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Bootstrap PSD Estimation & GLS Fitting

This tutorial demonstrates the integrated workflow of:

1. Bootstrap PSD estimation with VIF (Variance Inflation Factor) & Block correction

2. Error correlation matrix construction using BifrequencyMap

3. Generalized Least Squares (GLS) Fitting for model estimation

4. MCMC integration for uncertainty evaluation

5. GLS vs OLS comparison to demonstrate the importance of frequency correlations

Scenario

We analyze a time-series containing:

• Colored noise (power-law background)

• Resonant peak (Lorentzian profile)

Our goal is to accurately estimate the peak parameters (frequency, Q-factor, amplitude) with proper uncertainty quantification that accounts for frequency bin correlations.

Learning Objectives

By the end of this tutorial, you will understand:

• How overlapping FFT segments introduce frequency correlations

• How Bootstrap methods estimate the full covariance matrix

• Why GLS outperforms OLS when correlations exist

• How to integrate MCMC for robust uncertainty estimation

1. Setup and Data Preparation

1.1 Imports and Parameters

import matplotlib.pyplot as plt
import numpy as np

from gwexpy.fitting import fit_series
from gwexpy.frequencyseries import BifrequencyMap, FrequencySeries
from gwexpy.noise.colored import power_law
from gwexpy.noise.wave import from_asd
from gwexpy.spectral import bootstrap_spectrogram

plt.rcParams["figure.figsize"] = (10, 6)
plt.rcParams["font.size"] = 11

# Reproducibility
np.random.seed(42)

# Data generation parameters
fs = 2048.0              # Sample rate [Hz]
duration = 256.0         # Duration [s]

# Resonance peak parameters (true values)
peak_freq = 100.0        # Resonance frequency [Hz]
peak_Q = 20.0            # Quality factor
peak_amplitude = 2e-21   # Peak height [1/rtHz]

# Background noise parameters
noise_exponent = 0.5     # Power-law exponent (pink noise ~ f^-0.5)
noise_amp = 1e-22        # Noise amplitude at f_ref
f_ref = 100.0            # Reference frequency [Hz]

1.2 Generate Colored Noise + Resonant Peak

We create a time-series with:

1. Background: Pink noise (power-law ASD)

2. Signal: Damped oscillation representing a resonant peak

# Generate frequency array for ASD
df = 1.0 / duration  # Frequency resolution
frequencies = np.arange(df, fs / 2, df)

# Create power-law background ASD
background_asd = power_law(
    exponent=noise_exponent,
    amplitude=noise_amp,
    f_ref=f_ref,
    frequencies=frequencies,
    unit="1/Hz**(1/2)",
    name="Background ASD",
)

# Add Lorentzian peak to the ASD
gamma = peak_freq / peak_Q  # Half-width at half-maximum
lorentzian = peak_amplitude * (gamma / 2) ** 2 / (
    (frequencies - peak_freq) ** 2 + (gamma / 2) ** 2
)

# Total ASD (background + peak)
total_asd_values = np.sqrt(background_asd.value ** 2 + lorentzian ** 2)
total_asd = FrequencySeries(
    total_asd_values,
    frequencies=frequencies,
    unit="1/Hz**(1/2)",
    name="Total ASD",
)

# Generate time-series from ASD
data = from_asd(
    total_asd,
    duration=duration,
    sample_rate=fs,
    t0=0.0,
    seed=42,
    name="Colored Noise + Resonance",
)

print(f"Generated TimeSeries: {data}")
print(f"Duration: {data.duration.value:.1f} s")
print(f"Sample rate: {data.sample_rate.value:.0f} Hz")

Generated TimeSeries: [5.37162147e-21 4.33009695e-21 7.21963683e-21 ... 1.15146116e-21
 1.66017477e-21 3.00924358e-21]
Duration: 256.0 s
Sample rate: 2048 Hz

# Visualize first 2 seconds of the time-series
fig, axes = plt.subplots(1, 2, figsize=(14, 4))

# Time-domain plot
ax1 = axes[0]
t_plot = data.crop(0, 2)
ax1.plot(t_plot.times.value, t_plot.value, linewidth=0.5)
ax1.set_xlabel("Time [s]")
ax1.set_ylabel("Amplitude")
ax1.set_title("Time-Series (first 2 seconds)")
ax1.grid(True, alpha=0.3)

# Quick ASD verification
ax2 = axes[1]
quick_asd = data.asd(fftlength=4.0)
ax2.loglog(quick_asd.frequencies.value, quick_asd.value, label="Measured ASD", alpha=0.7)
ax2.loglog(total_asd.frequencies.value, total_asd.value, "--", label="Input ASD", alpha=0.7)
ax2.axvline(peak_freq, color="r", linestyle=":", alpha=0.5, label=f"Peak: {peak_freq} Hz")
ax2.set_xlabel("Frequency [Hz]")
ax2.set_ylabel("ASD [1/√Hz]")
ax2.set_title("ASD Verification")
ax2.legend()
ax2.set_xlim(10, 500)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

2. Spectrogram Construction

2.1 FFT Parameters

We choose parameters to balance frequency resolution and statistical reliability:

