kerneltiheyden

Kerneltiheyden, typically referred to in English as kernel density estimation (KDE), is a non-parametric method for estimating the probability density function of a random variable from observed data. It produces a smooth density curve without assuming a specific parametric form for the underlying distribution.

The standard estimator is f̂(x) = (1/(n h)) ∑_{i=1}^n K((x − X_i)/h), where X_i are the data, K is a kernel function, and h is the smoothing bandwidth. The kernel function is a nonnegative, integrates to one, and is usually symmetric around zero. Common choices include the Gaussian, Epanechnikov, and Uniform kernels. In practice, the Gaussian kernel is widely used because of its smoothness, but the choice of kernel often has less impact on the estimate than the bandwidth.

Bandwidth selection is central to KDE. A too small h yields a noisy, overfitted estimate; a too large h oversmooths by masking features. Methods for choosing h include rule-of-thumb formulas (such as Silverman’s rule), cross-validation, and plug-in approaches that aim to minimize estimated risk or bias-variance. In higher dimensions, KDE uses multivariate kernels and a bandwidth matrix, but the method suffers from the curse of dimensionality, requiring larger samples to achieve reliable estimates.

KDE complements histograms by providing a smooth, continuous density estimate and is used in data visualization, anomaly detection, and exploratory data analysis. It can be applied to univariate or multivariate data, with attention to bandwidth and dimensionality to ensure reliable results. In Finnish statistics, kerneltiheyden denotes this same density estimation approach.