User:Timothee Flutre/Notebook/Postdoc/2012/01/02: Difference between revisions
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== | ==About the multivariate Normal distribution== | ||
* '''Motivation''': when we measure things, we often have to measure several properties for each item. For instance, for each person, we measure the expression level of all genes in his sample. | |||
* '''Data''': we have N observations, noted <math>X = (x_1, x_2, ..., x_N)</math>, each being of dimension <math>P</math>. This means that each <math>x_i</math> is a vector belonging to <math>\mathbb{R}^P</math>. | |||
* '''Model''': we suppose that the <math>x_i</math> are independent and identically distributed according to a [http://en.wikipedia.org/wiki/Multivariate_normal_distribution multivariate Normal distribution] <math>N_P(\mu, \Sigma)</math>. <math>\mu</math> is the P-dimensional mean vector, and <math>\Sigma</math> the PxP covariance matrix. If <math>\Sigma</math> is [http://en.wikipedia.org/wiki/Positive-definite_matrix positive definite] (which we will assume), the density function for a given x is: <math>f(x/\mu,\Sigma) = (2 \pi)^{-P/2} |\Sigma|^{-1/2} exp(-\frac{1}{2} (x-\mu)^T \Sigma^{-1} (x-\mu))</math>, with <math>|M|</math> denoting the determinant of a matrix and <math>M^T</math> its transpose. | |||
* '''Likelihood''': as usual, we will start by writing down the likelihood of the data, the parameters being <math>\theta=(\mu,\Sigma)</math>: | |||
<math>L(\theta) = \mathbb{P}(X/\theta)</math> | |||
As the observations are independent: | |||
<math>L(\theta) = \prod_{i=1}^N f(x_i / \theta)</math> | |||
It is easier to work with the log-likelihood: | |||
<math>l(\theta) = ln(L(\theta)) = \sum_{i=1}^N ln( f(x_i / \theta) )</math> | |||
<math>l(\theta) = -\frac{NP}{2} ln(2\pi) - \frac{N}{2}ln(|\Sigma|) - \frac{1}{2} \sum_{i=1}^N (x_i-\mu)^T \Sigma^{-1} (x_i-\mu)</math> | |||
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Revision as of 09:35, 2 January 2012
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About the multivariate Normal distribution
[math]\displaystyle{ L(\theta) = \mathbb{P}(X/\theta) }[/math] As the observations are independent: [math]\displaystyle{ L(\theta) = \prod_{i=1}^N f(x_i / \theta) }[/math] It is easier to work with the log-likelihood: [math]\displaystyle{ l(\theta) = ln(L(\theta)) = \sum_{i=1}^N ln( f(x_i / \theta) ) }[/math] [math]\displaystyle{ l(\theta) = -\frac{NP}{2} ln(2\pi) - \frac{N}{2}ln(|\Sigma|) - \frac{1}{2} \sum_{i=1}^N (x_i-\mu)^T \Sigma^{-1} (x_i-\mu) }[/math] |