# Detecting Periodicity in Short Time Series

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This Demonstration compares the performance of the Fisher and Bartlett periodicity tests for the random sinusoid , where and is a random error that is independent and identically distributed with mean 0 and variance when . The plot shows the sinusoid curve and a simulated time series with parameter settings , , , and initial random seed 9379 for Gaussian white noise and Fisher's periodicity test. The time series plot does not show the vertical axis, since it is not important in this context. See Details for further discussion and examples.

Contributed by: Ian McLeod and Yuanhao Lai (October 2016)

Western University

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

White noise with a specified variance may also be generated from the Laplace distribution and the scaled distribution with five degrees of freedom. With "cubic distortion" selected, the tests are applied to . In actual applications, the errors may be correlated so an AR(1) model can be applied: , where is .

Notice that the signal-to-noise ratio depends inversely on the error variance.

Snapshot 1: When cubic distortion is applied to the series shown in the Thumbnail, periodicity is not statistically significant at 5%. Experiment by choosing the "Laplace" or "" error buttons as well as different random seeds using the slider.

Snapshot 2: When the "phase" slider of the underlying sinusoid is changed and errors remain the same as in the Thumbnail, the significance level periodicity test may change. In the case shown, Fisher's test is not significant even at 10%.

Snapshot 3: The log of the periodogram is plotted for the time series in the Thumbnail. The lower horizontal axis shows the frequency, while the upper axis shows the period.

Snapshot 4: Bartlett's test based on the cumulative periodogram is shown for the time series in the Thumbnail. The red lines show tests with significance limits corresponding to 1%, 5% and 10%. The test is only significant at about the 5% limit.

Snapshot 5: Setting "" to 0.8 with the slider introduces a major departure from the independence assumption, making periodicity much harder to detect in short time series.

Reference

[1] A. I. McLeod and Y. Lai, "Periodicity Detection in Short Time Series," working paper.

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