Pete Riegel, a well-known sport expert, proposed the formula of race time prediction in 1977. This article discusses whether it is also suitable for Taiwan marathoners. We compiled two hundred and four effective datum by questionary. Some variables possible to affect the running result are added in this work, namely: sex, age, the year of run, height, weight, the race number of marathon, the quantity and the frequency of practices each week. Next, we use multiple regression and sliced inverse regression to increase the accuracy of the running time prediction. The best model, found here has eighty percentage's player with predictive error within fifteen minuates, which is better than the original model by Riegel(1977) with only having sixty-two percentages.

Variable selection methods play important roles in modeling high-dimensional data and are key to data-driven scientific discoveries. In this thesis, we consider the problem of variable selection with interaction detection. Instead of building a predictive model of the response given combinations of predictors, we start by modeling the conditional distribution of predictors given partitions based on responses. We use this inverse modeling perspective as motivation to propose a stepwise procedure for effectively detecting interaction with few assumptions on parametric form. The proposed procedure is able to detect pairwise interactions among p predictors with a computational time of \(O(p)\) instead of \(O(p^2)\) under moderate conditions. We establish consistency of the proposed procedure in variable selection under a diverging number of predictors and sample size. We demonstrate its excellent empirical performance in comparison with some existing methods through simulation studies as well as real data examples. Next, we combine the forward and inverse modeling perspectives under the Bayesian framework to detect pleiotropic and epistatic effects in effects in expression quantitative loci (eQTLs) studies. We augment the Bayesian partition model proposed by Zhang et al. (2010) to capture complex dependence structure among gene expression and genetic markers. In particular, we propose a sequential partition prior to model the asymmetric roles played by the response and the predictors, and we develop an efficient dynamic programming algorithm for sampling latent individual partitions. The augmented partition model significantly improves the power in detecting eQTLs compared to previous methods in both simulations and real data examples pertaining to yeast. Finally, we study the application of Bayesian partition models in the unsupervised learning of transcription factor (TF) families based on protein binding microarray (PBM). The problem of TF subclass identification can be viewed as the clustering of TFs with variable selection on their binding DNA sequences. Our model provides simultaneous identification of TF families and their shared sequence preferences, as well as DNA sequences bound preferentially by individual members of TF families. Our analysis may aid in deciphering cis regulatory codes and determinants of protein-DNA binding specificity. / Statistics

Cette thèse propose trois extensions de la Régression linéaire par tranches (Sliced Inverse Regression, SIR), notamment Collaborative SIR, Student SIR et Knockoff SIR.Une des faiblesses de la méthode SIR est l’impossibilité de vérifier si la Linearity Design Condition (LDC) est respectée. Il est établi que, si x suit une distribution elliptique, la condition est vraie ; dans le cas d’une composition de distributions elliptiques il n y a aucune garantie que la condition soit vérifiée globalement, pourtant, elle est respectée localement.On va donc proposer une extension sur la base de cette considération. Étant donné une variable explicative x, Collaborative SIR réalise d’abord un clustering. Pour chaque cluster, la méthode SIR est appliquée de manière indépendante.Le résultat de chaque composant contribue à créer la solution finale.Le deuxième papier, Student SIR, dérive de la nécessité de robustifier la méthode SIR.Vu que cette dernière repose sur l’estimation de la covariance et contient une étape APC, alors elle est sensible au bruit.Afin d’étendre la méthode SIR on a utilisé une stratégie fondée sur une formulation inverse du SIR, proposée par R.D. Cook.Finalement, Knockoff SIR est une extension de la méthode SIR pour la sélection des variables et la recherche d’une solution sparse, ayant son fondement dans le papier publié par R.F. Barber et E.J. Candès qui met l’accent sur le false discovery rate dans le cadre de la régression. L’idée sous-jacente à notre papier est de créer des copies de variables d’origine ayant certaines proprietés.On va montrer que la méthode SIR est robuste par rapport aux copies et on va proposer une stratégie pour utiliser les résultats dans la sélection des variables et pour générer des solutions sparse / This thesis proposes three extensions of Sliced Inverse Regression namely: Collaborative SIR, Student SIR and Knockoff SIR.One of the weak points of SIR is the impossibility to check if the Linearity Design Condition (LDC) holds. It is known that if X follows an elliptic distribution thecondition holds true, in case of a mixture of elliptic distributions there are no guaranties that the condition is satisfied globally, but locally holds. Starting from this consideration an extension is proposed. Given the predictor variable X, Collaborative SIR performs initially a clustering. In each cluster, SIR is applied independently. The result from each component collaborates to give the final solution.Our second contribution, Student SIR, comes from the need to robustify SIR. Since SIR is based on the estimation of the covariance, and contains a PCA step, it is indeed sensitive to noise. To extend SIR, an approach based on a inverse formulation of SIR proposed by R.D. Cook has been used.Finally Knockoff SIR is an extension of SIR to perform variable selection and give sparse solution that has its foundations in a recently published paper by R. F. Barber and E. J. Candès that focuses on the false discovery rate in the regression framework. The underlying idea of this paper is to construct copies of the original variables that have some properties. It is shown that SIR is robust to this copies and a strategy is proposed to use this result for variable selection and to generate sparse solutions.

