A random walk model of fast-phase timing during optokinetic nystagmus

 Most vertebrate animals produce optokinetic nystagmus in response to rotation of their visual surround. Nystagmus consists of an alternation of slow-phase eye rotations, which follow the surround, and fast-phase eye rotations, which quickly reset eye position. The time intervals between fast phases vary stochastically, even during optokinetic nystagmus produced by constant velocity rotation of a uniform surround. The inter-fast-phase interval distribution has a long tail, and intervals that are long relative to the mode become even more likely as constant surround velocity is decreased. This paper provides insight into fast-phase timing by showing that the process of fast-phase generation during constant velocity optokinetic nystagmus is analogous to a random walk with drift toward a threshold. Neurophysiologically, the output of vestibular nucleus neurons, which drive the slow phase, would approximate a random walk with drift because they integrate the noisy, constant surround velocity signal they receive from the visual system. Burst neurons, which fire a burst to drive the fast phase and reset the slow phase, are brought to threshold by the vestibular nucleus neurons. Such a nystagmic process produces stochastically varying inter-fast-phase intervals, and long intervals emerge naturally because, as drift rate (related to surround velocity) decreases, it becomes more likely that any random walk can meander for a long time before it crosses the threshold. The theoretical probability density function of the first threshold crossing times of random walks with drift is known to be that of an inverse Gaussian distribution. This probability density function describes well the distributions of the intervals between fast phases that were either determined experimentally, or simulated using a neurophysiologically plausible neural network model of fast-phase generation, during constant velocity optokinetic nystagmus. Received: 1 June 1995/Accepted in revised form: 15 February 1996     

Model-checking techniques based on cumulative residuals

Residuals have long been used for graphical and numerical examinations of the adequacy of regression models. Conventional residual analysis based on the plots of raw residuals or their smoothed curves is highly subjective, whereas most numerical goodness-of-fit tests provide little information about the nature of model misspecification. In this paper, we develop objective and informative model-checking techniques by taking the cumulative sums of residuals over certain coordinates (e.g., covariates or fitted values) or by considering some related aggregates of residuals, such as moving sums and moving averages. For a variety of statistical models and data structures, including generalized linear models with independent or dependent observations, the distributions of these stochastic processes tinder the assumed model can be approximated by the distributions of certain zero-mean Gaussian processes whose realizations can be easily generated by computer simulation. Each observed process can then be compared, both graphically and numerically, with a number of realizations from the Gaussian process. Such comparisons enable one to assess objectively whether a trend seen in a residual plot reflects model misspecification or natural variation. The proposed techniques are particularly useful in checking the functional form of a covariate and the link function. Illustrations with several medical studies are provided.     

Predictability of stream insect distributions is dependent on niche position,but not on biological traits or taxonomic relatedness of species

Species distributions can be analysed under two perspectives: the niche‐based approach, which focuses on species–environment relationships; and the dispersal‐based approach, which focuses on metapopulation dynamics. The degree to which each of these two components affect species distributions may depend on habitat fragmentation, species traits and phylogenetic constraints. We analysed the distributions of 36 stream insect species across 60 stream sites in three drainage basins at high latitudes in Finland. We used binomial generalised linear models (GLMs) in which the predictor variables were environmental factors (E models), within‐basin spatial variables as defined by Moran's eigenvector maps (M models), among‐basin variability (B models), or a combination of the three (E + M + B models) sets of variables. Based on a comparative analysis, model performance was evaluated across all the species using Gaussian GLMs whereby the deviance accounted for by binomial GLMs was fitted on selected explanatory variables: niche position, niche breadth, site occupancy, biological traits and taxonomic relatedness. For each type of model, a reduced Gaussian GLM was eventually obtained after variable selection (Bayesian information criterion). We found that niche position was the only variable selected in all reduced models, implying that marginal species were better predicted than non‐marginal species. The influence of niche position was strongest in models based on environmental variables (E models) or a combination of all types of variables (E + M + B models), and weakest in spatial autocorrelation models (M models). This suggests that species–environment relationships prevail over dispersal processes in determining stream insect distributions at a regional scale. Our findings have clear implications for biodiversity conservation strategies, and they also emphasise the benefits of considering both the niche‐based and dispersal‐based approaches in species distribution modelling studies.     

