multinomial models
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2022 ◽  
pp. 004912412110675
Author(s):  
Michael Schultz

This paper presents a model of recurrent multinomial sequences. Though there exists a quite considerable literature on modeling autocorrelation in numerical data and sequences of categorical outcomes, there is currently no systematic method of modeling patterns of recurrence in categorical sequences. This paper develops a means of discovering recurrent patterns by employing a more restrictive Markov assumption. The resulting model, which I call the recurrent multinomial model, provides a parsimonious representation of recurrent sequences, enabling the investigation of recurrences on longer time scales than existing models. The utility of recurrent multinomial models is demonstrated by applying them to the case of conversational turn-taking in meetings of the Federal Open Market Committee (FOMC). Analyses are effectively able to discover norms around turn-reclaiming, participation, and suppression and to evaluate how these norms vary throughout the course of the meeting.


2021 ◽  
Author(s):  
◽  
Haizhen Wu

<p><b>Divisible statistics have been widely used in many areas of statistical analysis. For example, Pearson's Chi-square statistic and the log-likelihood ratio statistic are frequently used in goodness of fit (GOF) and categorical analysis; the maximum likelihood (ML) estimators of the Shannon's and Simpson's diversity indices are often used as measure of diversity; and the spectral statistic plays a key role in the theory of large number of rare events. In the classical multinomial model, where the number of disjoint events N and their probabilities are all fixed, limit distributions of many divisible statistics have gradually been established. However, most of the results are based on the asymptotic equivalence of these statistics to Pearson's Chi-square statistic and the known limit distribution of the latter. In fact, with deeper analysis, one can conclude that the key point is not the asymptotic behavior of the Chi-square statistic, but that of the normalized frequencies. Based on the asymptotic normality of the normalized frequencies in the classical model, a unified approach to the limit theorems of more general divisible statistics can be established, of which the case of the Chi-square statistic is simply a natural corollary.</b></p> <p>In many applications, however, the classical multinomial model is not appropriate, and an extension to new models becomes necessary. This new type of model, called "non-classical" multinomial models, considers the case when N increases and the {Pni} change as sample size n increases. As we will see, in these non-classical models, both the asymptotic normality of the normalized frequencies and the asymptotic equivalence of many divisible statistics to the Chi-square statistic are lost, and the limit theorems established in classical model are no longer valid in non-classical models.</p> <p>The extension to non-classical models not only met the demands of many real world applications, but also opened a new research area in statistical analysis, which has not been thoroughly investigated so far. Although some results on the limit distributions of the divisible statistics in non-classical models have been acquired, e.g., Holst (1972); Morris (1975); Ivchenko and Levin (1976); Ivchenko and Medvedev (1979), they are far from complete. Though not yet attracting much attention by many applied statisticians, another advanced approach, introduced by Khmaladze (1984), makes use of modern martingale theory to establish functional limit theorems of the partial sum processes of divisible statistics successfully. In the main part of this thesis, we show that this martingale approach can be extended to more general situations where both Gaussian and Poissonian frequencies exist, and further discuss the properties and applications of the limiting processes, especially in constructing distribution-free statistics.</p> <p>The last part of the thesis is about the statistical analysis of large number of rare events (LNRE), which is an important class of non-classical multinomial models and presented in numerous applications. In LNRE models, most of the frequencies are very small and it is not immediately clear how consistent and reliable inference can be achieved. Based on the definitions and key concepts firstly introduced by Khmaladze (1988), we discuss a particular model with the context of diversity of questionnaires. The advanced statistical techniques such as large deviation, contiguity and Edgeworth expansion used in establishing limit theorems underpin the potential of LNRE theory to become a fruitful research area in future.</p>


