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r2752 r2753 103 103 w_i} \f$. Hence, an estimator of the variance of \f$ X \f$ is 104 104 105 \f$ 106 s^2 = <X^2><X>^2= 105 \f$ 106 s^2 = <X^2><X>^2= 107 107 \f$ 108 108 … … 139 139 \f$ 140 140 <m<m>>^2+(<m>\mu)^2 141 \f$. 141 \f$. 142 142 143 143 In the case when weights are included in analysis due to varying … … 160 160 L(\sigma_0^2)=\prod\frac{1}{\sqrt{2\pi\sigma_0^2/w_i}}\exp{(\frac{w_i(xm)^2}{2\sigma_0^2})} 161 161 \f$ 162 and taking the derivity with respect to 163 \f$\sigma_o^2\f$, 162 and taking the derivity with respect to 163 \f$\sigma_o^2\f$, 164 164 165 165 \f$ … … 168 168 \f$ 169 169 \sum \frac{1}{2\sigma_0^2}+\frac{w_i(xm)^2}{2\sigma_0^2\sigma_o^2} 170 \f$ 170 \f$ 171 171 172 172 which … … 203 203 \subsection Correlation 204 204 205 As the mean is estimated as 206 \f$ 207 m_x=\frac{\sum w_xw_yx}{\sum w_xw_y} 205 As the mean is estimated as 206 \f$ 207 m_x=\frac{\sum w_xw_yx}{\sum w_xw_y} 208 208 \f$, 209 the variance is estimated as 210 \f$ 211 \sigma_x^2=\frac{\sum w_xw_y(xm_x)^2}{\sum w_xw_y} 212 \f$. 209 the variance is estimated as 210 \f$ 211 \sigma_x^2=\frac{\sum w_xw_y(xm_x)^2}{\sum w_xw_y} 212 \f$. 213 213 As in the nonweighted case we define the correlation to be the ratio 214 214 between the covariance and geometrical average of the variances … … 269 269 \f$ 270 270 \frac{\sum w(xm_x)^2+\sum w(ym_y)^2} 271 {\frac{\left(\sum w_x\right)^2}{\sum w_x^2}+ 271 {\frac{\left(\sum w_x\right)^2}{\sum w_x^2}+ 272 272 \frac{\left(\sum w_y\right)^2}{\sum w_y^2}2} 273 273 \f$ … … 281 281 \f$ 282 282 \frac{\sum w(xm_x)^2+\sum w(ym_y)^2} 283 {\frac{\left(\sum w_x\right)^2}{\sum w_x^2}+ 283 {\frac{\left(\sum w_x\right)^2}{\sum w_x^2}+ 284 284 \frac{\left(\sum w_y\right)^2}{\sum w_y^2}2} 285 285 \left(\frac{1}{\sum w_i}+\frac{1}{\sum w_i}\right), … … 289 289 \f$ 290 290 \frac{w\sum (xm_x)^2+w\sum (ym_y)^2} 291 {n_x+n_y2} 291 {n_x+n_y2} 292 292 \left(\frac{1}{wn_x}+\frac{1}{wn_y}\right), 293 293 \f$ 294 in other words the good old expression as for nonweighted. 294 in other words the good old expression as for nonweighted. 295 295 296 296 \subsection FoldChange … … 302 302 and calculating the weighted median of the distances. 303 303 304 \section Distance 304 \section Distance 305 305 306 306 A \ref concept_distance measures how far apart two ranges are. A Distance should … … 345 345 The polynomial kernel of degree \f$N\f$ is defined as \f$(1+<x,y>)^N\f$, where 346 346 \f$<x,y>\f$ is the linear kernel (usual scalar product). For the weighted 347 case we define the linear kernel to be 347 case we define the linear kernel to be 348 348 \f$<x,y>=\frac{\sum {w_xw_yxy}}{\sum{w_xw_y}}\f$ and the 349 349 polynomial kernel can be calculated as before 350 \f$(1+<x,y>)^N\f$. 350 \f$(1+<x,y>)^N\f$. 351 351 352 352 \subsection gaussian_kernel Gaussian Kernel … … 359 359 We have the model 360 360 361 \f$ 362 y_i=\alpha+\beta (xm_x)+\epsilon_i, 363 \f$ 361 \f$ 362 y_i=\alpha+\beta (xm_x)+\epsilon_i, 363 \f$ 364 364 365 365 where \f$\epsilon_i\f$ is the noise. The variance of the noise is … … 374 374 Taking the derivity with respect to \f$\alpha\f$ and \f$\beta\f$ yields two conditions 375 375 376 \f$ 376 \f$ 377 377 \frac{\partial Q_0}{\partial \alpha} = 2 \sum w_i(y_i  \alpha  378 \beta (x_im_x)=0 378 \beta (x_im_x)=0 379 379 \f$ 380 380 … … 382 382 383 383 \f$ \frac{\partial Q_0}{\partial \beta} = 2 \sum 384 w_i(x_im_x)(y_i\alpha\beta(x_im_x)=0 384 w_i(x_im_x)(y_i\alpha\beta(x_im_x)=0 385 385 \f$ 386 386 … … 394 394 395 395 \f$ \beta=\frac{\sum w_i(x_im_x)(ym_y)}{\sum 396 w_i(x_im_x)^2}=\frac{Cov(x,y)}{Var(x)} 396 w_i(x_im_x)^2}=\frac{Cov(x,y)}{Var(x)} 397 397 \f$ 398 398 … … 401 401 \f$\alpha\f$ and \f$\beta\f$. 402 402 403 \f$ 403 \f$ 404 404 \textrm{Var}(\alpha )=\frac{w_i^2\frac{\sigma^2}{w_i}}{(\sum w_i)^2}= 405 405 \frac{\sigma^2}{\sum w_i}
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