**Resonance
Detection**

Alexander Liss

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 [T_{0},T_{1}], which length
is a multiple of the period P and observation points, which fit in this
interval:

t_{1},…,t_{n}

where

t_{n} = T_{1}

For
a set of periods

p_{k} = P/k, k = 1,..K

one
computes

d_{i} = t_{i}
– t_{i-1}, where t_{0} = T_{0}

and
coefficients

a_{k} = sum( g(t_{i})*sin(t_{i}*k/P)*d_{i}_{ }) / (T_{1} –T_{0})

b_{k} = sum( g(t_{i})*cos(t_{i}*k/P)*d_{i}_{
}) ) / (T_{1} –T_{0})

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:

P_{1},…,P_{N}

The share of all components is

S(t) = s(t,P_{1}) + …
+ 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.