Bernard Ycart

Introductory Statistics and Random Phenomena: Uncertainty, Complexity and Chaotic Behavior in Engineering and Science

I Descriptive Statistics-Compressing Data.- 1 Why One Needs to Analyze Data.- 1.1 Coin tossing, lottery, and the stock market.- 1.2 Inventory problems in management.- 1.3 Battery life and quality control in manufacturing.- 1.4 Reliability of complex systems.- 1.5 Point processes in time and space.- 1.6 Polls-social sciences.- 1.7 Time series.- 1.8 Repeated experiments and testing.- 1.9 Simple chaotic dynamical systems.- 1.10 Complex dynamical systems.- 1.11 Pseudorandom number generators and the Monte-Carlo methods.- 1.12 Fractals and image reconstruction.- 1.13 Coding and decoding, unbreakable ciphers.- 1.14 Experiments, exercises, and projects.- 1.15 Bibliographical notes.- 2 Data Representation and Compression.- 2.1 Data types, categorical data.- 2.2 Numerical data: order statistics, median, quantiles.- 2.3 Numerical data: histograms, means, moments.- 2.4 Location, dispersion, and shape parameters.- 2.5 Probabilities: a frequentist viewpoint.- 2.6 Multidimensional data: histograms and other graphical representations.- 2.7 2-D data: regression and correlations.- 2.8 Fractal data.- 2.9 Measuring information content:entropy.- 2.10 Experiments, exercises, and projects.- 2.11 Bibliographical notes.- 3 Analytic Representation of Random Experimental Data.- 3.1 Repeated experiments and the law of large numbers.- 3.2 Characteristics of experiments: distribution functions, densities, means, variances.- 3.3 Uniform distributions, simulation of random quantities, the Monte Carlo method.- 3.4 Bernoulli and binomial distributions.- 3.5 Rescaling probabilities: Poisson approximation.- 3.6 Stability of Fluctuations Law: Gaussian approximation.- 3.7 How to estimate p in Bernoulli experiments.- 3.8 Other continuous distributions Gamma function calculus.- 3.9 Testing the fit of a distribution.- 3.10 Random vectors and multivariate distributions.- 3.11 Experiments, exercises, and projects.- 3.12 Bibliographical notes.- II Modeling Uncertainty.- 4 Algorithmic Complexity and Random Strings.- 4.1 Heart of randomness: when is random - random?.- 4.2 Computable strings and the Turing machine.- 4.3 Kolmogorov complexity and random strings.- 4.4 Typical sequences: Martin-Lof tests of randomness.- 4.5 Stability of subsequences: von Mises randomness.- 4.6 Computable framework of randomness: degrees of irregularity.- 4.7 Experiments, exercises, and projects.- 4.8 Bibliographical notes.- 5 Statistical Independence and Kolmogorovs Probability Theory.- 5.1 Description of experiments, random variables, and Kolmogorovs axioms.- 5.2 Uniform discrete distributions and counting.- 5.3 Statistical independence as a model for repeated experiments..- 5.4 Expectations and other characteristics of random variables.- 5.4.1 Expectations.- 5.4.2 Expectations of functions of random variables. Variance.- 5.4.3 Expectations of functions of vectors. Covariance.- 5.4.4 Expectation of the product. Variance of the sum of independent random variables.- 5.4.5 Moments and the moment generating function.- 5.4.6 Expectations of general random variables.- 5.5 Averages of independent random variables.- 5.6 Laws of large numbers and small deviations.- 5.7 Central limit theorem and large deviations.- 5.8 Experiments, exercises, and projects.- 5.9 Bibliographical Notes.- 6 Chaos in Dynamical Systems: How Uncertainty Arises in Scientific and Engineering Phenomena.- 6.1 Dynamical systems: general concepts and typical examples.- 6.2 Orbits and fixed points.- 6.3 Stability of frequencies and the ergodic theorem.- 6.4 Stability of fluctuations and the central limit theorem.- 6.5 Attractors, fractals, and entropy.- 6.6 Experiments, exercises, and projects.- 6.7 Bibliographical notes.- III Model Specification-Design of Experiments.- 7 General Principles of Statistical Analysis.- 7.1 Design of experiments and planning of investigation.- 7.2 Model selection.- 7.3 Determining the method of statistical inference.- 7.3.1 Maximum likelihood estimator (MLE).- 7.3.2 Least squares estimator (LSE).- 7.3.3 Method of moments (MM).- 7.3.4 Concluding remarks.- 7.4 Estimation of fractal dimension.- 7.5 Practical side of data collection and analysis.- 7.6 Experiments, exercises, and projects.- 7.7 Bibliographical notes.- 8 Statistical Inference for Normal Populations.- 8.1 Introduction parametric inference.- 8.2 Confidence intervals for one-sample model.