Python&Scipy：如何用分箱角度数据估计Von Mises尺度因子？

Hey there! Let's work through fixing that scaling mismatch between your Von Mises PDF and binned angle data. First, a quick clarification: the Von Mises distribution doesn't natively have a "scale parameter" like some linear distributions—what you're seeing is a mismatch between the PDF's probability density (which integrates to 1) and your binned data's count-based histogram height. Here are two solid approaches to fix this:

Approach 1: Scale the PDF Directly (Best for Large Weights) #

Since you're already using circstats.vmpar() to get loc (μ) and kappa (κ) with weighted data, you just need to calculate a scaling factor to align the PDF with your binned histogram. Here's how:

Step-by-Step Code #

import numpy as np
import circstats as cs
import matplotlib.pyplot as plt

# Your binned data (replace with your actual values)
binned_angles = np.array([0, np.pi/4, np.pi/2, 3*np.pi/4, np.pi])
bin_weights = np.array([5, 10, 15, 10, 5])

# Estimate Von Mises parameters using weighted binned data
loc, kappa = cs.vmpar(binned_angles, weights=bin_weights)

# Generate a smooth angle sequence for plotting the PDF
x = np.linspace(-np.pi, np.pi, 1000)

# Calculate raw Von Mises PDF (probability density, integrates to 1)
raw_pdf = cs.vmpdf(x, loc=loc, kappa=kappa)

# Calculate scaling factor to match histogram height:
total_samples = np.sum(bin_weights)
bin_width = binned_angles[1] - binned_angles[0]  # Assumes uniform bins
scale_factor = total_samples * bin_width

# Scale the PDF to align with your binned data
scaled_pdf = raw_pdf * scale_factor

# Plot the results
plt.bar(binned_angles, bin_weights, width=bin_width, alpha=0.5, label='Binned Angle Data')
plt.plot(x, scaled_pdf, color='orange', label='Scaled Von Mises PDF')
plt.legend()
plt.title('Binned Data vs. Scaled Von Mises PDF')
plt.show()

For Non-Uniform Bins #

If your bins aren't equal width, adjust the histogram to use density instead of raw counts, and scale the PDF by total samples to match the histogram's total area:

# Example non-uniform bins
bin_edges = np.array([-np.pi/4, 0, np.pi/4, np.pi/2, 3*np.pi/4, np.pi, 5*np.pi/4])
binned_angles = (bin_edges[:-1] + bin_edges[1:])/2  # Calculate bin centers
bin_weights = np.array([5, 10, 15, 10, 5, 3])
bin_widths = bin_edges[1:] - bin_edges[:-1]

# Estimate parameters
loc, kappa = cs.vmpar(binned_angles, weights=bin_weights)

# Generate PDF
x = np.linspace(-np.pi, np.pi, 1000)
raw_pdf = cs.vmpdf(x, loc=loc, kappa=kappa)

# Scale factor = total samples (matches PDF area to histogram area)
scale_factor = np.sum(bin_weights)
scaled_pdf = raw_pdf * scale_factor

# Plot density-based histogram and scaled PDF
plt.bar(binned_angles, bin_weights / bin_widths, width=bin_widths, alpha=0.5, label='Binned Data (Density)')
plt.plot(x, scaled_pdf, color='orange', label='Scaled Von Mises PDF')
plt.legend()
plt.show()

Approach 2: Expand Binned Data for Scipy's vonmises.fit() #

If you prefer using Scipy's fitting function (even though it doesn't support weights directly), you can "unpack" your binned data into a pseudo-raw dataset by repeating each bin's angle by its weight count. This works well if your weights aren't extremely large:

from scipy.stats import vonmises

# Expand binned data into individual sample points
expanded_angles = np.repeat(binned_angles, bin_weights)

# Fit Von Mises parameters (Scipy returns mu and kappa; no native scale parameter)
mu, kappa = vonmises.fit(expanded_angles, f0=1)  # f0 = initial guess for kappa

# Generate and scale the PDF the same way as Approach 1
raw_pdf_scipy = vonmises.pdf(x, kappa, loc=mu)
scaled_pdf_scipy = raw_pdf_scipy * total_samples * bin_width

# Plot to compare
plt.bar(binned_angles, bin_weights, width=bin_width, alpha=0.5, label='Binned Data')
plt.plot(x, scaled_pdf_scipy, color='green', label='Scipy-Fitted Scaled PDF')
plt.legend()
plt.show()

Key Takeaways #

• The "scale parameter" you're missing is just a manual scaling factor to bridge the gap between the PDF's probability density and your binned data's count/density values.
• circstats.vmpar() is ideal for weighted binned data—stick with it for parameter estimation, then scale the PDF using total samples and bin width.
• Unpacking binned data works for Scipy's fit, but avoid it if you have very large weights (it will create an unnecessarily huge array).

内容的提问来源于stack exchange，提问作者nn123