A multifractal Wavelet Packet approach is described for characterising one dimensional signals using the Hurst exponent. The Hurst exponent quantifies scaling behaviour between data points, and classifies these trends as persistent, antipersistent, or random walk. A multifractal approach provides additional information regarding local scaling behaviour. This algorithm uses commercially available software to perform the Wavelet Packet Transform, with additional manipulation carried out using an iterative spreadsheet function to determine the regression slope in lagged windows. The method is demonstrated in a biophysical reaction- diffusion system where image-analysis was used to collect raw data concerning the spatial distribution of local chemical (enzyme) concentrations. These one dimensional signals (Enzyme Walks) display antipersistent scaling at small lags, but random walk and persistent scaling at longer lags. This emphasises the importance of local coordination during enzyme secretion, which breaks down at longer lags as a function of the advancing diffusion front. |
This paper uses WPA to estimate the Hurst exponent. In addition, we demonstrate the method by examining a complex biophysical system which displays fractal-like scaling trends during enzyme secretion1. This gives rise to one dimensional signals called 'Enzyme Walks', which can be characterised using time series tools. Enzyme Walks are an example of a diffusion controlled spatial deposition process. From a practical viewpoint, we also extend the global method to estimate multifractal, or local scaling trends. This therefore extends the utility of this method to heterogeneous systems which display complex behaviours at multiple length or time scales. It is hoped that this method will find widespread application for the rapid analysis of complex, multifractal type signals encountered in physics, chemistry, biology or in other signal analysis situations.
Figure 1. Three day old colony of Pycnoporus cinnabarinus grown on a microporous membrane overlaid onto Malt Extract Agar, and stained to reveal the exo-enzyme laccase. |
Figure 2. Single digital line profile overlaid on the image [ALL]. |
Figure 3. Pseodocolour representation to reveal the radial double banding pattern of enzyme activity concentration. The arrow indicates the centre-point inoculum, and the letter a and b indicate the high laccase expression region. Scale bar == 10mm. |
Figure 4. (a) Single laccase enzyme walk for ALL. (b) Frequency partitioning for the Wavelet Packet Transform for ALL shown in (a), plotting the best basis (x-axis) versus scale level (y-axis). (c) Enzyme walk for HALF. (d) Wavelet Packet Transform frequency partition with the best basis coefficients identified as black blocks. |
To examine sensitivity to resolution scale changes, one can double the magnification and take a line profile from the centre-point to the right hand edge [HALF]. Figure 4c and 4d shows the respective 1-D profile and frequency partition with the d.4 wavelet packet.
(1)
The slope, can then be used to index the Hurst exponent. From here, a straightforward algebraic arrangement of terms can be used to extract the Hurst exponent following:(2)
Note that in 2-D, the following relationship holds between the Fractal Dimension, D and the Hurst exponent, H: D=2-H.
Figure 5. (a) Spreadsheet calculation of the n^{th} data set for specified lags. (b) x versus y plotting scheme between coefficient energy magnitude and index position for different lags. |
Figure 6. Sample plot for HALF showing the 12 regressions for each lag position. |
By taking the average slope for all data sets for the two treatments, it was possible to obtain mean statistics for a multifractal description of local enzyme activity concentration. This provides an overview of the scaling behaviour between nearest neighbour regions separated by a defined lag position (measured in pixel units). This data is plotted for both ALL and HALF in Figure 7.
Figure 7. Mean values for Hlocal for different lagged positions at each magnification scale (ALL or HALF). |
Here, the following cases can be viewed, corresponding to various time-dependences of the external magnetic field, H(t), at the surface of a flat superconductor (see also the figures):
time from 0 to 2500 images taken every 100 steps of iteration |
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time from 0 to 2500 images taken every 100 steps of iteration |
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time from 1000 to 2000 images taken every 20 steps of iteration |
CONST maxSTEP=5000; { number of loops, 50 iterations each } kappa=3; { parameter determining nonlinearity } PERIOD = 25000 { time-period of external AC field } a = 0.5; { 'a' must be less or equal than about} { 0.5, to have calculations stable } VAR f: array[0..200,0..50] of real; xs, ts, step : integer; BEGIN { magnetic field =0 inside of the sample } for xs:=1 to 199 do f[xs,0]:=0; step:=0; REPEAT FOR ts:=0 to 50 do BEGIN { field = sin(2*(*time/PERIOD) on both surfaces } { any other function could be placed there } f[0,ts]:= sin(2*3.14159*(50*step+ts)/PERIOD); f[200,ts]:=f[0,ts]; END; FOR ts:=0 to 49 do FOR xs:=1 to 199 DO f[xs,ts+1]:=f[xs,ts] +a *( pwr(f[xs+1,ts]-f[xs,ts],kappa+1) -pwr(f[xs,ts] - f[xs-1,ts], kappa+1) ); {The function pwr(x,z) is x to power z; this function is defined elsewhere } {Here, the data can be stored to the disk. Now,the magnetic} {field distribution at t=50 is assumed to be the starting } {condition for the next 50 steps of iteration } FOR xs:=0 to 200 do f[xs,0]:=f[xs,50]; inc(step); UNTIL step=maxSTEP; END.
A source code of a perl script for animation is also available at the same site.
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