MicroWavelet 26.7.6

CWT-based anomaly detector for multi-filter microlensing light curves

MicroWavelet

DOI: pending Zenodo release

MicroWavelet is a Python library for the detection of transient anomalies in multi-filter light curves using Continuous Wavelet Transform (CWT) methodologies. The package is optimized for microlensing signals but is applicable to general peak and dip detection in time-series data.

The library implements a scale-space search using Paczynski-profile wavelet kernels, integrated with robust Whittaker-Eilers (Smoothing Spline) detrending and multi-band chromaticity analysis.

Mathematical Framework

1. Continuous Wavelet Transform (CWT)

The library computes the CWT coefficients $W(s, \tau)$ by convolving the light curve flux $f(t)$ with a scaled and translated mother wavelet $\psi(t)$:

$$W(s, \tau) = \int_{-\infty}^{\infty} f(t) \frac{1}{s^p} \psi^*\left(\frac{t - \tau}{s}\right) dt$$

To achieve scale-invariance for the Paczynski profile, we implement a modified normalization where $p=1$ in the kernel construction and apply an $s^{-0.5}$ correction to the resulting coefficients.

2. Paczynski Kernels

The wavelet kernels are derived from the Paczynski magnification formula, which describes the flux excess $A(u) - 1$ for a point-source point-lens microlensing event:

$$A(u) = \frac{u^2 + 2}{u \sqrt{u^2 + 4}}, \quad u(t) = \sqrt{u_0^2 + \left(\frac{t - t_0}{t_E}\right)^2}$$

The library utilizes two distinct kernel morphologies for anomaly detection:

• Even (Symmetric) Kernel ($\psi_e$): Proportional to the negative second derivative of the magnification profile ($-\frac{d^2A}{dt^2}$). This kernel is optimized for detecting symmetric peaks and dips.
• Odd (Asymmetric) Kernel ($\psi_o$): Proportional to the first derivative of the magnification profile ($\frac{dA}{dt}$). This kernel serves as a diagnostic for asymmetric features and caustic crossings.

Both kernels are L1-normalized such that $\sum |\psi| = 1$, ensuring that the resulting Z-scores (SNR) are comparable across the Einstein timescale grid $t_E$.

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Methodological Implementation

1. Scale-Invariant Wavelet Normalization

The library utilizes L1-normalized kernels to ensure scale independence. For Paczynski templates, we apply an $s^{-0.5}$ scaling to the CWT coefficients to align the peak detection scale $t_{E,\text{scan}}$ with the physical Einstein crossing time, independent of the background noise power spectrum.

2. Systematic Bias Correction

Signal detection is performed using a template impact parameter of $u_0 = 0.05$. To account for the timescale inflation observed in events with larger $u_0$, a 5th-degree polynomial correction is applied:

$$r_{\text{peak}}(u_0) \approx 12.0707 u_0^5 - 29.2612 u_0^4 + 28.1550 u_0^3 - 18.9146 u_0^2 + 22.2832 u_0 - 0.0635$$

The estimated crossing time is then refined as $t_{E,\text{true}} = t_{E,\text{scan}} / r_{\text{peak}}(u_{0,\text{event}})$, maintaining estimation residuals below 2% across the parameter space.

3. Parametric Statistical Evaluation

For each candidate detection, the library performs a weighted linear least-squares fit of the Paczynski model ($y = F_s S + F_b$) within a local $t_0 \pm 5 t_E$ window. Test statistics provided include:

• $\Delta\chi^2 = \chi^2_{\text{null}} - \chi^2_{\text{lens}}$
• $\Delta\text{BIC} = \chi^2_{\text{lens}} - \chi^2_{\text{null}} + 2 \ln(N)$

4. Boundary and Artifact Mitigation

• Temporal Boundaries: Detections within $0.5 \cdot t_E$ of the observation limits are identified via an edge_flag to distinguish them from potential windowing artifacts.
• Interpolation: A Nadaraya-Watson local Gaussian kernel regression (weighted by $1/\sigma^2$) is available to minimize the impact of non-uniform sampling and outliers.

5. Multi-Band Chromaticity Analysis

The primary band signal is projected onto secondary filters using a local weighted linear fit. The resulting chromaticity_ratio ($\mathcal{R} = F_{s,\text{other}} / F_{s,\text{primary}}$) and chromatic_flag facilitate the identification of non-achromatic signals, such as stellar flares or instrumental systematic effects.