- 8.3 From confidence intervals to hypothesis testing.- 8.4 Statistical inference for two-sample normal models.- 8.5 Regression analysis for the normal model.- 8.6 Testing for goodness-of-fit.- 8.7 Experiments, exercises, and projects.- 8.8 Bibliographical notes.- 9 Analysis of Variance.- 9.1 Single-factor ANOVA.- 9.2 Two-factor ANOVA.- 9.3 Experiments, exercises, and projects.- 9.4 Bibliographical notes.- A Uncertainty Principle in Signal Processing and Quantum Mechanics.- B Fuzzy Systems and Logic.- C A Critique of Pure Reason.- D The Remarkable Bernoulli Family.- F Tables.Paraphrasing from the Preface, this book is an account of the major topics in the design of experiments, with emphasis on key concepts and their associated statistical structure. The authors’ goals in writing it are that in individual applications, an experimenter often cannot just use an out-of-the-box design, but instead must adapt existing concepts to x8e t the need. The book is aimed at a fairly general audience of readers “concerned with statistics in experimental x8e elds and with some knowledge and interest in theoretical issues” (from the Preface). The mathematical level is purportedly “mostly elementary.” Maybe so, but certainly not to the point of using notations sparingly. The book begins with a chapter on concepts, followed by one on avoidance of bias, which discusses randomization theory. Chapter 3 is on control of haphazard variation (blocking), and Chapter 4 covers more sophisticated blocking techniques. Next are two chapters on factorial designs, including replicated, saturated, fractional, split-plit, Taguchi, and other designs. Chapter 7 discusses optimal design, and Chapter 8 covers some additional topics, including scale of effort (or sample size) and adaptive, sequential, and spatial methods. Three appendices cover some basic statistical theory, algebra, and computation, the latter emphasizing S-PLUS. The reader will x8e nd something on almost anything that is design-related, but not a lot of exposition; there are only 224 pages of material, not counting the appendices. Topics are usually well referenced, however (the bibliography consumes 12 pages), so it is seldom hard to know where to look for more information. Although this is a book on theory, it offers a fair amount of practical material and mentions an impressively broad array of applications. Not much space is devoted to details of examples, however. I was disappointed to see a block design illustrated by an example (pp. 50–51) where blocking does not help much; then, when efx8e ciency is discussed on the next page, the authors do not bother to compute the estimated efx8e ciency (which is low) of the just-completed example. Given that the chapter is on haphazard variations, one would think that the authors would want to use an example that includes such variations to dramatize the efx8e ciency gained by blocking. The emphasis really is on design and not on analysis. A few topics are not mentioned or are not mentioned much, for example, multiple comparisons and variance-component estimation. I admit that I looked to see whether there is reference to my own meager contribution to the analysis of saturated designs; there is not, but more glaring omissions are evident in ignoring the important contributions of Lin (1993) to supersaturated designs and those of Shoemaker, Tsui, and Wu (1991) to Taguchi designs, even though there is ample mention of other work in these areas. I found a few errors. On page 270, for example, two illustrations are offered of linear models and their corresponding S-PLUS specix8e cations, but the two linear models are identical, and the S-PLUS models are not equivalent. Each chapter (and each of the x8e rst two appendices) ends with lists of exercises. A few of these exercises are routine, but many more are used to expand on what has not been developed explicitly in the text; for example, “construct a theory of orthogonal Latin cubes” (p. 265). Despite the presence of exercises, the book’s level and terseness do not make it a good choice for a x8e rst course in experimental design. It would be suitable for a graduate seminar where participants already have some exposure to experimental design and want to deepen their knowledge. In summary, The Theory of Design of Experiments will be a valuable addition to the library of one who is serious about design and is willing to x8e ll in some details by looking up references or working through them.

