G. P. Patil

Diversity as a Concept and its Measurement

Abstract This paper puts forth the view that diversity is an average property of a community and identifies that property as species rarity. An intrinsic diversity ordering of communities is defined and is shown to be equivalent to stochastic ordering. Also, the sensitivity of an index to rare species is developed, culminating in a crossing-point theorem and a response theory to perturbations. Diversity decompositions, analogous to the analysis of variance, are discussed for two-way classifications and mixtures. The paper concludes with a brief survey of genetic diversity, linguistic diversity, industrial concentration, and income inequality.

Weighted distributions and size-biased sampling with applications to wildlife populations and human families

When an investigator records an observation by nature according to a certain stochastic model, the recorded observation will not have the original distribution unless every observation is given an equal chance of being recorded. A number of papers have appeared during the last ten years implicitly using the concepts of weighted and size-biased sampling distributions. In this paper, we examine some general models leading to weighted distributions with weight functions not necessarily bounded by unity. The examples include: probability sampling in sample surveys, additive damage models, visibility bias dependent on the nature of data collection and two-stage sampling. Several important distributions and their size-biased forms are recorded. A few theorems are given on the inequalities between the mean values of two weighted distributions. The results are applied to the analysis of data relating to human populations and wildlife management. For human populations, the following is raised and discussed: Let us ascertain from each male student in a class the number of brothers, including himself, and sisters he has and denote by k the number of students and by B and S the total numbers of brothers and sisters. What would be the approximate values of B/(B+S), the ratio of brothers to the total number of children, and (B+S)/k, the average number of children per family? It is shown that B/(B+S) will be an overestimate of the proportion of boys among the children per family in the general population which is about half, and similarly (B+S)/k is biased upwards as an estimate of the average number of children per family in the general population. Some suggestions are offered for the estimation of these population parameters. Lastly, for the purpose of estimating wildlife population density, certain results are formulated within the framework of quadrat sampling involving visibility bias.

Upper level set scan statistic for detecting arbitrarily shaped hotspots

A declared need is around for geoinformatic surveillance statistical science and software infrastructure for spatial and spatiotemporal hotspot detection. Hotspot means something unusual, anomaly, aberration, outbreak, elevated cluster, critical resource area, etc. The declared need may be for monitoring, etiology, management, or early warning. The responsible factors may be natural, accidental, or intentional. This proof-of-concept paper suggests methods and tools for hotspot detection across geographic regions and across networks. The investigation proposes development of statistical methods and tools that have immediate potential for use in critical societal areas, such as public health and disease surveillance, ecosystem health, water resources and water services, transportation networks, persistent poverty typologies and trajectories, environmental justice, biosurveillance and biosecurity, among others. We introduce, for multidisciplinary use, an innovation of the health-area-popular circle-based spatial and spatiotemporal scan statistic. Our innovation employs the notion of an upper level set, and is accordingly called the upper level set scan statistic, pointing to a sophisticated analytical and computational system as the next generation of the present day popular SaTScan. Success of surveillance rests on potential elevated cluster detection capability. But the clusters can be of any shape, and cannot be captured only by circles. This is likely to give more of false alarms and more of false sense of security. What we need is capability to detect arbitrarily shaped clusters. The proposed upper level set scan statistic innovation is expected to fill this need

