Canadian Lynx Research Articles

Nonparametric Lag Selection for Time Series

A nonparametric version of the Final Prediction Error (FPE) is analysed for lag selection in nonlinear autoregressive time series under very general conditions including heteroskedasticity. We prove consistency and derive probabilities of incorrect selections that have been previously unavailable. Since it is more likely to overfit (have too many lags) than to underfit (miss some lags), a correction factor is proposed to reduce overfitting and hence increase correct fitting. For the FPE calculation, the local linear estimator is introduced in addition to the Nadaraya‐Watson estimator in order to cover a very broad class of processes. To achieve faster computation, a plug‐in band‐width is suggested for the local linear estimator. Our Monte‐Carlo study corroborates that the correction factor generally improves the probability of correct lag selection for both linear and nonlinear processes and that the plug‐in bandwidth works at least as well as its commonly used competitor. The proposed methods are applied to the Canadian lynx data and daily returns of DM/US‐Dollar exchange rates.

ASYMPTOTIC DISTRIBUTION OF THE ESTIMATED BDS STATISTIC AND RESIDUAL ANALYSIS OF AR MODELS ON THE CANADIAN LYNX DATA

The Brock–Dechert–Scheinkman (BDS) statistic is a nonparametric statistic based on correlation integral for testing independence. It has a special ability to identify dependence in a given time series generated by some simple dynamic systems when many conventional test statistics are not able to distinguish this type of time series from observations of independent, identically distributed (IID) random variables. Using the contiguity property derived from the local asymptotic normality for the log-likelihood ratio of nonlinear autoregressive processes, we prove the central limit theorem for the estimated BDS statistic on the residuals of fitting nonlinear autoregressive models. Comparative studies on the BDS statistic and some other nonparametric statistics on simulated time series and residuals from the AR models on the Canadian lynx are also provided in this paper.

Parameter estimation based on stacked regression and evolutionary algorithms

A new parameter-estimation algorithm, which minimises the cross-validated prediction error for linear-in-the-parameter models, is proposed, based on stacked regression and an evolutionary algorithm. It is initially shown that cross-validation is very important for prediction in linear-in-the-parameter models using a criterion called the mean dispersion error (MDE). Stacked regression, which can be regarded as a sophisticated type of cross-validation, is then introduced based on an evolutionary algorithm, to produce a new parameter-estimation algorithm, which preserves the parsimony of a concise model structure that is determined using the forward orthogonal least-squares (OLS) algorithm. The PRESS prediction errors are used for cross-validation, and the sunspot and Canadian lynx time series are used to demonstrate the new algorithms.

A Tale of Big Game and Small Bugs

Animal populations undergo repetitive cycles of rising and falling numbers. In a Perspective, [Ranta and colleagues][1] discuss the value of time-series analyses for examining the changing dynamics of animal populations. A large time series gathered by the Hudson Bay Company based on the fur trade has provided invaluable data on the rise and fall of the Canadian lynx population from 1821 to the present. Analysis of this time series reveals that climate as well as factors influencing birth and death rates are important in regulating the lynx population ([ Stenseth et al .][2]). In a separate study, mathematical modeling combined with fieldwork revealed the importance of predators in determining the population fluctuations of the southern pine beetle ([ Turchin et al .][3]) [1]: http://www.sciencemag.org/cgi/content/full/285/5430/1022 [2]: http://www.sciencemag.org/cgi/content/short/285/5430/1071 [3]: http://www.sciencemag.org/cgi/content/short/285/5430/1068

Variational approaches toward nonlinear time-series analysis.

