Lynx Data Research Articles

Central Mean Subspace in Time Series

We propose a notion of central mean dimension reduction subspace for time series {xt} which does not require specification of a model but seeks to find a p×d matrix Φd, d≤p, so that the d×1 vector ΦdTXt−1, where Xt−1=(xt−1, …, xt−p)T for some p≥1, includes all the information about xt that is available from E(xt|Xt−1). For known p and d, we estimate the mean central subspace through the Nadaraya–Watson kernel smoother and establish the strong consistency of our estimator. In addition, we propose estimation of d and p using a modified Schwarz Bayesian criterion, if either of d and p is unknown. Finally, we examine the performance of all the estimators extensively through a variety of simulations and provide a new analysis of the well-known Canadian lynx data. Supplemental materials for this article are available online.

Interlinking hare and lynx dynamics using a century's worth of annual data

AbstractThe classic fur trade records on Canadian lynx (Lynx canadensis) have rarely been analysed in direct conjunction with data on its principal prey, the snowshoe hare (Lepus americanus). Comparable long‐term data for hare exist only for a region south of Hudson Bay. We fitted a bivariate log‐linear time‐series model to this hare and lynx data to disentangle the within‐ and between‐population interactions of these species. To reduce problems with fur returns being non‐normal and non‐linearly related to abundance, we transformed the fur returns to a normal distribution based on sample quantiles. The estimated effect on next year's lynx abundance of a 1% increase in current hare abundance was a 0.23% (SE = 0.05) increase in lynx. Conversely, a 1% increase in current lynx abundance corresponded to a 0.46% (SE = 0.12) decrease in next year's hare abundance. This contrasts with some earlier studies. However, these studies mixed hare data from south of Hudson Bay with lynx totals for all of Canada. Despite this asymmetry of interaction strengths, coefficients of determination were similar for hare versus lynx and lynx versus hare, because hare abundance varies more than lynx. Both species showed clear intraspecific density‐dependence of about equal strength. A 1% increase in current abundance increased next year's abundance by about 0.75%.

Testing expert groups for a habitat suitability model for the lynx Lynx lynx in the Swiss Alps

Modelling species distribution is an important aspect of conservation ecology. Empirical models are most commonly used. However, collecting data for these models is time-consuming and expensive. Expert models may be a good alternative method, though previous studies have found mixed results. The purpose of our study was first to create an expert model and evaluate it with independent lynx data, and second to use two discrete types of experts to control for prior radio-tracking experience. Two habitat suitability expert models (scientific and local experts) were constructed in a Geographical Information System using the Analytical Hierarchy Process and Compromise Programming. The models were evaluated with lynx data, taken from the study area in the northwestern Swiss Alps, using Resource Selection Index and Spearman correlation. The correlations showed that both models fitted the data well. However, the local expert model was better (rs = 0.964, P < 0.001) than the scientific expert model (rs = 0.833, P < 0.001). The models were also evaluated in the Jura Mountains to test the local nature of the models. It was found that the local expert model performed less well (rs = 0.939, P < 0.001) than the scientific expert model (rs = 0.967, P < 0.001) as expected. Comparison between weights for each expert group revealed some interesting differences, which showed the local nature of answers and how personal experience and theoretical knowledge can lead to different answers. Our study shows that expert knowledge, and especially local knowledge, can be employed to create a good habitat suitability model. This has important implications for conservation and science because it shows not only that expert knowledge may be used when no other data exist, but also that local ‘ground workers’ should be employed more often in the development of habitat suitability models or conservation plans. However, there are limitations to the models and, as expert models are relatively new in ecology, more research is needed. Nevertheless, in a climate where there is pressure to keep up with human exploitation of natural resources and to adopt a more strategic approach to conservation, the findings of our study are encouraging.

Forecasting nonlinear time series with feed-forward neural networks: a case study of Canadian lynx data

The forecasting capabilities of feed-forward neural network (FFNN) models are compared to those of other competing time series models by carrying out forecasting experiments. As demonstrated by the detailed forecasting results for the Canadian lynx data set, FFNN models perform very well, especially when the series contains nonlinear and non-Gaussian characteristics. To compare the forecasting accuracy of a FFNN model with an alternative model, Pitman’s test is employed to ascertain if one model forecasts significantly better than another when generating one-step-ahead forecasts. Moreover, the residual-fit spread plot is utilized in a novel fashion in this paper to compare visually out-of-sample forecasts of two alternative forecasting models. Finally, forecasting findings on the lynx data are used to explain under what conditions one would expect FFNN models to furnish reliable and accurate forecasts. Copyright © 2005 John Wiley & Sons, Ltd.

Bayesian Subset Model Selection for Time Series

Abstract. This paper considers the problem of subset model selection for time series. In general, a few lags which are not necessarily continuous, explain lag structure of a time‐series model. Using the reversible jump Markov chain technique, the paper develops a fully Bayesian solution for the problem. The method is illustrated using the self‐exciting threshold autoregressive (SETAR), bilinear and AR models. The Canadian lynx data, the Wolfe's sunspot numbers and Series A of Box and Jenkins (1976) are analysed in detail.

Testing for common structures in a panel of threshold models.

We consider the problem of examining the extent of (partial) similarity in the dynamics of a panel of independent threshold autoregressive processes. We develop some tests for common structure via Wald's approach and by checking whether the parameter estimates of the unconstrained threshold models satisfy the constraints defining the common structure. One test concerns the equality of independent ratios of normal means, which is shown to have nonstandard asymptotic null distribution. These tests are illustrated with a modern panel of Canadian lynx data; our analysis suggests that the lynx data over Canada share similar dynamics in the decrease phase, but they appear to be different in the increase phase.

