Topological stability through tame retractions

A smooth map is said to be stable if small perturbations of the map only differ from the original one by a smooth change of coordinates. Smoothly stable maps are generic among the proper maps between given source and target manifolds when the source and target dimensions belong to the so-called nice dimensions, but outside this range of dimensions, smooth maps cannot generally be approximated by stable maps. This leads to the definition of topologically stable maps, where the smooth coordinate changes are replaced with homeomorphisms. The topologically stable maps are generic among proper maps for any dimensions of source and target. The purpose of this thesis is to investigate methods for proving topological stability by constructing extremely tame (E-tame) retractions onto the map in question from one of its smoothly stable unfoldings. In particular, we investigate how to use E-tame retractions from stable unfoldings to find topologically ministable unfoldings for certain weighted homogeneous maps or germs. Our first results are concerned with the construction of E-tame retractions and their relation to topological stability. We study how to construct the E-tame retractions from partial or local information, and these results form our toolbox for the main constructions. In the next chapter we study the group of right-left equivalences leaving a given multigerm f invariant, and show that when the multigerm is finitely determined, the group has a maximal compact subgroup and that the corresponding quotient is contractible. This means, essentially, that the group can be replaced with a compact Lie group of symmetries without much loss of information. We also show how to split the group into a product whose components only depend on the monogerm components of f. In the final chapter we investigate representatives of the E- and Z-series of singularities, discuss their instability and use our tools to construct E-tame retractions for some of them. The construction is based on describing the geometry of the set of points where the map is not smoothly stable, discovering that by using induction and our constructional tools, we already know how to construct local E-tame retractions along the set. The local solutions can then be glued together using our knowledge about the symmetry group of the local germs. We also discuss how to generalize our method to the whole E- and Z- series.

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.

Specific language impairment in pre-adolescence, adolescence, and adulthood with special emphasis on health-related quality of life

This clinical study focused on effects of childhood specific language impairment (SLI) on daily functioning in late life. SLI is a neurobiological disorder with genetic predisposition and manifests as poor language production or comprehension or both in a child with age-level non-verbal intelligence and no other known cause for deficient language development. The prevalence rate of around 7% puts it among the most prevalent developmental disorders in childhood. Negative long-term effects, such as problems in learning and behavior, are frequent. In follow-up studies the focus has seldom been on self-perception of daily functioning and participation, which are considered important in the International Classification of Functioning, Disability, and Health (ICF). To investigate the self-perceived aspects of everyday functioning in individuals with childhood receptive SLI compared with age- and gender-matched control populations, the 15D, 16D, and 17D health-related quality of life (HRQoL) questionnaires were applied. These generic questionnaires include 15, 16, and 17 dimensions, respectively, and give both a single index score and a profile with values on each dimension. Information on different life domains (rehabilitation, education, employment etc.) from each age-group was collected with separate questionnaires. The study groups comprised adults, adolescents (12-16 years), and pre-adolescents (8-11 years) who had received a diagnosis of receptive SLI and had been examined, usually before school age, at the Department of Phoniatrics of Helsinki University Central Hospital, where children with language deficits caused by various etiologies are examined and treated by a multidisciplinary team. The adult respondents included 33 subjects with a mean age of 34 years. Measured with 15D, the subjects perceived their HRQoL to be nearly as good as that of their controls, but on the dimensions of speech, usual activities, mental functioning, and distress they were significantly worse off. They significantly more often lived with their parents (19%) or were pensioned (26%) than the adult Finnish population on average. Adults with self-perceived problems in finding words and in remembering instructions, manifestations of persistent language impairment, showed inferior every day functioning to the rest of the study group. Of the adolescents and pre-adolescents, 48 and 51, respectively, responded. The majority in both groups had received special education or extra educational support at school. They all had attended speech therapy at some point; at the time of the study only one adolescent, but every third pre-adolescent still received speech therapy. The 16D score of the adolescent or the 17D score of the pre-adolescents did not differ from that of their controls. The 16D profiles differed on some dimensions; subjects were significantly worse off on the dimension of mental functioning, but better off on the dimension of vitality than controls. Of the 17D dimensions, the study group was significantly worse off on speech, whereas the control group reported significantly more problems in sleeping. Of the childhood performance measures investigated, low verbal intelligence quotient (VIQ), which is often considered to reflect receptive language impairment, was in adults subjects significantly associated with some of the self-perceived problems, such as problems in usual activities and mental functioning. The 15D, 16D, and 17D questionnaires served well in measuring self-perceived HRQoL. Such standardized measures with population values are especially important in confirming with the ICF guidelines. In the future these questionnaires could perhaps be used on a more individual level in follow-up of children in clinics, and even in special schools and classes, to detect those children at greatest risk of negative long-term effects and perhaps diminished well-being regarding daily functioning and participation.

The Effect of Intrauterine Growth Restriction on Long-Term Outcome in Very or Extremely Low Birth Weight Infants

The project consisted of two long-term follow-up studies of preterm children addressing the question whether intrauterine growth restriction affects the outcome. Assessment at 5 years of age of 203 children with a birth weight less than 1000 g born in Finland in 1996-1997 showed that 9% of the children had cognitive impairment, 14% cerebral palsy, and 4% needed a hearing aid. The intelligence quotient was lower (p<0.05) than the reference value. Thus, 20% exhibited major, 19% minor disabilities, and 61% had no functional abnormalities. Being small for gestational age (SGA) was associated with sub-optimal growth later. In children born before 27 gestational weeks, the SGA had more neuropsychological disabilities than those appropriate for gestational age (AGA). In another cohort with birth weight less than 1500 g assessed at 5 years of age, echocardiography showed a thickened interventricular septum and a decreased left ventricular end-diastolic diameter in both SGA and AGA born children. They also had a higher systolic blood pressure than the reference. Laser-Doppler flowmetry showed different endothelium-dependent and -independent vasodilation responses in the AGA children compared to those of the controls. SGA was not associated with cardio-vascular abnormalities. Auditory event-related potentials (AERPs) were recorded using an oddball paradigm with frequency deviants (standard tone 500 Hz and deviant 750-Hz with 10% probability). At term, the P350 was smaller in SGA and AGA infants than in controls. At 12 months, the automatic change detection peak (mismatch negativity, MMN) was observed in the controls. However, the pre-term infants had a difference positivity that correlated with their neurodevelopment scores. At 5 years of age, the P1-deflection, which reflects primary auditory processing, was smaller, and the MMN larger in the preterm than in the control children. Even with a challenging paradigm or a distraction paradigm, P1 was smaller in the preterm than in the control children. The SGA and AGA children showed similar AERP responses. Prematurity is a major risk factor for abnormal brain development. Preterm children showed signs of cardiovascular abnormality suggesting that prematurity per se may carry a risk for later morbidity. The small positive amplitudes in AERPs suggest persisting altered auditory processing in the preterm in-fants.

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.