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Successive Standardization of Rectangular Arrays.

Richard A Olshen1, Bala Rajaratnam

  • 1Department 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.

Algorithms
|January 29, 2013
PubMed
Summary
This summary is machine-generated.

This study details "Efron's algorithm," a successive standardization method for numerical arrays. The algorithm rapidly converges to a standardized array with zero means and unit standard deviations, with rare exceptions.

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Area of Science:

  • Numerical Analysis
  • Data Standardization
  • Statistical Computing

Background:

  • Successive standardization (normalization) is a technique applied to numerical arrays.
  • Previous work by the authors established the concept in [1] and [2].
  • The algorithm, termed "Efron's algorithm," was inspired by Bradley Efron.

Purpose of the Study:

  • To further illustrate and mathematically develop successive standardization using examples.
  • To present a corrected and rigorous proof for the convergence of the algorithm.
  • To demonstrate the rapid convergence properties of the algorithm through numerical examples.

Main Methods:

  • Applying iterative operations of subtracting column means and dividing by column standard deviations.
  • Successively applying row operations (subtracting row means and dividing by row standard deviations).
  • Repeating these four operations sequentially to complete one iteration, iterating multiple times.

Main Results:

  • The iterative process converges to a limiting array with row and column means of 0 and standard deviations of 1.
  • The set of arrays for which convergence fails has Lebesgue measure 0.
  • Convergence is demonstrated to be very rapid, often exponentially fast, with few exceptions.

Conclusions:

  • The corrected theorem confirms the convergence of "Efron's algorithm" for real-valued arrays.
  • The algorithm provides an efficient method for standardizing numerical data.
  • Numerical examples confirm the rapid convergence and practical utility of the algorithm.