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A note on smoothness measures for space–time surfaces

Abstract

The differentiability of a random field has a direct relationship with the differentiability of its covariance function. We review the concept of differentiability of space–time covariance models and random fields, and its implications on predictions. We analyze the change of behavior of the covariance function at the origin and at different space–time lags away from the origin, by using the concept of smoothness which can be considered the geometrical view of the differentiability. We propose a way to measure the smoothness of any covariance function, and apply it to purely spatial and space–time covariance functions.

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Acknowledgments

Work partially funded and supported by Grants MTM 2010-14961 and P1-1B2012-52.

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Correspondence to M. Bohorquez.

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Bohorquez, M., Mateu, J. & Diaz, L. A note on smoothness measures for space–time surfaces. Stoch Environ Res Risk Assess 28, 1011–1022 (2014). https://doi.org/10.1007/s00477-013-0797-8

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Keywords

  • Geostatistics
  • Random fields
  • General second-order processes
  • Continuity and differentiation questions
  • Surfaces in Euclidean space
  • Graphical methods