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== Objective and data-driven kernel selection ==
[[File:Empirical Characteristic Function.jpg|alt=An x-shaped region of empirical characteristic function in Fourier space.|thumb|Demonstration of the filter function <math>I_{\vec{A}}(\vec{t})</math>. The square of the empirical distribution function <math>|\hat{\varphi}|^2</math> from ''N''=10,000 samples of the ‘transition distribution’ discussed in Section 3.2 (and shown in Fig. 4), for <math>|\hat{\varphi}|^2 \ge 4(N-1)N^{-2}</math>. There are two color schemes present in this figure. The predominantly dark, multicolored colored ‘X-shaped’ region in the center corresponds to values of <math>|\hat{\varphi}|^2</math> for the lowest contiguous hypervolume (the area containing the origin); the colorbar at right applies to colors in this region. The lightly-colored, monotone areas away from the first contiguous hypervolume correspond to additional contiguous hypervolumes (areas) with <math>|\hat{\varphi}|^2 \ge 4(N-1)N^{-2}</math>. The colors of these areas are arbitrary and only serve to visually differentiate nearby contiguous areas from one another.]]
Recent research has shown that the kernel and its bandwidth can both be optimally and objectively chosen from the input data itself without making any assumptions about the form of the distribution.<ref name=":0">{{Cite journal|last = Bernacchia|first = Alberto|last2 = Pigolotti|first2 = Simone|date = 2011-06-01|title = Self-consistent method for density estimation|journal = Journal of the Royal Statistical Society, Series B|language = en|volume = 73|issue = 3|pages = 407–422|doi = 10.1111/j.1467-9868.2011.00772.x|issn = 1467-9868|arxiv = 0908.3856}}</ref> The resulting kernel density estimate converges rapidly to the true probability distribution as samples are added: at a rate close to the <math>n^{-1}</math> expected for parametric estimators.<ref name=":0" /><ref name=":1">{{Cite journal|last = O’Brien|first = Travis A.|last2 = Collins|first2 = William D.|last3 = Rauscher|first3 = Sara A.|last4 = Ringler|first4 = Todd D.|date = 2014-11-01|title = Reducing the computational cost of the ECF using a nuFFT: A fast and objective probability density estimation method|journal = Computational Statistics & Data Analysis|volume = 79|pages = 222–234|doi = 10.1016/j.csda.2014.06.002}}</ref><ref name=":22">{{Cite journal|last = O’Brien|first = Travis A.|last2 = Kashinath|first2 = Karthik|last3 = Cavanaugh|first3 = Nicholas R.|last4 = Collins|first4 = William D.|last5 = O’Brien|first5 = John P.|title = A fast and objective multidimensional kernel density estimation method: fastKDE|journal = Computational Statistics & Data Analysis|volume = 101|pages = 148–160|doi = 10.1016/j.csda.2016.02.014|year = 2016|url = https://escholarship.org/content/qt9g56181p/qt9g56181p.pdf?t=p7qvyp}}</ref> This kernel estimator works for univariate and multivariate samples alike. The optimal kernel is defined in Fourier space—as the optimal damping function <math>\hat{\psi_h}(\vec{t})</math> (the Fourier transform of the kernel <math>\hat{K}(\vec{x})</math> )-- in terms of the Fourier transform of the data <math>\hat{\varphi}(\vec{t})</math>, the ''[[Characteristic function (probability theory)|empirical characteristic function]]'' (see [[Kernel density estimation]]):
<math>\hat{\psi_h}(\vec{t}) \equiv \frac{N}{2(N-1)} \left[ 1 + \sqrt{1 - \frac{4(N-1)}{N^2 |\hat{\varphi}(\vec{t})|^2}} I_{\vec{A}}(\vec{t}) \right]</math> <ref name=":22"/>
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