Kernel Function

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A Kernel Function is a distance function that evaluates the similarity between two structures (in some metric space).



References

  • tbd
    • If X is the instance space, a kernel function is a mapping K:XxX->[0,infinity) such that given two instances [math]\displaystyle{ x }[/math] and [math]\displaystyle{ y }[/math], K(x,y) = SUM(i) ti(x) ti(y) = t(x)·t(y), where ti(x) is some feature function over the instance x.

2008

  • http://sfb649.wiwi.hu-berlin.de/fedc_homepage/xplore/tutorials/xlghtmlnode33.html
    • QUOTE: Assume we have [math]\displaystyle{ n }[/math] independent observations [math]\displaystyle{ x_1,\ldots,x_n }[/math] from the random variable [math]\displaystyle{ X }[/math]. The kernel density estimator [math]\displaystyle{ \widehat{f}_{h}(x) }[/math] for the estimation of the density value [math]\displaystyle{ f(x) }[/math] at point [math]\displaystyle{ x }[/math] is defined as : [math]\displaystyle{ \displaystyle \widehat{f}_{h}(x)= \frac{1}{nh}\sum_{i=1}^{n} K\left(\frac{x_i-x}{h}\right),\ 6.1) }[/math] [math]\displaystyle{ K(\bullet) }[/math] denoting a so-called kernel function, and $ h</math> denoting the bandwidth. A number of possible kernel functions is listed in the following table.

      Table 6.1: Kernel functions.

      • Kernel [math]\displaystyle{ K(u) }[/math]
      • Uniform [math]\displaystyle{ \frac{1}{2} {\boldsymbol{I}}(\vert u \vert \le 1) }[/math]
      • Triangle [math]\displaystyle{ (1-\vert u \vert) {\boldsymbol{I}}(\vert u \vert \le 1) }[/math]
      • Epanechnikov [math]\displaystyle{ \frac{3}{4}(1-u^{2}) {\boldsymbol{I}}(\vert u \vert \le 1) }[/math]
      • Quartic [math]\displaystyle{ \frac{15}{16}(1-u^{2})^{2} {\boldsymbol{I}}(\vert u \vert \le 1) }[/math]
      • Triweight [math]\displaystyle{ \frac{35}{32}(1-u^{2})^{3} {\boldsymbol{I}}(\vert u \vert \le 1) }[/math]
      • Gaussian [math]\displaystyle{ \frac{1}{\sqrt{2\pi}} \exp(-\frac{1}{2}u^2) }[/math]
      • Cosinus [math]\displaystyle{ \frac{\pi}{4}\cos(\frac{\pi}{2}u) {\boldsymbol{I}}(\vert u \vert \le 1) }[/math]

2004

1997

  • (Mitchell, 1997) ⇒ Tom M. Mitchell. (1997). “Machine Learning." McGraw-Hill.
    • Much of the literature on nearest-neighbor methods and weighted local regression uses a terminology that has arisen from the field of statistical pattern recognition....
      • Regression means approximating a real-valued target function.
      • Residual is the error f^(x) - [math]\displaystyle{ f }[/math](x) in approximating the target function.
      • Kernel function is the function of distance that is used to determine the weight of each training example. In other words, the kernel function is the function [math]\displaystyle{ K }[/math] such that wi = K(d(xi, xq)).