Gauss, Mandelbrot and the Reversion to Mean

Last time we talked about divergence, how life and nature is replete of divergence cases and the debate built around it. The funny part of all this debate is that we are somewhere still living the blind men and the elephant metaphor. We don’t see or may never see the big picture. But then seeing more of an elephant is still better that just seeing its tail.

Unfortunately that’s not how it works in society. We love tails, especially fat ones. There is a lot of literature on fat tails in statistics.

Carl Gauss, the prince of mathematics of the 18th century is challenged by contemporary mathematicians like fractal guru Benoit Mandelbrot, who claims that the Gaussian bell curve is nonsense. We at Orpheus don’t think Gauss was wrong. The law of frequency of error took the shape of a bell curve. The 1738 idea is now a part of societal faith and market modeling. Starting from Fundamental analysis, option pricing, statistics, the normal distribution curve is everywhere.

Mandelbrot said nature was about extreme divergences and the Gaussian bell curve had more fat tails than it could explain. The divergences from mean were large. Mandelbrot was correct, divergences in nature were large and divergences on occasions don't revert to mean. But the question to be asked is why did Gauss not see the large divergences from mean? There could be many reasons. Could it because Gauss was handling Astronomical data, which is more ordered than other data found in nature? Was it because data interpretation was a nascent science and hence the first obvious visual pattern Gauss could see was the mean line? Could it be that Gauss was more focused on data than on the divergence?

As time passed and data interpretation came off age, the size, quality and quantity of data enhanced. There was more data to fit, a lot of it. The attention moved from data to divergence in data. Larger the divergences became visible, fatter the tails became. Whatever was Gaussian suddenly started to look exponential. Could the question mathematicians fail to ask was whether it really was mean or the error that was more important?

Can we imagine world where the average mean is changing and dynamic? Is mean an overrated parameter? Average looks, average salary, average GDP is how the society understands. We were not designed to understand extremes, but mean and average. Average has safety, comfort, which extremes lack. How could nature ever be measured by an average mean? Small divergence around the mean was considered efficient and large divergence made everything inefficient. Can this debate be answered if we focus on the error and not on the mean in the first place? Is divergence cyclical?

John Murphy intermarket linkages are cyclical, as money moves from soft to hard assets and from commodities to bonds. Sam Stovall’s economic sectors cyclically move in and out of performance. There is repetitiveness in sector rotation, cyclicality in divergence. Robert Shiller’s case of bubble formations is cyclical. Mandelbrot's repeating extremities and Nassim Taleb's recurring random events are periodic. If extremities were not repeating there would be no fractal geometry, no butterfly effect.

Black swan is a subject of study because there is not only one black swan. The black swans come again and again, periodically. Robert Arnott’s investing out of growth into value and vice versa is cyclical. Robert Prechter’s social mood diverges cyclically positive mood to negative mood and back. Behavioral Finance talks about investor’s errors and repeating Long reversals. The subject would not be valuable if long reversals didn’t repeat.

Top performers underperform and vice versa. Dow Industrials underperforms S&P 500 and vice versa. Similar cyclicality can be seen between gold and copper. Gold underperforms and outperforms copper cyclically, the pairs diverging 8% to 10% cyclically in a quarter. Small and large divergences are cyclical. Divergence cyclicality changes the generation mindset built around mean. Because the focus was around mean, everything around mean became unpredictable, noisy and extreme. The idea of divergence cyclicality is a study of extremities, the study of cyclicality of error, which proves that mean was never static, but dynamic.

Simple computer programs, which attempt to find regularity in sequences, may not “See” the regularity in Champernowne’s number. This deficit reinforces the notion that statisticians must be very cautious when declaring a sequence to be random or patternless. Mandelbrot’s extremity somewhere robs us of the beauty of the bell shaped curve, and it’s the same bell curve, which is used today to disprove patterns in nature.

The paradox is really very sad. We are busy debating about randomness and order never even once thinking of the error that creates it all. Divergence cyclicality can reconcile the efficient and inefficient school of thought. It can explain that both Gauss and Mandelbrot are talking about the same mean. But it’s Mandelbrot who takes the argument to the next level by assuming larger divergence around mean as more important than the smaller divergence around the mean.

Both of them speak about mean reversion, one saying it works, while the other refutes it. Divergence cyclicality is more focused on divergence than defining an average. It is the new predictive model that takes data interpretation to the next level, by introducing time cyclicality into data and suggesting that the basic pattern of order (small divergence) or randomness (large divergence) was always cyclical.

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