naginterfaces.library.mv.gaussian_​mixture

naginterfaces.library.mv.gaussian_mixture(x, nvar, ng, sopt, isx=None, prob=None, niter=15, riter=0, tol=0.0)

gaussian_mixture performs a mixture of Normals (Gaussians) for a given (co)variance structure.

For full information please refer to the NAG Library document for g03ga

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g03/g03gaf.html

Parameters

xfloat, array-like, shape $(n,m)$

$x[\text{i}-1,\text{j}-1]$ must contain the value of the $\text{j}$th variable for the $\text{i}$th object, for $\text{j}=1,2,\dots ,m$, for $\text{i}=1,2,\dots ,n$.

nvarint

$p$, the number of variables included in the calculations.

ngint

$k$, the number of groups in the mixture model.

soptint

Determines the (co)variance structure:

$sopt=1$

Groupwise covariance matrices.

$sopt=2$

Pooled covariance matrix.

$sopt=3$

Groupwise variances.

$sopt=4$

Pooled variances.

$sopt=5$

Overall variance.

isxNone or int, array-like, shape $(:)$, optional

Note: the required length for this argument is determined as follows: if $nvar<m$: $m$; otherwise: $0$.

If $nvar=m$ all available variables are included in the model and $isx$ is not referenced; otherwise the $j$th variable will be included in the analysis if $isx[\text{j}-1]=1$ and excluded if $isx[\text{j}-1]=0$, for $\text{j}=1,2,\dots ,m$.

probNone or float, array-like, shape $(n,ng)$, optional

If $prob$ is not None, $prob[i-1,j-1]$ is the probability that the $i$th object belongs to the $j$th group. (These probabilities are normalised internally.) Otherwise, the initial membership probabilities are set internally.

niterint, optional

The maximum number of iterations.

riterint, optional

If $riter>0$, membership probabilities are rounded to $0.0$ or $1.0$ after the completion of the first $riter$ iterations.

tolfloat, optional

Iterations cease the first time an improvement in log-likelihood is less than $tol$. If $tol\le 0$ a value of ${10}^{-3}$ is used.

Returns

probfloat, ndarray, shape $(n,ng)$

$prob[i-1,j-1]$ is the probability of membership of the $i$th object to the $j$th group for the fitted model.

niterint

The number of completed iterations.

wfloat, ndarray, shape $\left(ng\right)$

$w_j$, the mixing probability for the $j$th group.

gfloat, ndarray, shape $(nvar,ng)$

$g[i-1,j-1]$ gives the estimated mean of the $i$th variable in the $j$th group.

sfloat, ndarray, shape $(:,:,:)$

If $sopt=1$, $s[i-1,j-1,k-1]$ gives the $(i,j)$th element of the $k$th group.

If $sopt=2$, $s[i-1,j-1,0]$ gives the $(i,j)$th element of the pooled covariance.

If $sopt=3$, $s[j-1,k-1,0]$ gives the $j$th variance in the $k$th group.

If $sopt=4$, $s[j-1,0,0]$ gives the $j$th pooled variance.

If $sopt=5$, $s[0,0,0]$ gives the overall variance.

ffloat, ndarray, shape $(n,ng)$

$f[i-1,j-1]$ gives the $p$-variate Normal (Gaussian) density of the $i$th object in the $j$th group.

loglikfloat

The log-likelihood for the fitted mixture model.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$ and $p=⟨value⟩$.

Constraint: $n>p$, the number of parameters, i.e., too few objects have been supplied for the model.

(errno $2$)

On entry, $m=⟨value⟩$.

Constraint: $m\ge 1$.

(errno $5$)

On entry, $nvar=⟨value⟩$ and $m=⟨value⟩$.

Constraint: $1\le nvar\le m$.

(errno $6$)

On entry, $nvar\ne m$ and $isx$ is invalid.

(errno $7$)

On entry, $ng=⟨value⟩$.

Constraint: $ng\ge 1$.

(errno $9$)

On entry, row $⟨value⟩$ of supplied $prob$ does not sum to $1$.

(errno $11$)

On entry, $niter=⟨value⟩$.

Constraint: $niter\ge 1$.

(errno $16$)

On entry, $sopt\ne 1$, $2$, $3$, $4$ or $5$.

(errno $44$)

A covariance matrix is not positive definite, try a different initial allocation.

(errno $45$)

An iteration cannot continue due to an empty group, try a different initial allocation.

Notes

A Normal (Gaussian) mixture model is a weighted sum of $k$ group Normal densities given by,

$$p(x|w,\mu ,\Sigma )=\sum _1^k{w}_{j}g(x|{\mu }_j,{\Sigma }_j)\text{,}x\in R^p$$

where:

$x$ is a $p$-dimensional object of interest;

$w_j$ is the mixture weight for the $j$th group and ${\sum }_1^k{w}_j=1$;

${\mu }_j$ is a $p$-dimensional vector of means for the $j$th group;

${\Sigma }_j$ is the covariance structure for the $j$th group;

$g(\cdot )$ is the $p$-variate Normal density:

$$g(x|{\mu }_j,{\Sigma }_j)=\frac{1}{{\left(2\pi \right)}^{p/2}{\left|{\Sigma }_j\right|}^{1/2}}exp[-\frac{1}{2}(x-{\mu }_j){\Sigma }_j^{-1}{(x-{\mu }_j)}^T]\text{.}$$

Optionally, the (co)variance structure may be pooled (common to all groups) or calculated for each group, and may be full or diagonal.

References

Hartigan, J A, 1975, Clustering Algorithms, Wiley

See also

naginterfaces.library.examples.mv.gaussian_mixture_ex.main()