Last Update 16 May 2005

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Impulse Events, with Exploratory Analyses

This is a preliminary look at a subset of the formal analyses, some 46 events, that are defined by an "impulse" at a specific moment in time, such as a terror attack or an earthquake. The formal hypothesis test typically is specified for a period beginning some time before the impulse and ending a few hours afterward. In the current exploration, all the impulse events are treated uniformly, with the standard Z^2, the Covar[z^2], and a combined "dispersion" statistic. Finally, a new set of "non-GCP" events is constructed by identifying all earthquakes with magnitude 7.0 or greater during the 6 years of the GCP data collection. This earthquake database is compared to the impulse subset. These analyses and most of the comments are by Peter Bancel.

For the Impulse events, the dispersion stat rises at t=0 increases for nearly 4 hrs, but it is not a significant trend. When we separate the Z^2 and Covar[z^2] statistics, a striking difference appears. The Z^2 shows a consistent negative trend, though it is only marginally significant. The Covar statistic on the other hand shows a strong positive trend that reaches significance at p=0.005. In both statistics the trends appear to begin 2 or 3 hours before the event.

Question: The timing and duration follow closely the average prediction: does this imply experimenter effect or methodological leak? Answer: Z^2 stat argues against this interpretation since its trend is opposite to the prediction; it is the stat for which most predictions are made. The covar stat also argues against this interpretation; it is non-correlated with Z^2 and its calculation post-dates the events; the covar also shows rise before t=0.

Roughly the same descriptions apply to the analysis with alignment to actual event start times, with some differences that probably are not significant.

The dispersion stat is weaker in this case. This is driven by the even stronger negative trend of the Z^2. The durations are similar for both component stats and again, the trends appear to begin at least 1 or two hours befor the nominal event.

The Z^2 stat 'may' be showing a negative effect, but the stats are weak for the formal start, though more impressive for the actual start times.

These analyses indicate that the Covar[z^2] statistic yields a significant deviation for the 46 impulse events. It deviates strongly for both the formal start times and the the real event start times. The deviation accumulates from t=0 out to about 4 hrs from the formal prediction times, and 3 hrs from the actual event times. There also may be a covar precursor starting at about t = -2hrs.

In any case, the difference between the two statistics is stark. If we simply calculate the composite for the formally defined events according to the Z^2 and Covar[z^2] recipes, over two subsets of events we get opposite results. The next figure compares 46 impulse against 122 non-impulse events. These are the 166 events that don't use epochs (or some other definition incompatible with the calculations). Recipe 1 is defined in the page on event-based analysis and the Recipe 10 is Covar[z^2].

Comparing Earthquakes with Impulse Events

A complement to the impulse event analysis looks at all earthquakes from 1999 - 2004 with Richter magnitudes M>7.0 There are 103 such events, and within these a subset of 95 that don't overlap any of the impulse events. The same analysis was performed as for the 46 impulse events: align the events to their t=0 and superpose them for the period t = -4 hrs thru' t = +7 hrs.

The result is curious: both the covar and the Z^2 stats yield cumdevs remarkably similar to the impulse events. See plot QksR7vsImp46.gif (below). I haven't yet done a curve-fitting correlation analysis ( as I did for the alt-sec analysis) but eyeballing it, I'd say that would give a significant result. The visual is impressive at any rate.

Digging further things get complicated. For example, separating the 95 quakes into equal sets of high and low M (roughly M between 7.5 and 9 compared with M between 7.0 and 7.5) shows the action is mostly in the lower M set. (note: M=7 is still pretty strong). But a simple division by quake magnitude doesn't take into account damage, geographic location or media intensity, so it's not a clear call.

Modeling Pair Separation

This does bring up again another question that is on the list of analyses to do: We know the Z^2 and covar stats both look at correlations between reg pairs. If there is any distance effect at all, then nearby pairs should respond differently than distant ones. The question we need to look at is if pair separation plays a role in the effect size. Recall another curiosity: we don't see the event effect size growing in an obvious way with the number of regs. I have generated 2 plots that indicate these two issues might be related.

A simple model would say that the signal to noise (s/n), and therefore the effect size, should grow with N, the number of regs in the network. However, if pair distances are important, then adding regs to the network may not contribute strongly to the s/n if the geographical distribution is widespread. That is because adding geographically isolated regs doesn't increase the number of nearby pairs, but does increase the number of distant (and thus noisier) pairs.

I made a toy model to see if this could, in principle, be a consideration for the network. The answer is yes. See the plots NodePairFraction.gif and NodePairSigtoNoise.gif (below). The smooth curve figure is a first approximation model of the node pair fraction.

As a toy model I take the NorthAm and Euro reg sets as being two groups of 'nearby' regs and assume that only nearby pairs contribute to a signal. I estimate the fraction of these 'signal-generating' pairs over the network lifetime. In this toy model, most of the time less than 1/2 of pairs contibute a signal . What about the s/n? The s/n depends on both the network N and the signal fraction. The plot NodePairSigtoNoise shows how the model s/n grows over the lifetime of the project. The red curve shows the s/n if all regs contribute equally. What is important here is that the model s/n increases only slightly from 2000 to the present.

So this gives at least a plausibility argument that, for certain models, the gcp network should show only a weak increase in effect size with N.