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A kernel is a probability density function (pdf) f(x) which is symmetric around the y axis, i.e. f(-x) = f(x).
A kernel density estimation (KDE) is a non-parametric method for estimating the pdf of a random variable based on a random sample using some kernel K and some smoothing parameter (aka bandwidth) h > 0.
Let {x1, x2, …, xn} be a random sample from some distribution whose pdf f(x) is not known. We estimate f(x) as follows:
The results are sensitive to the value chosen for h. Rules for choosing an optimum value for h are complex, but the following are some simple guidelines:
- You should use a larger bandwidth value when the sample size is small and the data are sparse. This results in a larger standard deviation; the estimate places more weight on the neighboring data values.
- You can use a smaller bandwidth value when the sample size is large and the data are densely packed. This results in a smaller standard deviation; the estimate places more weight on the specific data value and less on the neighboring data values.
Bandwidths that are too small results in a pdf that is too spiky, while bandwidths that are too large results in a pdf that is over-smoothed.
If f(x) follows a normal distribution then an optimal estimate for h is
where s is the standard deviation of the sample.
Silverman’s optimum estimate of h is
where s* = min(s, IQR/1.34) and IQR is the interquartile range of the sample data.
Some commonly used kernels are listed in Figure 1. Note that seven of the kernels restrict the domain to values |u| ≤ 1. The Epanechnikov kernel is the most efficient in some sense that we won’t go into here. The efficiency column in the figure displays the efficiency of each of the kernels choices as a percentage of the efficiency of the Epanechnikov kernel.
| Kernel name | Kernel pdf | restriction | efficiency |
| uniform | K(u) = 1/2 | |u| ≤ 1 | 92.9% |
| triangular | K(u) = 1 – |u| | |u| ≤ 1 | 98.6% |
| biweight | K(u) = 15(1–u2)2/16 | |u| ≤ 1 | 99.4% |
| triweight | K(u) = 35(1–u2)3/32 | |u| ≤ 1 | 98.7% |
| tricube | K(u) = 70(1–|u|3)3/81 | |u| ≤ 1 | 98.7% |
| Epanechnikov | K(u) = 3(1–u2)/4 | |u| ≤ 1 | 100% |
| cosine | K(u) = π·cos(1–π·u/2)/4 | |u| ≤ 1 | 99.9% |
| Gaussian | K(u) = exp(-u2/2) /√2π | 95.1% | |
| logistic | K(u) = 1/(eu + e-u + 2) | 88.7% | |
| sigmoid | K(u) = 2/[π(eu + e-u)] | 84.3% | |
| Silverman | K(u) = exp(-|u|/√2) · sin(|u|/√2 + π/4) | N/A |
Figure 1 – Kernels
Other Topics
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