Multinomial Assumption Research Articles

The asymptotic distribution of the permutation entropy.

Ordinal patterns serve as a robust symbolic transformation technique, enabling the unveiling of latent dynamics within time series data. This methodology involves constructing histograms of patterns, followed by the calculation of both entropy and statistical complexity-an avenue yet to be fully understood in terms of its statistical properties. While asymptotic results can be derived by assuming a multinomial distribution for histogram proportions, the challenge emerges from the non-independence present in the sequence of ordinal patterns. Consequently, the direct application of the multinomial assumption is questionable. This study focuses on the computation of the asymptotic distribution of permutation entropy, considering the inherent patterns' correlation structure. Furthermore, the research delves into a comparative analysis, pitting this distribution against the entropy derived from a multinomial law. We present simulation algorithms for sampling time series with prescribed histograms of patterns and transition probabilities between them. Through this analysis, we better understand the intricacies of ordinal patterns and their statistical attributes.

Noise response in monomolecular closed systems

AbstractOne of the primary interests in understanding biological systems is the interaction between noise and cellular regulation. Noise levels can be altered by rate constant perturbations, which may result in an imbalance of important biological functions. Although the theory of noise localization can estimate noise perturbation response, it is currently limited to open systems, where the majority of biological networks belong. However, during their lifetime, some components of biological systems participate in reactions that can be categorized under a closed system. Therefore, a counterpart theory for closed systems is desirable. In this work, steady‐state perturbations of monomolecular closed systems and ways to estimate the response of noise as (co)variances are explored. We extend the structural sensitivity analysis for analyzing noise response patterns by using additional clues derived from multinomial assumptions. We identified scenarios in which this combination might fail in noise prediction, which we call noise flip, arising from the nonlinear dependence of variance upon composition. Nevertheless, theoretical intuition and simulation show that noise flip diminishes with increasing system size. This work provides an efficient and scalable means to estimate noise responses with potential applications in model discrimination, network‐wide response scanning, and reconstruction.

Clonality: Point estimation

Assessments of biological complexity for populations that are of mixed species are central in many biological contexts, including microbiomes, tumor cell population structure, and immune cell populations. Here we address the problem of quantifying the population diversity in experiments where high throughput DNA sequencing is used to distinguish a large number of cell subpopulations. Our model assumes a list of clonal species and their observed frequencies in each of several replicate sequencing libraries. Though the underlying distribution of frequencies cannot be estimated well from data coming from only a small fraction of the total cell population, one can estimate well the population-level clonality, defined as the sum of squared underlying fractions of the respective clones, the complement of the Gini–Simpson index. Specifically, we proposed to adaptively combine multiple unbiased estimators of clonality derived from pairs of replicates to construct a single estimator without relying on the commonly used but restrictive multinomial assumption. The new estimator performs particularly well for replicates of unequal size. We further illustrate the proposed methods with extensive simulations and a small real data example.

Estimating cell probabilities in contingency tables with constraints on marginals/conditionals by geometric programming with applications

Contingency tables are often used to display the multivariate frequency distribution of variables of interest. Under the common multinomial assumption, the first step of contingency table analysis is to estimate cell probabilities. It is well known that the unconstrained maximum likelihood estimator (MLE) is given by cell counts divided by the total number of observations. However, in the presence of (complex) constraints on the unknown cell probabilities or their functions, the MLE or other types of estimators may often have no closed form and have to be obtained numerically. In this paper, we focus on finding the MLE of cell probabilities in contingency tables under two common types of constraints: known marginals and ordered marginals/conditionals, and propose a novel approach based on geometric programming. We present two important applications that illustrate the usefulness of our approach via comparison with existing methods. Further, we show that our GP-based approach is flexible, readily implementable, effort-saving and can provide a unified framework for various types of constrained estimation of cell probabilities in contingency tables.

