On probability-based inference under data missing by design

Whether a statistician wants to complement a probability model for observed data with a prior distribution and carry out fully probabilistic inference, or base the inference only on the likelihood function, may be a fundamental question in theory, but in practice it may well be of less importance if the likelihood contains much more information than the prior. Maximum likelihood inference can be justified as a Gaussian approximation at the posterior mode, using flat priors. However, in situations where parametric assumptions in standard statistical models would be too rigid, more flexible model formulation, combined with fully probabilistic inference, can be achieved using hierarchical Bayesian parametrization. This work includes five articles, all of which apply probability modeling under various problems involving incomplete observation. Three of the papers apply maximum likelihood estimation and two of them hierarchical Bayesian modeling. Because maximum likelihood may be presented as a special case of Bayesian inference, but not the other way round, in the introductory part of this work we present a framework for probability-based inference using only Bayesian concepts. We also re-derive some results presented in the original articles using the toolbox equipped herein, to show that they are also justifiable under this more general framework. Here the assumption of exchangeability and de Finetti's representation theorem are applied repeatedly for justifying the use of standard parametric probability models with conditionally independent likelihood contributions. It is argued that this same reasoning can be applied also under sampling from a finite population. The main emphasis here is in probability-based inference under incomplete observation due to study design. This is illustrated using a generic two-phase cohort sampling design as an example. The alternative approaches presented for analysis of such a design are full likelihood, which utilizes all observed information, and conditional likelihood, which is restricted to a completely observed set, conditioning on the rule that generated that set. Conditional likelihood inference is also applied for a joint analysis of prevalence and incidence data, a situation subject to both left censoring and left truncation. Other topics covered are model uncertainty and causal inference using posterior predictive distributions. We formulate a non-parametric monotonic regression model for one or more covariates and a Bayesian estimation procedure, and apply the model in the context of optimal sequential treatment regimes, demonstrating that inference based on posterior predictive distributions is feasible also in this case.

Bayesian Inference for Retrospective Population Genetics Models Using Markov Chain Monte Carlo Methods

Genetics, the science of heredity and variation in living organisms, has a central role in medicine, in breeding crops and livestock, and in studying fundamental topics of biological sciences such as evolution and cell functioning. Currently the field of genetics is under a rapid development because of the recent advances in technologies by which molecular data can be obtained from living organisms. In order that most information from such data can be extracted, the analyses need to be carried out using statistical models that are tailored to take account of the particular genetic processes. In this thesis we formulate and analyze Bayesian models for genetic marker data of contemporary individuals. The major focus is on the modeling of the unobserved recent ancestry of the sampled individuals (say, for tens of generations or so), which is carried out by using explicit probabilistic reconstructions of the pedigree structures accompanied by the gene flows at the marker loci. For such a recent history, the recombination process is the major genetic force that shapes the genomes of the individuals, and it is included in the model by assuming that the recombination fractions between the adjacent markers are known. The posterior distribution of the unobserved history of the individuals is studied conditionally on the observed marker data by using a Markov chain Monte Carlo algorithm (MCMC). The example analyses consider estimation of the population structure, relatedness structure (both at the level of whole genomes as well as at each marker separately), and haplotype configurations. For situations where the pedigree structure is partially known, an algorithm to create an initial state for the MCMC algorithm is given. Furthermore, the thesis includes an extension of the model for the recent genetic history to situations where also a quantitative phenotype has been measured from the contemporary individuals. In that case the goal is to identify positions on the genome that affect the observed phenotypic values. This task is carried out within the Bayesian framework, where the number and the relative effects of the quantitative trait loci are treated as random variables whose posterior distribution is studied conditionally on the observed genetic and phenotypic data. In addition, the thesis contains an extension of a widely-used haplotyping method, the PHASE algorithm, to settings where genetic material from several individuals has been pooled together, and the allele frequencies of each pool are determined in a single genotyping.

