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Multivariate linear regression methods are widely used statistical tools in data analysis, and were developed when some response variables are studied simultaneously, in which our aim is to study the relationship between predictor variables and response variables through the regression coefficient matrix. The rapid improvements of information technology have brought us a large number of large-scale data, but also brought us great challenges in data processing. When dealing with high dimensional data, the classical least squares estimation is not applicable in multivariate linear regression analysis. In recent years, some approaches have been developed to deal with high-dimensional data problems, among which dimension reduction is one of the main approaches. In some literature, random projection methods were used to reduce dimension in large datasets. In Chapter 2, a new random projection method, with low-rank matrix approximation, is proposed to reduce the dimension of the parameter space in high-dimensional multivariate linear regression model. Some statistical properties of the proposed method are studied and explicit expressions are then derived for the accuracy loss of the method with Gaussian random projection and orthogonal random projection. These expressions are precise rather than being bounds up to constants. In multivariate regression analysis, reduced rank regression is also a dimension reduction method, which has become an important tool for achieving dimension reduction goals due to its simplicity, computational efficiency and good predictive performance. In practical situations, however, the performance of the reduced rank estimator is not satisfactory when the predictor variables are highly correlated or the ratio of signal to noise is small. To overcome this problem, in Chapter 3, we incorporate matrix projections into reduced rank regression method, and then develop reduced rank regression estimators based on random projection and orthogonal projection in high-dimensional multivariate linear regression models. We also propose a consistent estimator of the rank of the coefficient matrix and achieve prediction performance bounds for the proposed estimators based on mean squared errors. Envelope technology is also a popular method in recent years to reduce estimative and predictive variations in multivariate regression, including a class of methods to improve the efficiency without changing the traditional objectives. Variable selection is the process of selecting a subset of relevant features variables for use in model construction. The purpose of using this technology is to avoid the curse of dimensionality, simplify models to make them easier to interpret, shorten training time and reduce overfitting. In Chapter 4, we combine envelope models and a group variable selection method to propose an envelope-based sparse reduced rank regression estimator in high-dimensional multivariate linear regression models, and then establish its consistency, asymptotic normality and oracle property. Tensor data are in frequent use today in a variety of fields in science and engineering. Processing tensor data is a practical but challenging problem. Recently, the prevalence of tensor data has resulted in several envelope tensor versions. In Chapter 5, we incorporate envelope technique into tensor regression analysis and propose a partial tensor envelope model, which leads to a parsimonious version for tensor response regression when some predictors are of special interest, and then consistency and asymptotic normality of the coefficient estimators are proved. The proposed method achieves significant gains in efficiency compared to the standard tensor response regression model in terms of the estimation of the coefficients for the selected predictors. Finally, in Chapter 6, we summarize the work carried out in the thesis, and then suggest some problems of further research interest. / Dissertation / Doctor of Philosophy (PhD)

Bankruptcy prediction is an important research topic. The cost of incorrect decision making in companies and financial institutions can be great and could affect large parts of society. But while it is indeed a major research area, there are few studies which consider the effects of feature selection. This is an important step that could improve the performance of bankruptcy prediction models. This thesis therefore aims to find which feature selection methods perform best for bankruptcy prediction. Five feature selection methods will be compared and used to create datasets with fewer redundant features. To test these methods, a LSTM model is used to train on both an unaltered dataset and datasets created by the mentioned models. The predictive performance of these are then compared with the metrics AUC, Type I error, and Type II error. This study finds that the forward selection algorithm from the Stepwise regression method performed best with an increase in AUC score and decrease in both Type I and Type II error rates compared to the model trained on the unaltered dataset.

