Parzen Window Density Estimation Using Gaussian Kernel Smoothing

Parzen window density estimation is a classical non-parametric probability density estimation method, particularly suitable for cases where the data distribution is unknown. When using Gaussian functions as smoothing factors (also known as kernel functions), this method generates continuous and smooth density curves.

The core concept involves placing a Gaussian window around each data point and then superimposing all windows to form the final density estimate. The bandwidth parameter of the Gaussian kernel (i.e., standard deviation) determines the window width: larger bandwidth results in smoother estimates but may lose details, while smaller bandwidth preserves more local features but may introduce noise.

Compared to other kernel functions (such as rectangular kernels), the Gaussian kernel offers advantages in infinite differentiability and robustness to outliers. In practical applications, bandwidth selection is typically optimized through cross-validation or empirical rules (like Silverman's rule). Implementationally, the Gaussian kernel is computed using the standard normal distribution formula: K(u) = (1/√(2π))exp(-u²/2), where u represents the normalized distance from data points. The method's computational complexity grows with data size, thus requiring approximation algorithms for large datasets, such as kernel density tree structures or fast Fourier transform-based implementations.