Lasso was considered the greatest composer of his generation with undisputable mastery of all sixteenth-century genres. The dynamism of the late sixteenth century saw the most sophisticated compositions in the continuation of the psalm motet tradition. However, more flexible applications for the psalms in the form of meditations, vernacular translations, and paraphrases opened the door for new and diverse interpretations. This dissertation is a study of Lasso's engagement with established musical traditions and new trends in psalmody. This study unfolds in two parts. First in the discussion of the Latin psalm motet genre and then ensuing with investigation of the German Lied. In each of the genres (1) there is a focus on the definition and classification of terms and older traditions, (2) the examination of the text and the discussion of ways in which the music engages with the prose and poetic forms, and (3) the evaluation of Lasso's interpretation of psalm texts. From negligible German Lieder to expansive motet cycles, the psalms afforded endless polyphonic inspiration and the diversity of which categorically points to the shifts and development of cultural and aesthetic traditions. The use of psalms to reflect devotion and confession amplifies the Catholic Reform implemented at the Bavarian court during Lasso's lifetime. This context along with Lasso's compositional innovation provides an interesting study for the stylistic development of psalm settings in the late sixteenth century.

本研究所利用的資料是來自美國東維吉尼亞醫學院所提供的攝護腺癌蛋白質質譜資料，其資料有原始資料和另一筆經過事前處理過的資料，而本研究是利用事前處理過的資料來作實証分析。由於此種資料通常都是屬於高維度資料，故變數間具有高度相關的現象也很常見，因此從大量的特徵變數中選取到重要的特徵變數來準確的判斷攝護腺的病變程度成為一個非常普遍且重要的課題。那麼本研究的目的是欲探討各(具有懲罰項)迴歸模型對於分類蛋白質質譜資料之變數選取結果，藉由LARS、Stagewise、LASSO、Group LASSO和Elastic Net各(具有懲罰項)迴歸模型將變數選入的先後順序當作其排序所產生的判別結果與利用「統計量排序」(t檢定、ANOVA F檢定以及Kruskal-Wallis檢定)以及SVM「分錯率排序」的判別結果相比較。而分析的結果顯示，Group LASSO對於六種兩兩分類的分錯率，其分錯率趨勢的表現都較其他方法穩定，並不會有大起大落的現象發生，且最小分錯率也幾乎較其他方法理想。此外Group LASSO在四分類的判別結果在與其他方法相較下也顯出此法可得出最低的分錯率，亦表示若須同時判別四種類別時，相較於其他方法之下Group LASSO的判別準確度最優。 / Our research uses the prostate proteomic spectra data which is offered by Eastern Virginia Medical School. The materials have raw data and preprocessed data. Our research uses the preprocessed data to do the analysis of real example. Because this kind of materials usually have high dimension, so it maybe has highly correlation between variables very common, therefore choose from a large number of characteristic variables to accurately determine the pathological change degree of the Prostate is become a very general and important subject. Then the purpose of our research wants to discuss every (penalized) regression model in variable selection results for classifying the proteomic spectra data. With LARS, Stagewise, LASSO, Group LASSO and Elastic Net, each variable is chosen successively by each (penalized) regression model, and it is regarded as each variable’s order then produce discrimination results. After that, we use their results to compare with using statistic order (t-test, ANOVA F-test and Kruskal-Wallis test) and SVM fault rate order. And the result of analyzing reveals Group LASSO to two by two of six kinds of rate by mistake that classify, the mistake rate behavior of trend is more stable than other ways, it doesn’t appear big rise or big fall phenomenon. Furthermore, this way’s mistake rate is almostly more ideal than other ways. Moreover, using Group LASSO to get the discrimination result of four classifications has the lowest mistake rate under comparing with other methods. In other words, when must distinguish four classifications in the same time, Group LASSO’s discrimination accuracy is optimum.

Variable Selection has always been a very important problem in statistics. We often meet situations where a huge data set is given and we want to find out the relationship between the response and the corresponding variables. With a huge number of variables, we often end up with a big model even if we delete those that are insignificant. There are two reasons why we are unsatisfied with a final model with too many variables. The first reason is the prediction accuracy. Though the prediction bias might be small under a big model, the variance is usually very high. The second reason is interpretation. With a large number of variables in the model, it's hard to determine a clear relationship and explain the effects of variables we are interested in. A lot of variable selection methods have been proposed. However, one disadvantage of variable selection is that different sizes of model require different tuning parameters in the analysis, which is hard to choose for non-statisticians. Xin and Zhu advocate variable ranking instead of variable selection. Once variables are ranked properly, we can make the selection by adopting a threshold rule. In this thesis, we try to rank the variables using Least Angle Regression (LARS). Some shrinkage methods like Lasso and LARS can shrink the coefficients to zero. The advantage of this kind of methods is that they can give a solution path which describes the order that variables enter the model. This provides an intuitive way to rank variables based on the path. However, Lasso can sometimes be difficult to apply to variable ranking directly. This is because that in a Lasso solution path, variables might enter the model and then get dropped. This dropping issue makes it hard to rank based on the order of entrance. However, LARS, which is a modified version of Lasso, doesn't have this problem. We'll make use of this property and rank variables using LARS solution path.