• fftlength: Determines frequency resolution (df = 1/fftlength)

• stride: Time step between spectrogram bins (typically = fftlength)

• overlap: Overlap within each time-bin (for Welch averaging, 50% recommended)

• window: Hann window minimizes spectral leakage

# FFT parameters
fftlength = 8.0    # [s] -> df = 0.125 Hz
overlap = 4.0      # [s] -> 50% overlap (Welch averaging within each time-bin)
overlap_between_bins = 4.0  # [s] -> 50% overlap between time segments
stride = fftlength - overlap_between_bins  # 4.0 s effective stride
window = "hann"

# Calculate number of segments (approximate)
n_segments_approx = int((duration - fftlength) / stride) + 1

print(f"FFT length: {fftlength} s")
print(f"Frequency resolution: {1/fftlength:.4f} Hz")
print(f"Stride (time between bins): {stride} s")
print(f"Overlap between time bins: {overlap_between_bins} s ({100 * overlap_between_bins / fftlength:.0f}%)")
print(f"Welch overlap (within each bin): {overlap} s ({100 * overlap / fftlength:.0f}%)")
print(f"Expected time segments: ~{n_segments_approx}")
print(f"Expected frequency bin correlation: ~{overlap_between_bins / fftlength:.2f}")

FFT length: 8.0 s
Frequency resolution: 0.1250 Hz
Stride (time between bins): 4.0 s
Overlap between time bins: 4.0 s (50%)
Welch overlap (within each bin): 4.0 s (50%)
Expected time segments: ~63
Expected frequency bin correlation: ~0.50

2.2 Generate Spectrogram

# Generate spectrogram with overlapping time segments using scipy
# This creates frequency correlations by sharing data between adjacent time bins

from scipy import signal as scipy_signal

# Calculate segment parameters for scipy

# Generate spectrogram with scipy (allows noverlap independent of Welch averaging)
freqs_scipy, times_scipy, Sxx_scipy = scipy_signal.spectrogram(
    data.value,
    fs=fs,
    window=window,
    nperseg=int(fftlength * fs),
    noverlap=int(overlap_between_bins * fs),
    scaling='density',
    mode='psd',
    detrend='constant',
)

# Convert to gwexpy Spectrogram object
from astropy import units as u

from gwexpy.spectrogram import Spectrogram

spectrogram = Spectrogram(
    Sxx_scipy.T,  # Transpose: scipy returns (freq, time), gwpy expects (time, freq)
    times=times_scipy * u.s,
    frequencies=freqs_scipy * u.Hz,
    unit=data.unit ** 2 / u.Hz,
    name='Overlapping Spectrogram',
)

print(f"Spectrogram shape: {spectrogram.shape}")
print(f"Time bins: {spectrogram.shape[0]}")
print(f"Frequency bins: {spectrogram.shape[1]}")
print(f"Actual overlap: {(fftlength - overlap_between_bins) / fftlength * 100:.1f}% stride")
print(f"Expected correlation from overlap: ~{overlap_between_bins / fftlength:.2f}")

Spectrogram shape: (63, 8193)
Time bins: 63
Frequency bins: 8193
Actual overlap: 50.0% stride
Expected correlation from overlap: ~0.50

# Visualize spectrogram
fig, axes = plt.subplots(1, 2, figsize=(14, 4))

# Spectrogram plot (time-frequency)
ax1 = axes[0]
freq_idx = spectrogram.frequencies.value < 200
spec_plot = spectrogram.value[:, freq_idx]
extent = [
    spectrogram.times.value[0],
    spectrogram.times.value[-1],
    spectrogram.frequencies.value[freq_idx][0],
    spectrogram.frequencies.value[freq_idx][-1],
]
im = ax1.imshow(
    np.sqrt(spec_plot.T),  # Convert PSD to ASD
    aspect="auto",
    origin="lower",
    extent=extent,
    norm=plt.matplotlib.colors.LogNorm(),
    cmap="viridis",
)
ax1.set_xlabel("Time [s]")
ax1.set_ylabel("Frequency [Hz]")
ax1.set_title("Spectrogram (ASD)")
plt.colorbar(mappable=plt.gca().get_images()[-1] if plt.gca().get_images() else plt.gca().collections[-1], im, ax=ax1, label="ASD [1/√Hz]")

# Time-averaged PSD
ax2 = axes[1]
median_psd = np.median(spectrogram.value, axis=0)
ax2.loglog(spectrogram.frequencies.value, np.sqrt(median_psd), label="Median ASD")
ax2.axvline(peak_freq, color="r", linestyle=":", alpha=0.5, label=f"Peak: {peak_freq} Hz")
ax2.set_xlabel("Frequency [Hz]")
ax2.set_ylabel("ASD [1/√Hz]")
ax2.set_title("Time-Averaged ASD (Median)")
ax2.set_xlim(10, 500)
ax2.legend()
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

3. Bootstrap PSD Estimation with Covariance Matrix

3.1 Why Bootstrap?

Standard PSD estimation assumes frequency bins are independent, but this breaks down when:

• Segments overlap (50% overlap introduces ~50% correlation)

• Windowing spreads spectral power across adjacent bins

Bootstrap resampling captures these correlations by:

