Computationally Efficient Methods for MDL-Optimal Density Estimation and Data Clustering

The Minimum Description Length (MDL) principle is a general, well-founded theoretical formalization of statistical modeling. The most important notion of MDL is the stochastic complexity, which can be interpreted as the shortest description length of a given sample of data relative to a model class. The exact definition of the stochastic complexity has gone through several evolutionary steps. The latest instantation is based on the so-called Normalized Maximum Likelihood (NML) distribution which has been shown to possess several important theoretical properties. However, the applications of this modern version of the MDL have been quite rare because of computational complexity problems, i.e., for discrete data, the definition of NML involves an exponential sum, and in the case of continuous data, a multi-dimensional integral usually infeasible to evaluate or even approximate accurately. In this doctoral dissertation, we present mathematical techniques for computing NML efficiently for some model families involving discrete data. We also show how these techniques can be used to apply MDL in two practical applications: histogram density estimation and clustering of multi-dimensional data.

Learning Nonlinear Visual Processing from Natural Images

The paradigm of computational vision hypothesizes that any visual function -- such as the recognition of your grandparent -- can be replicated by computational processing of the visual input. What are these computations that the brain performs? What should or could they be? Working on the latter question, this dissertation takes the statistical approach, where the suitable computations are attempted to be learned from the natural visual data itself. In particular, we empirically study the computational processing that emerges from the statistical properties of the visual world and the constraints and objectives specified for the learning process. This thesis consists of an introduction and 7 peer-reviewed publications, where the purpose of the introduction is to illustrate the area of study to a reader who is not familiar with computational vision research. In the scope of the introduction, we will briefly overview the primary challenges to visual processing, as well as recall some of the current opinions on visual processing in the early visual systems of animals. Next, we describe the methodology we have used in our research, and discuss the presented results. We have included some additional remarks, speculations and conclusions to this discussion that were not featured in the original publications. We present the following results in the publications of this thesis. First, we empirically demonstrate that luminance and contrast are strongly dependent in natural images, contradicting previous theories suggesting that luminance and contrast were processed separately in natural systems due to their independence in the visual data. Second, we show that simple cell -like receptive fields of the primary visual cortex can be learned in the nonlinear contrast domain by maximization of independence. Further, we provide first-time reports of the emergence of conjunctive (corner-detecting) and subtractive (opponent orientation) processing due to nonlinear projection pursuit with simple objective functions related to sparseness and response energy optimization. Then, we show that attempting to extract independent components of nonlinear histogram statistics of a biologically plausible representation leads to projection directions that appear to differentiate between visual contexts. Such processing might be applicable for priming, \ie the selection and tuning of later visual processing. We continue by showing that a different kind of thresholded low-frequency priming can be learned and used to make object detection faster with little loss in accuracy. Finally, we show that in a computational object detection setting, nonlinearly gain-controlled visual features of medium complexity can be acquired sequentially as images are encountered and discarded. We present two online algorithms to perform this feature selection, and propose the idea that for artificial systems, some processing mechanisms could be selectable from the environment without optimizing the mechanisms themselves. In summary, this thesis explores learning visual processing on several levels. The learning can be understood as interplay of input data, model structures, learning objectives, and estimation algorithms. The presented work adds to the growing body of evidence showing that statistical methods can be used to acquire intuitively meaningful visual processing mechanisms. The work also presents some predictions and ideas regarding biological visual processing.

Diagnostic Tests Based on Quantile Residuals for Nonlinear Time Series Models

This thesis studies quantile residuals and uses different methodologies to develop test statistics that are applicable in evaluating linear and nonlinear time series models based on continuous distributions. Models based on mixtures of distributions are of special interest because it turns out that for those models traditional residuals, often referred to as Pearson's residuals, are not appropriate. As such models have become more and more popular in practice, especially with financial time series data there is a need for reliable diagnostic tools that can be used to evaluate them. The aim of the thesis is to show how such diagnostic tools can be obtained and used in model evaluation. The quantile residuals considered here are defined in such a way that, when the model is correctly specified and its parameters are consistently estimated, they are approximately independent with standard normal distribution. All the tests derived in the thesis are pure significance type tests and are theoretically sound in that they properly take the uncertainty caused by parameter estimation into account. -- In Chapter 2 a general framework based on the likelihood function and smooth functions of univariate quantile residuals is derived that can be used to obtain misspecification tests for various purposes. Three easy-to-use tests aimed at detecting non-normality, autocorrelation, and conditional heteroscedasticity in quantile residuals are formulated. It also turns out that these tests can be interpreted as Lagrange Multiplier or score tests so that they are asymptotically optimal against local alternatives. Chapter 3 extends the concept of quantile residuals to multivariate models. The framework of Chapter 2 is generalized and tests aimed at detecting non-normality, serial correlation, and conditional heteroscedasticity in multivariate quantile residuals are derived based on it. Score test interpretations are obtained for the serial correlation and conditional heteroscedasticity tests and in a rather restricted special case for the normality test. In Chapter 4 the tests are constructed using the empirical distribution function of quantile residuals. So-called Khmaladze s martingale transformation is applied in order to eliminate the uncertainty caused by parameter estimation. Various test statistics are considered so that critical bounds for histogram type plots as well as Quantile-Quantile and Probability-Probability type plots of quantile residuals are obtained. Chapters 2, 3, and 4 contain simulations and empirical examples which illustrate the finite sample size and power properties of the derived tests and also how the tests and related graphical tools based on residuals are applied in practice.