Although there are many commercially available statistical software packages, only a few implement a competing risk analysis or a proportional hazards regression model with time-dependent covariates, which are necessary in studies on hematopoietic SCT. In addition, most packages are not clinician friendly, as they require that commands be written based on statistical languages. This report describes the statistical software 'EZR' (Easy R), which is based on R and R commander. EZR enables the application of statistical functions that are frequently used in clinical studies, such as survival analyses, including competing risk analyses and the use of time-dependent covariates, receiver operating characteristics analyses, meta-analyses, sample size calculation and so on, by point-and-click access. EZR is freely available on our website (http://www.jichi.ac.jp/saitama-sct/SaitamaHP.files/statmed.html) and runs on both Windows (Microsoft Corporation, USA) and Mac OS X (Apple, USA). This report provides instructions for the installation and operation of EZR.

/pdf/investigation-of-the-freely-available-easy-to-use-software-4qj9c0mudl.pdf

Investigation of the freely available easy-to-use software 'EZR' for medical statistics.

An overview of the self-organizing map algorithm, on which the papers in this issue are based, is presented in this article.

The Self-Organizing Map

Quantization is a process that maps a continous or discrete set of values into approximations that belong to a smaller set. Quantization is a lossy: some information about the original data is lost in the process. The key to a successful quantization is therefore the selection of an error criterion – such as entropy and signal-to-noise ratio – and the development of optimal quantizers for this criterion.

Quantization

This paper synthesizes the results, methodology, and research conducted concerning the K-means clustering method over the last fifty years. The K-means method is first introduced, various formulations of the minimum variance loss function and alternative loss functions within the same class are outlined, and different methods of choosing the number of clusters and initialization, variable preprocessing, and data reduction schemes are discussed. Theoretic statistical results are provided and various extensions of K-means using different metrics or modifications of the original algorithm are given, leading to a unifying treatment of K-means and some of its extensions. Finally, several future studies are outlined that could enhance the understanding of numerous subtleties affecting the performance of the K-means method.

K‐means clustering: A half‐century synthesis

Remarkable advances in computation and data storage and the ready availability of huge data sets have been the keys to the growth of the new disciplines of data mining and machine learning, while the enormous success of the Human Genome Project has opened up the field of bioinformatics. These exciting developments, which led to the introduction of many innovative statistical tools for high-dimensional data analysis, are described here in detail. The author takes a broad perspective; for the first time in a book on multivariate analysis, nonlinear methods are discussed in detail as well as linear methods. Techniques covered range from traditional multivariate methods, such as multiple regression, principal components, canonical variates, linear discriminant analysis, factor analysis, clustering, multidimensional scaling, and correspondence analysis, to the newer methods of density estimation, projection pursuit, neural networks, multivariate reduced-rank regression, nonlinear manifold learning, bagging, boosting, random forests, independent component analysis, support vector machines, and classification and regression trees. Another unique feature of this book is the discussion of database management systems. This book is appropriate for advanced undergraduate students, graduate students, and researchers in statistics, computer science, artificial intelligence, psychology, cognitive sciences, business, medicine, bioinformatics, and engineering. Familiarity with multivariable calculus, linear algebra, and probability and statistics is required. The book presents a carefully-integrated mixture of theory and applications, and of classical and modern multivariate statistical techniques, including Bayesian methods. There are over 60 interesting data sets used as examples in the book, over 200 exercises, and many color illustrations and photographs.

Modern Multivariate Statistical Techniques: Regression, Classification, and Manifold Learning

The problem of clustering functional data is addressed. Results on principal points (cluster means for probability distributions) are given for functional Gaussian distributions. Examples and simulations are provided to illustrate results.

Clustering Functional Data

The term "self-consistency" was introduced in 1989 by Hastie and Stuetzle to describe the property that each point on a smooth curve or surface is the mean of all points that project orthogonally onto it. We generalize this concept to self-consistent random vectors: a random vector Y is self-consistent for X if $\mathscr{E}[X|Y] = Y$ almost surely. This allows us to construct a unified theoretical basis for principal components, principal curves and surfaces, principal points, principal variables, principal modes of variation and other statistical methods. We provide some general results on self-consistent random variables, give examples, show relationships between the various methods, discuss a related notion of self-consistent estimators and suggest directions for future research.

https://corescholar.libraries.wright.edu/cgi/viewcontent.cgi?article=1047&context=math

Self-Consistency: A Fundamental Concept in Statistics

There is a well-known simple formula for computing prediction sum of squares (PRESS) residuals in a regression problem without having to refit the curve for each observation. This note shows that the same basic result holds for fitting a regression function when the regression coefficients are subject to linear constraints.

https://www.jstor.org/stable/info/2686028

A Note on the Prediction Sum of Squares Statistic for Restricted Least Squares

Functional data can be clustered by plugging estimated regression coefficients from individual curves into the k-means algorithm. Clustering results can differ depending on how the curves are fit to the data. Estimating curves using different sets of basis functions corresponds to different linear transformations of the data. k-means clustering is not invariant to linear transformations of the data. The optimal linear transformation for clustering will stretch the distribution so that the primary direction of variability aligns with actual differences in the clusters. It is shown that clustering the raw data will often give results similar to clustering regression coefficients obtained using an orthogonal design matrix. Clustering functional data using an L2 metric on function space can be achieved by clustering a suitable linear transformation of the regression coefficients. An example where depressed individuals are treated with an antidepressant is used for illustration.

/pdf/linear-transformations-and-the-k-means-clustering-algorithm-2ig3swyip4.pdf

Linear Transformations and the k-Means Clustering Algorithm: Applications to Clustering Curves

The $k$ principal points of a $p$-variate random vector $\mathbf{X}$ are those points $\xi_1, \ldots, \xi_k \in \mathbb{R}^p$ which approximate the distribution of $\mathbf{X}$ by minimizing the expected squared distance of $\mathbf{X}$ from the nearest of the $\xi_j$. Any set of $k$ points $\mathbf{y}_1, \ldots, \mathbf{y}_k$ partitions $\mathbb{R}^p$ into "domains of attraction" $D_1, \ldots, D_k$ according to minimal distance; following Hastie and Stuetzle we call $\mathbf{y}_1, \ldots, \mathbf{y}_k$ self-consistent if $E\lbrack\mathbf{X}\mid\mathbf{X} \in D_j\rbrack = \mathbf{y}_j$ for $j = 1, \ldots, k$. Principal points are a special case of self-consistent points. In this paper we study principal points and self-consistent points of $p$-variate elliptical distributions. The main results are the following: (1) If $k$ self-consistent points of $\mathbf{X}$ span a subspace of dimension $q < p$, then this subspace is also spanned by $q$ principal components, that is, self-consistent points of elliptical distributions exist only in principal component subspaces. (2) The subspace spanned by $k$ principal points of $\mathbf{X}$ is identical with the subspace spanned by the principal components associated with the largest roots. This proves a conjecture of Flury. We also discuss implications of our results for the computation and estimation of principal points.

/pdf/principal-points-and-self-consistent-points-of-elliptical-3dtg0zitdn.pdf

Principal Points and Self-Consistent Points of Elliptical Distributions

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