Linear autoregressive models (AR) have broad applications toward spectral analyses and digital filtering of signals, as well as identification, prediction, and control of dynamical systems. However, the symmetric Gaussian properties of linear AR presume independence between the first and second order moments of system outputs. Using nonlinear AR to quantify interactions between the mean and autocovariance of a signal, analysts can discriminate asymmetric frequency components, estimate the relative contribution of system nonlinearities toward mean output, evaluate relative structural stability of a system, and predict critical regions where bifurcations in its dynamics might occur. Quadratic analyses are performed on simulated outputs from a nonlinear model and various experimental time series (Canadian lynx population, respiratory volume dynamics in rat, Cheyne-Stokes respiration in patients with congestive heart failure, intracranial pressures in patient with cerebral hematoma). These case studies demonstrate the utility of the techniques for evaluating the qualitative behaviors of dynamical systems in the presence of slowly varying inputs.

The Moran effect: a cause of population synchrony

Progress in population ecology generally occurs through a continual synthesis of theory with empirical findings, but occasionally a study acts as a beacon and guides the discipline into exciting new areas of research. In the study of population fluctuations, Charles Elton’s first description of the regular cycles in the fur returns of Canadian lynx (Lynx canadensis) to the Hudson’s Bay Company1 was arguably one of the most influential. His description has stimulated more than 70 years of theoretical and empirical investigation into the possible causes of population cycles, why population fluctuations are spatially synchronized and the relative importance of intrinsic and extrinsic mechanisms. Although we still do not have a clear experimental demonstration of the mechanism that generates cycles in any natural system, a recent Nature paper by Grenfell and colleagues2 shows that there exists an interesting tension between the synchronizing effects of extrinsic environmental stochasticity and the desynchronizing effects of nonlinear density dependence. Random effects have a major role to play in a nonlinear world and their influence is becoming increasingly recognized in ecological theory3.

From patterns to processes: phase and density dependencies in the Canadian lynx cycle.

Across the boreal forest of North America, lynx populations undergo 10-year cycles. Analysis of 21 time series from 1821 to the present demonstrates that these fluctuations are generated by nonlinear processes with regulatory delays. Trophic interactions between lynx and hares cause delayed density-dependent regulation of lynx population growth. The nonlinearity, in contrast, appears to arise from phase dependencies in hunting success by lynx through the cycle. Using a combined approach of empirical, statistical, and mathematical modeling, we highlight how shifts in trophic interactions between the lynx and the hare generate the nonlinear process primarily by shifting functional response curves during the increase and the decrease phases.

Measuring Patterns in Population Fluctuations

Measuring Patterns in Population Fluctuations

Nonparametric and Non-Linear Models and Data Mining in time Series: A Case-Study on the Canadian Lynx Data

SUMMARY Nonparametric regression methods are used as exploratory tools for formulating, identifying and estimating non-linear models for the Canadian lynx data, which have attained bench-mark status in the time series literature since the work of Moran in 1953. To avoid the curse of dimensionality in the nonparametric analysis of this short series with 114 observations, we confine attention to the restricted class of additive and projection pursuit regression (PPR) models and rely on the estimated prediction error variance to compare the predictive performance of various (non-)linear models. A PPR model is found to have the smallest (in-sample) estimated prediction error variance of all the models fitted to these data in the literature. We use a data perturbation procedure to assess and adjust for the effect of data mining on the estimated prediction error variances; this renders most models fitted to the lynx data comparable and nearly equivalent. However, on the basis of the mean-squared error of out-of-sample prediction error, the semiparametric model Xt=1.08+1.37Xt−1+f(Xt−2)+et and Tong’s self-exciting threshold autoregression model perform much better than the PPR and other models known for the lynx data.

The spatial dimension in population fluctuations

Theoretical research into the dynamics of coupled populations has suggested a rich ensemble of spatial structures that are created and maintained either by external disturbances or self-reinforcing interactions among the populations. Long-term data of the Canadian lynx from eight Canadian provinces display large-scale spatial synchrony in population fluctuations. The synchronous dynamics are not time-invariant, however, as pairs of populations that are initially in step may drift out of phase and back into phase. These observations are in agreement with predictions of a spatially-linked population model and support contemporary population ecology theory.

