Using Moment Equations to Understand Stochastically Driven Spatial Pattern Formation in Ecological Systems

Spatial patterns in biological populations and the effect of spatial patterns on ecological interactions are central topics in mathematical ecology. Various approaches to modeling have been developed to enable us to understand spatial patterns ranging from plant distributions to plankton aggregation. We present a new approach to modeling spatial interactions by deriving approximations for the time evolution of the moments (mean and spatial covariance) of ensembles of distributions of organisms; the analysis is made possible by “moment closure,” neglecting higher-order spatial structure in the population. We use the growth and competition of plants in an explicitly spatial environment as a starting point for exploring the properties of second-order moment equations and comparing them to realizations of spatial stochastic models. We find that for a wide range of effective neighborhood sizes (each plant interacting with several to dozens of neighbors), the mean-covariance model provides a useful and analytically tractable approximation to the stochastic spatial model, and combines useful features of stochastic models and traditional reaction-diffusion-like models.     

Analytical methods for predicting the behaviour of population models with general spatial interactions

Many biologists use population models that are spatial, stochastic and individual based. Analytical methods that describe the behaviour of these models approximately are attracting increasing interest as an alternative to expensive computer simulation. The methods can be employed for both prediction and fitting models to data. Recent work has extended existing (mean field) methods with the aim of accounting for the development of spatial correlations. A common feature is the use of closure approximations for truncating the set of evolution equations for summary statistics. We investigate an analytical approach for spatial and stochastic models where individuals interact according to a generic function of their distance; this extends previous methods for lattice models with interactions between close neighbours, such as the pair approximation. Our study also complements work by Bolker and Pacala (BP) [Theor. Pop. Biol. 52 (1997) 179; Am. Naturalist 153 (1999) 575]: it treats individuals as being spatially discrete (defined on a lattice) rather than as a continuous mass distribution; it tests the accuracy of different closure approximations over parameter space, including the additive moment closure (MC) used by BP and the Kirkwood approximation. The study is done in the context of an susceptible-infected-susceptible epidemic model with primary infection and with secondary infection represented by power-law interactions. MC is numerically unstable or inaccurate in parameter regions with low primary infection (or density-independent birth rates). A modified Kirkwood approximation gives stable and generally accurate transient and long-term solutions; we argue it can be applied to lattice and to continuous-space models as a substitute for MC. We derive a generalisation of the basic reproduction ratio, R(0), for spatial models.     

Analytic approximation of spatial epidemic models of foot and mouth disease

The effect of spatial heterogeneity in epidemic models has improved with computational advances, yet far less progress has been made in developing analytical tools for understanding such systems. Here, we develop two classes of second-order moment closure methods for approximating the dynamics of a stochastic spatial model of the spread of foot and mouth disease. We consider the performance of such ‘pseudo-spatial’ models as a function of R₀, the locality in disease transmission, farm distribution and geographically-targeted control when an arbitrary number of spatial kernels are incorporated. One advantage of mapping complex spatial models onto simpler deterministic approximations lies in the ability to potentially obtain a better analytical understanding of disease dynamics and the effects of control. We exploit this tractability by deriving analytical results in the invasion stages of an FMD outbreak, highlighting key principles underlying epidemic spread on contact networks and the effect of spatial correlations.     

A chronology of plankton dynamics in silico: how computer models have been used to study marine ecosystems

Research on plankton ecology in the oceans has traditionally been conducted via two scientific approaches: in situ (in the field) and in vitro (in the laboratory). There is, however, a third approach: exploring plankton dynamics in silico, or using computer models as tools to study marine ecosystems. Models have been used for this purpose for over 60 years, and the innovations and implementations of historical studies provide a context for how future model applications can continue to advance our understanding. To that end, this paper presents a chronology of the in silico approach to plankton dynamics, beginning with modeling pioneers who worked in the days before computers. During the first 30 years of automated computation, plankton modeling focused on formulations for biological processes and investigations of community structure. The changing technological context and conceptual paradigms of the late-1970s and 1980s resulted in simulations becoming more widespread research tools for biological oceanographers. This period saw rising use of models as hypothesis-testing tools, and means of exploring the effects of circulation on spatial distributions of organisms. Continued computer advances and increased availability of data in the 1990s allowed old approaches to be applied to old and new problems, and led to developments of new approaches. Much of the modeling in the new millennium so far has incorporated these sophistications, and many cutting-edge applications have come from a new generation of plankton scientists who were trained by modeling gurus of previous eras. The future directions for modeling plankton dynamics are rooted in the historical studies.     