Comprehensive methodological analysis of long‐term changes in phenological extremes in Germany

This study reports on alterations in the magnitude and frequency of extremes in reproductive phenology using long‐term records (1951–2008) for plant species widely distributed across Germany. For each of fourteen indicator phases studied, time series of annual onset dates at up to 119 stations, providing 50–58 years of observation, were standardized by their station mean and standard deviation. Four alternative statistical models were applied and compared to derive probabilities of extreme early or late onset times for the phases: (1) Gaussian models were used to describe decadal probabilities of standardized anomalies, defined by data either falling below the 5th or exceeding the 95th percentile. (2) Semi‐parametric quantile regression was employed for flexible and robust modelling of trends in different quantiles of onset dates. (3) Generalized extreme value distributions (GEV) were fitted to annual detrended minima and maxima of standardized anomalies, and (4) Generalized Pareto distributions (GPD) were fitted to extremes defined as peaks over threshold. Probabilities of extreme early phenological events inferred from Gaussian models, increased on average from 3 to 12%, whereas probabilities of extreme late phenological events decreased from 6 to 2% over the study period. Based on quantile regressions, summer and autumn phases revealed a more pronounced advancing pattern than spring phases. Estimated return levels by GEV were similar for the GPD methods, indicating that extreme early phenological events of magnitudes 2.5, 2.8, and 3.6 on the detrended standardized anomaly scale would occur every 20 years for spring, summer and autumn phases, respectively. This corresponds to absolute onset advances of up to 2 months depending on the season and species. This study demonstrates how extreme phenological events can be accurately modelled even in cases of inherently small numbers of observations, and underlines the need for additional evaluation related to their impacts on ecosystem functioning.     

Central adaptation models of the vestibulo-ocular and optokinetic systems

A theoretical analysis of two models of the vestibulo-ocular and optokinetic systems was performed. Each model contains a filter element in the vestibular periphery to account for peripheral adaptation, and a filter element in the central vestibulooptokinetic circuit to account for central adaptation. Both models account for1 adaptation, i.e. a response decay to a constant angular acceleration input, in both peripheral vestibular afferent and vestibulo-ocular reflex (VOR) responses and2 the reversal phases of optokinetic after-nystagmus (OKAN) and the VOR and3 oscillatory behavior such as periodic alternating nystagmus. The two models differ regarding the order of their VOR transfer function. Also, they predict different OKAN patterns following a prolonged optokinetic stimulus. These models have behavioral implications and suggest future experiments.     

Predicting species distributions: a critical comparison of the most common statistical models using artificial species

Aim To test statistical models used to predict species distributions under different shapes of occurrence–environment relationship. We addressed three questions: (1) Is there a statistical technique that has a consistently higher predictive ability than others for all kinds of relationships? (2) How does species prevalence influence the relative performance of models? (3) When an automated stepwise selection procedure is used, does it improve predictive modelling, and are the relevant variables being selected? Location We used environmental data from a real landscape, the state of California, and simulated species distributions within this landscape. Methods Eighteen artificial species were generated, which varied in their occurrence response to the environmental gradients considered (random, linear, Gaussian, threshold or mixed), in the interaction of those factors (no interaction vs. multiplicative), and on their prevalence (50% vs. 5%). The landscape was then randomly sampled with a large (n = 2000) or small (n = 150) sample size, and the predictive ability of each statistical approach was assessed by comparing the true and predicted distributions using five different indexes of performance (area under the receiver‐operator characteristic curve, Kappa, correlation between true and predictive probability of occurrence, sensitivity and specificity). We compared generalized additive models (GAM) with and without flexible degrees of freedom, logistic regressions (general linear models, GLM) with and without variable selection, classification trees, and the genetic algorithm for rule‐set production (GARP). Results Species with threshold and mixed responses, additive environmental effects, and high prevalence generated better predictions than did other species for all statistical models. In general, GAM outperforms all other strategies, although differences with GLM are usually not significant. The two variable‐selection strategies presented here did not discriminate successfully between truly causal factors and correlated environmental variables. Main conclusions Based on our analyses, we recommend the use of GAM or GLM over classification trees or GARP, and the specification of any suspected interaction terms between predictors. An expert‐based variable selection procedure was preferable to the automated procedures used here. Finally, for low‐prevalence species, variability in model performance is both very high and sample‐dependent. This suggests that distribution models for species with low prevalence can be improved through targeted sampling.     