Computational Techniques for Haplotype Inference and for Local Alignment Significance

This thesis which consists of an introduction and four peer-reviewed original publications studies the problems of haplotype inference (haplotyping) and local alignment significance. The problems studied here belong to the broad area of bioinformatics and computational biology. The presented solutions are computationally fast and accurate, which makes them practical in high-throughput sequence data analysis. Haplotype inference is a computational problem where the goal is to estimate haplotypes from a sample of genotypes as accurately as possible. This problem is important as the direct measurement of haplotypes is difficult, whereas the genotypes are easier to quantify. Haplotypes are the key-players when studying for example the genetic causes of diseases. In this thesis, three methods are presented for the haplotype inference problem referred to as HaploParser, HIT, and BACH. HaploParser is based on a combinatorial mosaic model and hierarchical parsing that together mimic recombinations and point-mutations in a biologically plausible way. In this mosaic model, the current population is assumed to be evolved from a small founder population. Thus, the haplotypes of the current population are recombinations of the (implicit) founder haplotypes with some point--mutations. HIT (Haplotype Inference Technique) uses a hidden Markov model for haplotypes and efficient algorithms are presented to learn this model from genotype data. The model structure of HIT is analogous to the mosaic model of HaploParser with founder haplotypes. Therefore, it can be seen as a probabilistic model of recombinations and point-mutations. BACH (Bayesian Context-based Haplotyping) utilizes a context tree weighting algorithm to efficiently sum over all variable-length Markov chains to evaluate the posterior probability of a haplotype configuration. Algorithms are presented that find haplotype configurations with high posterior probability. BACH is the most accurate method presented in this thesis and has comparable performance to the best available software for haplotype inference. Local alignment significance is a computational problem where one is interested in whether the local similarities in two sequences are due to the fact that the sequences are related or just by chance. Similarity of sequences is measured by their best local alignment score and from that, a p-value is computed. This p-value is the probability of picking two sequences from the null model that have as good or better best local alignment score. Local alignment significance is used routinely for example in homology searches. In this thesis, a general framework is sketched that allows one to compute a tight upper bound for the p-value of a local pairwise alignment score. Unlike the previous methods, the presented framework is not affeced by so-called edge-effects and can handle gaps (deletions and insertions) without troublesome sampling and curve fitting.

Inference on Cointegration in Vector Autoregressive Models (summary section only)

In the thesis we consider inference for cointegration in vector autoregressive (VAR) models. The thesis consists of an introduction and four papers. The first paper proposes a new test for cointegration in VAR models that is directly based on the eigenvalues of the least squares (LS) estimate of the autoregressive matrix. In the second paper we compare a small sample correction for the likelihood ratio (LR) test of cointegrating rank and the bootstrap. The simulation experiments show that the bootstrap works very well in practice and dominates the correction factor. The tests are applied to international stock prices data, and the .nite sample performance of the tests are investigated by simulating the data. The third paper studies the demand for money in Sweden 1970—2000 using the I(2) model. In the fourth paper we re-examine the evidence of cointegration between international stock prices. The paper shows that some of the previous empirical results can be explained by the small-sample bias and size distortion of Johansen’s LR tests for cointegration. In all papers we work with two data sets. The first data set is a Swedish money demand data set with observations on the money stock, the consumer price index, gross domestic product (GDP), the short-term interest rate and the long-term interest rate. The data are quarterly and the sample period is 1970(1)—2000(1). The second data set consists of month-end stock market index observations for Finland, France, Germany, Sweden, the United Kingdom and the United States from 1980(1) to 1997(2). Both data sets are typical of the sample sizes encountered in economic data, and the applications illustrate the usefulness of the models and tests discussed in the thesis.

