本文使用Esscher轉換法推導狀態轉換模型、跳躍獨立風險下狀狀態轉換模型及跳躍相關風險下狀態轉換模型的選擇權定價公式。藉由1999年至2011年道瓊工業指數真實市場資料使用EM演算法估計模型參數並使用概似比檢定得到跳躍相關風險下狀態轉換模型最適合描述報酬率資料。接著進行敏感度分析得知，高波動狀態的機率、報酬率的整體波動度及跳躍頻率三者與買權呈現正相關。最後由市場驗證可知，跳躍相關風險下狀態轉換模型在價平及價外的定價誤差皆是最小，在價平的定價誤差則略高於跳躍獨立風險下狀態轉換模型。 / In this paper, we derive regime switching model, regime switching model with independent jump and regime switching model with dependent jump by Esscher transformation. We use the data from 1999 to 2011 Dow-Jones industrial average index market price to estimate the parameter by EM algorithm. Then we use likelihood ratio test to obtain that regime switching model with dependent jump is the best model to depict return data. Moreover, we do sensitivity analysis and find the result that the probability of the higher volatility state , the overall volatility of rate of return , and the jump frequency are positively correlated with call option value. Finally, we enhance the empirical value of regime switching model with dependent jump by means of calculating the price error.

No description available.

Submitted by Haroudo Xavier Filho (haroudo.xavierfo@ufpe.br) on 2016-04-05T14:28:43Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) dissertacao_Raphaela(CD).pdf: 1067677 bytes, checksum: 6d371901336a7515911aeffd9ee38c74 (MD5) / Made available in DSpace on 2016-04-05T14:28:43Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) dissertacao_Raphaela(CD).pdf: 1067677 bytes, checksum: 6d371901336a7515911aeffd9ee38c74 (MD5) Previous issue date: 2015-07-31 / CAPES / Este trabalho analisa propriedades da família de distribuições de probabilidade Composta N e propõe a sub-família Composta Poisson-Truncada como um meio de compor distribuições de probabilidade. Suas propriedades foram estudadas e uma nova distribuição foi investigada: a distribuição Composta Poisson-Truncada Normal. Esta distribuição possui três parâmetros e tem uma flexibilidade para modelar dados multimodais. Demonstramos que sua densidade é dada por uma mistura infinita de densidades normais em que os pesos são dados pela função de massa de probabilidade da Poisson-Truncada. Dentre as propriedades exploradas desta distribuição estão a função característica e expressões para o cálculo dos momentos. Foram analisados três métodos de estimação para os parâmetros da distribuição Composta Poisson-Truncada Normal, sendo eles, o método dos momentos, o da função característica empírica (FCE) e o método de máxima verossimilhança (MV) via algoritmo EM. Simulações comparando estes três métodos foram realizadas e, por fim, para ilustrar o potencial da distribuição proposta, resultados numéricos com modelagem de dados reais são apresentados. / This work analyzes properties of the Compound N family of probability distributions and proposes the sub-family Compound Poisson-Truncated as a means of composing probability distributions. Its properties were studied and a new distribution was investigated: the Compound Poisson-Truncated Normal distribution. This distribution has three parameters and has the flexibility to model multimodal data. We demonstrated that its density is given by an infinite mixture of normal densities where in the weights are given by the Poisson-Truncated probability mass function. Among the explored properties of this distribution are the characteristic function end expressions for the calculation of moments. Three estimation methods were analyzed for the parameters of the Compound Poisson-Truncated Normal distribution, namely, the method of moments, the empirical characteristic function (ECF) and the method of maximum likelihood (ML) by EM algorithm. Simulations comparing these three methods were performed and, finally, to illustrate the potential of the proposed distribution numerical results with real data modeling are presented.

