Residual and influence analysis

In this paragraph, influence and residual analysis is described. The focus is mainly on aspects particular to three-way models, whereas ordinary two-way influence and residual analysis can be found in the standard literature. 

It was explained in Chapters 2-4 that all models whether they are two-way or three-way, are built up as follows  

The main goal of the data analysis is usually to find X, but the residual E can give important clues to the quality of this model. Possibly, residuals obtained from a test-set or from cross-validation can be used instead of fitted residuals. Random noise or some symmetrical type of distribution for the elements of E is normally expected and this can be verified from plotting the residuals and by the use of diagnostics. A good description of the use of residuals in three-way analysis is given by Kroonenberg [1983], 

If X has e.g. I objects, /variables and K observation times, the study of s/ will indicate whether there are outliers among the objects, or whether there is a trend among the objects. In the same way, outliers among the variables and among e.g. the times may be found using s/and sk. If the variables are homogeneous, such as spectra, there may be discontinuities in certain parts of sj that indicate a chemical or physical artefact. A trend in the sk representing a time mode may point to a drift in time of an instrument. 

Multi-way Analysis With Applications in the Chemical Sciences 

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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. 

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