Changeset 2753 for trunk/doc


Ignore:
Timestamp:
Jun 24, 2012, 11:18:39 AM (9 years ago)
Author:
Peter
Message:

remove trailing WS

File:
1 edited

Legend:

Unmodified
Added
Removed
  • trunk/doc/Statistics.doxygen

    r2752 r2753  
    103103w_i} \f$. Hence, an estimator of the variance of \f$ X \f$ is
    104104
    105 \f$ 
    106 s^2 = <X^2>-<X>^2= 
     105\f$
     106s^2 = <X^2>-<X>^2=
    107107\f$
    108108
     
    139139\f$
    140140<m-<m>>^2+(<m>-\mu)^2
    141 \f$. 
     141\f$.
    142142
    143143In the case when weights are included in analysis due to varying
     
    160160L(\sigma_0^2)=\prod\frac{1}{\sqrt{2\pi\sigma_0^2/w_i}}\exp{(-\frac{w_i(x-m)^2}{2\sigma_0^2})}
    161161\f$
    162 and taking the derivity with respect to 
    163 \f$\sigma_o^2\f$, 
     162and taking the derivity with respect to
     163\f$\sigma_o^2\f$,
    164164
    165165\f$
     
    168168\f$
    169169\sum -\frac{1}{2\sigma_0^2}+\frac{w_i(x-m)^2}{2\sigma_0^2\sigma_o^2}
    170 \f$ 
     170\f$
    171171
    172172which
     
    203203\subsection Correlation
    204204
    205 As the mean is estimated as 
    206 \f$ 
    207 m_x=\frac{\sum w_xw_yx}{\sum w_xw_y} 
     205As the mean is estimated as
     206\f$
     207m_x=\frac{\sum w_xw_yx}{\sum w_xw_y}
    208208\f$,
    209 the variance is estimated as 
    210 \f$ 
    211 \sigma_x^2=\frac{\sum w_xw_y(x-m_x)^2}{\sum w_xw_y} 
    212 \f$. 
     209the variance is estimated as
     210\f$
     211\sigma_x^2=\frac{\sum w_xw_y(x-m_x)^2}{\sum w_xw_y}
     212\f$.
    213213As in the non-weighted case we define the correlation to be the ratio
    214214between the covariance and geometrical average of the variances
     
    269269\f$
    270270\frac{\sum w(x-m_x)^2+\sum w(y-m_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}+
    272272\frac{\left(\sum w_y\right)^2}{\sum w_y^2}-2}
    273273\f$
     
    281281\f$
    282282\frac{\sum w(x-m_x)^2+\sum w(y-m_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}+
    284284\frac{\left(\sum w_y\right)^2}{\sum w_y^2}-2}
    285285\left(\frac{1}{\sum w_i}+\frac{1}{\sum w_i}\right),
     
    289289\f$
    290290\frac{w\sum (x-m_x)^2+w\sum (y-m_y)^2}
    291 {n_x+n_y-2} 
     291{n_x+n_y-2}
    292292\left(\frac{1}{wn_x}+\frac{1}{wn_y}\right),
    293293\f$
    294 in other words the good old expression as for non-weighted. 
     294in other words the good old expression as for non-weighted.
    295295
    296296\subsection FoldChange
     
    302302and calculating the weighted median of the distances.
    303303
    304 \section Distance 
     304\section Distance
    305305
    306306A \ref concept_distance measures how far apart two ranges are. A Distance should
     
    345345The polynomial kernel of degree \f$N\f$ is defined as \f$(1+<x,y>)^N\f$, where
    346346\f$<x,y>\f$ is the linear kernel (usual scalar product). For the weighted
    347 case we define the linear kernel to be 
     347case we define the linear kernel to be
    348348\f$<x,y>=\frac{\sum {w_xw_yxy}}{\sum{w_xw_y}}\f$ and the
    349349polynomial kernel can be calculated as before
    350 \f$(1+<x,y>)^N\f$. 
     350\f$(1+<x,y>)^N\f$.
    351351
    352352\subsection gaussian_kernel Gaussian Kernel
     
    359359We have the model
    360360
    361 \f$ 
    362 y_i=\alpha+\beta (x-m_x)+\epsilon_i, 
    363 \f$ 
     361\f$
     362y_i=\alpha+\beta (x-m_x)+\epsilon_i,
     363\f$
    364364
    365365where \f$\epsilon_i\f$ is the noise. The variance of the noise is
     
    374374Taking the derivity with respect to \f$\alpha\f$ and \f$\beta\f$ yields two conditions
    375375
    376 \f$ 
     376\f$
    377377\frac{\partial Q_0}{\partial \alpha} = -2 \sum w_i(y_i - \alpha -
    378 \beta (x_i-m_x)=0 
     378\beta (x_i-m_x)=0
    379379\f$
    380380
     
    382382
    383383\f$ \frac{\partial Q_0}{\partial \beta} = -2 \sum
    384 w_i(x_i-m_x)(y_i-\alpha-\beta(x_i-m_x)=0 
     384w_i(x_i-m_x)(y_i-\alpha-\beta(x_i-m_x)=0
    385385\f$
    386386
     
    394394
    395395\f$ \beta=\frac{\sum w_i(x_i-m_x)(y-m_y)}{\sum
    396 w_i(x_i-m_x)^2}=\frac{Cov(x,y)}{Var(x)} 
     396w_i(x_i-m_x)^2}=\frac{Cov(x,y)}{Var(x)}
    397397\f$
    398398
     
    401401\f$\alpha\f$ and \f$\beta\f$.
    402402
    403 \f$ 
     403\f$
    404404\textrm{Var}(\alpha )=\frac{w_i^2\frac{\sigma^2}{w_i}}{(\sum w_i)^2}=
    405405\frac{\sigma^2}{\sum w_i}
Note: See TracChangeset for help on using the changeset viewer.