Monday, April 13, 2009

Sandpile model of earthquakes and economic instability

Dr. Steve Keen discusses this here and gives a clear description of a power law distribution:

"The pattern of movements you get from such a process can look superficially like a Normal distribution–the famous Bell Curve–but it differs from it in two fundamental ways. Firstly, there are many more movements near the average; secondly, there are also many more movements way, way away from the average–so many more that, in what is known as a pure “Power Law” distribution, the standard deviation is infinite: any scale event can occur, and will occur given enough time."

The standard deviation is's a picture of what this looks like from Wikipedia:
The main point is that if you are using a normal distribution to model something that is actually is a power law distribution, your model is going to give incorrect results. Key to the normal distribution is the idea that factors contributing to the frequency of results are independent. The dice rolling example is commonly used because it's obvious that each roll of the dice doesn't have any influence on the next roll of the dice. The financial results of companies listed on the NYSE are definitely not independent of each other; for example most are dependent on a relatively small number of banks for routine financing.

Also, generally 68% of values drawn from a normal distribution are within one standard deviation σ > 0 away from the mean μ; about 95% of the values are within two standard deviations and about 99.7% lie within three standard deviations. This is called the "68-95-99.7 rule" or the "empirical rule." That's clearly different from the power law distribution.

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