Characterizing activity-dependent processes with a piecewise exponential model

The response of some biological processes is dependent on the frequency of stimulation. With first-order processes, the response is driven exponentially to an equilibrium determined by the value of the driving function. When the stimulus or driving function is viewed as switching between constant values the resulting response is piecewise exponential. With periodic excitation, the time course of a point fixed in time relative to the initiation time of each stimulus is shown to be exponential with a rate and steady state that are linearly dependent on the rates and equilibria associated with each component exponential. This linearity can be exploited and leads to a simple estimation procedure for the apparent state-dependent rates.     

Incorporating spatial autocorrelation into species distribution models alters forecasts of climate‐mediated range shifts

Species distribution models (SDMs) are widely used to forecast changes in the spatial distributions of species and communities in response to climate change. However, spatial autocorrelation (SA) is rarely accounted for in these models, despite its ubiquity in broad‐scale ecological data. While spatial autocorrelation in model residuals is known to result in biased parameter estimates and the inflation of type I errors, the influence of unmodeled SA on species' range forecasts is poorly understood. Here we quantify how accounting for SA in SDMs influences the magnitude of range shift forecasts produced by SDMs for multiple climate change scenarios. SDMs were fitted to simulated data with a known autocorrelation structure, and to field observations of three mangrove communities from northern Australia displaying strong spatial autocorrelation. Three modeling approaches were implemented: environment‐only models (most frequently applied in species' range forecasts), and two approaches that incorporate SA; autologistic models and residuals autocovariate (RAC) models. Differences in forecasts among modeling approaches and climate scenarios were quantified. While all model predictions at the current time closely matched that of the actual current distribution of the mangrove communities, under the climate change scenarios environment‐only models forecast substantially greater range shifts than models incorporating SA. Furthermore, the magnitude of these differences intensified with increasing increments of climate change across the scenarios. When models do not account for SA, forecasts of species' range shifts indicate more extreme impacts of climate change, compared to models that explicitly account for SA. Therefore, where biological or population processes induce substantial autocorrelation in the distribution of organisms, and this is not modeled, model predictions will be inaccurate. These results have global importance for conservation efforts as inaccurate forecasts lead to ineffective prioritization of conservation activities and potentially to avoidable species extinctions.     

长白山东北红豆杉生境内针阔混交林直径结构分布研究

选择能够拟合长白山自然保护区东北红豆杉生境内针阔混交林的直径结构分布模型,为该区林分经营管理提供参考。以长白山自然保护区龙荒沟林场15块东北红豆杉林分内针阔混交林为研究对象,采用负指数函数和三参数Weibull分布函数2种直径分布模型拟合和χ^2检验,分析了15块样地的直径结构分布规律。15块样地的密度为400~981株·hm^-2,平均胸径变化范围为10.5~19.9 cm,各样地林分的直径分布的偏度均为正值,直径分布函数曲线均往左偏,除了9~11号样地的直径分布用三参数Weibull分布模型效果较好外,其他样地均适合用负指数模型拟合。两种模型均能有效描述直径结构分布规律,三参数Weibull分布模型用来拟合不规则直径分布时效果较好,而负指数分布模型拟合近似“反J”型曲线时效果好。该区东北红豆杉林分内针阔混交林直径结构分布不合理,应加强该林区经营技术措施。     

Nonlinear Mixed-Effects (NLME) Diameter Growth Models for Individual China-Fir (Cunninghamia lanceolata) Trees in Southeast China

An individual-tree diameter growth model was developed for Cunninghamia lanceolata in Fujian province, southeast China. Data were obtained from 72 plantation-grown China-fir trees in 24 single-species plots. Ordinary non-linear least squares regression was used to choose the best base model from among 5 theoretical growth equations; selection criteria were the smallest absolute mean residual and root mean square error and the largest adjusted coefficient of determination. To account for autocorrelation in the repeated-measures data, we developed one-level and nested two-level nonlinear mixed-effects (NLME) models, constructed on the selected base model; the NLME models incorporated random effects of the tree and plot. The best random-effects combinations for the NLME models were identified by Akaike''s information criterion, Bayesian information criterion and −2 logarithm likelihood. Heteroscedasticity was reduced with two residual variance functions, a power function and an exponential function. The autocorrelation was addressed with three residual autocorrelation structures: a first-order autoregressive structure [AR(1)], a combination of first-order autoregressive and moving average structures [ARMA(1,1)] and a compound symmetry structure (CS). The one-level (tree) NLME model performed best. Independent validation data were used to test the performance of the models and to demonstrate the advantage of calibrating the NLME models.     