Dynamics of Canadian lynx populations in space and time1

The records of Canadian lynx fur returns indicate rather regular self‐repeating pattern with 9–11 yr cycles That the lynx populations over a vast geographical range fluctuate in synchrony appears to be well established The finding that the degree of synchrony levels off with increasing distance among provinces compared to increase again between the furthest–away provinces (O–shape) has largely escaped notice in previous analyses Moreover, the pattern of synchrony against distance varies depending on the period of time used m the analysis, sometimes there is O–shape, sometimes mere negative correlation, and sometimes no relationship whatsoever The entire rich pattern of lynx dynamics in space and time can be emulated by sets of local subpopulations acknowledging dispersal among populations This all suggest that local dynamics of populations cannot be fully understood unless the significance of spatial linkage of subpopulations via dispersal of individuals is acknowledged

Population regulation in snowshoe hare and Canadian lynx: asymmetric food web configurations between hare and lynx.

The snowshoe hare and the Canadian lynx in the boreal forests of North America show 9- to 11-year density cycles. These are generally assumed to be linked to each other because lynx are specialist predators on hares. Based on time series data for hare and lynx, we show that the dominant dimensional structure of the hare series appears to be three whereas that of the lynx is two. The three-dimensional structure of the hare time series is hypothesized to be due to a three-trophic level model in which the hare may be seen as simultaneously regulated from below and above. The plant species in the hare diet appear compensatory to one another, and the predator species may, likewise, be seen as an internally compensatory guild. The lynx time series are, in contrast, consistent with a model of donor control in which their populations are regulated from below by prey availability. Thus our analysis suggests that the classic view of a symmetric hare-lynx interaction is too simplistic. Specifically, we argue that the classic food chain structure is inappropriate: the hare is influenced by many predators other than the lynx, and the lynx is primarily influenced by the snowshoe hare.

Limit cycles in Norwegian lemmings: tensions between phase–dependence and density-dependence

Ever since Elton, the 3–5 year density cycles in lemmings (and other microtines) in Fennoscandia have troubled scientists. Explanations have involved intrinsic regulation and trophic interactions. We have analysed yearly changes in fall abundances for lemmings over 25 years from two local mountain sites in South Norway. These time series appear to have an underlying nonlinear structure of order two. Fitting a piece-wise linear threshold model of maximum order two, the most parsimonious model was, however, of first order for both series. The resulting dynamics from this model is a limit cycle. Reformulating the model in terms of abundances yields a model which combines (delayed) density-dependent effects and the influence of the cyclic phase. The delayed density-dependence of one part of the model is consistent with an effect of specialist predators during the peak and crash phases of the cycle, although other trophic interactions cannot be excluded.

Feeding enrichment and behavioural changes in Canadian lynx Lynx Canadensis at Louisville Zoo

A study at Louisville Zoo examined whether feeding enrichment significantly increased the level and diversity of activity in a 4 year‐old ♂ and a 15 year‐old ♀ Canadian lynx Lynx canadensis. Dead prey items, such as day‐old chicks, trout and mice, were hidden around the outdoor exhibit. Previously, these items were fed in the holding area with processed diet. Activity levels and enclosure use were recorded for a total of 37 hours under three feeding conditions: baseline data prior to the addition of the hidden prey; prey items offered on a daily basis; prey items offered only twice in one week. Follow‐up observations of behavioural changes were made 26 months later. The ♂ lynx showed a significant reduction in time spent sleeping and resting, particularly when the hidden prey was offered daily. Because a new 2 had been introduced by the time of the follow‐up observation it was not possible to assess the long‐term changes in the study ♂'s behaviour.

Minimum distance estimators for random coefficient autoregressive models

This paper discusses a class of minimum distance (MD) estimators for a class of pth-order random coefficient autoregressive (RCAR( p)) models. These estimators are defined via certain weighted empiricals as in Koul (1986). The class of estimators considered includes the least absolute deviation estimator and an analogue of the Hodges-Lehmann estimator. The paper contains a proof of the asymptotic normality of these estimators and a simulation study. It is observed that the RCAR(2) model with the Hodges-Lehmann type estimator fits the Canadian lynx data at least as well as with the least square estimator.