Paradigms in Statistical Inference for Finite Populations; Up to the 1950s

Modern sample surveys started to spread after statistician at the U.S. Bureau of the Census in the 1940s had developed a sampling design for the Current Population Survey (CPS). A significant factor was also that digital computers became available for statisticians. In the beginning of 1950s, the theory was documented in textbooks on survey sampling. This thesis is about the development of the statistical inference for sample surveys. For the first time the idea of statistical inference was enunciated by a French scientist, P. S. Laplace. In 1781, he published a plan for a partial investigation in which he determined the sample size needed to reach the desired accuracy in estimation. The plan was based on Laplace s Principle of Inverse Probability and on his derivation of the Central Limit Theorem. They were published in a memoir in 1774 which is one of the origins of statistical inference. Laplace s inference model was based on Bernoulli trials and binominal probabilities. He assumed that populations were changing constantly. It was depicted by assuming a priori distributions for parameters. Laplace s inference model dominated statistical thinking for a century. Sample selection in Laplace s investigations was purposive. In 1894 in the International Statistical Institute meeting, Norwegian Anders Kiaer presented the idea of the Representative Method to draw samples. Its idea was that the sample would be a miniature of the population. It is still prevailing. The virtues of random sampling were known but practical problems of sample selection and data collection hindered its use. Arhtur Bowley realized the potentials of Kiaer s method and in the beginning of the 20th century carried out several surveys in the UK. He also developed the theory of statistical inference for finite populations. It was based on Laplace s inference model. R. A. Fisher contributions in the 1920 s constitute a watershed in the statistical science He revolutionized the theory of statistics. In addition, he introduced a new statistical inference model which is still the prevailing paradigm. The essential idea is to draw repeatedly samples from the same population and the assumption that population parameters are constants. Fisher s theory did not include a priori probabilities. Jerzy Neyman adopted Fisher s inference model and applied it to finite populations with the difference that Neyman s inference model does not include any assumptions of the distributions of the study variables. Applying Fisher s fiducial argument he developed the theory for confidence intervals. Neyman s last contribution to survey sampling presented a theory for double sampling. This gave the central idea for statisticians at the U.S. Census Bureau to develop the complex survey design for the CPS. Important criterion was to have a method in which the costs of data collection were acceptable, and which provided approximately equal interviewer workloads, besides sufficient accuracy in estimation.

Language change as an evolutionary process

In the thesis it is discussed in what ways concepts and methodology developed in evolutionary biology can be applied to the explanation and research of language change. The parallel nature of the mechanisms of biological evolution and language change is explored along with the history of the exchange of ideas between these two disciplines. Against this background computational methods developed in evolutionary biology are taken into consideration in terms of their applicability to the study of historical relationships between languages. Different phylogenetic methods are explained in common terminology, avoiding the technical language of statistics. The thesis is on one hand a synthesis of earlier scientific discussion, and on the other an attempt to map out the problems of earlier approaches in addition to finding new guidelines in the study of language change on their basis. Primarily literature about the connections between evolutionary biology and language change, along with research articles describing applications of phylogenetic methods into language change have been used as source material. The thesis starts out by describing the initial development of the disciplines of evolutionary biology and historical linguistics, a process which right from the beginning can be seen to have involved an exchange of ideas concerning the mechanisms of language change and biological evolution. The historical discussion lays the foundation for the handling of the generalised account of selection developed during the recent few decades. This account is aimed for creating a theoretical framework capable of explaining both biological evolution and cultural change as selection processes acting on self-replicating entities. This thesis focusses on the capacity of the generalised account of selection to describe language change as a process of this kind. In biology, the mechanisms of evolution are seen to form populations of genetically related organisms through time. One of the central questions explored in this thesis is whether selection theory makes it possible to picture languages are forming populations of a similar kind, and what a perspective like this can offer to the understanding of language in general. In historical linguistics, the comparative method and other, complementing methods have been traditionally used to study the development of languages from a common ancestral language. Computational, quantitative methods have not become widely used as part of the central methodology of historical linguistics. After the fading of a limited popularity enjoyed by the lexicostatistical method since the 1950s, only in the recent years have also the computational methods of phylogenetic inference used in evolutionary biology been applied to the study of early language history. In this thesis the possibilities offered by the traditional methodology of historical linguistics and the new phylogenetic methods are compared. The methods are approached through the ways in which they have been applied to the Indo-European languages, which is the most thoroughly investigated language family using both the traditional and the phylogenetic methods. The problems of these applications along with the optimal form of the linguistic data used in these methods are explored in the thesis. The mechanisms of biological evolution are seen in the thesis as parallel in a limited sense to the mechanisms of language change, however sufficiently so that the development of a generalised account of selection is deemed as possibly fruiful for understanding language change. These similarities are also seen to support the validity of using phylogenetic methods in the study of language history, although the use of linguistic data and the models of language change employed by these models are seen to await further development.

