| Uncorrelated variables require no covariance estimation | All these parameters must be estimated and interpreted. That is a herculean task, to say the least. Now, if we could transform the data so that we obtain a vector of uncorrelated variables, life becomes much more bearable, since there are no covariances. | ||
