Bivariate wavelet coherency is a measure of correlation between two variables in the location–scale (spatial data) or time–frequency (time series) domain. It is particularly suited to geoscience, where relationships between multiple variables differ with locations (times) and/or scales (frequencies) because of the various processes involved. However, it is well-known that bivariate relationships can be misleading when both variables are dependent on other variables. Partial wavelet coherency (PWC) has been proposed to detect scale-specific and localized bivariate relationships by excluding the effects of other variables but is limited to one excluding variable and provides no phase information. We aim to develop a new PWC method that can deal with multiple excluding variables and provide phase information. Both stationary and non-stationary artificial datasets with the response variable being the sum of five cosine waves at 256 locations are used to test the method. The new method was also applied to a free water evaporation dataset. Our results verified the advantages of the new method in capturing phase information and dealing with multiple excluding variables. Where there is one excluding variable, the new PWC implementation produces higher and more accurate PWC values than the previously published PWC implementation that mistakenly considered bivariate real coherence rather than bivariate complex coherence. We suggest the PWC method is used to untangle scale-specific and localized bivariate relationships after removing the effects of other variables in geosciences. The PWC implementations were coded with Matlab and are freely accessible (

Geoscience data, such as the spatial distribution of soil moisture in undulating terrains and time series of climatic variables, usually consist of a variety of transient processes with different scales or frequencies that may be localized in space or time (Torrence and Compo, 1998; Si, 2008; Graf et al., 2014). For example, time series of air temperature usually fluctuate periodically at different scales (e.g., daily and yearly), but abrupt changes in air temperature (e.g., extremely high or low) may occur at certain time points as a result of extreme weather and climate events (e.g., heat and rain). Wavelet methods are widely used to detect localized features of geoscience data.

Wavelet analyses are based on the wavelet transform using the mother wavelet function, which expands spatial data (or time series) into location–scale (or time–frequency) space for identification of localized intermittent scales (or frequencies). For convenience, we will mainly refer to location and scale irrespective of spatial or time series data unless otherwise mentioned. Bivariate wavelet coherency (BWC) is widely accepted as a tool for detecting scale-specific and localized bivariate relationships in a range of areas in geoscience (Lakshmi et al., 2004; Si and Zeleke, 2005; Das and Mohanty, 2008; Polansky et al., 2010; Biswas and Si, 2011). The BWC partitions correlation between two variables into different locations and scales, which are different from the overall relationships at the sampling scale as shown by the traditional correlation coefficient. For example, BWC analysis indicated that soil water content of a hummocky landscape in the Canadian Prairies was negatively correlated with soil organic carbon content at a slope scale (50 m), but they were positively correlated at a watershed scale (120 m) in summer because of the different processes involved at different scales (Hu et al., 2017b). Because the positive correlation may cancel out with the negative one at different scales and/or locations, the traditional correlation coefficient between soil water content and soil organic carbon content does not differ significantly from zero, which can be misleading.

Recently, Hu and Si (2016) extended BWC to multiple wavelet coherence (MWC) that can be used to untangle multivariate (

BWC was extended to PWC by Mihanoviæ et al. (2009). Their method has been widely used in the areas of marine science (Ng and Chan, 2012a, b), meteorology (Tan et al., 2016; Rathinasamy et al., 2017), and economics (Aloui et al., 2018; Altarturi et al., 2018; Wu et al., 2020), as well as in the study of greenhouse gas emissions (Jia et al., 2018; Li et al., 2018; Mutascu and Sokic, 2020), among others. For example, PWC analysis indicated that the Southern Oscillation Index and Pacific Decadal Oscillation did not affect precipitation across India, while this was misinterpreted by the BWC analysis because of their interdependence on Niño 3.4, which affects precipitation (Rathinasamy et al., 2017). Unfortunately, the PWC implementation in many previous studies (Ng and Chan, 2012b; Rathinasamy et al., 2017; Aloui et al., 2018; Altarturi et al., 2018; Jia et al., 2018; Li et al., 2018; Mutascu and Sokic, 2020; Wu et al., 2020) was based on an incorrect Matlab code developed by Ng and Chan (2012a), who might have misinterpreted the equation of Mihanoviæ et al. (2009) and mistakenly used bivariate real coherence rather than bivariate complex coherence for calculating PWC. Moreover, Mihanoviæ et al. (2009) considered only one excluding variable (i.e., the variable that influences the response variable is excluded) and did not include the phase angle difference between response and predictor variables. The PWC values between response and predictor variables can still be misleading if more than one variable is interdependent with the predictor variable. This is especially true if these variables are correlated with the predictor variable at different locations and/or scales. Without phase information, it is hard to tell whether the correlation at a location and scale is positive or negative.