Multivariate environmental statistics

Preface. Multivariate environmental statistics in agriculture (V. Barnett). A study of the relationship between diversity indices of Benthic communities and heavy metal concentrations of northwest Atlantic sediments (N.C. Bolgiano, R.N. Reid, G.P. Patil). Multivariate non-normal statistics in site characterization and evaluation (J.H. Carson, Jr., A.K. Gupta). Spatial prediction in a multivariate setting (N. Cressie). Needs and developments in multivariate environmental monitoring statistics (G.T. Flatman). Certain multivariate considerations in ranked set sampling and composite sampling designs (S.D. Gore et al.). Relating two sets of variables in environmental studies (R.H. Green). Statistical analysis of biological monitoring data (P. Guttorp). Using entropy in the redesign of an environmental monitoring network (P. Guttorp et al.). Graphical techniques for enhancing the utility of multivariate environmental statistics (W.A. Huber). Carcinogenicity tests involving multiple tumor sites (R.L. Kodell, E.O. George). Errors-in-variables analysis of extrapolation procedures in environmental toxicology (E. Linder, G.P. Patil, G. Suter II). A unified linear model for estimation with composite sample data (G. Lovison). Change-point methods for spatial data (I.B. MacNeill, V.K. Jandkyala). A review of computer intensive multivariate methods in ecology (B.F.J. Manly). Spatial-temporal statistical analysis of multivariate environmental monitoring data (K.V. Mardia, C.R. Goodall). Correspondence analysis applied to environmental data sets (D.E. Myers). A meta-network approach to higher-order spatial constructs and scale effects (W.L. Myers). Estimation of trend in Chesapeake Bay water quality data (N.K. Nagaraj, S.L. Brunenmeister). The complexities and scenarios of ecosystem analysis (L. Orloci). Multivariate methods in nuclear waste remediation. Needs and applications (B.A. Pulsipher). Omnibus robust procedures for assessment of multivariate normality and detection of multivariate outliers (A. Singh). Multivariate assessment of trend in environmental variables (E.P. Smith, S. Rheem, G.I. Holtzman). Multivariate chemometrics - a case study. Applying and developing receptor models for the 1990 El Paso winter PM10 receptor modeling scoping study (C.H. Spiegelman, S. Dattner). Weighted averaging partial least squares regression (WA-PLS): Definition and comparison with other methods for species-environment calibration (C.J.F. Ter Braak et al.). Multivariate aspects of adaptive cluster sampling (S.K. Thompson). Approximating multivariate distributions in stochastic models of insect population dynamics (T.E. Wehrly, J.H. Matis, G.W. Otis).

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.

The gamma distribution and weighted multimodal gamma distributions as models of population abundance

Abstract The gamma probability distribution is a general model of a population fluctuating around a steady state. We show this using stochastic differential equations (SDEs), constructed by adding white noise to the specific growth rate in deterministic models of population abundance. The gamma is an approximate stationary solution for almost any SDE having an underlying deterministic equilibrium. If the deterministic model possesses multiple stable and unstable equilibria, the approximate stationary solution to the stochastic case is a weighted gamma distribution. Modes of the stationary distribution roughly correspond to the equilibria of the deterministic model. Stochastic forces have effects similar to harvesting. These findings provide: (1) a theoretical basis for certain descriptive uses of the gamma in statistical ecology, (2) a concise graphical summary of the interactions between density dependent and density independent population regulation, (3) a statistical framework for fitting catastrophe-theoretic models to ecological data sets.

Ranked set sampling: an annotated bibliography

The paper provides an up-to-date annotated bibliography of the literature on ranked set sampling. The bibliography includes all pertinent papers known to the authors, and is intended to cover applications as well as theoretical developments. The annotations are arranged in chronological order and are intended to be sufficiently complete and detailed that a reading from beginning to end would provide a statistically mature reader with a state-of-the-art survey of ranked set sampling, including historical development, current status, and future research directions and applications. A final section of the paper gives a listing of all annotated papers, arranged in alphabetical order by author.