Plugging space into predator-prey models: an empirical approach

Extrapolating ecological processes from small-scale experimental systems to scales of natural populations usually entails a considerable increase in spatial heterogeneity, which may affect process rates and, ultimately, population dynamics. We demonstrate how information on the heterogeneity of natural populations can be taken into account when scaling up laboratory-derived process functions, using the technique of moment approximation. We apply moment approximation to a benthic crustacean predator-prey system, where a laboratory-derived functional response is made spatial by including correction terms for the variance in prey density and the covariance between prey and predator densities observed in the field. We also show how moment approximation may be used to incorporate spatial information into a dynamic model of the system. While the nonspatial model predicts stable dynamics, its spatial equivalent also produces bounded fluctuations, in agreement with observed dynamics. A detailed analysis shows that predator-prey covariance, but not prey variance, destabilizes the dynamics. We conclude that second-order moment approximation may provide a useful technique for including spatial information in population models. The main advantage of the method is its conceptual value: by providing explicit estimates of variance and covariance effects, it offers the possibility of understanding how heterogeneity affects ecological processes.     

Emergence of Holling type III zooplankton functional response: Bringing together field evidence and mathematical modelling

Food-web population models are rather sensitive to parameterization of functional response in predation terms. Theoretical studies predict enhancing of ecosystems’ stability for a functional response of sigmoid type (Holling type III). The choice of a correct type of response is especially important for modelling outcome of grazing control of algal blooms by zooplankton in nutrient-rich ecosystems. Extensive experiments on zooplankton feeding in laboratories show non-sigmoid nature of response for most herbivorous zooplankton species. As a consequence, there is a strong opinion in literature that the implementation of Holling III type grazing in plankton models is biologically meaningless. I argue, however, that such an ‘evident’ claim might be wrong and sigmoid functional responses in real plankton communities would emerge more often than was suggested earlier. Especially, this concerns plankton models without vertical resolution, which ignore heterogeneity in vertical distribution of species. Having conducted extensive literature search of data on zooplankton feeding in situ, I show that vertical heterogeneity in food distribution as well as active food searching behaviour of zooplankton can modify the type of functional response. In particular, the rate of food intake by the whole zooplankton population in the column, as a function of total amount of food, often exhibits a sigmoid behaviour, instead of a non-sigmoid one postulated previously based on laboratory experiments. This conceptual discrepancy is due to the ability of zooplankton to feed mostly in layers with high algal density. I propose a generic model explaining the observed alteration of type between overall and local functional responses. I show that emergence of Holling type III in plankton systems is due to mechanisms different from those well known in the ecological literature (e.g. food search learning, existence of alternative food, refuge for prey).     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

Towards a correct description of zooplankton feeding in models: Taking into account food-mediated unsynchronized vertical migration

Complex nature of foraging behaviour of zooplankton makes it difficult to describe adequately zooplankton grazing in models with vertical space. In mean-field models (based on systems of PDEs or coupled ODEs), zooplankton feeding at a given depth is normally computed as the product of the local functional response and the zooplankton density at this depth. Such simplification is often at odds with field observations which show the absence of clear relationship between intake rates of organisms and the ambient food density. The observed discrepancy is generic and is often caused by fast non-synchronous vertical migration of organisms with different nutrition status. In this paper, we suggest a simple way of incorporating unsynchronized short-term vertical migration of zooplankton into the mean-field modelling framework. We compute grazing of zooplankton in each layer depending on feeding activity of organisms in the layer. We take into account grazing impact of animals which are in the active phase of foraging cycle at the given moment of time but neglect the impact of animals which are in the non-active phase of the cycle (e.g. digesting food). Unsynchronized vertical migration determines the vertical distribution of actively feeding animals in layers depending on vertical distribution of food. In this paper, we compare two generic plankton models: (i) a model based on ‘classical’ grazing approach and (ii) a model incorporating food-mediated unsynchronized vertical migration of zooplankton. We show that including unsynchronized food-mediated migration would make the behaviour of a plankton model more realistic. This would imply a significant enhancement of ecosystem's stability and some additional mechanisms of regulation of algal blooms. In the system with food-mediated unsynchronized vertical migration, the control of phytoplankton by herbivorous becomes possible even for very large concentrations of nutrients in the water (formally, when the system's carrying capacity tends to infinity).     

