arXiv subjects

Simon N. Wood

Publications and source records attributed to Simon N. Wood.

At least 19 recordsLinked to original sources

arXiv2026

Integrating Complex Covariate Transformations in Generalized Additive Models

Transformations of covariates are widely used in applied statistics to improve interpretability and to satisfy assumptions required for valid inference. More broadly, feature engineering encompasses a wider set of practices aimed at enhancing predictive performance, and is typically performed as part of a data pre-processing step. In contrast, this paper integrates a substantial component of the feature engineering process directly into the modelling stage. This is achieved by introducing a novel general framework for embedding interpretable covariate transformations within multi-parameter Generalised Additive Models (GAMs). Our framework accommodates any sufficiently differentiable scalar-valued transformation of potentially high-dimensional and complex covariates. These transformations are treated as integral model components, with their parameters estimated jointly with regression coefficients via maximum a posteriori (MAP) methods, and joint uncertainty quantified via approximate Bayesian techniques. Smoothing parameters are selected in an empirical Bayes framework using a Laplace approximation to the marginal likelihood, supported by efficient computation based on implicit differentiation methods. We demonstrate the flexibility and practical value of the proposed methodology through applications to forecasting electricity net-demand in Great Britain and to modelling house prices in London. Methods for building and fitting GAMs with nested transformations are provided by the gamFactory R package, available at https://github.com/mfasiolo/gamFactory, while the code for reproducing the results in this paper is available at https://doi.org/10.5281/zenodo.19239350.

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arXiv2025

Some statistical aspects of the Covid-19 response

This paper discusses some statistical aspects of the U.K. Covid-19 pandemic response, focussing particularly on cases where we believe that a statistically questionable approach or presentation has had a substantial impact on public perception, or government policy, or both. We discuss the presentation of statistics relating to Covid risk, and the risk of the response measures, arguing that biases tended to operate in opposite directions, overplaying Covid risk and underplaying the response risks. We also discuss some issues around presentation of life loss data, excess deaths and the use of case data. The consequences of neglect of most individual variability from epidemic models, alongside the consequences of some other statistically important omissions are also covered. Finally the evidence for full stay at home lockdowns having been necessary to reverse waves of infection is examined, with new analyses provided for a number of European countries.

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arXiv2025

On Neighbourhood Cross Validation

Many varieties of cross validation would be statistically appealing for the estimation of smoothing and other penalized regression hyperparameters, were it not for the high cost of evaluating such criteria. Here it is shown how to efficiently and accurately compute and optimize a broad variety of cross validation criteria for a wide range of models estimated by minimizing a quadratically penalized loss. The leading order computational cost of hyperparameter estimation is made comparable to the cost of a single model fit given hyperparameters. In many cases this represents an $O(n)$ computational saving when modelling $n$ data. This development makes if feasible, for the first time, to use leave-out-neighbourhood cross validation to deal with the wide spread problem of un-modelled short range autocorrelation which otherwise leads to underestimation of smoothing parameters. It is also shown how to accurately quantifying uncertainty in this case, despite the un-modelled autocorrelation. Practical examples are provided including smooth quantile regression, generalized additive models for location scale and shape, and focussing particularly on dealing with un-modelled autocorrelation.

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arXiv2025

Scalable Fitting Methods for Multivariate Gaussian Additive Models with Covariate-dependent Covariance Matrices

We propose efficient computational methods to fit multivariate Gaussian additive models, where the mean vector and the covariance matrix are allowed to vary with covariates, in an empirical Bayes framework. To guarantee the positive-definiteness of the covariance matrix, we model the elements of an unconstrained parametrisation matrix, focussing particularly on the modified Cholesky decomposition and the matrix logarithm. A key computational challenge arises from the fact that, for the model class considered here, the number of parameters increases quadratically with the dimension of the response vector. Hence, here we discuss how to achieve fast computation and low memory footprint in moderately high dimensions, by exploiting parsimonious model structures, sparse derivative systems and by employing block-oriented computational methods. Methods for building and fitting multivariate Gaussian additive models are provided by the SCM R package, available at https://github.com/VinGioia90/SCM, while the code for reproducing the results in this paper is available at https://github.com/VinGioia90/SACM.

