Multiple indicators, partially ordered sets, and linear extensions: Multi-criterion ranking and prioritization

This paper is concerned with the question of ranking a finite collection of objects when a suite of indicator values is available for each member of the collection. The objects can be represented as a cloud of points in indicator space, but the different indicators (coordinate axes) typically convey different comparative messages and there is no unique way to rank the objects while taking all indicators into account. A conventional solution is to assign a composite numerical score to each object by combining the indicator information in some fashion. Consciously or otherwise, every such composite involves judgments (often arbitrary or controversial) about tradeoffs or substitutability among indicators. Rather than trying to combine indicators, we take the view that the relative positions in indicator space determine only a partial ordering and that a given pair of objects may not be inherently comparable. Working with Hasse diagrams of the partial order, we study the collection of all rankings that are compatible with the partial order (linear extensions). In this way, an interval of possible ranks is assigned to each object. The intervals can be very wide, however. Noting that ranks near the ends of each interval are usually infrequent under linear extensions, a probability distribution is obtained over the interval of possible ranks. This distribution, called the rank-frequency distribution, turns out to be unimodal (in fact, log-concave) and represents the degree of ambiguity involved in attempting to assign a rank to the corresponding object. Stochastic ordering of probability distributions imposes a partial order on the collection of rank-frequency distributions. This collection of distributions is in one-to-one correspondence with the original collection of objects and the induced ordering on these objects is called the cumulative rank-frequency (CRF) ordering; it extends the original partial order. Although the CRF ordering need not be linear, it can be iterated to yield a fixed point of the CRF operator. We hypothesize that the fixed points of the CRF operator are exactly the linear orderings. The CRF operator treats each linear extension as an equal voter in determining the CRF ranking. It is possible to generalize to a weighted CRF operator by giving linear extensions differential weights either on mathematical grounds (e.g., number of jumps) or empirical grounds (e.g., indicator concordance). Explicit enumeration of all possible linear extensions is computationally impractical unless the number of objects is quite small. In such cases, the rank-frequencies can be estimated using discrete Markov chain Monte Carlo (MCMC) methods.     

基于MCMC法的水质模型参数不确定性研究

参数识别是数学模型应用的前提.鉴于常用贝叶斯离散化方法在搜索复杂模型参数后验分布时的计算限制的原因,本文引入了MCMC采样法.为考察MCMC法对参数后验分布的搜索性能和效率,进行了2个案例研究.结果表明,MCMC法对参数后验分布的搜索,无论是搜索性能还是搜索效率,均表现出了独特的优越性.同时,Gelman收敛判别准则计算表明,MCMC采样序列均能稳定收敛到参数的后验分布上.可见,MCMC法适用于复杂环境模型的参数识别和不确定分析研究.     

Spatiotemporal trends of illegal activities from ranger‐collected data in a Ugandan national park

Within protected areas, biodiversity loss is often a consequence of illegal resource use. Understanding the patterns and extent of illegal activities is therefore essential for effective law enforcement and prevention of biodiversity declines. We used extensive data, commonly collected by ranger patrols in many protected areas, and Bayesian hierarchical models to identify drivers, trends, and distribution of multiple illegal activities within the Queen Elizabeth Conservation Area (QECA), Uganda. Encroachment (e.g., by pastoralists with cattle) and poaching of noncommercial animals (e.g., snaring bushmeat) were the most prevalent illegal activities within the QECA. Illegal activities occurred in different areas of the QECA. Poaching of noncommercial animals was most widely distributed within the national park. Overall, ecological covariates, although significant, were not useful predictors for occurrence of illegal activities. Instead, the location of illegal activities in previous years was more important. There were significant increases in encroachment and noncommercial plant harvesting (nontimber products) during the study period (1999–2012). We also found significant spatiotemporal variation in the occurrence of all activities. Our results show the need to explicitly model ranger patrol effort to reduce biases from existing uncorrected or capture per unit effort analyses. Prioritization of ranger patrol strategies is needed to target illegal activities; these strategies are determined by protected area managers, and therefore changes at a site‐level can be implemented quickly. These strategies should also be informed by the location of past occurrences of illegal activity: the most useful predictor of future events. However, because spatial and temporal changes in illegal activities occurred, regular patrols throughout the protected area, even in areas of low occurrence, are also required.     

Hierarchical Bayesian space-time models

Space-time data are ubiquitous in the environmental sciences. Often, as is the case with atmo- spheric and oceanographic processes, these data contain many different scales of spatial and temporal variability. Such data are often non-stationary in space and time and may involve many observation/prediction locations. These factors can limit the effectiveness of traditional space- time statistical models and methods. In this article, we propose the use of hierarchical space-time models to achieve more flexible models and methods for the analysis of environmental data distributed in space and time. The first stage of the hierarchical model specifies a measurement- error process for the observational data in terms of some 'state' process. The second stage allows for site-specific time series models for this state variable. This stage includes large-scale (e.g. seasonal) variability plus a space-time dynamic process for the anomalies'. Much of our interest is with this anomaly proc ess. In the third stage, the parameters of these time series models, which are distributed in space, are themselves given a joint distribution with spatial dependence (Markov random fields). The Bayesian formulation is completed in the last two stages by speci- fying priors on parameters. We implement the model in a Markov chain Monte Carlo framework and apply it to an atmospheric data set of monthly maximum temperature.     