As an extension of previous studies (Mihanoviæ et al., 2009; Hu and Si, 2016), this paper aims to develop a PWC method that considers more than one excluding variable and provides phase information. This new method reveals the magnitude and type of bivariate relationships after removing the effects from all potentially interdependent variables. We expect that the new method will produce more accurate PWC values than the implementation of Ng and Chan (2012a), where there is one excluding variable. The new method is an extension of the multivariate partial coherency in the frequency (scale) domain (Koopmans, 1974). The proposed method is first tested with artificial datasets following Yan and Gao (2007) and Hu and Si (2016) to demonstrate its capability of capturing the known relationships of the artificial data. Then it is applied to a real dataset, i.e., time series of free water evaporation at the Changwu site in China (Hu and Si, 2016). Finally, the advantages and weaknesses of the new method are discussed by comparing it with the previous PWC method (Mihanoviæ et al., 2009) and implementation (Ng and Chan, 2012a).

Wavelet analysis is based on the wavelet transform, which includes continuous wavelet transform and discrete wavelet transform. While the discrete wavelet transform is mainly used for data compression and noise reduction, the continuous wavelet transform is widely used for extracting scale-specific and localized features, as in the case of this study (Grinsted et al., 2004). The wavelet transform decomposes the spatial data (or time series) into a set of location- and scale-specific wavelet coefficients, which are scaled (contracted or expanded) and shifted versions of mother wavelets. Different mother wavelets are available for wavelet transform, among which the Morlet wavelet, composed of a complex exponential multiplied by a Gaussian window, provides a good balance between location and scale localization. Therefore, continuous wavelet transform with the Morlet wavelet is suitable for transforming spatial data (or time series) into a location–scale (or time–frequency) domain, which allows us to identify both location-specific amplitude and phase information of wavelet coefficients at different scales (Torrence and Compo, 1998). Wavelet coefficients and their complex conjugates are used to calculate auto-wavelet power spectra and cross-wavelet power spectra. BWC is calculated as the ratio of smoothed cross-wavelet power spectra of two variables to the product of their auto-wavelet power spectra (Grinsted et al., 2004). Hu and Si (2016) extended wavelet coherence from two to multiple (

Similarly to BWC and MWC, PWC is calculated from auto- and cross-wavelet power spectra, for the response variable

The squared PWC (hereinafter referred to as PWC) at scale

When only one variable (e.g.,

The widely used Monte Carlo method (Torrence and Compo, 1998; Grinsted et al., 2004; Si and Farrell, 2004) is used to calculate PWC at the 95 % confidence level. In brief, the PWC calculation is repeated for a sufficient number (i.e., minimum number required) of times using data generated by Monte Carlo simulations based on the first-order autocorrelation coefficient (r1). The first-order autoregressive model (AR(1)) is chosen because most geoscience data can be effectively simulated by it (Wendroth et al., 1992; Grinsted et al., 2004; Si and Farrell, 2004), although we recognize that a time series with long-range dependence is also common in many areas such as hydrology (Szolgayová et al., 2014). Different combinations of r1 values (i.e., 0.0, 0.5, and 0.9) were used to generate 10 to 10 000 AR(1) series with three, four, and five variables. Our results indicate that the noise combination has little impact on the PWC values at the 95 % confidence level as also found by Grinsted et al. (2004) for the BWC case (data not shown). The relative difference of PWC at the 95 % confidence level compared with that calculated from the 10 000 AR(1) series decreases with the increase in number of AR(1) series (Fig. S1 of Sect. S3 in the Supplement). When the number of AR(1) is above 300, a very low maximum relative difference (e.g.,

In the case of one excluding variable (

PWC is first tested using the cosine-like artificial dataset produced following Yan and Gao (2007). The cosine-like artificial datasets are suitable for testing the new method because they mimic many spatial or time series data in geoscience such as climatic variables, hydrologic fluxes, seismic signals, El Niño–Southern Oscillation, land surface topography, ocean waves, and soil moisture. The procedures to test PWC are largely based on Hu and Si (2016), where the same dataset has been used to test the MWC method (refer to Hu and Si, 2016, for a detailed description of the artificial dataset). The response variable (

First, PWC between response variable

Second, PWC between response variable

The merit of the artificial data is that we know the exact scale-specific and localized bivariate relationships after the effect of excluding variables is removed. Theoretically, we expect that (a) PWC is 1 at scales corresponding to the relative complement of excluding variable scales in predictor variable scales and 0 at other scales. For example, PWC between

Figure 1 shows PWC between response variable

Partial wavelet coherency (PWC) between response variable

Compared with the case of the excluding variable of

When the second half of the excluding variable series is replaced by 0, the PWC values in that half are close to 1, while those in the first half of the data series are 0 at scales corresponding to the predictor variable (Fig. 1i and m). For the stationary case, after excluding the effect of

