Sequential decision making in general state space models

Many problems in control and signal processing can be formulated as sequential decision problems for general state space models. However, except for some simple models one cannot obtain analytical solutions and has to resort to approximation. In this thesis, we have investigated problems where Sequential Monte Carlo (SMC) methods can be combined with a gradient based search to provide solutions to online optimisation problems. We summarise the main contributions of the thesis as follows. Chapter 4 focuses on solving the sensor scheduling problem when cast as a controlled Hidden Markov Model. We consider the case in which the state, observation and action spaces are continuous. This general case is important as it is the natural framework for many applications. In sensor scheduling, our aim is to minimise the variance of the estimation error of the hidden state with respect to the action sequence. We present a novel SMC method that uses a stochastic gradient algorithm to find optimal actions. This is in contrast to existing works in the literature that only solve approximations to the original problem. In Chapter 5 we presented how an SMC can be used to solve a risk sensitive control problem. We adopt the use of the Feynman-Kac representation of a controlled Markov chain flow and exploit the properties of the logarithmic Lyapunov exponent, which lead to a policy gradient solution for the parameterised problem. The resulting SMC algorithm follows a similar structure with the Recursive Maximum Likelihood(RML) algorithm for online parameter estimation. In Chapters 6, 7 and 8, dynamic Graphical models were combined with with state space models for the purpose of online decentralised inference. We have concentrated more on the distributed parameter estimation problem using two Maximum Likelihood techniques, namely Recursive Maximum Likelihood (RML) and Expectation Maximization (EM). The resulting algorithms can be interpreted as an extension of the Belief Propagation (BP) algorithm to compute likelihood gradients. In order to design an SMC algorithm, in Chapter 8 uses a nonparametric approximations for Belief Propagation. The algorithms were successfully applied to solve the sensor localisation problem for sensor networks of small and medium size.

Learning in Network Games

We report the findings of an experiment designed to study how people learn and make decisions in network games. Network games offer new opportunities to identify learning rules, since on networks (compared to e.g. random matching) more rules differ in terms of their information requirements. Our experimental design enables us to observe both which actions participants choose and which information they consult before making their choices. We use this information to estimate learning types using maximum likelihood methods. There is substantial heterogeneity in learning types. However, the vast majority of our participants' decisions are best characterized by reinforcement learning or (myopic) best-response learning. The distribution of learning types seems fairly stable across contexts. Neither network topology nor the position of a player in the network seem to substantially affect the estimated distribution of learning types.

Automated, objective texture segmentation of multibeam echosounder data - Seafloor survey and substrate maps from James Island to Ozette Lake, Washington Outer Coast

Without knowledge of basic seafloor characteristics, the ability to address any number of critical marine and/or coastal management issues is diminished. For example, management and conservation of essential fish habitat (EFH), a requirement mandated by federally guided fishery management plans (FMPs), requires among other things a description of habitats for federally managed species. Although the list of attributes important to habitat are numerous, the ability to efficiently and effectively describe many, and especially at the scales required, does not exist with the tools currently available. However, several characteristics of seafloor morphology are readily obtainable at multiple scales and can serve as useful descriptors of habitat. Recent advancements in acoustic technology, such as multibeam echosounding (MBES), can provide remote indication of surficial sediment properties such as texture, hardness, or roughness, and further permit highly detailed renderings of seafloor morphology. With acoustic-based surveys providing a relatively efficient method for data acquisition, there exists a need for efficient and reproducible automated segmentation routines to process the data. Using MBES data collected by the Olympic Coast National Marine Sanctuary (OCNMS), and through a contracted seafloor survey, we expanded on the techniques of Cutter et al. (2003) to describe an objective repeatable process that uses parameterized local Fourier histogram (LFH) texture features to automate segmentation of surficial sediments from acoustic imagery using a maximum likelihood decision rule. Sonar signatures and classification performance were evaluated using video imagery obtained from a towed camera sled. Segmented raster images were converted to polygon features and attributed using a hierarchical deep-water marine benthic classification scheme (Greene et al. 1999) for use in a geographical information system (GIS). (PDF contains 41 pages.)

