Prediction Intervals in Nonlinear Regression

By treating the nonlinear model as if it were linear in the parameterization θ in the neighbourhood of the least squares estimate θ, we construct two-sided nominally-q-prediction intervals by applying the usual linear model theory. The derivation of the truncated series expansion of the expected coverage of the prediction intervals at a feasible value of the parameter vector is described. The quadratic approximation of the expected coverage is then obtained for a two-parameter nonlinear model. Finally we show how we may construct the prediction intervals when a certain type of nonlinear transformation of the parameter vector has been applied.     

The potential of generalized additive modelling for the prediction of radial growth of Norway spruce from Central Germany

During the past decades managed forest ecosystems in Central Europe underwent vast changes, induced by extreme climate conditions and occasionally adverse forest management. Tree ring width patterns mirror these changes and thus have been widely examined as environmental archives and reliable empirical data sources in ‘tree growth modelling’. Dendrochronologists often suppose linear co-variation among the covariates, variable independence and homoscedasticity. Conventionally, these assumptions were achieved by eliminating biological age trends (detrending) and removing the autocorrelation from the time series (pre-whitening). Particularly detrending might be biased according to the scientific problem and sometimes inflexible age models. In this study, we tackle these issues and examine the suitability of a flexible Generalized Additive Model (GAM) on recently developed tree ring width time series of 30 Norway spruce stands (Picea abies [L.] H. Karst) from Central Germany.The model was established to simultaneously cope with the mentioned detrending issue, to unravel nonlinear climate-growth relationships and to predict mean ring width time series for spruce stands in the region. Particularly the latter was of primary interest, since recent forest planning relies on static yield tables that often underestimate the actual growth.The model reliably captured the empirical data, indicated by a small Generalized Cross Validation criterion (GCV = 0.045) and a deviance explained of 88.6 %. The flexible additive smoothers accounted for the social status of individual trees, captured low frequency variations of changing growth conditions adequately and displayed a rather flat biological age trend. The radial increment responded positively to summer season precipitation of the current and previous year. Positive temperature responses were found during the early vegetation period, whereas high summer season temperatures negatively affected the radial growth. The seasonal transition from spring to summer in June induced a shift in the climate response of the linear predictor, leading to a distinct negative effect of temperature and a no-role of precipitation on the linear predictor.Most important, utilizing the calibrated GAM for the purely climate-driven prediction of mean ring width time series from five independent spruce sites revealed proper coherencies. Herein, the mean ring width for sites located within the climatic-optimum for spruce growth were more exactly predicted than for sites with adverse spruce growth conditions. In addition, large mean ring widths were systematically underestimated, whereas small mean ring widths were precisely predicted. Overall, we strongly recommend GAMs as a powerful tool for the investigation of nonlinear climate-growth relationships and for the prediction of radial growth in managed forest ecosystems.     

A Parameter-Elimination Method for Nonlinear Regression with Linear Parameters and Autocorrelated Errors

In this study, we are interested in the problem of estimating the parameters in a nonlinear regression model when the error terms are correlated. Throughout this work, we restrict ourselves to the special case when the error terms follow a pth order stationary autoregressive model (AR(p)). Following the idea of LAWTON and SYLVESTRE (1971) and GALLANT and GOEBEL (1976), a parameter-elimination method is proposed, which has the advantages that it is not sensitive to the initial values and convergence of the procedure may be more stable because of the reduced dimension of the problem. The parameter-elimination method is compared with the methods by GALLANT and GOEBEL (1976) and GLASBEY (1980) by Monte Carlo Simulation, and the results of applying the first two methods to the real data obtained from the Environmental Protection Administration of the Executive Yuan of the Republic of China are presented.     

