Analysis of a fractal dimension of REG behaviour

William C. Treurniet, December, 2008

An exploratory analysis of Random Event Generator (REG) behaviour during the global financial crisis beginning in October of 2008 was presented on the GCP website. The figures in the report show the Stouffer_Z fluctuations as a function of time for a period of 10 hours and for a longer period of five days. The shapes of the two plots are strikingly similar and suggest that the function has a fractal structure. Perhaps the response of the REGs to meaningful societal events includes alteration of the fractal dimension of the graph representing the cumulative Stouffer_Z statistic. This possibility is examined here using the historical data in the GCP database.

The hypothesis that the fractal dimension of the Stouffer_Z function changes significantly following a meaningful event was evaluated using 20 events selected from the list of GCP formal analyses. To be included, the Stouffer_Z score for an event had to exceed at some point the p=.05 level of significance.

The fractal dimension was calculated using the box counting method (for example, see discussions here or here), as well as a method proposed by Carlos Sevcik (1998). The two methods give different but highly correlated results. The source code implementing the box counting method for this project may be seen here, and the implementation of Sevcik's method may be seen here. The results for the two methods are presented separately in the following tables. The Event name column identifies the events, and the Event date and Event time columns indicate when the events occurred. The fractal dimension in the Event column was calculated using the data for the 24-hr period starting at the event onset. The value in the Pre-event column was calculated using the data from the 24-hr period beginning two days before the event. Finally, the Difference column shows the difference between the Pre-event and Event fractal dimensions. The BC and CS prefixes to the column headings indicate the box counting method and the Carlos Sevcik method, respectively, for calculating the fractal dimension.

Table 1 shows the list of events that achieved a p-value for the Stouffer_Z that was above 0.05 at some point. These events presumably affected a large number of people who, in turn, affected the behaviour of the REGs.

 
Table 1. Experimental pre and post-event fractal dimension
Event name  Event date     Event time     BCPre-event     BCEvent         BCDifference     CSPre-event     CSEvent         CSDifference    
BaghdadBridge 2005-08-31 06:30:00 1.568026 1.495369 0.072657 1.418503 1.390541 0.027962
BaliBomb2 2005-10-01 11:30:00 1.612066 1.567712 0.044354 1.434802 1.413225 0.021577
Beslan 2004-09-03 05:00:00 1.628362 1.449743 0.178619 1.429263 1.369722 0.059541
IsrealAssass 2001-10-17 05:00:00 1.601758 1.480483 0.121275 1.423557 1.383203 0.040354
Lysistrata 2003-03-03 07:00:00 1.646745 1.505113 0.141632 1.441556 1.391719 0.049837
ObamaAccept 2008-08-29 02:00:00 1.561223 1.407336 0.153887 1.411635 1.357569 0.054066
ObamaNom 2008-06-04 00:00:00 1.548592 1.443682 0.10491 1.410837 1.366173 0.044664
QanaBomb 2006-07-30 01:00:00 1.554814 1.572236 -0.017422 1.409787 1.418137 -0.00835
QuakeIndonesia 2006-05-26 22:54:00 1.550158 1.457637 0.092521 1.408133 1.372797 0.035336
QuakePakistan 2005-10-08 03:20:00 1.478686 1.521171 -0.042485 1.38137 1.400274 -0.018904
Sept11 2001-09-11 12:00:00 1.564466 1.53074 0.033726 1.411428 1.396562 0.014866
JordanBomb 2005-11-09 18:00:00 1.529467 1.541536 -0.012069 1.40242 1.399993 0.002427
MadridBomb 2004-03-11 00:00:00 1.491695 1.471334 0.020361 1.385717 1.376457 0.00926
Buddhas 2001-03-12 10:00:00 1.536004 1.475535 0.060469 1.402252 1.379182 0.02307
Live8 2005-06-30 00:00:00 1.527742 1.501148 0.026594 1.404483 1.386184 0.018299
CentralAmerQuake 2001-01-13 17:18:39 1.58385 1.544106 0.039744 1.423943 1.409229 0.014714
IraqElection 2005-01-30 04:00:00 1.600469 1.510896 0.089573 1.426783 1.390965 0.035818
PhillipinesLandslide 2006-02-17 02:00:00 1.515077 1.413975 0.101102 1.397291 1.359162 0.038129
ArafatDead 2004-11-11 02:30:00 1.597539 1.510826 0.086713 1.426128 1.393812 0.032316
SpainDemo 2004-03-12 11:00:00 1.540506 1.518644 0.021862 1.404644 1.394355 0.010289
Mean     1.561862 1.4959611 0.0659011 1.4127266 1.3874630 0.0252635
StDev     0.043987 0.046142 0.059070 0.015529 0.017009 0.020540