2021 ◽  
Author(s):  
◽  
Haizhen Wu

<p><b>Divisible statistics have been widely used in many areas of statistical analysis. For example, Pearson's Chi-square statistic and the log-likelihood ratio statistic are frequently used in goodness of fit (GOF) and categorical analysis; the maximum likelihood (ML) estimators of the Shannon's and Simpson's diversity indices are often used as measure of diversity; and the spectral statistic plays a key role in the theory of large number of rare events. In the classical multinomial model, where the number of disjoint events N and their probabilities are all fixed, limit distributions of many divisible statistics have gradually been established. However, most of the results are based on the asymptotic equivalence of these statistics to Pearson's Chi-square statistic and the known limit distribution of the latter. In fact, with deeper analysis, one can conclude that the key point is not the asymptotic behavior of the Chi-square statistic, but that of the normalized frequencies. Based on the asymptotic normality of the normalized frequencies in the classical model, a unified approach to the limit theorems of more general divisible statistics can be established, of which the case of the Chi-square statistic is simply a natural corollary.</b></p> <p>In many applications, however, the classical multinomial model is not appropriate, and an extension to new models becomes necessary. This new type of model, called "non-classical" multinomial models, considers the case when N increases and the {Pni} change as sample size n increases. As we will see, in these non-classical models, both the asymptotic normality of the normalized frequencies and the asymptotic equivalence of many divisible statistics to the Chi-square statistic are lost, and the limit theorems established in classical model are no longer valid in non-classical models.</p> <p>The extension to non-classical models not only met the demands of many real world applications, but also opened a new research area in statistical analysis, which has not been thoroughly investigated so far. Although some results on the limit distributions of the divisible statistics in non-classical models have been acquired, e.g., Holst (1972); Morris (1975); Ivchenko and Levin (1976); Ivchenko and Medvedev (1979), they are far from complete. Though not yet attracting much attention by many applied statisticians, another advanced approach, introduced by Khmaladze (1984), makes use of modern martingale theory to establish functional limit theorems of the partial sum processes of divisible statistics successfully. In the main part of this thesis, we show that this martingale approach can be extended to more general situations where both Gaussian and Poissonian frequencies exist, and further discuss the properties and applications of the limiting processes, especially in constructing distribution-free statistics.</p> <p>The last part of the thesis is about the statistical analysis of large number of rare events (LNRE), which is an important class of non-classical multinomial models and presented in numerous applications. In LNRE models, most of the frequencies are very small and it is not immediately clear how consistent and reliable inference can be achieved. Based on the definitions and key concepts firstly introduced by Khmaladze (1988), we discuss a particular model with the context of diversity of questionnaires. The advanced statistical techniques such as large deviation, contiguity and Edgeworth expansion used in establishing limit theorems underpin the potential of LNRE theory to become a fruitful research area in future.</p>


2021 ◽  
Author(s):  
◽  
Anna Friedlander

<p>The sheer volume of data to be produced by the next generation of radio telescopes—exabytes of data on hundreds of millions of objects—makes automated methods for the detection of astronomical objects ("sources") essential. Of particular importance are low surface brightness objects, which are not well found by current automated methods.  This thesis explores Bayesian methods for source detection that use Dirichlet or multinomial models for pixel intensity distributions in discretised radio astronomy images. A novel image discretisation method that incorporates uncertainty about how the image should be discretised is developed. Latent Dirichlet allocation — a method originally developed for inferring latent topics in document collections — is used to estimate source and background distributions in radio astronomy images. A new Dirichlet-multinomial ratio, indicating how well a region conforms to a well-specified model of background versus a loosely-specified model of foreground, is derived. Finally, latent Dirichlet allocation and the Dirichlet-multinomial ratio are combined for source detection in astronomical images.   The methods developed in this thesis perform source detection well in comparison to two widely-used source detection packages and, importantly, find dim sources not well found by other algorithms.</p>


2021 ◽  
Author(s):  
◽  
Anna Friedlander

<p>The sheer volume of data to be produced by the next generation of radio telescopes—exabytes of data on hundreds of millions of objects—makes automated methods for the detection of astronomical objects ("sources") essential. Of particular importance are low surface brightness objects, which are not well found by current automated methods.  This thesis explores Bayesian methods for source detection that use Dirichlet or multinomial models for pixel intensity distributions in discretised radio astronomy images. A novel image discretisation method that incorporates uncertainty about how the image should be discretised is developed. Latent Dirichlet allocation — a method originally developed for inferring latent topics in document collections — is used to estimate source and background distributions in radio astronomy images. A new Dirichlet-multinomial ratio, indicating how well a region conforms to a well-specified model of background versus a loosely-specified model of foreground, is derived. Finally, latent Dirichlet allocation and the Dirichlet-multinomial ratio are combined for source detection in astronomical images.   The methods developed in this thesis perform source detection well in comparison to two widely-used source detection packages and, importantly, find dim sources not well found by other algorithms.</p>


Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2173
Author(s):  
Isaac Akoto ◽  
João T. Mexia ◽  
Filipe J. Marques

In this work, we derived new asymptotic results for multinomial models. To obtain these results, we started by studying limit distributions in models with a compact parameter space. This restriction holds since the key parameter whose components are the probabilities of the possible outcomes have non-negative components that add up to 1. Based on these results, we obtained confidence ellipsoids and simultaneous confidence intervals for models with normal limit distributions. We then studied the covariance matrices of the limit normal distributions for the multinomial models. This was a transition between the previous general results and on the inference for multinomial models in which we considered the chi-square tests, confidence regions and non-linear statistics—namely log-linear models with two numerical applications to those models. Namely, our approach overcame the hierarchical restrictions assumed to analyse the multidimensional contingency table.


