Many geophysical time series possess nonlinear characteristics that reflect the underlying physics of the phenomena the time series describe. The nonlinear character of times series can change with time, so it is important to quantify time series nonlinearity without assuming stationarity. A common way of quantifying the time evolution of time series nonlinearity is to compute sliding skewness time series, but it is shown here that such an approach can be misleading when time series contain periodicities. To remedy this deficiency of skewness, a new waveform skewness index is proposed for quantifying local nonlinearities embedded in time series. A waveform skewness spectrum is proposed for determining the frequency components that are contributing to time series waveform skewness. The new methods are applied to the El Niño–Southern Oscillation (ENSO) and the Indian monsoon to test a recently proposed hypothesis that states that changes in the ENSO–Indian monsoon relationship are related to ENSO nonlinearity. We show that the ENSO–Indian rainfall relationship weakens during time periods of high ENSO waveform skewness. The results from two different analyses suggest that the breakdown of the ENSO–Indian monsoon relationship during time periods of high ENSO waveform skewness is related to the more frequent occurrence of strong central Pacific El Niño events, supporting arguments that changes in the ENSO–Indian rainfall relationship are not solely related to noise.

Many geophysical time series such as the solar cycle (Rusu, 2007), Quasi-biennial Oscillation (QBO; Hamilton and Hsieh, 2002; Lu et al., 2009), and El Niño–Southern Oscillation (ENSO; Timmermann, 2003) are nonlinear. From a time series analysis perspective, the nonlinearities in the time series manifest as the tendency for the time series to rise more quickly than they fall or as the propensity for positive deviations above a horizonal axis (zero axis in the case of zero-mean time series) to be greater than negative deviations below the same horizonal axis. Understanding these nonlinear time series features is important because the nonlinear characteristics of the time series reflect the underlying physics of the phenomena in question. In the case of ENSO, the tendency for El Niño events to be stronger than La Niña events (i.e., ENSO asymmetry) is related to the propagation characteristics of equatorial Pacific SST anomalies and nonlinear dynamical heating (NDH; An and Jin, 2004; Santoso et al., 2013), where strong El Niño events are associated with eastward-propagating SST anomalies and enhanced NDH. On the other hand, weak ENSO events are associated with westward-propagating SST anomalies and minimal NDH. Understanding ENSO nonlinearity is also important because it is related to ENSO diversity (Duan et al., 2017) and the associated diversity of teleconnection responses.

Another reason why quantifying time series nonlinearity is important is that changing time series nonlinear characteristics is related to fluctuating time-domain correlations between two time series. As shown by Schulte et al. (2020), the well-documented weakening ENSO–Indian monsoon relationship (Kumar et al., 2006) around the 1970s could be related to the transition of ENSO from a linear regime to a more nonlinear regime. More specifically, they showed that the ENSO–Indian monsoon relationship weakens during time periods when ENSO evolves more nonlinearly because ENSO nonlinearity contributes to the occurrence of distinct ENSO flavors (Johnson, 2013) that differentially influence the Indian summer monsoon (Fan et al., 2017). Thus, it may not be a coincidence that ENSO transitioned to a nonlinear regime (An, 2004; An and Jin 2004; An, 2009) around the same time the ENSO–Indian monsoon began to weaken in the 1970s (Kumar et al., 1999). The results from that study oppose those of other studies that suggest that changes in the ENSO–Indian monsoon relationship are related to noise (Gershunov et al., 2001), statistical undersampling (Cash et al., 2017), the Indian Ocean Dipole (Ashok et al., 2001, 2004), and Atlantic SSTs (Kucharski et al., 2007, 2009; Chen et al., 2010). Other recent works indicate that the ENSO–Indian monsoon relationship may be influenced by volcanic radiative forcing (Maraun and Kurths, 2005; Singh et al., 2020). Maraun and Kurths (2005) found that distinct epochs of phase coherence between ENSO and the Indian monsoon may be related to volcanic radiative forcing because the identified epochs did not arise from stochastic fluctuations. In a more recent study, Singh et al. (2020) found that large volcanic eruptions alter the angular frequency of ENSO and consequently enhance the phase coherence between ENSO and the Indian monsoon. Given the ongoing debate about why the ENSO–Indian monsoon relationship changes, an additional study that examines the possible relationship between ENSO nonlinearity and the ENSO–Indian monsoon relationship is warranted.