Submitted by Rafael Santana (rafael.silvasantana@ufpe.br) on 2017-04-26T17:07:37Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5) / Made available in DSpace on 2017-04-26T17:07:38Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Carolina Marchant.pdf: 5792192 bytes, checksum: adbd82c79b286d2fe2470b7955e6a9ed (MD5) Previous issue date: 2016-10-31 / CAPES; BOLSA DO CHILE. / In the last decades, univariate Birnbaum-Saunders models have received considerable attention in the literature. These models have been widely studied and applied to fatigue, but they have also been applied to other areas of the knowledge. In such areas, it is often necessary to model several variables simultaneously. If these variables are correlated, individual analyses for each variable can lead to erroneous results. Multivariate regression models are a useful tool of the multivariate analysis, which takes into account the correlation between variables. In addition, diagnostic analysis is an important aspect to be considered in the statistical modeling. Furthermore, multivariate quality control charts are powerful and simple visual tools to determine whether a multivariate process is in control or out of control. A multivariate control chart shows how several variables jointly affect a process. First, we propose, derive and characterize multivariate generalized logarithmic Birnbaum-Saunders distributions. Also, we propose new multivariate generalized Birnbaum-Saunders regression models. We use the method of maximum likelihood estimation to estimate their parameters through the expectation-maximization algorithm. We carry out a simulation study to evaluate the performance of the corresponding estimators based on the Monte Carlo method. We validate the proposed models with a regression analysis of real-world multivariate fatigue data. Second, we conduct a diagnostic analysis for multivariate generalized Birnbaum-Saunders regression models. We consider the Mahalanobis distance as a global influence measure to detect multivariate outliers and use it for evaluating the adequacy of the distributional assumption. Moreover, we consider the local influence method and study how a perturbation may impact on the estimation of model parameters. We implement the obtained results in the R software, which are illustrated with real-world multivariate biomaterials data. Third and finally, we develop a robust methodology based on multivariate quality control charts for generalized Birnbaum-Saunders distributions with the Hotelling statistic. We use the parametric bootstrap method to obtain the distribution of this statistic. A Monte Carlo simulation study is conducted to evaluate the proposed methodology, which reports its performance to provide earlier alerts of out-of-control conditions. An illustration with air quality real-world data of Santiago-Chile is provided. This illustration shows that the proposed methodology can be useful for alerting episodes of extreme air pollution. / Nas últimas décadas, o modelo Birnbaum-Saunders univariado recebeu considerável atenção na literatura. Esse modelo tem sido amplamente estudado e aplicado inicialmente à modelagem de fadiga de materiais. Com o passar dos anos surgiram trabalhos com aplicações em outras áreas do conhecimento. Em muitas das aplicações é necessário modelar diversas variáveis simultaneamente incorporando a correlação entre elas. Os modelos de regressão multivariados são uma ferramenta útil de análise multivariada, que leva em conta a correlação entre as variáveis de resposta. A análise de diagnóstico é um aspecto importante a ser considerado no modelo estatístico e verifica as suposições adotadas como também sua sensibilidade. Além disso, os gráficos de controle de qualidade multivariados são ferramentas visuais eficientes e simples para determinar se um processo multivariado está ou não fora de controle. Este gráfico mostra como diversas variáveis afetam conjuntamente um processo. Primeiro, propomos, derivamos e caracterizamos as distribuições Birnbaum-Saunders generalizadas logarítmicas multivariadas. Em seguida, propomos um modelo de regressão Birnbaum-Saunders generalizado multivariado. Métodos para estimação dos parâmetros do modelo, tal como o método de máxima verossimilhança baseado no algoritmo EM, foram desenvolvidos. Estudos de simulação de Monte Carlo foram realizados para avaliar o desempenho dos estimadores propostos. Segundo, realizamos uma análise de diagnóstico para modelos de regressão Birnbaum-Saunders generalizados multivariados. Consideramos a distância de Mahalanobis como medida de influência global de detecção de outliers multivariados utilizando-a para avaliar a adequacidade do modelo. Além disso, desenvolvemos medidas de diagnósticos baseadas em influência local sob alguns esquemas de perturbações. Implementamos a metodologia apresentada no software R, e ilustramos com dados reais multivariados de biomateriais. Terceiro, e finalmente, desenvolvemos uma metodologia robusta baseada em gráficos de controle de qualidade multivariados para a distribuição Birnbaum-Saunders generalizada usando a estatística de Hotelling. Baseado no método bootstrap paramétrico encontramos aproximações da distribuição desta estatística e obtivemos limites de controle para o gráfico proposto. Realizamos um estudo de simulação de Monte Carlo para avaliar a metodologia proposta indicando seu bom desempenho para fornecer alertas precoces de processos fora de controle. Uma ilustração com dados reais de qualidade do ar de Santiago-Chile é fornecida. Essa ilustração mostra que a metodologia proposta pode ser útil para alertar sobre episódios de poluição extrema do ar, evitando efeitos adversos na saúde humana.