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In many hypothesis testing problems such as one-sample and two-sample test problems, the test statistics are degenerate U-statistics. One of the challenges in practice is the computation of U-statistics for a large sample size. Besides, for degenerate U-statistics, the limiting distribution is a mixture of weighted chi-squares, involving the eigenvalues of the kernel of the U-statistics. As a result, it’s not straightforward to construct the rejection region based on this asymptotic distribution. In this research, we aim to reduce the computation complexity of degenerate U-statistics and propose an easy-to-calibrate test statistic by using the divide-and-conquer method. Specifically, we randomly partition the full n data points into kn even disjoint groups, and compute U-statistics on each group and combine them by averaging to get a statistic Tn. We proved that the statistic Tn has the standard normal distribution as the limiting distribution. In this way, the running time is reduced from O(n^m) to O( n^m/km_n), where m is the order of the one sample U-statistics. Besides, for a given significance level , it’s easy to construct the rejection region. We apply our method to the goodness of fit test and two-sample test. The simulation and real data analysis show that the proposed test can achieve high power and fast running time for both one and two-sample tests.

<p dir="ltr">Generating images from training samples solves the challenge of imbalanced data. It provides the necessary data to run machine learning algorithms for image classification, anomaly detection, and pattern recognition tasks. In medical settings, having imbalanced data results in higher false negatives due to a lack of positive samples. Generative Adversarial Networks (GANs) have been widely adopted for image generation. GANs allow models to train without computing intractable probability while producing high-quality images. However, evaluating GANs has been challenging for the researchers due to a need for an objective function. Most studies assess the quality of generated images and the variety of classes those images cover. Overfitting of training images, however, has received less attention from researchers. When the generated images are mere copies of the training data, GAN models will overfit and will not generalize well. This study examines the ability to detect overfitting of popular metrics: Maximum Mean Discrepancy (MMD) and Fréchet Inception Distance (FID). We investigate the metrics on two types of data: handwritten digits and chest x-ray images using Analysis of Variance (ANOVA) models.</p>

The advent of Next Generation Sequencing (NGS), a little over a decade ago, has led to a vast and rapid increase in the generation of genomic data. The drastically reduced cost has in turn enabled powerful modifications that can be used to investigate not just genetic, but epigenetic, phenomena. Epigenetics refers to the study of mechanisms effecting gene expression other than the genetic code itself and thus, at the transcription level, incorporates DNA methylation, transcription factor binding and histone modifications amongst others. This thesis outlines and tackles two major challenges in the computational analysis of such data using techniques from machine learning. Firstly, I address the problem of testing for differential methylation between groups of bisulfite sequencing data sets. DNA methylation plays an important role in genomic imprinting, X-chromosome inactivation and the repression of repetitive elements, as well as being implicated in numerous diseases, such as cancer. Bisulfite sequencing provides single nucleotide resolution methylation data at the whole genome scale, but a sensitive analysis of such data is difficult. I propose a solution that uses a powerful kernel-based machine learning technique, the Maximum Mean Discrepancy, to leverage well-characterised spatial correlations in DNA methylation, and adapt the method for this particular use. I use this tailored method to analyse a novel data set from a study of ageing in three different tissues in the mouse. This study motivates further modifications to the method and highlights the utility of the underlying measure as an exploratory tool for methylation analysis. Secondly, I address the problem of predictive and explanatory modelling of chromatin immunoprecipitation sequencing data (ChIP-Seq). ChIP-Seq is typically used to assay the binding of a protein of interest, such as a transcription factor or histone, to the DNA, and as such is one of the most widely used sequencing assays. While peak callers are a powerful tool in identifying binding sites of sparse and clean ChIPSeq profiles, more broad signals defy analysis in this framework. Instead, generative models that explain the data in terms of the underlying sequence can help uncover mechanisms that predicting binding or the lack thereof. I explore current problems with ChIP-Seq analysis, such as zero-inflation and the use of the control experiment, known as the input. I then devise a method for representing k-mers that enables the use of longer DNA sub-sequences within a flexible model development framework, such as generalised linear models, without heavy programming requirements. Finally, I use these insights to develop an appropriate Bayesian generative model that predicts ChIP-Seq count data in terms of the underlying DNA sequence, incorporating DNA methylation information where available, fitting the model with the Expectation-Maximization algorithm. The model is tested on simulated data and real data pertaining to the histone mark H3k27me3. This thesis therefore straddles the fields of bioinformatics and machine learning. Bioinformatics is both plagued and blessed by the plethora of different techniques available for gathering data and their continual innovations. Each technique presents a unique challenge, and hence out-of-the-box machine learning techniques have had little success in solving biological problems. While I have focused on NGS data, the methods developed in this thesis are likely to be applicable to future technologies, such as Third Generation Sequencing methods, and the lessons learned in their adaptation will be informative for the next wave of computational challenges.

The need to quantify distance between two groups of objects is prevalent throughout the signal processing world. The difference of group means computed using the Euclidean, or L2 distance, is one of the predominant distance measures used to compare feature vectors and groups of vectors, but many problems arise with it when high data dimensionality is present. Maximum mean discrepancy (MMD) is a recent unsupervised kernel-based pattern recognition method which may improve differentiation between two distinct populations over many commonly used methods such as the difference of means, when paired with the proper feature representations and kernels. MMD-based distance computation combines many powerful concepts from the machine learning literature, such as data distribution-leveraging similarity measures and kernel methods for machine learning. Due to this heritage, we posit that dissimilarity-based classification and changepoint detection using MMD can lead to enhanced separation between different populations. To test this hypothesis, we conduct studies comparing MMD and the difference of means in two subareas of image analysis and understanding: first, to detect scene changes in video in an unsupervised manner, and secondly, in the biomedical imaging field, using clinical ultrasound to assess tumor response to treatment. We leverage effective computer vision data descriptors, such as the bag-of-visual-words and sparse combinations of SIFT descriptors, and choose from an assessment of several similarity kernels (e.g. Histogram Intersection, Radial Basis Function) in order to engineer useful systems using MMD. Promising improvements over the difference of means, measured primarily using precision/recall for scene change detection, and k-nearest neighbour classification accuracy for tumor response assessment, are obtained in both applications.

Regression is a central problem in statistics and machine learning with applications everywhere in science and technology. In probabilistic regression the relationship between a set of features and a real-valued target variable is modelled as a conditional probability distribution. There are cases where this distribution is very complex and not properly captured by simple approximations, such as assuming a normal distribution. This thesis investigates how conditional Generative Adversarial Networks (GANs) can be used to properly capture more complex conditional distributions. GANs have seen great success in generating complex high-dimensional data, but less work has been done on their use for regression problems. This thesis presents experiments to better understand how conditional GANs can be used in probabilistic regression. Different versions of GANs are extended to the conditional case and evaluated on synthetic and real datasets. It is shown that conditional GANs can learn to estimate a wide range of different distributions and be competitive with existing probabilistic regression models.