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Randomness, Risk, and Financial Markets
Pi, the ratio of a circle's circumference to its diameter, is known as an irrational number because it can't be exactly expressed as a ratio of whole numbers. It would take an infinite number of digits to write it out in full as a decimal or, in binary form, as a string of 1s and 0s. The square root of 2, the square root of 3, and the constant e (the base of the natural logarithms) fall into the same category.

The known digits of these numbers appear patternless. According to one novel method of assessing the randomness of a sequence of numbers, however, the digits of pi turn out to be somewhat more irregular than the digits of the other irrational numbers.

The measure used to determine the irregularity or degree of disorder (entropy) of these sequences is called the approximate entropy. Invented by Steve Pincus of Guilford, Conn., and developed in cooperation with Burton H. Singer of Princeton University, this measure characterizes the randomness of a sequence of numbers. ...

...Because the approximate entropy method does not depend on any assumptions about the process involved in generating a sequence of numbers, it can be applied to biological, medical, or financial data and to physical measurements, such as the number of alpha particles emitted by a radioactive element in specified time intervals, as readily as to the digits of irrational numbers.

For example, Pincus has looked at stock market performance, as measured by Standard and Poor's index of 500 stocks. His calculations show that fluctuations in the index's value are generally quite far from being completely irregular, or random.

One striking exception occurred during the 2-week period immediately preceding the stock market crash of 1987, when the approximate entropy indicated nearly complete irregularity. That change flagged the incipient collapse.

Now, Pincus and Rudolf E. Kalman of the Swiss Federal Institute of Technology in Zurich have applied approximate entropy to the analysis of a wide range of other financial data. They describe their findings in the Sept. 21 Proceedings of the National Academy of Sciences.

Approximate entropy "appears to be a potentially useful marker of system stability, with rapid increases possibly foreshadowing significant changes in a financial variable," Pincus and Kalman contend.

To provide another example of such foreshadowing, Pincus and Kalman examined fluctuations in Hong Kong's Hang Seng index from 1992 to 1998. In this case, the approximate entropy value rose sharply to its highest observed value immediately before this market crashed in November 1997. ...