Brain Monoamines and Optokinetic Performances in Pigmented and Albino Rats

The aim of this study was two-fold: 1) To provide in DA-HAN rats the basic brain monoamine data useful for later investigations of the neurochemical effects of sensory alterations and 2) to assess whether there is a relationship between the monoaminergic pattern in medial vestibular nuclei and optokinetic performances. We comparatively studied the regional brain monoamine distribution and the optokinetic performances in pigmented DA-HAN and albino Sprague-Dawley rats. As expected, the optokinetic responses and vestibulo-ocular reflex gain were by far more efficient in DA-HAN rats. Norepinephrine (NE), dopamine (DA), serotonin (5-HT) and their metabolites were determined in retina, brainstem nuclei and dopaminergic areas. DA-HAN rats exhibited an increased noradrenergic activity in the medial vestibular nuclei, locus cœruleus and anteroventral cochlear nucleus, an extended decrease of serotonergic activity in brainstem nuclei and increased DA stores with a reduced dopaminergic activity in most dopaminergic areas. These data confirm and extend the general findings that biochemical data obtained in one strain cannot be extrapolated to another strain. The possible role of the morphological neuronal abnormalities and functional impairment induced by albinism has been discussed especially in medial vestibular nucleus, cochlear nuclei and retina. Alternatively, behavioral factors may also explain some of the observed neurochemical differences.     

On the distribution of developmental errors: comparing the normal, gamma, and log-normal distribution

The amount of between‐individual variation in the unobservable developmental instability (DI) has been the subject of intense recent debates. The unexpectedly high estimates of between‐individual variation in DI based on distributional characteristics of observable asymmetry values (of on average bilaterally symmetric traits) rely on statistical models that assume an underlying normal distribution of developmental errors. This prompted doubts on the assumption of the Gaussian nature of developmental errors. However, when applying other candidate distributions [log‐normal and gamma (γ)], recent analyses of empirical datasets have indicated that estimates remain generally high. Yet, all estimates were based on bilaterally symmetric traits, which did not allow for a formal comparison of the alternative distributions. In the present study, we extend a recent statistical model to allow statistical comparison of the different distributions based on traits that developed repeatedly under the same conditions, such as flower traits and regrown feathers. We analyse simulated and empirical data and show that: (1) it is statistically difficult to differentiate among the three alternatives when variances are small relative to the mean, as is often the case with DI; (2) the normal distribution fits the log‐normal or γ relatively well under those circumstances; (3) the deviance information criterion (DIC) is able to pick up differences in model fit among the three alternative distributions, yet more strongly so when levels of DI were high; (4) empirical datasets show a better fit of the normal over the log‐normal and γ‐distributions as judged by the DIC; and (5) estimates of between‐individual variation in DI in the three empirical datasets were relatively high (> 50%) under each distributional assumption. In conclusion, and based on our three datasets, the normal approximation appears to be a reasonable choice for statistical models of DI and the remarkably high estimates of variation in DI cannot be attributed to non‐normal developmental noise. Nevertheless, our method should be applied to a broad range of traits and organisms to evaluate the generality of this result. We argue that there is an urgent need for studies that reveal the underlying mechanisms of developmental noise and stability, as well as the role of developmental selection, in order to be able to determine the biological importance of the highly skewed distributions of developmental instability often observed. © 2007 The Linnean Society of London, Biological Journal of the Linnean Society, 2007, 92 , 197–210.     