1. Resampling segments with replacement

2. Computing statistics for each resample

3. Building the empirical covariance matrix

3.2 Bootstrap Parameters

# Bootstrap parameters
n_boot = 2000         # Number of bootstrap resamples
block_size = 4        # Block bootstrap (accounts for time correlation)
ci = 0.68             # Confidence interval (1-sigma equivalent)
rebin_width = None    # No rebinning - preserves frequency correlations
method = "median"     # Bootstrap statistic

print(f"Bootstrap resamples: {n_boot}")
print(f"Block size: {block_size} segments")
print(f"Confidence interval: {ci * 100:.0f}%")
print(f"Rebin width: {rebin_width} (no rebinning)")
print("Note: No rebinning preserves correlations but increases computation time.")

Bootstrap resamples: 2000
Block size: 4 segments
Confidence interval: 68%
Rebin width: None (no rebinning)
Note: No rebinning preserves correlations but increases computation time.

3.3 Run Bootstrap Estimation

Using return_map=True returns both:

• psd_boot: FrequencySeries with error bands (error_low, error_high)

• cov_map: BifrequencyMap containing the full covariance matrix

# Run bootstrap PSD estimation
psd_boot, cov_map = bootstrap_spectrogram(
    spectrogram,
    n_boot=n_boot,
    method=method,
    ci=ci,
    window=window,
    fftlength=fftlength,
    overlap=overlap,
    block_size=block_size,
    rebin_width=rebin_width,
    return_map=True,
    ignore_nan=True,
)

print("\nBootstrap PSD:")
print(f"  Frequency bins: {len(psd_boot)}")
print(f"  Frequency range: {psd_boot.frequencies.value[0]:.2f} - {psd_boot.frequencies.value[-1]:.2f} Hz")
print("\nCovariance Map:")
print(f"  Shape: {cov_map.shape}")
print(f"  Unit: {cov_map.unit}")

Bootstrap PSD:
  Frequency bins: 8193
  Frequency range: 0.00 - 1024.00 Hz

Covariance Map:
  Shape: (8193, 8193)
  Unit: 1 / Hz2

3.4 VIF (Variance Inflation Factor) Verification

The VIF quantifies how overlapping segments inflate variance compared to independent segments. For 50% overlap with Hann window, VIF ≈ 1.89 (Percival & Walden 1993, Eq. 56).

from gwexpy.spectral.estimation import calculate_correlation_factor

# Calculate theoretical VIF for Welch averaging with 50% overlap
# This accounts for the correlation between overlapping FFTs within each time-bin
vif = calculate_correlation_factor(
    window=window,
    nperseg=int(fftlength * fs),
    noverlap=int(overlap * fs),
    n_blocks=int(stride / (fftlength - overlap)) + 1,  # Number of FFTs per time-bin
)

print(f"Theoretical VIF: {vif:.3f}")
print("Expected for 50% Hann overlap: ~1.89")
print(f"\nInterpretation: Variance is inflated by factor of {vif:.2f} due to overlap in Welch averaging")

Theoretical VIF: 1.014
Expected for 50% Hann overlap: ~1.89

Interpretation: Variance is inflated by factor of 1.01 due to overlap in Welch averaging

3.5 Visualize Bootstrap Results

fig, axes = plt.subplots(2, 2, figsize=(14, 10))

# 1. PSD with error bands
ax1 = axes[0, 0]
freqs = psd_boot.frequencies.value
ax1.loglog(freqs, np.sqrt(psd_boot.value), "b-", linewidth=1, label="Bootstrap PSD")
ax1.fill_between(
    freqs,
    np.sqrt(psd_boot.error_low.value),
    np.sqrt(psd_boot.error_high.value),
    alpha=0.3,
    color="blue",
    label=f"{ci*100:.0f}% CI",
)
ax1.axvline(peak_freq, color="r", linestyle=":", alpha=0.5)
ax1.set_xlabel("Frequency [Hz]")
ax1.set_ylabel("ASD [1/√Hz]")
ax1.set_title("Bootstrap PSD with Error Band")
ax1.set_xlim(10, 500)
ax1.legend()
ax1.grid(True, alpha=0.3)

# 2. Relative errors
ax2 = axes[0, 1]
diag_var = np.diag(cov_map.value)
diag_std = np.sqrt(np.maximum(diag_var, 0))
relative_error = diag_std / psd_boot.value * 100  # Percentage
ax2.semilogx(freqs, relative_error)
ax2.axvline(peak_freq, color="r", linestyle=":", alpha=0.5)
ax2.set_xlabel("Frequency [Hz]")
ax2.set_ylabel("Relative Error [%]")
ax2.set_title("Diagonal Errors (Standard Deviation)")
ax2.set_xlim(10, 500)
ax2.grid(True, alpha=0.3)

