Modèle mixte

Résumé

Un modèle mixte est un modèle statistique qui comporte à la fois des effets fixes et des effets aléatoires. Ce type de modèle est utile dans une grande variété de domaines, tels que la physique, la biologie ou encore les sciences sociales. Les modèles mixtes sont particulièrement utiles dans les situations où des mesures répétées sont effectuées sur les mêmes variables (étude longitudinale). Ils sont souvent préférés à d'autres approches telle que rANOVA, dans la mesure où ils peuvent être utilisés dans le cas où le jeu de données présente des valeurs manquantes. Historique La notion de modèle à effets aléatoires fut introduite par Ronald Fisher, dans son étude des corrélations entre traits de caractère entre parents. Définition En notation matricielle,un modèle mixte peut être représenté comme suit : : \boldsymbol{y} = X \boldsymbol{\beta} + Z \boldsymbol{u} + \boldsymbol{\epsilon} avec

\boldsymbol{y} un vecteur connu d'observatio

Source officielle

https://fr.wikipedia.org/wiki/Mod%C3%A8le_mixte

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Time series modeling and analysis is central to most financial and econometric data modeling. With increased globalization in trade, commerce and finance, national variables like gross domestic productivity (GDP) and unemployment rate, market variables like indices and stock prices and global variables like commodity prices are more tightly coupled than ever before. This translates to the use of multivariate or vector time series models and algorithms in analyzing and understanding the relationships that these variables share with each other. Autocorrelation is one of the fundamental aspects of time series modeling. However, traditional linear models, that arise from a strong observed autocorrelation in many financial and econometric time series data, are at times unable to capture the rather nonlinear relationship that characterizes many time series data. This necessitates the study of nonlinear models in analyzing such time series. The class of bilinear models is one of the simplest nonlinear models. These models are able to capture temporary erratic fluctuations that are common in many financial returns series and thus, are of tremendous interest in financial time series analysis. Another aspect of time series analysis is homoscedasticity versus heteroscedasticity. Many time series data, even after differencing, exhibit heteroscedasticity. Thus, it becomes important to incorporate this feature in the associated models. The class of conditional heteroscedastic autoregressive (ARCH) models and its variants form the primary backbone of conditional heteroscedastic time series models. Robustness is a highly underrated feature of most time series applications and models that are presently in use in the industry. With an ever increasing amount of information available for modeling, it is not uncommon for the data to have some aberrations within itself in terms of level shifts and the occasional large fluctuations. Conventional methods like the maximum likelihood and least squares are well known to be highly sensitive to such contaminations. Hence, it becomes important to use robust methods, especially in this age with high amounts of computing power readily available, to take into account such aberrations. While robustness and time series modeling have been vastly researched individually in the past, application of robust methods to estimate time series models is still quite open. The central goal of this thesis is the study of robust parameter estimation of some simple vector and nonlinear time series models. More precisely, we will briefly study some prominent linear and nonlinear models in the time series literature and apply the robust S-estimator in estimating parameters of some simple models like the vector autoregressive (VAR) model, the (0, 0, 1, 1) bilinear model and a simple conditional heteroscedastic bilinear model. In each case, we will look at the important aspect of stationarity of the model and analyze the asymptotic behavior of the S-estimator.

EPFL2010

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Biological oscillators are pervasive in biology, covering all aspects of life from enzyme kinetics reactions to population dynamics. Although their behaviour has been intensively studied in the last decades, the recent advances of high-throughput experimental technologies in the fields of omics and microscopy has called for the development of new analysis methods. Among the many types of models and quantitative analyses, parametric approaches are promising as they enable for a mechanistic or physical explanation of the phenomena under study. In particular, dynamical systems theory seems particularly adapted as the vast majority of oscillators can be modelled through differential equations. Dynamical systems parameters can also be easily opti-mized via maximum likelihood approaches. The validity of the inferred model can then be assessed from the quality of its predictions. We here present three different scientific questions regarding noisy biological oscillators, which are answered using maximum-likelihood inference approaches applied to parametric models. We first take interest in the characterisation of the influence of the cell-cycle over the circadi-an clock in individual mammalian cells. To this end, we develop a method combining a Hidden Markov Model with an Expectation-Maximization algorithm to infer their coupling from single-cell microscopy traces. We show that this coupling predicts multiple phase-locked states exhib-iting different degrees of robustness against molecular fluctuations inherent to cellular scale bio-logical oscillators. We then try to understand how the mammalian transcriptome behaves in the liver. Thence, we use single-cell RNA sequencing (scRNA-seq) along with mixed-models to investigate the interplay between gene regulation in space and time. Categorising mRNA expression profiles using mixed-effect models and smFISH validations, we find that many genes in the liver are both zonated and rhythmic, most of them showing multiplicative space-time effects. Finally, we look more closely at the cell-cycle, as it is one of the main drivers of gene expres-sion cell-to-cell heterogeneity in otherwise homogeneous cell populations. Here, we would like to understand if and how cell-cycle velocity changes depending on the phase of the cycling cells. To that end, we formulate the problem in terms of an autonomous dynamical system and use this to infer consistent dynamics for the cell-cycle from scRNA-seq data. Phase inference being paramount in all of these three studies, a short technical review on the topic is also provided at the end of this thesis, along with Julia scripts for the main inference methods presented. Various computational tools assisting the understanding of the scientific questions at stake are also presented, including a Python Dash web-app, a D3 widget and many Matplotlib animations and widgets.

EPFL2020

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A dimension reduction technique for estimation in linear mixed models

This paper proposes a dimension reduction technique for estimation in linear mixed models. Specifically, we show that in a linear mixed model, the maximum-likelihood (ML) problem can be rewritten as a substantially simpler optimization problem which presents at least two main advantages: the number of variables in the simplified problem is lower and the search domain of the simplified problem is a compact set. Whereas the former advantage reduces the computational burden, the latter permits the use of stochastic optimization methods well qualified for closed bounded domains. The developed dimension reduction technique makes the computation of ML estimates, for fixed effects and variance components, feasible with large computational savings. Computational experience is reported here with the results evidencing an overall good performance of the proposed technique.

Taylor & Francis2012

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EPFL2020

EPFL2010