6. Periodic Baseline Detrending

For observations of variable stars (e.g., RR Lyrae, eclipsing binaries), the library provides a highly optimized, multi-stage Whittaker-Eilers (Smoothing Spline) detrending pipeline:

1. Robust Period Search: Performs an initial Lomb-Scargle search on a "cleaned" version of the light curve where large positive transients (like microlensing events) are masked using a 2.5-sigma MAD threshold.

2. Bayesian Period Selection: Implements an iterative halving strategy to find the fundamental period. Candidates ($P/2, P, 2P$) are evaluated using a binned Gaussian Log-Likelihood proxy:

   $$\ln L = -0.5 \left( RSS + \sum_{k \in \text{valid}} \ln(2\pi \sigma_k^2) \right)$$

   This incorporates a sum-of-log-variances term that mirrors the noise determinant of Gaussian Processes to penalize poorly-folded periods (which exhibit high phase scatter and large bin errors $\sigma_k$) without complexity-penalty bias.

3. Iterative Whittaker Optimization: The period is fine-tuned using a non-linear scalar optimizer that fits a robust Whittaker-Eilers smoother with circular boundary conditions and Generalized Cross-Validation (GCV) optimal smoothing parameter $\lambda$ selection at each iteration.

4. Normalization: The raw flux is divided by the converged periodic Whittaker baseline model to isolate the stationary transient signal for CWT analysis.

7. Robust Non-Periodic GP Detrending

For non-periodic light curves, the library provides a robust Gaussian Process (GP) detrending method that prevents transient signals from being absorbed into the baseline.

• Top Panel: Comparison of the Naive GP baseline (red) vs. the Robust GP baseline (green). The naive baseline attempts to smooth through the anomaly, while the robust baseline stays true to the quiescent signal.
• Middle Panel: Naive residuals, where the signal is significantly dampened.
• Bottom Panel: Robust residuals, where the transient signal is preserved with high significance.

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8. Noise Characterisation & Multi-Band Analysis

• Top Panel: Estimated spectral indices ($\beta$) for different photometric bands.
• Bottom Panel: Wavelet coherence map between two bands, showing correlated noise across scales and time.

from microwavelet import characterize_multiband_noise

# 1. Input data (multi-band residuals)
bands_data = {
    "W146": {"t": t1, "y": y1, "y_err": e1},
    "W184": {"t": t2, "y": y2, "y_err": e2},
}

# 2. Run multi-band noise analysis
results = characterize_multiband_noise(bands_data)

# 3. Access metrics
print(results["individual_metrics"]["W146"]["beta"])
print(results["coherence_matrix"]["W146_W184"])

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9. Cumulative Sum (CUSUM) Anomaly Detection

To detect abrupt shifts or persistent deviations from a flat baseline (e.g., weak or fast microlensing signals, sharp magnification peaks), the library provides cumulative sum (CUSUM) detection routines:

• Linear CUSUM (run_linear_cusum): Accumulates positive residual deviations, particularly useful for detecting magnification peaks.
• Quadratic CUSUM (run_quadratic_cusum): Accumulates variance excess above the expected baseline noise level.
• CUSUM-based Parameter Seeding (seed_by_flat_cusum): Analyzes the standardized residuals from a flat baseline median fit to trigger anomaly signals. If the threshold is exceeded, it determines the peak time (t0) and estimates the event duration to seed the Einstein crossing time (tE).
• CUSUM-based Anomaly Detection (find_anomalies_cusum): Runs quadratic CUSUM on standardized residuals and extracts anomaly properties, including the peak time (t0), trigger status, onset/end times, signal duration, and local noise standard deviation.

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Installation

From PyPI (Standard)

You can install the stable release of microwavelet directly from PyPI:

pip install microwavelet

From Source (Local Development)

Clone the repository and install in editable mode:

git clone https://github.com/dylannpaterson/MicroWavelet.git
cd MicroWavelet
pip install -e ".[dev]"

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Usage Example

import numpy as np
from microwavelet import analyze_lightcurve

# 1. Input data structure (multi-band relative flux)
data = {
    "F146": {
        "t": np.arange(0, 100, 0.1),
        "y": np.random.normal(1.0, 0.02, 1000),
        "y_err": np.ones(1000) * 0.02
    },
    "F087": {
        "t": np.arange(0, 100, 0.5),
        "y": np.random.normal(1.0, 0.03, 200),
        "y_err": np.ones(200) * 0.03
    }
}

