## Dust flux, Vostok ice core

Two dimensional phase space reconstruction of dust flux from the Vostok core over the period 186-4 ka using the time derivative method. Dust flux on the x-axis, rate of change is on the y-axis. From Gipp (2001).

## Wednesday, April 2, 2014

### From the small to the big: earthquakes, avalanches, and high-frequency trading

I've been talking about scale invariance a lot lately. I became interested in the topic quite a few years ago in the context of geological phenomena like earthquakes and avalanches. The Gutenberg-Richter law describing the size-frequency relationship for earthquakes was one of the first natural laws based on scale invariance, but interest in the topic really picked up with the Bak et al. paper in 1987 (pdf - may only be a temporary link).

The cause for this relationship is still foggy, as is the physical mechanism between the small and large earthquakes. The best proposed explanation is that the scale-invariant distibution of events allows for the most efficient flow of energy (and information) through the system (but it isn't clear why that should be so).

So back in the early '90s I was estimating recurrence intervals estimates for certain hazardous events and I started trying to work out a methodology for detecting scale invariance in the geologic record. Using the Gutenberg-Richter Law, you can estimate the likelihood of a large earthquake in an area based on the number of small earthquakes. There were interesting implications for areas where the recurrence interval of large earthquakes is longer than the local recorded history (as in much of Canada). At the time, there were seismic hazard maps produced by the USGS which showed significant earthquake risk in zones which mysteriously ended right at the Canadian border.

One of my classmates in my undergrad days (we're back in the 80s, now) studied the correlation between microquakes and fluid injection at oil extraction operations in southwestern Ontario. The oil companies were surprisingly cooperative until they understood the point of the research, after which they started to withhold data.

And here is the mystery. The principle of scale invariance in earthquakes would suggest that increasing the number of small earthquakes should increase the number of large earthquakes at least in the short term. Yet our understanding of the dynamics of earthquakes tells us that lubricating the fault should allow stresses to be relieved through the small earthquakes, which in the long-run should reduce the chance of a large quake in the longer term. (This idea has been proposed at various times over the past fifty years, but for various obvious reasons, it has never been deliberately pursued).

By the early 2000s, other geophysicists (notably Didier Sornette, but there were others) had moved a portion of their data processing expertise into studying econometric time series. I made this move later as I gradually came to appreciate the key problem with developing quantitative techniques when the data were suspect. First of all, the measurements themselves are inaccurate. More importantly, our estimate of the timing of each observation was just that--an estimate. Most quantitative methods assume that the observations are evenly spaced in time. Failing that, they assume you know the timing of your observations. The consequences of errors in the timing are terrible, and frequently underestimated. The point is that it is difficult to develop excellent quantitative methods when the data are terrible.

The big advantage of working with economic time series--pricing data, in particular, is the elimination of the observational errors. When a transaction occurs, there is no doubt about either the price of the time--right down to the millisecond scale.

I started looking at market macrostructure--because (several years ago) nothing interesting ever happened on a scale of less than about an hour. Until just the past few years. Suddenly, strange, rich, unusual behaviours began to occur in individual stock prices, and even indices, on the millisecond scale. I didn't know what was causing it--but it sure was interesting.

Three seconds on the tilt-a-whirl.

This was the signature of onset of HFT. I was initially interested in it for entirely different reasons than most of you. After Crutchfield's (1994) paper (pdf) on emergence, I had been pondering the idea of how to recognize a fundamental change in a complex system. Again, my interest was in the earth system as a whole, and how to recognize whether or not new observations were pointing to a fundamental change in its mode of operation.

Given our understanding that the number of large avalanches is positively correlated to the number of small avalanches, it seems pretty clear that (as Nanex and Zerohedge has been saying) the damaged market microstructure is mirrored in the increasing number of flash crashes since Reg NMS. Unfortunately, our murky understanding of how the microstructure causes the macrostructural changes can be used by the regulatory authorities to avoid investigation. They can't see a smoking gun.

We would normally expect the micro-crashes to eventually relieve imbalances in the system, improving its long-run stability. (Perhaps this is how the SEC justifies the practice). But unlike earthquakes and avalanches, these uncountably many small crashes are not reducing the imbalances. One reason is that the cause of the imbalances is separate from HFT--the dollars keep being shoveled to the top of the mountain as fast as, if not faster, than HFT brings them cascading down. Another reason is that the trades (mostly) get unwound--so the exchanges push most of the snow back to the mountaintop after the avalanche.

HFT certainly benefits unfairly from the system, but isn't responsible for it. If anything, it is a symptom of corruption--but the cause of the corruption is elsewhere.

Accordingly, my modest proposal for dealing with HFT is this--nothing. Don't bust trades--let them stand. I'd be curious to see the response of the various Ivy-League endowment funds and pension funds when they suffer brutal, near-instantaneous, multi-billion-dollar losses. At a guess, I would probably hear the screaming up here. How would real companies, producing real products, react to a sudden monkey-hammering of their stock price, especially if it triggered debt covenants? Maybe they would all exit the market en masse. It might even force a real change.

1. Does scale invariance depend at all on the presence or absence of negative feedbacks? Or does my sentence even make sense?

Emergence (from the VERY LITTLE I'VE READ) seems to depend on negative feedbacks, and you seem to be suggesting emergence and scale invariance are related somehow.

BTW, I'd be interested in a "beginner's reading list on emergence", if you had any books to suggest.

1. Depending on what you believe, scale invariance is an example of emergence. Also, depending on what you believe, emergent behaviour may depend on both negative and positive feedbacks.

I tend to fall into the camp that supposes the interesting emergent properties arise from the presence of improperly nested heirarchical feedbacks--kind of like if in computer programming you had improperly nested do-loops. Computer languages are designed to khex when they hit something like that, so programmers naturally avoid that sort of structure--but I think that is one of the deficiencies in the modeling community (computer modeling, that is, not the other kind).

For beginners on emergence, may I suggest you go and register for one or more free courses on Complexity Explorer

http://www.complexityexplorer.org/

The courses include lectures and exercises, and the amount you can benefit from them depends on how much you put into them.

2. At the Complexity Explorer site, even if you don't complete the course, you can still view materials, and they have an extensive reading list. Plus a forum where other students contribute their ideas.

For the record, I am not involved with and not affiliated with Complexity Explorer, but did take one of their courses.