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Fractional Differentiation on Long-Memory Time Series: A Case of Study in Fractional Brownian Motion Processes

Frank Salvador Ygnacio Rosas

Author

Frank Salvador Ygnacio Rosas

Title

Fractional Differentiation on Long-Memory Time Series: A Case of Study in Fractional Brownian Motion Processes

Description

Fractional Differentiation method on Fractional Brownian motion processes.

Fractional Differentiation on Long-Memory Time Series: A Case of Study in Fractional Brownian Motion Processes

Frank Salvador Ygnacio Rosas

Co-founder and Quantitative researcher at Quantmoon Technologies

The aim of this project is the implementation of an optimal fractional differentiation procedure for time series with ‘long-memory’. Moreover, since there is a trade-off between the informative dependence of the time series and the stationarity needed to make statistical inferences, this project proposes a recursive binary search method to get the optimal (minimum) non-integer order of differentiation to make the time series (weakly) stationary, while preserving as much as memory as possible. This procedure will be tested on several Fractional Brownian motion simulations using different values for the Hurst exponent between 1/2 and 1. This will make possible to get an empirical relationship between the latter as a measure of the degree of dependency, and the optimal non-integer order of differentiation. The hypothesis is that this relationship tends to be linear, being the classical d(1) order of differentiation only optimal mostly when H is closer to 1. Thus, the project starts by implementing the non-integer differentiation method for time series developed by Lopez de Prado (2018). Then, a new recursive binary search algorithm to find the optimal non-integer “d” is proposed, due to its efficient computational time, degree of precision and simplicity. This will be tested over the Fractional Brownian Motion simulations, proving the linear relationship between the Hurst exponent and the optimal “d” developed in this project. Finally, some general considerations about the differentiation procedure implemented are outlined, as well as possible improvements.

Why Fractional Differentiation for Time Series?
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Part I: The Fractional Differentiation Method
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Part II: The Fractional Brownian Motion (fBm) Experiment
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Concluding remarks
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Keywords
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Acknowledgment
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References
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Cite this as: Frank Salvador Ygnacio Rosas, "Fractional Differentiation on Long-Memory Time Series: A Case of Study in Fractional Brownian Motion Processes" from the Notebook Archive (2022), https://notebookarchive.org/2022-07-9nbew3e

Fractional Differentiation on Long-Memory Time Series: A Case of Study in Fractional Brownian Motion Processes

Why Fractional Differentiation for Time Series?

Part I: The Fractional Differentiation Method

Part II: The Fractional Brownian Motion (fBm) Experiment

Concluding remarks

Keywords

Acknowledgment

References

Why Fractional Differentiation for Time Series?


Part I: The Fractional Differentiation Method


Part II: The Fractional Brownian Motion (fBm) Experiment


Concluding remarks


Keywords


Acknowledgment


References