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arXiv2024

Modelling tree survival for investigating climate change effects

Using German forest health monitoring data we investigate the main drivers leading to tree mortality and the association between defoliation and mortality; in particular (a) whether defoliation is a proxy for other covariates (climate, soil, water budget); (b) whether defoliation is a tree response that mitigates the effects of climate change and (c) whether there is a threshold of defoliation which could be used as an early warning sign for irreversible damage. Results show that environmental drivers leading to tree mortality differ by species, but some are always required in the model. The defoliation effect on mortality differs by species but it is always strong and monotonic. There is some evidence that a defoliation threshold exists for spruce, fir and beech. We model tree survival with a smooth additive Cox model allowing for random effects taking care of dependence between neighbouring trees and non-linear functions of spatial time varying and functional predictors on defoliation, climate, soil and hydrology characteristics. Due to the large sample size and large number of parameters, we use parallel computing combined with marginal discretization of covariates. We propose a 'boost forward penalise backward' model selection scheme based on combining component-wise gradient boosting with integrated backward selection.

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arXiv2021

Inferring UK COVID-19 fatal infection trajectories from daily mortality data: were infections already in decline before the UK lockdowns?

The number of new infections per day is a key quantity for effective epidemic management. It can be estimated relatively directly by testing of random population samples. Without such direct epidemiological measurement, other approaches are required to infer whether the number of new cases is likely to be increasing or decreasing: for example, estimating the pathogen effective reproduction number, R, using data gathered from the clinical response to the disease. For Covid-19 (SARS-CoV-2) such R estimation is heavily dependent on modelling assumptions, because the available clinical case data are opportunistic observational data subject to severe temporal confounding. Given this difficulty it is useful to retrospectively reconstruct the time course of infections from the least compromised available data, using minimal prior assumptions. A Bayesian inverse problem approach applied to UK data on first wave Covid-19 deaths and the disease duration distribution suggests that fatal infections were in decline before full UK lockdown (24 March 2020), and that fatal infections in Sweden started to decline only a day or two later. An analysis of UK data using the model of Flaxman et al. (2020, Nature 584) gives the same result under relaxation of its prior assumptions on R, suggesting an enhanced role for non pharmaceutical interventions (NPIs) short of full lock down in the UK context. Similar patterns appear to have occurred in the subsequent two lockdowns. Estimates from the main UK Covid statistical surveillance surveys, available since original publication, support these results. Replication code for the paper is available in the supporting information of doi/10.1111/biom.13462.

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arXiv2020

Spatial+: a novel approach to spatial confounding

In spatial regression models, collinearity between covariates and spatial effects can lead to significant bias in effect estimates. This problem, known as spatial confounding, is encountered modelling forestry data to assess the effect of temperature on tree health. Reliable inference is difficult as results depend on whether or not spatial effects are included in the model. The mechanism behind spatial confounding is poorly understood and methods for dealing with it are limited. We propose a novel approach, spatial+, in which collinearity is reduced by replacing the covariates in the spatial model by their residuals after spatial dependence has been regressed away. Using a thin plate spline model formulation, we recognise spatial confounding as a smoothing-induced bias identified by Rice (1986), and through asymptotic analysis of the effect estimates, we show that spatial+ avoids the bias problems of the spatial model. This is also demonstrated in a simulation study. Spatial+ is straight-forward to implement using existing software and, as the response variable is the same as that of the spatial model, standard model selection criteria can be used for comparisons. A major advantage of the method is also that it extends to models with non-Gaussian response distributions. Finally, while our results are derived in a thin plate spline setting, the spatial+ methodology transfers easily to other spatial model formulations.