Permutationally invariant state reconstruction

Feasible tomography schemes for large particle numbers must possess, besides an appropriate data acquisition protocol, an efficient way to reconstruct the density operator from the observed finite data set. Since state reconstruction typically requires the solution of a nonlinear large-scale optimization problem, this is a major challenge in the design of scalable tomography schemes. Here we present an efficient state reconstruction scheme for permutationally invariant quantum state tomography. It works for all common state-of-the-art reconstruction principles, including, in particular, maximum likelihood and least squares methods, which are the preferred choices in today's experiments. This high efficiency is achieved by greatly reducing the dimensionality of the problem employing a particular representation of permutationally invariant states known from spin coupling combined with convex optimization, which has clear advantages regarding speed, control and accuracy in comparison to commonly employed numerical routines. First prototype implementations easily allow reconstruction of a state of 20 qubits in a few minutes on a standard computer

Efficient learning of decomposable models with a bounded clique size

The learning of probability distributions from data is a ubiquitous problem in the fields of Statistics and Artificial Intelligence. During the last decades several learning algorithms have been proposed to learn probability distributions based on decomposable models due to their advantageous theoretical properties. Some of these algorithms can be used to search for a maximum likelihood decomposable model with a given maximum clique size, k, which controls the complexity of the model. Unfortunately, the problem of learning a maximum likelihood decomposable model given a maximum clique size is NP-hard for k > 2. In this work, we propose a family of algorithms which approximates this problem with a computational complexity of O(k · n^2 log n) in the worst case, where n is the number of implied random variables. The structures of the decomposable models that solve the maximum likelihood problem are called maximal k-order decomposable graphs. Our proposals, called fractal trees, construct a sequence of maximal i-order decomposable graphs, for i = 2, ..., k, in k − 1 steps. At each step, the algorithms follow a divide-and-conquer strategy based on the particular features of this type of structures. Additionally, we propose a prune-and-graft procedure which transforms a maximal k-order decomposable graph into another one, increasing its likelihood. We have implemented two particular fractal tree algorithms called parallel fractal tree and sequential fractal tree. These algorithms can be considered a natural extension of Chow and Liu’s algorithm, from k = 2 to arbitrary values of k. Both algorithms have been compared against other efficient approaches in artificial and real domains, and they have shown a competitive behavior to deal with the maximum likelihood problem. Due to their low computational complexity they are especially recommended to deal with high dimensional domains.

Equilibrium in dynamic economic models

The main theme running through these three chapters is that economic agents are often forced to respond to events that are not a direct result of their actions or other agents actions. The optimal response to these shocks will necessarily depend on agents' understanding of how these shocks arise. The economic environment in the first two chapters is analogous to the classic chain store game. In this setting, the addition of unintended trembles by the agents creates an environment better suited to reputation building. The third chapter considers the competitive equilibrium price dynamics in an overlapping generations environment when there are supply and demand shocks.

The first chapter is a game theoretic investigation of a reputation building game. A sequential equilibrium model, called the "error prone agents" model, is developed. In this model, agents believe that all actions are potentially subjected to an error process. Inclusion of this belief into the equilibrium calculation provides for a richer class of reputation building possibilities than when perfect implementation is assumed.

In the second chapter, maximum likelihood estimation is employed to test the consistency of this new model and other models with data from experiments run by other researchers that served as the basis for prominent papers in this field. The alternate models considered are essentially modifications to the standard sequential equilibrium. While some models perform quite well in that the nature of the modification seems to explain deviations from the sequential equilibrium quite well, the degree to which these modifications must be applied shows no consistency across different experimental designs.

The third chapter is a study of price dynamics in an overlapping generations model. It establishes the existence of a unique perfect-foresight competitive equilibrium price path in a pure exchange economy with a finite time horizon when there are arbitrarily many shocks to supply or demand. One main reason for the interest in this equilibrium is that overlapping generations environments are very fruitful for the study of price dynamics, especially in experimental settings. The perfect foresight assumption is an important place to start when examining these environments because it will produce the ex post socially efficient allocation of goods. This characteristic makes this a natural baseline to which other models of price dynamics could be compared.

Essays in Behavioral Decision Theory

This thesis studies decision making under uncertainty and how economic agents respond to information. The classic model of subjective expected utility and Bayesian updating is often at odds with empirical and experimental results; people exhibit systematic biases in information processing and often exhibit aversion to ambiguity. The aim of this work is to develop simple models that capture observed biases and study their economic implications.

