Methods for detecting and quantifying nonlinearities in nonstationary time series are introduced and developed. In particular, higher-order wavelet analysis was applied to an ideal time series and the quasi-biennial oscillation (QBO) time series. Multiple-testing problems inherent in wavelet analysis were addressed by controlling the false discovery rate. A new local autobicoherence spectrum facilitated the detection of local nonlinearities and the quantification of cycle geometry. The local autobicoherence spectrum of the QBO time series showed that the QBO time series contained a mode with a period of 28 months that was phase coupled to a harmonic with a period of 14 months. An additional nonlinearly interacting triad was found among modes with periods of 10, 16 and 26 months. Local biphase spectra determined that the nonlinear interactions were not quadratic and that the effect of the nonlinearities was to produce non-smoothly varying oscillations. The oscillations were found to be skewed so that negative QBO regimes were preferred, and also asymmetric in the sense that phase transitions between the easterly and westerly phases occurred more rapidly than those from westerly to easterly regimes.

Spectral analysis is a tool for extracting embedded structures in a time series. In particular, Fourier analysis has been used extensively by researchers for extracting deterministic structures from time series but is incapable of detecting nonstationary features often present in geophysical time series. Wavelet analysis can extract transient features embedded in time series, with a wavelet power spectrum representing variance (power) of a time series as a function of time and period. Since the seminal work of Torrence and Compo (1998), wavelet analysis has been applied extensively to geophysical time series such as the indices for the North Atlantic Oscillation (Olsen et al., 2012), Arctic Oscillation (Jevrejeva et al., 2003), Pacific Decadal Oscillation (Macdonald and Case, 2005; Newmann et al., 2003), El Niño–Southern Oscillation (ENSO; Torrence and Webster, 1999), Pacific–North American pattern, and west Pacific pattern (Gan et al., 2007). The application of wavelet coherence and cross-wavelet analyses (Grinsted et al., 2004), moreover, has proven useful in relating geophysical time series to other time series (Jevrejeva et al., 2003; Gan et al., 2007; Labat, 2010; Lee and Lwiza, 2008).

Many statistical methods, including power and cross-spectral analyses, rely on the assumption that the variable in question is Gaussian distributed (King, 1996). For a linear system in which the output is proportional to the input, the first- and second-order moments (the mean and variance) can fully describe the distribution of a process. In the frequency domain, by analogy, the variable can be fully described by the power spectrum, the decomposition of variance as a function of frequency. Suppose, however, that the distribution is non-Gaussian so that higher-order moments such as skewness and kurtosis exist. In this case, the mean and variance, while useful, are unable to fully describe the distribution in question. In a time series context, non-Gaussian distributions can arise from nonlinear systems, systems for which the output is no longer simply proportional to the input. For a nonlinear system, if the input is the sum of two sinusoids with different frequency components the output will contain additional frequency components representing the sum and difference of the input frequencies (King, 1996). In such cases, it is necessary to examine the decomposition of higher-order moments in frequency space.

The frequency decomposition of the third-order moment, for example, results in a bispectrum or skewness function that measure deviations from Gaussianity (Nikias and Raghuveer, 1987; King, 1996). In fact, Hinich (1985) developed a bispectral test to determine if a time series is non-Gaussian and nonlinear. In some situations, higher-order nonlinearities such as cubic nonlinearities may exist, in which case the trispectrum or other polyspectra would have to be used (Collis et al., 1998).

Another advantage of higher-order spectral analysis is that the cycle geometry of oscillations, such as asymmetry with respect to a horizontal axis (skewed oscillation) or with respect to a vertical axis (asymmetric oscillation) can be quantified using the biphase. A pure sine wave, for example, is neither skewed nor asymmetric, whereas a time series resembling a sawtooth is asymmetric. Skewed and asymmetric cycle geometry can identify, for example, abrupt climatic shifts, sudden shifts in the climate system that exceed the magnitude of the background variability (King, 1996). Abrupt climate shifts have occurred numerous times in the past and have dire impacts on ecological and economic systems (Alley et al., 2003). An understanding of past abrupt climate shifts is essential to understanding future climate change and so there is a need to quantify nonlinearities present in climatic oscillations.

The quasi-biennial oscillation (QBO), as another example, has been shown to behave nonlinearly, transitioning from easterly phases to westerly phases more rapidly than from westerly to easterly phases (Lu et al., 2009). Another source of asymmetry in the QBO time series arises from the westerly shear zone descending more regularly than the easterly shear zone. Asymmetries in the QBO time series are not well captured by linear methods such as linear principal component and singular spectrum analyses (Lu et al., 2009) but are better captured using, for example, nonlinear principal component analysis (Hamilton and Hsieh, 2002). Another example of a nonlinear time series is the sunspot cycle. Solar activity undergoes an 11-year oscillation characterized by asymmetric cycle geometry, with solar maxima generally rising faster than they fall, indicating the presence of nonlinearities (Moussas et al., 2005; Rusu, 2007). ENSO, a climate phenomenon with regional- to global-scale impacts, has also been shown to exhibit nonlinearities (Timmermann, 2003). The presence of nonlinearities and possible nonstationarities in the QBO, ENSO and sunspot time series makes traditional Fourier and wavelet analysis inadequate for feature extraction, underscoring the need to develop methods for quantifying nonlinearities in a nonstationary geophysical setting.

