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| − | <br> for which | + | <br> for which the expected value of ''x'' is |
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| − | where '''x''' is a ''d''-component column vector, '''μ''' is the ''d''-component mean vector, '''Σ''' is the ''d''-by-''d'' covariance matrix, and '''|Σ|''' and '''Σ<sup>-1< | + | where '''x''' is a ''d''-component column vector, '''μ''' is the ''d''-component mean vector, '''Σ''' is the ''d''-by-''d'' covariance matrix, and '''|Σ|''' and '''Σ<sup>-1<\sup>''' are its determinant and inverse respectively. Also, ('''x - μ''')<sup>t<\sup> denotes the transpose of ('''x - μ'''). |
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Revision as of 18:09, 4 April 2013
Discriminant Functions For The Normal Density
Lets begin with the continuous univariate normal or Gaussian density.
$ f_x = \frac{1}{\sqrt{2 \pi} \sigma} \exp \left [- \frac{1}{2} \left ( \frac{x - \mu}{\sigma} \right)^2 \right ] $
for which the expected value of x is
$ \mu = \mathcal{E}[x] =\int\limits_{-\infty}^{\infty} xp(x)\, dx $
and where the expected squared deviation or variance is
$ \sigma^2 = \mathcal{E}[(x- \mu)^2] =\int\limits_{-\infty}^{\infty} (x- \mu)^2 p(x)\, dx $
The univariate normal density is completely specified by two parameters; its mean μ and variance σ2. The function fx can be written as N(μ,σ) which says that x is distributed normally with mean μ and variance σ2. Samples from normal distributions tend to cluster about the mean with a spread related to the standard deviation σ.
For the multivariate normal density in d dimensions, fx is written as
$ f_x = \frac{1}{(2 \pi)^ \frac{d}{2} |\boldsymbol{\Sigma}|^\frac{1}{2}} \exp \left [- \frac{1}{2} (\mathbf{x} -\boldsymbol{\mu})^t\boldsymbol{\Sigma}^{-1} (\mathbf{x} -\boldsymbol{\mu}) \right] $
where x is a d-component column vector, μ is the d-component mean vector, Σ is the d-by-d covariance matrix, and |Σ| and Σ-1<\sup> are its determinant and inverse respectively. Also, (x - μ)t<\sup> denotes the transpose of (x - μ).
and
$ \boldsymbol{\Sigma} = \mathcal{E} \left [(\mathbf{x} - \boldsymbol{\mu})(\mathbf{x} - \boldsymbol{\mu})^t \right] = \int(\mathbf{x} - \boldsymbol{\mu})(\mathbf{x} - \boldsymbol{\mu})^t p(\mathbf{x})\, dx $
where the expected value of a vector or a matrix is found by taking the expected value of the individual components. i.e if xi<\sub> is the ith component of x, μi<\sub> the ith component of μ, and σij the ijth component of Σ, then
$ \mu_i = \mathcal{E}[x_i] $
and
$ \sigma_ij = \mathcal{E}[(x_i - \mu_i)(x_j - \mu_j)] $
