On the condition number anomaly of Gaussian correlation matrices

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Abstract

Spatial correlation matrices appear in a large variety of applications.
For example, they are an essential component of} spatial Gaussian processes,
also known as spatial linear models or Kriging estimators,
which are powerful and well-established tools for a multitude of engineering applications such as
the design and analysis of computer experiments, geostatistical problems
and meteorological tasks.

In radial basis function interpolation, Gaussian correlation matrices arise
frequently as interpolation matrices from the Gaussian radial kernel function.
In the field of data assimilation in numerical weather prediction, such matrices arise
as background error covariances.

Over the past thirty years, it was observed by several authors from several fields that the Gaussian
correlation model is exceptionally prone to suffer from ill-conditioning,
but a quantitative theoretical explanation for this anomaly was lacking.
In this paper, a proof for the special position of the Gaussian
correlation matrix is given.
The theoretical findings are illustrated by numerical experiment.
Original languageEnglish
JournalLinear Algebra and Its Applications
Volume466
Pages (from-to)512-526
ISSN0024-3795
DOIs
Publication statusPublished - 2015
Externally publishedYes

Keywords

  • Condition number
  • Correlation matrix
  • Data assimilation
  • Design and analysis of computer experiments
  • Euclidean distance matrix
  • Gaussian process
  • Kriging
  • Radial basis functions
  • Spatial linear model

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