Soft computing, as an engineering science, and statistics, as a classical branch of mathematics, emphasize different aspects of data analysis. Soft computing focuses on obtaining working solutions quickly, accepting approximations and unconventional approaches. Its strength lies in its flexibility to create models that suit the needs arising in applications. In addition, it emphasizes the need for intuitive and interpretable models, which are tolerant to imprecision and uncertainty. Statistics is more rigorous and focuses on establishing objective conclusions based on experimental data by analyzing the possible situations and their (relative) likelihood. It emphasizes the need for mathematical methods and tools to assess solutions and guarantee performance. Combining the two fields enhances the robustness and generalizability of data analysis methods, while preserving the flexibility to solve real-world problems efficiently and intuitively.

Fuzzy logic is a computational paradigm capable of modelling the own uncertainness of human beings. This wide-ranging book focuses in-depth on the VLSI CMOS implementation and application of programmable analogue Fuzzy Logic Controllers following a mixed-signal philosophy.

Soft computing, as an engineering science, and statistics, as a classical branch of mathematics, emphasize different aspects of data analysis.
Soft computing focuses on obtaining working solutions quickly, accepting approximations and unconventional approaches. Its strength lies in its flexibility to create models that suit the needs arising in applications. In addition, it emphasizes the need for intuitive and interpretable models, which are tolerant to imprecision and uncertainty.
Statistics is more rigorous and focuses on establishing objective conclusions based on experimental data by analyzing the possible situations and their (relative) likelihood. It emphasizes the need for mathematical methods and tools to assess solutions and guarantee performance.
Combining the two fields enhances the robustness and generalizability of data analysis methods, while preserving the flexibility to solve real-world problems efficiently and intuitively.

Similarity-based learning methods have a great potential as an intuitive and ?exible toolbox for mining, visualization,and inspection of largedata sets. They combine simple and human-understandable principles, such as distance-based classi?cation, prototypes, or Hebbian learning, with a large variety of di?erent, problem-adapted design choices, such as a data-optimum topology, similarity measure, or learning mode. In medicine, biology, and medical bioinformatics, more and more data arise from clinical measurements such as EEG or fMRI studies for monitoring brain activity, mass spectrometry data for the detection of proteins, peptides and composites, or microarray pro?les for the analysis of gene expressions. Typically, data are high-dimensional, noisy, and very hard to inspect using classic (e. g. , symbolic or linear) methods. At the same time, new technologies ranging from the possibility of a very high resolution of spectra to high-throughput screening for microarray data are rapidly developing and carry thepromiseofane?cient,cheap,andautomaticgatheringoftonsofhigh-quality data with large information potential. Thus, there is a need for appropriate - chine learning methods which help to automatically extract and interpret the relevant parts of this information and which, eventually, help to enable und- standingofbiologicalsystems,reliablediagnosisoffaults,andtherapyofdiseases such as cancer based on this information. Moreover, these application scenarios pose fundamental and qualitatively new challenges to the learning systems - cause of the speci?cs of the data and learning tasks. Since these characteristics are particularly pronounced within the medical domain, but not limited to it and of principled interest, this research topic opens the way toward important new directions of algorithmic design and accompanying theory.

Fuzzy logic is a computational paradigm capable of modelling the own uncertainness of human beings. This wide-ranging book focuses in-depth on the VLSI CMOS implementation and application of programmable analogue Fuzzy Logic Controllers following a mixed-signal philosophy.

Methods of dimensionality reduction provide a way to understand and visualize the structure of complex data sets. Traditional methods like principal component analysis and classical metric multidimensional scaling suffer from being based on linear models. Until recently, very few methods were able to reduce the data dimensionality in a nonlinear way. However, since the late nineties, many new methods have been developed and nonlinear dimensionality reduction, also called manifold learning, has become a hot topic. New advances that account for this rapid growth are, e.g. the use of graphs to represent the manifold topology, and the use of new metrics like the geodesic distance. In addition, new optimization schemes, based on kernel techniques and spectral decomposition, have lead to spectral embedding, which encompasses many of the

Methods of dimensionality reduction provide a way to understand and visualize the structure of complex data sets. Traditional methods like principal component analysis and classical metric multidimensional scaling suffer from being based on linear models. Until recently, very few methods were able to reduce the data dimensionality in a nonlinear way. However, since the late nineties, many new methods have been developed and nonlinear dimensionality reduction, also called manifold learning, has become a hot topic. New advances that account for this rapid growth are, e.g. the use of graphs to represent the manifold topology, and the use of new metrics like the geodesic distance. In addition, new optimization schemes, based on kernel techniques and spectral decomposition, have lead to spectral embedding, which encompasses many of the recently developed methods.

This book describes existing and advanced methods to reduce the dimensionality of numerical databases. For each method, the description starts from intuitive ideas, develops the necessary mathematical details, and ends by outlining the algorithmic implementation. Methods are compared with each other with the help of different illustrative examples.

The purpose of the book is to summarize clear facts and ideas about well-known methods as well as recent developments in the topic of nonlinear dimensionality reduction. With this goal in mind, methods are all described from a unifying point of view, in order to highlight their respective strengths and shortcomings.

The book is primarily intended for statisticians, computer scientists and data analysts. It is also accessible to other practitioners having a basic background in statistics and/or computational learning, like psychologists (in psychometry) and economists.