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Database with missing values it is an occurrence often found in the real world, beiging of this problem caused by several reasons (equipment failure that transmits and stores the data, handler failure, failure who provides information, etc.). This may make the data inconsistent and unable to be analyzed, leading to very skewed conclusions. This dissertation aims to explore the use of Multilayer Perceptron Artificial Neural Network (ANN MLP), with new activation functions, considering two approaches (single imputation and multiple imputation). First, we propose the use of Maximum Likelihood Estimation Method (MLE) in each network neuron activation function, against the approach currently used, which is without the use of such a method or when is used only in the cost function (network output). It is then analyzed the results of these approaches compared with the Expectation Maximization algorithm (EM) is that the state of the art to treat missing data. The results indicate that when using the Artificial Neural Network MLP with Maximum Likelihood Estimation Method, both in all neurons and only in the output function, lead the an imputation with lower error. These experimental results, evaluated by metrics such as MAE (Mean Absolute Error) and RMSE (Root Mean Square Error), showed that the better results in most experiments occured when using the MLP RNA addressed in this dissertation to single imputation and multiple. / Base de dados com valores faltantes é uma ocorrência frequentemente encontrada no mundo real, sendo as causas deste problema são originadas por motivos diversos (falha no equipamento que transmite e armazena os dados, falha do manipulador, falha de quem fornece a informação, etc.). Tal situação pode tornar os dados inconsistentes e inaptos de serem analisados, conduzindo às conclusões muito enviesadas. Esta dissertação tem como objetivo explorar o emprego de Redes Neurais Artificiais Multilayer Perceptron (RNA MLP), com novas funções de ativação, considerando duas abordagens (imputação única e imputação múltipla). Primeiramente, é proposto o uso do Método de Estimativa de Máxima Verossimilhança (EMV) na função de ativação de cada neurônio da rede, em contrapartida à abordagem utilizada atualmente, que é sem o uso de tal método, ou quando o utiliza é apenas na função de custo (na saída da rede). Em seguida, são analisados os resultados destas abordagens em comparação com o algoritmo Expectation Maximization (EM) que é o estado da arte para tratar dados faltantes. Os resultados obtidos indicam que ao utilizar a Rede Neural Artificial MLP com o Método de Estimativa de Máxima Verossimilhança, tanto em todos os neurônios como apenas na função de saída, conduzem a uma imputação com menor erro. Os resultados experimentais foram avaliados via algumas métricas, sendo as principais o MAE (Mean Absolute Error) e RMSE (Root Mean Square Error), as quais apresentaram melhores resultados na maioria dos experimentos quando se utiliza a RNA MLP abordada neste trabalho para fazer imputação única e múltipla.

Esta tesis presenta diversas contribuciones en los campos de la clasificación automática de texto, traducción automática y traducción asistida por ordenador bajo el marco estadístico. En clasificación automática de texto, se propone una nueva aplicación llamada clasificación de texto bilingüe junto con una serie de modelos orientados a capturar dicha información bilingüe. Con tal fin se presentan dos aproximaciones a esta aplicación; la primera de ellas se basa en una asunción naive que contempla la independencia entre las dos lenguas involucradas, mientras que la segunda, más sofisticada, considera la existencia de una correlación entre palabras en diferentes lenguas. La primera aproximación dió lugar al desarrollo de cinco modelos basados en modelos de unigrama y modelos de n-gramas suavizados. Estos modelos fueron evaluados en tres tareas de complejidad creciente, siendo la más compleja de estas tareas analizada desde el punto de vista de un sistema de ayuda a la indexación de documentos. La segunda aproximación se caracteriza por modelos de traducción capaces de capturar correlación entre palabras en diferentes lenguas. En nuestro caso, el modelo de traducción elegido fue el modelo M1 junto con un modelo de unigramas. Este modelo fue evaluado en dos de las tareas más simples superando la aproximación naive, que asume la independencia entre palabras en differentes lenguas procedentes de textos bilingües. En traducción automática, los modelos estadísticos de traducción basados en palabras M1, M2 y HMM son extendidos bajo el marco de la modelización mediante mixturas, con el objetivo de definir modelos de traducción dependientes del contexto. Asimismo se extiende un algoritmo iterativo de búsqueda basado en programación dinámica, originalmente diseñado para el modelo M2, para el caso de mixturas de modelos M2. Este algoritmo de búsqueda n / Civera Saiz, J. (2008). Novel statistical approaches to text classification, machine translation and computer-assisted translation [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/2502 / Palancia