In sufficient dimension reduction (Li, 1991; Cook, 1998b), original predictors are replaced by their low-dimensional linear combinations while preserving all of the conditional information of the response given the predictors. Sliced inverse regression [SIR; Li, 1991] and principal Hessian directions [PHD; Li, 1992] are two popular sufficient dimension reduction methods, and both SIR and PHD estimators involve all of the original predictor variables. To deal with the cases when the linear combinations involve only a subset of the original predictors, we propose a Bayesian model averaging (Raftery et al., 1997) approach to achieve sparse sufficient dimension reduction. We extend both SIR and PHD under the Bayesian framework. The superior performance of the proposed methods is demonstrated through extensive numerical studies as well as a real data analysis. / Statistics

Existing sufficient dimension reduction (SDR) methods typically consider cases with no missing data. The dissertation aims to propose methods to facilitate the SDR methods when the response can be missing. The first part of the dissertation focuses on the seminal sliced inverse regression (SIR) approach proposed by Li (1991). We show that missing responses generally affect the validity of the inverse regressions under the mechanism of missing at random. We then propose a simple and effective adjustment with inverse probability weighting that guarantees the validity of SIR. Furthermore, a marginal coordinate test is introduced for this adjusted estimator. The proposed method share the simplicity of SIR and requires the linear conditional mean assumption. The second part of the dissertation proposes two new estimating equation procedures: the complete case estimating equation approach and the inverse probability weighted estimating equation approach. The two approaches are applied to a family of dimension reduction methods, which includes ordinary least squares, principal Hessian directions, and SIR. By solving the estimating equations, the two approaches are able to avoid the common assumptions in the SDR literature, the linear conditional mean assumption, and the constant conditional variance assumption. For all the aforementioned methods, the asymptotic properties are established, and their superb finite sample performances are demonstrated through extensive numerical studies as well as a real data analysis. In addition, existing estimators of the central mean space have uneven performances across different types of link functions. To address this limitation, a new hybrid SDR estimator is proposed that successfully recovers the central mean space for a wide range of link functions. Based on the new hybrid estimator, we further study the order determination procedure and the marginal coordinate test. The superior performance of the hybrid estimator over existing methods is demonstrated in simulation studies. Note that the proposed procedures dealing with the missing response at random can be simply adapted to this hybrid method. / Statistics

There are two main parts in this thesis. The first one concerns valvometry, which is here the study of the distance between both parts of the shell of an oyster, over time. The health status of oysters can be characterized using valvometry in order to obtain insights about the quality of their environment. We consider that a renewal process with four states underlies the behaviour of the studied oysters. Such a hidden process can be retrieved from a valvometric signal by assuming that some probability density function linked with this signal, is bimodal. We then compare several estimators which take this assumption into account, including kernel density estimators.In another chapter, we compare several regression approaches, aiming at analysing transcriptomic data. To understand which explanatory variables have an effect on gene expressions, we apply a multiple testing procedure on these data, through the linear model FAMT. The SIR method may find nonlinear relations in such a context. It is however more commonly used when the response variable is univariate. A multivariate version of SIR was then developed. Procedures to measure gene expressions can be expensive. The sample size n of the corresponding datasets is then often small. That is why we also studied SIR when n is less than the number of explanatory variables p.

