Multiple-descriptor data sets and quality analysis

1 Multiple-descriptor data sets and quality analysis 

In the previous section there were only two sets of data, the y data, we wanted to model and the variable x, each being a vector of dimension A x 1. In many cases, however, one has several sets of x variables (xl, x2, x3. xM), each of which can potentially describe some of the variation in the y data set. There may also be several different sets of y data that we want to model with the same x descriptors but for simplicity we will only consider a single set of y data. The x variables can be arranged into an A x M matrix. 

The X descriptors are often derived from many different sources and may have different units, means and variances. Prior to any correlation analysis, each x vector is usually 

Each X vector may also be scaled with a suitable factor to take into account for example different units for the variables. This, however, is non-trivial and requires careful consideration. A common procedure, which avoids a user decision, is to normalize each X vector to have a variance of 1, a procedure called autoscaling. Antoscsil-ing equalizes the variance of each descriptor and can thus amplify random noise in the sample data and reduce the importance of a variable having a large response and a good correlation with the y data. 

Analogous to the correlation coefficient in eq. (17.10), we want a measure of the quality of fit produced by a given correlation model. Two commonly used quantities are the Predicted REsidual Sum of Squares (PRESS) and the correlation coefficient R defined by the normalized PRESS value and the variance of the y data (c/). 

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