Resonance Detection

Alexander Liss

03/21/2009

Some resonances are well known, as tides with lunar and solar cycles, other are less known as weather with the cycles of Sun rotation and cycles of solar activity.

There many cyclical are pseudo-cyclical processes, where even relatively small reduction of uncertainty of their future movement could be beneficial, as stock market, where detection of patterns in its movement could be profitably exploited. For such processes, detection of resonance with some other known leading cyclical processes is important.

When a leading process is periodic with a period P, its influence on a given process could be evaluated through detection of periodic component with the period P in a given process g(t).

There are observations of the process g(t) in a discrete moments in an given time interval.

Inside this interval one selects an interval [T₀,T₁], which length is a multiple of the period P and observation points, which fit in this interval:

t₁,…,t_n

where

t_n = T₁

For a set of periods

p_k = P/k, k = 1,..K

one computes

d_i = t_i – t_i-1, where t₀ = T₀

and coefficients

a_k = sum( g(t_i)*sin(t_i*k/P)*d_i ) / (T₁ –T₀)

b_k = sum( g(t_i)*cos(t_i*k/P)*d_i ) ) / (T₁ –T₀)

These coefficients show how important is the periodic component with period P in overall oscillation of the process g(t).

The share of this component is computed as

s(t,P) = sum( a_k*sin(t*k/P) + b_k*cos(t*k/P) )

This procedure is repeated for all N leading processes with periods:

P₁,…,P_N

The share of all components is

S(t) = s(t,P₁) + … + s(t,P_N)

When

g(t) – S(t)

is small, this is a sign that the resonance with selected processes is a main factor determining the shape of the given oscillation.