本文以維度縮減(dimension reduction)為主題，介紹其定義以及四種目前較被廣為討論的處理方式。文中首先針對Li (1991)所使用的維度縮減定義型式y = g(x,ε) = g1(βx,ε)，與Cook (1994)所採用的定義型式「條件密度函數f(y | x)=f(y |βx)」作探討，並就Cook (1994)對最小維度縮減子空間的相關討論作介紹。此外文中也試圖提出另一種適用於pHd的可能定義(E(y | x)=E(y |βx)，亦即縮減前後y的條件期望值不變)，並發現在此一新定義下所衍生而成的子空間會包含於Cook (1994)所定義的子空間。 有關現有四種維度縮減方法(SIR、SAVE、SIR-II、pHd)的理論架構，則重新予以說明並作必要的補充證明，並以兩個機率模式(y = bx +ε及y = |z| +ε)為例，分別測試四種方法能否縮減出正確的方向。文中同時也分別找出對應於這四種方法的等價條件，並利用這些等價條件相互比較，得到彼此間的關係。我們發現當解釋變數x為多維常態情形下，四種方法理論上都不會保留可以被縮減的方向，而該保留住的方向卻不一定能夠被保留住，但是使用SAVE所可以保留住的方向會比單獨使用其他三者之一來的多(或至少一樣多)，而如果SIR與SIR-II同時使用則恰好等同於使用SAVE。另外使用pHd似乎時並不需要「E(y│x)二次可微分」這個先決條件。 / The focus of the study is on the dimension reduction and the over-view of the four methods frequently cited in the literature, i.e. SIR, SAVE, SIR-II, and pHd. The definitions of dimension reduction proposed by Li (1991)(y = g( x,ε) = g1(βx,ε)), and by Cook (1994)(f(y | x)=f(y|βx)) are briefly reviewed. Issues on minimum dimension reduction subspace (Cook (1994)) are also discussed. In addition, we propose a possible definition (E(y | x)=E(y |βx)), i.e. the conditional expectation of y remains the same both in the original subspace and the reduced subspace), which seems more appropriate when pHd is concerned. We also found that the subspace induced by this definition would be contained in the subspace generated based on Cook (1994). We then take a closer look at basic ideas behind the four methods, and supplement some more explanations and proofs, if necessary. Equivalent conditions related to the four methods that can be used to locate "right" directions are presented. Two models (y = bx +ε and y = |z| +ε) are used to demonstrate the methods and to see how good they can be. In order to further understand the possible relationships among the four methods, some comparisons are made. We learn that when x is normally distributed, directions that are redundant will not be preserved by any of the four methods. Directions that contribute significantly, however, may be mistakenly removed. Overall, SAVE has the best performance in terms of saving the "right" directions, and applying SIR along with SIR-II performs just as well. We also found that the prerequisite, 「E(y | x) is twice differentiable」, does not seem to be necessary when pHd is applied.