Exponentiated Exponential Family: An Alternative to Gamma and Weibull Distributions

In this article we study some properties of a new family of distributions, namely Exponentiated Exponential distribution, discussed in Gupta , Gupta , and Gupta (1998). The Exponentiated Exponential family has two parameters (scale and shape) similar to a Weibull or a gamma family. It is observed that many properties of this new family are quite similar to those of a Weibull or a gamma family, therefore this distribution can be used as a possible alternative to a Weibull or a gamma distribution. We present two real life data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to Weibull or gamma distribution and in the other data set Weibull has a better fit than exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes.     

Fluctuation analysis in small chemically reacting systems

A theory of noise fluctuations is developed which is applicable to systems of any size in which unimolecular or bimolecular reactions are occurring. The main difference between small and large reacting systems is that in the former the probability of finding a particle in a particular state does not obey a Gaussian distribution, but satisfies a distribution which reflects the mechanism of the chemical reaction. This difference is reflected in the main result of the theory: an autocorrelation function that is expressible as a sum of exponentials, the amplitudes of which are explicit functions of the moments of the distribution. Thus, by using small systems, the autocorrelation function,in principle, allows the elucidation of reaction mechanisms. Numerical simulations indicate that for reacting systems having ten or fewer particles, the deviation of the autocorrelation function from a single exponential should be easily detectable, and that estimates of the first four moments of the distribution should be possible. Accurate inference of the distribution, however, will require further mathematical and experimental advances.     

Spatio-temporal ecology and management of cabbage maggot

This study analyzes the spatio-temporal dispersion patterns of the cabbage maggot (Delia radicum L.) (Diptera: Anthomyiidae) infestation in rutabagas and turnips in Oregon and suggests ways to exploit the spatial and temporal ecology of the cabbage maggot to improve management of the pest. The patchy distribution of cabbage maggots arises from a combination of first-order effects driven by spatial heterogeneity and second-order effects driven by spatial autocorrelation. The intensity of cabbage maggot infestations varied from year to year. Within a given year, damage rates tended to be higher in rutabagas than turnips, in crops planted earlier in the season, and in fields near nurseries and houses. Nonsignificant first-order effects included soil texture, distance from river, proximity to maggot sources (other than cultivated fields), type of vegetation on field borders, field manager, field area, and perimeter. Second-order effects were processes intrinsic to the population and would give rise to patchiness even in a homogeneous environment. For example, adults may be attracted to others of their species or eggs may be deposited in batches. The locations of patches arising from second-order effects cannot be predicted from knowledge of environmental covariates. However, cabbage maggot does not tend to disperse far, and existing patches tend to give rise to other patches nearby at a later time. We found elevated damage rates in spring fields planted near fields that were heavily damaged the previous fall and in fields planted late in the season near fields that had heavy damage early in the season.     

Inference on Risk Rates Based on Mortality Data Under Censoring and Competing Risks Using Parametric Models

In this paper we consider the competing risks model where the risks may not be independent. We assume both fixed and random censoring. The random censoring mechanism could have either a parametric or a non-parametric form. The life distributions and the parametric censoring distribution considered are exponential or Weibull. The expressions for the asymptotic confidence intervals for various parameters of interest under different models, using the estimated Fisher information matrix and parametric bootstrap techniques have been derived. Monte Carlo simulation studies for some of these cases have been carried out.     

Repeated significance tests for clinical trials with a fixed number of patients and variable follow-up

Group-sequential tests may be applied to trials in which the number of patients is fixed, a response variable is measured for each patient at successive follow-up visits, and the accumulated responses are compared across treatment groups. The standard theory is inapplicable because the increments in the accumulated responses are no longer independent. An adjustment can be made to allow for the ratio of between-patient to within-patient variance and for possible first-order autocorrelation. The method is illustrated by reference to a dental trial.     

