Changeset 1153 for trunk/doc/Statistics.doxygen
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 Feb 26, 2008, 3:01:02 AM (16 years ago)
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trunk/doc/Statistics.doxygen
r1125 r1153 302 302 and calculating the weighted median of the distances. 303 303 304 \section Distance 305 304 306 \section Kernel 305 307 \subsection polynomial_kernel Polynomial Kernel 306 308 The polynomial kernel of degree \f$N\f$ is defined as \f$(1+<x,y>)^N\f$, where 307 309 \f$<x,y>\f$ is the linear kernel (usual scalar product). For the weighted 308 case we define the linear kernel to be \f$<x,y>=\sum {w_xw_yxy}\f$ and the 310 case we define the linear kernel to be 311 \f$<x,y>=\frac{\sum {w_xw_yxy}}{\sum{w_xw_y}\f$ and the 309 312 polynomial kernel can be calculated as before 310 \f$(1+<x,y>)^N\f$. Is this kernel a proper kernel (always being semi 311 positive definite). Yes, because \f$<x,y>\f$ is obviously a proper kernel 312 as it is a scalar product. Adding a positive constant to a kernel 313 yields another kernel so \f$1+<x,y>\f$ is still a proper kernel. Then also 314 \f$(1+<x,y>)^N\f$ is a proper kernel because taking a proper kernel to the 315 \f$Nth\f$ power yields a new proper kernel (see any good book on SVM). 313 \f$(1+<x,y>)^N\f$. 314 316 315 \subsection gaussian_kernel Gaussian Kernel 317 We define the weighted Gaussian kernel as \f$\exp\left(\frac{\sum 318 w_xw_y(xy)^2}{\sum w_xw_y}\right)\f$, which fulfills the conditions 319 listed in the introduction. 320 321 Is this kernel a proper kernel? Yes, following the proof of the 322 nonweighted kernel we see that \f$K=\exp\left(\frac{\sum 323 w_xw_yx^2}{\sum w_xw_y}\right)\exp\left(\frac{\sum w_xw_yy^2}{\sum 324 w_xw_y}\right)\exp\left(\frac{\sum w_xw_yxy}{\sum w_xw_y}\right)\f$, 325 which is a product of two proper kernels. \f$\exp\left(\frac{\sum 326 w_xw_yx^2}{\sum w_xw_y}\right)\exp\left(\frac{\sum w_xw_yy^2}{\sum 327 w_xw_y}\right)\f$ is a proper kernel, because it is a scalar product and 328 \f$\exp\left(\frac{\sum w_xw_yxy}{\sum w_xw_y}\right)\f$ is a proper 329 kernel, because it a polynomial of the linear kernel with positive 330 coefficients. As product of two kernel also is a kernel, the Gaussian 331 kernel is a proper kernel. 332 333 \section Distance 316 We define the weighted Gaussian kernel as \f$\exp\left(N\frac{\sum 317 w_xw_y(xy)^2}{\sum w_xw_y}\right)\f$. 334 318 335 319 \section Regression
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