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arXiv2020

qgam: Bayesian non-parametric quantile regression modelling in R

Generalized additive models (GAMs) are flexible non-linear regression models, which can be fitted efficiently using the approximate Bayesian methods provided by the mgcv R package. While the GAM methods provided by mgcv are based on the assumption that the response distribution is modelled parametrically, here we discuss more flexible methods that do not entail any parametric assumption. In particular, this article introduces the qgam package, which is an extension of mgcv providing fast calibrated Bayesian methods for fitting quantile GAMs (QGAMs) in R. QGAMs are based on a smooth version of the pinball loss of Koenker (2005), rather than on a likelihood function, hence jointly achieving satisfactory accuracy of the quantile point estimates and coverage of the corresponding credible intervals requires adopting the specialized Bayesian fitting framework of Fasiolo, Wood, Zaffran, Nedellec, and Goude (2020b). Here we detail how this framework is implemented in qgam and we provide examples illustrating how the package should be used in practice.

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arXiv2020

Additive stacking for disaggregate electricity demand forecasting

Future grid management systems will coordinate distributed production and storage resources to manage, in a cost effective fashion, the increased load and variability brought by the electrification of transportation and by a higher share of weather dependent production. Electricity demand forecasts at a low level of aggregation will be key inputs for such systems. We focus on forecasting demand at the individual household level, which is more challenging than forecasting aggregate demand, due to the lower signal-to-noise ratio and to the heterogeneity of consumption patterns across households. We propose a new ensemble method for probabilistic forecasting, which borrows strength across the households while accommodating their individual idiosyncrasies. In particular, we develop a set of models or 'experts' which capture different demand dynamics and we fit each of them to the data from each household. Then we construct an aggregation of experts where the ensemble weights are estimated on the whole data set, the main innovation being that we let the weights vary with the covariates by adopting an additive model structure. In particular, the proposed aggregation method is an extension of regression stacking (Breiman, 1996) where the mixture weights are modelled using linear combinations of parametric, smooth or random effects. The methods for building and fitting additive stacking models are implemented by the gamFactory R package, available at https://github.com/mfasiolo/gamFactory.

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arXiv2020

COVID-19 and the difficulty of inferring epidemiological parameters from clinical data

Knowing the infection fatality ratio (IFR) is of crucial importance for evidence-based epidemic management: for immediate planning; for balancing the life years saved against the life years lost due to the consequences of management; and for evaluating the ethical issues associated with the tacit willingness to pay substantially more for life years lost to the epidemic, than for those to other diseases. Against this background Verity et al. (2020, Lancet Infections Diseases) have rapidly assembled case data and used statistical modelling to infer the IFR for COVID-19. We have attempted an in-depth statistical review of their approach, to identify to what extent the data are sufficiently informative about the IFR to play a greater role than the modelling assumptions, and have tried to identify those assumptions that appear to play a key role. Given the difficulties with other data sources, we provide a crude alternative analysis based on the Diamond Princess Cruise ship data and case data from China, and argue that, given the data problems, modelling of clinical data to obtain the IFR can only be a stop-gap measure. What is needed is near direct measurement of epidemic size by PCR and/or antibody testing of random samples of the at risk population.

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arXiv2019

Scalable visualisation methods for modern Generalized Additive Models

In the last two decades the growth of computational resources has made it possible to handle Generalized Additive Models (GAMs) that formerly were too costly for serious applications. However, the growth in model complexity has not been matched by improved visualisations for model development and results presentation. Motivated by an industrial application in electricity load forecasting, we identify the areas where the lack of modern visualisation tools for GAMs is particularly severe, and we address the shortcomings of existing methods by proposing a set of visual tools that a) are fast enough for interactive use, b) exploit the additive structure of GAMs, c) scale to large data sets and d) can be used in conjunction with a wide range of response distributions. All the new visual methods proposed in this work are implemented by the mgcViz R package, which can be found on the Comprehensive R Archive Network.