Ranked Set Sampling: a Bibliography

Abu-Dayyeh, H., and Muttlak, H.A. (1996). Using ranked set sampling for hypothesis tests on the scale parameter of the exponential and uniform distributions. Pakistan Journal of Statistics, 12, 131±138. Aragon, E., Gore, S.D., and Patil, G.P. (1994). Environmental sampling, observational economy, and extreme values. Parisankhyan Samikkha, 1, 71±81. Aragon, E., Patil, G.P., and Taillie, C. (1999). A performance indicator for ranked set sampling using ranking error probability matrix. Environmental and Ecological Statistics, 6. (To appear). Balakrishnan, N, and Rao, C.R. (1997). A note on the best linear unbiased estimation based on order statistics. American Statistician, 51, 181±185. Balasubramanian, K., and Balakrishnan, N. (1993). Duality principle in order statistics. Journal of the Royal Statistical Society B , 687±691. Barabesi, L. (1998). The computation of the distribution of the sign test statistics for ranked set sampling. Communication in Statistics, Simulation and Computation, 27, 833±842. Barabesi, L., and Fattorini, L. (1994). Kernel estimators for the intensity of a spatial point process by point-to-plant distances: Random sampling vs. ranked set sampling. TR 94±0402, Center for Statistical Ecology and Environmental Statistics, Department of Statistics, Penn State University, University Park, PA. Barnett, V. (1999). Ranked set sample design for environmental investigations. Environmental and Ecological Statistics, 6. (To appear). Barnett, V., and Moore, K. (1997). Best linear unbiased estimates in ranked set sampling with particular reference to imperfect ordering. Journal of Applied Statistics, 24, 697±710. Barreto, M.C.M., and Barnett, V. (1999). Best linear unbiased estimators for the simple linear regression model using ranked-set sampling. Environmental and Ecological Statistics, 6. (To appear). Bhoj, D.S. (1997). Estimation of parameters of the extreme value distribution using ranked set sampling. Communications in Statistics ± Theory and Methods, 26, 653±. Bhoj, D.S., and Ahsanullah, M. (1996). Estimation of parameters of the generalized geometric distribution using ranked set sampling. Biometrics, 52, 685±694. Bohn, L.L. (1993). A two-sample procedure for ranked-set samples. Technical Report Number 426, Department of Statistics, University of Florida, Gainesville, FL. Bohn, L.L. (1996). A review of nonparametric ranked set sampling methodology. Communication in Statistics ± Theory and Methods, 25, 2675±. Bohn, L.L., and Wolfe, D.A. (1992) Nonparametric two-sample procedures for ranked-set samples data. Journal of the American Statistical Association, 87, 552±561. Bohn, L.L., and Wolfe, D.A. (1994). The effect of imperfect judgment rankings on properties of procedures based on the ranked-set samples analog of the Mann-Whitney-Wilcoxon statistics. Journal of the American Statistical Association, 89, 168±176. Boyles, R.A., and Samaniego, F.J. (1986). Estimating a distribution function based on nomination sampling. Journal of the American Statistical Association, 81, 1039±1045. Environmental and Ecological Statistics 6, 91±98 (1999)

Concepts of aggregation and their quantification: A critical review with some new results and applications

Various indices of spatial patterns based on plot counts are reviewed for theoretical population models appropriate to ecological studies. It is seen that many of the indices proposed in the literature are essentially equivalent to either the index ω= σ2/μ or to the index γ= (σ2-μ)/μ2, thus providing a variety of motiviations and interpretations of these two indices as measures of spatial patterns. A vector approach to measuring spatial patterns which suggests both a unifying relationship between these indices and an extension of them is proposed. This leads to an interpretation of the measures of spatial patterns in terms of the transition probabilities of a pure birth process.

Multicriteria prioritization and partial order in environmental sciences

The complexity of the present data-centric world finds its expression in the increasing number of multi-indicator systems. This has led to the development of multicriteria ranking systems based on partial orders. Order theory is a main pillar of structural mathematics. Partial orders help to reveal why an object of interest holds a certain ranking position and how much it is subject to change if a composite indicator is upgraded. Order theory helps to derive linear or weak orders without indicator weighting schemes. Hence, rankings obtained from decision support systems (DSS) which depend on many parameters beyond the data matrix can be checked and discrepancies can lead to examine the parameters of the DSS. Order theory helps discover association and implication structures derived from formal concept lattices. Association and implication networks among the attributes of the data matrix allow more insights into multi-indicator systems and lead to new hypotheses and motivate further research. Some new and innovative concepts, like separated subsets, antagonistic indicators, ranking stability fields are rendered. Separated subsets are the typical outcome of a partial order analysis; their identification leads to antagonistic indicators, which are responsible for the separatedness of object’s subsets. Numerical aggregation can be performed step-by-step and the question which values of a weight lead to an order inversion is of high interest. The concept of stability fields is one possible answer, discussed in this paper. After an outline of partial order theory some more specific theoretical results are shown, then we discuss the role of composite indicators in the light of partial order and give some examples of interesting applications of partial order. Finally examples are selected from real life case studies of watersheds, environmental performance evaluations, child well being, geographic and administrative regions and more.