Changeset 1153 for trunk/doc/Statistics.doxygen

Timestamp:

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

Author:

Peter

Message:

Removing text not being true anymore

File:

• trunk/doc/Statistics.doxygen (modified) (1 diff)

trunk/doc/Statistics.doxygen

@@ -302 +302 @@
 and calculating the weighted median of the distances.
 
+\section Distance
+
 \section Kernel
 \subsection polynomial_kernel Polynomial Kernel
 The polynomial kernel of degree \f$N\f$ is defined as \f$(1+<x,y>)^N\f$, where
 \f$<x,y>\f$ is the linear kernel (usual scalar product). For the weighted
-case we define the linear kernel to be \f$<x,y>=\sum {w_xw_yxy}\f$ and the
+case we define the linear kernel to be 
+\f$<x,y>=\frac{\sum {w_xw_yxy}}{\sum{w_xw_y}\f$ and the
 polynomial kernel can be calculated as before
-\f$(1+<x,y>)^N\f$. Is this kernel a proper kernel (always being semi
-positive definite). Yes, because \f$<x,y>\f$ is obviously a proper kernel
-as it is a scalar product. Adding a positive constant to a kernel
-yields another kernel so \f$1+<x,y>\f$ is still a proper kernel. Then also
-\f$(1+<x,y>)^N\f$ is a proper kernel because taking a proper kernel to the
-\f$Nth\f$ power yields a new proper kernel (see any good book on SVM).
+\f$(1+<x,y>)^N\f$. 
+
 \subsection gaussian_kernel Gaussian Kernel
-We define the weighted Gaussian kernel as \f$\exp\left(-\frac{\sum
-w_xw_y(x-y)^2}{\sum w_xw_y}\right)\f$, which fulfills the conditions
-listed in the introduction.
-
-Is this kernel a proper kernel? Yes, following the proof of the
-non-weighted kernel we see that \f$K=\exp\left(-\frac{\sum
-w_xw_yx^2}{\sum w_xw_y}\right)\exp\left(-\frac{\sum w_xw_yy^2}{\sum
-w_xw_y}\right)\exp\left(\frac{\sum w_xw_yxy}{\sum w_xw_y}\right)\f$,
-which is a product of two proper kernels. \f$\exp\left(-\frac{\sum
-w_xw_yx^2}{\sum w_xw_y}\right)\exp\left(-\frac{\sum w_xw_yy^2}{\sum
-w_xw_y}\right)\f$ is a proper kernel, because it is a scalar product and
-\f$\exp\left(\frac{\sum w_xw_yxy}{\sum w_xw_y}\right)\f$ is a proper
-kernel, because it a polynomial of the linear kernel with positive
-coefficients. As product of two kernel also is a kernel, the Gaussian
-kernel is a proper kernel.
-
-\section Distance
+We define the weighted Gaussian kernel as \f$\exp\left(-N\frac{\sum
+w_xw_y(x-y)^2}{\sum w_xw_y}\right)\f$.
 
 \section Regression