# Direct determination approach for the multifractal detrending moving average analysis

## TL;DR

This paper introduces a new direct method for multifractal analysis using the detrending moving average approach, which simplifies computation while maintaining accuracy, and demonstrates its effectiveness on synthetic and real financial data.

## Contribution

The paper presents a novel direct determination approach for multifractal analysis with MF-DMA, reducing computational cost and confirming its applicability to financial time series.

## Key findings

- Both approaches accurately reveal fractal and multifractal features.
- The new method reduces computation time without sacrificing accuracy.
- Multifractality is confirmed in stock market volatility data.

## Abstract

In the canonical framework, we propose an alternative approach for the multifractal analysis based on the detrending moving average method (MF-DMA). We define a canonical measure such that the multifractal mass exponent $\tau(q)$ is related to the partition function and the multifractal spectrum $f(\alpha)$ can be directly determined. The performances of the direct determination approach and the traditional approach of the MF-DMA are compared based on three synthetic multifractal and monofractal measures generated from the one-dimensional $p$-model, the two-dimensional $p$-model and the fractional Brownian motions. We find that both approaches have comparable performances to unveil the fractal and multifractal nature. In other words, without loss of accuracy, the multifractal spectrum $f(\alpha)$ can be directly determined using the new approach with less computation cost. We also apply the new MF-DMA approach to the volatility time series of stock prices and confirm the presence of multifractality.