On the smoothing of multinomial estimates using Liouville mixture models and applications

There has been major progress in recent years in statistical model-based pattern recognition, data mining and knowledge discovery. In particular, generative models are widely used and are very reliable in terms of overall performance. Success of these models hinges on their ability to construct a representation which captures the underlying statistical distribution of data. In this article, we focus on count data modeling. Indeed, this kind of data is naturally generated in many contexts and in different application domains. Usually, models based on the multinomial assumption are used in this case that may have several shortcomings, especially in the case of high-dimensional sparse data. We propose then a principled approach to smooth multinomials using a mixture of Beta-Liouville distributions which is learned to reflect and model prior beliefs about multinomial parameters, via both theoretical interpretations and experimental validations, we argue that the proposed smoothing model is general and flexible enough to allow accurate representation of count data.

Effects of process and observation errors on effective sample size of fishery and survey age and length composition using variance ratio and likelihood methods

Abstract Hulson, P-J. F., Hanselman, D. H., and Quinn, T. J. II. 2011. Effects of process and observation errors on effective sample size of fishery and survey age and length composition using variance ratio and likelihood methods. – ICES Journal of Marine Science, 68: 1548–1557. Observations of age or length composition from fisheries or research surveys are modelled frequently with the multinomial distribution. Violations of multinomial assumptions in data collection usually cause overdispersion of observations and consequent underestimation of uncertainty. This has led to the adoption of an effective sample size less than the actual sample size to approximate the likelihood function for age or length composition better in, for example, fishery stock assessment models. The behaviour of effective sample size is examined under different scenarios for population age distribution and sampling design. Effective sample size was approximated with three approaches: (i) the ratio of multinomial to empirical variance; (ii) sampling estimation; and (iii) the Dirichlet likelihood. The most significant changes in effective sample size were attributable to process error involving aggregation of ages within schools. In terms of observation error, effective sample size can be increased by increasing the number of tows from which samples are obtained for age or length composition, then, because of the reduced uncertainty in effective sample size, the Dirichlet likelihood can be integrated into the objective function of fishery stock assessment models to estimate the effective sample size in future assessments.

Review and evaluation of likelihood functions for composition data in stock-assessment models: Estimating the effective sample size

Catch-at-age (or catch-at-length) data are one of the major components of most modern statistical stock assessment methods. Catch-at-age data provide, among other things, information about gear selectivity and recruitment strength. Catch-at-age data can also have a large influence on the estimates of fishing mortality, absolute abundance, and trends in abundance. The multinomial distribution describes the theoretical sampling process that is used to collect catch-at-age data, but only under the assumption of random sampling. Sampling designs generally employed to collect fishery-related data lead to age–composition estimates that depart from the strict theoretical multinomial probability distribution. Lack of independence can, for example, be due to size- or age-specific schooling or aggregating, causing positive correlations among individuals, and overdispersion. An additional cause of inadequacy of the multinomial assumption is model misspecification. Therefore, the effective sample size that should be used in an assessment model can be much smaller than the actual sample size. This can cause inappropriate weighting among data sets and negatively biased estimates of uncertainty. I use simulation analysis to evaluate five methods to estimate the effective sample size for catch-at-age data: (1) iterative multinomial likelihood; (2) normal approximation, using binomial variance; (3) lognormal likelihood with variance proportional to the inverse of the proportion; (4) Dirichlet likelihood; and (5) a multivariate normal approximation. The results show that all five methods perform similarly, but do not reduce estimation error relative to using the actual sample size unless the effective sample size is about one fifth of the actual sample size. All but (4) produced positively biased estimates of the effective sample size. If the effective sample size is not known within half an order of magnitude, I recommend using the lognormal likelihood with variance proportional to the inverse of the proportion and a regression against the actual sample size. This method is less computationally intense than (1), more robust than (2) and (4), and produces the least biased estimates of effective sample size, except for (4). Unlike (1), it can be included in Bayesian analysis.

On Testing Departure from the Binomial and Multinomial Assumptions

This paper is concerned with testing the multinomial (binomial) assumption against the Dirichlet-multinomial (beta-binomial) alternatives. In particular, we discuss the distribution of the asymptotic likelihood ratio (LR) test and obtain the C(alpha) goodness-of-fit test statistic. The inadequacy of the regular chi-square approximation to the LR test is supported by some Monte Carlo experiments. The C(alpha) test is recommended based on empirical significance level and power and also computational simplicity. Two examples are given.