Variable Selection has always been a very important problem in statistics. We often meet situations where a huge data set is given and we want to find out the relationship between the response and the corresponding variables. With a huge number of variables, we often end up with a big model even if we delete those that are insignificant. There are two reasons why we are unsatisfied with a final model with too many variables. The first reason is the prediction accuracy. Though the prediction bias might be small under a big model, the variance is usually very high. The second reason is interpretation. With a large number of variables in the model, it's hard to determine a clear relationship and explain the effects of variables we are interested in. A lot of variable selection methods have been proposed. However, one disadvantage of variable selection is that different sizes of model require different tuning parameters in the analysis, which is hard to choose for non-statisticians. Xin and Zhu advocate variable ranking instead of variable selection. Once variables are ranked properly, we can make the selection by adopting a threshold rule. In this thesis, we try to rank the variables using Least Angle Regression (LARS). Some shrinkage methods like Lasso and LARS can shrink the coefficients to zero. The advantage of this kind of methods is that they can give a solution path which describes the order that variables enter the model. This provides an intuitive way to rank variables based on the path. However, Lasso can sometimes be difficult to apply to variable ranking directly. This is because that in a Lasso solution path, variables might enter the model and then get dropped. This dropping issue makes it hard to rank based on the order of entrance. However, LARS, which is a modified version of Lasso, doesn't have this problem. We'll make use of this property and rank variables using LARS solution path.

This thesis considers the problem of learning the structure of gene regulatory networks using gene expression time series data. A more realistic scenario where the state space model representing a gene network evolves nonlinearly is considered while a linear model is assumed for the microarray data. To capture the nonlinearity, a particle filter based state estimation algorithm is studied instead of the contemporary linear approximation based approaches. The parameters signifying the regulatory relations among various genes are estimated online using a Kalman filter. Since a particular gene interacts with a few other genes only, the parameter vector is expected to be sparse. The state estimates delivered by the particle filter and the observed microarray data are then fed to a LASSO based least squares regression operation, which yields a parsimonious and efficient description of the regulatory network by setting the irrelevant coefficients to zero. The performance of the aforementioned algorithm is compared with Extended Kalman filtering (EKF), employing Mean Square Error as fidelity criterion using synthetic data and real biological data. Extensive computer simulations illustrate that the particle filter based gene network inference algorithm outperforms EKF and therefore, it can serve as a natural framework for modeling gene regulatory networks.

Inaug.-Diss.--Würzburg. / Lebenslauf. "Literatur-verzeichnis": p. [85]-90.

Introduction: Alzheimer’s disease is an irreversible brain disorder characterized by distortion of memory and other mental functions. Although, several psychometric tests are available for diagnosis of Alzheimer’s, there is a great concern about the validity of these tests at recognizing the early onset of the disease. Currently, brain magnetic resonance imaging is not commonly utilized in the diagnosis of Alzheimer’s, because researchers are still puzzled by the association of brain regions with the disease status and its progress. Moreover, MRI data tend to be of high dimensional nature requiring advanced statistical methods to accurately analyze them. In the past decade, the application of Least Absolute Shrinkage and Selection Operator (LASSO) has become increasingly popular in the analysis of high dimensional data. With LASSO, only a small number of the regression coefficients are believed to have a non-zero value, and therefore allowed to enter the model; other coefficients are while others are shrunk to zero. Aim: Determine the non-zero regression coefficients in models predicting patients’ classification (Normal, mild cognitive impairment (MCI), or Alzheimer’s) using both non-ordinal and ordinal LASSO. Methods: Pre-processed high dimensional MRI data of the Alzheimer’s Disease Neuroimaging Initiative was analyzed. Predictors of the following model were differentiated: Alzheimer’s vs. normal, Alzheimer’s vs. normal and MCI, Alzheimer’s and MCI vs. Normal. Cross-validation followed by ordinal LASSO was executed on these same sets of models. Results: Results were inconclusive. Two brain regions, frontal lobe and putamen, appeared more frequently in the models than any other region. Non-ordinal multinomial models performed better than ordinal multinomial models with higher accuracy, sensitivity, and specificity rates. It was determined that majority of the models were best suited to predict MCI status than the other two statues. Discussion: In future research, the other stages of the disease, different statistical analysis methods, such as elastic net, and larger samples sizes should be explored when using brain MRI for Alzheimer’s disease classification.

