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Topic: Ratio involving 2nd order numerical derivatives
Replies: 0

 Paul Posts: 764 Registered: 7/12/10
Ratio involving 2nd order numerical derivatives
Posted: Oct 1, 2012 6:20 AM

I would like to understand the maths behind the mathworks implementation of the Hurst exponent. That is perhaps more involved than it sounds because my main problem is to interpret the meaning behind the process of taking a ratio involving different 2nd order numerical derivatives.

The mathworks wfbmesti function seems to involve the following:
Let y be a function of t where t represents time.
Consider numerical computations of the 2nd derivative of y.
Let s2 be the second derivative of y computed by central differences with a timestep of 2*delta_t
Let s1 be the second derivative of y computed by central differences with a timestep of delta_t.

s2 and s1 are both evaluated at every timestep (except for some points at or near t = 0 where the data is not available).

The Hurst exponent is 0.5 * log to base 2 of
( 16 * mean(s2 ^ 2) / mean(s1 ^ 2) ).

Could anyone explain the maths behind this methodology?

A good starting point would be to understand the idea behind the ratio for which the log is taken. I understand all the definitions involved and I also understand what the end result (Hurst exponent) indicates.

I don't understand why the above algorithm leads to a computation of the Hurst exponent or the physical intuition behind the above algorithm.

Thank you very much for your help,

Paul Epstein