In the first chapter I present an axiomatic model of cognitive dissonance, in which an agent's response to information explicitly depends upon past actions. I introduce novel behavioral axioms and derive a representation in which beliefs are directionally updated. The agent twists the information and overweights states in which his past actions provide a higher payoff. I then characterize two special cases of the representation. In the first case, the agent distorts the likelihood ratio of two states by a function of the utility values of the previous action in those states. In the second case, the agent's posterior beliefs are a convex combination of the Bayesian belief and the one which maximizes the conditional value of the previous action. Within the second case a unique parameter captures the agent's sensitivity to dissonance, and I characterize a way to compare sensitivity to dissonance between individuals. Lastly, I develop several simple applications and show that cognitive dissonance contributes to the equity premium and price volatility, asymmetric reaction to news, and belief polarization.

The second chapter characterizes a decision maker with sticky beliefs. That is, a decision maker who does not update enough in response to information, where enough means as a Bayesian decision maker would. This chapter provides axiomatic foundations for sticky beliefs by weakening the standard axioms of dynamic consistency and consequentialism. I derive a representation in which updated beliefs are a convex combination of the prior and the Bayesian posterior. A unique parameter captures the weight on the prior and is interpreted as the agent's measure of belief stickiness or conservatism bias. This parameter is endogenously identified from preferences and is easily elicited from experimental data.

The third chapter deals with updating in the face of ambiguity, using the framework of Gilboa and Schmeidler. There is no consensus on the correct way way to update a set of priors. Current methods either do not allow a decision maker to make an inference about her priors or require an extreme level of inference. In this chapter I propose and axiomatize a general model of updating a set of priors. A decision maker who updates her beliefs in accordance with the model can be thought of as one that chooses a threshold that is used to determine whether a prior is plausible, given some observation. She retains the plausible priors and applies Bayes' rule. This model includes generalized Bayesian updating and maximum likelihood updating as special cases.

Wireless optical communication-based spatial pattern

We propose a novel communication technique which utilizes a set of mutually distinguishable optical patterns instead of convergent facula to transmit information. The communication capacity is increased by exploiting the optical spatial bandwidth resources. An optimum detector for this communication is proposed based on maximum-likelihood decision. The fundamental rule of designing signal spatial pattern is formulated from analysis of the probability of error decision. Finally, we present a typical electro-optical system scheme of the proposed communication. (c) 2006 Elsevier GmbH. All rights reserved.

The global optimization of phase-incoherent signals

The problem of global optimization of M phase-incoherent signals in N complex dimensions is formulated. Then, by using the geometric approach of Landau and Slepian, conditions for optimality are established for N = 2 and the optimal signal sets are determined for M = 2, 3, 4, 6, and 12.

The method is the following: The signals are assumed to be equally probable and to have equal energy, and thus are represented by points ṡ_i, i = 1, 2, …, M, on the unit sphere S₁ in C^N. If W_ik is the halfspace determined by ṡ_i and ṡ_k and containing ṡ_i, i.e. W_ik = {ṙϵC^N:| ≥ | ˂ṙ, ṡ_k˃|}, then the Ʀ_i = ∩/k≠i W_ik, i = 1, 2, …, M, the maximum likelihood decision regions, partition S₁. For additive complex Gaussian noise ṅ and a received signal ṙ = ṡ_ie^jϴ + ṅ, where ϴ is uniformly distributed over [0, 2π], the probability of correct decoding is P_C = 1/π^N ∞/ʃ/0 r^2N-1e^-(r2+1)U(r)dr, where U(r) = 1/M M/Ʃ/i=1 Ʀ_i ʃ/∩ S₁ I₀(2r | ˂ṡ, ṡ_i˃|)dσ(ṡ), and r = ǁṙǁ.

For N = 2, it is proved that U(r) ≤ ʃ/C_α I₀(2r|˂ṡ, ṡ_i˃|)dσ(ṡ) – 2K/M. h(1/2K [Mσ(C_α)-σ(S₁)]), where C_α = {ṡϵS₁:|˂ṡ, ṡ_i˃| ≥ α}, K is the total number of boundaries of the net on S₁ determined by the decision regions, and h is the strictly increasing strictly convex function of σ(C_α∩W), (where W is a halfspace not containing ṡ_i), given by h = ʃ/C_α∩W I₀ (2r|˂ṡ, ṡ_i˃|)dσ(ṡ). Conditions for equality are established and these give rise to the globally optimal signal sets for M = 2, 3, 4, 6, and 12.

