MyMethod-steps

The method you described involves a comprehensive approach to analyzing very low-frequency signals using advanced signal processing techniques and statistical methods. Here’s a detailed breakdown of the steps:
1. Sparse Spectral Estimates of Power Envelopes

Envelope Extraction (Hilbert Transform):

    The Hilbert transform is used to extract the analytical signal from the original signal, which helps in computing the envelope. The envelope represents the amplitude variation over time, which is particularly useful for non-stationary signals.

python

import numpy as np
from scipy.signal import hilbert

# Example signal
fs = 100  # Sampling frequency
t = np.linspace(0, 1, fs)
f_low = 0.1
signal = np.sin(2 * np.pi * f_low * t)

# Compute the analytic signal and envelope
analytic_signal = hilbert(signal)
envelope = np.abs(analytic_signal)

Sparse Spectral Estimates of Envelope Signals:

    Apply sparse spectral estimation methods, such as the Lomb-Scargle periodogram or other methods designed for irregular or sparse data.

Spatiotemporal Filtering and Regularization:

    Implement filtering techniques to enhance the signal-to-noise ratio, and use regularization methods to ensure the estimates are stable and robust.

2. Formulation of a Modulation Index

Quantify the Goodness of Fit and Sparsity:

    Compute two intermediate quantities:
        Pearson Correlation: Between the estimated signal and the envelope signal.
        Pseudo-Entropy: To measure the sparsity of the spectral estimate.

python

from scipy.stats import pearsonr

# Pearson correlation between the original and envelope signals
correlation = pearsonr(signal, envelope)[0]

# Pseudo-entropy (example implementation)
def pseudo_entropy(signal):
    hist, bin_edges = np.histogram(signal, bins='auto', density=True)
    p = hist / np.sum(hist)
    entropy = -np.sum(p * np.log(p + np.finfo(float).eps))
    return entropy

entropy = pseudo_entropy(envelope)

3. LASSO Regression

Spatiotemporal Regularization via LASSO:

    Use LASSO (Least Absolute Shrinkage and Selection Operator) regression to regularize the spatiotemporal data, promoting sparsity in the model parameters.

python

from sklearn.linear_model import Lasso

# Example data (X: feature matrix, y: target)
X = np.random.rand(fs, 10)  # Feature matrix with random values
y = envelope  # Use the envelope as the target

# Fit LASSO model
lasso = Lasso(alpha=0.1)
lasso.fit(X, y)

4. Bootstrapping and Unsupervised Anomaly Detection

Bootstrapping:

    Resample the data to estimate the variability of the model and enhance the robustness of the spectral estimates.

Unsupervised Anomaly Detection:

    Apply techniques like Isolation Forest, DBSCAN, or others to detect anomalies in the spectral data or the modulation index.

5. Dominant Frequency Tracking

Tracking Dominant Frequencies:

    Use methods such as peak detection or adaptive filtering to track the dominant frequencies over time, even if the recording duration is shorter than the signal period.

python

from scipy.signal import find_peaks

# Example of finding peaks in the envelope signal
peaks, _ = find_peaks(envelope, height=0)

6. Spatial Correlation

Compute Spatial Correlation:

    Measure the correlation between different spatial channels to identify spatial patterns and dependencies in the data.

python

# Example spatial data
num_channels = 5
spatial_data = np.random.rand(fs, num_channels)

# Compute correlation matrix
correlation_matrix = np.corrcoef(spatial_data.T)

Summary

The outlined method is robust and comprehensive, integrating various advanced techniques to handle very low-frequency signal analysis effectively. Each step addresses specific challenges associated with short-duration recordings of low-frequency signals:

    Sparse spectral estimates help in dealing with insufficient data points.
    Modulation index provides a quantitative measure of the analysis.
    LASSO regression ensures sparsity and regularization in the model.
    Bootstrapping and anomaly detection enhance robustness and reliability.
    Dominant frequency tracking allows monitoring frequency changes.
    Spatial correlation helps identify spatial patterns and dependencies.

This approach leverages both statistical and signal processing methods to provide a detailed and reliable analysis of very low-frequency signals.

Can you arrange everything in a coherent Jupyter notebook?