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arXiv2017

An Extended Empirical Saddlepoint Approximation for Intractable Likelihoods

The challenges posed by complex stochastic models used in computational ecology, biology and genetics have stimulated the development of approximate approaches to statistical inference. Here we focus on Synthetic Likelihood (SL), a procedure that reduces the observed and simulated data to a set of summary statistics, and quantifies the discrepancy between them through a synthetic likelihood function. SL requires little tuning, but it relies on the approximate normality of the summary statistics. We relax this assumption by proposing a novel, more flexible, density estimator: the Extended Empirical Saddlepoint approximation. In addition to proving the consistency of SL, under either the new or the Gaussian density estimator, we illustrate the method using two examples. One of these is a complex individual-based forest model for which SL offers one of the few practical possibilities for statistical inference. The examples show that the new density estimator is able to capture large departures from normality, while being scalable to high dimensions, and this in turn leads to more accurate parameter estimates, relative to the Gaussian alternative. The new density estimator is implemented by the esaddle R package, which can be found on the Comprehensive R Archive Network (CRAN).

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arXiv2016

A generalized Fellner-Schall method for smoothing parameter estimation with application to Tweedie location, scale and shape models

We consider the estimation of smoothing parameters and variance components in models with a regular log likelihood subject to quadratic penalization of the model coefficients, via a generalization of the method of Fellner (1986) and Schall (1991). In particular: (i) we generalize the original method to the case of penalties that are linear in several smoothing parameters, thereby covering the important cases of tensor product and adaptive smoothers; (ii) we show why the method's steps increase the restricted marginal likelihood of the model, that it tends to converge faster than the EM algorithm, or obvious accelerations of this, and investigate its relation to Newton optimization; (iii) we generalize the method to any Fisher regular likelihood. The method represents a considerable simplification over existing methods of estimating smoothing parameters in the context of regular likelihoods, without sacrificing generality: for example, it is only necessary to compute with the same first and second derivatives of the log-likelihood required for coefficient estimation, and not with the third or fourth order derivatives required by alternative approaches. Examples are provided which would have been impossible or impractical with pre-existing Fellner-Schall methods, along with an example of a Tweedie location, scale and shape model which would be a challenge for alternative methods.

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arXiv2016

P-splines with derivative based penalties and tensor product smoothing of unevenly distributed data

The P-splines of Eilers and Marx (1996) combine a B-spline basis with a discrete quadratic penalty on the basis coefficients, to produce a reduced rank spline like smoother. P-splines have three properties that make them very popular as reduced rank smoothers: i) the basis and the penalty are sparse, enabling efficient computation, especially for Bayesian stochastic simulation; ii) it is possible to flexibly `mix-and-match' the order of B-spline basis and penalty, rather than the order of penalty controlling the order of the basis as in spline smoothing; iii) it is very easy to set up the B-spline basis functions and penalties. The discrete penalties are somewhat less interpretable in terms of function shape than the traditional derivative based spline penalties, but tend towards penalties proportional to traditional spline penalties in the limit of large basis size. However part of the point of P-splines is not to use a large basis size. In addition the spline basis functions arise from solving functional optimization problems involving derivative based penalties, so moving to discrete penalties for smoothing may not always be desirable. The purpose of this note is to point out that the three properties of basis-penalty sparsity, mix-and-match penalization and ease of setup are readily obtainable with B-splines subject to derivative based penalization. The penalty setup typically requires a few lines of code, rather than the two lines typically required for P-splines, but this one off disadvantage seems to be the only one associated with using derivative based penalties. As an example application, it is shown how basis-penalty sparsity enables efficient computation with tensor product smoothers of scattered data.

