[Olena-patches] 4466: Correct english writing in the documentation file.

* green/doc/formulae/formulae.tex : Correct file. --- trunk/milena/sandbox/ChangeLog | 6 + .../milena/sandbox/green/doc/formulae/formulae.tex | 116 ++++++++++---------- 2 files changed, 64 insertions(+), 58 deletions(-) diff --git a/trunk/milena/sandbox/ChangeLog b/trunk/milena/sandbox/ChangeLog index 6572258..923875c 100644 --- a/trunk/milena/sandbox/ChangeLog +++ b/trunk/milena/sandbox/ChangeLog @@ -1,3 +1,9 @@ +2009-09-11 Yann Jacquelet <jacquelet(a)lrde.epita.fr> + + Correct english writing in the documentation file. + + * green/doc/formulae/formulae.tex : Correct file. + 2009-09-10 Yann Jacquelet <jacquelet(a)lrde.epita.fr> Try to fix order problem between points and vectors. diff --git a/trunk/milena/sandbox/green/doc/formulae/formulae.tex b/trunk/milena/sandbox/green/doc/formulae/formulae.tex index 52af7a7..655cb1a 100644 --- a/trunk/milena/sandbox/green/doc/formulae/formulae.tex +++ b/trunk/milena/sandbox/green/doc/formulae/formulae.tex @@ -41,7 +41,7 @@ \newcommand{\fm}[1]{\mathbb{#1}} %% \df -%% input : the set which the dim we are intering in +%% input : the set whose dimension we are interested in %% \newcommand{\df}[1]{|{\mathbb{#1}}|} @@ -65,20 +65,20 @@ in the development. \section{Notations} \begin{itemize} - \item Use lowercase and normal font for the scalar variables. - \item Use lowercase and bold font for the vector variables. - \item Use uppercase and normal font for the matrix variables. - \item Use uppercase and double font for the set variables. + \item Use lowercase and normal font for scalar variables. + \item Use lowercase and bold font for vector variables. + \item Use uppercase and normal font for matrix variables. + \item Use uppercase and double font for set variables. \end {itemize} %================================================================= \subsection{Sets} -There is three particular sets that we use every time: +There are three particular sets that we use every time: \begin{itemize} - \item The color space $\fm{C}$ in which the pixels take their value. - \item The dataset $\fm{P}$ which contains every pixel we take care. - \item The group set $\fm{G}$ that define any splitting of the dataset. + \item The color space $\fm{C}$ in which the pixels take their values. + \item The dataset $\fm{P}$ which contains every pixel we care about. + \item The group set $\fm{G}$ that defines any splitting of the dataset. \end{itemize} \begin{tabular}{|c|l|l|c|c|} @@ -98,15 +98,15 @@ There is three particular sets that we use every time: %================================================================= \subsection{Color space} -We use the euclidian distance. +We use the Euclidian distance. -$$ +\[ d(a,b) = \sqrt{\sum_{i=0}^q (a_i - b_i)^2} = \sqrt{(a_x - b_x)^2 + (a_y - b_y)^2 + (a_z - b_z)^2} -$$ +\] -$$ +\[ d(a,b)^2 = \left[\begin{array}{ccc} a_x - b_x & a_y - b_y & a_z - b_z @@ -117,13 +117,13 @@ d(a,b)^2 = a_y - b_y \\ a_y - b_z \end{array}\right] -$$ +\] %================================================================= \subsection{Data points in $\mathbb{R}^3$} We present the four points of the dataset with their vector representation. -$$ +\[ \bm{a} = \left[\begin{array}{c} a_x \\ @@ -143,9 +143,9 @@ $$ p_{13} \end{array}\right] = \bm{p}_1 -$$ +\] -$$ +\[ \bm{b} = \left[\begin{array}{c} b_x \\ @@ -187,9 +187,9 @@ $$ p_{33} \end{array}\right] = \bm{p}_3 -$$ +\] -$$ +\[ \mbox{\boldmath$d$} = \left[\begin{array}{c} d_x \\ @@ -209,11 +209,11 @@ $$ p_{43} \end{array}\right] = \bm{p}_4 -$$ +\] One may group the four points in one matrix P : -$$ +\[ P = \left[\begin{array}{c} \bm{a}^t \\ @@ -249,12 +249,12 @@ P = \bm{p}_3^t \\ \bm{p}_4^t \end{array}\right] -$$ +\] %================================================================= \subsection{The group} -We can define the group set $\fm{G}$ in thow context: +We can define the group set $\fm{G}$ in this context: \begin{itemize} \item First, the group set is a partition. \item Second, the group set if a fuzzy set. @@ -321,7 +321,7 @@ Let's have a look to the first three moments. %================================================================= \subsection{The mean} -$$ +\[ \mbox{\boldmath$m$} = \left[\begin{array}{c} m_x \\ @@ -353,8 +353,8 @@ $$ \mbox{\boldmath$d$}) = \frac{1}{4}\sum_{i=1}^{4}\mbox{\boldmath$p$}_i -$$ -$$ +\] +\[ \mbox{\boldmath$m$} = \frac{1}{4} \left[\begin{array}{cccc} @@ -382,10 +382,10 @@ $$ \end{array}\right] = \frac{1}{4} P^t \mbox{\boldmath$ 1$} -$$ +\] The mean matrix: -$$ +\[ M = \left[\begin{array}{c} \mbox{\boldmath$m$}^t \\ @@ -407,10 +407,10 @@ M = m_1 & m_2 & m_3 \\ m_1 & m_2 & m_3 \end{array}\right] -$$ +\] We define the difference between a point $\mbox{\boldmath$p$}_i$ and the mean: -$$ +\[ (\mbox{\boldmath$p$}_i - \mbox{\boldmath$m$}) = \left[\begin{array}{c} p_{ix} - m_x \\ @@ -423,11 +423,11 @@ $$ p_{i2} - m_2 \\ p_{i3} - m_3 \end{array}\right] -$$ +\] And for all the dataset: -$$ +\[ (P - M) = \left[\begin{array}{ccc} a_x & a_y & a_z \\ @@ -449,12 +449,12 @@ $$ c_x - m_x & c_y - m_y & c_z - m_z \\ d_x - m_x & d_y - m_y & d_z - m_z \end{array}\right] -$$ +\] %================================================================= \subsection{The variance} -$$ +\[ \begin{array}{lcl} V & = & \left[\begin{array}{ccc} @@ -501,12 +501,12 @@ V & = & \frac{1}{4} (P - M)^t (P - M) \end{array} -$$ +\] %################################################################# \section{Splitting into groups} -When we study some mixed population, the total variance can be splitted in the -the variance between the groups and in the variance whithin each group. +When we study some mixed population, the total variance can be split in +the variance between the groups and in the variance within each group. We define two groups in the population. Each group owns its moments of the second order. @@ -514,51 +514,51 @@ second order. %================================================================= \subsection{Decomposing the count} -$$ +\[ \begin{array}{lcl} n_t & = & n_1 + n_2 \\ & = & \sum_{i=1}^{2} n_i \end{array} -$$ +\] %================================================================= \subsection{Decomposing the mean} -$$ +\[ \begin{array}{lcl} \mbox{\boldmath$m_t$} & = & \frac{1}{n_t}(n_1 \mbox{\boldmath$m_1$} + n_2 \mbox{\boldmath$m_2$}) \\ & = & \frac{1}{n_t}\sum_{i=1}^{2} n_i \mbox{\boldmath$m_i$} \end{array} -$$ +\] %================================================================= \subsection{Decomposing the variance} -When we study some mixed population, the total variance can be splitted in the -the variance between the groups and in the variance whithin each group. +When we study some mixed population, the total variance can be split in +the variance between the groups and in the variance within each group. -$$ +\[ V_t = V_i + V_b -$$ +\] -$$ +\[ \begin{array}{lcl} V_i & = & \frac{1}{n_t}(n_1 V_1 + n_2 V_2) \\ & = & \frac{1}{n_t}\sum_{i=1}^2 n_i V_i \end{array} -$$ +\] -$$ +\[ \begin{array}{lcl} V_b & = & \frac{1}{n_t}(n_1 (\mbox{\boldmath$m_1$} - \mbox{\boldmath$m_t$})^2 + n_2 (\mbox{\boldmath$m_2$} - \mbox{\boldmath$m_t$})^2 \\ & = & \frac{1}{n_t} \sum_{i=1}^2 n_i (\mbox{\boldmath$m_i$} - \mbox{\boldmath$m_t$})^2 \end{array} -$$ +\] %################################################################# \section{Basis} @@ -566,7 +566,7 @@ $$ %%================================================================= \subsection{Determinant of a square matrix 3x3} -$$ +\[ \det{V} = \left|\begin{array}{ccc} v_{11} & v_{12} & v_{13} \\ @@ -577,12 +577,12 @@ $$ v_{11}(v_{22}v_{33} - v_{32}v_{23}) - v_{12}(v_{21}v_{33} - v_{31}v_{23}) + v_{13}(v_{21}v_{32} - v_{31}v_{22}) -$$ +\] %%================================================================= \subsection{Transpose} -$ +\[ V^t = \left[\begin{array}{ccc} v_{11} & v_{12} & v_{13} \\ @@ -595,7 +595,7 @@ V^t = v_{12} & v_{22} & v_{32} \\ v_{13} & v_{23} & v_{33} \end{array}\right] -$ +\] %%================================================================= \subsection{Inverse of a square matrix 3x3} @@ -759,8 +759,8 @@ $$ %%================================================================= \subsection{Eigenvalues and eigenvectors} -We assume that we work on variance/covariance matrix which is real and symetric. -In this case, all the three eigen values are real. +We assume that we work on a variance/covariance matrix which is real and +symmetric. In this case, all the three eigenvalues are real. $$ V \bm{x} = \lambda \bm{x} @@ -1089,14 +1089,14 @@ polynomia. The difficulties disappear when we find one of the three roots. A planar regression allows us to reach the equation of the plane. From the equation, we can determine its normal vector $\bm{w}$. It satisfies the following equation $A\bm{w} = \lambda_3\bm{w}$. Thus we know $\lambda_3$. -By the way, as far as $trace(A) = \lambda_1 + \lambda_2 + \lambda_3$ and -$det(A) = \lambda_1 \lambda_2 \lambda_3$, then we can access to the value of +By the way, $trace(A) = \lambda_1 + \lambda_2 + \lambda_3$ and +$det(A) = \lambda_1 \lambda_2 \lambda_3$, then we can determine the value of the two others eigenvalues (just solve $\lambda^2 - (trace(A) - \lambda_3) \lambda + \frac{det(A)}{\lambda_3} = 0$). The knowledge of $\lambda_1$ and $\lambda_2$ let us find $\bm{u}$ and $\bm{v}$ by the equations $A\bm{u} = \lambda_1\bm{u}$ and $A\bm{v} = \lambda_2\bm{v}$. -Let's center the points by susbtracting their center of mass. Now, we have +Let's center the points by subtracting their center of mass. Now, we have three equivalent ways to estimate the coefficients of the plane: \begin{itemize} \item if not $c = 0$, then $\frac{a}{c}x + \frac{b}{c}y + z = 0$, -- 1.5.6.5