Recognizing the importance of understanding nonlinear time series characteristics, many researchers have quantified the nonlinearity of ENSO using a variety of approaches. Commonly, traditional skewness is used to measure ENSO nonlinearity (Burgers and Stephenson, 1999) because it captures the propensity for El Niño events to be stronger than La Niña events (An and Jin, 2004; An, 2004). One drawback of this skewness metric, however, is that it measures the skewness of a distribution of ENSO index values and does not measure the skewness of specific El Niño or La Niña events. Another metric called the maximum potential intensity index proposed by An and Jin (2004) is a proxy for event skewness but with two caveats. The first caveat is that the index only quantifies the amplitude of an event and therefore cannot distinguish two events that have the same amplitude but differing nonlinear characteristics. The second caveat is that the index can only be applied to ENSO and not to arbitrary geophysical time series. Given these deficiencies, there is a clear need to construct a quantity that can measure the skewness of individual time series events regardless of the chosen study topic.

Another approach to quantifying time series nonlinearity is Fourier or wavelet-based higher-order spectral analysis. Using these methods, the cycle geometry of time series and the frequency components contributing to time-domain nonlinearity can be quantified. For example, Schulte et al. (2020) used the methods to show how the nonlinear character of ENSO has evolved from 1871 to 2016, with ENSO being especially nonlinear in recent decades. In an earlier study, Timmermann (2003) applied Fourier-based bispectral methods to identify quadratically phase-dependent oscillators embedded in ENSO time series. In another study, Pires and Hannachi (2021) evaluated the nonlinearity of ENSO by examining the standardized difference between the bispectrum of the Niño 3.4 index and the bispectrum of a linear non-Gaussian process that was fitted to the Niño 3.4 index and coerced to have same bispectral properties and skewness of ENSO. They found that interacting Fourier components of the Niño 3.4 index on typical ENSO timescales of 2 to 7 years contribute to the overall skewness of the Niño 3.4 index. While these approaches can quantify time series nonlinearity, they cannot measure the nonlinearity of individual time series events like the other methods mentioned above. These limitations further highlight the need to develop a method that can quantify time series event skewness.

In this study, we develop a nonlinear index that can be used to measure the nonlinearity of specific events embedded in arbitrary time series. More specifically, the three objectives of the paper are as follows. (1) Create a waveform skewness index to quantify local nonlinearity of time series. (2) Demonstrate the importance of the index through the application of the waveform skewness index to ENSO time series. (3) Test the hypothesis that ENSO nonlinearity is related to the weakening ENSO–Indian monsoon relationship, contributing to the current debate regarding the mechanism behind the ENSO–Indian monsoon relationship changes.

The Niño 1

Also considered in this study was the trans-Niño index (TNI; Trenberth and Stephenaik, 2001), which quantifies the SST gradient across the
equatorial Pacific. The TNI was used in this study because the trans-Niño pattern has been implicated as an SST pattern contributing to changes in the ENSO–AIR (All-India rainfall) relationship (Kumar et al., 2006). We defined the trans-Niño index as the standardized Niño 4 minus the standardized Niño 1

The AIR (Parthasarathy et al., 1994) time series was used to characterize changes in the Indian summer monsoon system. The AIR time series was created by averaging representative rain gauges at various locations across India. To remove the annual cycle, the AIR time series was converted into anomaly time series by subtracting the 1871–2016 long-term mean for each month from the individual monthly values. The AIR anomaly (AIR hereafter) time series was subsequently standardized by dividing it by its 1871–2016 standard deviation. An early (JJ) monsoon season and late monsoon (August–September) season time series were constructed in the same way as they were created for the ENSO time series.