拟似然非线性模型中的置信域：几何法

对拟似然非线性模型在欧氏内积实间建立了修改的Ｂａｔｅｓ＆Ｗａｔｔｓ几何结构，基于此几何结构，导出了参数和子集系数的与统计曲率有关的三种近似置信域，进一步推广和发展了Ｈａｍｉｌｔｏｎ ｅｔ ａｌ．（１９８２）。Ｈａｍｉｌｔｏｎ（１９８６）和Ｗｅｉ（１９９４，１９９８）等人的相应结果。     

Hopf bifurcation in a stomatal oscillator

Stomata are microscopic openings in leaves of green plants which permit gas exchange. This paper presents a parameter study of a model of a stomatal oscillator first derived by Delwiche and Cooke in 1977. We prove the existence of an unstable limit cycle by using the theory of the Hopf bifurcation. Other bifurcations exhibited by the model are also discussed.     

Discretizing a Normal Prior for Change Point Estimation in Switching Regressions

Bayes decision procedures are considered for change point estimation in the simple bilinear segmented model. A discretized normal prior density is employed as the prior distribution for the change point index. Posterior probability functions are developed for this index under a vague prior formulation on the regression parameters. The procedure is applied to an example involving mercury toxicity data.     

Gait stability,variability and complexity on inclined surfaces

This study evaluated the gait stability, variability, and complexity of healthy young adults on inclined surfaces. A total of 49 individuals walked on a treadmill at their preferred speed for 4 min at inclinations of 6%, 8%, and 10% in upward (UP) and downward (DOWN) conditions, and in horizontal (0%) condition. Gait variability was assessed using average standard deviation trunk acceleration between strides (VAR), gait stability was assessed using margin of stability (MoS) and maximum Lyapunov exponent (λs), and gait complexity was assessed using sample entropy (SEn). Trunk variability (VAR) increased in the medial-lateral (ML), anterior-posterior, and vertical directions for all inclined conditions. The SEn values indicated that movement complexity decreased almost linearly from DOWN to UP conditions, reflecting changes in gait pattern with longer and slower steps as inclination increased. The DOWN conditions were associated with the highest variability and lowest stability in the MoS ML, but not in λs. Stability was lower in UP conditions, which exhibited the largest λs values. The overall results support the hypothesis that inclined surfaces decrease gait stability and alter gait variability, particularly in UP conditions.     

A methodological basis for description and analysis of systems with complex switch-like interactions

 A wide range of complex systems appear to have switch-like interactions, i.e. below (or above) a certain threshold x has no or little influence on y, while above (or below) this threshold the effect of x on y saturates rapidly to a constant level. Switching functions are frequently described by sigmoid functions or combinations of these. Within the context of ordinary differential equations we present a very general methodological basis for designing and analysing models involving complicated switching functions together with any other non-linearities. A procedure to determine position and stability properties of all stationary points lying close to a threshold for one or several variables, so-called singular stationary points, is developed. Such points may represent homeostatic states in models, and are therefore of considerable interest. The analysis provides a profound insight into the generic effects of steep sigmoid interactions on the dynamics around homeostatic points. It leads to qualitative as well as quantitative predictions without using advanced mathematical methods. Thus, it may have an important heuristic function in connection with numerical simulations aimed at unfolding the predictive potential of realistic models. Received 25 January 1996; received in revised form 29 June 1997     

Effects of acorn abundance on density dependence in a Japanese wood mouse (Apodemus speciosus) population

We analysed the effects of Quercus crispula acorn abundance on the density dependence of the large Japanese wood mouse Apodemus speciosus using time series data (1992–2007). The data were obtained in a forest in northern Hokkaido, Japan, by live-trapping rodents and directly counting acorns on the ground. Acorn abundance in one year clearly influenced the abundance of wood mice in the following year in all models examined based on the Gompertz and Ricker model; in addition, the abundance of wood mice had effects on the population. Acorn abundance influenced the strength of density dependence (intraspecific competition) of the wood mouse population. When the abundance of acorns was high, density dependence was relaxed, and as a result the equilibrium density at which the population growth rate decreased to zero became higher. Those effects of acorn abundance were regarded as a nonlinear perturbation effect (sensu Royama 1992). The nonlinearity of density dependence was also detected; higher densities had stronger effects on population growth rates. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.