When both

PWC between response variable

The free water evaporation dataset was used to test MWC (Hu and Si, 2016). In brief, this dataset includes monthly free water evaporation (

The PWC analysis indicates that the correlations between

PWC between evaporation (

PWC between

The relationships between

In general, PASC decreased after excluding the effects of more factors (data not shown). The correlations between

PWC between evaporation (

We extend the partial coherence method from the frequency (scale) domain (Koopmans, 1974) to the time–frequency (location–scale) domain. The new method is an extension of previous work on PWC and MWC (Mihanoviæ et al., 2009; Hu and Si, 2016). The method test and application have verified that it has the advantage of dealing with more than one excluding variable and providing the phase information associated with PWC. In the case of one excluding variable, Mihanoviæ et al. (2009) have suggested calculating PWC by using an equation analogous to the traditional partial correlation squared (Eq. 14), which can be derived from our Eq. (9). However, their equation was, unfortunately, widely used by replacing the complex coherence in Eq. (14) with real coherence as expressed in Eq. (15) (Ng and Chan, 2012b, a; Rathinasamy et al., 2017; Aloui et al., 2018; Altarturi et al., 2018; Jia et al., 2018; Li et al., 2018; Mutascu and Sokic, 2020; Wu et al., 2020). This mistake is corrected in this paper.

The differences between the new (Eq. 14) and classical (Eq. 15) implementations are compared in the case of one excluding variable using both the artificial and real datasets. Except for the phase information, the two implementations generally produce comparable coherence for the artificial dataset (Fig. S5 of Sect. S3 in the Supplement). However, the new implementation produces consistently and slightly higher coherence than the classical implementation. For example, their mean PWCs between

PWC between evaporation (

Compared with the Mihanoviæ et al. (2009) method, the additional phase information from the new PWC is another advantage of this new method. This is because phase information is directly related to the type of correlation, i.e., in-phase and out-of-phase indicating positive and negative correlation, respectively. Different types of correlations were usually found at different locations and scales (Hu et al., 2017b). The phase information helps understand the differences in associated mechanisms or processes at different locations and scales. In addition, the phase information will allow us to detect the changes in not only the degree of correlation (i.e., coherence), but also the type of correlation after excluding the effect of other variables. For example,

Moreover, our new PWC method applies to cases with more than one excluding variable, which is a knowledge gap. When multiple variables are correlated with both the predictor and response variables, the correlations between predictor and response variables may be misleading if the effects of all these multiple variables were not removed. For example, at the dominant scale (i.e., 1 year) of

The new method has the risk of producing spurious high correlations after excluding the effect from other variables. Take the artificial dataset for example: at the scale of 32, PWC values between

Similar to BWC and MWC, the confidence level of PWC calculated from the Monte Carlo simulation is based on a single hypothesis testing, but in reality, the confidence level of PWC values at all locations and scales needs to be tested simultaneously. Therefore, the significance test has the problem of multiple testing; i.e., more than one individual hypothesis is tested simultaneously (Schaefli et al., 2007; Schulte et al., 2015). The new method may benefit from a better statistical significance testing method. Options for multiple testing can be the Bonferroni adjusted

Partial wavelet coherency (PWC) is improved to investigate scale-specific and localized bivariate relationships after excluding the effect of one or more variables in geoscience. Method tests using stationary and non-stationary artificial datasets verified the known scale and localized bivariate relationships after eliminating the effects of other variables. Compared with the previous PWC method, the new PWC method has the advantage of dealing with more than one excluding variable and providing the phase information (i.e., correlation type) associated with PWC. In the case of one excluding variable, the PWC implementation provided here (in the paper and the published code) produces more accurate coherence than the previously published PWC implementation that considered wrongly real coherence rather than complex coherence between every two variables. Application of the new method to the real dataset has further proved its robustness in untangling the bivariate relationships after removing the effects of all other variables in multiple location–scale domains. The new method provides a much needed data-driven tool for unraveling underlying mechanisms in both temporal and spatial data. Thus, combined with wavelet transform, BWC, and MWC, the new PWC method can be used to analyze various processes in geoscience, such as streamflow, droughts, greenhouse gas emissions (e.g.,

The Matlab codes for calculating PWC, along with the updated MWC codes, are freely accessible (

The supplement related to this article is available online at:

WH wrote the paper, developed the Matlab code, and analyzed the data. Both authors conceived the study, interpreted the results, and revised the paper.

The authors declare that they have no conflict of interest.

This research was supported by the New Zealand Institute for Plant and Food Research Limited under the Sustainable Agro-ecosystems programme.

This paper was edited by Bettina Schaefli and reviewed by three anonymous referees.