Abstract
Abstract. River water-quality time series often exhibit fractal scaling, which here refers to autocorrelation that decays as a power law over some range of scales. Fractal scaling presents challenges to the identification of deterministic trends because (1) fractal scaling has the potential to lead to false inference about the statistical significance of trends and (2) the abundance of irregularly spaced data in water-quality monitoring networks complicates efforts to quantify fractal scaling. Traditional methods for estimating fractal scaling – in the form of spectral slope (β) or other equivalent scaling parameters (e.g., Hurst exponent) – are generally inapplicable to irregularly sampled data. Here we consider two types of estimation approaches for irregularly sampled data and evaluate their performance using synthetic time series. These time series were generated such that (1) they exhibit a wide range of prescribed fractal scaling behaviors, ranging from white noise (β = 0) to Brown noise (β = 2) and (2) their sampling gap intervals mimic the sampling irregularity (as quantified by both the skewness and mean of gap-interval lengths) in real water-quality data. The results suggest that none of the existing methods fully account for the effects of sampling irregularity on β estimation. First, the results illustrate the danger of using interpolation for gap filling when examining autocorrelation, as the interpolation methods consistently underestimate or overestimate β under a wide range of prescribed β values and gap distributions. Second, the widely used Lomb–Scargle spectral method also consistently underestimates β. A previously published modified form, using only the lowest 5 % of the frequencies for spectral slope estimation, has very poor precision, although the overall bias is small. Third, a recent wavelet-based method, coupled with an aliasing filter, generally has the smallest bias and root-mean-squared error among all methods for a wide range of prescribed β values and gap distributions. The aliasing method, however, does not itself account for sampling irregularity, and this introduces some bias in the result. Nonetheless, the wavelet method is recommended for estimating β in irregular time series until improved methods are developed. Finally, all methods' performances depend strongly on the sampling irregularity, highlighting that the accuracy and precision of each method are data specific. Accurately quantifying the strength of fractal scaling in irregular water-quality time series remains an unresolved challenge for the hydrologic community and for other disciplines that must grapple with irregular sampling.
Highlights
1.1 Autocorrelations in time seriesIt is well known that time series from natural systems often exhibit autocorrelation; that is, observations at each time step are correlated with observations one or more time stepsPublished by Copernicus Publications on behalf of the European Geosciences Union.Q
Zhang et al.: Evaluation of statistical methods for quantifying fractal scaling in the past. This property is usually characterized by the autocorrelation function (ACF), which is defined as follows for a process Xt at lag k: γ (k) = cov (Xt, Xt+k)
Because traditional spectral estimation methods are generally not applicable to irregularly sampled time series, we have examined two broad types of estimation approaches and evaluated their performances against synthetic data with a wide range of prescribed β values and gap www.hydrol-earth-syst-sci.net/22/1175/2018/
Summary
It is well known that time series from natural systems often exhibit autocorrelation; that is, observations at each time step are correlated with observations one or more time steps. Autocorrelation has been frequently modeled with classical techniques such as autoregressive (AR) or autoregressive moving-average (ARMA) models (Darken et al, 2002; Yue et al, 2002; Box et al, 2008) These models assume that the underlying process has short-term memory; i.e., the ACF decays exponentially with lag k (Box et al, 2008). The short-term memory assumption holds sometimes, it cannot adequately describe many time series whose ACFs decay as a power law ( much slower than exponentially) and may not reach zero even for large lags, which implies that the ACF is non-summable. As stressed by Cohn and Lins (2005), it is “surprising that nearly every assessment of trend significance in geophysical variables published during the past few decades has failed [to do so]”, and a similar tendency is evident in the decade following that statement as well
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