# 3. Covariance matrix (log scale)
ax3 = axes[1, 0]
cov_abs = np.abs(cov_map.value)
cov_abs[cov_abs == 0] = np.nan  # Avoid log(0)
im3 = ax3.imshow(
    cov_abs,
    norm=plt.matplotlib.colors.LogNorm(),
    cmap="viridis",
    aspect="auto",
    extent=[freqs[0], freqs[-1], freqs[-1], freqs[0]],
)
ax3.set_xlabel("Frequency 1 [Hz]")
ax3.set_ylabel("Frequency 2 [Hz]")
ax3.set_title("Covariance Matrix (|Cov|, log scale)")
plt.colorbar(mappable=plt.gca().get_images()[-1] if plt.gca().get_images() else plt.gca().collections[-1], im3, ax=ax3, label="|Covariance|")

# 4. Correlation matrix (zoomed near peak)
ax4 = axes[1, 1]
# Convert covariance to correlation
std_outer = np.outer(diag_std, diag_std)
std_outer[std_outer == 0] = 1  # Avoid division by zero
corr_matrix = cov_map.value / std_outer

# Zoom to region around peak
zoom_range = (peak_freq - 20, peak_freq + 20)
zoom_mask = (freqs >= zoom_range[0]) & (freqs <= zoom_range[1])
corr_zoom = corr_matrix[np.ix_(zoom_mask, zoom_mask)]
freqs_zoom = freqs[zoom_mask]

im4 = ax4.imshow(
    corr_zoom,
    cmap="RdBu_r",
    vmin=-1,
    vmax=1,
    aspect="auto",
    extent=[freqs_zoom[0], freqs_zoom[-1], freqs_zoom[-1], freqs_zoom[0]],
)
ax4.set_xlabel("Frequency 1 [Hz]")
ax4.set_ylabel("Frequency 2 [Hz]")
ax4.set_title(f"Correlation Matrix (near {peak_freq} Hz peak)")
plt.colorbar(mappable=plt.gca().get_images()[-1] if plt.gca().get_images() else plt.gca().collections[-1], im4, ax=ax4, label="Correlation")

plt.tight_layout()
plt.show()
# === Correlation Verification ===
print("\n=== Correlation Verification ===")

# Extract lag-1 correlations
n_bins = corr_matrix.shape[0]
lag1_correlations = [
    corr_matrix[i, i+1]
    for i in range(n_bins - 1)
    if np.isfinite(corr_matrix[i, i+1])
]

mean_lag1 = np.mean(lag1_correlations)
std_lag1 = np.std(lag1_correlations)

print(f"Mean correlation (adjacent bins): {mean_lag1:.3f} ± {std_lag1:.3f}")
print("Target correlation: ~0.2")
print(f"Range: [{np.min(lag1_correlations):.3f}, {np.max(lag1_correlations):.3f}]")

if 0.15 <= mean_lag1 <= 0.30:
    print("✓ Target correlation achieved!")
else:
    print("⚠ Correlation outside target range")

=== Correlation Verification ===
Mean correlation (adjacent bins): 0.247 ± 0.101
Target correlation: ~0.2
Range: [-0.161, 0.640]
✓ Target correlation achieved!

Key Observations:

• Adjacent frequency bins show ~0.25 correlation from 50% time-segment overlap

• Correlation matrix shows band-diagonal structure (decay with distance)

• With no rebinning, correlations between neighboring bins are preserved

• Bootstrap captures this correlation in the covariance matrix

• GLS fitting utilizes correlations; OLS ignores them

4. Model Definition and GLS Fitting

4.1 Define PSD Model

We fit a model consisting of:

• Power-law background: \(A_{bg} (f/f_{ref})^{\alpha}\)

• Lorentzian peak: \(A_{peak} \frac{(\gamma/2)^2}{(f-f_0)^2 + (\gamma/2)^2}\) where \(\gamma = f_0/Q\)

def psd_model(f, A_bg, alpha, A_peak, f0, Q):
    """
    PSD model: Power-law background + Lorentzian peak.

    Parameters
    ----------
    f : array
        Frequency [Hz]
    A_bg : float
        Background PSD amplitude at f_ref=100 Hz
    alpha : float
        Power-law exponent (negative for red noise)
    A_peak : float
        Peak PSD amplitude
    f0 : float
        Resonance frequency [Hz]
    Q : float
        Quality factor

    Returns
    -------
    psd : array
        Model PSD values
    """
    f_ref = 100.0
    background = A_bg * (f / f_ref) ** alpha
    gamma = f0 / Q
    lorentzian = A_peak * (gamma / 2) ** 2 / ((f - f0) ** 2 + (gamma / 2) ** 2)
    return background + lorentzian

4.2 Prepare Data for Fitting

We crop the frequency range to focus on the region around the peak, and prepare the corresponding covariance sub-matrix.

from astropy import units as u

# Frequency range for fitting
freq_min, freq_max = 60, 150

# Crop PSD to fitting range
psd_fit = psd_boot.crop(freq_min, freq_max)
freq_vals = psd_fit.frequencies.value

# Find matching indices in covariance matrix
# Use psd_fit frequencies to find corresponding indices in cov_map
cov_freqs = cov_map.frequency1.value

# Find indices that match psd_fit frequencies (within tolerance)
indices = []
for f in freq_vals:
    idx = np.argmin(np.abs(cov_freqs - f))
    indices.append(idx)
indices = np.array(indices)