Multi‐variate t Autoregressions: Innovations, Prediction Variances and Exact Likelihood Equations

Abstract. The multi‐variate t distribution provides a viable framework for modelling volatile time‐series data; it includes the multi‐variate Cauchy and normal distributions as special cases. For multi‐variate t autoregressive models, we study the nature of the innovation distribution and the prediction error variance; the latter is nonconstant and satisfies a kind of generalized autoregressive conditionally heteroscedastic model. We derive the exact likelihood equations for the model parameters, they are related to the Yule–Walker equations and involve simple functions of the data, the model parameters and the autocovariances up to the order of the model. The maximum likelihood estimators are obtained by alternately solving two linear systems and illustrated using the lynx data. The simplicity of these equations contributes greatly to our theoretical understanding of the likelihood function and the ensuing estimators. Their range of applications are not limited to the parameters of autoregressive models; in fact, they are applicable to the parameters of ARMA models and covariance matrices of stochastic processes whose finite‐dimensional distributions are multi‐variate t.

A note on testing for nonlinearity with partially observed time series

SUMMARY We have implemented a Lagrange multiplier test for the alternative hypothesis of a nonlinear continuous-time autoregressive model with the instantaneous mean having multiple degrees of nonlinearity. This test is an extension of a Lagrange multiplier test proposed by Tsai & Chan (2000), with the alternative model analogous to the model used in Tsay's (1986) discrete-time work. The performance of the test in the finite-sample case is compared with several existing tests for nonlinearity including Keenan's (1985) test, Petruccelli & Davies' (1986) test, Tsay's (1986, 1989) tests and Tsai & Chan's (2000) test. The comparison is based on simulated data from some linear autoregressive models, self-exciting threshold autoregressive models, bilinear models and the nonlinear continuous-time autoregressive models for which the Lagrange multiplier test is designed. In general, the test is more powerful than all the other tests. The test is further illustrated with the annual sunspot data and the lynx data.

PRESENT STATUS AND DISTRIBUTION OF THE LYNX IN THE GERMAN ALPS

The period I995 to 1999 did not yield any confirmed lynx data in the German part of the Alpine Arc. Only from the area of Berchtesgaden, 6 Q3-data exist. Outside of the Alpine arc Germany hosts some areas with lynx evidence. For a better understanding of the real situation in the Alps it would be necessary to have a few skilled persons who could check any lynx signs quickly and/or interview people that suspect lynx presence. Because of possible lynx immigrations, especially from eastern Switzerland, and its central location, it would be important that Germany starts to play a more active role in lynx conservation in the Alps. Conservation strategies and possible actions need to be assessed in the near future.

A generalization of some classical time series tools

In classical time series analysis the sample autocorrelation function ( SACF) and the sample partial autocorrelation function ( SPACF) has gained wide application for structural identification of linear time series models. We suggest generalizations, founded on smoothing techniques, applicable for structural identification of non-linear time series models. A similar generalization of the sample cross correlation function is discussed. Furthermore, a measure of the departure from linearity is suggested. It is shown how bootstrapping can be applied to construct confidence intervals under independence or linearity. The generalizations do not prescribe a particular smoothing technique. In fact, when the smoother is replaced by a linear regression the generalizations reduce to close approximations of SACF and SPACF. For this reason a smooth transition from the linear to the non-linear case can be obtained by varying the bandwidth of a local linear smoother. By adjusting the flexibility of the smoother, the power of the tests for independence and linearity against specific alternatives can be adjusted. The generalizations allow for graphical presentations, very similar to those used for SACF and SPACF. In this paper the generalizations are applied to some simulated data sets and to the Canadian lynx data. The generalizations seem to perform well and the measure of the departure from linearity proves to be an important additional tool.

A Hierarchical Approach to Covariance Function Estimation for Time Series

The covariance function in time series models is typically modelled via a parametric family. This ensures straightforward best linear prediction while maintaining positive‐definiteness of the covariance function. We suggest an alternative approach, which will result in data‐determined shrinkage towards this parametric model. Positive‐definiteness is maintained by carrying out the shrinkage in the spectral domain. We offer both a fully Bayesian hierarchical approach and an approximate hierarchical approach that will be much simpler computationally. These are implemented on the frequently analysed Canadian lynx data and compared to other models that have been fitted to these data.

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.

Bootstrap‐assisted Goodness‐of‐fit Tests in the Frequency Domain

The validity of a stationary time series model may be measured by the goodness of fit of the spectral distribution function. Anderson (Technical Report 27, 1991; Technical Report 309, 1995; Stanford University) has worked out the closed‐form characteristic functions for the Cramer–von Mises criterion for general linear processes, under the condition that all values of the parameters are specified. The asymptotic approach is not easily implemented and usually requires a case by case analysis. In this paper we propose a bootstrap goodness‐of‐fit test in the frequency domain. By properly resampling the residuals, we can consistently estimate the p values for many weakly dependent semiparametric models with unspecified parameter values. This is the content of the main theorem that we try to explain. A group of simulations is conducted, showing consistent significance level and good power. The special tests are applied to the lynx data and reveal structure unexplained by the AR(1) model fitted by Tong (J. R. Stat. Soc. A 140 (1977), 432–36). A possible generalization with application to financial data analysis is also discussed.

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.

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.

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.

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.

Testing for independence in heavy tailed and positive innovation time series

For time series with positive innovations, a test is given to distinguish between data coming from a stationary process where the variables are dependent versus the model being independent identically distributed random variables. The techniques are suitable either when the innovation distribution has heavy right tails or regularly varying left tails and is based on linear programming estimators given by Feigin and Resnick (1992, 1994). Examples using teletraffic data and the lynx data are discussed. The method has applications to model confirmation where the fit of a model is also examined by gauging whether the residuals are independent or not