If I calculate the period of one series (S1) as T1, and the other (S2) as T2, and then I calculate the ratio of T1 and T2. What is the deficits when S1 and S2 are not in the identical time period and with unequal length?
Probably. can you convert the shorter span series to the span of the longer? E. G. monthly to quarterly? Is there some filtering involved, say averaging? The later would confound comparisons.
Is there something like Zellner's Seeming unrelated equations for seemingly unrelated time spans?
take annual discharge data as an example. I want to compare the periodic time of the annual discharge of two rivers: river 1 and river 2. We have the data
S1: annual discharge of river 1 from 1961 to 2000 (40 year). T1 is its periodic time.
S2: annual discharge of river 2 from 1921-1960 (40 year). T2 is its periodic time.
S3: annual discharge of river 2 from 1961-2000 (40 year). T3 is its periodic time.
The method to determine periodic time could be periodogram/maximum entropy spectra analysis/wavelet analysis or whatever. What is the deficit of using T1/T2 compared to using T1/T3.
Also, if S4: annual discharge of river 2 from 1941-2000 (60 years), T4 is its periodic time. Then what is the deficit of using T1/T4 compared to using T1/T3.
Both S1 and S2 must have enough samples to represent monthly or quarterly properties.
Assuming your process is stationary, you do need neither the identical time period nor an equal length. I agree with Xiquan that the processes just need to be sampled sufficiently.
first I agree with Xiquan and Cedrick.
To answer in detail to your detailed question:
If you are performing a Discrete Fourier Transform the length of the time series you are considering affects the resolution of the frequencies, so the use of 60 years gives you 1/60, the other 1/40.
It allows you to see more in detail the peaks.
The number of samples instead will allow you to see more high frequencies.
Basically using the time coverage and the number of samples you can estimate the uncertainties on the estimated peaks, and so you can compare different series using Welsh t-test and see how much they are confident.
You can look to any text book about wavelet.
Probably you are interested in short time Fourier transform STFT and uncertainty principle.
There are many approaches to your task:
raw estimation:
the uncertainty on the frequency is given by the resolution of the axis +-1/T (T is the period of observations)
empirical:
The uncertainty for the localization of a peak in the spectrum when assumed to be a Gaussian curve can be estimated fitting with a Gaussian curve. This is used widely, also when the peak is not clearly a Gaussian.
direct calculus:
Being the DFT the sum of the sin component and the cos component, you could propagate the errors of each measurement to the sin and cos components.
Then you can propagate to power spectra.
This last method does not find the uncertainty on the location, but the one on the peak value.
However, the way to approach can be much different depending on the thing you want to study. A check on STFT and signal analysis is the best suggestion _I can give you.
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