星载激光对水下目标通信可行性研究

The characteristics of media in communication channel are analyzed briefly and the reasonable optical parameters of media are adopted. With certain communication system parameters the temporal and spatial distributions of the received signal from submerged platform are simulated using Monte Carlo method. The upper limit of the ratio of Monte Carlo estimated error to averaged value is about 0.3%. From the simulated results, the optimized sampling timing of receiver and field of view of telescope are obtained. Also the signal-to-noise ratio of the receiver is calculated. Based on this, the error probability of the communication system is deduced from laser pulse position modulation and maximum likelihood detection. The results show that under severe environment robust laser communication from a satellite to a submerged platform can be achieved.

Caracterização de imagens de microtomografia de raios X baseada em descritores de textura

A microtomografia computadorizada (computed microtomography - μCT) permite uma análise não destrutiva de amostras, além de possibilitar sua reutilização. A μCT permite também a reconstrução de objetos tridimensionais a partir de suas seções transversais que são obtidas interceptando a amostra através de planos paralelos. Equipamentos de μCT oferecem ao usuário diversas opções de configurações que alteram a qualidade das imagens obtidas afetando, dessa forma, o resultado esperado. Nesta tese foi realizada a caracterização e análise de imagens de μCT geradas pelo microtomógrafo SkyScan1174 Compact Micro-CT. A base desta caracterização é o processamento de imagens. Foram aplicadas técnicas de realce (brilho, saturação, equalização do histograma e filtro de mediana) nas imagens originais gerando novas imagens e em seguida a quantificação de ambos os conjuntos, utilizando descritores de textura (probabilidade máxima, momento de diferença, momento inverso de diferença, entropia e uniformidade). Os resultados mostram que, comparadas às originais, as imagens que passaram por técnicas de realce apresentaram melhoras quando gerados seus modelos tridimensionais.

A-SCALA: an age-structured statistical catch-at-length analysis for assessing tuna stocks in the eastern tropical Pacific Ocean

English: We describe an age-structured statistical catch-at-length analysis (A-SCALA) based on the MULTIFAN-CL model of Fournier et al. (1998). The analysis is applied independently to both the yellowfin and the bigeye tuna populations of the eastern Pacific Ocean (EPO). We model the populations from 1975 to 1999, based on quarterly time steps. Only a single stock for each species is assumed for each analysis, but multiple fisheries that are spatially separate are modeled to allow for spatial differences in catchability and selectivity. The analysis allows for error in the effort-fishing mortality relationship, temporal trends in catchability, temporal variation in recruitment, relationships between the environment and recruitment and between the environment and catchability, and differences in selectivity and catchability among fisheries. The model is fit to total catch data and proportional catch-at-length data conditioned on effort. The A-SCALA method is a statistical approach, and therefore recognizes that the data collected from the fishery do not perfectly represent the population. Also, there is uncertainty in our knowledge about the dynamics of the system and uncertainty about how the observed data relate to the real population. The use of likelihood functions allow us to model the uncertainty in the data collected from the population, and the inclusion of estimable process error allows us to model the uncertainties in the dynamics of the system. The statistical approach allows for the calculation of confidence intervals and the testing of hypotheses. We use a Bayesian version of the maximum likelihood framework that includes distributional constraints on temporal variation in recruitment, the effort-fishing mortality relationship, and catchability. Curvature penalties for selectivity parameters and penalties on extreme fishing mortality rates are also included in the objective function. The mode of the joint posterior distribution is used as an estimate of the model parameters. Confidence intervals are calculated using the normal approximation method. It should be noted that the estimation method includes constraints and priors and therefore the confidence intervals are different from traditionally calculated confidence intervals. Management reference points are calculated, and forward projections are carried out to provide advice for making management decisions for the yellowfin and bigeye populations. Spanish: Describimos un análisis estadístico de captura a talla estructurado por edad, A-SCALA (del inglés age-structured statistical catch-at-length analysis), basado en el modelo MULTIFAN- CL de Fournier et al. (1998). Se aplica el análisis independientemente a las poblaciones de atunes aleta amarilla y patudo del Océano Pacífico oriental (OPO). Modelamos las poblaciones de 1975 a 1999, en pasos trimestrales. Se supone solamente una sola población para cada especie para cada análisis, pero se modelan pesquerías múltiples espacialmente separadas para tomar en cuenta diferencias espaciales en la capturabilidad y selectividad. El análisis toma en cuenta error en la relación esfuerzo-mortalidad por pesca, tendencias temporales en la capturabilidad, variación temporal en el reclutamiento, relaciones entre el medio ambiente y el reclutamiento y entre el medio ambiente y la capturabilidad, y diferencias en selectividad y capturabilidad entre pesquerías. Se ajusta el modelo a datos de captura total y a datos de captura a talla proporcional condicionados sobre esfuerzo. El método A-SCALA es un enfoque estadístico, y reconoce por lo tanto que los datos obtenidos de la pesca no representan la población perfectamente. Además, hay incertidumbre en nuestros conocimientos de la dinámica del sistema e incertidumbre sobre la relación entre los datos observados y la población real. El uso de funciones de verosimilitud nos permite modelar la incertidumbre en los datos obtenidos de la población, y la inclusión de un error de proceso estimable nos permite modelar las incertidumbres en la dinámica del sistema. El enfoque estadístico permite calcular intervalos de confianza y comprobar hipótesis. Usamos una versión bayesiana del marco de verosimilitud máxima que incluye constreñimientos distribucionales sobre la variación temporal en el reclutamiento, la relación esfuerzo-mortalidad por pesca, y la capturabilidad. Se incluyen también en la función objetivo penalidades por curvatura para los parámetros de selectividad y penalidades por tasas extremas de mortalidad por pesca. Se usa la moda de la distribución posterior conjunta como estimación de los parámetros del modelo. Se calculan los intervalos de confianza usando el método de aproximación normal. Cabe destacar que el método de estimación incluye constreñimientos y distribuciones previas y por lo tanto los intervalos de confianza son diferentes de los intervalos de confianza calculados de forma tradicional. Se calculan puntos de referencia para el ordenamiento, y se realizan proyecciones a futuro para asesorar la toma de decisiones para el ordenamiento de las poblaciones de aleta amarilla y patudo.

