Robust detection of periodic time series measured from biological systems

Abstract

Background: Periodic phenomena are widespread in biology. The problem of finding periodicity in biological time series can be viewed as a multiple hypothesis testing of the spectral content of a given time series. The exact noise characteristics are unknown in many bioinformatics applications. Furthermore, the observed time series can exhibit other non-idealities, such as outliers, short length and distortion from the original wave form. Hence, the computational methods should preferably be robust against such anomalies in the data.

Results: We propose a general-purpose robust testing procedure for finding periodic sequences in multiple time series data. The proposed method is based on a robust spectral estimator which is incorporated into the hypothesis testing framework using a so-called g-statistic together with correction for multiple testing. This results in a robust testing procedure which is insensitive to heavy contamination of outliers, missing-values, short time series, nonlinear distortions, and is completely insensitive to any monotone nonlinear distortions. The performance of the methods is evaluated by performing extensive simulations. In addition, we compare the proposed method with another recent statistical signal detection estimator that uses Fisher's test, based on the Gaussian noise assumption. The results demonstrate that the proposed robust method provides remarkably better robustness properties. Moreover, the performance of the proposed method is preferable also in the standard Gaussian case. We validate the performance of the proposed method on real data on which the method performs very favorably.

Conclusion: As the time series measured from biological systems are usually short and prone to contain different kinds of non-idealities, we are very optimistic about the multitude of possible applications for our proposed robust statistical periodicity detection method.

Figure 1

Examples of time series. An example of a time series composed of a sine and (a) additive standard Gaussian noise, (b) additive standard Gaussian and impulsive noise, and (c) additive standard Gaussian noise and cubic distortion.

Figure 2

Examples of spectral estimates. The spectral estimates for the time series in Figures 1 (a)-(c), respectively, using both the standard periodogram and the proposed robust method.

Figure 3

Power of the test. The power of the tests (y-axis) for the three different test cases as the function of the time series length and varying noise parameters (x-axis). The solid (resp. dashed) line corresponds to the proposed robust method (resp. Fisher's test). Three different types of non-idealities are considered, namely, pure standard Gaussian noise (the first row), standard Gaussian and impulsive noise (the second row), and standard Gaussian noise and x³ distortion (the third row). The left (resp. right) column shows the results for different time series lengths (resp. different values of the noise parameters).

Figure 4

Benchmark results. The fraction of the benchmark set that is identified (y-axis) as the function of the highest ranked genes (x-axis). The solid (resp. dashed) line corresponds the robust detection having fixed (resp. unknown) frequency. The dotted line shows the performance of the random gene selection. The columns from left to right correspond to the Alpha, the Cdcl5 and the Cdc28 experiment by [6]. The rows, from top to bottom, correspond to the three different benchmark gene sets B1, B2 and B3. for more details about the benchmark gene sets, see [24].