How real are observed trends in small correlated datasets?

Abstract

The eye may perceive a significant trend in plotted time-series data, but if the model errors of nearby data points are correlated, the trend may be an illusion. We examine generalized least-squares (GLS) estimation, finding that error correlation may be underestimated in highly correlated small datasets by conventional techniques. This risks indicating a significant trend when there is none. A new correlation estimate based on the Durbin–Watson statistic is developed, leading to an improved estimate of autoregression with highly correlated data, thus reducing this risk. These techniques are generalized to randomly located data points in space, through the new concept of the nearest new neighbour path. We describe tests on the validity of the GLS schemes, allowing verification of the models employed. Examples illustrating our method include a 40-year record of atmospheric carbon dioxide, and Antarctic ice core data. While more conservative than existing techniques, our new GLS estimate finds a statistically significant increase in background carbon dioxide concentration, with an accelerating trend. We conclude with an example of a worldwide empirical climate model for radio propagation studies, to illustrate dealing with spatial correlation in unevenly distributed data points over the surface of the Earth. The method is generally applicable, not only to climate-related data, but to many other kinds of problems (e.g. biological, medical and geological data), where there are unequally (or randomly) spaced observations in temporally or spatially distributed datasets.

Universal Kriging Prediction of Line-of-Sight Microwave Fading

“…It was suspected that the statistical significance attributed to some of the parameters in this new model by the t-statistic may have been inflated, so we investigated taking spatial correlation into account, by using generalised least squares (GLS) regression instead of OLS regression [33]. This analysis confirmed that parameter dN 1ERAI , nominally the same parameter as dN 1 but from a more recent and much more extensive ECMWF reanalysis [34], was not significant after all.…”

“…In the case of the N sA0.1 parameter, the mean difference is -0.14 N-units, but an unweighted mean is an OLS estimate, which ignores spatial correlation that may exist between nearby stations. This correlation may be taken into account with a GLS estimate, using the methods described in [33], or later in this paper, as the mean difference is a regression model with only an intercept and no other parameters. In the case of N sA0.1 , the GLS estimate of the difference over 20 years is −0.40 N-units, but this is only marginally significant, with a 95% confidence interval from −0.84 to +0.04 Nunits.…”

“…A detailed desciption of OLS and GLS estimation is given in [33], but briefly, OLS assumes uncorrelated errors, and the regression coefficients b OLS are given by b OLS = (X X) −1 X y, (24) where y is the column vector of observed responses, corresponding to the rows of matrix X, whose columns are the prediction parameters, the first column being all ones, to estimate the intercept, the first element of b OLS . The regression model estimateŷ = Xb OLS minimises the sum of squared residuals e =ŷ − y.…”

“…The suitability of the GLS scheme may be judged by a correlation test on the transformed residuals Pe. We do this with a spatial equivalent [33] of the Durbin-Watson statistic d w , the sum of squared residual forward differences divided by the sum of squared residuals. A value of d w close to 2 is expected for uncorrelated residuals; d w < 2 for a positive correlation, or d w > 2 for a negative correlation.…”

Probing the Brain–Body Connection Using Transcranial Magnetic Stimulation (TMS): Validating a Promising Tool to Provide Biomarkers of Neuroplasticity and Central Nervous System Function

“…The order in which the TMS variables were added was based on the Pearson’s correlation results; TMS variables with strongest correlation with the dependent variable (MS symptoms severity) were added first. Independence of observations was tested with Durbin-Watson test (~2.0) [ 104 ] and strength of the relationship was evaluated using R 2 [ 105 ]. The size of the coefficient of determination (regression R 2 ) was interpreted as ‘very weak’ < 0.3, ‘weak’ 0.3–0.49, ‘moderate’ 0.5–0.69, and ‘strong’ > 0.7 [ 106 ].…”