bandwidthmatrix

Bandwidth matrix, often denoted H, is a smoothing parameter used in multivariate kernel methods such as kernel density estimation and kernel regression. It is a symmetric positive definite matrix that determines the scale, orientation, and interaction of smoothing across variables.

In kernel density estimation, the estimator at a point x is f_hat(x) = (1/n) sum |H|^{-1/2} K(H^{-1/2}(x - X_i)). Here K is a kernel, and H controls smoothing in every direction. If H is diagonal, smoothing is independent across coordinates; if H is full, the kernel can adapt to correlations between variables. This allows anisotropic smoothing that reflects the data structure.

Bandwidth selection: Choosing H involves bias-variance trade-off and is more complex in higher dimensions. Common strategies include cross-validation, likelihood cross-validation, and plug-in methods. A simple isotropic choice uses H = h^2 I_d, with scalar h and I_d the identity matrix. General rules of thumb extend Scott's or Silverman's guidelines to H ≈ n^{-1/(d+4)} times a covariance proxy such as the sample covariance Σ.

Properties and interpretation: For kernels with Gaussian form, smoothing with H is equivalent to convolving the data with a Gaussian of covariance H, producing a smoothed density or regression surface. The SPD nature guarantees a valid smoothing operator, and the rank and conditioning of H affect the shape of the estimator.

Applications: bandwidth matrices are used in nonparametric density estimation, regression, conditional density estimation, mode finding, and anomaly detection in multivariate data.