The focus of this study was quadratic phase nonlinearities that give rise to time series skewness. Quadratic nonlinearities were associated with
quadratic phase dependence among oscillators with periods

The biphase is closely linked to the skewness of a distribution, which was computed using

To measure the time evolution of skewness, we first partitioned a time series into overlapping segments and then computed the skewness for each
individual segment. The segment length used in the calculations had to be
chosen in advance, meaning that the results of the analysis depended on the
chosen segment length. To see the segment length dependence, a sliding
skewness analysis was applied to

In this situation, an appropriate measure of quadratic nonlinearity was
constrained to be zero and constant because the cosine function is linear
and stationary. Yet Fig. 2a shows that skewness is a function of time and

Sliding skewness and waveform time series associated with

It was also found that periodicities will also impact how one can interpret
the skewness of truly nonlinear time series. For example, we considered the
nonlinear time series given by

As shown in Fig. 1b, the quadratically phase-coupled oscillators composing

The deficiencies of traditional skewness motivated the construction of a new
waveform skewness index that was more weakly influenced by periodicities.
The construction of the waveform skewness index was also motivated by Fig. 1b that shows how positive deviations are larger than negative ones in the
case of zero biphase. The waveform skewness index was constructed as
follows. First standardized anomaly time series were decomposed into
positive and negative events using an event decomposition approach (Schulte
and Lee, 2019), where positive (negative) events are contiguous strings of
positive (negative) anomalies. The peak intensity of a positive (negative)
event was defined as the maximum (minimum) value obtained by the data points
associated with the event. The waveform skewness of a positive time series
event with peak intensity

Using the waveform skewness index, a waveform skewness time series was created by assigning to each time point the waveform skewness of the event to which the time point belongs. Performing this step for positive and negative events resulted in a waveform skewness time series whose length was nearly equal to that of the original time series, where the length inequality occurred because the waveform skewness index of events at the end of the time series could not be computed. By construction, the waveform skewness index measured time series asymmetry with respect to a horizonal axis at a moment in time.

The waveform time series is a transformed version of the original time series that need not be correlated with the original time series. Indeed, we found that on average a realization of a white noise process and its waveform skewness time series have a 0.4 correlation (not shown), meaning much of the information of the original time series is lost. For linear time series and nonlinear time series, waveform skewness can have no correlation with the original time series because waveform skewness could be constant even though the corresponding time series fluctuates (see below).

Although large negative or positive values of the waveform index were suggestive of time series nonlinearity, it is important to note that the waveform index could also be large for linear stochastic processes driven by non-Gaussian noise or even Gaussian white noise. For these reasons, we also evaluated the statistical significance of the waveform skewness (see below).

Unlike traditional skewness, the sliding waveform skewness time series for the linear cosine time series shown in Fig. 1a is zero and independent of time (Fig. 2a). The waveform skewness of zero is consistent with how the time series is stationary and linear so that the waveform skewness index is a more appropriate measure of quadratic nonlinearity in this situation. Similarly, the sliding waveform skewness time series corresponding to the nonlinear time series shown in Fig. 1b are nearly constant and always positive (Fig. 2b), reflecting the constant biphase of 0 and the constant degree of nonlinearity. The index appears to be slightly dependent on the period of the cosine function, though the dependence is not as strong as it is for traditional skewness. Thus, the waveform skewness index is a more theoretically consistent measure of quadratic nonlinearity given that these nonlinearities are less impacted by periodicities. Away from the edges of the time series, the sliding waveform skewness is not impacted by the chosen segment length (not shown), contrasting with sliding skewness time series whose depiction of nonlinearity depends on what segment length is chosen.

Despite these benefits of waveform skewness, the waveform skewness index at
individual time points is highly influenced by noise because waveform skewness is only a function of three peak values. The sensitivity of
waveform skewness to noise was confirmed by generating nonlinear time series
like

Although the waveform skewness index measures local nonlinearity, it cannot determine the frequency components of the time series that are contributing to the time-domain waveform skewness. This frequency information is important because the frequency components determine how often positively or negatively skewed events will occur, as Fig. 1b suggests.