Appartenir à une catégorie socio-économique moins élevée est généralement associé à une mortalité plus élevée pour de nombreuses causes de décès. De précédentes études ont déjà montré l’importance de la prise en compte des différentes dimensions des trajectoires socio-économiques au cours de la vie. L’analyse des trajectoires professionnelles constitue une étape importante pour mieux comprendre ces phénomènes. L’enjeu pour mesurer l’association entre les parcours de vie des trajectoires socio-économiques et la mortalité est de décomposer la part respective de ces facteurs dans l’explication du niveau de survie des individus. La complexité de l’interprétation de cette association réside dans la causalité bidirectionnelle qui la sous-tend: Les différentiels de mortalité sont-ils dus à des différentielsd’état de santé initial influençant conjointement la situation professionnelle et la mortalité, ou l’évolution professionnelle influence-t-elle directement l’état de santé puis la mortalité?Les méthodes usuelles ne tiennent pas compte de l’interdépendance des changements de situation professionnelle et de la bidirectionnalité de la causalité qui conduit à un biais important dans l’estimation du lien causale entre situation professionnelle et mortalité. Par conséquent, il est nécessaire de proposer des méthodes statistiques qui prennent en compte des mesures répétées (les professions) simultanément avec les variables de survie. Cette étude est motivée par la base de données Cosmop-DADS qui est un échantillon de la population salariée française.Le premier objectif de cette thèse était d’examiner l’ensemble des trajectoires professionnelles avec une classification professionnelle précise, au lieu d’utiliser un nombre limité d’états dans un parcours professionnel qui a été considéré précédemment. A cet effet, nous avons défini des variables dépendantes du temps afinde prendre en compte différentes dimensions des trajectoires professionnelles, à travers des modèles dits de "life-course", à savoir critical period, accumulation model et social mobility model, et nous avons mis en évidence l’association entre les trajectoires professionnelles et la mortalité par cause en utilisant ces variables dans un modèle de Cox.Le deuxième objectif a consisté à intégrer les épisodes professionnel comme un sous-modèle longitudinal dans le cadre des modèles conjoints pour réduire le biais issude l’inclusion des covariables dépendantes du temps endogènes dans le modèle de Cox. Nous avons proposé un modèle conjoint pour les données longitudinales nominaleset des données de risques concurrents dans une approche basée sur la vraisemblance. En outre, nous avons proposé une approche de type méta-analyse pour résoudre les problèmes liés au temps des calculs dans les modèles conjoints appliqués à l’analyse des grandes bases de données. Cette approche consiste à combiner les résultats issus d’analyses effectuées sur les échantillons stratifiés indépendants. Dans la même perspective de l’utilisation du modèle conjoint sur les grandes bases de données, nous avons proposé une procédure basée sur l’avantage computationnel de la régression de Poisson.Cette approche consiste à trouver les trajectoires typesà travers les méthodes de la classification, et d’appliquerle modèle conjoint sur ces trajectoires types. / Being in low socioeconomic position is associated with increased mortality risk from various causes of death. Previous studies have already shown the importance of considering different dimensions of socioeconomic trajectories across the life-course. Analyses of professional trajectories constitute a crucial step in order to better understand the association between socio-economic position and mortality. The main challenge in measuring this association is then to decompose the respectiveshare of these factors in explaining the survival level of individuals. The complexity lies in the bidirectional causality underlying the observed associations:Are mortality differentials due to differences in the initial health conditions that are jointly influencing employment status and mortality, or the professional trajectory influences directly health conditions and then mortality?Standard methods do not consider the interdependence of changes in occupational status and the bidirectional causal effect underlying the observed association and that leads to substantial bias in estimating the causal link between professional trajectory and mortality. Therefore, it is necessary to propose statistical methods that consider simultaneously repeated measurements (careers) and survivalvariables. This study was motivated by the Cosmop-DADS database, which is a sample of the French salaried population.The first aim of this dissertation was to consider the whole professional trajectories and an accurate occupational classification, instead of using limitednumber of stages during life course and a simple occupational classification that has been considered previously. For this purpose, we defined time-dependent variables to capture different life course dimensions, namely critical period, accumulation model and social mobility model, and we highlighted the association between professional trajectories and cause-specific mortality using the definedvariables in a Cox proportional hazards model.The second aim was to incorporate the employment episodes in a longitudinal sub-model within the joint model framework to reduce the bias resulting from the inclusion of internal time-dependent covariates in the Cox model. We proposed a joint model for longitudinal nominal outcomes and competing risks data in a likelihood-based approach. In addition, we proposed an approach mimicking meta-analysis to address the calculation problems in joint models and large datasets, by extracting independent stratified samples from the large dataset, applying the joint model on each sample and then combining the results. In the same objective, that is fitting joint model on large-scale data, we propose a procedure based on the appeal of the Poisson regression model. This approach consist of finding representativetrajectories by means of clustering methods and then applying the joint model on these representative trajectories.

