Using Wavelet Coefficients to Explore Fractal Behavior in Time Series

Summary

The document considers ways to investigate fractal behavior and harmonic patterns in stock time series, with the broader aim of building trading systems. Its concrete suggestion is to fit a Daubechies wavelet and examine coefficients across scales at a fixed time location. Similar coefficients across those scales may indicate fractal-like behavior in that local region of the series.

The response also points toward stochastic calculus and econometric estimators that use multiple frequencies as useful areas to review. It does not give a validated detection procedure, define a threshold for deciding that a series is fractal, or explain how to identify harmonic trading patterns. Nor does it provide evidence of predictive power or a complete system-design method. The wavelet idea is therefore exploratory: scale similarity may motivate further analysis, but by itself it does not establish a tradable signal or distinguish structure from noise.

Key ideas

• A Daubechies wavelet can represent a time series at different scales and locations.
• Comparing coefficients across scales at a fixed location may help explore local fractal-like behavior.
• The response suggests studying multi-frequency econometric methods and stochastic calculus.
• The proposed diagnostic is exploratory and does not establish a predictive trading signal.

Tags

# How to look for fractals/harmonics patterns in time series?


# How to look for fractals/harmonics patterns in time series?












I want to build trading systems based on two things:

1)Fractal Theory

2)Harmonics Pattern

I have read the book : The Misbehavior of Markets: A Fractal View of Financial Turbulence By Mandelbrot but he didn’t show how to find fractal in time series. Moreover I don’t have time to go through all books with this topic, because I have work afterhours. So my question to you is:

-Do you know best measures for fractal theory or harmonics patterns in Time Series of Stock?

-Do you know some helpful publications/books connected with this topic – how to look for patterns/measures?

-Do you have any suggestions about construction of such trading systems?

## Answer by Jase (score 2, accepted)

https://quant.stackexchange.com/a/8550

Perhaps to detect fractal behaviour you could fit something like a Daubechies wavelet. That is, $W(a,b) := \frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} f(t) \phi((t-b)/a)dt$. Then you want to check the set $\{W(a,b) : a \in \mathbb{R}_+\}$ where $a$ is the scale, for some fixed $b$. If all the coefficients are similar then this might be some indication of fractal like behaviour at least in that time localisation.

Good to review stochastic calculus or econometric estimators that take in multiple frequencies for possible solutions.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.