Analyzing Resonant Manifolds

Given a dataset or signal exhibiting periodic/field-like behavior, extract its resonant modes and map them onto a manifold representation:

Python
import numpy as np
from scipy.signal import find_peaks
from sklearn.decomposition import PCA

def extract_resonant_modes(signal, sample_rate, n_components=3):
    # 1. Frequency domain transform
    spectrum = np.abs(np.fft.rfft(signal))
    freqs = np.fft.rfftfreq(len(signal), 1/sample_rate)

    # 2. Find resonant peaks
    peaks, props = find_peaks(spectrum, prominence=spectrum.max() * 0.05)
    resonant_freqs = freqs[peaks]
    resonant_amps = spectrum[peaks]

    # 3. Build phase-space embedding (manifold candidate)
    embedding = np.column_stack([
        np.roll(signal, i) for i in range(n_components)
    ])[n_components:]

    # 4. Reduce to low-dim manifold
    manifold = PCA(n_components=n_components).fit_transform(embedding)

    return {
        "resonant_freqs": resonant_freqs,
        "resonant_amps": resonant_amps,
        "manifold": manifold
    }

Output: a set of resonant frequencies/amplitudes plus a low-dimensional manifold embedding ready for visualization or further tuning.

Progress:

• Step 1: Characterize the raw signal/dataset (sample rate, dimensionality, noise floor)
• Step 2: Transform into frequency or phase space
• Step 3: Detect resonant peaks/modes with prominence filtering
• Step 4: Construct a time-delay or feature embedding
• Step 5: Reduce embedding to a manifold (PCA/UMAP/t-SNE depending on scale)
• Step 6: Validate stability of modes (re-run on sub-windows, check consistency)
• Step 7: Flag anomalies — modes that appear/disappear or drift significantly
• Step 8: Report resonant structure with confidence/stability scores

Step 1 — Characterize input. Confirm sample rate, units, and whether data is 1D (audio/signal) or multi-channel (field/sensor array). Note noise floor via a quiet/baseline segment.

Step 2 — Transform. Use FFT for periodic signals; use delay embedding directly for chaotic/nonlinear systems where frequency content alone is insufficient.

Step 3 — Detect modes. Prominence-based peak detection avoids false positives from noise. Prominence threshold should scale with the noise floor, not a fixed constant.

Step 4–5 — Embed and reduce. Time-delay embedding dimension should exceed estimated system dimensionality by at least 2 (Takens' theorem heuristic). Use PCA for speed, UMAP for nonlinear manifold structure when PCA explains <80% variance in first 3 components.

Step 6 — Validate. Split signal into overlapping windows; recompute modes. Modes present in >70% of windows are "stable resonances." Others are transient.

Step 7 — Anomaly flagging. Compare mode amplitude/frequency drift across windows using z-scores relative to the stable-mode population.