Metapopulation moments: coupling, stochasticity and persistence

1.  Spatial heterogeneity has long been viewed as a reliable means of increasing persistence. Here, an analytical model is developed to consider the variation and, hence, the persistence of stochastic metapopulations. This model relies on a novel moment closure technique, which is equivalent to assuming log-normal distributions for the population sizes.
2.  Single-species models show the greatest persistence when the mixing between subpopulations is large, so spatial heterogeneity is of no benefit. This result is confirmed by stochastic simulation of the full metapopulation.
3.  In contrast, natural-enemy models exhibit the greatest persistence for intermediate levels of coupling. When the coupling is too low, there are insufficient rescue effects between the subpopulations to sustain the dynamics, whereas when the coupling is too high all spatial heterogeneity is lost.
4.  The difference in behaviour between the one- and two-species models can be attributed to the oscillatory nature of the natural-enemy system.     

Morphologic variability of the coccolithophorid Calcidiscus leptoporus in the plankton, surface sediments and from the Early Pleistocene

On a global scale, morphological variability of the extant coccolithophorid Calcidiscus leptoporus (Murray and Blackman, 1898) Loeblich and Tappan was investigated in surface sediments and plankton samples and from an Early Pleistocene time-slice (1.8 Ma to 1.6 Ma). In the bivariate space coccolith diameter versus number of rays in the distal shield, Holocene samples follow a single, unimodal morphocline. Sample means of coccolith size and number of elements group in three clusters, I, II and III, which are of biogeographic significance. Clusters II and III coccoliths (mean coccolith size of 5.0 μm and 20.9 elements, and 6.6 μm and 25.6 elements, respectively) are found in a tropical belt extending from 11 °N to 17 °S with an annual minimum sea-surface temperature above 23.5 °C. Cluster I coccoliths (5.8 μm, 20.7 elements) are found in samples outside that belt. The distribution of coccoliths in the surface sediments is tentatively interpreted to be a result of mixing to a varying degree of at least three different morphotypes (‘small’, ‘intermediate’ and ‘large’), which were identified in the living plankton, and which are separated from each other at 5 μm and 8 μm mean coccolith diameter, respectively. A comparison of the surface sediments with the Early Pleistocene assemblages revealed that between 1.6 Ma and 1.8 Ma two morphoclines A and B existed, the first of which persisted until the Holocene in the form of C. leptoporus, while the second comprises only extinct morphotypes including Calcidiscus macintyrei as one end-member. During the Early Pleistocene morphocline A was more homogeneous and no clusters were evident.Morphocline B shows a clear bimodality with a separation of morphotypes at 9.5 μm. Our observations suggest that morphoclines are subsets within the total stratigraphical range of a single species, and represent the global variability of that species in a particular time interval. Morphotypes, which belong to a morphocline, represent the infra-specific variability of that species within the biogeographic and stratigraphic limits of that species.     

A multiscale maximum entropy moment closure for locally regulated space–time point process models of population dynamics

The prevalence of structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based only on mean densities (local or global). Individual-based models (IBMs) were introduced during the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite useful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space–time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. This approach has shown that simplified representations of fecundity, local dispersal and density-dependent mortality weighted by the local competitive environment are sufficient to generate spatial patterns that mimic field observations. Continuum approximations of these stochastic processes try to distill their fundamental properties, and they keep track of not only mean densities, but also higher order spatial correlations. However, due to the non–linearities involved they result in infinite hierarchies of moment equations. This leads to the problem of finding a ‘moment closure’; that is, an appropriate order of (lower order) truncation, together with a method of expressing the highest order density not explicitly modelled in the truncated hierarchy in terms of the lower order densities. We use the principle of constrained maximum entropy to derive a closure relationship for truncation at second order using normalisation and the product densities of first and second orders as constraints, and apply it to one such hierarchy. The resulting ‘maxent’ closure is similar to the Kirkwood superposition approximation, or ‘power-3’ closure, but it is complemented with previously unknown correction terms that depend mainly on the avoidance function of an associated Poisson point process over the region for which third order correlations are irreducible. This domain of irreducible triplet correlations is found from an integral equation associated with the normalisation constraint. This also serves the purpose of a validation check, since a single, non-trivial domain can only be found if the assumptions of the closure are consistent with the predictions of the hierarchy. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the truncated hierarchy to predict equilibrium values for mildly aggregated spatial patterns. However, the maxent closure performs comparatively poorly in segregated ones. Although the closure is applied in the context of point processes, the method does not require fixed locations to be valid, and can in principle be applied to problems where the particles move, provided that their correlation functions are stationary in space and time.     