Shapes and functions of species–area curves: a review of possible models

Aim This paper reviews possible candidate models that may be used in theoretical modelling and empirical studies of species–area relationships (SARs). The SAR is an important and well‐proven tool in ecology. The power and the exponential functions are by far the models that are best known and most frequently applied to species–area data, but they might not be the most appropriate. Recent work indicates that the shape of species–area curves in arithmetic space is often not convex but sigmoid and also has an upper asymptote. Methods Characteristics of six convex and eight sigmoid models are discussed and interpretations of different parameters summarized. The convex models include the power, exponential, Monod, negative exponential, asymptotic regression and rational functions, and the sigmoid models include the logistic, Gompertz, extreme value, Morgan–Mercer–Flodin, Hill, Michaelis–Menten, Lomolino and Chapman–Richards functions plus the cumulative Weibull and beta‐P distributions. Conclusions There are two main types of species–area curves: sample curves that are inherently convex and isolate curves, which are sigmoid. Both types may have an upper asymptote. A few have attempted to fit convex asymptotic and/or sigmoid models to species–area data instead of the power or exponential models. Some of these or other models reviewed in this paper should be useful, especially if species–area models are to be based more on biological processes and patterns in nature than mere curve fitting. The negative exponential function is an example of a convex model and the cumulative Weibull distribution an example of a sigmoid model that should prove useful. A location parameter may be added to these two and some of the other models to simulate absolute minimum area requirements.     

Assessment of land use changes on woody cover and landscape fragmentation in the Orinoco savannas using fractal distributions

The effect of land use changes on the woody cover and patchiness of the Orinoco lowlands was analyzed from 1951 to 1997 by using a stretched exponential model in terms of the patch size distribution. We selected five production systems on the basis of the following components: policy option, technology and environment. They are: (a) high-input pasture in deep soil (DL-HIGraz), (b) low-input cropping in deep soil (DH-LOCrop), (c) high-input cropping in deep soil (DH-HICrop), (d) low-input cropping in shallow soil (SL-LOCrop) and (e) extensive cattle grazing in shallow soil (SL-LOGraz). For comparison, stands with similar soil depth and woody density (DL-Fallow, DH-Fallow and SL-Fallow) were selected at the protected Biological Reserve in the Calabozo region. Results indicate that the distribution of the patches was described by the stretched exponential distribution model with c as the exponent distribution. The magnitude of c reflected the effect of land use changes over time. Thus, the parameter c for human impacted stands was higher than that for the fallow stands. The stretched exponential model implies that the human impact had a multiscalar effect on the Orinoco mosaic, in terms of the multiplicative processes. There is a generic mechanism exhibiting an anomalous distribution of the patches such as has been proposed for physical systems. Therefore, the number of generation of the multiplicative processes (i.e. inverse of c) for impacted stands was lower than that in the fallow stands. As the time from fallow establishment proceeded, there was a tendency to patch re-organization by biotic effects.     

Using radius frequency distribution functions as a metric for quantifying root systems

Root radius frequency distributions have been measured to quantify the effect of plant type, environment and methodology on root systems, however, to date the results of such studies have not been synthesised. We propose that cumulative frequency distribution functions can be used as a metric to describe root systems because (1) statistical properties of the frequency distribution can be determined, (2) the parameters for these can be used as a means of comparison, and (3) the analytical expressions can be easily incorporated into models that are dependent upon root geometry. We collated a database of 96 root radii frequency distributions and botanical and methodology traits for each distribution. To determine if there was a frequency distribution function that was best suited to root radii measurements we fitted the exponential, Rayleigh, normal, log-normal, logistic and Weibull cumulative distribution functions to each distribution in our database. We found that the log-normal function provided the best fit to these distributions and that none of the distribution functions was better or worse suited to particular shapes. We derived analytical expressions for root surface and volume and found that they are a valid, and simpler method for incorporating root architectural traits into analytical models. We also found that growth habit and growth media had a significant effect on mean root radius.     

Time trends of smoking cessation analyzed with six mathematical survival models

In this paper, six mathematical models were applied to model time trends of smoking cessation. Both statistical and non-statistical methods were used and included the exponential, ideodynamic, log-logistic, Pareto, sickle and Weibull models. All models included the possibilities of both permanent abstinence and relapse to smoking. Time trends from all models were compared with data from the Multiple Risk Factor Intervention Trial (MRFIT) program. The Pareto, log-logistic, Weibull and ideodynamic models yielded satisfactory fits to the data while the sickle and exponential models did not. Even though the data used in this paper were not sufficient to distinguish among these four models, the methodology will be useful for further narrowing the model choices as additional data for the testing become available.     

