# algorithms.clustering.ggmixture¶

## Module: algorithms.clustering.ggmixture¶

Inheritance diagram for nipy.algorithms.clustering.ggmixture: One-dimensional Gamma-Gaussian mixture density classes : Given a set of points the algo provides approcumate maximum likelihood estimates of the mixture distribution using an EM algorithm.

Author: Bertrand Thirion and Merlin Keller 2005-2008

## Classes¶

### GGGM¶

class nipy.algorithms.clustering.ggmixture.GGGM(shape_n=1, scale_n=1, mean=0, var=1, shape_p=1, scale_p=1, mixt=array([ 0.33333333, 0.33333333, 0.33333333]))

Bases: object

The basic one dimensional Gamma-Gaussian-Gamma Mixture estimation class, where the first gamma has a negative sign, while the second one has a positive sign.

7 parameters are used: - shape_n: negative gamma shape - scale_n: negative gamma scale - mean: gaussian mean - var: gaussian variance - shape_p: positive gamma shape - scale_p: positive gamma scale - mixt: array of mixture parameter (weights of the n-gamma,gaussian and p-gamma)

__init__(shape_n=1, scale_n=1, mean=0, var=1, shape_p=1, scale_p=1, mixt=array([ 0.33333333, 0.33333333, 0.33333333]))

Constructor

Parameters: shape_n : float, optional scale_n: float, optional parameters of the nehative gamma; must be positive mean : float, optional var : float, optional parameters of the gaussian ; var must be positive shape_p : float, optional scale_p : float, optional parameters of the positive gamma; must be positive mixt : array of shape (3,), optional the mixing proportions; they should be positive and sum to 1
Estep(x)

Update probabilistic memberships of the three components

Parameters: x: array of shape (nbitems,) the input data z: ndarray of shape (nbitems, 3) probabilistic membership

Notes

z[0,:] is the membership the negative gamma z[1,:] is the membership of the gaussian z[2,:] is the membership of the positive gamma

Mstep(x, z)

Mstep of the estimation: Maximum likelihood update the parameters of the three components

Parameters: x: array of shape (nbitem,) input data z: array of shape (nbitems,3) probabilistic membership
component_likelihood(x)

Compute the likelihood of the data x under the three components negative gamma, gaussina, positive gaussian

Parameters: x: array of shape (nbitem,) the data under evaluation ng,y,pg: three arrays of shape(nbitem) The likelihood of the data under the 3 components
estimate(x, niter=100, delta=0.0001, bias=0, verbose=0, gaussian_mix=0)

Whole EM estimation procedure:

Parameters: x: array of shape (nbitem) input data niter: integer, optional max number of iterations delta: float, optional increment in LL at which convergence is declared bias: float, optional lower bound on the gaussian variance (to avoid shrinkage) gaussian_mix: float, optional if nonzero, lower bound on the gaussian mixing weight (to avoid shrinkage) verbose: 0, 1 or 2 verbosity level z: array of shape (nbitem, 3) the membership matrix
init(x, mixt=None)

initialization of the differnt parameters

Parameters: x: array of shape(nbitems) the data to be processed mixt : None or array of shape(3), optional prior mixing proportions. If None, the classes have equal weight
init_fdr(x, dof=-1, copy=True)

Initilization of the class based on a fdr heuristic: the probability to be in the positive component is proportional to the ‘positive fdr’ of the data. The same holds for the negative part. The point is that the gamma parts should model nothing more that the tails of the distribution.

Parameters: x: array of shape(nbitem) the data under consideration dof: integer, optional number of degrees of freedom if x is thought to be a student variate. By default, it is handeled as a normal copy: boolean, optional If True, copy the data.
parameters()

Print the parameters

posterior(x)

Compute the posterior probability of the three components given the data

Parameters: x: array of shape (nbitem,) the data under evaluation ng,y,pg: three arrays of shape(nbitem) the posteriori of the 3 components given the data

Notes

ng + y + pg = np.ones(nbitem)

show(x, mpaxes=None)

Visualization of mixture shown on the empirical histogram of x

Parameters: x: ndarray of shape (nditem,) data mpaxes: matplotlib axes, optional axes handle used for the plot if None, new axes are created.

### GGM¶

class nipy.algorithms.clustering.ggmixture.GGM(shape=1, scale=1, mean=0, var=1, mixt=0.5)

Bases: object

This is the basic one dimensional Gaussian-Gamma Mixture estimation class Note that it can work with positive or negative values, as long as there is at least one positive value. NB : The gamma distribution is defined only on positive values.

5 scalar members - mean: gaussian mean - var: gaussian variance (non-negative) - shape: gamma shape (non-negative) - scale: gamma scale (non-negative) - mixt: mixture parameter (non-negative, weight of the gamma)

__init__(shape=1, scale=1, mean=0, var=1, mixt=0.5)
Estep(x)

E step of the estimation: Estimation of ata membsership

Parameters: x: array of shape (nbitems,) input data z: array of shape (nbitems, 2) the membership matrix
Mstep(x, z)

Mstep of the model: maximum likelihood estimation of the parameters of the model

Parameters: x : array of shape (nbitems,) input data z array of shape(nbitrems, 2) the membership matrix
estimate(x, niter=10, delta=0.0001, verbose=False)

Complete EM estimation procedure

Parameters: x : array of shape (nbitems,) the data to be processed niter : int, optional max nb of iterations delta : float, optional criterion for convergence verbose : bool, optional If True, print values during iterations LL, float average final log-likelihood
parameters()

print the paramteres of self

posterior(x)

Posterior probability of observing the data x for each component

Parameters: x: array of shape (nbitems,) the data to be processed y, pg : arrays of shape (nbitem) the posterior probability
show(x)

Visualization of the mm based on the empirical histogram of x

Parameters: x : array of shape (nbitems,) the data to be processed

### Gamma¶

class nipy.algorithms.clustering.ggmixture.Gamma(shape=1, scale=1)

Bases: object

Basic one dimensional Gaussian-Gamma Mixture estimation class

Note that it can work with positive or negative values, as long as there is at least one positive value. NB : The gamma distribution is defined only on positive values. 5 parameters are used: - mean: gaussian mean - var: gaussian variance - shape: gamma shape - scale: gamma scale - mixt: mixture parameter (weight of the gamma)

__init__(shape=1, scale=1)
check(x)
estimate(x, eps=1e-07)

ML estimation of the Gamma parameters

parameters()