On the Origin of Ideas : An Abductivist Approach to Discovery

The purpose of this study is to analyze and develop various forms of abduction as a means of conceptualizing processes of discovery. Abduction was originally presented by Charles S. Peirce (1839-1914) as a "weak", third main mode of inference -- besides deduction and induction -- one which, he proposed, is closely related to many kinds of cognitive processes, such as instincts, perception, practices and mediated activity in general. Both abduction and discovery are controversial issues in philosophy of science. It is often claimed that discovery cannot be a proper subject area for conceptual analysis and, accordingly, abduction cannot serve as a "logic of discovery". I argue, however, that abduction gives essential means for understanding processes of discovery although it cannot give rise to a manual or algorithm for making discoveries. In the first part of the study, I briefly present how the main trend in philosophy of science has, for a long time, been critical towards a systematic account of discovery. Various models have, however, been suggested. I outline a short history of abduction; first Peirce's evolving forms of his theory, and then later developments. Although abduction has not been a major area of research until quite recently, I review some critiques of it and look at the ways it has been analyzed, developed and used in various fields of research. Peirce's own writings and later developments, I argue, leave room for various subsequent interpretations of abduction. The second part of the study consists of six research articles. First I treat "classical" arguments against abduction as a logic of discovery. I show that by developing strategic aspects of abductive inference these arguments can be countered. Nowadays the term 'abduction' is often used as a synonym for the Inference to the Best Explanation (IBE) model. I argue, however, that it is useful to distinguish between IBE ("Harmanian abduction") and "Hansonian abduction"; the latter concentrating on analyzing processes of discovery. The distinctions between loveliness and likeliness, and between potential and actual explanations are more fruitful within Hansonian abduction. I clarify the nature of abduction by using Peirce's distinction between three areas of "semeiotic": grammar, critic, and methodeutic. Grammar (emphasizing "Firstnesses" and iconicity) and methodeutic (i.e., a processual approach) especially, give new means for understanding abduction. Peirce himself held a controversial view that new abductive ideas are products of an instinct and an inference at the same time. I maintain that it is beneficial to make a clear distinction between abductive inference and abductive instinct, on the basis of which both can be developed further. Besides these, I analyze abduction as a part of distributed cognition which emphasizes a long-term interaction with the material, social and cultural environment as a source for abductive ideas. This approach suggests a "trialogical" model in which inquirers are fundamentally connected both to other inquirers and to the objects of inquiry. As for the classical Meno paradox about discovery, I show that abduction provides more than one answer. As my main example of abductive methodology, I analyze the process of Ignaz Semmelweis' research on childbed fever. A central basis for abduction is the claim that discovery is not a sequence of events governed only by processes of chance. Abduction treats those processes which both constrain and instigate the search for new ideas; starting from the use of clues as a starting point for discovery, but continuing in considerations like elegance and 'loveliness'. The study then continues a Peircean-Hansonian research programme by developing abduction as a way of analyzing processes of discovery.