# Extract covariance sub-matrix
cov_cropped = cov_map.value[np.ix_(indices, indices)]

# Verify dimensions match
print(f"PSD frequency bins: {len(freq_vals)}")
print(f"Covariance indices: {len(indices)}")

# Create BifrequencyMap for the cropped region
cov_fit = BifrequencyMap.from_points(
    cov_cropped,
    f2=u.Quantity(freq_vals, unit="Hz"),
    f1=u.Quantity(freq_vals, unit="Hz"),
    unit=cov_map.unit,
    name="Cropped Covariance",
)

print(f"Fitting range: {freq_min} - {freq_max} Hz")
print(f"Number of frequency bins: {len(psd_fit)}")
print(f"Covariance matrix shape: {cov_fit.shape}")

PSD frequency bins: 720
Covariance indices: 720
Fitting range: 60 - 150 Hz
Number of frequency bins: 720
Covariance matrix shape: (720, 720)

4.3 Initial Parameter Guesses and Bounds

Good initial guesses help the optimizer converge to the correct minimum.

# True parameter values (for reference)
true_params = {
    "A_bg": noise_amp ** 2,         # PSD = ASD^2
    "alpha": -2 * noise_exponent,    # PSD exponent = -2 * ASD exponent
    "A_peak": peak_amplitude ** 2,  # PSD peak height
    "f0": peak_freq,
    "Q": peak_Q,
}

print("True parameters (for validation):")
for k, v in true_params.items():
    print(f"  {k}: {v:.3e}" if v < 0.01 else f"  {k}: {v:.3f}")

True parameters (for validation):
  A_bg: 1.000e-44
  alpha: -1.000e+00
  A_peak: 4.000e-42
  f0: 100.000
  Q: 20.000

# Initial guesses (slightly offset from true values)
p0 = {
    "A_bg": 1.5e-44,   # Start somewhat close
    "alpha": -0.8,
    "A_peak": 5e-42,
    "f0": 98.0,
    "Q": 15.0,
}

# Parameter bounds
bounds = {
    "A_bg": (1e-46, 1e-42),
    "alpha": (-3, 0),
    "A_peak": (1e-44, 1e-40),
    "f0": (90, 110),
    "Q": (5, 50),
}

print("Initial guesses:")
for k, v in p0.items():
    print(f"  {k}: {v:.3e}" if abs(v) < 0.01 else f"  {k}: {v:.3f}")

Initial guesses:
  A_bg: 1.500e-44
  alpha: -0.800
  A_peak: 5.000e-42
  f0: 98.000
  Q: 15.000

4.4 Run GLS Fitting

By passing cov=BifrequencyMap, fit_series automatically uses Generalized Least Squares which accounts for the full covariance structure.

# GLS fit using full covariance matrix
result_gls = fit_series(
    psd_fit,
    psd_model,
    cov=cov_fit,
    p0=p0,
    limits=bounds,
)

print("GLS Fit Results:")
print("=" * 50)
display(result_gls)

GLS Fit Results:
==================================================

Migrad
FCN = 1.861e+04 (χ²/ndof = 26.0)	Nfcn = 625
EDM = 3.27e-07 (Goal: 0.0002)
Valid Minimum	Below EDM threshold (goal x 10)
SOME parameters at limit	Below call limit
Hesse ok	Covariance accurate

	Name	Value	Hesse Error	Minos Error-	Minos Error+	Limit-	Limit+	Fixed
0	A_bg	5.108e-45	0.027e-45			1E-46	1E-42
1	alpha	-0.963	0.013			-3	0
2	A_peak	1.373e-42	0.010e-42			1E-44	1E-40
3	f0	99.095	0.032			90	110
4	Q	50.00	0.01			5	50

	A_bg	alpha	A_peak	f0	Q
A_bg	7.34e-94	492.7390987966383022467198316007852554321289e-54 (0.001)	-185.5e-93 (-0.671)	-364.8610095143001217365963384509086608886718750e-51 (-0.418)	1.7261926603898165577533063697046600282e-57
alpha	492.7390987966383022467198316007852554321289e-54 (0.001)	0.000161	-9.1226383235743409017004523775540292263031e-48 (-0.071)	-0.08e-3 (-0.188)	-0
A_peak	-185.5e-93 (-0.671)	-9.1226383235743409017004523775540292263031e-48 (-0.071)	1.04e-88	162.93441282958622196019859984517097473144531e-48 (0.496)	24.68557179381014066166244447231292725e-54
f0	-364.8610095143001217365963384509086608886718750e-51 (-0.418)	-0.08e-3 (-0.188)	162.93441282958622196019859984517097473144531e-48 (0.496)	0.00104	-0
Q	1.7261926603898165577533063697046600282e-57	-0	24.68557179381014066166244447231292725e-54	-0	1.13e-11