On the Origins of Suboptimality in Human Probabilistic Inference

Humans have been shown to combine noisy sensory information with previous experience (priors), in qualitative and sometimes quantitative agreement with the statistically-optimal predictions of Bayesian integration. However, when the prior distribution becomes more complex than a simple Gaussian, such as skewed or bimodal, training takes much longer and performance appears suboptimal. It is unclear whether such suboptimality arises from an imprecise internal representation of the complex prior, or from additional constraints in performing probabilistic computations on complex distributions, even when accurately represented. Here we probe the sources of suboptimality in probabilistic inference using a novel estimation task in which subjects are exposed to an explicitly provided distribution, thereby removing the need to remember the prior. Subjects had to estimate the location of a target given a noisy cue and a visual representation of the prior probability density over locations, which changed on each trial. Different classes of priors were examined (Gaussian, unimodal, bimodal). Subjects'' performance was in qualitative agreement with the predictions of Bayesian Decision Theory although generally suboptimal. The degree of suboptimality was modulated by statistical features of the priors but was largely independent of the class of the prior and level of noise in the cue, suggesting that suboptimality in dealing with complex statistical features, such as bimodality, may be due to a problem of acquiring the priors rather than computing with them. We performed a factorial model comparison across a large set of Bayesian observer models to identify additional sources of noise and suboptimality. Our analysis rejects several models of stochastic behavior, including probability matching and sample-averaging strategies. Instead we show that subjects'' response variability was mainly driven by a combination of a noisy estimation of the parameters of the priors, and by variability in the decision process, which we represent as a noisy or stochastic posterior.     

There Is No Need To Be Normal: Generalized Linear Models of Natural Variation

Both ecological field studies and attempts to extrapolate from laboratory experiments to natural populations generally encounter the high degree of natural variability and chaotic behavior that typify natural ecosystems. Regardless of this variability and non-normal distribution, most statistical models of natural systems use normal error which assumes independence between the variance and mean. However, environmental data are often random or clustered and are better described by probability distributions which have more realistic variance to mean relationships. Until recently statistical software packages modeled only with normal error and researchers had to assume approximate normality on the original or transformed scale of measurement and had to live with the consequences of often incorrectly assuming independence between the variance and mean. Recent developments in statistical software allow researchers to use generalized linear models (GLMs) and analysis can now proceed with probability distributions from the exponential family which more realistically describe natural conditions: binomial (even distribution with variance less than mean), Poisson (random distribution with variance equal mean), negative binomial (clustered distribution with variance greater than mean). GLMs fit parameters on the original scale of measurement and eliminate the need for obfuscating transformations, reduce bias for proportions with unequal sample size, and provide realistic estimates of variance which can increase power of tests. Because GLMs permit modeling according to the non-normal behavior of natural systems and obviate the need for normality assumptions, they will likely become a widely used tool for analyzing toxicity data. To demonstrate the broad-scale utility of GLMs, we present several examples where the use of GLMs improved the statistical power of field and laboratory studies to document the rapid ecological recovery of Prince William Sound following the Exxon Valdez oil spill.     

On the quantitative genetics of mixture characters

Finite mixture models are helpful for uncovering heterogeneity due to hidden structure. Quantitative genetics issues of continuous characters having a finite mixture of Gaussian components as statistical distribution are explored in this article. The partition of variance in a mixture, the covariance between relatives under the supposition of an additive genetic model, and the offspring-parent regression are derived. Formulas for assessing the effect of mass selection operating on a mixture are given. Expressions for the genetic and phenotypic correlations between mixture and Gaussian traits and between two mixture traits are presented. It is found that, if there is heterogeneity in a population at the genetic or environmental level, then genetic parameters based on theory treating distributions as homogeneous can lead to misleading interpretations. Some peculiarities of mixture characters are: heritability depends on the mean values of the component distributions, the offspring-parent regression is nonlinear, and genetic or phenotypic correlations cannot be interpreted devoid of the mixture proportions and of the parameters of the distributions mixed.     

Fluorescence lifetime distributions of diphenylhexatriene-labeled phosphatidylcholine as a tool for the study of phospholipid-cholesterol interactions.

