Kernel Density Estimation

Notes

• Most popular technique for estimating probability densities of random variables
• A chosen kernel function is applied for each point in the finite dataset
• Many possible kernel functions, such as, but not limited to:
  • Gaussian (normal)
  • Triangular
  • Rectangular (uniform/box)
  • Epanechnikov (parabolic)
  • Biweight (quartic)
  • Triweight
  • Tricube
  • Cosine
• One can choose different kernel widths
• Adaptive KDE

Formula

where $n$ is the number of sample, $h$ is the bandwidth and $K$ is the kernel function.

References

• https://en.wikipedia.org/wiki/Kernel_density_estimation
• https://en.wikipedia.org/wiki/Kernel_(statistics)#Kernel_functions_in_common_use
• http://www.mit.edu/~talw/publications/kde_space_full.pdf
• https://towardsdatascience.com/modality-tests-and-kernel-density-estimations-3f349bb9e595
• https://bmcproc.biomedcentral.com/articles/10.1186/1753-6561-7-S7-S7
• http://faculty.washington.edu/yenchic/18W_425/Lec6_hist_KDE.pdf
• https://stats.stackexchange.com/questions/219833/density-estimation-for-large-dataset

Implementations

• https://cran.r-project.org/web/packages/kedd/vignettes/kedd.pdf
• https://stat.ethz.ch/R-manual/R-devel/library/stats/html/density.html
• https://stat.ethz.ch/R-manual/R-devel/library/KernSmooth/html/bkde.html
• https://jakevdp.github.io/PythonDataScienceHandbook/05.13-kernel-density-estimation.html

Articles & Papers

• https://aakinshin.net/posts/kde-bw/
• https://aakinshin.net/posts/discrete-sample-jittering/
• https://www.cs.utah.edu/~lifeifei/papers/kernelsigmod13.pdf
• http://www.econ.upf.edu/~lugosi/l1testrev.pdf

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Backlinks

• Violin Plot (vault)