Determination of Thermal Inactivation Kinetics of Hepatitis A Virus in Blue Mussel (Mytilus edulis) Homogenate

Hepatitis A virus (HAV) is a food-borne enteric virus responsible for outbreaks of hepatitis associated with shellfish consumption. The objectives of this study were to determine the thermal inactivation behavior of HAV in blue mussels, to compare the first-order and Weibull models to describe the data, to calculate Arrhenius activation energy for each model, and to evaluate model efficiency by using selected statistical criteria. The times required to reduce the population by 1 log cycle (D-values) calculated from the first-order model (50 to 72°C) ranged from 1.07 to 54.17 min for HAV. Using the Weibull model, the times required to destroy 1 log unit (t_{D = 1}) of HAV at the same temperatures were 1.57 to 37.91 min. At 72°C, the treatment times required to achieve a 6-log reduction were 7.49 min for the first-order model and 8.47 min for the Weibull model. The z-values (changes in temperature required for a 90% change in the log D-values) calculated for HAV were 15.88 ± 3.97°C (R², 0.94) with the Weibull model and 12.97 ± 0.59°C (R², 0.93) with the first-order model. The calculated activation energies for the first-order model and the Weibull model were 165 and 153 kJ/mol, respectively. The results revealed that the Weibull model was more appropriate for representing the thermal inactivation behavior of HAV in blue mussels. Correct understanding of the thermal inactivation behavior of HAV could allow precise determination of the thermal process conditions to prevent food-borne viral outbreaks associated with the consumption of contaminated mussels.     

长白山原始阔叶红松林径级结构模拟

以原始阔叶红松林为研究对象,设置了2块1hm2样地,利用Weibull分布函数、负指数函数和q值理论对其林分径级结构进行模拟。结果表明:2块样地林分的径级分布均呈倒"J"形,森林更新良好;Weibull分布函数和负指数函数都取得了较好的模拟效果,但Weibull分布函数更好地预测了大径阶株数,因此,Weibull分布函数比负指数函数更适合原始阔叶红松林径级结构的模拟;2块样地的q值偏小(分别为1.74和1.45),说明径级结构曲线较平缓,大径材数量多。q值法则对于阔叶红松林径阶株数密度的表达效果较好,可以被用于描述阔叶红松林的胸径分布。     

Conformational fluctuations and protein reactivity. Determination of the rate-constant spectrum and consequences in elementary biochemical processes

When a protein's active site happens to be strongly coupled with the protein structure, the rate constant of the reaction may eventually be modulated by the conformational fluctuations. Evidence for this effect has long been provided by extensive flash photolysis investigations of liganded hemoproteins and more recently of the non-heme respiratory protein hemerythrin in hydro-organic solvents. Within a given protein conformational substate, an elementary reaction step is characterized by one single free energy barrier and by a first-order rate constant, k, which changes with temperature according to an Arrhenius law. At physiological temperature and low viscosity, ultrafast conformational relaxation causes efficient averaging of the reaction rates and the protein displays exponential kinetics with an average rate constant (k). Under sufficiently general conditions, it can be shown that (k) also follows a simple Arrhenius law with 'effective' values of the pre-exponential factor Aeff and activation enthalpy Heff. It is found that Aeff strongly depends on the overall shape of the rate constant distribution and that Heff actually corresponds to the lower limit of the enthalpy of activation, i.e. the value associated with the highest possible reaction rate. The underlying distribution of rate constants can be reconstructed from a set of experiments in which the kinetics depart from an exponential, i.e. at low temperature and high viscosity. The most probable distribution of exponentials consistent with the observed kinetics of the geminate recombinations of oxygen with photodissociated hemerythrin has been determined by using a new approach, known as the maximum entropy method. The results are consistent with a single pre-exponential value and a distributed enthalpy spectrum. As expected, Heff does not coincide either with the most probable nor with the average value of the enthalpy. The most salient findings are that the probability for any protein molecule to have an enthalpy of activation equal to the effective value Heff vanishes and that Aeff differs by nearly three orders of magnitude from the true value A0. Biochemical reaction rates are actually average values, since protein reactions are measured under physiological conditions, where conformational relaxation is always fast. Our understanding of the significance of Aeff and Heff is therefore entirely dependent on the knowledge of the distribution function of the rate constants. In particular, enthalpy and entropy terms of similar reactions performed by different proteins cannot be compared as long as the distribution of the rate constants remains unknown.     

