Predictor variables single-response regression

Correlation models differ from regression models in that each variable (y,s and XiS) plays a symmetrical role, with neither variable designated as a response or predictor variable. They are viewed as relational, instead of predictive in this process. Correlation models can be very useful for making inferences about any one variable relative to another, or to a group of variables. We use the correlation models in terms of y and single or multiple x,s. 

The basic parameter estimation, or regression, problem involves fitting the parameters of a proposed model to agree with the observed behavior of a system (Figure 8.1). We assume that, in any particular measurement of the system behavior, there is some set of predictor variables jc e 91 that fiiUy determines the behavior of the system (in the absence of any random noise or error). For each experiment, we measure some set of response variables yf.r) g fp— single-response data mAilL >, multiresponse data. t NxiXe... 

We first discuss the regression, or estimation, of parameters in linear models tfom singleresponse data. Let us say that we have performed a set of TV experiments in which for each experiment A = 1, 2,..., W, the set of predictor variables xfis known a priori, and a measurement is made of the single-response variable We assume that this single-response variable depends linearly upon the predictors,... 

We can go one step further, however. Each of the above multiple regression relations is between a single variable (response) of one data set and a linear combination of the variables (predictors) from the other set. Instead, one may consider the multiple-multiple correlation, i.e. the correlation of a linear combination from one set with a linear combination of the other set. Such linear combinations of the original variables are variously called factors, components, latent variables, canonical variables or canonical variates (also see Chapters 9,17, 29, and 31). 

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