Fluorescence lifetimes of 1-palmitoyl-2-diphenylhexatrienylpro-pionyl-phosphatidylc hol ine in vesicles of palmitoyloleoyl phosphatidylcholine (POPC) (1:300, mol/mol) in the liquid crystalline state were determined by multifrequency phase fluorometry. On the basis of statistic criteria (chi 2red) the measured phase angles and demodulation factors were equally well fitted to unimodal Lorentzian, Gaussian, or uniform lifetime distributions. No improvement in chi 2red could be observed if the experimental data were fitted to bimodal Lorentzian distributions or a double exponential decay. The unimodal Lorentzian lifetime distribution was characterized by a lifetime center of 6.87 ns and a full width at half maximum of 0.57 ns. Increasing amounts of cholesterol in the phospholipid vesicles (0-50 mol% relative to POPC) led to a slight increase of the lifetime center (7.58 ns at 50 mol% sterol) and reduced significantly the distributional width (0.14 ns at 50 mol% sterol). Lifetime distributions of POPC-cholesterol mixtures containing greater than 20 mol% sterol were within the resolution limit and could not be distinguished from monoexponential decays on the basis of chi 2red. Cholesterol stabilizes and rigidifies phospholipid bilayers in the fluid state. Considering its effect on lifetime distributions of fluorescent phospholipids it may also act as a membrane homogenizer.     

Weak climatic associations among British plant distributions

Aim Species distribution models (SDMs) are used to infer niche responses and predict climate change‐induced range shifts. However, their power to distinguish real and chance associations between spatially autocorrelated distribution and environmental data at continental scales has been questioned. Here this is investigated at a regional (10 km) scale by modelling the distributions of 100 plant species native to the UK. Location UK. Methods SDMs fitted using real climate data were compared with those utilizing simulated climate gradients. The simulated gradients preserve the exact values and spatial structure of the real ones, but have no causal relationships with any species and so represent an appropriate null model. SDMs were fitted as generalized linear models (GLMs) or by the Random Forest machine‐learning algorithm and were either non‐spatial or included spatially explicit trend surfaces or autocovariates as predictors. Results Species distributions were significantly but erroneously related to the simulated gradients in 86% of cases (P < 0.05 in likelihood‐ratio tests of GLMs), with the highest error for strongly autocorrelated species and gradients and when species occupied 50% of sites. Even more false effects were found when curvilinear responses were modelled, and this was not adequately mitigated in the spatially explicit models. Non‐spatial SDMs based on simulated climate data suggested that 70–80% of the apparent explanatory power of the real data could be attributable to its spatial structure. Furthermore, the niche component of spatially explicit SDMs did not significantly contribute to model fit in most species. Main conclusions Spatial structure in the climate, rather than functional relationships with species distributions, may account for much of the apparent fit and predictive power of SDMs. Failure to account for this means that the evidence for climatic limitation of species distributions may have been overstated. As such, predicted regional‐ and national‐scale impacts of climate change based on the analysis of static distribution snapshots will require re‐evaluation.     

Effects of incorporating spatial autocorrelation into the analysis of species distribution data

Aim  Spatial autocorrelation (SAC) in data, i.e. the higher similarity of closer samples, is a common phenomenon in ecology. SAC is starting to be considered in the analysis of species distribution data, and over the last 10 years several studies have incorporated SAC into statistical models (here termed 'spatial models'). Here, I address the question of whether incorporating SAC affects estimates of model coefficients and inference from statistical models.
Methods  I review ecological studies that compare spatial and non-spatial models.
Results  In all cases coefficient estimates for environmental correlates of species distributions were affected by SAC, leading to a mis-estimation of on average c . 25%. Model fit was also improved by incorporating SAC.
Main conclusions  These biased estimates and incorrect model specifications have implications for predicting species occurrences under changing environmental conditions. Spatial models are therefore required to estimate correctly the effects of environmental drivers on species present distributions, for a statistically unbiased identification of the drivers of distribution, and hence for more accurate forecasts of future distributions.     