Avec l'essor des objets connectés pour fournir un suivi systématique, objectif et fiable aux sportifs et à leur entraineur, de plus en plus de paramètres sont collectés pour un même individu. Une alternative aux méthodes d'évaluation en laboratoire est l'utilisation de capteurs inertiels qui permettent de suivre la performance sans l'entraver, sans limite d'espace et sans procédure d'initialisation fastidieuse. Les données collectées par ces capteurs peuvent être vues comme des données fonctionnelles multivariées : se sont des entités quantitatives évoluant au cours du temps de façon simultanée pour un même individu statistique. Cette thèse a pour objectif de chercher des paramètres d'analyse de la locomotion du cheval athlète à l'aide d'un capteur positionné dans la selle. Cet objet connecté (centrale inertielle, IMU) pour le secteur équestre permet de collecter l'accélération et la vitesse angulaire au cours du temps, dans les trois directions de l'espace et selon une fréquence d'échantillonnage de 100 Hz. Une base de données a ainsi été constituée rassemblant 3221 foulées de galop, collectées en ligne droite et en courbe et issues de 58 chevaux de sauts d'obstacles de niveaux et d'âges variés. Nous avons restreint notre travail à la prédiction de trois paramètres : la vitesse par foulée, la longueur de foulée et la qualité de saut. Pour répondre aux deux premiers objectifs nous avons développé une méthode de clustering fonctionnelle multivariée permettant de diviser notre base de données en sous-groupes plus homogènes du point de vue des signaux collectés. Cette méthode permet de caractériser chaque groupe par son profil moyen, facilitant leur compréhension et leur interprétation. Mais, contre toute attente, ce modèle de clustering n'a pas permis d'améliorer les résultats de prédiction de vitesse, les SVM restant le modèle ayant le pourcentage d'erreur inférieur à 0.6 m/s le plus faible. Il en est de même pour la longueur de foulée où une précision de 20 cm est atteinte grâce aux Support Vector Machine (SVM). Ces résultats peuvent s'expliquer par le fait que notre base de données est composée uniquement de 58 chevaux, ce qui est un nombre d'individus très faible pour du clustering. Nous avons ensuite étendu cette méthode au co-clustering de courbes fonctionnelles multivariées afin de faciliter la fouille des données collectées pour un même cheval au cours du temps. Cette méthode pourrait permettre de détecter et prévenir d'éventuels troubles locomoteurs, principale source d'arrêt du cheval de saut d'obstacle. Pour finir, nous avons investigué les liens entre qualité du saut et les signaux collectés par l'IMU. Nos premiers résultats montrent que les signaux collectés par la selle seuls ne suffisent pas à différencier finement la qualité du saut d'obstacle. Un apport d'information supplémentaire sera nécessaire, à l'aide d'autres capteurs complémentaires par exemple ou encore en étoffant la base de données de façon à avoir un panel de chevaux et de profils de sauts plus variés / With the growth of smart devices market to provide athletes and trainers a systematic, objective and reliable follow-up, more and more parameters are monitored for a same individual. An alternative to laboratory evaluation methods is the use of inertial sensors which allow following the performance without hindering it, without space limits and without tedious initialization procedures. Data collected by those sensors can be classified as multivariate functional data: some quantitative entities evolving along time and collected simultaneously for a same individual. The aim of this thesis is to find parameters for analysing the athlete horse locomotion thanks to a sensor put in the saddle. This connected device (inertial sensor, IMU) for equestrian sports allows the collection of acceleration and angular velocity along time in the three space directions and with a sampling frequency of 100 Hz. The database used for model development is made of 3221 canter strides from 58 ridden jumping horses of different age and level of competition. Two different protocols are used to collect data: one for straight path and one for curved path. We restricted our work to the prediction of three parameters: the speed per stride, the stride length and the jump quality. To meet the first to objectives, we developed a multivariate functional clustering method that allow the division of the database into smaller more homogeneous sub-groups from the