Simon N. Wood

Generalized Additive Models: An Introduction with R

LINEAR MODELS A simple linear model Linear models in general The theory of linear models The geometry of linear modelling Practical linear models Practical modelling with factors General linear model specification in R Further linear modelling theory Exercises GENERALIZED LINEAR MODELS The theory of GLMs Geometry of GLMs GLMs with R Likelihood Exercises INTRODUCING GAMS Introduction Univariate smooth functions Additive models Generalized additive models Summary Exercises SOME GAM THEORY Smoothing bases Setting up GAMs as penalized GLMs Justifying P-IRLS Degrees of freedom and residual variance estimation Smoothing Parameter Estimation Criteria Numerical GCV/UBRE: performance iteration Numerical GCV/UBRE optimization by outer iteration Distributional results Confidence interval performance Further GAM theory Other approaches to GAMs Exercises GAMs IN PRACTICE: mgcv Cherry trees again Brain imaging example Air pollution in Chicago example Mackerel egg survey example Portuguese larks example Other packages Exercises MIXED MODELS and GAMMs Mixed models for balanced data Linear mixed models in general Linear mixed models in R Generalized linear mixed models GLMMs with R Generalized additive mixed models GAMMs with R Exercises APPENDICES A Some matrix algebra B Solutions to exercises Bibliography Index

Making mistakes when predicting shifts in species range in response to global warming

Many attempts to predict the biotic responses to climate change rely on the ‘climate envelope’ approach, in which the current distribution of a species is mapped in climate-space and then, if the position of that climate-space changes, the distribution of the species is predicted to shift accordingly. The flaw in this approach is that distributions of species also reflect the influence of interactions with other species, so predictions based on climate envelopes may be very misleading if the interactions between species are altered by climate change. An additional problem is that current distributions may be the result of sources and sinks, in which species appear to thrive in places where they really persist only because individuals disperse into them from elsewhere,. Here we use microcosm experiments on simple but realistic assemblages to show how misleading the climate envelope approach can be. We show that dispersal and interactions, which are important elements of population dynamics, must be included in predictions of biotic responses to climate change.

Stable and Efficient Multiple Smoothing Parameter Estimation for Generalized Additive Models

Representation of generalized additive models (GAMs) using penalized regression splines allows GAMs to be employed in a straightforward manner using penalized regression methods. Not only is inference facilitated by this approach, but it is also possible to integrate model selection in the form of smoothing parameter selection into model fitting in a computationally efficient manner using well founded criteria such as generalized cross-validation. The current fitting and smoothing parameter selection methods for such models are usually effective, but do not provide the level of numerical stability to which users of linear regression packages, for example, are accustomed. In particular the existing methods cannot deal adequately with numerical rank deficiency of the GAM fitting problem, and it is not straightforward to produce methods that can do so, given that the degree of rank deficiency can be smoothing parameter dependent. In addition, models with the potential flexibility of GAMs can also present practical fitting difficulties as a result of indeterminacy in the model likelihood: Data with many zeros fitted by a model with a log link are a good example. In this article it is proposed that GAMs with a ridge penalty provide a practical solution in such circumstances, and a multiple smoothing parameter selection method suitable for use in the presence of such a penalty is developed. The method is based on the pivoted QR decomposition and the singular value decomposition, so that with or without a ridge penalty it has good error propagation properties and is capable of detecting and coping elegantly with numerical rank deficiency. The method also allows mixtures of user specified and estimated smoothing parameters and the setting of lower bounds on smoothing parameters. In terms of computational efficiency, the method compares well with existing methods. A simulation study compares the method to existing methods, including treating GAMs as mixed models.

Modelling and smoothing parameter estimation with multiple quadratic penalties

Penalized likelihood methods provide a range of practical modelling tools, including spline smoothing, generalized additive models and variants of ridge regression. Selecting the correct weights for penalties is a critical part of using these methods and in the single‐penalty case the analyst has several well‐founded techniques to choose from. However, many modelling problems suggest a formulation employing multiple penalties, and here general methodology is lacking. A wide family of models with multiple penalties can be fitted to data by iterative solution of the generalized ridge regression problem minimize ||W1/2 (Xp−y) ||2ρ+Σi=1m θip′Sip (p is a parameter vector, X a design matrix, Si a non‐negative definite coefficient matrix defining the ith penalty with associated smoothing parameter θi, W a diagonal weight matrix, y a vector of data or pseudodata and ρ an ‘overall’ smoothing parameter included for computational efficiency). This paper shows how smoothing parameter selection can be performed efficiently by applying generalized cross‐validation to this problem and how this allows non‐linear, generalized linear and linear models to be fitted using multiple penalties, substantially increasing the scope of penalized modelling methods. Examples of non‐linear modelling, generalized additive modelling and anisotropic smoothing are given.

