Surrogate Data with Correlations, Trends, and Nonstationarities

Data Description

The data in this collection include: (1) 6 surrogate stationary signals with different correlations; (2) 7 surrogate correlated signals with linear, sinusoidal and power-law trends; and (3) 15 surrogate correlated signals with different types of nonstationarities. Each data file contains one column of data in ASCII format. Results on correlated signals with trends are discussed in Physical Review E 64, 011114 (2001). Results on correlated signals with different types of nonstationarities are discussed in Physical Review E 65, 041107 (2002). The parameter "alpha" (see below) is an exponent measuring the degree of correlations in a signal, and Nmax is the signal length. A detailed description of these signals can be found in the original articles.

Correlations in these signals can be quantified using Detrended Fluctuation Analysis (DFA). Limitations of the DFA method are discussed in the articles cited above. In particular, the second paper notes that

... for anti-correlated signals, the scaling exponent obtained from the DFA method overestimates the true correlations at small scales. To avoid this problem, one needs first to integrate the original anti-correlated signal and then apply the DFA method. The correct scaling exponent can thus be obtained from the relation between n [the DFA box length] and F(n)/n instead of F(n) ... In order to provide a more accurate estimate of F(n), the largest box size n we use is Nmax/10, where Nmax is the total number of points in the signal.

Since these files are quite large, they are provided as gzip-compressed text.

1. Correlated stationary signals

• noise0117.txt.gz: alpha = 0.1, Nmax = 217;
• noise0217.txt.gz: alpha = 0.2, Nmax = 217;
• noise0517.txt.gz: alpha = 0.5, Nmax = 217;
• noise0817.txt.gz: alpha = 0.8, Nmax = 217;
• noise0917.txt.gz: alpha = 0.9, Nmax = 217;
• noise1517.txt.gz: alpha = 1.5, Nmax = 217.

2. Surrogate signals with trends

    2a) Signals with linear trends

• trlina1.txt.gz: alpha = 0.1, Nmax = 217, slope of linear trend Al = 2-16 / index;
• trlina2.txt.gz: alpha = 0.1, Nmax = 217, slope of linear trend Al = 2-12 / index;
• trlina3.txt.gz: alpha = 0.1, Nmax = 217, slope of linear trend Al = 2-8 / index.

    2b) Signals with sinusoidal trends

• trsin1.txt.gz: alpha = 0.9, Nmax = 217, Amplitude of trend As = 2, period T = 128;
• trsin2.txt.gz: alpha = 0.1, Nmax = 217, Amplitude of trend As = 2, period T = 128.

    2c) Signals with power-law trends

• trpow1.txt.gz: alpha = 0.9, Nmax = 217, power lambda = 0.4, Amplitude Ap = 1000 / (Nmax) lambda;
• trpow2.txt.gz: alpha = 1.5, Nmax = 217, power lambda = -0.7, Amplitude Ap = 0.01 / (Nmax) lambda.

3. Surrogate nonstationary signals

    3a) Signals with cutout segments (discontinuities)

• cut0117w20p95.txt.gz: alpha = 0.1, seg. cutout probability p = 0.05, Width W = 20, Nmax = 217;
• ut0117w20p50.txt.gz: alpha = 0.1, seg. cutout probability p = 0.50, Width W = 20, Nmax = 217;
• cut0917w20p95.txt.gz: alpha = 0.9, seg. cutout probability p = 0.05, Width W = 20, Nmax = 217;
• cut0917w20p50.txt.gz: alpha = 0.9, seg. cutout probability p = 0.50, Width W = 20, Nmax = 217.

    3b) Signals with spikes

• sp02p05a1.txt.gz: spikes probability p = 0.05, Amplitude Asp = 1, Nmax = 217;
• sp02p05a1sp.txt.gz: spikes signal only, spikes probability p = 0.05, Amplitude Asp = 1, Nmax = 217;
• sp08p05a10.txt.gz: spikes probability p = 0.05, Amplitude Asp = 10, Nmax = 217;
• sp08p05a10sp.txt.gz: spikes signal only, spikes probability p = 0.05, Amplitude Asp = 10, Nmax = 217.

    3c) Signals with different local standard deviation

• d2h4pd050118s.txt.gz: alpha = 0.1, sigma1 = 1, sigma2 = 4 (probability p = 0.05), Nmax = 218;
• d2h4pd950118s.txt.gz: alpha = 0.1, sigma1 = 1, sigma2 = 4 (probability p = 0.95), Nmax = 218;
• d2h4pd050918s.txt.gz: alpha = 0.9, sigma1 = 1, sigma2 = 4 (probability p = 0.05), Nmax = 218;
Nmax = 218.

    3d) Signals with different local correlations

• cut010917p90w20_sum.txt.gz: (mixed signal)          alpha1 = 0.1 (90%), alpha2 = 0.9(10%), Width = 20, Nmax = 217;
• cut010917p90w20_comp1.txt.gz: (component 1)     alpha1 = 0.1 (90%) only, Width W = 20, Nmax = 217;
• cut010917p90w20_comp2.txt.gz: (component 2)     alpha2 = 0.9 (10%) only, Width W = 20, Nmax = 217.

Contributors

These data were contributed by Plamen Ch. Ivanov, Zhi Chen and Kun Hu, who used them in:

• Hu K, Ivanov PCh, Chen Z, Carpena P, Stanley HE. Effects of trends on detrended fluctuation analysis. Phys Rev E 2001; 64:011114.
• Chen Z, Ivanov PCh, Hu K, Stanley HE. Effects of nonstationarities on detrended fluctuation analysis. Phys Rev E 2002; 65:041107.