Multifractal characterization for bivariate data

Multifractal analysis is a reference tool for the analysis of data based on local regularity, which has been proven useful in an increasing number of applications. However, in its current formulation, it remains a fundamentally univariate tool, while being confronted with multivariate data in an increasing number of applications. Recent contributions have explored a first multivariate theoretical grounding for multifractal analysis and shown that it can be effective in capturing and quantifying transient higher-order dependence beyond correlation. Building on these first fundamental contributions, this work proposes and studies the use of a quadratic model for the joint multifractal spectrum of bivariate time series. We obtain expressions for the Pearson correlation in terms of the random walk and a multifractal cascade dependence parameters under this model, provide complete expressions for the multifractal parameters and propose a transformation of these parameters into natural coordinates that allows to effectively summarize the information they convey. Finally, we propose estimators for these parameters and assess their statistical performance through numerical simulations. The results indicate that the bivariate multifractal parameter estimates are accurate and effective in quantifying non-linear, higher-order dependencies between time series.