The Effective Number of Spatial Degrees of Freedom of a Time-Varying Field

The authors systematically investigate two easily computed measures of the effective number of spatial degrees of freedom (ESDOF), or number of independently varying spatial patterns, of a time-varying field of data. The first measure is based on matching the mean and variance of the time series of the spatially integrated squared anomaly of the field to a chi-squared distribution. The second measure, which is equivalent to the first for a long time sample of normally distributed field values, is based on the partitioning of variance between the EOFs. Although these measures were proposed almost 30 years ago, this paper aims to provide a comprehensive discussion of them that may help promote their more widespread use. The authors summarize the theoretical basis of the two measures and considerations when estimating them with a limited time sample or from nonnormally distributed data. It is shown that standard statistical significance tests for the difference or correlation between two realizations of a field (e.g., a forecast and an observation) are approximately valid if the number of degrees of freedom is chosen using an appropriate combination of the two ESDOF measures. Also described is a method involving ESDOF for deciding whether two time-varying fields are significantly correlated to each other. A discussion of the parallels between ESDOF and the effective sample size of an autocorrelated time series is given, and the authors review how an appropriate measure of effective sample size can be computed for assessing the significance of correlations between two time series.

[1]  C. Schär,et al.  A principal component and long‐term trend analysis of daily precipitation in Switzerland , 1997 .

[2]  Eduardo Damasio Da Costa,et al.  A Qualitatively Realistic Low-Order Model of the Extratropical Low-Frequency Variability Built from Long Records of Potential Vorticity. , 1997 .

[3]  V. Dymnikov,et al.  On the structure of the attractors of the finite-dimensional approximations of the barotropic vorticity equation on a rotating sphere , 1997 .

[4]  Huug van den Dool,et al.  Analysis of model-calculated soil moisture over the United States (1931-1993) and applications to long-range temperature forecasts , 1996 .

[5]  Z. Tóth,et al.  Degrees of freedom in Northern Hemisphere circulation data , 1995 .

[6]  Klaus Fraedrich,et al.  Estimates of Spatial Degrees of Freedom , 1995 .

[7]  John M. Wallace,et al.  The Signature of ENSO in Global Temperature and Precipitation Fields Derived from the Microwave Sounding Unit , 1994 .

[8]  Donald B. Percival,et al.  Spectral Analysis for Physical Applications , 1993 .

[9]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[10]  E. Lorenz.  Dimension of weather and climate attractors , 1991, Nature.

[11]  J. Wallace,et al.  Does low-frequency atmospheric variability exhibit regime-like behavior? , 1991 .

[12]  G. der Megreditchian.  Meteorological networks optimization from a statistical point of view , 1989 .

[13]  R. Vautard,et al.  Singular spectrum analysis in nonlinear dynamics, with applications to paleoclimatic time series , 1989 .

[14]  Masayuki Ishida,et al.  Number of degrees of freedom, correlation times, and equivalent bandwidths of a random process , 1988, IEEE Trans. Inf. Theory.

[15]  H. M. Dool.  An Empirical Study on the Parameterization of Precipitation in a Model of the Time Mean Atmosphere , 1987 .

[16]  R. Chervin,et al.  A Comparison of Month-to-Month Persistence of Anomalies in a General Circulation Model and in the Earth's Atmosphere , 1986 .

[17]  F. Zwiers,et al.  The interpretation and estimation of effective sample size , 1984 .

[18]  R. E. Livezey,et al.  Statistical Field Significance and its Determination by Monte Carlo Techniques , 1983 .

[19]  Robert F. Cahalan,et al.  Sampling Errors in the Estimation of Empirical Orthogonal Functions , 1982 .

[20]  G. Reinsel,et al.  Introduction to Mathematical Statistics (4th ed.). , 1980 .

[21]  R. Laha.  Probability Theory , 1979 .

[22]  Russ E. Davis,et al.  Predictability of Sea Surface Temperature and Sea Level Pressure Anomalies over the North Pacific Ocean , 1976 .

[23]  Richard H. Jones.  Estimating the Variance of Time Averages , 1975 .

[24]  C. E. Leith,et al.  The Standard Error of Time-Average Estimates of Climatic Means , 1973 .

[25]  J. W. Tukey,et al.  The Measurement of Power Spectra from the Point of View of Communications Engineering , 1958 .

[26]  G. Orcutt,et al.  TESTING THE SIGNIFICANCE OF CORRELATION BETWEEN TIME SERIES , 1948 .

[27]  J. M. Hammersley,et al.  The “Effective” Number of Independent Observations in an Autocorrelated Time Series , 1946 .

[28]  M. S. Bartlett,et al.  Some Aspects of the Time-Correlation Problem in Regard to Tests of Significance , 1935 .

[29]  Geoffrey Ingram Taylor,et al.  Diffusion by Continuous Movements , 1922 .