To determine the frequency components that are contributing to waveform
skewness, we computed the waveform skewness of nonlinear modes embedded in
time series, resulting in a waveform skewness spectrum. Following Schulte et
al. (2020), a nonlinear mode was defined as the sum

After the computation of all possible nonlinear modes, the global waveform
skewness of

The global waveform skewness represented the average waveform skewness index
of a nonlinear mode. A positive value meant that a nonlinear mode was
positively skewed, and a negative value indicated that a nonlinear mode was
negatively skewed. In other words, positive (negative) values implied that
the nonlinear mode was contributing to the positive (negative) waveform
skewness of

It was found that even realizations of a red-noise process had large global waveform skewness even though they are linear. Thus, it was necessary to
implement statistical significance tests. The statistical significance of
global waveform skewness was assessed using a two-sided

The global waveform skewness spectrum is like the auto-bicoherence spectrum used by Schulte et al. (2020) but with a few notable differences. Unlike auto-bicoherence, high values of global waveform skewness will only occur for skewed waveforms defined with respect to a horizontal axis. On the other hand, auto-bicoherence can be high (close to 1) even if there is no skewness because the method detects waveforms that are asymmetric with respect to vertical and horizonal axes.

The second difference is that statistical significance of global waveform
skewness can be assessed using a two-sided

To better understand what the global wavelet waveform spectrum measures, we considered the nonlinear and nonstationary time series given by

As shown in Fig. 4a, the global waveform skewness is statistically
significant around the point (32, 32), which indicates that there is
quadratic phase dependence between oscillators with periods of

The local waveform skewness time series corresponding to

The above experiment shows that phase synchronization among frequency modes produces positive waveform skewness. In another example shown in the Supplement, we showed that large waveform skewness can also arise if there is covariance between amplitude and phase, a finding consistent with how nonlinearity can arise from such covariance (Pires and Hannachi, 2021).

As shown in Fig. 5, the ENSO time series comprise fluctuations of various
magnitudes. For both the Niño 1

The waveform skewness time series associated with the ENSO time series
better illustrate the temporal changes in nonlinearity. As shown in Fig. 5c, the two most skewed Niño 1

The standardized

The transition of ENSO from a linear regime to a nonlinear one is evident
from an inspection of the (standardized) sliding skewness and sliding
waveform skewness time series. As shown in Figs. 6 and 7, the skewness and
waveform skewness of the Niño 3 and Niño 1

10-year sliding skewness and waveform skewness time series corresponding to the

20-year sliding skewness and waveform skewness time series corresponding to the

The 1940s and 1950s appear to correspond to relatively low skewness and waveform skewness. In fact, the 10-year sliding skewness and waveform
skewness time series associated with the Niño 1

The waveform and auto-bicoherence spectra indicate that the nonlinearity of
both ENSO indices is mainly the result of quadratic phase dependence among
oscillators, with periods ranging from 24 to 64 months (Fig. 8). For example, a statistically significant peak is located at (60, 60) in both
spectra for the Niño 3 and Niño 1

The waveform skewness spectra corresponding to the

As shown in Fig. 9, the relationship between seasonally averaged Niño 3 time series and AIR anomalies fluctuates on interdecadal timescales. The 20-year sliding intervals, which are nearly equal in length to the 21-year sliding intervals used in previous studies (Yun and Timmermann, 2018), especially highlight the interdecadal variability. The 10-year sliding analysis emphasizes the shorter timescale variations. Choosing other interval lengths produced curves with the general features of those depicted in Fig. 9.

Sliding correlation between the Niño 3 index and the AIR anomalies for the JJ (blue curve) and AS (orange curve) seasons. The horizonal dashed line represents the 5 % significance bound.

