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Fractional Differentiation

ML Features ml stationarity prado feature-engineering fractional

Applies Prado-style fractional differencing to preserve memory while reducing non-stationarity in price series.

Visual Example

Fractional Differentiation — annotated preview mapping to core implementation

Synthetic ideal per library logic. Generated 2026-06-28 IST via docs/generate_all_previews.py (reproducible; maps to core Next<T> implementation).

Description

Fractional differentiation transforms a price or return series into a partially stationary feature while preserving long memory — the key insight from Marcos López de Prado's Advances in Financial Machine Learning. Integer differencing (d=1) removes autocorrelation but erases predictive structure; raw levels (d=0) are non-stationary. Fractional order d ∈ (0,1) sits between those extremes.

Practitioners typically apply this to log-prices or cumulative returns before tree models, neural nets, or cross-sectional factor research. Lower d (0.3–0.5) keeps more history in each bar; higher d approaches a first difference. The threshold parameter truncates the weight tail so the convolution window stays finite and streaming-friendly.

Compared to ad-hoc rolling z-scores or winsorization alone, frac-diff is a principled stationarity transform with a published weight recurrence. Pair with regime filters (Hurst, HMM) when building ML feature matrices.

QuantWave implements this indicator via the universal Next<T> trait, guaranteeing bit-identical results between Rust streaming, Python streaming, and Polars batch (.ta.frac_diff()) surfaces.

Formula / Specification

Implementation (frac_diff):

\[ w_0 = 1,\quad w_k = -w_{k-1}\frac{d - k + 1}{k},\quad \tilde{X}_t = \sum_{k=0}^{K} w_k X_{t-k} \]

Gold-standard parity vectors: quantwave-core/tests/gold_standard/frac_diff.json.

Parameters

Parameter Default Description
d 0.4 Fractional differentiation order (0 = identity, 1 = full integer diff)
threshold 1e-5 Truncate weights when

Usage Examples

Streaming (Rust)

use quantwave_core::indicators::FracDiff;
use quantwave_core::traits::Next;

let mut ind = FracDiff::new(0.4, 1e-5);
for price in &prices {
    let value = ind.next(price);
}

Streaming (Python)

from quantwave import FracDiff

ind = FracDiff(0.4, 1e-5)
for price in prices:
    value = ind.next(price)

Polars Batch (Python)

import polars as pl
import quantwave as qw  # registers LazyFrame.ta when using quantwave-polars plugin

df = (
    pl.read_csv('ohlcv.csv')
    .lazy()
    .ta.frac_diff("close", 0.4, 1e-5)
    .collect()
)

All surfaces are bit-identical via the single Next<T> implementation and proptests.

Edge Cases & Limitations

  • Warm-up: output is NaN until the full weight window fills (length depends on d and threshold; e.g. d=0.4, threshold=1e-5 may require 200+ bars).
  • Very small threshold increases window length and memory; very large threshold truncates weights early and may under-difference.
  • d outside [0,1] is clamped; d=0 approaches identity, d=1 approaches integer differencing.
  • NaN inputs propagate NaN output for that bar; do not forward-fill before frac-diff without documenting bias.
  • Sudden gaps or bad ticks distort the weighted sum — winsorize or filter outliers on illiquid names.
  • Validated via proptests and quantwave-core/tests/gold_standard/frac_diff.json.
  • No look-ahead bias; streaming and Polars batch paths are bit-identical.

Boundary Behavior

Condition Behavior
Warm-up Leading bars return NaN until warmup_bars is satisfied.
period > len When period exceeds series length, output is all NaN.
NaN inputs NaN in input propagates to output (NaN out).
Invalid params Non-positive period or missing required params raise ValueError.
Empty data Empty input returns an empty result series.

Sources & References

Primary Source: https://www.wiley.com/en-us/Advances+in+Financial+Machine+Learning-p-9781119482086

Implementation: quantwave-core/src/indicators/frac_diff (FracDiff / _METADATA). Parity: quantwave-core/tests/gold_standard/frac_diff.json

Provenance: Standards bulk upgrade 2026-06-28 IST — see docs/DOCUMENTATION_STANDARDS.md.