GAMs with integrated model selection using penalized regression splines and applications to environmental modelling

Generalized Additive Models (GAMs) have been popularized by the work of Hastie and Tibshirani (1990) and the availability of user friendly GAM software in Splus. However, whilst it is flexible and ecient, the GAM framework based on backfitting with linear smoothers presents some diculties when it comes to model selection and

WHY DO POPULATIONS CYCLE? A SYNTHESIS OF STATISTICAL AND MECHANISTIC MODELING APPROACHES

Population cycles have long fascinated ecologists. Even in the most-studied populations, however, scientists continue to dispute the relative importance of various potential causes of the cycles. Over the past three decades, theoretical ecologists have cataloged a large number of mechanisms that are capable of generating cycles in population models. At the same time, statisticians have developed new techniques both for characterizing time series and for fitting population models to time-series data. Both disciplines are now sufficiently advanced that great gains in understanding can be made by synthesizing these complementary, and heretofore mostly independent, quantitative approaches. In this paper we demonstrate how to apply this synthesis to the problem of population cycles, using both long-term population time series and the often-rich observational and experimental data on the ecology of the species in question. We quantify hypotheses by writing mathematical models that embody the interactions and forces that might cause cycles. Some hypotheses can be rejected out of hand, as being unable to generate even qualitatively appropriate dynamics. We finish quantifying the remaining hypotheses by estimating parameters, both from independent experiments and from fitting the models to the time-series data using modern statistical techniques. Finally, we compare simulated time series generated by the models to the observed time series, using a variety of statistical descriptors, which we refer to collectively as “probes.” The model most similar to the data, as measured by these probes, is considered to be the most likely candidate to represent the mechanism underlying the population cycles. We illustrate this approach by analyzing one of Nicholson’s blowfly populations, in which we know the “true” governing mechanism. Our analysis, which uses only a subset of the information available about the population, uncovers the correct answer, suggesting that this synthetic approach might be successfully applied to field populations as well.

Statistical inference for noisy nonlinear ecological dynamic systems

Chaotic ecological dynamic systems defy conventional statistical analysis. Systems with near-chaotic dynamics are little better. Such systems are almost invariably driven by endogenous dynamic processes plus demographic and environmental process noise, and are only observable with error. Their sensitivity to history means that minute changes in the driving noise realization, or the system parameters, will cause drastic changes in the system trajectory. This sensitivity is inherited and amplified by the joint probability density of the observable data and the process noise, rendering it useless as the basis for obtaining measures of statistical fit. Because the joint density is the basis for the fit measures used by all conventional statistical methods, this is a major theoretical shortcoming. The inability to make well-founded statistical inferences about biological dynamic models in the chaotic and near-chaotic regimes, other than on an ad hoc basis, leaves dynamic theory without the methods of quantitative validation that are essential tools in the rest of biological science. Here I show that this impasse can be resolved in a simple and general manner, using a method that requires only the ability to simulate the observed data on a system from the dynamic model about which inferences are required. The raw data series are reduced to phase-insensitive summary statistics, quantifying local dynamic structure and the distribution of observations. Simulation is used to obtain the mean and the covariance matrix of the statistics, given model parameters, allowing the construction of a ‘synthetic likelihood’ that assesses model fit. This likelihood can be explored using a straightforward Markov chain Monte Carlo sampler, but one further post-processing step returns pure likelihood-based inference. I apply the method to establish the dynamic nature of the fluctuations in Nicholson’s classic blowfly experiments.

Habitat structure and population persistence in an experimental community.

Understanding spatial population dynamics is fundamental for many questions in ecology and conservation. Many theoretical mechanisms have been proposed whereby spatial structure can promote population persistence, in particular for exploiter–victim systems (host–parasite/pathogen, predator–prey) whose interactions are inherently oscillatory and therefore prone to extinction of local populations. Experiments have confirmed that spatial structure can extend persistence, but it has rarely been possible to identify the specific mechanisms involved. Here we use a model-based approach to identify the effects of spatial population processes in experimental systems of bean plants (Phaseolus lunatus), herbivorous mites (Tetranychus urticae) and predatory mites (Phytoseiulus persimilis). On isolated plants, and in a spatially undivided experimental system of 90 plants, prey and predator populations collapsed; however, introducing habitat structure allowed long-term persistence. Using mechanistic models, we determine that spatial population structure did not contribute to persistence, and spatially explicit models are not needed. Rather, habitat structure reduced the success of predators at locating prey outbreaks, allowing between-plant asynchrony of local population cycles due to random colonization events.

Practical variable selection for generalized additive models

The problem of variable selection within the class of generalized additive models, when there are many covariates to choose from but the number of predictors is still somewhat smaller than the number of observations, is considered. Two very simple but effective shrinkage methods and an extension of the nonnegative garrote estimator are introduced. The proposals avoid having to use nonparametric testing methods for which there is no general reliable distributional theory. Moreover, component selection is carried out in one single step as opposed to many selection procedures which involve an exhaustive search of all possible models. The empirical performance of the proposed methods is compared to that of some available techniques via an extensive simulation study. The results show under which conditions one method can be preferred over another, hence providing applied researchers with some practical guidelines. The procedures are also illustrated analysing data on plasma beta-carotene levels from a cross-sectional study conducted in the United States.

Dynamical effects of plant quality and parasitism on population cycles of larch budmoth

Population cycles have been remarkably resistant to explanation, in part because crucial experiments are rarely possible on appropriate spatial and temporal scales. Here we show how new approaches to nonlinear time-series analysis can distinguish between competing hypotheses for population cycles of larch budmoth in the Swiss Alps: delayed effects of budmoth density on food quality, and budmoth-parasitoid interactions. We re- examined data on budmoth density, plant quality, and parasitism rates. Our results suggest that the effect of plant quality on budmoth density is weak. By contrast, a simple model of budmoth-parasitoid interaction accounts for 90% of the variance in budmoth population growth rates. Thus, contrary to previous studies, we find that parasitoid-budmoth interaction