Large-scale microarray profiling reveals four stages of immune escape in non-Hodgkin lymphomas

ABSTRACT Non-Hodgkin B-cell lymphoma (B-NHL) are aggressive lymphoid malignancies that develop in patients due to oncogenic activation, chemo-resistance, and immune evasion. Tumor biopsies show that B-NHL frequently uses several immune escape strategies, which has hindered the development of checkpoint blockade immunotherapies in these diseases. To gain a better understanding of B-NHL immune editing, we hypothesized that the transcriptional hallmarks of immune escape associated with these diseases could be identified from the meta-analysis of large series of microarrays from B-NHL biopsies. Thus, 1446 transcriptome microarrays from seven types of B-NHL were downloaded and assembled from 33 public Gene Expression Omnibus (GEO) datasets, and a method for scoring the transcriptional hallmarks in single samples was developed. This approach was validated by matching scores to phenotypic hallmarks of B-NHL such as proliferation, signaling, metabolic activity, and leucocyte infiltration. Through this method, we observed a significant enrichment of 33 immune escape genes in most diffuse large B-cell lymphoma (DLBCL) and follicular lymphoma (FL) samples, with fewer in mantle cell lymphoma (MCL) and marginal zone lymphoma (MZL) samples. Comparing these gene expression patterns with overall survival data evidenced four stages of cancer immune editing in B-NHL: non-immunogenic tumors (stage 1), immunogenic tumors without immune escape (stage 2), immunogenic tumors with immune escape (stage 3), and fully immuno-edited tumors (stage 4). This model complements the standard international prognostic indices for B-NHL and proposes that immune escape stages 3 and 4 (76% of the FL and DLBCL samples in this data set) identify patients relevant for checkpoint blockade immunotherapies.

Probability Distributions on Indexed Dendrograms and Related Problems of Classifiability

This paper studies the dendrograms produced by algorithms of classification such as the Single Link Algorithm. We introduce probability distributions on dendrograms corresponding to distinct non classifiability hypotheses. The distributions of the height of a random dendrogram under these hypotheses are studied and their asymptotics explicitly computed. This leads to statistical tests for non-classifiability.

RENEWAL-TYPE BEHAVIOR OF ABSORPTION TIMES IN MARKOV CHAINS

This paper studies the absorption time of an integer-valued Markov chain with a lower-triangular transition matrix. The main results concern the asymptotic behavior of the absorption time when the starting point tends to infinity (asymptotics of moments and central limit theorem). Mney are obtained using stochastic comparison for Markov chains and the classical theorems of renewal theory. Applications to the description of large random chains of partitions and large random ordered partitions are given.

Statistical Inference for Normal Populations

In this chapter the general assumption is that the statistical model is normal. We begin by discussing the general issue of parametric inference and then quickly move to construction of confidence intervals for one-sample models and the related hypothesis testing issues. A few remarks on the two-sample model follow and the chapter concludes with the regression analysis for the normal model and a goodness-of-fit test.

Random dendrograms for classifiability testing

We propose statistical tests to decide between an hypothesis of non classifiability of the data against the presence of a classification in some simple situations. We consider only the single link algorithm and two null hypotheses of non classifiability, according to whether the dissimilarities or the objects themselves are i.i.d. random variables. Each choice for the distribution of the input of the single link algorithm induces a different probability distribution on the output, which is a random indexed dendrogram. Certain characteristics of these random indexed dendrograms are studied, and their asymptotic distributions computed under each null hypothesis. All these random variables can be used to define statistical tests. Explicit examples of such tests are provided.

General Principles of Statistical Analysis

The exploration of experimental data and the reliability of the statistical inference based on these data depend heavily on the selection of the mathematical model and on the design of the data collection method.

Algorithmic Complexity and Random Strings

In this chapter we will try to get to the heart of the notion of randomness by showing its fundamental connection with several concepts of algorithmic and computational complexity. Although the discussion illuminates the philosophical underpinnings of the concept of randomness for a concrete string of data, the conclusions are sobering: perfectly random strings cannot be produced by any finite algorithms (read, computers). A practical way out of this dilemma is suggested.