Statistical analysis of the associations of hereditary factors with the risk of Type 1 diabetes and Diabetic Nephropathy based on the familial data collected through ascertaiment from population-based registers

In genetic epidemiology, population-based disease registries are commonly used to collect genotype or other risk factor information concerning affected subjects and their relatives. This work presents two new approaches for the statistical inference of ascertained data: a conditional and full likelihood approaches for the disease with variable age at onset phenotype using familial data obtained from population-based registry of incident cases. The aim is to obtain statistically reliable estimates of the general population parameters. The statistical analysis of familial data with variable age at onset becomes more complicated when some of the study subjects are non-susceptible, that is to say these subjects never get the disease. A statistical model for a variable age at onset with long-term survivors is proposed for studies of familial aggregation, using latent variable approach, as well as for prospective studies of genetic association studies with candidate genes. In addition, we explore the possibility of a genetic explanation of the observed increase in the incidence of Type 1 diabetes (T1D) in Finland in recent decades and the hypothesis of non-Mendelian transmission of T1D associated genes. Both classical and Bayesian statistical inference were used in the modelling and estimation. Despite the fact that this work contains five studies with different statistical models, they all concern data obtained from nationwide registries of T1D and genetics of T1D. In the analyses of T1D data, non-Mendelian transmission of T1D susceptibility alleles was not observed. In addition, non-Mendelian transmission of T1D susceptibility genes did not make a plausible explanation for the increase in T1D incidence in Finland. Instead, the Human Leucocyte Antigen associations with T1D were confirmed in the population-based analysis, which combines T1D registry information, reference sample of healthy subjects and birth cohort information of the Finnish population. Finally, a substantial familial variation in the susceptibility of T1D nephropathy was observed. The presented studies show the benefits of sophisticated statistical modelling to explore risk factors for complex diseases.

Urban energy fluxes in Helsinki and their high frequency spectral corrections

Inadvertent climate modification has led to an increase in urban temperatures compared to the surrounding rural area. The main reason for the temperature rise is the altered energy portioning of input net radiation to heat storage and sensible and latent heat fluxes in addition to the anthropogenic heat flux. The heat storage flux and anthropogenic heat flux have not yet been determined for Helsinki and they are not directly measurable. To the contrary, turbulent fluxes of sensible and latent heat in addition to net radiation can be measured, and the anthropogenic heat flux together with the heat storage flux can be solved as a residual. As a result, all inaccuracies in the determination of the energy balance components propagate to the residual term and special attention must be paid to the accurate determination of the components. One cause of error in the turbulent fluxes is the fluctuation attenuation at high frequencies which can be accounted for by high frequency spectral corrections. The aim of this study is twofold: to assess the relevance of high frequency corrections to water vapor fluxes and to assess the temporal variation of the energy fluxes. Turbulent fluxes of sensible and latent heat have been measured at SMEAR III station, Helsinki, since December 2005 using the eddy covariance technique. In addition, net radiation measurements have been ongoing since July 2007. The used calculation methods in this study consist of widely accepted eddy covariance data post processing methods in addition to Fourier and wavelet analysis. The high frequency spectral correction using the traditional transfer function method is highly dependent on relative humidity and has an 11% effect on the latent heat flux. This method is based on an assumption of spectral similarity which is shown not to be valid. A new correction method using wavelet analysis is thus initialized and it seems to account for the high frequency variation deficit. Anyhow, the resulting wavelet correction remains minimal in contrast to the traditional transfer function correction. The energy fluxes exhibit a behavior characteristic for urban environments: the energy input is channeled to sensible heat as latent heat flux is restricted by water availability. The monthly mean residual of the energy balance ranges from 30 Wm-2 in summer to -35 Wm-2 in winter meaning a heat storage to the ground during summer. Furthermore, the anthropogenic heat flux is approximated to be 50 Wm-2 during winter when residential heating is important.

