Causal Modeling, Regression, and Calibration

In the following, several approaches to constructing a regression model will be tested on some simple example data sets. In these examples, traditional second-order calibration is not possible, as there is no causal or direct relationship between specific measured variables and the responses. The purpose of this example is mainly to illustrate the relative merits of different calibration approaches on different types of data, whereas the model building (residual and influence analysis) itself is not treated in detail. 

The underlying notion in bilinear modeling is that something causes the systematic variabilities in the X data. But we may not correctly know what it is there may be surprises in the data due to unexpected interferents, chemical interactions, nonlinear responses, etc. An approximate model of the subspace spanned by these phenomena in X is created. This X model is used for stabilizing the calibration modeling. The PLS regression primarily models the most dominant and most y-relevant of these X phenomena. Thus neither the manifest measured variables nor our causal assumptions about physical laws are taken for granted. Instead we tentatively look for systematic patterns in the data, and if they seem reasonable, we use them in the final calibration model. 

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