fractional differentiation for engineering stationary features from financial time series

financial/fractional_differentiation.py

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+"""Implementation of fractional differentiation from the book
+`Advances in Financial Machine Learning` by Marcos Lopez de Prado.
+
+Fractional differentiation is a technique to make time series stationary
+while preserving the memory of the original data.
+
+This implementation utilizes a fixed window size to calculate the
+fractional differentiation. Note that this implementation is a simplified
+version compared to the one presented in the book by Marcos Lopez de Prado.
+Here, we do not consider the weight loss. In the book, weight loss is
+calculated to account for the fact that the initial points in
+the series carry different amounts of information than the final points.
+
+In the calculation of fractional differentiation,
+the price is convolved with the weights to obtain the
+fractional differentiated series. This process enables the
+transformation of the time series into a stationary form while
+retaining the memory of the original data.
+
+To determine the optimal degree of differentiation,
+one needs to find the minimum value of the differentiation degree,
+a value between 0 and 1, that renders the time series stationary.
+
+Reference
+---------
+https://www.wiley.com/en-us/Advances+in+Financial+Machine+Learning-p-9781119482086
+
+https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3257419
+
+"""
+
+from collections.abc import Sequence
+from math import nan
+
+
+def calculate_weights(degree: float, length: int) -> list[float]:
+    r"""Calculate the weights for fractional differentiation.
+
+    .. math::
+
+        w_{0} = 1
+        w_{k} = -w_{k-1} * (d - k + 1) / k
+
+    Parameters
+    ----------
+    degree : float
+        The degree of differentiation.
+    length : int
+        The length of the weights.
+
+    Returns
+    -------
+    list[float]
+        The weights for fractional differentiation.
+
+    Examples
+    --------
+    >>> calculate_weights(0.5, 3)
+    [1.0, -0.5, -0.125]
+    >>> calculate_weights(0.5, 4)
+    [1.0, -0.5, -0.125, -0.0625]
+    """
+    weights = [1.0]
+    for k in range(1, length):
+        weights.append(-1 * weights[-1] * (degree - k + 1) / k)
+    return weights
+
+
+def fracdiff_fixedwindow(
+    price_series: Sequence[float], degree: float, window_size: int
+) -> list[float]:
+    """Calculate the fractional differentiation with a fixed window size.
+
+    Parameters
+    ----------
+    price_series : Sequence[float]
+        The price series to calculate the fractional differentiation.
+    degree : float
+        The degree of differentiation.
+
+    Returns
+    -------
+    list[float]
+        The fractional differentiated series.
+
+    Examples
+    --------
+    >>> price_series = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
+    >>> fracdiff_fixedwindow(price_series, 0.5, 3)
+    [nan, nan, nan, 2.25, 2.625, 3.0, 3.375, 3.75, 4.125, 4.5]
+    """
+    weights = calculate_weights(degree=degree, length=len(price_series))
+    frac_diff_series = [nan] * window_size
+    for i in range(window_size, len(price_series)):
+        frac_diff_series.append(
+            sum(weights[j] * price_series[i - j] for j in range(window_size))
+        )
+    return frac_diff_series
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()