I am interested to find out more on timeseries analysis
methods. Power Spectral Density analysis seems to be a useful tool to analyze
variations along different scales. The following is clippings from various
websites.
What is power spectral density
function?
Power spectral density function (PSD) shows the
strength of the variations(energy) as a function of frequency. In other words,
it shows at which frequencies variations are strong and at which frequencies
variations are weak. The unit of PSD is energy per frequency(width) and you can
obtain energy within a specific frequency range by integrating PSD within that
frequency range. Computation of PSD is done directly by the method called FFT or
computing autocorrelation function and then transforming it.
What you can do with power spectral density
function?
PSD is a very useful tool if you want to identify
oscillatory signals in your time series data and want to know their amplitude.
For example let assume you are operating a factory with many machines and some
of them have motors inside. You detect unwanted vibrations from somewhere. You
might be able to get a clue to locate offending machines by looking at PSD which
would give you frequencies of vibrations. PSD is still useful even if data do
not contain any purely oscillatory signals. For example, if you have sales data
from an ice-cream parlor, you can get rough estimate of summer sales peak by
looking at PDF of your data. We quite often compute and plot PSD to get a "feel"
of data at an early stage of time series analysis. Looking at PSD is like
looking at simple time series plot except that we look at time series as a
function of frequency instead of time. Here, we could say that frequency is a
transformation of time and looking at variations in frequency domain is just
another way to look at variations of time series data. PSD tells us at which
frequency ranges variations are strong and that might be quite useful for
further analysis.
What is cross spectral density function?
When we have two sets of time series data at hand and we want to know the relationships between them, we compute coherency function and some other functions
computed from cross spectral density function (CSD) of two time series
data and power spectral density functions of both time series data. If
we have more than two sets of time series data, we might compute
frequency domain complex empirical orthogonal functions from cross
spectral density function to know the relationships among those data.
The cross spectral density function is a Fourier transform of cross
correlation function but we can compute CSD directly using a method
called FFT.
What is coherency function, phase function and gain function?
Coherency
function shows the degree how much two sets of time series are
resembled each other by numbers ranging from 0(not resembled at all) to
1(perfectly resembled). The advantage of coherency function over
correlation coefficient is that Coherency function is a function of
frequency. In other word, it can show that at which frequencies two
sets of time series data are coherent and at which frequencies they are
not. Numerically, one might interpret coherency function as a cross
spectral density function normalized by the product of power spectral
density functions of both time series.
The
phase function, which is usually computed with coherency function,
shows phase difference as a function of frequency between two sets of
time series data. One note about the phase difference is that it is not
the same as time difference. For example, if the variations in time
series data B are constantly lagged behind those in time series A, say
by 2 hours, at all the frequencies, the phase lag at a period of 4
hours is 180 degrees but it is 90 degrees at a period of 8 hours.
The gain factor of the frequency response function shows the amplitude
relationship between two sets of time series data as a function of
frequency. The gain factor combined with coherency function and phase
function would give us fairy clear picture about the relationships
between two sets of time series data if they have common variations. We
think that correlation coefficient is not that much useful if time lag
of coherent variations between two sets of time series data is
different at different frequencies. The caveat here is that, even if
coherency function indicates there are common variations, they might be
utterly unimportant because the amplitudes of those variations might be
rather minuscule. It is still necessary to look at a power spectral
density function of, at least, one of the time series data to know the
amplitude of those coherent variations for this reason. Also, we
implicitly assume that the processes at work are linear (output is
proportional to input) when we use these functions to analyze data. We
can evaluate confidence interval of these functions similar to those of
spectral density functions. The computations of coherency function,
phase function and gain factor are not any more expensive than
computation of a simple correlation coefficient at CRInternational.
Resources
[1] Cygres