Inge S. Helland

ON THE STRUCTURE OF PARTIAL LEAST SQUARES REGRESSION

We prove that the two algorithms given in the literature for partial least squares regression are equivalent, and use this equivalence to give an explicit formula for the resulting prediction equation. This in turn is used to investigate the regression method from several points of view. Its relation to principal component regression is clearified, and some heuristic arguments are given to explain why partial least squares regression often needs fewer factors to give its optimal prediction.

Related versions of the multiplicative scatter correction method for preprocessing spectroscopic data

Abstract Various multiplicative transformations of spectral variables have been used with some success as preprocessing methods for diffuse near-infrared spectroscopy data. We discuss first which additive/multiplicative transformations conserve the area under the spectral curve. Next we look at the relationship between the Multiplicative Scatter and the Standard Normal Variate transformations, and use a minimizing norm formulation to isolate three specific transformations that seem to be singled out as particularly interesting. In an empirical investigation using 8 different data sets, each with several constituents, the different transformation methods are compared.

Some theoretical aspects of partial least squares regression

Abstract We give a survey of partial least squares regression with one y variable from a theoretical point of view. Some general comments are made on the motivation as seen by a statistician to study particular chemometric methods, and the concept of soft modelling is criticized from the same angle. Various aspects of the PLS algorithm are considered and the population PLS model is defined. Asymptotic properties of the prediction error are briefly discussed and the relation to other regression methods are commented upon. Results indicating positive and negative properties of PLSR are mentioned, in particular the recent result of Butler, Denham and others which seem to show that PLSR can not be an optimal regression method in any reasonable way. The only possible path left towards some kind of optimality, it seems, is by first trying to find a good motivation for the population model and then possibly finding an optimal estimator under this model. Some results on this are sketched.

Comparison of Prediction Methods when Only a Few Components are Relevant

Abstract We consider prediction in a multiple regression model where we also look on the explanatory variables as random. If the number of explanatory variables is large, then the common least squares multiple regression solution may not be the best one. We give a methodology for comparing certain alternative prediction methods by asymptotic calculations and perform such a comparisons for four specific methods. The results indicate that none of these methods dominates the others, and that the difference between the methods typically (but not always) is small when the number of observations is large. In particular, principal component regression does well when the eigenvalues corresponding to components not correlated with the dependent variables (i.e., the irrelevant eigenvalues) are extremely small or extremely large. Partial least squares regression does well for intermediate irrelevant eigenvalues. A maximum likelihood-type method dominates the others asymptotically, at least in the case of one relevan...

On the Interpretation and Use of R 2 in Regression Analysis

The coefficient of determination (or squared multiple correlation coefficient) R2 is a common output from computer regression packages. We argue first that this statistic can be interpreted as an estimator of a population parameter only when the regressors are random. In such a model the variation of R2 is discussed, and a simple approximate confidence interval for the population coefficient of determination is proposed. Its use is illustrated on data from computerized tomography investigation of pigs.

Attraction of bark beetles (Coleoptera: Scolytidae) to a pheromone trap Experiment and mathematical models.

The movement of bark beetles near an attractive pheromone source is described in terms of mathematical models of the diffusion type. To test the models, two release experiments involving 47,000 marked spruce bark beetles [Ips typographus (L.)] were performed. The attractive source was a pheromone trap, surrounded by eight concentric rings with eight passive trap stations on each ring. Captures were recorded every 2-10 minutes for the pheromone trap and once for the passive traps. The models were fitted to the distribution in time of the central pheromone trap catch and to the spatial distribution of catch among the passive traps. The first model that gives a reasonable fit consists of two phases: Phase one-After release the beetles move according to a diffusion process with drift towards the pheromone trap. The strength of the drift is inversely proportional to the distance from the traps. Phase two-those beetles attracted to, but not caught by, the pheromone trap are no longer influenced by the pheromone, and their movement is described by a diffusion process without drift. In phase two we work with a loss of beetles, whereas the experiment seems to indicate that the loss of beetles in phase one is negligible. As a second model, the following modification of phase one is considered: After release the beetles move according to a diffusion process without drift, until they start responding to the pheromone (with constant probability per unit time), whereafter they start moving according to a diffusion process with drift. This study, like other release experiments, shows that the efficiency of the pheromone trap is rather low. What is specific for the present investigation is that we try to explain this low efficiency in terms of dynamic models for insect movement. Two factors seem to contribute: Some beetles do not respond to pheromone at all, and some beetles disappear again after having been close to the pheromone trap. It also seems that the motility of the beetles decreased after they ceased responding to the pheromone. Furthermore, the data lend some support to the hypothesis that flight exercise increases the response of the beetles to pheromone.

Attraction of bark beetles (Coleoptera: Scolytidae) to a pheromone trap

The movement of bark beetles near an attractive pheromone source is described in terms of mathematical models of the diffusion type. To test the models, two release experiments involving 47,000 marked spruce bark beetles [Ips typographus (L.)] were performed. The attractive source was a pheromone trap, surrounded by eight concentric rings with eight passive trap stations on each ring. Captures were recorded every 2–10 minutes for the pheromone trap and once for the passive traps. The models were fitted to the distribution in time of the central pheromone trap catch and to the spatial distribution of catch among the passive traps. The first model that gives a reasonable fit consists of two phases: Phase one—After release the beetles move according to a diffusion process with drift towards the pheromone trap. The strength of the drift is inversely proportional to the distance from the traps. Phase two—those beetles attracted to, but not caught by, the pheromone trap are no longer influenced by the pheromone, and their movement is described by a diffusion process without drift. In phase two we work with a loss of beetles, whereas the experiment seems to indicate that the loss of beetles in phase one is negligible. As a second model, the following modification of phase one is considered: After release the beetles move according to a diffusion process without drift, until they start responding to the pheromone (with constant probability per unit time), whereafter they start moving according to a diffusion process with drift. This study, like other release experiments, shows that the efficiency of the pheromone trap is rather low. What is specific for the present investigation is that we try to explain this low efficiency in terms of dynamic models for insect movement. Two factors seem to contribute: Some beetles do not respond to pheromone at all, and some beetles disappear again after having been close to the pheromone trap. It also seems that the motility of the beetles decreased after they ceased responding to the pheromone. Furthermore, the data lend some support to the hypothesis that flight exercise increases the response of the beetles to pheromone.

Continuity of a class of random time transformations

A random time change is defined as a map from one function space to another. The continuity of this map is investigated. Applications are made to weak limit theorems of random processes.

ON A GENERAL RANDOM EXCHANGE MODEL

Two independent i.i.d. sequences of random variables {U,,} and {D,,} generate a Markov process {X,,} by X, = max(X, ,- D,, U,,), n = 1,2,-..

Simple Counterexamples against the Conditionality Principle

Abstract The famous theorem of Birnbaum, stating that the likelihood principle follows from the conditionality principle together with the sufficiency principle, has caused much discussion among statisticians. Briefly, many writers dislike the consequences of the likelihood principle (among other things, confidence coefficients and levels of tests are dismissed as meaningless), but at the same time they feel that both the conditionality principle and the sufficiency principle are intuitively obvious. In the present article we give examples to show that the conditionality principle should not be taken to be of universal validity, and we discuss some consequences of these examples.