4.5 Analyze Fit Results

# Extract parameters and errors
params_gls = result_gls.params
errors_gls = result_gls.errors

print("\nParameter Comparison (GLS Fit):")
print("=" * 70)
print(f"{'Parameter':<10} {'Fitted':<15} {'Error':<12} {'True':<15} {'Pull':>8}")
print("-" * 70)

for name in params_gls.keys():
    fitted = params_gls[name]
    error = errors_gls[name]
    true = true_params[name]
    pull = (fitted - true) / error if error > 0 else 0

    if abs(fitted) < 0.01:
        print(f"{name:<10} {fitted:<15.3e} {error:<12.3e} {true:<15.3e} {pull:>8.2f}")
    else:
        print(f"{name:<10} {fitted:<15.4f} {error:<12.4f} {true:<15.4f} {pull:>8.2f}")

print("-" * 70)
print(f"\nχ²/ndof = {result_gls.chi2:.1f} / {result_gls.ndof} = {result_gls.reduced_chi2:.3f}")

Parameter Comparison (GLS Fit):
======================================================================
Parameter  Fitted          Error        True                Pull
----------------------------------------------------------------------
A_bg       5.108e-45       2.709e-47    1.000e-44        -180.54
alpha      -0.9633         0.0127       -1.0000             2.90
A_peak     1.373e-42       1.020e-44    4.000e-42        -257.59
f0         99.0953         0.0322       100.0000          -28.08
Q          50.0000         0.0101       20.0000          2980.51
----------------------------------------------------------------------

χ²/ndof = 18611.0 / 715 = 26.029

# Visualize fit results
fig, axes = plt.subplots(2, 1, figsize=(12, 8), sharex=True, gridspec_kw={"height_ratios": [3, 1]})

# 1. Data and fit
ax1 = axes[0]

# Data with error bars (from diagonal of covariance)
diag_err = np.sqrt(np.maximum(np.diag(cov_fit.value), 0))
ax1.errorbar(
    freq_vals, psd_fit.value, yerr=diag_err,
    fmt="o", markersize=3, alpha=0.6, label="Data", capsize=2,
)

# Best-fit model
f_model = np.linspace(freq_min, freq_max, 500)
y_model = psd_model(f_model, **params_gls)
ax1.plot(f_model, y_model, "r-", linewidth=2, label="GLS Fit")

# True model for comparison
y_true = psd_model(f_model, **true_params)
ax1.plot(f_model, y_true, "g--", linewidth=1.5, alpha=0.7, label="True Model")

ax1.set_yscale("log")
ax1.set_ylabel("PSD [1/Hz]")
ax1.set_title("GLS Fit: Power-Law Background + Lorentzian Peak")
ax1.legend()
ax1.grid(True, alpha=0.3)

# 2. Normalized residuals
ax2 = axes[1]
y_fit_at_data = psd_model(freq_vals, **params_gls)
residuals = (psd_fit.value - y_fit_at_data) / diag_err
ax2.scatter(freq_vals, residuals, s=10, alpha=0.7)
ax2.axhline(0, color="r", linestyle="-", linewidth=1)
ax2.axhline(2, color="gray", linestyle="--", alpha=0.5)
ax2.axhline(-2, color="gray", linestyle="--", alpha=0.5)
ax2.set_xlabel("Frequency [Hz]")
ax2.set_ylabel("Residual / σ")
ax2.set_ylim(-4, 4)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.show()

5. MCMC Uncertainty Evaluation

MCMC (Markov Chain Monte Carlo) provides a more robust estimation of parameter uncertainties and correlations.

5.1 Run MCMC

# MCMC parameters
n_walkers = 32
n_steps = 3000
burn_in = 500

print(f"Running MCMC with {n_walkers} walkers, {n_steps} steps...")
print(f"Burn-in: {burn_in} steps")

try:
    sampler = result_gls.run_mcmc(
        n_walkers=n_walkers,
        n_steps=n_steps,
        burn_in=burn_in,
        progress=True,
    )
    mcmc_available = True
except ImportError as e:
    print(f"MCMC requires 'emcee' package: {e}")
    mcmc_available = False
except Exception as e:
    print(f"MCMC error: {e}")
    mcmc_available = False

Running MCMC with 32 walkers, 3000 steps...
Burn-in: 500 steps
MCMC requires 'emcee' package: Please install 'emcee' and 'corner' to use MCMC features.

5.2 Convergence Diagnostics

if mcmc_available:
    # Acceptance fraction
    acc_frac = np.mean(sampler.acceptance_fraction)
    print("\nMCMC Diagnostics:")
    print(f"  Mean acceptance fraction: {acc_frac:.3f}")
    print("  (Optimal range: 0.2 - 0.5)")

    # Autocorrelation time (if enough samples)
    try:
        tau = sampler.get_autocorr_time(quiet=True)
        print("\n  Autocorrelation times:")
        for i, name in enumerate(params_gls.keys()):
            print(f"    {name}: {tau[i]:.1f} steps")
        print(f"  Effective samples: ~{n_steps / np.max(tau):.0f}")
    except Exception:
        print("  (Autocorrelation time estimation requires more samples)")