The application of higher-order wavelet analysis has been rather limited compared to traditional wavelet analysis (van Millagan et al., 1995; Elsayed, 2006). One geophysical application of higher-order wavelet analysis is to oceanic waves (Elsayed, 2006), which was found to be capable of identifying nonlinearities in wind–wave interactions. However, the study lacked rigorous statistical significance testing, which is problematic because even a Gaussian process of finite length can produce nonzero bicoherence. Therefore, the first aspect of this paper is to apply significance testing methods for higher-order wavelet analysis to aid physical interpretation of results.

The number of bicoherence estimates to which the statistical test is applied will be large and multiple artifacts will result. The multiple-testing problem was already identified for traditional wavelet analysis (Maraun et al., 2007; Schulte et al., 2015; Schulte, 2016). The first objective of this paper will be therefore to apply statistical methods controlling false positive detection. It is also noted that the bicoherence spectra calculated are only sample estimates of the true bicoherence spectra. The second objective of this paper will be to develop a procedure for calculating confidence intervals corresponding to the sample estimates, which represent a range of plausible values for the sample estimates.

Another problem with the application of higher-order wavelet analysis is selection of a time interval on which to calculate the high-order wavelet quantities. Such an approach is subjective and the result of the analysis may depend on the time interval chosen. Objective three of this paper will address the time interval selection problem. Such an approach has already been adopted in wavelet coherence analysis (Grinsted et al., 2004).

Additionally, properties of the biphase have only been examined for Fourier-based bispectral analysis (Elgar and Sebert, 1989; Maccarone, 2013) and its usefulness in higher-order wavelet analysis has yet to be examined. For nonstationary time series, the biphase and cycle geometry corresponding to the time series may change with time and thus objective four of this paper will be to introduce a local wavelet-based biphase spectrum.

In this paper, higher-order wavelet analysis is put in a statistical framework and applied to the QBO time series to demonstrate the insights afforded by the methods. Before describing higher-wavelet analysis, a brief overview of wavelet analysis is first presented in Sect. 2. Higher-order wavelet analysis is described in Sect. 3 and a new local autobicoherence spectrum is introduced, eliminating the selection of a time interval on which to calculate nonlinear properties of time series. The new and existing methods are applied to an ideal time series and the QBO index. In Sect. 4, a new procedure for estimating confidence intervals of global autobicoherence quantities is developed to estimate uncertainties in the sample autobicoherence spectra. The application of the new procedure to the sample autobicoherence spectrum of the QBO time series is then used to further assess confidence in results.

The idea behind wavelet analysis is to convolve a time series with a
function satisfying certain conditions. Such functions are called wavelets,
of which the most widely used is the Morlet wavelet, a sinusoid damped by a
Gaussian envelope:

Shown in Fig. 1a is the time series of the QBO index and shown in Fig. 1b is
the corresponding wavelet power spectrum. The QBO data from 1950 to 2013 were
obtained from the National Oceanic Atmospheric Administration Earth System
Research Laboratory (available at:

There are also secondary features located at a period of approximately 14 months, primarily from 1985 to 2013. The appearance of significant power at a period of 14 months also coincides with most of the largest negative phases of the QBO. Such a correspondence may not have been a coincidence; the 14-month mode and the 28-month mode may have interacted constructively to generate large negative events but interacted destructively to create smaller positive events. However, additional tools are needed to confirm if the periodicities are interacting and to understand how the interactions were related to the behavior of the QBO.

Higher-order spectral analysis provides the opportunity to quantify
nonlinearities and allows for the detection of interacting oscillatory modes
within a time series. More specifically, nonlinearities are quantified using
bicoherence, a tool for measuring quadratic nonlinearities, where quadratic
nonlinearities imply that for frequencies

Unlike the power spectrum, which is the Fourier transform of the
second-order moment of a time series, the bispectrum is defined as the
double Fourier transform of the third-order moment, or, more generally, the
third-order cumulant, i.e.,

Phase information and cycle geometry can be obtained from the biphase, which
is given by

For asymmetric waveforms, a biphase of 90

According to Elsayed (2006), the wavelet-based autobicoherence is defined as

In practice, the autobicoherence is computed discretely so that Eq. (16) can
be written as

The Monte Carlo approach to pointwise significance testing is adopted in this
paper and is similar to that used in wavelet coherence (Grinsted et al.,
2004). To estimate the significance of wavelet-based autobicoherence at each
point

Let

The procedure can be described as follows: suppose that

To demonstrate the features of a time series that can be extracted using
higher-order wavelet analysis, an idealized nonstationary time series will
first be considered. Consider the quadratically nonlinear time series

To determine if the oscillations are quadratically interacting, the
autobicoherence of

Shown in Fig. 6 is the wavelet-based autobicoherence spectrum for the QBO
time series. A large region of significance was identified, which contained
the local maximum at (28, 28) months. The peak represents the phase coupling
of the primary frequency component with its harmonic with a period of 14 months. The power at

The wavelet-based autobicoherence spectrum of the QBO index for the period 1950–2013. Thick contours enclose regions of 5 % pointwise significance.