Deux parties principales composent cette thèse. La première d'entre elles est consacrée à la valvométrie, c'est-à-dire ici l'étude de la distance entre les deux parties de la coquille d'une huître au cours du temps. La valvométrie est utilisée afin de déterminer si de tels animaux sont en bonne santé, pour éventuellement tirer des conclusions sur la qualité de leur environnement. Nous considérons qu'un processus de renouvellement à quatre états sous-tend le comportement des huîtres étudiées. Afin de retrouver ce processus caché dans le signal valvométrique, nous supposons qu'une densité de probabilité reliée à ce signal est bimodale. Nous comparons donc plusieurs estimateurs qui prennent en compte ce type d'hypothèse, dont des estimateurs à noyau.Dans un second temps, nous comparons plusieurs méthodes de régression, dans le but d'analyser des données transcriptomiques. Pour comprendre quelles variables explicatives influent sur l'expression de gènes, nous avons réalisé des tests multiples grâce au modèle linéaire FAMT. La méthode SIR peut être envisagée pour trouver des relations non-linéaires. Toutefois, elle est principalement employée lorsque la variable à expliquer est univariée. Une version multivariée de cette approche a donc été développée. Le coût d'acquisition des données transcriptomiques pouvant être élevé, la taille n des échantillons correspondants est souvent faible. C'est pourquoi, nous avons également étudié la méthode SIR lorsque n est inférieur au nombre de variables explicatives p. / There are two main parts in this thesis. The first one concerns valvometry, which is here the study of the distance between both parts of the shell of an oyster, over time. The health status of oysters can be characterized using valvometry in order to obtain insights about the quality of their environment. We consider that a renewal process with four states underlies the behaviour of the studied oysters. Such a hidden process can be retrieved from a valvometric signal by assuming that some probability density function linked with this signal, is bimodal. We then compare several estimators which take this assumption into account, including kernel density estimators.In another chapter, we compare several regression approaches, aiming at analysing transcriptomic data. To understand which explanatory variables have an effect on gene expressions, we apply a multiple testing procedure on these data, through the linear model FAMT. The SIR method may find nonlinear relations in such a context. It is however more commonly used when the response variable is univariate. A multivariate version of SIR was then developed. Procedures to measure gene expressions can be expensive. The sample size n of the corresponding datasets is then often small. That is why we also studied SIR when n is less than the number of explanatory variables p.

Cette thèse est consacrée au problème de la réduction de dimension. Cette thématique centrale en Statistique vise à rechercher des sous-espaces de faibles dimensions tout en minimisant la perte d'information contenue dans les données. Tout d'abord, nous nous intéressons à des méthodes de statistique multidimensionnelle dans le cas de variables qualitatives. Nous abordons la question de la rotation en Analyse des Correspondances Multiples (ACM). Nous définissons l'expression analytique de l'angle de rotation planaire optimal pour le critère de rotation choisi. Lorsque le nombre de composantes principales retenues est supérieur à deux, nous utilisons un algorithme de rotations planaires successives de paires de facteurs. Nous proposons également différents algorithmes de classification de variables qualitatives qui visent à optimiser un critère de partitionnement basé sur la notion de rapports de corrélation. Un jeu de données réelles illustre les intérêts pratiques de la rotation en ACM et permet de comparer empiriquement les différents algorithmes de classification de variables qualitatives proposés. Puis nous considérons un modèle de régression semiparamétrique, plus précisément nous nous intéressons à la méthode de régression inverse par tranchage (SIR pour Sliced Inverse Regression). Nous développons une approche basée sur un partitionnement de l'espace des covariables, qui est utilisable lorsque la condition fondamentale de linéarité de la variable explicative est violée. Une seconde adaptation, utilisant le bootstrap, est proposée afin d'améliorer l'estimation de la base du sous-espace de réduction de dimension. Des résultats asymptotiques sont donnés et une étude sur des données simulées démontre la supériorité des approches proposées. Enfin les différentes applications et collaborations interdisciplinaires réalisées durant la thèse sont décrites. / This thesis concentrates on dimension reduction approaches, that seek for lower dimensional subspaces minimizing the lost of statistical information. First we focus on multivariate analysis for categorical data. The rotation problem in Multiple Correspondence Analysis (MCA) is treated. We give the analytic expression of the optimal angle of planar rotation for the chosen criterion. If more than two principal components are to be retained, this planar solution is used in a practical algorithm applying successive pairwise planar rotations. Different algorithms for the clustering of categorical variables are also proposed to maximize a given partitioning criterion based on correlation ratios. A real data application highlights the benefits of using rotation in MCA and provides an empirical comparison of the proposed algorithms for categorical variable clustering. Then we study the semiparametric regression method SIR (Sliced Inverse Regression). We propose an extension based on the partitioning of the predictor space that can be used when the crucial linearity condition of the predictor is not verified. We also introduce bagging versions of SIR to improve the estimation of the basis of the dimension reduction subspace. Asymptotic properties of the estimators are obtained and a simulation study shows the good numerical behaviour of the proposed methods. Finally applied multivariate data analysis on various areas is described.