Determining alternative models for vegetation response analysis: a non-parametric approach

Abstract. Vegetation models based on multiple logistic regression are of growing interest in environmental studies and decision making. The relatively simple sigmoid Gaussian optimum curves are most common in current vegetation models, although several different other response shapes are known. However, improvements in the technical means for handling statistical data now facilitate fast and interactive calculation of alternative complex, more data-related, non-parametric models. The aim in this study was to determine whether, and if so how often, a complex response shape could be more adequate than a linear or quadratic one. Using the framework of Generalized Additive Models, both parametric (linear and quadratic) and non-parametric (smoothed) stepwise multiple logistic regression techniques were applied to a large data set on wetlands and water plants and to six environmental variables: pH, chloride, orthophosphate, inorganic nitrogen, thickness of the sapropelium layer and depth of the water-body. All models were tested for their goodness-of-fit and significance. Of all 156 generalized additive models calculated, 77 % were found to contain at least one smoothed predictor variable, i.e. an environmental variable with a response better fitted by a complex, non-parametric, than by a linear or quadratic parametric curve. Chloride was the variable with the highest incidence of smoothed responses (48 %). Generally, a smoothed curve was preferable in 23 % of all species-variable correlations calculated, compared to 25 % and 18 % for sigmoid and Gaussian shaped curves, respectively. Regression models of two plant species are presented in detail to illustrate the potential of smoothers to produce good fitting and biologically sound response models in comparison to linear and polynomial regression models. We found Generalized Additive Modelling a useful and practical technique for improving current regression-based vegetation models by allowing for alternative, complex response shapes.     

Circular distributions of fallen logs as an indicator of forest disturbance regimes

We present a retrospective method for studying forest disturbance regimes, and especially the role of windthrows, based on circular statistical models of directions of fallen logs. This approach was applied to fallen log data from three areas of pristine Picea abies-dominated boreal forests in northern Europe. The data consisted of 5 plots from each of the three areas, totaling 15 plots and covering an area of 24 ha. The disturbance history of the plots, which varied from area to area, was known from previous detailed studies. We selected and fitted the most suitable circular model for each plot, based on goodness-of-fit and the Akaike information criterion. For uneven-aged forests, the symmetric von Mises distribution, was best fitted, while for the even-aged forest the sine-skewed wrapped Cauchy distribution was selected. The degree of concentration around the mean direction of fallen trees was strongest for the late-successional even-aged forest most exposed to windthrow, while an uneven-aged forest with drought-driven mortality had the lowest concentration and the greatest variance over the mean directions. For the third area, characterized by an uneven age structure and tree mortality driven by heart-rot fungi in old trees in interaction with wind, an intermediate between these two were derived. Our results suggested the utility of circular distributions of fallen logs and their statistical models for retrospective assessments of forest disturbance regimes.     

Generalized Seasonal Autoregressive Integrated Moving Average Models for Count Data with Application to Malaria Time Series with Low Case Numbers

Introduction

With the renewed drive towards malaria elimination, there is a need for improved surveillance tools. While time series analysis is an important tool for surveillance, prediction and for measuring interventions’ impact, approximations by commonly used Gaussian methods are prone to inaccuracies when case counts are low. Therefore, statistical methods appropriate for count data are required, especially during “consolidation” and “pre-elimination” phases.

Methods

Generalized autoregressive moving average (GARMA) models were extended to generalized seasonal autoregressive integrated moving average (GSARIMA) models for parsimonious observation-driven modelling of non Gaussian, non stationary and/or seasonal time series of count data. The models were applied to monthly malaria case time series in a district in Sri Lanka, where malaria has decreased dramatically in recent years.

Results

The malaria series showed long-term changes in the mean, unstable variance and seasonality. After fitting negative-binomial Bayesian models, both a GSARIMA and a GARIMA deterministic seasonality model were selected based on different criteria. Posterior predictive distributions indicated that negative-binomial models provided better predictions than Gaussian models, especially when counts were low. The G(S)ARIMA models were able to capture the autocorrelation in the series.

Conclusions

G(S)ARIMA models may be particularly useful in the drive towards malaria elimination, since episode count series are often seasonal and non-stationary, especially when control is increased. Although building and fitting GSARIMA models is laborious, they may provide more realistic prediction distributions than do Gaussian methods and may be more suitable when counts are low.