Discussing the article: "Beyond GARCH (Part V): Fitting the Multifractal Spectrum in MQL5"

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Check out the new article: Beyond GARCH (Part V): Fitting the Multifractal Spectrum in MQL5.

This article builds the Spectrum Fitter: from tau(q) we compute f(alpha) with a discrete Legendre transform, then fit Normal, Binomial, Poisson, and Gamma spectra under box constraints using BLEIC. The best model by SSE is selected, and its parameters (eg, alpha min, alpha max or alpha_0, gamma) become the cascade inputs for multifractal simulation.

In Part 4, we built the partition analysis engine: the module that takes raw price data, computes the scaling function τ(q), estimates the Hurst exponent, and determines whether the data exhibits multifractal behavior. The output of that module is a curve, τ(q), that encodes how different statistical moments scale across time. But a curve alone is not enough to generate synthetic price paths. To build the multiplicative cascade that creates multifractal trading time, we need a specific distribution and its parameters.

This article bridges that gap. We will implement the Spectrum Fitter, the module that transforms the raw τ(q) curve into a fitted probability distribution ready for simulation. The process has two stages. First, we apply the Legendre transform to convert τ(q) into the singularity spectrum f(α), a more physically interpretable representation of the multifractal structure. Second, we fit four candidate distributions to the empirical spectrum using bounded optimization (BLEIC from ALGLIB). We then select the model with the lowest sum of squared errors. The winner provides the exact parameters that the cascade generator will use in Part 6.

Author: Muhammad Minhas Qamar