cfad — Characteristic Function Anomaly Detector

cfad is a research-oriented Python package for detecting changes in the distributional shape of financial returns with empirical characteristic functions (ECFs) and sequential CUSUM monitoring.

For each rolling window, CFAD compares the empirical characteristic function with the Gaussian characteristic function fitted to that window's sample mean and variance. The normalized real-frequency discrepancy is the anomaly score:

$$ D_t = \left[ \frac{1}{\xi_{\max}-\xi_{\min}} \int_{\xi_{\min}}^{\xi_{\max}} \left| \widehat\varphi_t(\xi)

\varphi_{\mathcal N(\widehat\mu_t,\widehat\sigma_t^2)}(\xi) \right|^2 ,d\xi \right]^{1/2}. $$

Because location and scale are fitted within each window, the score is aimed at higher-order shape changes such as tail and skewness changes. A two-sided Page-CUSUM then converts the score sequence into sequential alarms.

Scientific scope. The finite-sample ECF $\widehat\varphi_n(z)=n^{-1}\sum_j e^{izx_j}$ is a finite sum of entire functions and is itself entire. Therefore its exact closed-contour integral is zero. CFAD does not infer population-CF branch cuts or poles from an empirical contour residue. Complex contour integration remains available as a diagnostic helper for parametric characteristic functions evaluated at complex arguments, but it is not the empirical anomaly statistic.

Status

The repository is currently a research/development project. The codebase contains release scaffolding, documentation, benchmarks, notebooks, and a draft software paper. GitHub releases are archival research-software snapshots; no PyPI distribution is claimed unless a separate PyPI publication is explicitly performed and verified.

Current validation boundary

Two frozen validation programmes have now tested the corrected method rather than the retired contour-residue interpretation.

The v2 sequential benchmark showed that Monte Carlo calibration can control the Gaussian-null false-alarm rate, but the Gaussian-reference score was not robust to a stable Student-t in-control law and had weak first-alarm power once false alarms were controlled. The v3 score-level ablation then removed CUSUM and separated frequency scaling from reference-law choice. Standardizing each window corrected the legacy score's sensitivity to pure variance changes, and a frozen empirical in-control ECF produced strong null-law stability. However, the empirical-reference score achieved AUC 0.646 for a Gaussian-to-Student-t shape change and 0.783 for a Gaussian-to-skew change, for an average of 0.715 versus 0.738 for a simple kurtosis-distance comparator.

Accordingly, the current evidence does not establish that CFAD is a validated sequential detector or that its ECF score outperforms simpler moment summaries. The negative v2 and v3 results are retained as reproducible evidence in benchmarks/ and are treated as design constraints rather than tuned away.

Installation

From the default branch:

git clone https://github.com/DiogoRibeiro7/cfad
cd cfad
git switch main
python -m pip install -e ".[dev]" --no-build-isolation

The package includes optional Cython acceleration for rolling ECF evaluation and CUSUM updates. The statistical score itself is implemented once in NumPy, so installing the extensions changes performance rather than the definition of the statistic.

Quick start

import numpy as np
from cfad import detect

rng = np.random.default_rng(42)
returns = np.concatenate(
    [
        rng.normal(0.0, 0.01, 300),
        rng.standard_t(df=3.0, size=150) * 0.01 / np.sqrt(3.0),
    ]
)

report = detect(
    returns,
    window=60,
    xi_range=(-10.0, 10.0),
    step=1,
    calibration_frac=0.4,
    k=0.5,
    h=5.0,
)
print(report.summary())

For market data, fetch the series explicitly with the provider of your choice, then pass returns to detect. Keeping data acquisition outside the core example makes the detector reproducible without relying on network access.

Detection pipeline

returns
  │
  ├─ rolling ECF on a real-frequency grid
  │
  ├─ fitted Gaussian CF in each window
  │
  ├─ normalized ECF L2 shape distance D_t
  │
  ├─ calibration of score mean/std on an in-control prefix
  │
  └─ two-sided Page-CUSUM → alarms

Parametric CF models

CFAD also implements characteristic-function models for descriptive model comparison and goodness-of-fit work:

Model	Main use
`GaussianCF`	location/scale baseline
`NIGCF`	semi-heavy tails and skewness
`CGMYCF`	jump/tail-shape modelling
`LevyStableCF`	power-law tail modelling

compare_models() compares fitted models by real-frequency ECF discrepancy and AIC. A better-fitting non-Gaussian model is evidence of distributional fit, not a direct empirical test for complex singularities.

Utilities

The repository includes:

rolling_gof, cf_distance, and epps_pulley_test for ECF goodness of fit;
WalkForwardBacktest for temporal evaluation;
bootstrap and score-stability diagnostics;
window_sensitivity, frequency_sensitivity, and threshold sensitivity;
multivariate and market-oriented helpers;
a Streamlit dashboard under apps/;
reproducible notebooks and benchmark scripts.

Reproducibility

The main scientific validation target is not "does an alarm fire on one famous market event?" but how the detector behaves under controlled null and alternative data-generating processes. The benchmark layer reports false-positive behaviour, power/discrimination under prespecified shape changes, specificity to location/scale changes, and comparison against simpler baselines.

Frozen failed experiments are part of the evidence record. In particular, benchmarks/v2_failed_calibration_record.json records the failed sequential screen and benchmarks/v3_failed_score_validation_record.json records the failed score-level screen. Those failures are not retroactively reclassified after parameter or method changes.

Documentation and paper

Sphinx sources: docs/source/
Draft software paper: paper/paper.md
Reproducible notebooks: notebooks/
Benchmarks: benchmarks/

The manuscript is a draft companion to the software. Citation metadata should only advertise a journal DOI after an actual accepted/published record exists.

Author

Diogo Ribeiro
Faculty of Media Arts and Design, Technical University of Porto
ORCID: 0009-0001-2022-7072

License

MIT

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This release

0.2.2 This release

Sep 1, 2026 10 files

cfad 0.2.2

cfad — Characteristic Function Anomaly Detector

$$ D_t = \left[ \frac{1}{\xi_{\max}-\xi_{\min}} \int_{\xi_{\min}}^{\xi_{\max}} \left| \widehat\varphi_t(\xi)

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Current validation boundary

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