Social signals generated from users tends to follow a diurnal pattern where there’s activity at certain hours of the day and lulls at others.  This pattern is interesting but often it is other patterns that are more salient, like spikes  in traffic.  When the spikes are combined with the diurnal fluctuations, we have a signal separation problem.  In this framing of the problem, the observed web traffic is modeled as an underlying trend signal multiplied by the harmonic noise of daily fluctuations.  Multiplication in the time domain is equivalent to addition in the frequency domain.  So to recover the underlying trend signal we subtract the average frequency spectrum from the daily observed frequency spectrum.

It turned out that this was an easy problem to solve in R, so I wanted to share my experience.  First, to visualize the process below is an example of a snippet of the observed signal over about 3 weeks.

hourly_app_status_snip_raw

you can see that there are several big spikes as well as smaller daily bumps.  The goal is to remove the  daily bumps so that we can see underlying trends.  To do that, we use the FFT to convert the signal to the frequency spectral domain, average the the daily spectra, then subtract the average from the observed daily spectra.  One of the cool things about R is that the FFT is a built in function.  However, we had to import a library for the Hamming window.  The daily spectra correspond to sliding windows of one day length.  The edges of the window need to be rounded to prevent artifacts, which is what the Hamming window is needed for. library(‘e1071’) (line 5) imports the Hamming window and library(zoo) imports ‘rollapply’ (line 2), the function that applies the window across the signal (line 14).

The output looks like this:

hourly_app_status_snip_remAvgSpectrum

One question that comes up is how this compares to just plain old median smoothing.  Median smoothing is already a part of the code to generate the output because it acts as a low frequency filter to remove high frequency artifacts from cutting up and reassembling the signal.  But what if we just apply plain old median smoothing to the original smoothing without the FFT.  The result of just using median smoothing looks like this.

hourly_app_status_snip_onlySmoothing

Subjectively, it seems to me that the FFT frequency subtracted signal shows a bit more resolution and does a better job of removing the DC (constant) component of the signal.