Non‐parametric habitat models with automatic interactions

Questions: Can a statistical model be designed to represent more directly the nature of organismal response to multiple interacting factors? Can multiplicative kernel smoothers be used for this purpose? What advantages does this approach have over more traditional habitat modelling methods? Methods: Non‐parametric multiplicative regression (NPMR) was developed from the premises that: the response variable has a minimum of zero and a physiologically‐determined maximum, species respond simultaneously to multiple ecological factors, the response to any one factor is conditioned by the values of other factors, and that if any of the factors is intolerable then the response is zero. Key features of NPMR are interactive effects of predictors, no need to specify an overall model form in advance, and built‐in controls on overfitting. The effectiveness of the method is demonstrated with simulated and real data sets. Results: Empirical and theoretical relationships of species response to multiple interacting predictors can be represented effectively by multiplicative kernel smoothers. NPMR allows us to abandon simplistic assumptions about overall model form, while embracing the ecological truism that habitat factors interact.     

On partial local smoothing rules for curve estimation

We compare the performances of local and global rules for smoothingparameter choice, in terms of asymptotic mean squared errorsof the resulting estimators. In some instances there is surprisinglylittle to choose between local and global approaches; our analysisidentifies contexts where the differences are small or large.This work motivates development of smoothing rules that forma ‘half-way house’ between local and global smoothing.There, interpolation provides a basis for partial local smoothing.A key result shows that interpolation on even a coarse gridcan produce a very good approximation to full local smoothing.Our theoretical and numerical results lead us to suggest linearinterpolation of a bandwidth obtained by integral approximationson discrete intervals.     

Two-stage functional mixed models for evaluating the effect of longitudinal covariate profiles on a scalar outcome

The Daily Hormone Study, a substudy of the Study of Women's Health Across the Nation (SWAN) consisting of more than 600 pre- and perimenopausal women, includes a scalar measure of total hip bone mineral density (BMD) together with repeated measures of creatinine-adjusted follicle stimulating hormone (FSH) assayed from daily urine samples collected over one menstrual cycle. It is of scientific interest to investigate the effect of the FSH time profile during a menstrual cycle on total hip BMD, adjusting for age and body mass index. The statistical analysis is challenged by several features of the data: (1) the covariate FSH is measured longitudinally and its effect on the scalar outcome BMD may be complex; (2) due to varying menstrual cycle lengths, subjects have unbalanced longitudinal measures of FSH; and (3) the longitudinal measures of FSH are subject to considerable among- and within-subject variations and measurement errors. We propose a measurement error partial functional linear model, where repeated measures of FSH are modeled using a functional mixed effects model and the effect of the FSH time profile on BMD is modeled using a partial functional linear model by treating the unobserved true subject-specific FSH time profile as a functional covariate. We develop a two-stage nonparametric regression calibration method using period smoothing splines. Using the connection between smoothing splines and mixed models, we show that a key feature of our approach is that estimation at both stages can be conveniently cast into a unified mixed model framework. A simple testing procedure for constant functional covariate effect is also proposed. The proposed methods are evaluated using simulation studies and applied to the SWAN data.     

Reducing variability of crossvalidation for smoothing-parameter choice

One of the attractions of crossvalidation, as a tool for smoothing-parameterchoice, is its applicability to a wide variety of estimatortypes and contexts. However, its detractors comment adverselyon the relatively high variance of crossvalidatory smoothingparameters, noting that this compromises the performance ofthe estimators in which those parameters are used. We show thatthe variability can be reduced simply, significantly and reliablyby employing bootstrap aggregation or bagging. We establishthat in theory, when bagging is implemented using an adaptivelychosen resample size, the variability of crossvalidation canbe reduced by an order of magnitude. However, it is arguablymore attractive to use a simpler approach, based for exampleon half-sample bagging, which can reduce variability by approximately50%.     

The effects of interpolation error and location quality on animal track reconstruction

The Global Positioning System (GPS) gives precise estimates of location. However, the investigation of animal movement and behavior often requires interpolation to examine events between such fixes. We obtained 6,288 GPS locations from an electronic tag deployed for 170 d on an adult male gray seal ( Halichoerus grypus ) that ranged freely off the east coast of Scotland, and interpolated between subsamples of these data to investigate the growth of uncertainty within the intervals between observations. Average uncertainty over the path increased linearly as the interval between interpolating locations increased, reaching 12 km in longitude and 6 km in latitude at 2-d separation. The decrease in precision caused by duty-cycling, only collecting data in part of the day, was demonstrated. Adding noise to the GPS locations to simulate data from the ARGOS satellite system had little effect on the total errors for observations separated by more than 12 h. While the rate of growth in interpolation error is likely to vary between species, these results suggest that frequent, and preferably evenly spaced, location fixes are required to take full advantage of the precision of GPS data in the reconstruction of animal tracks.