In vielen Anwendungen ist es notwendig, die stochastische Schwankungen der maximalen Abweichungen der nichtparametrischen Schätzer von Quantil zu wissen, zB um die verschiedene parametrische Modelle zu überprüfen. Einheitliche Konfidenzbänder sind daher für nichtparametrische Quantil Schätzungen der Regressionsfunktionen gebaut. Die erste Methode basiert auf der starken Approximation der empirischen Verfahren und Extremwert-Theorie. Die starke gleichmäßige Konsistenz liegt auch unter allgemeinen Bedingungen etabliert. Die zweite Methode beruht auf der Bootstrap Resampling-Verfahren. Es ist bewiesen, dass die Bootstrap-Approximation eine wesentliche Verbesserung ergibt. Der Fall von mehrdimensionalen und diskrete Regressorvariablen wird mit Hilfe einer partiellen linearen Modell behandelt. Das Verfahren wird mithilfe der Arbeitsmarktanalysebeispiel erklärt. Hoch-dimensionale Zeitreihen, die nichtstationäre und eventuell periodische Verhalten zeigen, sind häufig in vielen Bereichen der Wissenschaft, zB Makroökonomie, Meteorologie, Medizin und Financial Engineering, getroffen. Der typische Modelierungsansatz ist die Modellierung von hochdimensionalen Zeitreihen in Zeit Ausbreitung der niedrig dimensionalen Zeitreihen und hoch-dimensionale zeitinvarianten Funktionen über dynamische Faktorenanalyse zu teilen. Wir schlagen ein zweistufiges Schätzverfahren. Im ersten Schritt entfernen wir den Langzeittrend der Zeitreihen durch Einbeziehung Zeitbasis von der Gruppe Lasso-Technik und wählen den Raumbasis mithilfe der funktionalen Hauptkomponentenanalyse aus. Wir zeigen die Eigenschaften dieser Schätzer unter den abhängigen Szenario. Im zweiten Schritt erhalten wir den trendbereinigten niedrig-dimensionalen stochastischen Prozess (stationär). / In many applications it is necessary to know the stochastic fluctuation of the maximal deviations of the nonparametric quantile estimates, e.g. for various parametric models check. Uniform confidence bands are therefore constructed for nonparametric quantile estimates of regression functions. The first method is based on the strong approximations of the empirical process and extreme value theory. The strong uniform consistency rate is also established under general conditions. The second method is based on the bootstrap resampling method. It is proved that the bootstrap approximation provides a substantial improvement. The case of multidimensional and discrete regressor variables is dealt with using a partial linear model. A labor market analysis is provided to illustrate the method. High dimensional time series which reveal nonstationary and possibly periodic behavior occur frequently in many fields of science, e.g. macroeconomics, meteorology, medicine and financial engineering. One of the common approach is to separate the modeling of high dimensional time series to time propagation of low dimensional time series and high dimensional time invariant functions via dynamic factor analysis. We propose a two-step estimation procedure. At the first step, we detrend the time series by incorporating time basis selected by the group Lasso-type technique and choose the space basis based on smoothed functional principal component analysis. We show properties of this estimator under the dependent scenario. At the second step, we obtain the detrended low dimensional stochastic process (stationary).

The large potential of big data and how it has brought value into various industries have been established in research. Since big data has such large potential if handled and analyzed in the right way, revealing information to support decision making in an organization, this thesis is conducted as a case study at an automotive manufacturer with access to large amounts of customer usage data of their vehicles. The reason for performing an analysis of this kind of data is based on the cornerstones of Total Quality Management with the end objective of increasing customer satisfaction of the concerned products or services. The case study includes a data analysis exploring how and if patterns about what affects fuel consumption can be revealed from aggregated customer usage data of trucks linked to truck applications. Based on the case study, conclusions are drawn about how a company can use this type of analysis as well as how to handle the data in order to turn it into business value. The data analysis reveals properties describing truck usage using Factor Analysis and Principal Component Analysis. Especially one property is concluded to be important as it appears in the result of both techniques. Based on these properties the trucks are clustered using k-means and Hierarchical Clustering which shows groups of trucks where the importance of the properties varies. Due to the homogeneity and complexity of the chosen data, the clusters of trucks cannot be linked to truck applications. This would require data that is more easily interpretable. Finally, the importance for fuel consumption in the clusters is explored using model estimation. A comparison of Principal Component Regression (PCR) and the two regularization techniques Lasso and Elastic Net is made. PCR results in poor models difficult to evaluate. The two regularization techniques however outperform PCR, both giving a higher and very similar explained variance. The three techniques do not show obvious similarities in the models and no conclusions can therefore be drawn concerning what is important for fuel consumption. During the data analysis many problems with the data are discovered, which are linked to managerial and technical issues of big data. This leads to for example that some of the parameters interesting for the analysis cannot be used and this is likely to have an impact on the inability to get unanimous results in the model estimations. It is also concluded that the data was not originally intended for this type of analysis of large populations, but rather for testing and engineering purposes. Nevertheless, this type of data still contains valuable information and can be used if managed in the right way. From the case study it can be concluded that in order to use the data for more advanced analysis a big-data plan is needed at a strategic level in the organization. The plan summarizes the suggested solution for the managerial issues of the big data for the organization. This plan describes how to handle the data, how the analytic models revealing the information should be designed and the tools and organizational capabilities needed to support the people using the information.

