In statistics, **kernel density estimation** (**KDE**) is a non-parametric way to estimate the probability density function of a random variable. Kernel density estimation is a fundamental data smoothing problem where inferences about the population are made, based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the **Parzen–Rosenblatt window** method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form.[1][2] One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier,[3][4] which can improve its prediction accuracy.[3]

## . . . Kernel density estimation . . .

Let (*x*_{1}, *x*_{2}, …, *x _{n}*) be independent and identically distributed samples drawn from some univariate distribution with an unknown density

*ƒ*at any given point

*x*. We are interested in estimating the shape of this function

*ƒ*. Its

*kernel density estimator*is

where *K* is the kernel — a non-negative function — and *h* > 0 is a smoothing parameter called the bandwidth. A kernel with subscript *h* is called the *scaled kernel* and defined as *K _{h}*(

*x*) = 1/

*h K*(

*x*/

*h*). Intuitively one wants to choose

*h*as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below.

A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov, normal, and others. The Epanechnikov kernel is optimal in a mean square error sense,[5] though the loss of efficiency is small for the kernels listed previously.[6] Due to its convenient mathematical properties, the normal kernel is often used, which means *K*(*x*) = *ϕ*(*x*), where *ϕ* is the standard normal density function.

The construction of a kernel density estimate finds interpretations in fields outside of density estimation.[7] For example, in thermodynamics, this is equivalent to the amount of heat generated when heat kernels (the fundamental solution to the heat equation) are placed at each data point locations *x _{i}*. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning (e.g. diffusion map).

## . . . Kernel density estimation . . .

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