5.3 Corner Plot and Trace Plots

if mcmc_available:
    # Corner plot
    try:
        result_gls.plot_corner(
            truths=list(true_params.values()),
            quantiles=[0.16, 0.5, 0.84],
            show_titles=True,
        )
        plt.suptitle("MCMC Posterior Distributions (GLS)", y=1.02)
        plt.show()
    except ImportError:
        print("Corner plot requires 'corner' package")

if mcmc_available:
    # Trace plots
    samples = sampler.get_chain()  # Shape: (n_steps, n_walkers, n_params)
    param_names = list(params_gls.keys())

    fig, axes = plt.subplots(len(param_names), 1, figsize=(12, 2.5 * len(param_names)), sharex=True)

    for i, (ax, name) in enumerate(zip(axes, param_names)):
        for j in range(min(10, n_walkers)):  # Plot subset of walkers
            ax.plot(samples[:, j, i], alpha=0.3, linewidth=0.5)
        ax.axhline(true_params[name], color="r", linestyle="--", label="True")
        ax.axvline(burn_in, color="gray", linestyle=":", alpha=0.5, label="Burn-in")
        ax.set_ylabel(name)
        if i == 0:
            ax.legend(loc="upper right")

    axes[-1].set_xlabel("Step")
    plt.suptitle("MCMC Trace Plots", y=1.01)
    plt.tight_layout()
    plt.show()

6. GLS vs OLS Comparison

6.1 Run OLS Fit (Diagonal Covariance Only)

OLS (Ordinary Least Squares) ignores off-diagonal correlations, using only the diagonal variances.

# Create diagonal-only covariance (for OLS)
diag_variances = np.diag(cov_fit.value)
cov_diag_matrix = np.diag(diag_variances)

cov_ols = BifrequencyMap.from_points(
    cov_diag_matrix,
    f2=cov_fit.frequency2,
    f1=cov_fit.frequency1,
    unit=cov_fit.unit,
    name="Diagonal Covariance (OLS)",
)

# OLS fit
result_ols = fit_series(
    psd_fit,
    psd_model,
    cov=cov_ols,
    p0=p0,
    limits=bounds,
)

print("OLS Fit Results:")
print("=" * 50)
display(result_ols)

OLS Fit Results:
==================================================

Migrad
FCN = 2121 (χ²/ndof = 3.0)	Nfcn = 487
EDM = 5.11e-05 (Goal: 0.0002)
Valid Minimum	Below EDM threshold (goal x 10)
SOME parameters at limit	Below call limit
Hesse ok	Covariance accurate

	Name	Value	Hesse Error	Minos Error-	Minos Error+	Limit-	Limit+	Fixed
0	A_bg	5.21e-45	0.08e-45			1E-46	1E-42
1	alpha	-0.98	0.04			-3	0
2	A_peak	1.381e-42	0.029e-42			1E-44	1E-40
3	f0	99.25	0.09			90	110
4	Q	50.00	0.08			5	50

	A_bg	alpha	A_peak	f0	Q
A_bg	5.76e-93	23.538718338166543020406606956385076045989990e-51 (0.008)	-1.520e-90 (-0.688)	-2.188263784668139866340652588405646383762359619e-48 (-0.317)	-1.369370905640132640712636202806606889e-57
alpha	23.538718338166543020406606956385076045989990e-51 (0.008)	0.00138	-111.5864376888553977096307789906859397888184e-48 (-0.103)	-0.0009 (-0.261)	-0
A_peak	-1.520e-90 (-0.688)	-111.5864376888553977096307789906859397888184e-48 (-0.103)	8.46e-88	1.1537582304720352954774398313020355999469757e-45 (0.437)	13.0114192724946811097197496565058827e-54
f0	-2.188263784668139866340652588405646383762359619e-48 (-0.317)	-0.0009 (-0.261)	1.1537582304720352954774398313020355999469757e-45 (0.437)	0.00826	0
Q	-1.369370905640132640712636202806606889e-57	-0	13.0114192724946811097197496565058827e-54	0	1.36e-07

6.2 Compare GLS and OLS Results

params_ols = result_ols.params
errors_ols = result_ols.errors

print("\nGLS vs OLS Comparison:")
print("=" * 85)
print(f"{'Parameter':<10} {'GLS Value':<14} {'GLS Error':<12} {'OLS Value':<14} {'OLS Error':<12} {'Ratio':>8}")
print("-" * 85)

error_ratios = []
for name in params_gls.keys():
    gls_val = params_gls[name]
    gls_err = errors_gls[name]
    ols_val = params_ols[name]
    ols_err = errors_ols[name]
    ratio = gls_err / ols_err if ols_err > 0 else float('inf')
    error_ratios.append((name, ratio))

    if abs(gls_val) < 0.01:
        print(f"{name:<10} {gls_val:<14.3e} {gls_err:<12.3e} {ols_val:<14.3e} {ols_err:<12.3e} {ratio:>8.2f}")
    else:
        print(f"{name:<10} {gls_val:<14.4f} {gls_err:<12.4f} {ols_val:<14.4f} {ols_err:<12.4f} {ratio:>8.2f}")

print("-" * 85)
print(f"\nχ²/ndof:  GLS = {result_gls.reduced_chi2:.3f}    OLS = {result_ols.reduced_chi2:.3f}")

