On the trend, detrending, and variability of nonlinear and nonstationary time series

Abstract

Determining trend and implementing detrending operations are important steps in data analysis. Yet there is no precise definition of "trend" nor any logical algorithm for extracting it. As a result, various ad hoc extrinsic methods have been used to determine trend and to facilitate a detrending operation. In this article, a simple and logical definition of trend is given for any nonlinear and nonstationary time series as an intrinsically determined monotonic function within a certain temporal span (most often that of the data span), or a function in which there can be at most one extremum within that temporal span. Being intrinsic, the method to derive the trend has to be adaptive. This definition of trend also presumes the existence of a natural time scale. All these requirements suggest the Empirical Mode Decomposition (EMD) method as the logical choice of algorithm for extracting various trends from a data set. Once the trend is determined, the corresponding detrending operation can be implemented. With this definition of trend, the variability of the data on various time scales also can be derived naturally. Climate data are used to illustrate the determination of the intrinsic trend and natural variability.

Fig. 1.

The annual GSTA from 1856 to 2003.

Fig. 2.

Means (black lines) and standard deviations (gray lines) of IMFs of 10 different siftings corresponding to S numbers from 4 to 13.

Fig. 3.

The annual GSTA (thin black line) and its trends (linear trend, thin gray line; overall adaptive trend, thick black line; and multidecadal trend, thick gray line).

Fig. 4.

Anomalies with respect to various trends (linear trend, thin gray line; overall adaptive trend, thick black line; and multidecadal trend, thick gray line).

Fig. 5.

Rates of change (temporal derivative of a trend, in degrees Kelvin per year) for the overall trend (thick solid line) and the multidecadal trend (the sum of C5 and C6, thick dashed line).

Fig. 6.

Time scale of the trend (thick solid line) defined by C5 and C6, its spreads marked by the significance level (dashed lines), and its mean (thin solid line).

Fig. 7.

Statistical significance test for the annual global surface temperature anomalies data. For the residual trend, because its time scale (usually marked by the period of a wave) is not determinable but longer than the length of the annual GSTA, the length of the annual GSTA is selected. The gray solid line is the expectation of variance of IMFs of the white noise with its first IMF containing the same variance as that of the annual GSTA (the expected case); the upper (lower) gray dashed line is the expectation of variance of IMFs of white noise of three (one-third) times that of the expected case. The variance of IMFs of the annual GSTA is plotted as a function of the mean periods, marked by circles. An IMF with its variance labeled as “noise” (“signal”) implies that the IMF is not (is) distinguishable from the corresponding IMF of a pure white noise series.