On a class of nonlinear time series models for biological population abundance data

A class of nonlinear time series models are developed based on the Ricker biological population abundance model. These models are useful in providing some description to aid understanding of the complex underlying population dynamics. They are proved to have some desirable probabilistic and statistical properties like geometric ergodicity and asymptotic stationarity. 'Conditional least squares' (CLS) is used to estimate the parameters of these models. Consistency and asymptotic normality of the CLS estimators are established. Tests of hypotheses for subsets of the parameters are examined. Simulation studies suggest that the asymptotic results are applicable to small samples. Finally, we apply the methodology to analyse two real data sets. One of these is the classic Canadian lynx data. One of the models developed fits the lynx data quite well and is parsimonious in that it involves estimation of only three parameters; as many as 24 parameters have been used.

Comparison study of AR models of the Canadian lynx data: A close look at BDS statistic

In this paper, several linear and nonlinear autoregressive models on the classic Canadian lynx data are compared through analyzing their residuals by some conventional nonparametric test statistics as well as by a new test statistic, called BDS statistic. The BDS statistic is originated from chaos theory and has been shown very effective in identifying nonlinear structure in dynamic systems from their observations.

DISTRIBUTION OF RESIDUAL AUTOCORRELATIONS IN NONSTATIONARY AUTOREGRESSIVE PROCESSES

Abstract. The residual autocorrelations in nonstationary autoregressive processes with autoregressive characteristic roots on the unit circle are considered. Limiting distributions of the residual autocovariances and the residual autocorrelations are shown to be the same as the limiting distributions when parameters are estimated with all roots on the unit circle known. The portmanteau statistic is shown to have a x2 limiting distribution. The Canadian lynx data set is analysed to illustrate our theory. The portmanteau test seems also useful when the characteristic roots are close to the unit circle.

PEAK‐INSENSITIVE NON‐PARAMETRIC SPECTRUM ESTIMATION

Abstract. We study the problem of non‐parametric spectrum estimation of a stationary time series that might contain periodic components. In this case the periodogram ordinates have a significant amplitude at frequencies near the frequencies of the periodic components. These can be regarded as outliers in an asymptotically exponential sample. We develop a non‐parametric estimator for the spectral density that is insensitive to these outliers in the frequency domain. This is done by robustifying the usual kernel estimator (smoothed periodogram) by means of M‐estimation in the frequency domain. We propose to use data‐tapered periodograms, which yield a drastic improvement of the procedure, typically for the contaminated situation. This is both shown theoretically and supported by means of simulation. We show consistency of the resulting estimator in the general case, and asymptotic normality in the special case of a Gaussian time series, whether contamination is present or not. Finally we illustrate the finite sample performance of the estimating procedure by some simulation results and by application to the Canadian lynx trappings data.

On continuous-time threshold ARMA processes

A recent paper of Brockwell and Hyndman ( Internat. J. Forecasting (1992)) considers the use of continuous- time threshold autoregressive ( CTAR) processes in the modelling and forecasting of time series data. As in the linear case, the continuous-time model is particularly advantageous for dealing with irregularly spaced data. In this paper we consider an analogous continuous-time threshold ARMA ( p, q) process with 0⩽ q< p, expressing the process in terms of an underlying p-dimensional diffusion process. Recursions are derived for the likelihood of observations { y( t 1),…, y( t n )} in terms of the transition probabilities of the diffusion process. In the case when the underlying white noise and moving average coefficients are constant, the characteristic function of the transition probability distribution of the underlying diffusion process is expressed, via the Cameron-Martin-Girsanov formula, as an explicit functional of standard Brownian motion. Approximate numerical techniques for computing Gaussian likelihoods are examined and applied to the modelling of the Canadian Lynx Data.