Fastest Gaussian Addition
This article explores the most efficient methods to cumulatively add Gaussian distributions.
Understanding the Problem
Gaussian distributions, often called normal distributions, are ubiquitous in statistics and data analysis. Adding Gaussians is a common operation, especially when combining measurements or modeling uncertainties.
The Challenge
Directly adding Gaussians in their standard form involves complex integrations. This can be computationally expensive, especially when dealing with many distributions. The goal is to find a method that minimizes computation time.
Efficient Methods
1. Moment Matching
This approach leverages the fact that the sum of Gaussian distributions is also Gaussian. We can determine the mean and variance of the resulting distribution by adding the corresponding moments of the individual Gaussians.
Algorithm
- Calculate the mean (µ) and variance (σ²) of each individual Gaussian.
- Sum the means to get the mean of the combined distribution: µsum = Σµi
- Sum the variances to get the variance of the combined distribution: σ²sum = Σσ²i
Example
| Gaussian | Mean (µ) | Variance (σ²) |
|---|---|---|
| G1 | 1 | 2 |
| G2 | 3 | 4 |
| G3 | 5 | 1 |
Mean of the combined distribution: µsum = 1 + 3 + 5 = 9
Variance of the combined distribution: σ²sum = 2 + 4 + 1 = 7
2. Convolution
Convolution is a mathematical operation that provides a precise way to add probability distributions. It involves integrating the product of the two distributions at different shifts.
Algorithm
- Define the probability density functions (PDFs) of the individual Gaussians.
- Perform convolution on the PDFs. This typically involves numerical integration.
Example
“`python import numpy as np from scipy.signal import convolve # Define PDFs of two Gaussians mu1, sigma1 = 1, 2 mu2, sigma2 = 3, 4 x = np.linspace(-10, 10, 1000) gaussian1 = np.exp(-(x – mu1)**2 / (2 * sigma1**2)) gaussian2 = np.exp(-(x – mu2)**2 / (2 * sigma2**2)) # Convolution combined_gaussian = convolve(gaussian1, gaussian2, mode=’same’) # Normalize to obtain a probability distribution combined_gaussian /= np.sum(combined_gaussian) “`
3. Fast Fourier Transform (FFT)
The FFT is a highly efficient algorithm for computing convolutions. By transforming the PDFs into the frequency domain, we can multiply them and then inverse transform back to obtain the convolved distribution.
Algorithm
- Calculate the FFT of the PDFs of individual Gaussians.
- Multiply the FFTs element-wise.
- Perform inverse FFT to obtain the combined distribution.
Example
“`python import numpy as np from scipy.fftpack import fft, ifft # Define PDFs of two Gaussians (same as previous example) # FFT fft1 = fft(gaussian1) fft2 = fft(gaussian2) # Multiply FFTs combined_fft = fft1 * fft2 # Inverse FFT combined_gaussian = ifft(combined_fft) # Normalize combined_gaussian /= np.sum(combined_gaussian) “`
Comparison
Moment matching is the fastest method but offers an approximation. Convolution provides precise results but can be computationally intensive, especially for a large number of Gaussians. The FFT method offers a balance between speed and accuracy and is generally the most efficient for a moderate number of distributions.
Conclusion
The most efficient way to cumulatively add Gaussians depends on the specific application and desired accuracy. Moment matching is ideal for quick estimations, convolution provides precision, and the FFT offers a good compromise for most scenarios.