Contribuição à confiabilidade de estacas cravadas através de um estudo de caso com aplicação da teoria Bayesiana.

Nos diversos segmentos da Geotecnia e em especial na área de fundações, o engenheiro se depara com uma série de incertezas. Algumas destas incertezas são inerentes à variabilidade local do solo, às condições de carregamento, aos efeitos do tempo, às diferenças nos processos executivos, erros de sondagens, que influenciam diretamente a estimativa da capacidade de carga da fundação, seja por ocasião de seu carregamento estático, seja durante ou logo após a cravação. O objetivo desta dissertação é a adaptação, a estacas em terra (onshore), de um procedimento concebido originalmente para emprego em estacas offshore, que trata da atualização da estimativa da resistência durante a cravação, com base em registros documentados durante a execução. Neste procedimento a atualização é feita através da aplicação dos conceitos da análise Bayesiana, assumindo que os parâmetros da distribuição probabilística utilizada sejam variáveis randômicas. A incerteza dos parâmetros é modelada por distribuições a priori e a posteriori. A distribuição a posteriori é calculada pela atualização da distribuição a priori, utilizando uma função de máxima verossimilhança, que contém a observação obtida dos registros de cravação. O procedimento é aplicado a um conjunto de estacas de um extenso estaqueamento executado na Zona Oeste do Rio de Janeiro. As estimativas atualizadas são posteriormente comparadas aos resultados dos ensaios de carregamento dinâmico. Várias aplicações podem surgir com o emprego deste procedimento, como a seleção das estacas que, por apresentarem reduzido valor de estimativa atualizada de resistência, ou uma maior incerteza desta estimativa, devam ser submetidas a provas de carga. A extensão deste estudo a diferentes tipos de estacas em perfis de solo de natureza distintos poderá levar ao desenvolvimento de sistemas mais adequados de controle de execução, capazes de identificar as principais incertezas presentes nos diferentes tipos de execução de estacas, contribuindo assim para a otimização de futuros projetos de fundações.

