Significance tests for the wavelet cross spectrum and wavelet linear coherence

Abstract

Abstract. This work attempts to develop significance tests for the wavelet cross spectrum and the wavelet linear coherence as a follow-up study on Ge (2007). Conventional approaches that are used by Torrence and Compo (1998) based on stationary background noise time series were used here in estimating the sampling distributions of the wavelet cross spectrum and the wavelet linear coherence. The sampling distributions are then used for establishing significance levels for these two wavelet-based quantities. In addition to these two wavelet quantities, properties of the phase angle of the wavelet cross spectrum of, or the phase difference between, two Gaussian white noise series are discussed. It is found that the tangent of the principal part of the phase angle approximately has a standard Cauchy distribution and the phase angle is uniformly distributed, which makes it impossible to establish significance levels for the phase angle. The simulated signals clearly show that, when there is no linear relation between the two analysed signals, the phase angle disperses into the entire range of [−π, π] with fairly high probabilities for values close to ±π to occur. Conversely, when linear relations are present, the phase angle of the wavelet cross spectrum settles around an associated value with considerably reduced fluctuations. When two signals are linearly coupled, their wavelet linear coherence will attain values close to one. The significance test of the wavelet linear coherence can therefore be used to complement the inspection of the phase angle of the wavelet cross spectrum.The developed significance tests are also applied to actual data sets, simultaneously recorded wind speed and wave elevation series measured from a NOAA buoy on Lake Michigan. Significance levels of the wavelet cross spectrum and the wavelet linear coherence between the winds and the waves reasonably separated meaningful peaks from those generated by randomness in the data set. As with simuCorrespondence to: Z. Ge (gezhfu@yahoo.com) lated signals, nearly constant phase angles of the wavelet cross spectrum are found to coincide with large values in the wavelet linear coherence between the winds and the waves. Not limited to geophysics, the significance tests developed in the present work can also be applied to many other quantitative studies using the continuous wavelet transform.

Wavelet Co-movement Significance Testing with Respect to Gaussian White Noise Background

“…We follow work of Torrence and Compo [21] and Ge [22]. Firstly, we present definition of wavelet cross spectrum (WCS).…”

“…Firstly, we present definition of wavelet cross spectrum (WCS). Consequently, we describe algorithm for statistical testing according to Ge [22] and according to MC simulations.…”

“…Both works propose algorithm for testing wavelet power spectrum, cross-spectrum and linear coherence. Ge [22,23] proved that the sampling period factor should be incorporated into testing to achieve numerically stable results. In all cases, authors' uses testing statistic with devoted distribution corrected with respect to the background spectrum distribution and sampling period.…”

“…The basic works, giving instrument for significance testing, gives Torrence and Compo [21] followed by Ge [22]. Both works propose algorithm for testing wavelet power spectrum, cross-spectrum and linear coherence.…”

“…We use both the statistical significance testing according Ge [22], and the identification of critical values using MC simulations. We deal with the advantages and disadvantages and provide recommendations for usage of TF co-movement significance testing.…”