Feb 26, 2008, 3:01:02 AM (16 years ago)

Removing text not being true anymore

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  • trunk/doc/Statistics.doxygen

    r1125 r1153  
    302302and calculating the weighted median of the distances.
     304\section Distance
    304306\section Kernel
    305307\subsection polynomial_kernel Polynomial Kernel
    306308The polynomial kernel of degree \f$N\f$ is defined as \f$(1+<x,y>)^N\f$, where
    307309\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
     310case we define the linear kernel to be
     311\f$<x,y>=\frac{\sum {w_xw_yxy}}{\sum{w_xw_y}\f$ and the
    309312polynomial 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).
    316315\subsection gaussian_kernel Gaussian Kernel
    317 We define the weighted Gaussian kernel as \f$\exp\left(-\frac{\sum
    318 w_xw_y(x-y)^2}{\sum w_xw_y}\right)\f$, which fulfills the conditions
    319 listed in the introduction.
    321 Is this kernel a proper kernel? Yes, following the proof of the
    322 non-weighted 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.
    333 \section Distance
     316We define the weighted Gaussian kernel as \f$\exp\left(-N\frac{\sum
     317w_xw_y(x-y)^2}{\sum w_xw_y}\right)\f$.
    335319\section Regression
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