A versatile ODE approximation to a network model for the spread of sexually transmitted diseases

 We develop a moment closure approximation (MCA) to a network model of sexually transmitted disease (STD) spread through a steady/casual partnership network. MCA has been used previously to approximate static, regular lattices, whereas application to dynamic, irregular networks is a new endeavour, and application to sociologically-motivated network models has not been attempted. Our goals are 1) to investigate issues relating to the application of moment closure approximations to dynamic and irregular networks, and 2) to understand the impact of concurrent casual partnerships on STD transmission through a population of predominantly steady monogamous partnerships. We are able to derive a moment closure approximation for a dynamic irregular network representing sexual partnership dynamics, however, we are forced to use a triple approximation due to the large error of the standard pair approximation. This example underscores the importance of doing error analysis for moment closure approximations. We also find that a small number of casual partnerships drastically increases the prevalence and rate of spread of the epidemic. Finally, although the approximation is derived for a specific network model, we can recover approximations to a broad range of network models simply by varying model parameters which control the structure of the dynamic network. Thus our moment closure approximation is very flexible in the kinds of network models it can approximate. Received: 26 August 2001 / Revised version: 15 March 2002 / Published online: 23 August 2002 C.T.B. was supported by the NSF. Key words or phrases: Moment closure approximation – Network model – Pair approximation – Sexually transmitted diseases – Steady/casual partnership network     

Estimating gap lifetime and memory from a simple model of forest canopy dynamics

Gap formation and closure represent important disturbance events in forests, but the processes involved are still poorly understood. We use models, which we and others previously developed, to make long-term predictions of tropical forest gap dynamics based on Barro Colorado Island data. We first fit the models to the data by comparing their discrete Fourier transforms, and we propose a definition for the lifetime of a gap and predict a large-gap lifetime typically to be less than 50 years. We find that the gap lifetime diverges logarithmically for large-gap sizes. We examine the ‘memory’ of spatial gap patterns via spatiotemporal correlations and find a correlation time of about 160 years, suggesting that present gap patterns could have long-lasting effects on forest spatial patterns.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Emergent Features Due to Grid-Cell Biology: Synchronisation in Biophysical Models

Modelling studies of upper ocean phenomena, such as that of the spatial and temporal patchiness in plankton distributions, typically employ coupled biophysical models, with biology in each grid-cell represented by a plankton ecosystem model. It has not generally been considered what impact the choice of grid-cell ecosystem model, from the many developed in the literature, might have upon the results of such a study. We use the methods of synchronisation theory, which is concerned with ensembles of interacting oscillators, to address this question, considering the simplest possible case of a chain of identically represented interacting plankton grid-cells. It is shown that the ability of the system to exhibit stably homogeneous (fully synchronised) dynamics depends crucially upon the choice of biological model and number of grid-cells, with dynamics changing dramatically at a threshold strength of mixing between grid-cells. Consequently, for modelling studies of the ocean the resolution chosen, and therefore number of grid-cells used, could drastically alter the emergent features of the model. It is shown that chaotic ecosystem dynamics, in particular, should be used with care.     

Spatial scaling in a benthic population model with density-dependent disturbance.

This work investigates approaches to simplifying individual-based models in which the rate of disturbance depends on local densities. To this purpose, an individual-based model for a benthic population is developed that is both spatial and stochastic. With this model, three possible ways of approximating the dynamics of mean numbers are examined: a mean-field approximation that ignores space completely, a second-order approximation that represents spatial variation in terms of variances and covariances, and a patch-based approximation that retains information about the age structure of the patch population. Results show that space is important and that a temporal model relying on mean disturbance rates provides a poor approximation to the dynamics of mean numbers. It is possible, however, to represent relevant spatial variation with second-order moments, particularly when recruitment rates are low and/or when disturbances are large and weak. Even better approximations are obtained by retaining patch age information.     

A Many-Body Field Theory Approach to Stochastic Models in Population Biology

Background

Many models used in theoretical ecology, or mathematical epidemiology are stochastic, and may also be spatially-explicit. Techniques from quantum field theory have been used before in reaction-diffusion systems, principally to investigate their critical behavior. Here we argue that they make many calculations easier and are a possible starting point for new approximations.

Methodology

We review the many-body field formalism for Markov processes and illustrate how to apply it to a ‘Brownian bug’ population model, and to an epidemic model. We show how the master equation and the moment hierarchy can both be written in particularly compact forms. The introduction of functional methods allows the systematic computation of the effective action, which gives the dynamics of mean quantities. We obtain the 1-loop approximation to the effective action for general (space-) translation invariant systems, and thus approximations to the non-equilibrium dynamics of the mean fields.

Conclusions

The master equations for spatial stochastic systems normally take a neater form in the many-body field formalism. One can write down the dynamics for generating functional of physically-relevant moments, equivalent to the whole moment hierarchy. The 1-loop dynamics of the mean fields are the same as those of a particular moment-closure.     