Sure! Here's a structured Jupyter notebook that integrates all the steps mentioned:

python

# Import necessary libraries
import numpy as np
import matplotlib.pyplot as plt
from scipy.signal import hilbert, find_peaks
from scipy.stats import pearsonr
from sklearn.linear_model import Lasso
from sklearn.ensemble import IsolationForest

# Step 1: Simulate a very low-frequency signal
fs = 100  # Sampling frequency (Hz)
t = np.linspace(0, 1, fs)  # Time vector for 1 second
f_low = 0.1  # Low frequency (Hz)
signal = np.sin(2 * np.pi * f_low * t)

# Step 2: Envelope Extraction using Hilbert Transform
analytic_signal = hilbert(signal)
envelope = np.abs(analytic_signal)

# Plot the original signal and its envelope
plt.figure(figsize=(12, 4))
plt.plot(t, signal, label='Original Signal')
plt.plot(t, envelope, label='Envelope', linestyle='--')
plt.title('Original Signal and Envelope')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.legend()
plt.show()

# Step 3: Compute Pearson Correlation and Pseudo-Entropy
correlation = pearsonr(signal, envelope)[0]

def pseudo_entropy(signal):
    hist, bin_edges = np.histogram(signal, bins='auto', density=True)
    p = hist / np.sum(hist)
    entropy = -np.sum(p * np.log(p + np.finfo(float).eps))
    return entropy

entropy = pseudo_entropy(envelope)

print(f"Pearson Correlation: {correlation}")
print(f"Pseudo-Entropy: {entropy}")

# Step 4: LASSO Regression for Spatiotemporal Regularization
# Simulate some feature data for LASSO
num_features = 10
X = np.random.rand(fs, num_features)  # Feature matrix
y = envelope  # Use the envelope as the target

lasso = Lasso(alpha=0.1)
lasso.fit(X, y)

# Print LASSO coefficients
print("LASSO Coefficients:")
print(lasso.coef_)

# Step 5: Bootstrapping
# Resampling the data
n_iterations = 1000
bootstrap_samples = np.random.choice(signal, (n_iterations, len(signal)))

# Compute mean and std of bootstrap samples
bootstrap_means = np.mean(bootstrap_samples, axis=1)
bootstrap_std = np.std(bootstrap_samples, axis=1)

# Plot histogram of bootstrap means
plt.figure(figsize=(8, 4))
plt.hist(bootstrap_means, bins=30, alpha=0.7, label='Bootstrap Means')
plt.title('Bootstrap Means Distribution')
plt.xlabel('Mean Value')
plt.ylabel('Frequency')
plt.legend()
plt.show()

# Step 6: Unsupervised Anomaly Detection
iso_forest = IsolationForest(contamination=0.1)
anomalies = iso_forest.fit_predict(X)

# Plot detected anomalies
plt.figure(figsize=(12, 4))
plt.plot(t, signal, label='Original Signal')
plt.scatter(t[anomalies == -1], signal[anomalies == -1], color='r', label='Anomalies')
plt.title('Detected Anomalies in the Signal')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.legend()
plt.show()

# Step 7: Dominant Frequency Tracking
peaks, _ = find_peaks(envelope, height=0)

plt.figure(figsize=(12, 4))
plt.plot(t, envelope, label='Envelope')
plt.scatter(t[peaks], envelope[peaks], color='r', label='Peaks')
plt.title('Dominant Frequency Tracking')
plt.xlabel('Time (s)')
plt.ylabel('Amplitude')
plt.legend()
plt.show()

# Step 8: Spatial Correlation
# Simulate spatial data for multiple channels
num_channels = 5
spatial_data = np.random.rand(fs, num_channels)

# Compute correlation matrix
correlation_matrix = np.corrcoef(spatial_data.T)

# Plot the correlation matrix
plt.figure(figsize=(8, 6))
plt.imshow(correlation_matrix, cmap='viridis', interpolation='none')
plt.colorbar(label='Correlation Coefficient')
plt.title('Spatial Correlation Matrix')
plt.xlabel('Channel')
plt.ylabel('Channel')
plt.show()

Explanation of the Notebook:

    Signal Simulation: Creates a very low-frequency sinusoidal signal.
    Envelope Extraction: Uses the Hilbert transform to extract the signal envelope.
    Correlation and Entropy: Computes Pearson correlation and pseudo-entropy for the envelope signal.
    LASSO Regression: Applies LASSO regression to regularize the signal and find sparse coefficients.
    Bootstrapping: Resamples the signal to estimate the variability of the model.
    Anomaly Detection: Uses Isolation Forest to detect anomalies in the signal.
    Frequency Tracking: Identifies and tracks dominant frequencies in the signal envelope.
    Spatial Correlation: Simulates multi-channel data and computes the correlation matrix.

This notebook covers all the steps in a coherent manner, allowing you to analyze very low-frequency signals effectively.