We present an approach to analyse time series with unequal spacing. The approach enables the identification of significant periodic fluctuations and the derivation of time-resolved periods and amplitudes of these fluctuations. It is based on the classical Lomb–Scargle periodogram (LSP), a method that can handle unequally spaced time series. Here, we additionally use the idea of a moving window. The significance of the results is analysed with the typically used false alarm probability (FAP). We derived the dependencies of the FAP levels on different parameters that either can be changed manually (length of the analysed time interval, frequency range) or that change naturally (number of data gaps). By means of these dependencies, we found a fast and easy way to calculate FAP levels for different configurations of these parameters without the need for a large number of simulations. The general performance of the approach is tested with different artificially generated time series and the results are very promising. Finally, we present results for nightly mean

Many time series in atmospheric sciences are characterised by an unequal spacing of the data points, e.g. due to data gaps. OH or other airglow observations often have such data gaps in the measured time series

A second important point with respect to the analysis of periodicities is the variation of these periodicities with time, i.e. the period is not stable during the complete analysed time interval or the amplitude varies. In such cases many methods as the FFT and the LSP will lead to results of a mean state only. The wavelet transform is a method that is very useful as it delivers time-resolved information on the periodicities of the analysed time series and it is used in several studies analysing the temporal evolution of periodic signals in airglow observations

The main intention of the paper is to describe the approach from a user perspective and to illustrate the capabilities of the approach with examples of artificial data sets as well as observations. The paper is structured as follows. In Sect.

The Lomb–Scargle periodogram (LSP) was developed by

An advantage compared to other methods such as the FFT is that the LSP can handle unequally spaced time series. A prerequisite is that the time series has zero mean before the calculation of the periodogram powers. With the given definition, the LSP has two useful properties: (1) it is invariant to a shift of the origin of time, and (2) it is equivalent to the least squares fitting of sinusoids

The approach used in the following analyses is based on the classical LSP, but the whole time series is analysed sequentially. The procedure is as follows.

A window size (time interval), which is typically much smaller than the length of the whole time series, is defined. Then the procedure starts at the beginning of the time series:

calculate LSP for the data points within the window (time interval),

move the window by one time step (minimum possible sampling step),

move to step one until the end of the times series is reached.

By executing this procedure, one single LSP is calculated for each possible part of the time series with the length of the window (time interval). By contrast to the LSP for the whole time series at once, this procedure delivers time-resolved information on the periodicities and amplitudes.

There are different ways to normalise the periodogram: sample variance (or sum of squares), known variance of data, and variance of the residuals

Alternatively, one can determine the amplitude of the sinusoid at each frequency. This is also based on the equivalence of the periodogram power and the reduction in sum of squares. Furthermore, the variance of a sinusoid is given by

In total the LSP delivers information on the periodicities together with a measure of the explained variance when a sinusoid is fitted to the data and the corresponding amplitude of the sinusoid. An example periodogram is shown in Fig.

Example LSP for a time series composed of two sinusoids. The first one has a period of 10 d and an amplitude of 1 K and the second has a period of 35 d and an amplitude of 0.5 K. The normalised power is shown as a black curve and the amplitude as a red curve with a second axis to the right.

An important quantity with respect to the LSP is the so-called false alarm probability (FAP). It gives the probability that a peak with a height above a certain level can occur just by chance, e.g. due to noise. The distribution of the periodogram powers and thus the description of the false alarm probability depends on the type of normalisation

The procedure to determine

An example for the results of this procedure is shown in Fig.

False alarm probability (FAP) and

The number of independent frequencies

Firstly, we analysed the dependency of

In a second analysis we varied the frequency range and repeated the analysis that was done before. The frequency ranges lay between

Dependency of

In order to study the performance of the approach we analysed different time series of artificial data. In this section we present selected examples of these time series. The total length of the time series was always 1 year (365 d) and the sampling was 1 d

The analysis of a single sinusoid is a very trivial problem and the approach delivers the expected results (not shown). As the approach shall be used in the case of non-stable periodicities, we focus here on such problems. The first example shows a time series of a periodic signal with a period that increases with time from approximately 8 to 16 d and an amplitude of 1 K. The time series is shown in Fig.

We additionally present two further examples. The time series and the results of the analyses are shown in Fig.

In summary, the applied method is able to detect periodic signals that vary with time, i.e. the amplitude or the period changes with time. In cases where changes occur on much smaller timescales than the used time window, the results show some kind of averaging. Then the maximum values of the amplitude or the explained variance cannot be obtained and a mean value in the analysed time window is derived. The method is also very useful when noise is added to the time series and additionally data gaps are introduced. Although about 30 % of the data points have been removed, the results are very good and still reflect the behaviour of the signals. Thus, the presented method is well suited to analyse time-varying periodicities even in the case of unequally spaced time series.

The

Figure

Nightly mean

The results for the normalised power and the amplitude are shown in Fig.

Results for the normalised power and amplitude for the analysis of the temperature residual of the GRIPS observations in 1989. The results are displayed at the centre day of the corresponding time window. The length of the time window was 60 d. The white contours mark the significant results.

We present an approach to analyse time series with unequal spacing with respect to significant period fluctuations. The
approach is also able to derive time-resolved information on the periods and amplitudes of the detected fluctuations. It is
based on the classical Lomb–Scargle periodogram (LSP), a method that can handle
unequally spaced time series. Additionally, it uses the idea of a moving window to enable the determination
of time-resolved periods and amplitudes. The significance of the results is
analysed with the typically used false alarm probability (FAP). As the determination of the FAP levels needs many simulations,
we derived the dependencies of the FAP levels on the length of the analysed time interval

The approach was tested with different artificially
generated time series. These time series include variations of the period and amplitude with time, and,
additionally, noise is added and data gaps have been introduced. In all cases, the approach shows very good results and thus
the approach is a suitable method for the time-resolved detection of periodic fluctuations, even in the case of unequal
spacing. Finally, we analysed the nightly mean
OH

The nightly mean

CK conceptualised the method. CK and RR performed the simulations and did the analyses under intensive discussion with RK.
PK provided the

The authors declare that they have no conflict of interest.

We thank the two reviewers for their useful comments that helped to improve the paper.

This research has been supported by the Bundesministerium für Bildung und Forschung (ROMIC project MALODY (grant no. 01LG1207A)).

This paper was edited by Lars Hoffmann and reviewed by two anonymous referees.