Wavelet Cross-correlation in Bivariate Time-Series Analysis

Eniuce Menezes de Souza, Vinícius Basseto Félix


The estimation of the correlation between independent data sets using classical estimators, such as the Pearson coefficient, is well established in the literature. However, such estimators are inadequate for analyzing the correlation among dependent data. There are several types of dependence, the most common being the serial (temporal) and spatial dependence, which are inherent to the data sets from several fields. Using a bivariate time-series analysis, the relation between two series can be assessed. Further, as one time series may be related to an other with a time offset (either to the past or to the future), it is essential to also consider lagged correlations. The cross-correlation function (CCF), which assumes that each series has a normal distribution and is not autocorrelated, is used frequently. However, even when a time series is normally distributed, the autocorrelation is still inherent to one or both time series, compromising the estimates obtained using the CCF and their interpretations. To address this issue, analysis using the wavelet cross-correlation (WCC) has been proposed. WCC is based on the non-decimated wavelet transform (NDWT), which is translation invariant and decomposes dependent data into multiple scales, each representing the behavior of a different frequency band. To demonstrate the applicability of this method, we analyze simulated and real time series from different stochastic processes. The results demonstrated that analyses based on the CCF can be misleading; however, WCC can be used to correctly identify the correlation on each scale. Furthermore, the confidence interval (CI) for the results of the WCC analysis was estimated to determine the statistical significance.


Multiscale Analysis, Time Series, Cross Correlation, Non-Decimated Wavelet Transform

Full Text:



P. Abry, P. Flandrin. On the initialization of the discrete wavelet transform algorithm. IEEE Signal Processing Letters, 1(2), 32-34, 1994.

G. O. N. Brassarote. Análise multiescala de séries temporais do efeito da cintilação ionosférica nos sinais de satélite GPS a partir de wavelets não decimadas. 84 p. Master’s thesis. São Paulo State University (UNESP), 2014. Available in : .

G. O. N. Brassarote, E. M. Souza, J. F. G. Monico. Multiscale analysis of GPS time series from non-decimated wavelet to investigate the effects of ionospheric scintillation. TEMA (São Carlos), v. 16, n. 2, p. 119-130, 2015.

C. Chatfield. The Analysis of Time Series: An Introduction. London: Chapman and Hall, 1996.

T. Conlon, H. J. Ruskin, M. Crane. Cross-correlation dynamics in financial time series. Physica A: Statistical Mechanics and its Applications, 388(5), 705-714, 2009.

H. Li, T. Nozaki. Application of wavelet cross-correlation analysis to a plane turbulent jet. JSME International Journal Series B, v. 40, n. 1, p. 58-66, 1997.

G. Nason. “Wavelet methods in statistics with R”. Springer Science & Business Media, Bristol, United Kingdom, 2010.

D. B. Percival, H. O. Mofjeld. Analysis of subtidal coastal sea level fluctuations using wavelets. Journal of the American Statistical Association, 92(439), 868-880, 1997.

D. B. Percival, A. T. Walden. Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistics Mathematics, New York, USA, 2000.

R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria, 2016. Available in: .

A. B. Rowley et al. Synchronization between arterial blood pressure and cerebral oxyhaemoglobin concentration investigated by wavelet cross-correlation. Physiological measurement, v. 28, n. 2, p. 161, 2007.

G. Turbelin, P. Ngae, M. Grignon. Wavelet cross-correlation analysis of wind speed series generated by ANN based models. Renewable Energy, v. 34, n. 4, p. 024-1032, 2009.

P. Vielva, E. Martínez-González, M. Tucci. Cross-correlation of the cosmic microwave background and radio galaxies in real, harmonic and wavelet spaces: detection of the integrated Sachs-Wolfe effect and dark energy constraints. Monthly Notices of the Royal Astronomical Society, v. 365, n. 3, p. 891-901, 2006.

B. Whitcher, P. Guttorp, D. B. Percival. Mathematical background for wavelet estimators for cross covariance and cross-correlation, TR38, Natl. Res. Cent. Stat. and Environ., Seattle, 1999.

B. Whitcher, P. Guttorp, D. B. Percival. Wavelet analysis of covariance with applications to atmospheric time series. Journal of Geophysical Research: Atmospheres (1984-2012), v.105, n. D11, p. 14941-14962, 2000.

DOI: https://doi.org/10.5540/tema.2018.019.03.391

Article Metrics

Metrics Loading ...

Metrics powered by PLOS ALM


  • There are currently no refbacks.

Trends in Computational and Applied Mathematics

A publication of the Brazilian Society of Applied and Computational Mathematics (SBMAC)


Indexed in:



Desenvolvido por:

Logomarca da Lepidus Tecnologia