The means for the Event and Pre-event conditions in Table 1 were compared using a t-test for paired comparisons. The t-value for the box counting method was significant at p=0.00008, while that for the Sevcik method was significant at p=0.00003. Figure 1 shows histograms of the differences for the two methods.


   
Figure 1. Differences between experimental Pre-event and Event fractal dimensions
(BC - Box counting method; CS - Carlos Sevcik method)

Additionally, 10 events were selected where the Stouffer_Z statistic did not meet the criterion of p= .05 at any point in the analysis period. These events are listed in Table 2 and were evaluated as a control condition.

 
Table 2. Control pre and post-event fractal dimension
Event name  Event date     Event time     BCPre-event     BCEvent         BCDifference     CSPre-event     CSEvent         CSDifference    
Mumbai 2008-11-26 16:30:00 1.55434 1.574919 -0.020579 1.411275 1.411604 -0.000329
IslamabadBomb 2008-09-20 13:00:00 1.510196 1.570973 -0.060777 1.397032 1.415685 -0.018653
McCainAccept 2008-09-05 2:00:00 1.595397 1.561457 0.03394 1.432235 1.414461 0.017774
MyanmarCyclone 2008-05-03 0:00:00 1.522603 1.570458 -0.047855 1.39862 1.419927 -0.021307
KandaharBomb 2008-02-17 5:30:00 1.5294 1.55271 -0.02331 1.398561 1.408913 -0.010352
AlgiersBomb 2007-12-11 8:00:00 1.467664 1.562803 -0.095139 1.386131 1.415063 -0.028932
InternationalPeace07 2007-09-21 0:00:00 1.635268 1.524054 0.111214 1.445647 1.3975 0.048147
GeorgiaWarEnds 2008-08-12 10:00:00 1.503614 1.617089 -0.113475 1.390259 1.435136 -0.044877
IndiaTrainFire 2007-02-18 17:30:00 1.60166 1.462766 0.138894 1.42776 1.369383 0.058377
SaddamExecution 2006-12-30 3:00:00 1.513032 1.596768 -0.083736 1.398634 1.423379 -0.024745
Mean     1.543317 1.559399 -0.016082 1.408615 1.411105 -0.002489
StDev     0.052312 0.041991 0.085710 0.019944 0.017590 0.033915

The means for the control Event and Pre-event conditions in Table 2 were also compared using a t-test for paired comparisons. In this case, the t-values were not significant for both methods of calculating the fractal dimension. Figure 2 shows histograms of the differences.


   
Figure 2. Differences between control Pre-event and Event fractal dimensions
(BC - Box counting method; CS - Carlos Sevcik method)

Discussion

When the Stouffer_Z statistic exceeded a significance level of 0.05 at some point following an event, the average fractal dimension of the array was lowered significantly as well. In the control condition, where the Stouffer_Z was never significant, the average change in fractal dimension was also not significant. These observations suggest that the degree of self-similarity of the data structure at different scales was changed by a meaningful event. The implied change in structure goes beyond the mere departure from chance behaviour measured by the Stouffer_Z statistic alone. A Stouffer_Z array should be able to reach a significant value without requiring a change in its degree of self-similarity.

However, the obtained results may yet be an artifact of the data selection criterion employed. It is possible that the fractal dimension of the Stouffer_Z array is not independent of its maximum value. The calculation includes normalization of the array using the maximum and minimum values, and the magnitude of the fractal dimension could be affected by this transformation. Further investigation is needed to examine whether such an artifact is responsible for the apparent relationship between statistical significance and temporal structure.