Fracdiff: Super-fast Fractional Differentiation

Documentation

Fracdiff performs fractional differentiation of time-series, a la "Advances in Financial Machine Learning" by M. Prado. Fractional differentiation processes time-series to a stationary one while preserving memory in the original time-series. Fracdiff features super-fast computation and scikit-learn compatible API.

What is fractional differentiation?

See M. L. Prado, "Advances in Financial Machine Learning".

Installation

pip install fracdiff

Features

Functionalities

fdiff: A function that extends numpy.diff to fractional differentiation.
sklearn.Fracdiff: A scikit-learn transformer to compute fractional differentiation.
sklearn.FracdiffStat: Fracdiff plus automatic choice of differentiation order that makes time-series stationary.
torch.fdiff: A functional that extends torch.diff to fractional differentiation.
torch.Fracdiff: A module that computes fractional differentiation.

Speed

Fracdiff is blazingly fast.

The following graphs show that Fracdiff computes fractional differentiation much faster than the "official" implementation.

It is especially noteworthy that execution time does not increase significantly as the number of time-steps (n_samples) increases, thanks to NumPy engine.

The following tables of execution times (in unit of ms) show that Fracdiff can be ~10000 times faster than the "official" implementation.

n_samples	fracdiff	official
100	0.675 +-0.086	20.008 +-1.472
1000	5.081 +-0.426	135.613 +-3.415
10000	50.644 +-0.574	1310.033 +-17.708
100000	519.969 +-8.166	13113.457 +-105.274

n_features	fracdiff	official
1	5.081 +-0.426	135.613 +-3.415
10	6.146 +-0.247	1350.161 +-15.195
100	6.903 +-0.654	13675.023 +-193.960
1000	13.783 +-0.700	136610.030 +-540.572

(Run on Ubuntu 20.04, Intel(R) Xeon(R) CPU E5-2673 v3 @ 2.40GHz. See fracdiff/benchmark for details.)

How to use

Fractional differentiation

A function fdiff calculates fractional differentiation. This is an extension of numpy.diff to a fractional order.

import numpy as np
from fracdiff import fdiff

a = np.array([1, 2, 4, 7, 0])
fdiff(a, 0.5)
# array([ 1.       ,  1.5      ,  2.875    ,  4.6875   , -4.1640625])
np.array_equal(fdiff(a, n=1), np.diff(a, n=1))
# True

a = np.array([[1, 3, 6, 10], [0, 5, 6, 8]])
fdiff(a, 0.5, axis=0)
# array([[ 1. ,  3. ,  6. , 10. ],
#        [-0.5,  3.5,  3. ,  3. ]])
fdiff(a, 0.5, axis=-1)
# array([[1.    , 2.5   , 4.375 , 6.5625],
#        [0.    , 5.    , 3.5   , 4.375 ]])

Scikit-learn API

Preprocessing by fractional differentiation

A transformer class Fracdiff performs fractional differentiation by its method transform.

from fracdiff.sklearn import Fracdiff

X = ...  # 2d time-series with shape (n_samples, n_features)

f = Fracdiff(0.5)
X = f.fit_transform(X)

For example, 0.5th differentiation of S&P 500 historical price looks like this:

Fracdiff is compatible with scikit-learn API. One can imcorporate it into a pipeline.

from sklearn.linear_model import LinearRegression
from sklearn.preprocessing import StandardScaler
from sklearn.pipeline import Pipeline

X, y = ...  # Dataset

pipeline = Pipeline([
    ('scaler', StandardScaler()),
    ('fracdiff', Fracdiff(0.5)),
    ('regressor', LinearRegression()),
])
pipeline.fit(X, y)

Fractional differentiation while preserving memory

A transformer class FracdiffStat finds the minumum order of fractional differentiation that makes time-series stationary. Differentiated time-series with this order is obtained by subsequently applying transform method. This series is interpreted as a stationary time-series keeping the maximum memory of the original time-series.

from fracdiff.sklearn import FracdiffStat

X = ...  # 2d time-series with shape (n_samples, n_features)

f = FracdiffStat()
X = f.fit_transform(X)
f.d_
# array([0.71875 , 0.609375, 0.515625])

The result for Nikkei 225 index historical price looks like this:

PyTorch API

One can fracdiff a PyTorch tensor. One can enjoy strong GPU acceleration.

from fracdiff.torch import fdiff

input = torch.tensor(...)
output = fdiff(input, 0.5)

from fracdiff.torch import Fracdiff

module = Fracdiff(0.5)
module
# Fracdiff(0.5, dim=-1, window=10, mode='same')

input = torch.tensor(...)
output = module(input)

More Examples

More examples are provided here.

Example solutions of exercises in Section 5 of "Advances in Financial Machine Learning" are provided here.

Contributing

Any contributions are more than welcome.

The maintainer (simaki) is not making further enhancements and appreciates pull requests to make them. See Issue for proposed features. Please take a look at CONTRIBUTING.md before creating a pull request.

References

Marcos Lopez de Prado, "Advances in Financial Machine Learning", Wiley, (2018).

jmrichardson/fracdiff