Le contexte de cette thèse est la sélection de variables en grande dimension à l'aide de procédures de régression régularisée en présence de redondance entre variables explicatives. Parmi les variables candidates, on suppose que seul un petit nombre est réellement pertinent pour expliquer la réponse. Dans ce cadre de grande dimension, les approches classiques de type Lasso voient leurs performances se dégrader lorsque la redondance croît, puisqu'elles ne tiennent pas compte de cette dernière. Regrouper au préalable ces variables peut pallier ce défaut, mais nécessite usuellement la calibration de paramètres supplémentaires. L'approche proposée combine regroupement et sélection de variables dans un souci d'interprétabilité et d'amélioration des performances. D'abord une Classification Ascendante Hiérarchique (CAH) fournit à chaque niveau une partition des variables en groupes. Puis le Group-lasso est utilisé à partir de l'ensemble des groupes de variables des différents niveaux de la CAH à paramètre de régularisation fixé. Choisir ce dernier fournit alors une liste de groupe candidats issus potentiellement de différents niveaux. Le choix final des groupes est obtenu via une procédure de tests multiples. La procédure proposée exploite la structure hiérarchique de la CAH et des pondérations dans le Group-lasso. Cela permet de réduire considérablement la complexité algorithmique induite par la flexibilité. / This thesis takes place in the context of variable selection in the high dimensional setting using penalizedregression in presence of redundancy between explanatory variables. Among all variables, we supposethat only a few number is relevant for predicting the response variable. In this high dimensional setting,performance of classical lasso-based approaches decreases when redundancy increases as they do not takeit into account. Firstly aggregating variables can overcome this problem but generally requires calibrationof additional parameters. The proposed approach combines variables aggregation and selection in order to improve interpretabilityand performance. First, a hierarchical clustering procedure provides at each level a partition of the variablesinto groups. Then the Group-lasso is used with the set of groups of variables from the different levels ofthe hierarchical clustering and a fixed regularization parameter. Choosing this parameter provides a list ofcandidates groups potentially coming from different levels. The final choice of groups is done by a multipletesting procedure. The proposed procedure exploits the hierarchical structure from hierarchical clustering and some weightsin Group-lasso. This allows to greatly reduce the algorithm complexity induced by the possibility to choosegroups coming from different levels of the hierarchical clustering.

Yes / penalized is a exible, extensible, and e cient MATLAB toolbox for penalized maximum likelihood. penalized allows you to t a generalized linear model (gaussian, logistic, poisson, or multinomial) using any of ten provided penalties, or none. The toolbox can be extended by creating new maximum likelihood models or new penalties. The toolbox also includes routines for cross-validation and plotting.

La thèse contient 6 chapitres. Le premier chapitre contient une introduction à la régression linéaire et aux problèmes Lasso et Lasso bayésien. Le chapitre 2 rappelle les algorithmes d’optimisation convexe et présente l’algorithme FISTA pour calculer l’estimateur Lasso. La statistique de la convergence de cet algorithme est aussi donnée dans ce chapitre en utilisant l’entropie et l’estimateur de Pitman-Yor. Le chapitre 3 est consacré à la comparaison des méthodes quasi-Monte Carlo et Monte Carlo dans les calculs numériques du Lasso bayésien. Il sort de cette comparaison que les points de Hammersely donne les meilleurs résultats. Le chapitre 4 donne une interprétation géométrique de la fonction de partition du Lasso bayésien et l’exprime en fonction de la fonction Gamma incomplète. Ceci nous a permis de donner un critère de convergence pour l’algorithme de Metropolis Hastings. Le chapitre 5 présente l’estimateur bayésien comme la loi limite d’une équation différentielle stochastique multivariée. Ceci nous a permis de calculer le Lasso bayésien en utilisant les schémas numériques semi implicite et explicite d’Euler et les méthodes de Monte Carlo, Monte Carlo à plusieurs couches (MLMC) et l’algorithme de Metropolis Hastings. La comparaison des coûts de calcul montre que le couple (schéma semi-implicite d’Euler, MLMC) gagne contre les autres couples (schéma, méthode). Finalement dans le chapitre 6 nous avons trouvé la vitesse de convergence du Lasso bayésien vers le Lasso lorsque le rapport signal/bruit est constant et le bruit tend vers 0. Ceci nous a permis de donner de nouveaux critères pour la convergence de l’algorithme de Metropolis Hastings. / The thesis contains 6 chapters. The first chapter contains an introduction to linear regression, the Lasso and the Bayesian Lasso problems. Chapter 2 recalls the convex optimization algorithms and presents the Fista algorithm for calculating the Lasso estimator. The properties of the convergence of this algorithm is also given in this chapter using the entropy estimator and Pitman-Yor estimator. Chapter 3 is devoted to comparison of Monte Carlo and quasi-Monte Carlo methods in numerical calculations of Bayesian Lasso. It comes out of this comparison that the Hammersely points give the best results. Chapter 4 gives a geometric interpretation of the partition function of the Bayesian lasso expressed as a function of the incomplete Gamma function. This allowed us to give a convergence criterion for the Metropolis Hastings algorithm. Chapter 5 presents the Bayesian estimator as the law limit a multivariate stochastic differential equation. This allowed us to calculate the Bayesian Lasso using numerical schemes semi-implicit and explicit Euler and methods of Monte Carlo, Monte Carlo multilevel (MLMC) and Metropolis Hastings algorithm. Comparing the calculation costs shows the couple (semi-implicit Euler scheme, MLMC) wins against the other couples (scheme method). Finally in chapter 6 we found the Lasso convergence rate of the Bayesian Lasso when the signal / noise ratio is constant and when the noise tends to 0. This allowed us to provide a new criteria for the convergence of the Metropolis algorithm Hastings.