How to identify a non-periodic time-series?

Question

I am working on a problem where I have to first classify whether a time-series is periodic or not and then, if it is periodic identify its period(s) (the time-series could have multiple periodicities, also).

I need a reliable approach to say that a time-series is not periodic. I have been searching about this topic and these are some leads:

• Low values in the autocorrelation function (this leads to missing some multi-periodic time-series). Based on section 3.2 of the paper Characteristic-Based Clustering for Time Series Data.
• The time-series values are Gaussian and its power spectral density is exponential (this also leads to missing time-series with periodicities). Based on section 5.1 of the paper Identifying Similarities, Periodicities and Bursts for Online Search Queries.

Are there some other properties of time-series that can be used?

Answer 1

I would approach the problem "check if a time-series is not periodic" as follows

• Compute the fft of the series
• test the significance of the related harmonics (the classic reference is "Tests of significance in harmonic analysis" by Ronald Aylmer Fisher)

Maybe this is a bit naive approach but for, at least a simple case, it seems to work (see example below).

Example

As minimal test for this 2-stages approach we could see how it works if we have the following 2 series:

• Periodic series: $y_{i}^{(1)} = \sin(x_{i}) + \epsilon_{i}$
• Non periodic series: $y_{i}^{(2)} = \alpha + \beta x_{i} + \epsilon_{i}$

where $x_{i} = [-2\pi, 2\pi]$ and $\epsilon_{i} \sim N(0, \sigma^{2})$ and $i = 1,...,N$.

Below you will find a R code for these settings. The harmonic test is performed for the two series using the PML package.

library(PML)
N  = 100
x  = seq(-2*pi, 2*pi, len = N)
ep = rnorm(N) * 0.25
y1 = sin(x) + ep
y2 =  x + 0.5 + ep 

re1 = test.harmonic(y1, p = 0.025/(N - 1)) # p: correct for multiple freq 
re1$sig
       
        frequency     prop (g)  p-value     p-threshold
      1 50.000000 0.3565748 3.148508e-08 0.0002525253
      2  2.857143 0.0316902 1.000000e+00 0.0002525253

re2 = test.harmonic(y2, p = 0.025/(N-1))
re2$sig 
      
      frequency  prop (g)      p-value  p-threshold
      1 100     0.2077127   0.0006863734 0.0002525253

From the results above we notice that for $y^{(1)}$ we have one significant frequency while for the linear function we do not find any.

I hope this answer helps (maybe it does not solve your problem but suggests a possible line of investigation).

Answer 2

Periodicity in a time-series generally manifests itself in one or more "spikes" in the intensity of the signal when represented in the frequency domain. In fact, from the theory of Fourier series representation we know that a periodic signal can be well approximated by a linear combination of one or more sinusoidal signals that will appear as "spikes" in the frequency domain. Now, this is complicated by the fact that random data will also give you some spikiness in the frequency domain, so the goal is to distinguish between spikiness that is due to randomness and spikiness that is due to periodicity. To do this we would typically use the maximum signal intensity as a test statistic for testing periodicity --- the larger the maximum signal intensity the more evidence there is for at least one sinusoidal signal in the data (and therefore at least some periodicity).

One useful formal test for this purpose is the "permutation spectrum test" which tests the maximum signal intensity against its null distribution under the assumption of exchangeability of the values in the signal (see O'Neill 2020). This particular test does not make any assumption about the marginal distribution of the data, so it is not restricted to testing Gaussian time-series. The null hypothesis for the test is that the time-series values are exchangeable and the alternative hypothesis is that there is at least one periodic signal in the time-series. (It may be useful to remove trends before applying this test.)

---

Implementation in R: You can use the ts.extend package in R to produce and plot the signal intensity for a time-series or conduct the permutation-spectrum-test. To show you an example of this, let's first produce a time-series with a periodic component.

#Generate periodic part and random part of time-series
set.seed(1)
n <- 1000
A <- rep(1:20, 50)
E <- rgamma(n, shape = 2, scale = 30)

#Generate time-series with periodic part
a <- 1
X <- a*A + E

It is simple to produce and plot the intensity of the series in the frequency domain to see if there are any "spikes" giving evidence of a periodic component.

#Show intensity of time-series
library(ts.extend)
INTENSITY <- intensity(X, scaled = TRUE)
plot(INTENSITY)

We can see that there are some spikes at particular frequencies in the Fourier domain, but are they big enough to falsify the assumption that this is exchangeable noise? To test this we implement the permutation-spectrum test and produce an appropriate plot. In the present case the test correctly identifies strong evidence that there is at least one signal in the data. (The p-value for the test is $p=0.001385$.)

#Implement the permutation-spectrum test
TEST <- spectrum.test(X)
plot(TEST)
TEST

        Permutation-Spectrum Test

data:  real time-series vector X with 1000 values
maximum scaled intensity = 3.5762, p-value = 0.001385
alternative hypothesis: distribution of time-series vector is not exchangeable 
(at least one periodic signal is present)