Workshop 24

Workshop 24 - CHEN 3010 Applied Data Analysis Fall 2004...

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CHEN 3010 Applied Data Analysis Fall 2004 Names:__________________________ __________________________ Workshop #24 Polynomial Linear Regression: Model Building, Selection & Adequacy Objective Algebraic polynomial models are the most common used to represent engineering and scientific phenomena. In addition to fitting a polynomial model to the data, other issues need to be addressed: ¾ confidence intervals on the model parameters ¾ confidence interval on a prediction made by the fitted model ¾ analysis of residuals to determine model adequacy ¾ selection among different orders of polynomial In this workshop, you will deal with the second two of these using the example from Workshop #23. Background on Scale and Co-linearity When building models via regression techniques, many practical considerations arise. Two of these are scale and co-linearity . By scale , we mean when the values of the components of a model terms (ß parameter values and f(x) function values) differ greatly in magnitude. This can easily happen in higher-order polynomial models. If our independent variable (x) is a temperature in Kelvins and we use a 10 th -order polynomial, x 10 will be on the order of 10 20 and model parameters may be 10 -20 or smaller. This scale difference can cause numerical problems (round-off errors) in the regression calculation. By co-linearity , we mean that the function terms ( f(x) ) tend to behave in a similar fashion, that is, they are correlated. This is perhaps most obvious in a polynomial model, where, as the x value changes, x 2 , x 3 , etc., will change in similar fashion. In essence, this means that the variations in the f(x) terms in the data set are not independent one from the other. The implication is that this causes a lot of interaction in the estimation of the model parameters (ß). This interferes with our ability to tell whether an individual parameter is significant or not, so it interferes with model building and our ability to come up with the best model.
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Workshop 24 - CHEN 3010 Applied Data Analysis Fall 2004...

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