successive standardization of rectangular arrays连续的标准化矩形数组.pdf

Algorithms 2012, 5, 98-112; doi:10.3390/a5010098 OPEN ACCESS algorithms ISSN 1999-4893 /journal/algorithms Article Successive Standardization of Rectangular Arrays Richard A. Olshen ,,,⋆ and Bala Rajaratnam ,,,⋆ Department of Health Research and Policy—Biostatistics, HRP Redwood Building, Stanford University School of Medicine, Stanford, CA 94305-5405, USA Department of Electrical Engineering, Stanford University, Packard Electrical Engineering Building, 350 Serra Mall, Stanford, CA 94305, USA Department of Statistics, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, CA 94305-4065, USA Department of Environmental Earth System Science, Yang and Yamazaki Environment Energy Building, 473 Via Ortega, Suite 140 Stanford, CA 94305, USA The Woods Institute for the Environment, Jerry Yang Akiko Yamazaki Environment Energy Building—MC 4205, 473 Via Ortega, Stanford, CA 94305, USA * Authors to whom correspondence should be addressed; E-Mail: olshen@; Tel.:+1-650-725-2241; Fax: +1-650-725-8977 (R.A.O.); E-Mail: brajarat@; Tel.: +1-650-721-6404; Fax:+1-650-725-8977 (B.R.) Received: 17 November 2011; in revised form: 20 February 2012 / Accepted: 20 February 2012 / Published: 29 February 2012 Abstract: 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 [1] and [2]. 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