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

Abstract

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.

Co-registration and bias corrections of satellite elevation data sets for quantifying glacier thickness change

“…Standard principles of error propagation are used for estimating errors between two DEMs (Burrough et al, 1998). For example, if one DEM has a random error, σ 1 , and the second DEM, σ 2 , then the resulting error of a statistically independent elevation difference point or pixel is defined as:…”

“…where r is the correlation between σ 1 and σ 2 (Burrough et al, 1998;Etzelmüller, 2000). Determination of r requires semi-variogram analysis and advanced statistical procedures (Bretherton et al, 1999;Rolstad et al, 2009).…”

“…Determination of r requires semi-variogram analysis and advanced statistical procedures (Bretherton et al, 1999;Rolstad et al, 2009). When analyzing and quantifying glacier elevation changes, not just the spread of elevation changes is desired but rather the mean of the elevation changes over a particular area, e.g.…”

On the skill of seasonal sea surface temperature forecasts in the California Current System and its connection to ENSO variability

“…Forecast anomalies are calculated based on a lead-dependent climatology to account for model drift. We evaluate significance of forecast ACC using the method of Bretherton et al (1999), which corrects for autocorrelation in the sample. Significance is tested relative to zero (no forecast skill) and relative to a persistence forecast, which assumes that SST anomalies from the month prior to initialization will persist across all lead times.…”

Arctic Oscillation and ice severity in the Bohai Sea, East Asia

“…To avoid any overestimation, we applied the (Bretherton et al, 1999) method to calculate the effective degree of freedom by computing the auto-correlation of the time series. By this method, the number of degrees of freedom is shorter by about eight than the sample length of the ice severity index.…”