Construção e Validação da Escala de Crenças Conjugais (ECC)

As crenças conjugais podem ser entendidas como um conjunto de ideias acerca de como o casamento e o cônjuge devem ser. Observa-se que os padrões de crenças mantidos em relação ao casamento interferem na qualidade conjugal, de modo que crenças realistas estariam vinculadas a relações mais satisfatórias e crenças irrealistas estariam associadas à insatisfação com o casamento. Assim, evidencia-se a importância de conhecer quais os estilos de crenças mantidos no casamento, o que traz à tona a questão da avaliação. Em âmbito internacional foram identificados alguns instrumentos criados e validados para a avaliação de elementos cognitivos influentes nas relações entre casais. Todavia, no cenário nacional verificou-se a ausência de instrumentos sobre crenças no casamento construídos e validados para a população brasileira. Dessa lacuna metodológica surgiu o interesse de criar um instrumento de avaliação das crenças no casamento. Desse modo, o presente estudo objetivou a construção e a validação da Escala de Crenças Conjugais (ECC). Para tanto, esta pesquisa foi composta por dois estudos: 1) Estudo I Construção da Escala de Crenças Conjugais (ECC); 2) Estudo II Validação da Escala de Crenças Conjugais (ECC). O Estudo I correspondeu à criação da ECC, com a realização de entrevistas e pesquisa na literatura para obtenção das crenças mais comuns sobre o casamento, seguida pela avaliação da validade de conteúdo dos itens da ECC. Desse primeiro estudo, resultou uma versão da ECC composta por 33 itens, dentre crenças conjugais realistas e irrealistas. O Estudo II correspondeu à validação de construto da ECC. Para tanto a versão da escala resultante do Estudo I foi aplicada numa amostra de 333 participantes, com escolaridade a partir do ensino fundamental completo, dentre homens e mulheres, solteiros e casados. Para verificar a validade de construto da ECC foram realizados testes de análise fatorial (AF), em que o método escolhido para este estudo foi o método de máxima verossimilhança com rotação quartimax. Os resultados da AF revelaram a existência de dois Fatores para a ECC identificados como Comunicação Interpessoal e Compromisso (CIC) e Papéis Sociais (PS). Após a obtenção dos Fatores foi realizada a análise da consistência interna de cada Fator por meio do cálculo do coeficiente alfa de Cronbach (α), em que constatou-se que ambos os Fatores apresentaram níveis satisfatórios de confiabilidade: CIC (α = 0,81) e PS (α = 0,70). Testes adicionais acerca de eventuais contrastes nos resultados de acordo com as características da amostra também foram realizados, por meio do Teste t de Student, verificando-se que na amostra deste estudo, os indivíduos mais jovens apresentaram níveis mais elevados no Fator 2 (PS), os indivíduos que haviam concluído um curso superior possuíam níveis mais elevados no Fator 1 (CIC) e os indivíduos solteiros apresentaram níveis mais elevados no Fator 2 (PS). Os dois estudos resultaram numa medida inédita de avaliação das crenças conjugais a ser utilizada em pesquisas brasileiras. Espera-se que a ECC possa ser útil no estudo das relações entre casais, possibilitando a realização de novas pesquisas e quiçá novas intervenções terapêuticas.

Spatial and temporal variability in the relative contribution of king mackerel (Scomberomorus cavalla) stocks to winter mixed fisheries off South Florida

King mackerel (Scomberomorus cavalla) are ecologically and economically important scombrids that inhabit U.S. waters of the Gulf of Mexico (GOM) and Atlantic Ocean (Atlantic). Separate migratory groups, or stocks, migrate from eastern GOM and southeastern U.S. Atlantic to south Florida waters where the stocks mix during winter. Currently, all winter landings from a management-defined south Florida mixing zone are attributed to the GOM stock. In this study, the stock composition of winter landings across three south Florida sampling zones was estimated by using stock-specific otolith morphological variables and Fourier harmonics. The mean accuracies of the jackknifed classifications from stepwise linear discriminant function analysis of otolith shape variables ranged from 66−76% for sex-specific models. Estimates of the contribution of the Atlantic stock to winter landings, derived from maximum likelihood stock mixing models, indicated the contribution was highest off southeastern Florida (as high as 82.8% for females in winter 2001−02) and lowest off southwestern Florida (as low as 14.5% for females in winter 2002−03). Overall, results provided evidence that the Atlantic stock contributes a certain, and perhaps a significant (i.e., ≥50%), percentage of landings taken in the management-defined winter mixing zone off south Florida, and the practice of assigning all winter mixing zone landings to the GOM stock should