Modern medical practice and science involve complex phenotypic definitions. Understanding patterns of association across this range of phenotypes requires co-analysis of high-dimensional association studies in order to characterise shared and distinct elements. In this thesis I address several problems in this area, with a general linking aim of making more efficient use of available data. The main application of these methods is in the analysis of genome-wide association studies (GWAS) and similar studies. Firstly, I developed methodology for a Bayesian conditional false discovery rate (cFDR) for levering GWAS results using summary statistics from a related disease. I extended an existing method to enable a shared control design, increasing power and applicability, and developed an approximate bound on false-discovery rate (FDR) for the procedure. Using the new method I identified several new variant-disease associations. I then developed a second application of shared control design in the context of study replication, enabling improvement in power at the cost of changing the spectrum of sensitivity to systematic errors in study cohorts. This has application in studies on rare diseases or in between-case analyses. I then developed a method for partially characterising heterogeneity within a disease by modelling the bivariate distribution of case-control and within-case effect sizes. Using an adaptation of a likelihood-ratio test, this allows an assessment to be made of whether disease heterogeneity corresponds to differences in disease pathology. I applied this method to a range of simulated and real datasets, enabling insight into the cause of heterogeneity in autoantibody positivity in type 1 diabetes (T1D). Finally, I investigated the relation of subtypes of juvenile idiopathic arthritis (JIA) to adult diseases, using modified genetic risk scores and linear discriminants in a penalised regression framework. The contribution of this thesis is in a range of methodological developments in the analysis of high-dimensional association study comparison. Methods such as these will have wide application in the analysis of GWAS and similar areas, particularly in the development of stratified medicine.

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Doctorat en philosophie et lettres / info:eu-repo/semantics/nonPublished

<p>Sequential decision-making is a fundamental class of problem that motivates algorithm designs of online machine learning and reinforcement learning. Arguably, the resulting online algorithms have supported modern online service industries for their data-driven real-time automated decision making. The applications span across different industries, including dynamic pricing (Marketing), recommendation (Advertising), and dosage finding (Clinical Trial). In this dissertation, we contribute fundamental statistical design advances for sequential decision-making algorithms, leaping progress in theory and application of online learning and sequential decision making under uncertainty including online sparse learning, finite-armed bandits, and high-dimensional online decision making. Our work locates at the intersection of decision-making algorithm designs, online statistical machine learning, and operations research, contributing new algorithms, theory, and insights to diverse fields including optimization, statistics, and machine learning.</p> <p><br></p> <p>In part I, we contribute a theoretical framework of continuous risk monitoring for regularized online statistical learning. Such theoretical framework is desirable for modern online service industries on monitoring deployed model's performance of online machine learning task. In the first project (Chapter 1), we develop continuous risk monitoring for the online Lasso procedure and provide an always-valid algorithm for high-dimensional dynamic pricing problems. In the second project (Chapter 2), we develop continuous risk monitoring for online matrix regression and provide new algorithms for rank-constrained online matrix completion problems. Such theoretical advances are due to our elegant interplay between non-asymptotic martingale concentration theory and regularized online statistical machine learning.</p> <p><br></p> <p>In part II, we contribute a bootstrap-based methodology for finite-armed bandit problems, termed Residual Bootstrap exploration. Such a method opens a possibility to design model-agnostic bandit algorithms without problem-adaptive optimism-engineering and instance-specific prior-tuning. In the first project (Chapter 3), we develop residual bootstrap exploration for multi-armed bandit algorithms and shows its easy generalizability to bandit problems with complex or ambiguous reward structure. In the second project (Chapter 4), we develop a theoretical framework for residual bootstrap exploration in linear bandit with fixed action set. Such methodology advances are due to our development of non-asymptotic theory for the bootstrap procedure.</p> <p><br></p> <p>In part III, we contribute application-driven insights on the exploration-exploitation dilemma for high-dimensional online decision-making problems. Such insights help practitioners to implement effective high-dimensional statistics methods to solve online decisionmaking problems. In the first project (Chapter 5), we develop a bandit sampling scheme for online batch high-dimensional decision making, a practical scenario in interactive marketing, and sequential clinical trials. In the second project (Chapter 6), we develop a bandit sampling scheme for federated online high-dimensional decision-making to maintain data decentralization and perform collaborated decisions. These new insights are due to our new bandit sampling design to address application-driven exploration-exploitation trade-offs effectively. </p>

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.