Loren Cobb

Estimation and Moment Recursion Relations for Multimodal Distributions of the Exponential Family

Abstract Multimodal generalizations of the normal, gamma, inverse gamma, and beta distributions are introduced within a unified framework. These multimodal distributions, belonging to the exponential family, require fewer parameters than corresponding mixture densities and have unique maximum likelihood estimators. Simple moment recursion relations, which make maximum likelihood estimation feasible, also yield easily computed estimators that themselves are shown to be consistent and asymptotically normal. Lastly, a statistic for bimodality, based on Cardans discriminant for a cubic shape polynomial, is introduced.

Applications of Catastrophe Theory for Statistical Modeling in the Biosciences

Abstract Although catastrophe theory has been applied with mixed success to many problems in the biosciences, very few of these applications have used any form of statistical modeling. We present examples of the applications of statistical catastrophe theory in the analysis of experimental data. These include examples of hysteresis effects, bifurcation effects, and the full cusp catastrophe model. The methods of statistical catastrophe theory draw upon the theories of parameter estimation for multiparameter exponential families, nonlinear time-series analysis, and stochastic differential equations. We discuss the application of these methods to both canonical and noncanonical catastrophe models.

Statistical catastrophe theory: An overview

This paper is a summary of an address given to the Conference on Frontiers of Applied Geometry. A stochastic version of catastrophe theory is presented, using stochastic differential equations. We show that there is a nontrivial relationship between the potential functions of the deterministic models and the stationary probability density functions of the stochastic models. In the second part of the paper, we use maximum likelihood theory to derive estimators for the stationary densities, and we demonstrate how to test statistical hypotheses for these models.

On the convergence of the ensemble Kalman filter

Convergence of the ensemble Kalman filter in the limit for large ensembles to the Kalman filter is proved. In each step of the filter, convergence of the ensemble sample covariance follows from a weak law of large numbers for exchangeable random variables, the continuous mapping theorem gives convergence in probability of the ensemble members, and Lp bounds on the ensemble then give Lp convergence.

Computer reconstruction of serial sections

A computer graphics system is described which reconstructs three-dimensional images from serial sectional data. Microscopic sectional tracings are first digitized and coded with a microcomputer (APPLE II-Plus). The data are transferred to a main frame facility for reconstruction and the final result is displayed on a high-resolution color monitor (Hewlett-Packard 9845C). Depth cueing of the image is enhanced by edge ribboning and area filling. The system is very simple to operate and yet flexible enough to allow selective portions of the tissue sample to be reconstructed and displayed in various orientations.

The Multimodal Exponential Families of Statistical Catastrophe Theory

This paper reviews recent developments in statistical catastrophe theory. A connection is established between a class of stochastic catastrophe models (the ‘cuspoid’ catastrophes, with Weiner input) and a class of regular exponential families, which are the stationary probability densities of the stochastic catastrophe models. These are called the exponential catastrophe densities. Parameter estimation is examined from the point of view of three methods: maximum likelihood, moments, and approximation theory. Special attention is given to the cusp densities, and a comparative example is presented. Then an inferential theory is presented, based on the likelihood ratio test. This test can be used on a hierarchy of catastrophe densities. At the base of the heirarchy are the familiar normal, gamma, and beta densities, while at the top are complex multimodal forms. The theory as presented has none of the topolological flavor of catastrophe theory, but the principle of invariance up to diffeomorphism is discussed in relation to the inferential theory.

Power analysis of statistical methods for comparing treatment differences from limiting dilution assays.

SummarySix different statistical methods for comparing limiting dilution assays were evaluated, using both real data and a power analysis of simulated data. Simulated data consisted of a series of 12 dilutions for two treatment groups with 24 cultures per dilution and 1,000 independent replications of each experiment. Data within each replication were generated by Monte Carlo simulation, based on a probability model of the experiment. Analyses of the simulated data revealed that the type I error rates for the six methods differed substantially, with only likelihood ratio and Taswells weighted mean methods approximating the nominal 5% significance level. Of the six methods, likelihood ratio and Taswells minimum Chi-square exhibited the best power (least probability of type II errors). Taswells weighted mean test yielded acceptable type I and type II error rates, whereas the regression method was judged unacceptable for scientific work.

Study of discontinuities in hydrological data using catastrophe theory.

Abstract Modelling and prediction of hydrological processes (e.g. rainfall–runoff) can be influenced by discontinuities in observed data, and one particular case may arise when the time scale (i.e. resolution) is coarse (e.g. monthly). This study investigates the application of catastrophe theory to examine its suitability to identify possible discontinuities in the rainfall–runoff process. A stochastic cusp catastrophe model is used to study possible discontinuities in the monthly rainfall–runoff process at the Aji River basin in Azerbaijan, Iran. Monthly-averaged rainfall and flow data observed over a period of 20 years (1981–2000) are analysed using the Cuspfit program. In this model, rainfall serves as a control variable and runoff as a behavioural variable. The performance of this model is evaluated using four measures: correlation coefficient, log-likelihood, Akaike information criterion (AIC) and Bayesian information criterion (BIC). The results indicate the presence of discontinuities in the rainfall–runoff process, with a significant sudden jump in flow (cusp signal) when rainfall reaches a threshold value. The performance of the model is also found to be better than that of linear and logistic models. The present results, though preliminary, are promising in the sense that catastrophe theory can play a possible role in the study of hydrological systems and processes, especially when the data are noisy. Citation Ghorbani, M. A., Khatibi, R., Sivakumar, B. & Cobb, L. (2010) Study of discontinuities in hydrological data using catastrophe theory. Hydrol. Sci. J. 55(7), 1137–1151.

Comparison of statistical methods for the analysis of limiting dilution assays.

SummaryThis study reports the results of a critical comparison of five statistical methods for estimating the density of viable cells in a limiting dilution assay (LDA). Artificial data were generated using Monte Carlo simulation. The performance of each statistical method was examined with respect to the accuracy of its estimator and, most importantly, the accuracy of its associated estimated standard error (SE). The regression method was found to perform at a level that is unacceptable for scientific research, due primarily to gross underestimation of the SE. The maximum likelihood method exhibited the best overall performance. A corrected version of Taswells weighted-mean method, which provides the best performance among all noniterative methods examined, is also presented.

Bayesian tracking of emerging epidemics using ensemble optimal statistical interpolation

We present a preliminary test of the Ensemble Optimal Statistical Interpolation (EnOSI) method for the statistical tracking of an emerging epidemic, with a comparison to its popular relative for Bayesian data assimilation, the Ensemble Kalman Filter (EnKF). The spatial data for this test was generated by a spatial susceptible-infectious-removed (S-I-R) epidemic model of an airborne infectious disease. Both tracking methods in this test employed Poisson rather than Gaussian noise, so as to handle epidemic data more accurately. The EnOSI and EnKF tracking methods worked well on the main body of the simulated spatial epidemic, but the EnOSI was able to detect and track a distant secondary focus of infection that the EnKF missed entirely.