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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. ==Definition== Let (''x''1, ''x''2, …, ''xn'') be an independent and identically distributed sample drawn from some distribution with an unknown density ''ƒ''. 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 that integrates to one and has mean zero — and is a smoothing parameter called the ''bandwidth''. A kernel with subscript ''h'' is called the ''scaled kernel'' and defined as . Intuitively one wants to choose ''h'' as small as the data allow, however there is always a trade-off between the bias of the estimator and its variance; more on the choice of bandwidth 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, though the loss of efficiency is small for the kernels listed previously, and due to its convenient mathematical properties, the normal kernel is often used , where ''ϕ'' is the standard normal density function. The construction of a kernel density estimate finds interpretations in fields outside of density estimation. 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 ''xi''. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning. Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. To see this, we compare the construction of histogram and kernel density estimators, using these 6 data points: ''x''1 = −2.1, ''x''2 = −1.3, ''x''3 = −0.4, ''x''4 = 1.9, ''x''5 = 5.1, ''x''6 = 6.2. For the histogram, first the horizontal axis is divided into sub-intervals or bins which cover the range of the data. In this case, we have 6 bins each of width 2. Whenever a data point falls inside this interval, we place a box of height 1/12. If more than one data point falls inside the same bin, we stack the boxes on top of each other. For the kernel density estimate, we place a normal kernel with variance 2.25 (indicated by the red dashed lines) on each of the data points ''xi''. The kernels are summed to make the kernel density estimate (solid blue curve). The smoothness of the kernel density estimate is evident compared to the discreteness of the histogram, as kernel density estimates converge faster to the true underlying density for continuous random variables. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「kernel density estimation」の詳細全文を読む スポンサード リンク
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