From 1871 to 1970, the Niño 3–AIR relationship for the JJ and AS seasons is negative, consistent with the well-established idea that El Niño events are associated with Indian monsoon failures. However, after 1970, the AS Niño 3–AIR relationship weakens and becomes nearly positive. The weakening is not seen for the season JJ, which suggests that the processes influencing the AS Niño 3–AIR relationship are different from those influencing the relationship in the JJ season.

A comparison of Figs. 7b and 9 reveals that the weakest AS Niño 3–AIR correlation coincides with the greatest AS Niño 3 waveform skewness.
Moreover, the AS relationship is seen to weaken when the Niño 3 waveform
skewness increases after the 1970s but strengthens when the Niño 3
waveform skewness declines around the 1990s. These results support the idea
that Niño 3 waveform skewness could be related to the weakening AS
Niño 3–AIR relationship. Similar arguments hold for the Niño 1

To gain confidence that the sliding time series shown in Figs. 7 and 9 are
related, the sliding Niño 3 and Niño 1

As shown in Fig. 10a, there is a positive correlation between time series
for AS Niño 1

Repeating the analysis for the JJ season revealed weaker relationships between ENSO nonlinearity and the AIR–ENSO relationship. Nevertheless, positive correlations were identified, which agrees with the idea that time periods of greater ENSO nonlinearity coincide with a weaker ENSO–AIR relationship, as originally suggested by Schulte et al. (2020). Furthermore, the stronger association for the AS season could explain why the Niño 3–AIR relationship weakens after the 1970s while the strength of the JJ relationship is more stable (Fig. 9).

A possible explanation for the relationships between ENSO nonlinearity and the strength of the ENSO–AIR relationship was determined by compositing AS ENSO indices and AIR anomalies based on AS ENSO waveform skewness bins (Appendix B). For example, we identified the years for which the AS waveform skewness was greater than the 95th percentile and computed the mean AS AIR anomaly for those years.

Figure 11a indicates that the intensity of Niño 3 anomalies is related
to Niño 3 waveform skewness. For waveform skewness values less than
0.25, negative Niño 3 indices are preferred, whereas positive indices
are preferred for waveform skewness values greater than 0.25. These results
highlight the strong linear relationship between the Niño 3 index and
Niño 3 waveform skewness. A similar analysis using the Niño 1

Composite mean

Consistent with a linear negative correlation between AIR and the Niño 3
index, AIR anomalies are preferentially positive for Niño 3 waveform
skewness values less than 0 and negative for waveform skewness values
greater than 0 (Fig. 11b). For the JJ season, the relationship appears to
be generally linear, but the relationship between AS Niño 3 waveform
skewness and AS AIR anomalies is slightly more complicated. For waveform
skewness values ranging from

To diagnose why the Niño 3–AIR relationship breaks down for high Niño 3 waveform skewness, we composited the magnitude of the TNI index based on Niño 3 waveform skewness. In this analysis, statistical significance of the composite means was assessed relative to the smallest composite mean TNI magnitude (Appendix B).

The results shown in Fig. 11c indicate that there is a propensity for the
TNI magnitude to increase with increasing Niño 3 waveform skewness. For
both seasons, the magnitude of the TNI appears to be preferentially small
for Niño 3 waveform skewness, ranging from

A similar analysis conducted between Niño 1

As shown in Fig. 12, there is a statistically significant relationship
between the waveform skewness of the Niño 1

Correlation between waveform skewness and TNI magnitude.

Correlation between skewness and TNI magnitude.

A new waveform skewness index was developed based on principles from nonlinear time series analysis. The waveform skewness allowed the waveform skewness of individual time series events to be quantified. Using statistical significance tests and waveform spectra, waveform skewness distinguishable from background noise could be identified. Practical applications of the waveform skewness index to ENSO time series highlighted its importance to geophysics. The practical applications led to the identification of numerous positively skewed ENSO events that generally coincide with the strongest El Niño events on record. The analysis also revealed that ENSO cycles between periods of high and low waveform skewness, with the 1950s being a time period when waveform skewness was relatively low. In contrast, after the 1970s, the waveform skewness of ENSO increased dramatically to a maximum around the 1990s. The fluctuation of waveform skewness is generally consistent with prior work identifying interdecadal changes in ENSO skewness and a prominent regime shift in the 1970s (An, 2009).

