Here is a summary of some of the autocorrelation analysis that I've been
doing on the GCP data from September 11. It is rough and preliminary,
intended to give a flavor this approach to the data. Enjoy.
September 11 data. Cumulative Sum of the second by second autocorrelation using Stouffer z scores calculated across regs. The plot extends to a lag time of 10800 seconds (3 hrs).One data point corresponds to a 5 second lag. The red curves are .05 probability envelopes.
The Event. This autocorrelation uses a 12 hour window around the disaster (7am to 7pm EDT). The window excludes any data before 7am.One possibility in exploring autocorrelations is to get some insight into time structure in the data that corresponds to different stages of the event. Two things to note in the plot are the strong rising trend out to about 1100 on the plot. This suggests a relatively persistent positive autocorrelation in the data for a period of at least 92 minutes (one data pt is 5 secs), which roughly corresponds to the length of the disaster event. After a lag of 92 minutes the autocorrelation sum falls off and the descent is fairly smooth. Second, the rising portion contains visibly more structure than the later part of the plot. In particular, there seem to be 'impulses' at lag times of roughly 22.5, 42, 58 and 83 minutes (270, 500, 700, 1000, on the plot).
To look at the structure more closely , I focus on the four sub-events that were broadcast live, since these (as a working hypothesis) are likely to provide the strongest "impulse" registered by the reg network.The events are the second strike on the WTC , the pentagon strike and the collapses of towers 1 and 2.These occured at 9:03,9:43,10:05 and 10:28, according to the European press. These four events define six lag times of 22,23,40,45,62 and 85minutes. If you look at the autocorrelation sum you can see that these times correspond roughly to regions where the sum makes a marked upward movement.
To see this more clearly I fit an analytic trig series to the data and differentiate. The series is truncated to get a smoothing effect. The differentiated function has peaks where the cumulative sum of the autocorrelation has a strong rising slope. The figure below shows the positive peaks of the fit for the 12 hour window from 7am to 7pm overlaid with the six lag times mentioned above. The result looks pretty striking.
things are not as simple as they seem. I've looked at successive 8 hour
windows to try to zero in on the effect. Oddly, the close match of
autocorrelation derivative peaks to the empirical lag times is maintained
even for later windows that exclude the event (ie 8 hour data windows from
(11am - 7pm) out to (1pm - 9pm). Below is the result for 8 hours of data
windowed from 11am to 7pm. (The x-axis is now in minutes of lag time;
multiply by 12 to convert to the scale of the fig. above). You can see that
removing the portion of the raw data around the event (7-11am) has not
altered the situation much.
match of autocorrelation structure with the timing of events may be a
fluke. A couple of more plots make for a cautionary tale. Here is the plot
if I take the 8 hour window from 7am to 3pm, that is, including the event.
The matches are not nearly as good and some extra peaks occur.
window from midnight to 8 am there is a mix of matches and misses:
So caution is the byword here. One could also quibble about the timing of events (the pentagon strike in particular- is it better to use the moment of the crash or the time the news broke on the networks a few minutes later?).Nevertheless, if there is any validity to this approach, it might be that 'impulses' have long tails so that fine scale correlations in the data are observable even for windows that exclude the event, in a temporal sense. Perhaps the windows centered on the event itself contain correlations from both the event and prior, premonitory influences. Windows just beyond the event might be cleanly registering the tails of the event in a region where more distant "pre-event" correlations have nearly completely decayed. I will be playing with this more. The first thing is to look at data from a few days before the 11th.
To finish, I've plotted below the cumulative sums of autocorrelations
for a selection of windows. It is particularly interesting to contemplate
the series of 4-hour windows.The lag times are in seconds. The 4-hour
window autocorrelations are extended out to 4000 secs = 67 minutes. This is
pushing things a bit for the window length of only 14,400 secs but the
plots are pretty...