For many data approximation problems in metrology, there are a num- ber of competing models which can potentially fit the observed data. A crucial task is to quantify the extent to which one model performs better than others, taking into account the influence of random effects associated with the data. For example, for a given data set, we can use a series of polynomials of various degrees to fit the data using a least squares criterion. The residual sum of squares is a measure of how well the model fits the data. However, it is generally required to balance goodness of fit with minimising the model complexity. We consider a number of criteria that aim to do this: the Akaike information criterion (AIC), the Bayesian/Schwarz in- formation criterion, and the AIC with a correction for small sample size (AICc). In this paper, we compare the performance of these criteria for polynomial regression and show that for the examples tested the AICc criterion performs best. A second element of model selection is to determine from a set of feature vectors, the sub- set that defines a model space most suitable for describing the observed response. Since there are 2
possible model spaces defined by
feature vectors, for even a modest number of feature vectors it is necessary to reduce or prioritise the number of candidate models. Partial least squares and the least angle regression algorithms can be used as model reduction tools. We describe these algorithms in the context of feature selection and how they can be used with a model selection criterion such as AICc and illustrate their performance using simulations and on an application from human sensory perception.