We found that Niño 3 waveform skewness is related to the Niño 3–AIR and Niño 1

A possible explanation for the breakdown of the Niño 3–AIR relationship during time periods of high Niño 3 waveform skewness is the presence of
central equatorial Pacific El Niño events during high waveform skewness
time periods. More specifically, the largest TNI values tend to occur during
time periods of high ENSO skewness and waveform skewness. We interpret these
findings as a failure of the Niño 1

Our findings support the hypothesis proposed by Schulte et al. (2020) that states that the ENSO–AIR relationship weakens during time periods of high ENSO nonlinearity because the skewness of AIR anomalies is weakly correlated with ENSO skewness. However, our results indicate that the association between ENSO waveform skewness and the ENSO–AIR relationship mainly exists during the AS season. Nevertheless, our findings together with those from Schulte et al. (2020) support the idea that the occurrences of ENSO flavors are related to the nonlinearity of ENSO.

While some studies suggest that changes in the ENSO–Indian monsoon relationship are related to statistical undersampling and stochastic processes (Gershunov et al., 2001; Cash et al., 2017; Yun and Timmermann, 2018), we found robust evidence that changes in the Niño 3 skewness–rainfall anomaly and Niño 3 peak intensity–rainfall anomaly relationships are related to Niño 3 waveform skewness. Thus, changes in the ENSO–Indian monsoon teleconnection may be predictable to the extent that ENSO flavors can be forecast. Our results agree with previous work showing a link between ENSO flavors and the Indian monsoon (Kumar et al., 1999; Fan et al., 2017) and suggest that ENSO-based forecast should not only include ENSO intensity information, but also information about ENSO flavor. However, our results indicate that the ENSO flavor information may only be useful for Indian monsoon prediction during August and September.

Although we found evidence that the AIR–ENSO relationship changes are related to ENSO nonlinearity, there are two limitations of the present study that are worth noting. The first limitation is that the spatial variability of ENSO teleconnections was not considered. As ENSO teleconnections vary spatially (Roy et al., 2017), future work could include understanding how ENSO nonlinearity impacts the relationship between Indian rainfall and ENSO across subregions of the Indian subcontinent. Such an analysis could help improve the current understanding of how ENSO teleconnections vary spatially. The second drawback of the current study is that linear correlation analyses were used to relate ENSO to AIR despite how other methods for quantifying time series relationships exist. For example, transfer entropy and mutual information (Song et al., 2012) may be a more suitable method for quantifying ENSO–AIR relationships. Future work could include applying these methods to ideal and geophysical time series to assess how nonstationarities in skewness can impact relationships calculated using those methods.

Although the study focused on ENSO waveform skewness, the generality of the waveform skewness index means that it can be applied to arbitrary time series. For example, the index could be readily applied to other nonlinear geophysical time series such as the QBO and solar cycle, possibly improving the current understanding of the physics driving the nonlinearity of those time series. Furthermore, because the waveform skewness index is derived using an event decomposition approach, lag composites could be made to identify physical processes associated with the development and decay of nonlinear events. Another application of the waveform skewness index could be in model evaluation, because assessing the ability of a numerical model to capture time series waveform skewness could provide insight into model deficiencies. The waveform skewness index provides new future directions for research focused on understanding nonlinear climate phenomena, numerical model performance, and nonlinear time series in general.

Data for the Indian rainfall can be accessed through

The supplement related to this article is available online at:

JS conceived the idea of waveform skewness and conducted the experiments. JS and BZ wrote the manuscript, and FP provided feedback throughout the manuscript preparation process.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We are thankful for the helpful suggestions provided by the anonymous reviewers.

This research has been supported by the National Aeronautics and Space Administration (grant no. NNX26AN38G).

This paper was edited by Norbert Marwan and reviewed by Rathinasamy Maheswaran and one anonymous referee.