Significance of trends in long-term correlated records

Abstract

We study the distribution P(x;α,L) of the relative trend x in long-term correlated records of length L that are characterized by a Hurst exponent α between 0.5 and 1.5. The relative trend x is the ratio between the strength of the trend Δ in the record measured by linear regression and the standard deviation σ around the regression line. We consider L between 400 and 2200, which is the typical length scale of monthly local and annual reconstructed global climate records. Extending previous work by Lennartz and Bunde [S. Lennartz and A. Bunde, Phys. Rev. E 84, 021129 (2011)] we show explicitly that x follows the Student's t distribution P∝[1+(x/a)2/l]-(l+1)/2, where the scaling parameter a depends on both L and α, while the effective length l depends, for α below 1.15, only on the record length L. From P we can derive an analytical expression for the trend significance S(x;α,L)=∫(-x)xP(x';α,L)dx' and the border lines of the 95% significance interval. We show that the results are nearly independent of the distribution of the data in the record, holding for Gaussian data as well as for highly skewed non-Gaussian data. For an application, we use our methodology to estimate the significance of central west Antarctic warming.

Statistical significance of seasonal warming/cooling trends

“…If H [k * ]) is not rejected, then all H [k] with rank k > k * are also not rejected. When a record is fully characterized by a certain Hurst exponent h, the significance S of a relative trend x depends only on h and the record length N. It has been shown recently by Tamazian et al (54) that S(x, N) also follows Eq. 4, but with different parameters a and l. These parameters depend on h and N and have been tabulated (for 0.5 ≤ h ≤ 1.5 and N ≥ 400) in ref.…”

“…4, but with different parameters a and l. These parameters depend on h and N and have been tabulated (for 0.5 ≤ h ≤ 1.5 and N ≥ 400) in ref. 54. Accordingly, for given h and N, the trend significance can be obtained straightforwardly from ref.…”

“…Accordingly, for given h and N, the trend significance can be obtained straightforwardly from ref. 54, making the trend estimation as easy as for short-term persistent processes.…”

Detrending moving average algorithm: Frequency response and scaling performances

“…In the simplest operational definition, trends are observed when a regression estimated over a data subset has a not negligible slope. For example in climatology, based on regular measurements from weather stations and satellite data, temperature trends are estimated both locally and globally [1][2][3][4]. In finance and economics, technical rules and visualization tools based on moving average trends are under continuous investigation and improvement [5][6][7][8].…”

“…Another critical aspect is the ability to distinguish whether the trend or other stochastic component embedded in a nonstationary time series arises from the intrinsic system dynamics or from external forcing drives (see e.g. [2]). Hence, the development of techniques (having the simultaneous ability of simulating trends and estimating longand short-range correlations of stochastic data sets) and the assessment of their performances is of relevance to diverse scientific communities.…”