# 2. Execute analysis pipeline with periodic detrending enabled
results = analyze_lightcurve(
    data,
    detrend_periodic=True,    # Enable baseline removal
    min_period=1.0,
    max_period=10.0,
    interpolator="weighted",
    cwt_threshold=12.0,
    stamp_dir="plots/"        # Optional: Save a detailed stamp plot of all peaks
)

# 3. Access detection parameters
for anomaly in results["anomalies"]:
    print(f"Candidate t0: {anomaly['t0']:.3f}")
    print(f"Timescale tE: {anomaly['tE']:.2f} (u0: {anomaly['u0']:.3f})")

CUSUM-based Anomaly Detection

The library provides several CUSUM-based tools for detecting deviations in light curves. These can be used for both event detection (finding the main microlensing peak) and anomaly detection (finding planetary deviations or stellar flares).

Detection Modes

• Forward CUSUM: Accumulates deviations from the start of the time series. Excellent for detecting the onset of a magnification event.
• Backward CUSUM: Accumulates deviations from the end of the time series (running backwards). Useful for verifying the symmetry of an event or detecting late-time anomalies.
• Bidirectional CUSUM: The minimum of the forward and backward passes. This is the most robust mode for localizing the exact peak and duration of an anomaly, as it prevents the "drift" that occurs when a single-pass CUSUM accumulates too much signal from a long-duration event.

Multi-Stage CUSUM Workflow Demonstration

The following plot demonstrates a complete multi-stage pipeline:

1. Event Detection: Using bidirectional CUSUM on residuals relative to a flat baseline to identify the onset of a microlensing event.
2. PSPL Fitting: Fitting a Point Source Point Lens (PSPL) model to the observed light curve.
3. Anomaly Detection: Using bidirectional CUSUM on the residuals of the PSPL fit to isolate planetary or other transient deviations.

from microwavelet import find_anomalies_cusum

# 1. Detect the main event (vs baseline)
event = find_anomalies_cusum(t, residuals_baseline, threshold=25.0, bidirectional=True)

# 2. Detect the anomaly (vs PSPL model)
anomaly = find_anomalies_cusum(t, residuals_pspl, threshold=12.0, bidirectional=True)
```residuals = (y - y_base) / y_err

# 1. Run linear CUSUM
cusum_scores = run_linear_cusum(residuals, k=1.0)

# 2. Seed parameters using the flat CUSUM routine
t0_seed, tE_seed, triggered = seed_by_flat_cusum(
    t, y, y_err, method='linear', k=1.0, threshold=10.0
)
if triggered:
    print(f"CUSUM anomaly triggered! Seed t0={t0_seed:.3f}, tE={tE_seed:.3f}")

# 3. Detect and extract anomaly details using quadratic CUSUM
anomaly_details = find_anomalies_cusum(t, residuals, threshold=25.0, k=2.0)
if anomaly_details["triggered"]:
    print(f"Anomaly detected! Score: {anomaly_details['score']:.2f}")
    print(f"t0: {anomaly_details['t0']:.3f}, Duration: {anomaly_details['duration']:.3f}")

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Pipeline Diagnostics

The following plot illustrates the CWT detection process and subsequent parametric fit on a synthetic dataset:

• Top Panel: Observed flux across multiple bands with the error-weighted Gaussian interpolation and the analytical Paczynski fit.
• Bottom Panel: Scale-space consensus SNR for symmetric (even) and asymmetric (odd) morphologies relative to the detection threshold.

Periodic Baseline Removal

For variable sources, the iterative Whittaker detrending pipeline isolates the periodic modulation to recover the underlying transient signal:

• Step 1: Robust Period Search: Raw data with identified transient points highlighted.
• Step 2: Whittaker Modeling: The converged baseline model in phase-folded space.
• Step 3: Recovered Signal: The final detrended light curve with the isolated microlensing event.

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Verification

The core logic and timescale estimation accuracy can be verified using the provided test scripts:

python -m pytest
python -m ruff check --fix .
python -m ruff format .

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Requirements

• Python >= 3.9
• numpy
• scipy
• pandas
• astropy
• scikit-learn

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Contributing

Contributions should be made from a fork via pull request. Before opening a pull request, install the development extras and run the test and lint checks:

python -m pip install -e ".[dev]"
python -m pytest
python -m ruff check --fix .
python -m ruff format .

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Attributing

If you use MicroWavelet in published work, cite the archived release DOI listed above or use the repository citation metadata in CITATION.cff.

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Maintainer Release Notes

Release and workflow instructions are documented in docs/RELEASE.md.