Time-varying models for extreme values

We propose a new approach for modeling extreme values that are measured in time and space. First we assume that the observations follow a Generalized Extreme Value (GEV) distribution for which the location, scale or shape parameters define the space–time structure. The temporal component is defined through a Dynamic Linear Model (DLM) or state space representation that allows to estimate the trend or seasonality of the data in time. The spatial element is imposed through the evolution matrix of the DLM where we adopt a process convolution form. We show how to produce temporal and spatial estimates of our model via customized Markov Chain Monte Carlo (MCMC) simulation. We illustrate our methodology with extreme values of ozone levels produced daily in the metropolitan area of Mexico City and with rainfall extremes measured at the Caribbean coast of Venezuela.     

Modeling and spatial prediction of pre-settlement patterns of forest distribution using witness tree data

At the time of European settlement, land surveys were conducted progressively westward throughout the United States. Outside of the original 13 colonies, surveys generally followed the Public Land Survey system in which trees, called witness trees, were regularly recorded at 1 mi by 1 mi grid intersections. This unintentional sampling provides insight into the composition and structure of pre-European settlement forests, which is used as baseline data to assess forest change following settlement. In this paper, a model for the Public Land Surveys of east central Alabama is developed. Assuming that the locations of trees of each species are realized from independent Poisson processes whose respective log intensities are linear functions of environmental covariates (i.e., elevation, landform, and physiographic province), the species observed at the survey grid intersections are independently sampled from a generalized logistic regression model. If all 68 species found in the survey were included, the model would be highly over-parameterized, so only the distribution of the most common taxon, pines, will be considered at this time. To assess the impact of environmental factors not included in the model, a hidden Gaussian random field shall be added as a random effect. A Markov Chain Monte Carlo algorithm is developed for Bayesian inference on model parameters, and for Bayes posterior prediction of the spatial distribution of pines in east central Alabama. Received: June 2004 / Revised: November 2004     

Hierarchical modeling of count data with application to nuclear fall-out

We propose a method for a Bayesian hierarchical analysis of count data that are observed at irregular locations in a bounded domain of R². We model the data as having been observed on a fine regular lattice, where we do not have observations at all the sites. The counts are assumed to be independent Poisson random variables whose means are given by a log Gaussian process. In this article, the Gaussian process is assumed to be either a Markov random field (MRF) or a geostatistical model, and we compare the two models on an environmental data set. To make the comparison, we calibrate priors for the parameters in the geostatistical model to priors for the parameters in the MRF. The calibration is obtained empirically. The main goal is to predict the hidden Poisson-mean process at all sites on the lattice, given the spatially irregular count data; to do this we use an efficient MCMC. The spatial Bayesian methods are illustrated on radioactivity counts analyzed by Diggle et al. (1998).     

Modelling predation by transient leopard seals for an ecosystem-based management of Southern Ocean fisheries

Correctly quantifying the impacts of rare apex marine predators is essential to ecosystem-based approaches to fisheries management, where harvesting must be sustainable for targeted species and their dependent predators. This requires modelling the uncertainty in such processes as predator life history, seasonal abundance and movement, size-based predation, energetic requirements, and prey vulnerability. We combined these uncertainties to evaluate the predatory impact of transient leopard seals on a community of mesopredators (seals and penguins) and their prey at South Georgia, and assess the implications for an ecosystem-based management. The mesopredators are highly dependent on Antarctic krill and icefish, which are targeted by regional fisheries. We used a state-space formulation to combine (1) a mark-recapture open-population model and individual identification data to assess seasonally variable leopard seal arrival and departure dates, numbers, and residency times; (2) a size-based bioenergetic model; and (3) a size-based prey choice model from a diet analysis. Our models indicated that prey choice and consumption reflected seasonal changes in leopard seal population size and structure, size-selective predation and prey vulnerability. A population of 104 (90–125) leopard seals, of which 64% were juveniles, consumed less than 2% of the Antarctic fur seal pup production of the area (50% of total ingested energy, IE), but ca. 12–16% of the local gentoo penguin population (20% IE). Antarctic krill (28% IE) were the only observed food of leopard seal pups and supplemented the diet of older individuals. Direct impacts on krill and fish were negligible, but the “escapement” due to leopard seal predation on fur seal pups and penguins could be significant for the mackerel icefish fishery at South Georgia. These results suggest that: (1) rare apex predators like leopard seals may control, and may depend on, populations of mesopredators dependent on prey species targeted by fisheries; and (2) predatory impacts and community control may vary throughout the predator's geographic range, and differ across ecosystems and management areas, depending on the seasonal abundance of the prey and the predator's dispersal movements. This understanding is important to integrate the predator needs as natural mortality of its prey in models to set prey catch limits for fisheries. Reliable estimates of the variability of these needs are essential for a precautionary interpretation in the context of an ecosystem-based management.