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arXiv2016

Smoothing parameter and model selection for general smooth models

This paper discusses a general framework for smoothing parameter estimation for models with regular likelihoods constructed in terms of unknown smooth functions of covariates. Gaussian random effects and parametric terms may also be present. By construction the method is numerically stable and convergent, and enables smoothing parameter uncertainty to be quantified. The latter enables us to fix a well known problem with AIC for such models. The smooth functions are represented by reduced rank spline like smoothers, with associated quadratic penalties measuring function smoothness. Model estimation is by penalized likelihood maximization, where the smoothing parameters controlling the extent of penalization are estimated by Laplace approximate marginal likelihood. The methods cover, for example, generalized additive models for non-exponential family responses (for example beta, ordered categorical, scaled t distribution, negative binomial and Tweedie distributions), generalized additive models for location scale and shape (for example two stage zero inflation models, and Gaussian location-scale models), Cox proportional hazards models and multivariate additive models. The framework reduces the implementation of new model classes to the coding of some standard derivatives of the log likelihood.

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arXiv2016

Autocorrelated errors in experimental data in the language sciences: Some solutions offered by Generalized Additive Mixed Models

A problem that tends to be ignored in the statistical analysis of experimental data in the language sciences is that responses often constitute time series, which raises the problem of autocorrelated errors. If the errors indeed show autocorrelational structure, evaluation of the significance of predictors in the model becomes problematic due to potential anti-conservatism of p-values. This paper illustrates two tools offered by Generalized Additive Mixed Models (GAMMs) (Lin and Zhang, 1999; Wood, 2006, 2011, 2013) for dealing with autocorrelated errors, as implemented in the current version of the fourth author's mgcv package (1.8.9): the possibility to specify an ar(1) error model for Gaussian models, and the possibility of using factor smooths for random-effect factors such as subject and item. These factor smooths are set up to have the same smoothing parameters, and are penalized to yield the non-linear equivalent of random intercepts and random slopes in the classical linear framework. Three case studies illustrate these issues.

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arXiv2015

A comparison of inferential methods for highly non-linear state space models in ecology and epidemiology

Highly non-linear, chaotic or near chaotic, dynamic models are important in fields such as ecology and epidemiology: for example, pest species and diseases often display highly non-linear dynamics. However, such models are problematic from the point of view of statistical inference. The defining feature of chaotic and near chaotic systems is extreme sensitivity to small changes in system states and parameters, and this can interfere with inference. There are two main classes of methods for circumventing these difficulties: information reduction approaches, such as Approximate Bayesian Computation or Synthetic Likelihood and state space methods, such as Particle Markov chain Monte Carlo, Iterated Filtering or Parameter Cascading. The purpose of this article is to compare the methods, in order to reach conclusions about how to approach inference with such models in practice. We show that neither class of methods is universally superior to the other. We show that state space methods can suffer multimodality problems in settings with low process noise or model mis-specification, leading to bias toward stable dynamics and high process noise. Information reduction methods avoid this problem but, under the correct model and with sufficient process noise, state space methods lead to substantially sharper inference than information reduction methods. More practically, there are also differences in the tuning requirements of different methods. Our overall conclusion is that model development and checking should probably be performed using an information reduction method with low tuning requirements, while for final inference it is likely to be better to switch to a state space method, checking results against the information reduction approach.

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arXiv2015

Approximate methods for dynamic ecological models

This document is due to appear as a chapter of the forthcoming Handbook of Approximate Bayesian Computation (ABC) by S. Sisson, L. Fan, and M. Beaumont. Here we describe some of the circumstances under which statistical ecologists might benefit from using methods that base statistical inference on a set of summary statistics, rather than on the full data. We focus particularly on one such approach, Synthetic Likelihood, and we show how this method represents an alternative to particle filters, for the purpose of fitting State Space Models of ecological interest. As an example application, we consider the prey-predator model of Turchin and Ellner (2000), and we use it to analyse the observed population dynamics of Fennoscandian voles.

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