2021 ◽  
pp. 1-19
Author(s):  
Wendy Leutert ◽  
Samantha A. Vortherms

Abstract State-owned enterprises (SOEs) retain a strong presence in many economies around the world. How do governments manage these firms given their dual economic and political nature? Many states use authority over executive appointments as a key means of governing SOEs. We analyze the nature of this “personnel power” by assessing patterns in SOE leaders’ political mobility in China, the country with the largest state-owned sector. Using logit and multinomial models on an original dataset of central SOE leaders’ attributes and company information from 2003 to 2017, we measure the effects of economic performance and political connectedness on leaders’ likelihood of staying in power. We find that leaders of well-performing firms and those with patronage ties to elites in charge of their evaluation are more likely to stay in office. These findings suggest that states can leverage personnel power in pursuit of economic and political stability when SOE management is highly politically integrated.


2021 ◽  
pp. 1-13
Author(s):  
Shama D. Karanth ◽  
Frederick A. Schmitt ◽  
Peter T. Nelson ◽  
Yuriko Katsumata ◽  
Richard J. Kryscio ◽  
...  

Background: Late-life cognitive function is heterogeneous, ranging from no decline to severe dementia. Prior studies of cognitive trajectories have tended to focus on a single measure of global cognition or individual tests scores, rather than considering longitudinal performance on multiple tests simultaneously. Objective: The current study aimed to examine cognitive trajectories from two independent datasets to assess whether similar patterns might describe longitudinal cognition in the decade preceding death, as well as what participant characteristics were associated with trajectory membership. Methods: Data were drawn from autopsied longitudinally followed participants of two cohorts (total N = 1,346), community-based cohort at the University of Kentucky Alzheimer’s Disease Research Center (n = 365) and National Alzheimer’s Coordinating Center (n = 981). We used group-based multi-trajectory models (GBMTM) to identify cognitive trajectories over the decade before death using Mini-Mental State Exam, Logical Memory-Immediate, and Animal Naming performance. Multinomial logistic and Random Forest analyses assessed characteristics associated with trajectory groups. Results: GBMTM identified four similar cognitive trajectories in each dataset. In multinomial models, death age, Braak neurofibrillary tangles (NFT) stage, TDP-43, and α-synuclein were associated with declining trajectories. Random Forest results suggested the most important trajectory predictors were Braak NFT stage, cerebral atrophy, death age, and brain weight. Multiple pathologies were most common in trajectories with moderate or accelerated decline. Conclusion: Cognitive trajectories associated strongly with neuropathology, particularly Braak NFT stage. High frequency of multiple pathologies in trajectories with cognitive decline suggests dementia treatment and prevention efforts must consider multiple diseases simultaneously.


2021 ◽  
Author(s):  
Tyler Bonnell ◽  
Robert Michaud ◽  
Angelique Dupuch ◽  
Veronique Lesage

Photo identification of individuals within a population is a common data source that is becoming more common given technological advances and the use of computer vision and machine learning to re-identify individuals. These data are collected through hand-held cameras, drones, and camera traps, and often come with biases in terms of sampling effort and distribution. In spite of these biases, a common goal of collecting these datasets is to better understand the habitat use pattern of individuals and populations. Here, we examine the potential for multilevel multinomial models to generate socio-spatial networks that capture the similarities in individual users across the spatial distribution of a species. We use this approach with 18 years of photo-ID data to better understand population structuring of beluga whales in the St. Lawrence River. We show using permuted and simulated data that this approach can identify community network structures within populations in a way that accounts for biases in collections methods. Applying this method to the entire 18 years dataset for SLE beluga, we found three spatially distinct communities. These results suggest that within the population's summer range individuals are moving within restricted areas (i.e., home ranges), and have implications for the estimated impacts of localized anthropogenic stressors, such as chemical pollution or acoustic disturbances on animal populations. We conclude that multilevel multinomial models can be effective at estimating socio-spatial networks that describe community structuring within wildlife populations.


2021 ◽  
Author(s):  
Alexandra Sarafoglou ◽  
Frederik Aust ◽  
Maarten Marsman ◽  
Eric-Jan Wagenmakers ◽  
Julia M. Haaf

The multibridge R package allows a Bayesian evaluation of informed hypotheses H_r applied to frequency data from an independent binomial or multinomial distribution. multibridge uses bridge sampling to efficiently compute Bayes factors for the following hypotheses concerning the latent category proportions theta: (a) hypotheses that postulate equality constraints (e.g., theta_1 = theta_2 = theta_3); (b) hypotheses that postulate inequality constraints (e.g., theta_1 &lt; theta_2 &lt; theta_3) or (theta_1 &gt; theta_2 &gt; theta_3); (c) hypotheses that postulate mixtures of inequality constraints and equality constraints (e.g., theta_1 &lt; theta_2 = theta_3); and (d) hypotheses that postulate mixtures of (a)--(c) (e.g., theta_1 &lt; theta_2 = theta_3, theta_4). Any informed hypothesis H_r may be compared against the encompassing hypothesis H_e that all category proportions vary freely, or against the null hypothesis H_0 that all category proportions are equal. multibridge facilitates the fast and accurate comparison of large models with many constraints and models for which relatively little posterior mass falls in the restricted parameter space. This paper describes the underlying methodology and illustrates the use of multibridge through fully reproducible examples.


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