Coucou Yann, Quelques typos YJ> + YJ> +%% \df YJ> +%% input : the set which the dim we are intering in the set whose dimension we are interested in? YJ> +\begin{itemize} YJ> + \item Use lowercase and normal font for the scalar variables. YJ> + \item Use lowercase and bold font for the vector variables. YJ> + \item Use uppercase and normal font for the matrix variables. YJ> + \item Use uppercase and double font for the set variables. YJ> +\end {itemize} s/the// YJ> + YJ> +%================================================================= YJ> +\subsection{Sets} YJ> + YJ> +There is three particular sets that we use every time: s/is/are/ YJ> +\begin{itemize} YJ> + \item The color space $\fm{C}$ in which the pixels take their value. YJ> + \item The dataset $\fm{P}$ which contains every pixel we take care. we care about. YJ> + \item The group set $\fm{G}$ that define any splitting of the dataset. defines YJ> +We use the euclidian distance. *E*uclidian YJ> +$$ Use the backslash/bracket pair ("\[ ...\]") instead of a double dollar construct ("$$ ... $$") for single line displays. The double dollar sign as a begin/end marker for displayed math material is a relic from plain TeX, but has been officially deprecated in LaTeX. In fact, many LaTeX books don't even mention it, since it is not supposed to be used. While it works just fine in most situations, and many authors use it with impunity, there are a few situations where it causes problems, and it may not work under future versions of TeX. If you come from a plain TeX/Amstex background and are in the habit of typing $$'s, try to gradually switch over to using the backslash/bracket pair. If you are new to Tex/LaTeX, learn it right from the start (i.e., use the backslash/bracket method). http://www.math.uiuc.edu/~hildebr/tex/tips-mathdisplays.html YJ> +%================================================================= YJ> +\subsection{The group} YJ> + YJ> +We can define the group set $\fm{G}$ in thow context: in this context? YJ> +When we study some mixed population, the total variance can be splitted in the YJ> +the variance between the groups and in the variance whithin each group. s/splitted/split/ s/in the the/in the/ s/whithin/within/ YJ> + YJ> +%================================================================= YJ> +\subsection{Decomposing the variance} YJ> +When we study some mixed population, the total variance can be splitted in the YJ> +the variance between the groups and in the variance whithin each group. s/splitted/split/ s/in the the/in the/ s/whithin/within/ YJ> +%%================================================================= YJ> +\subsection{Eigenvalues and eigenvectors} YJ> + YJ> +We assume that we work on variance/covariance matrix which is real and symetric. s/on/on a/ s/symetric/symmetric/ YJ> +In this case, all the three eigen values are real. s/eigen values/eigenvalues/ YJ> +\subsection{MCO for eigenvalues} YJ> + YJ> +In 3d, it's difficult to extract the cubic roots from the characteristic YJ> +polynomia. The difficulties disappear when we find one of the three roots. YJ> +A planar regression allows us to reach the equation of the plane. From the YJ> +equation, we can determine its normal vector $\bm{w}$. It satisfies the YJ> +following equation $A\bm{w} = \lambda_3\bm{w}$. Thus we know $\lambda_3$. YJ> +By the way, as far as $trace(A) = \lambda_1 + \lambda_2 + \lambda_3$ and YJ> +$det(A) = \lambda_1 \lambda_2 \lambda_3$, then we can access to the value of Je ne comprends pas le "as far as". N'est-ce pas toujours vrai ? s/access to the value of/determine/ YJ> +the two others eigenvalues (just solve $\lambda^2 - (trace(A) - \lambda_3) YJ> +\lambda + \frac{det(A)}{\lambda_3} = 0$). The knowledge of $\lambda_1$ and YJ> +$\lambda_2$ let us find $\bm{u}$ and $\bm{v}$ by the equations $A\bm{u} = YJ> +\lambda_1\bm{u}$ and $A\bm{v} = \lambda_2\bm{v}$. YJ> + YJ> +Let's center the points by susbtracting their center of YJ> mass. Now, we have s/susbtracting/subtracting/ -- Alexandre Duret-Lutz