Remembering Leo Breiman

I published an interview of Leo Breiman in Statistical Science [Olshen (2001)], and also the solution to a problem concerning almost sure convergence of binary tree-structured estimators in regression [Olshen (2007)]. The former summarized much of my thinking about Leo up to five years before his death. I discussed the latter with Leo and dedicated that paper to his memory. Therefore, this note is on other topics. In preparing it I am reminded how much I miss this man of so many talents and interests. I miss him not because I always agreed with him, but instead because his comments about statistics in particular and life in general always elicited my substantial reflection.Comment: Published in at http://dx.doi.org/10.1214/10-AOAS385 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Successive Standardization of Rectangular Arrays

In this note we illustrate and develop further with mathematics and examples, the work on successive standardization (or normalization) that is studied earlier by the same authors in Olshen and Rajaratnam (2010) and Olshen and Rajaratnam (2011). Thus, we deal with successive iterations applied to rectangular arrays of numbers, where to avoid technical difficulties an array has at least three rows and at least three columns. Without loss, an iteration begins with operations on columns: first subtract the mean of each column; then divide by its standard deviation. The iteration continues with the same two operations done successively for rows. These four operations applied in sequence completes one iteration. One then iterates again, and again, and again,.... In Olshen and Rajaratnam (2010) it was argued that if arrays are made up of real numbers, then the set for which convergence of these successive iterations fails has Lebesgue measure 0. The limiting array has row and column means 0, row and column standard deviations 1. A basic result on convergence given in Olshen and Rajaratnam (2010) is true, though the argument in Olshen and Rajaratnam (2010) is faulty. The result is stated in the form of a theorem here, and the argument for the theorem is correct. Moreover, many graphics given in Olshen and Rajaratnam (2010) suggest that but for a set of entries of any array with Lebesgue measure 0, convergence is very rapid, eventually exponentially fast in the number of iterations. Because we learned this set of rules from Bradley Efron, we call it "Efron's algorithm". More importantly, the rapidity of convergence is illustrated by numerical examples

Successive normalization of rectangular arrays

Standard statistical techniques often require transforming data to have mean $0$ and standard deviation $1$. Typically, this process of "standardization" or "normalization" is applied across subjects when each subject produces a single number. High throughput genomic and financial data often come as rectangular arrays where each coordinate in one direction concerns subjects who might have different status (case or control, say), and each coordinate in the other designates "outcome" for a specific feature, for example, "gene," "polymorphic site" or some aspect of financial profile. It may happen, when analyzing data that arrive as a rectangular array, that one requires BOTH the subjects and the features to be "on the same footing." Thus there may be a need to standardize across rows and columns of the rectangular matrix. There arises the question as to how to achieve this double normalization. We propose and investigate the convergence of what seems to us a natural approach to successive normalization which we learned from our colleague Bradley Efron. We also study the implementation of the method on simulated data and also on data that arose from scientific experimentation.Comment: Published in at http://dx.doi.org/10.1214/09-AOS743 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org). With Correction

A Generalized Unimodality

Generalization of unimodality for random objects taking values in finite dimensional vector spac

Almost surely consistent nonparametric regression from recursive partitioning schemes

AbstractPresented here are results on almost sure convergence of estimators of regression functions subject to certain moment restrictions. Two somewhat different notions of almost sure convergence are studied: unconditional and conditional given a training sample. The estimators are local means derived from certain recursive partitioning schemes

Gene Expression Differences between Enriched Normal and Chronic Myelogenous Leukemia Quiescent Stem/Progenitor Cells and Correlations with Biological Abnormalities

In comparing gene expression of normal and CML CD34+ quiescent (G0) cell, 292 genes were downregulated and 192 genes upregulated in the CML/G0 Cells. The differentially expressed genes were grouped according to their reported functions, and correlations were sought with biological differences previously observed between the same groups. The most relevant findings include the following. (i) CML G0 cells are in a more advanced stage of development and more poised to proliferate than normal G0 cells. (ii) When CML G0 cells are stimulated to proliferate, they differentiate and mature more rapidly than normal counterpart. (iii) Whereas normal G0 cells form only granulocyte/monocyte colonies when stimulated by cytokines, CML G0 cells form a combination of the above and erythroid clusters and colonies. (iv) Prominin-1 is the gene most downregulated in CML G0 cells, and this appears to be associated with the spontaneous formation of erythroid colonies by CML progenitors without EPO

A classification model for distinguishing copy number variants from cancer-related alterations

<p>Abstract</p> <p>Background</p> <p>Both somatic copy number alterations (CNAs) and germline copy number variants (CNVs) that are prevalent in healthy individuals can appear as recurrent changes in comparative genomic hybridization (CGH) analyses of tumors. In order to identify important cancer genes CNAs and CNVs must be distinguished. Although the Database of Genomic Variants (DGV) contains a list of all known CNVs, there is no standard methodology to use the database effectively.</p> <p>Results</p> <p>We develop a prediction model that distinguishes CNVs from CNAs based on the information contained in the DGV and several other variables, including segment's length, height, closeness to a telomere or centromere and occurrence in other patients. The models are fitted on data from glioblastoma and their corresponding normal samples that were collected as part of The Cancer Genome Atlas project and hybridized to Agilent 244 K arrays.</p> <p>Conclusions</p> <p>Using the DGV alone CNVs in the test set can be correctly identified with about 85% accuracy if the outliers are removed before segmentation and with 72% accuracy if the outliers are included, and additional variables improve the prediction by about 2-3% and 12%, respectively. Final models applied to data from ovarian tumors have about 90% accuracy with all the variables and 86% accuracy with the DGV alone.</p