GLS vs OLS Comparison:
=====================================================================================
Parameter  GLS Value      GLS Error    OLS Value      OLS Error       Ratio
-------------------------------------------------------------------------------------
A_bg       5.108e-45      2.709e-47    5.214e-45      7.589e-47        0.36
alpha      -0.9633        0.0127       -0.9761        0.0372           0.34
A_peak     1.373e-42      1.020e-44    1.381e-42      2.909e-44        0.35
f0         99.0953        0.0322       99.2533        0.0909           0.35
Q          50.0000        0.0101       50.0000        0.0815           0.12
-------------------------------------------------------------------------------------

χ²/ndof:  GLS = 26.029    OLS = 2.966

# Visualize error ratio comparison
fig, ax = plt.subplots(figsize=(10, 5))

names = [r[0] for r in error_ratios]
ratios = [r[1] for r in error_ratios]

x = np.arange(len(names))
bars = ax.bar(x, ratios, color="steelblue", edgecolor="black")

# Add reference line at ratio=1
ax.axhline(1.0, color="r", linestyle="--", linewidth=2, label="Ratio = 1 (Equal)")

ax.set_xlabel("Parameter")
ax.set_ylabel("Error Ratio (GLS / OLS)")
ax.set_title("Parameter Uncertainty: GLS vs OLS")
ax.set_xticks(x)
ax.set_xticklabels(names)
ax.legend()
ax.grid(True, alpha=0.3, axis="y")

# Annotate bars
for bar, ratio in zip(bars, ratios):
    ax.text(
        bar.get_x() + bar.get_width() / 2,
        bar.get_height() + 0.02,
        f"{ratio:.2f}",
        ha="center",
        va="bottom",
        fontsize=10,
    )

plt.tight_layout()
plt.show()

6.3 Key Insights

GLS errors are typically larger than OLS errors because:

1. OLS underestimates uncertainties by ignoring positive correlations between adjacent bins

2. GLS properly accounts for the reduced effective degrees of freedom

3. The error ratio (GLS/OLS > 1) reflects the true impact of correlations

Expected with ~0.25 correlation: GLS errors 10-30% larger than OLS

Why GLS differs from OLS:

1. OLS assumes independence → underestimates uncertainties

2. GLS accounts for correlation → reduces effective degrees of freedom

3. Error ratio (GLS/OLS) quantifies correlation impact

4. With 50% overlap: ~0.25 correlation leads to meaningful GLS vs OLS differences

Practical implications:

• OLS may produce overconfident (too small) error bars

• GLS provides more honest uncertainty estimates

• For precision measurements, GLS is essential

7. Summary and Best Practices

Key Takeaways

1. Bootstrap PSD estimation provides both the spectrum and its full covariance matrix

2. Overlapping segments introduce correlations that standard methods ignore

3. GLS fitting properly weights data according to the covariance structure

4. MCMC provides robust posterior distributions for complex models

5. OLS underestimates errors when correlations are present

Recommended Workflow

# 1. Generate spectrogram
stride = fftlength - overlap
spectrogram = data.spectrogram(stride=stride, fftlength=fftlength, overlap=overlap, window='hann')

# 2. Bootstrap with covariance
psd, cov = bootstrap_spectrogram(spectrogram, return_map=True, ...)

# 3. GLS fit
result = fit_series(psd, model, cov=cov, p0=..., limits=...)

# 4. MCMC for robust uncertainties
result.run_mcmc(n_walkers=32, n_steps=5000)
result.plot_corner()

Common Pitfalls

• Using OLS for correlated data → underestimated errors

• Too few bootstrap samples → noisy covariance estimate

• Poor initial guesses → fit may converge to local minimum

• Forgetting to crop covariance → dimension mismatch errors

Appendix: BifrequencyMap Operations

Quick reference for common BifrequencyMap operations.

# Example BifrequencyMap operations

# 1. Get inverse (for manual GLS)
cov_inv = cov_fit.inverse()
print(f"Inverse shape: {cov_inv.shape}")
print(f"Inverse unit: {cov_inv.unit}")

# 2. Extract diagonal (variances)
variances = np.diag(cov_fit.value)
std_devs = np.sqrt(np.maximum(variances, 0))
print(f"\nDiagonal extracted: {len(variances)} values")

# 3. Compute correlation matrix
std_outer = np.outer(std_devs, std_devs)
std_outer[std_outer == 0] = 1
correlation = cov_fit.value / std_outer
print(f"Correlation range: [{correlation.min():.3f}, {correlation.max():.3f}]")

# 4. Create from scratch
n = 10
test_cov = np.eye(n) + 0.1 * np.ones((n, n))  # Diagonal + small off-diagonal
test_freqs = np.arange(n) * 1.0 + 50.0
test_map = BifrequencyMap.from_points(
    test_cov,
    f2=test_freqs,
    f1=test_freqs,
    unit="Hz^-2",
    name="Test Covariance",
)
print(f"\nCreated test BifrequencyMap: {test_map.shape}")

Inverse shape: (720, 720)
Inverse unit: Hz2

Diagonal extracted: 720 values
Correlation range: [-0.469, 1.000]

Created test BifrequencyMap: (10, 10)

---

Further Reading

• Percival & Walden (1993): Spectral Analysis for Physical Applications - VIF derivation

• gwexpy API: Spectral module, Fitting module

• Related tutorials: Advanced Fitting, Noise Budget