Fractional Differentiation on Time Series

Corrected and sanitized version of code from Advances of Machine Learning by Marcos Prado.

Comparison to Prado's original code

In [176]: import mlfinlab.features.fracdiff as fd
     ...: import fracdiff.fracdiff as fd_
     ...: x = np.random.randn(10000)
     ...: s = pd.DataFrame(x)

In [177]: %timeit a = fd.frac_diff_ffd(s, 0.5, thresh=1e-4)
     ...:
1.31 s ± 16.5 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)

In [178]: %timeit b = fd_.frac_diff_ffd(x, 0.5, thres=1e-4)
     ...:
1.77 ms ± 10.4 µs per loop (mean ± std. dev. of 7 runs, 1000 loops each)

In [179]:

In [179]: a[1] = b
     ...: np.abs(a[0] - a[1]).max()
     ...:
Out[179]: 8.881784197001252e-16

In [180]: a.tail()
Out[180]:
             0         1
9995  2.072307  2.072307
9996 -0.504402 -0.504402
9997  0.095372  0.095372
9998  0.296584  0.296584
9999  1.216419  1.216419

Best degree of diff

AAPL

FB

GS

IBM

V

Diffs

SP500 returns

F(X) = X and its (frac) derivatives/antiderivates

The animation shows the derivative operator oscillating between the antiderivative (α=−1: y = ​1⁄2⋅x2) and the derivative (α = +1: y = 1) of the simple function y = x continuously.

Get Started

git clone [email protected]:philipperemy/fractional-differentiation-time-series.git && cd fractional-differentiation-time-series
virtualenv -p python3 venv
source venv/bin/activate
pip install . --upgrade
python frac_diff_sp500.py

References:

• https://www.wiley.com/en-us/Advances+in+Financial+Machine+Learning-p-9781119482086
• https://wwwf.imperial.ac.uk/~ejm/M3S8/Problems/hosking81.pdf
• https://en.wikipedia.org/wiki/Fractional_calculus