gaussiankde

GaussianKDE, or Gaussian kernel density estimation, is a non-parametric method for estimating the probability density function of a continuous random variable. It constructs a smooth density by placing a Gaussian kernel centered at each observed data point and summing these contributions. This approach makes few assumptions about the underlying distribution and can reveal multimodal structure in the data.

The estimator f_hat(x) is typically written as a normalized sum of Gaussian kernels: f_hat(x) = (1/(n h)) sum_{i=1}^n φ((x - x_i)/h), where φ is the standard normal density and h is a bandwidth parameter that controls smoothness. In multiple dimensions, a multivariate Gaussian kernel is used, often with a bandwidth matrix H that determines the scale and orientation of the kernels. The choice of bandwidth is crucial: too small a bandwidth yields a jagged estimate, while too large a bandwidth oversmooths important features. Common methods for selecting bandwidth include rule-of-thumb formulas (such as Silverman’s or Scott’s rules), cross-validation, and plug-in approaches.

GaussianKDE can be extended to higher dimensions, but performance and accuracy deteriorate with increasing dimensionality due to the curse of dimensionality. Boundary bias can also arise for data confined to finite ranges. Computationally, naive implementations scale poorly with sample size, but there are efficient algorithms using fast Fourier transforms, kd-trees, or approximation techniques.

Software implementations widely support GaussianKDE. In Python, libraries offer Gaussian kernel density estimation through functions such as gaussian_kde in SciPy or KernelDensity in scikit-learn. Similar functionality exists in other languages, with varying interfaces. GaussianKDE remains a common tool for exploratory data analysis, density visualization, and probabilistic modeling where parametric assumptions are undesirable.