Spatial heterogeneity as a multiscale characteristic of zooplankton community

Zooplankton spatial heterogeneity has profound effects on understanding and modelling of zooplankton population dynamics and interactions with other planktonic compartments, and consequently, on the structure and function of planktonic ecosystems. On the one hand, zooplankton heterogeneity at spatial and temporal scales of ecological interest is an important focus of aquatic ecology research because of its implications in models of productivity, herbivory, nutrient cycling and trophic interactions in planktonic ecosystems. On the other hand, estimating zooplankton spatial variation at the scale of an ecosystem, is a powerful tool to achieve accurate sampling design. This review concentrates on the spatial heterogeneity of marine and freshwater zooplankton with respect to scale. First to be examined are the concept of spatial heterogeneity, the sampling and statistical methods used to estimate zooplankton heterogeneity, and the scales at which marine and freshwater zooplankton heterogeneity occurs. Then, the most important abiotic and biotic processes driving zooplankton heterogeneity over a range of spatial scales are presented and illustrated by studies conducted over large and fine scales in both oceans and lakes. Coupling between abiotic and biotic processes is finally discussed in the context of the multiple driving forces hypothesis.Studies of zooplankton spatial heterogeneity refer both to the quantification of the degree of heterogeneity (measured heterogeneity) and to the estimation of the heterogeneity resulting from the interactions between the organisms and their environment (functional heterogeneity) (Kolasa & Rollo, 1991). To resolve the problem of measuring zooplankton patchiness on a wide range of spatial scales, advanced technologies (acoustic devices, the Optical Plankton Counter (OPC), and video systems) have been developed and tested in marine and freshwater ecosystems. A comparison of their potential applications and limitations is presented. Furthermore, many statistical tools have been developed to estimate the degree of measured heterogeneity; the three types most commonly used are indices of spatial aggregation, variance: mean ratio, and spatial analysis methods. The variance partitioning method proposed by Borcard et al. (1992) is presented as a promising tool to assess zooplankton functional heterogeneity.Nested patchiness is a common feature of zooplankton communities and spatial heterogeneity occurs on a hierarchical continuum of scales in both marine and freshwater environments. Zooplankton patchiness is the product of many physical processes interacting with many biological processes. In marine systems, patterns of zooplankton patchiness at mega- to macro-scales are mostly linked to large advective vectorial processes whereas at coarse-, fine- and micro-scales, physical turbulence and migratory, reproductive and swarm behaviors act together to structure zooplankton distribution patterns. In freshwater environments, physical advective forces related to currents of various energy levels, and vertical stratification of lake interact with biological processes, especially with vertical migration, to structure zooplankton community over large to fine- and micro-scales. Henceforth, the zooplankton community must be perceived as a spatially well-structured and dynamic system that requires a combination of both abiotic and biotic explanatory factors for a better comprehension and more realistic and reliable predictions of its ecology.     

Predictive models in ecology: Comparison of performances and assessment of applicability

Ecological systems are governed by complex interactions which are mainly nonlinear. In order to capture the inherent complexity and nonlinearity of ecological, and in general biological systems, empirical models recently gained popularity. However, although these models, particularly connectionist approaches such as multilayered backpropagation networks, are commonly applied as predictive models in ecology to a wide variety of ecosystems and questions, there are no studies to date aiming to assess the performance, both in terms of data fitting and generalizability, and applicability of empirical models in ecology. Our aim is hence to provide an overview for nature of the wide range of the data sets and predictive variables, from both aquatic and terrestrial ecosystems with different scales of time-dependent dynamics, and the applicability and robustness of predictive modeling methods on such data sets by comparing different empirical modeling approaches. The models used in this study range from predicting the occurrence of submerged plants in shallow lakes to predicting nest occurrence of bird species from environmental variables and satellite images. The methods considered include k-nearest neighbor (k-NN), linear and quadratic discriminant analysis (LDA and QDA), generalized linear models (GLM) feedforward multilayer backpropagation networks and pseudo-supervised network ARTMAP.Our results show that the predictive performances of the models on training data could be misleading, and one should consider the predictive performance of a given model on an independent test set for assessing its predictive power. Moreover, our results suggest that for ecosystems involving time-dependent dynamics and periodicities whose frequency are possibly less than the time scale of the data considered, GLM and connectionist neural network models appear to be most suitable and robust, provided that a predictive variable reflecting these time-dependent dynamics included in the model either implicitly or explicitly. For spatial data, which does not include any time-dependence comparable to the time scale covered by the data, on the other hand, neighborhood based methods such as k-NN and ARTMAP proved to be more robust than other methods considered in this study. In addition, for predictive modeling purposes, first a suitable, computationally inexpensive method should be applied to the problem at hand a good predictive performance of which would render the computational cost and efforts associated with complex variants unnecessary.