Fractal Dimension

The Fractal Dimension option in the Time menu or the Time toolbar is included for checking spectra which produce a flat frequency response. A flat frequency spectrum is not necessarily indicative of white noise. There may be order within the randomness, a "memory" effect where a subsequent point is dependent on a long term series of those data elements preceding it.

The Hurst exponent measures the fractal dimension of a data series. AutoSignal uses the growth in cumulative range algorithm. A Hurst exponent of 0.5 indicates no long-term memory effect. Higher values indicate an increasing presence of such an effect. The duration of such a memory effect is often visible in the form of a threshold. This algorithm is useful for determining if a data set is truly distinguishable from Gaussian or white noise.

The Hurst fit is shown in an AutoSignal Graph.

Growth in Range Algorithm

The cumulative of white (normally distributed or Gaussian) noise is known as regular Brownian motion or a random walk. The range or distance that a variable covers in regular Brownian motion will increase in proportion to the square root of time. To compute the growth in range of a time series, a time scale index is used by partitioning n obs (number of observations) elements and averaging over these groups. The range (R) must be divided by the standard deviation (S) of the n data elements to produce a normalized R/S or rescaled range value.

The Hurst equation is R/S = k*(n obs)^H. The parameter H is known as the Hurst exponent. If H is 0.5, the cumulative of the data series is a random walk or pure Brownian motion. The data series is thus composed of true white noise in that each observation is fully independent of all prior observations and the estimated autocorrelation series is essentially zero everywhere except at lag zero.

If H is less than 0.5, a data series is said to be anti-persistent. Each data value is more likely to have a negative correlation with preceding values. These data series reverse signs more frequently than would be true for white noise. Such systems are rare in nature.

Far more common are H values above 0.5. These are persistent series which contain a memory effect. Each data value is related to some number of preceding values. These data series reverse signs less frequently than would be true for white noise. AR modeling depends on exactly this effect. For a persistent series, the autocorrelation series will have a decay to zero. Both the R/S analysis and the autocorrelation map the memory effect.

Fractal Dimension

The "fractal dimension" can be measured in one of two ways. One is geometric and measures to what extent the series fills 2D space. This measure of the fractal dimension is 2-H. By this definition, the cumulative of white noise would have a fractal dimension of 1.5. Only a highly anti-persistent series with an H near 0 would have a fractal dimension of 2 and fill up the applicable 2D space. A line would have a fractal dimension of 1.

Perhaps more useful to signal analysis is the definition of fractal dimension that uses probability space rather than geometric space. Here the fractal dimension is 1/H. By this definition, a white noise cumulative would have a fractal dimension of 2.0, completely filling the expectation space. A series with a memory effect will have a fractal dimension between 1.0 and 2.0.

Log Regression for H

The R/S vs. n obs graph is presented with log x and log y scales since this linearizes the relationship. The H value is found by using an inverse variance-weighted linear least-squares curve fit using the logarithms of the R/S and n obs values. Because of the large number of samples averaged when n obs is low, these values will exhibit low variance and will exert a major influence in the overall fit.

If Full Range is checked, the maximum time scale is fitted. Otherwise, the n init and n final fields are used to specify the beginning and ending n obs values for the regression. Because the memory effect often disappears after a certain characteristic time scale, the R/S curve often presents two separate slopes.

The power law fit can either be Linear, where lny = a + b lnx is fitted, or Non-Linear, where y=ax^b is fitted directly. The linear fit is given a secondary (y^2) weighting to approximate the results from a linear fit. Because of how the regression is done, the AutoSignal results are not likely to match other programs that compute the Hurst coefficient. The AutoSignal fits should offer a greater accuracy.

There are two fit options. Normally, the initial R/S values will consist of a lower variance. To give a greater influence in the fit to the lower magnitude R/S points, the Weight by SD option can be used. Each point in the linear or non-linear fit is weighted by the inverse variance associated with the computation of the R/S value. The Robust Fit applies only to non-linear fitting. Instead of least-squares, a robust Lorentzian maximum likelihood minimization is made. A robust fit is less influenced by outliers and a wide dynamic range on the Y-variable.

Reported Values

In addition to reporting H, the coefficient standard error derived from the fit is also reported in the SD H informational field. The rē goodness of fit index is also reported. Note that the rē value cannot be compared between a weighted and non-weighted fit. Also, the rē is based on a sum of squares criterion as such, the robust fit will always report a lower rē value, even though the fit may be more appropriate to the data.

Sunspot Example

The following Hurst evaluation uses the monthly sunspot numbers data set in Sunspot!A,Sunspot!B in sample.xls. Note the high degree of persistence in the series:

The Fourier multitaper spectrum readily finds the 11 year period peak. There is also a prominent peak at a 5.4 year period, about half the 11 year primary. There is also low frequency signal content with a significant power at an 80-90 year period.

The R/S curve that follows displays an obvious threshold near the 11.2 year (12*11.2=134) cycle. The curve also changes slope near the 5.4 year (12*5.4=64) cycle. The H value, for a full range fit between 3 and 1470 n, varies from 0.8 to 0.9 depending on the fitting method used. When there are portions of the R/S curve that have different slopes, the H value computed can be quite dependent on the fitting procedure.

List

The List Data option lists the index, n obs, and R/S values in a three column table. The listing uses the AutoSignal text viewer facility.

Copy

The Copy Data to Clipboard option copies the number of observations and R/S values to the clipboard. Formats include full precision binary (for spreadsheets such as Excel) and ASCII (for pasting into text editors).

Save

The Save Data to Disk option writes the number of observations and R/S values to a supported file format. These formats include ASCII, Excel 97, Excel 95, Lotus WK3, Lotus WK1, SPSS, or Systat.

Production Facility

The AutoSignal Automation facility allows unattended processing of large numbers of data sets. The data sets can be consolidated in an Excel file or acquired using a DLL. The graphs can be exported to an MS Word RTF file, while the processed data can be exported to an Excel 95 or Excel 97 file.

Local Options

A local option changes the data set for the duration of the current procedure only. The main data table is not altered. AutoSignal offers four local options in most of the spectral procedures.

Section the data to isolate specific regions for processing.

Detrend for removing mean or subtracting a least-squares trend model.

Fourier Filtration for isolating spectral components by frequency.

Eigendecomposition Filtration for isolating spectral components by signal strength.

The Reset button restores the data to its state when first entering the procedure. Note that if you implement sequential local procedures, all of the revisions are discarded upon reset. If an Automation Session is in progress, the Reset button can be used to terminate the automated processing.