It may also be desirable to see how autobicoherence along slices of the full
autobicoherence spectrum changes with time. To compute local autobicoherence,
apply a smoothing operator

It is important to mention that the numerator of Eq. (22) contains a term
with wavelet coefficients at two different scales so that the choice of
smoothing is not as straightforward as for wavelet coherence. Smoothing
autobicoherence estimates with respect to

The advantage of using Eq. (22) is that transient quadratic
nonlinearities can now be detected and the need for choosing an integration
time interval has been eliminated. If

Same as Fig. 7 except at (28, 28) in the autobicoherence spectrum of
the QBO index. Biphases differing from 90

One can also compute a local biphase from the smoothed bispectrum by taking
the four-quadrant inverse tangent of the smoothed imaginary part divided by
the smoothed real part. The local biphase, for example, was computed for the
skewed time series shown in Fig. 2a. As expected, the biphase fluctuates
regularly around 0

Same as Fig. 8 except at the point (16, 26).

The procedure for the estimation of the statistical significance of local
autobicoherence is the following: generate red-noise time series with the
same lag-1 autocorrelation coefficients as the input time series and use the
local autobicoherence estimates outside the COI to generate a null
distribution of

The local autobicoherence spectrum of

In order to determine if the peaks in autobicoherence are associated with a
quadratic nonlinearity, it is important to compute the biphase, which is
shown in Fig. 7b. From

Same as Fig. 5b except for the QBO index for the period 1950–2013.

The local autobicoherence spectrum of the QBO index at the point (28, 28) in the full autobicoherence spectrum is shown in Fig. 8. From 1950 to 1970 the magnitude of the autobicoherence fluctuated and consisted of one local significant peak at 1965. Significant autobicoherence was also found from 1975 to 1998, contrasting with the autobicoherence after 1998, which was not found to be significant until 2010.

To determine if the peaks indicated in the autobicoherence spectrum are associated
with a quadratic nonlinearity, the local biphase was computed. Figure 8a shows
the local biphase for the autobicoherence peak at (28, 28). For most of the
study period, the biphase was found to vary considerably, particularly
during the 1950–1970 and 1995–2013 periods. On the other hand, the biphase
varied smoothly from 1970 to 1995, consistent with how the autobicoherence
during that period was large and stable (Fig. 8a). Also, during that period
the biphase was nonzero; in fact, the mean biphase during the period was

The local autobicoherence and biphase corresponding to the peak (16, 26) was
also computed (Fig. 9). The mean of the absolute value of the biphase for the
period 1950–2013 was 130

Bootstrapping is a widely used technique to estimate the variance or uncertainty of a sample estimate. For independent data, one samples with individual replacement data points (Efron, 1979); for dependent data one must sample with replacement blocks of data to preserve the autocorrelation structure of the data (Kunsch, 1989). The latter technique is called block bootstrapping and should be used for variance estimation of global wavelet quantities, as wavelet coefficients are known to be autocorrelated in both time and scale. The use of traditional bootstrapping techniques would result in confidence intervals that are too narrow. It is expected, however, that the choice of the bootstrapping technique is more critical at larger scales, as the decorrelation length of the mother wavelet increases with scale.

A brief overview of the procedure is provided below but a more detailed
discussion can be found in Schulte et al. (2016). To find the approximate

As noted by Schulte et al. (2016), the appropriate block length to use can be
determined by Monte Carlo methods. In that study, it was determined from a
Monte Carlo experiment that a block length of

Figure 5b shows the application of the block bootstrap procedure to the
diagonal slice

The application of the block bootstrap procedure to the diagonal slice

Higher-order wavelet analysis together with significance testing procedures were used to detect nonlinearities embedded in an ideal time series and the QBO time series. The autobicoherence spectrum of the QBO index revealed phase coupling of the 28-month mode with a higher-frequency mode with period 14 months. A local autobicoherence spectrum of the QBO index showed that the strength of the nonlinearities varied temporally. Furthermore, the local biphase spectrum indicated that a statistical dependence among frequency components resulted in waveforms that were both skewed and asymmetric, indicating that the strength of negative QBO events were stronger than positive events, and that transitions between events occurred at different rates. The author has written a software package (Schulte, 2015) to implement all higher-order wavelet analysis methods presented in the paper.

The data for the Quasi-biennial Oscillation used in this paper are freely
available at the National Oceanic Atmospheric Administration's Earth System
Research Laboratory Physical Science Division website available at:

Support for this research was provided by the National Science Foundation Physical Oceanography Program (award no. 0961423) and the Hudson River Foundation (award no. GF/02/14). Edited by: J. Kurths Reviewed by: two anonymous referees