Bayesian fisheries stock assessment: integrating and updating knowledge

In this thesis the use of the Bayesian approach to statistical inference in fisheries stock assessment is studied. The work was conducted in collaboration of the Finnish Game and Fisheries Research Institute by using the problem of monitoring and prediction of the juvenile salmon population in the River Tornionjoki as an example application. The River Tornionjoki is the largest salmon river flowing into the Baltic Sea. This thesis tackles the issues of model formulation and model checking as well as computational problems related to Bayesian modelling in the context of fisheries stock assessment. Each article of the thesis provides a novel method either for extracting information from data obtained via a particular type of sampling system or for integrating the information about the fish stock from multiple sources in terms of a population dynamics model. Mark-recapture and removal sampling schemes and a random catch sampling method are covered for the estimation of the population size. In addition, a method for estimating the stock composition of a salmon catch based on DNA samples is also presented. For most of the articles, Markov chain Monte Carlo (MCMC) simulation has been used as a tool to approximate the posterior distribution. Problems arising from the sampling method are also briefly discussed and potential solutions for these problems are proposed. Special emphasis in the discussion is given to the philosophical foundation of the Bayesian approach in the context of fisheries stock assessment. It is argued that the role of subjective prior knowledge needed in practically all parts of a Bayesian model should be recognized and consequently fully utilised in the process of model formulation.

Discovering hidden structures in molecular data using a Bayesian partition model approach

Advancements in the analysis techniques have led to a rapid accumulation of biological data in databases. Such data often are in the form of sequences of observations, examples including DNA sequences and amino acid sequences of proteins. The scale and quality of the data give promises of answering various biologically relevant questions in more detail than what has been possible before. For example, one may wish to identify areas in an amino acid sequence, which are important for the function of the corresponding protein, or investigate how characteristics on the level of DNA sequence affect the adaptation of a bacterial species to its environment. Many of the interesting questions are intimately associated with the understanding of the evolutionary relationships among the items under consideration. The aim of this work is to develop novel statistical models and computational techniques to meet with the challenge of deriving meaning from the increasing amounts of data. Our main concern is on modeling the evolutionary relationships based on the observed molecular data. We operate within a Bayesian statistical framework, which allows a probabilistic quantification of the uncertainties related to a particular solution. As the basis of our modeling approach we utilize a partition model, which is used to describe the structure of data by appropriately dividing the data items into clusters of related items. Generalizations and modifications of the partition model are developed and applied to various problems. Large-scale data sets provide also a computational challenge. The models used to describe the data must be realistic enough to capture the essential features of the current modeling task but, at the same time, simple enough to make it possible to carry out the inference in practice. The partition model fulfills these two requirements. The problem-specific features can be taken into account by modifying the prior probability distributions of the model parameters. The computational efficiency stems from the ability to integrate out the parameters of the partition model analytically, which enables the use of efficient stochastic search algorithms.

Joint Regression and Association Models for Repeated Categorical Responses

The focus of this study is on statistical analysis of categorical responses, where the response values are dependent of each other. The most typical example of this kind of dependence is when repeated responses have been obtained from the same study unit. For example, in Paper I, the response of interest is the pneumococcal nasopharengyal carriage (yes/no) on 329 children. For each child, the carriage is measured nine times during the first 18 months of life, and thus repeated respones on each child cannot be assumed independent of each other. In the case of the above example, the interest typically lies in the carriage prevalence, and whether different risk factors affect the prevalence. Regression analysis is the established method for studying the effects of risk factors. In order to make correct inferences from the regression model, the associations between repeated responses need to be taken into account. The analysis of repeated categorical responses typically focus on regression modelling. However, further insights can also be gained by investigating the structure of the association. The central theme in this study is on the development of joint regression and association models. The analysis of repeated, or otherwise clustered, categorical responses is computationally difficult. Likelihood-based inference is often feasible only when the number of repeated responses for each study unit is small. In Paper IV, an algorithm is presented, which substantially facilitates maximum likelihood fitting, especially when the number of repeated responses increase. In addition, a notable result arising from this work is the freely available software for likelihood-based estimation of clustered categorical responses.

Mathematical Aspects of Financial Markets with Frictions

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property.

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.