Probability / Statistics

Procrustean methods are used to transform one set of data to represent another set of data as closely as possible. This book unifies several strands in the literature and contains new algorithms. It focuses on matching two or more configurations by using orthogonal, projection, and oblique axes transformations. Group-average summaries play an important part, and links with other group-average methods are discussed. The text is multi-disciplinary and also presents a unifying ANOVA framework.

This book sets out a body of rigorous mathematical theory for finite graphs with nodes placed randomly in Euclidean d-space according to a common probability density, and edges added to connect points that are close to each other. As an alternative to classical random graph models, these geometric graphs are relevant to the modelling of real networks having spatial content, arising for example in wireless communications, parallel processing, classification, epidemiology, astronomy, and the internet. Their study illustrates numerous techniques of modern stochastic geometry, including Stein's method, martingale methods, and continuum percolation. Typical results in the book concern properties of a graph G on n random points with edges included for interpoint distances up to r, with the parameter r dependent on n and typically small for large n. Asymptotic distributional properties are derived for numerous graph quantities. These include the number of copies of a given finite graph embedded in G, the number of isolated components isomorphic to a given graph, the empirical distributions of vertex degrees, the clique number, the chromatic number, the maximum and minimum degree, the size of the largest component, the total number of components, and the connectivity of the graph.

The book examines the distinguishing features of the various contending approaches to statistical inference (including decision-making) that are currently available in statistical literature, and traces the historical evolution of the concepts underlying these approaches and their applications. The first part entitled, Perspective, shows that statistical inference is really a prolongation of the philosophical problem of induction, and in it, probability is involved both in the input (in the form of model) and the output (for quantifying uncertainty). Four different approaches (behavioural, instantial, pro-subjective Bayesian, and purely subjective) to such statistical induction arise due to the invocation of different conceptions of probability (objective and subjective) at the two stages of the process. The comparative characteristics, advantages, and disadvantages of the different approaches are considered, and it is concluded that each is appropriate in its natural setting. The second part entitled, History, discusses how the different types of probability originated and evolved, and how their application to statistical induction gave rise to the variety of concepts and principles associated with the different approaches. After some reference to pre-history, the developments made by the principal contributors to probability and statistics during 17th-20th centuries (from Cardano, Pascal, Fermat, Huygens, and James Bernoulli through to Daniel Bernoulli, Bayes, Laplace, Gauss, to Galton, Karl Pearon, Fisher, Jeffreys, de Finetti, Neyman, E. S. Pearson, Wald, and their successors) are delineated.

This book discusses novel advances in informatics and statistics in molecular cancer research. Through eight chapters it discusses specific topics in cancer research, talks about how the topics give rise to development of new informatics and statistics tools, and explains how the tools can be applied. The focus of the book is to provide an understanding of key concepts and tools, rather than focusing on technical issues. A main theme is the extensive use of array technologies in modern cancer research — gene expression and exon arrays, SNP and copy number arrays and methylation arrays — to derive quantitative and qualitative statements about cancer, its progression and aetiology, and to understand how these technologies at one hand allow us learn about cancer tissue as a complex system and at the other hand allow us to pinpoint key genes and events as crucial for the development of the disease. Cancer is characterized by genetic and genomic alterations that influence all levels of the cell's machinery and function.

Systems evolve and interact, and when they do this in a continuous way, differential equations can be used to provide accurate mathematical models for the interaction or reaction of the controlled system to external stimuli or forcing. If the forcing is not smooth on normal scales, one cannot use classical calculus. The theory of rough paths provides a rigorous mathematical extension of Newtonian calculus, allowing one to model the responses of systems subject to more wildly oscillating or ‘rough’ stimuli. This book defines rough paths as the completion of the piecewise smooth paths under a p-variation rough path metric. Building on the earlier work of K.T. Chen on the iterated integrals of paths, and of Young on the integration of paths with p-variation < 2, the book proves that the Itô functional, taking stimuli to responses, is uniformly continuous in the p-rough path metric; from which it is elementary to make sense of these differential equations when the external stimuli are rough paths. One important initial application of these results is to stochastic differential equations. Almost every Brownian path is a p-rough path for every p > 2. In addition, the theory allows one to consider stimuli outside of the classes traditionally treated by the Itô calculus, for example the book explains how fractional Brownian motion can often be regarded as a rough path. The basic estimates are uniform without regard to dimension, and apply to infinite dimensional noise sources.

Thorvald Nicolai Thiele was a brilliant Danish researcher of the 19th century. He was a professor of Astronomy at the University of Copenhagen and the founder of Hafnia, the first Danish private insurance company. Thiele worked in astronomy, mathematics, actuarial science, and statistics, his most spectacular contributions were in the latter two areas, where his published work was far ahead of his time. This book is concerned with his statistical work. It evolves around his three main statistical masterpieces, which are now translated into English for the first time: 1) his article from 1880 where he derives the Kalman filter; 2) his book from 1889, where he lays out the subject of statistics in a highly original way, derives the half-invariants (today known as cumulants), the notion of likelihood in the case of binomial experiments, the canonical form of the linear normal model, and develops model criticism via analysis of residuals; and 3) an article from 1899 where he completes the theory of the half-invariants. This book also contains three chapters, written by A. Hald and S. L. Lauritzen, which describe Thiele's statistical work in modern terms and puts it into an historical perspective.