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Problem: Detecting cyclical patterns in daily data using periodogram and FFT.

The issue is how to code in R the periodogram to detect monthly, quarterly, semi-annual, annual..etc cyclical patterns in the data. In other words I need to detect the existence of cyclical patterns for low frequencies (i.e., 1 year => 2*pi/365, 6 months => 4*pi/365, etc.).

Reproducible Example:

library(weatherData)
w2009 = getWeatherForYear("sfo", 2009)
w2010 = getWeatherForYear("sfo", 2010)
w2011 = getWeatherForYear("sfo", 2011)
w2012 = getWeatherForYear("sfo", 2012)
w2013 = getWeatherForYear("sfo", 2013)
w2014 = getWeatherForYear("sfo", 2014)
w = rbind(w2009, w2010); w=rbind(w, w2011); w=rbind(w, w2012) 
w = rbind(w, w2013); w=rbind(w, w2014)

# Next we analyze the periodograms
# This is IMAGE 1
TSA::periodogram(w$Max_TemperatureF)
# Next: I dont really know to use this information
GeneCycle::periodogram(w$Max_TemperatureF)
# Next THIS IS IMAGE 2
stats::spectrum(w$Max_TemperatureF)

How do you interpret these images?

TSA Periodogram

Spectrum Periodogram

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  • $\begingroup$ The part of your question concerning 'how do you interpret these images' is on-topic here, but the part asking 'how do you code...' is off-topic. Be aware that you may not get answers to that part. $\endgroup$ – gung Mar 3 '15 at 1:08
  • $\begingroup$ Of course, but I will be glad to at least have a partial answer than no answer ! Thank you $\endgroup$ – Oniropolo Mar 3 '15 at 2:15
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The periodogram (a scaled squared FFT) shows strong concentrations of variance in frequencies close to zero --exactly which cannot be ascertained from your plots.

The periodogram is an inconsistent estimator of the spectrum of a stationary time series, hence the very erratic behaviour you see in your second plot. In order to gain a further insight in your series, you should either smooth the periodogram (look at argument spans= and taper= of spec.pgram()) or resort to a parametric estimate of the spectrum (look at spectrum(...,method="ar") and choose the autoregressive order using AIC for instance).

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