collected signals point of view. This method allows the characterization of each group by it average profile, which ease the data understanding and interpretation. But surprisingly, this clustering model did not improve the results of speed prediction, Support Vector Machine (SVM) is the model with the lowest percentage of error above 0.6 m/s. The same applied for the stride length where an accuracy of 20 cm is reached thanks to SVM model. Those results can be explained by the fact that our database is build from 58 horses only, which is a quite low number of individuals for a clustering method. Then we extend this method to the co-clustering of multivariate functional data in order to ease the datamining of horses’ follow-up databases. This method might allow the detection and prevention of locomotor disturbances, main source of interruption of jumping horses. Lastly, we looked for correlation between jumping quality and signals collected by the IMU. First results show that signals collected by the saddle alone are not sufficient to differentiate finely the jumping quality. Additional information will be needed, for example using complementary sensors or by expanding the database to have a more diverse range of horses and jump profiles

Indiana University-Purdue University Indianapolis (IUPUI) / Joint models of longitudinal and survival outcomes have been used with increasing frequency in clinical investigations. Correct specification of fixed and random effects, as well as their functional forms is essential for practical data analysis. However, no existing methods have been developed to meet this need in a joint model setting. In this dissertation, I describe a penalized likelihood-based method with adaptive least absolute shrinkage and selection operator (ALASSO) penalty functions for model selection. By reparameterizing variance components through a Cholesky decomposition, I introduce a penalty function of group shrinkage; the penalized likelihood is approximated by Gaussian quadrature and optimized by an EM algorithm. The functional forms of the independent effects are determined through a procedure for structural discovery. Specifically, I first construct the model by penalized cubic B-spline and then decompose the B-spline to linear and nonlinear elements by spectral decomposition. The decomposition represents the model in a mixed-effects model format, and I then use the mixed-effects variable selection method to perform structural discovery. Simulation studies show excellent performance. A clinical application is described to illustrate the use of the proposed methods, and the analytical results demonstrate the usefulness of the methods.

Different road users follow different behaviors and intentions in the trajectories that they traverse. Predicting the intent of these road users at intersections would not only help increase the comfort of drive in autonomous vehicles, but also help detect potential accidents. In this thesis, the research objective is to build models that predicts future positions of road users (pedestrians,cyclists and autonomous shuttles) by capturing behaviors endemic to different road users.  Firstly, a constant velocity state space model is used as a benchmark for intent prediction, with a fresh approach to estimate parameters from the data through the EM algorithm. Then, a neural network based LSTM sequence modeling architecture is used to better capture the dynamics of road user movement and their dependence on the spatial area. Inspired by the recent success of transformers and attention in text mining, we then propose a mechanism to capture the road users' social behavior amongst their neighbors. To achieve this, past trajectories of different road users are forward propagated through the LSTM network to obtain representative feature vectors for each road users' behaviour. These feature vectors are then passed through an attention-layer to obtain representations that incorporate information from other road users' feature vectors, which are in-turn used to predict future positions for every road user in the frame. It is seen that the attention based LSTM model slightly outperforms the plain LSTM models, while both substantially outperform the constant velocity model. A comparative qualitative analysis is performed to assess the behaviors that are captured/missed by the different models. The thesis concludes with a dissection of the behaviors captured by the attention module.