The overall distribution of survival times for U.K. AIDS patients

Data on the survival times of 997 U.K. AIDS patients are analysed with the aim of deriving a simple form for the overall survival distribution. The exponential and Weibull distributions are modified to accommodate specific features of the data, in particular, the recording of survival times to the nearest month and the occurrence of a significant proportion of cases recorded as having zero time on study. The final model has a probability 0.08 of underlying survival time being zero and, given non-zero survival time, takes the form of an exponential distribution with mean of 14.95 months. The results are in close agreement with those of a study of New York City patients as well as the empirical data.     

On the existence of stable pairing distributions

We formulate and analyze pair-formation models for multiple groups with general pairing rates and arbitrary mixing probabilities. Under the assumption of constant recruitment rates and equal average duration of all types of partnerships, we have shown that the dynamics are relatively simple because of the monotonicity properties of the dynamical system associated with the pairing/mixing of heterogeneous populations of male and female individuals. In fact, we have shown that the corresponding asymptotic stable paired distribution is given precisely by the asymptotic values of the matrices that prescribe the mixing/contact structure. In other words, if the sizes of the mixing subpopulations of males and females are asymptotically constant and if the average durations of partnerships are about the same regardless of type, then the matrices that describe the mixing between subpopulations also characterize the distribution of paired types. Alternatively, if the distribution of the average duration of relationships between individuals has a large variance then it may be impossible to detect any relationship between the mixing/contact structure and the observed distribution of paired types. The study of models with constant per-capita recruitment rates give rise to homogeneous systems of degree one. The analysis of the dynamics of pairs for models with exponentially growing populations of singles is complicated. So far, we are only able to classify the stability of all non-strictly positive boundary exponential solutions. From our incomplete analysis, it is not possible to detect necessary and sufficient conditions for the existence and stability of strictly interior exponential solutions. We cannot rule out the possibility of oscillations. The mathematical problems associated with the stability of exponential solutions of dynamical systems of degree one are of relevance in demography, epidemiology, and population dynamics.On leave from University of Alabama in Huntsville     

Goodness of Fit of Probability Distributions for Sightings as Species Approach Extinction

Estimating the probability that a species is extinct and the timing of extinctions is useful in biological fields ranging from paleoecology to conservation biology. Various statistical methods have been introduced to infer the time of extinction and extinction probability from a series of individual sightings. There is little evidence, however, as to which of these models provide adequate fit to actual sighting records. We use L-moment diagrams and probability plot correlation coefficient (PPCC) hypothesis tests to evaluate the goodness of fit of various probabilistic models to sighting data collected for a set of North American and Hawaiian bird populations that have either gone extinct, or are suspected of having gone extinct, during the past 150 years. For our data, the uniform, truncated exponential, and generalized Pareto models performed moderately well, but the Weibull model performed poorly. Of the acceptable models, the uniform distribution performed best based on PPCC goodness of fit comparisons and sequential Bonferroni-type tests. Further analyses using field significance tests suggest that although the uniform distribution is the best of those considered, additional work remains to evaluate the truncated exponential model more fully. The methods we present here provide a framework for evaluating subsequent models.     

Dealing with spatial autocorrelation when learning predictive clustering trees

Spatial autocorrelation is the correlation among data values which is strictly due to the relative spatial proximity of the objects that the data refer to. Inappropriate treatment of data with spatial dependencies, where spatial autocorrelation is ignored, can obfuscate important insights. In this paper, we propose a data mining method that explicitly considers spatial autocorrelation in the values of the response (target) variable when learning predictive clustering models. The method is based on the concept of predictive clustering trees (PCTs), according to which hierarchies of clusters of similar data are identified and a predictive model is associated to each cluster. In particular, our approach is able to learn predictive models for both a continuous response (regression task) and a discrete response (classification task). We evaluate our approach on several real world problems of spatial regression and spatial classification. The consideration of the autocorrelation in the models improves predictions that are consistently clustered in space and that clusters try to preserve the spatial arrangement of the data, at the same time providing a multi-level insight into the spatial autocorrelation phenomenon. The evaluation of SCLUS in several ecological domains (e.g. predicting outcrossing rates within a conventional field due to the surrounding genetically modified fields, as well as predicting pollen dispersal rates from two lines of plants) confirms its capability of building spatial aware models which capture the spatial distribution of the target variable. In general, the maps obtained by using SCLUS do not require further post-smoothing of the results if we want to use them in practice.