Quantile Detrending Moving Average Algorithm for Multifractal Time Series and Surfaces

Abstract We introduce the multifractal quantile detrending moving average (MFQDMA) algorithm for one-dimensional (1D) time series and two-dimensional (2D) surfaces. MFQDMA replaces the moving average of the multifractal detrending moving average (MFDMA) method by a moving $\lam$-quantile, the local-constant solution of Koenker--Bassett quantile regression, and thereby produces a one-parameter family of generalized Hurst exponents $h(q,\lam)$ for backward, centered, and forward windows. We derive exact properties of the estimator. For a non-negative measure the quantile residual is exactly the mass of a gliding box of size $\ell=(1-\theta-\lam)(n-1)$; consequently $h(q,\lam)$ is asymptotically independent of $\lam$, and the algorithm degenerates on the line $\theta+\lam=1$, which contains the centered median. For signals, $h(q,\lam)$ is affine invariant and satisfies $h(q,\lam;-x)=h(q,1-\lam;x)$. No box-sum reduction exists in 2D. On benchmarks with analytically known spectra, backward MFQDMA matches MFDMA on clean 1D cascades and fractional Gaussian noises, and lowers the error of the mass exponent $\tau(q)$ of $256\times256$ multiplicative cascades by 29\% at $\lam=1/4$. When 2\%--10\% of a 1D cascade is replaced by outliers, Student-$t$, or L\'evy-stable values, low-$\lam$ MFQDMA reduces the $\tau(q)$ error of MFDMA by 22\%--46\% and that of multifractal detrended fluctuation analysis by a factor of 2--4, whereas $\lam\ge3/4$ is harmful. MFQDMA does not remove the spurious multifractality of $\alpha$-stable noise, does not confer invariance under monotone transforms, and gives only marginal gains for contaminated surfaces. Self-affine surfaces must be analyzed through their mixed increments; otherwise $h(2)\approx2$ irrespective of the Hurst index. We apply the method to daily returns of 18 financial assets (12 stock indices, three exchange rates, gold, oil, Bitcoin) and 29 Dow Jones stocks. There the multifractality is mainly distributional, and low-$\lam$ MFQDMA is the estimator least sensitive to isolated bad ticks (less sensitive than MFDMA for 28 of 29 stocks, and about half as sensitive as MFDFA) and to a few crash days: removing 19 October 1987 changes the S\&P 500 spectral width by $0.010$, versus $0.093$ for MFDMA. MFDMA remains preferable for dense microstructure noise. In 500 electroencephalogram segments and a terrain elevation model, the multifractality is dominated by linear correlations, and the $\lam$-resolved exponents carry class information not contained in $h(q)$. A jointly submitted companion paper extends the method to cross-correlations.