mvnfactory

mvnfactory

Construct a multi-variate normal distribution structure

Syntax

D = mvnfactory(datadim)

Description

D = mvnfactory(datadim) returns a structure representing a datadim-dimensional normal distribution.

Distribution Parameters

  • mu (datadim-by-1 vector) : The mean vector.
  • sigma (datadim-by-datadim matrix) : The covariance matrix (or the variance, $\sigma^2$, when datadim=1).

Probability Density Function

The distribution has the following density:

$$ f(x;\mu,\Sigma)=
(2\pi)^{-\frac{n}{2}}|\Sigma|^{-\frac{1}{2}}
\exp\left(-\frac{1}{2}({x}-{\mu})^T{\Sigma}^{-1}({x}-{\mu})
\right) $$

where $n$ is the data space dimensions, $\mu$ is the mean vector and $\Sigma$ is the covariance matrix.

Example

% Construct a bivariate normal distribution:
D = mvnfactory(2);
% Build a parameter structure for it:
theta = struct('mu', [0; 0], 'sigma', [2 0; 0 2]);
% Plot the PDF:
x = -5:0.2:5;
y = -5:0.2:5;
[X, Y] = meshgrid(x, y);
data = [X(:) Y(:)]';
f = D.pdf(theta, data);
surf(X, Y, reshape(f, size(X)));

name

See distribution structure common members.

Flag to control the memory usage (resulting code will be slower)

M

See distribution structure common members.

dim

See distribution structure common members.

datadim

See distribution structure common members.

ll

See distribution structure common members.

llvec

See distribution structure common members.

llgrad

See distribution structure common members.

llgraddata

See distribution structure common members.

cdf

See distribution structure common members.

pdf

See distribution structure common members.

sample

See distribution structure common members.

randparam

See distribution structure common members.

init

See distribution structure common members.

estimatedefault

Default estimation function for multi-variate normal distribution. This function implements the maximum likelihood method.

Syntax

theta = D.estimatedefault(data)
theta = D.estimatedefault(data, options)
[theta, D] = D.estimatedefault(...)
[theta, D, info] = D.estimatedefault(...)
[theta, D, info, options] = D.estimatedefault(...)

Description

theta = D.estimatedefault(data) returns estimated parameters for the distribution D, using data.

theta = D.estimatedefault(data, options) utilizes applicable options from the options structure in the estimation procedure.

[theta, D] = D.estimatedefault(...) also returns D, the distribution structure for which theta is applicable. (This is the same as the distribution structure D from which you called estimate, and so it should not normally be used. The purpose of including it in the output is to maintain compatibility with other estimation functions).

[theta, D, info] = D.estimatedefault(...) also returns info, a structure array containing information about successive iterations performed by iterative estimation functions.

[theta, D, info, options] = D.estimatedefault(...) also returns the effective options used, so you can see what default values the function used on top of the options you possibly specified.

For information about the output theta, see Distribution Parameters Structure. The input argument data is described in Data Input Argument to Functions. You may also want to read about options or info arguments.

Available Options

Currently no options are available for this function.

Returned info fields

The method used is not iterative and so the returned info is empty.

Example

% create a Gaussian distribution
D = mvnfactory(1);
% generate 1000 random data points
data = randn(1,1000) .* 2 + 1;
% estimate distribution parameters to fit the data
theta = D.estimatedefault(data)

penalizerparam

See distribution structure common members.

Penalizer Info

The default penalizer for this distribution is the Inverse-Wishart distribution for covariance and Normal distribution for mean

Inverse-Wishart prior on covariance has the following form:

$$ f(\Sigma) =
|\Sigma|^{-(nu+d+1)/2} \exp(-0.5 trace(\Sigma^{-1} invLambda)) $$

where

  • nu (scalar) : Degrees of freedom.
  • invLambda (datadim-by-datadim matrix) : The inverse scale matrix.

Normal prior on the mean has the following form:

$$ f(\mu|\Sigma) =
|\Sigma|^{-1/2} exp(-kappa/2 (\mu-\mu_p)^T \Sigma^{-1} (\mu-\mu_p)) $$

where

  • mu_p (datadim-by-1 vector) : The mean vector.
  • kappa (scalar) : the shrinkage parameter

penalizercost

See distribution structure common members.

penalizergrad

See distribution structure common members.

sumparam

See distribution structure common members.

scaleparam

See distribution structure common members.

sumgrad

See distribution structure common members.

scalegrad

See distribution structure common members.

entropy

See distribution structure common members.

kl

See distribution structure common members.

AICc

See distribution structure common members.

BIC

See distribution structure common members.

display

See distribution structure common members.

selfsplit

See distribution structure common members.

selfmerge

See distribution structure common members.

visualize

Syntax

handle_array = D.visualize(D, theta, vis_options)