Multivariate Gaussian

Data Type: Sequence[float]

The probability density function of a d-dimensional multivariate Gaussian random variable \((X_1, X_2, ..., X_d)\) with mean \(\boldsymbol{\mu}\) and postive definite covariance matrix \(\Sigma\) is given by

\[f(\boldsymbol{x} | \boldsymbol{\mu}, \Sigma) = \left(\frac{1}{2\pi}\right)^{d/2} \vert\Sigma\vert^{d/2} \exp{\left(-frac{1}{2}\left(\boldsymbol{x} - \boldsymbol{\mu}\right)^{t} \Sigma^{-1} \left(\boldsymbol{x} - \boldsymbol{\mu}\right)\right)}.\]

If you are assuming \(Cov(X_i, X_j) = 0 \; i \neq j\), a faster and more efficient option is the Diagonal Multivariate Gaussian.

For more info see Multivariate Normal Distribution.

MultivariateGaussianDistribution

class dml.stats.mvn.MultivariateGaussianDistribution(mu, covar, name=None, keys=None)

MultivariateGaussianDistribution object for multivariate Gaussian with mean mu and covaraince ‘covar’.

dim

N is the dim of multivariate normal.

Type:: int

mu

Length N numpy array

Type:: np.ndarray

covar

N by N numpy array for Covariance matrix.

Type:: np.ndarray

chol

Cholesky decomposition of covar.

Type:: np.ndarray

lower

Flag for lower (False for upper)

Type:: bool

name

Set name to object.

Type:: Optional[str]

keys

Set keys for distribution.

Type:: Optional[str]

self.use_lstsq

Cholesky does not exist so use least squares approx.

Type:: bool

self.chol_const

det from covar if lstsq is to be used.

Type:: float

__init__(mu, covar, name=None, keys=None)

MultivariateGaussianDistribution object.

Parameters:

mu (Union[List[float], np.ndarray]) – N-dimensional mean.
covar (Union[List[List[float]], np.ndarray]) – Covariance matrix, should be N by N and positive definite.
name (Optional[str]) – Set name to object.
keys (Optional[str]) – Set keys for distribution.

density(x)

Evaluate the density at x.

Parameters:: x (np.ndarray) – Observation from multivariate Gaussian distribution.
Returns:: Density at x.
Return type:: float

dist_to_encoder()

Create DataSequenceEncoder object for SequenceEncodableProbabilityDistribution instance.

Return type:: MultivariateGaussianDataEncoder
Returns:: DataSequenceEncoder

estimator(pseudo_count=None)

Create a ParameterEstimator for corresponding SequenceEncodableProbabilityDistribution.

Parameters:: pseudo_count (Optional[float]) – Regularize sufficient statistics in estimation step.
Returns:: ParameterEstimator

log_density(x)

Evaluate the log-density at x.

Parameters:: x (np.ndarray) – Observation from multivariate Gaussian distribution.
Returns:: Log-density at x.
Return type:: float

sampler(seed=None)

Create a DistributionSampler object for a given ProbabilityDistribution.

Parameters:: seed (Optional[int]) – Set seed for drawing samples from distribution.

seq_log_density(x)

Vectorized evaluation of the log density.

Parameters:: x (EncodedDataSequence) – EncodedDataSequence for corresponding SequenceEncodedProbabilityDistribution.
Return type:: ndarray
Returns:: np.ndarray

MultivariateGaussianEstimator

class dml.stats.mvn.MultivariateGaussianEstimator(dim=None, pseudo_count=(None, None), suff_stat=(None, None), name=None, keys=None)

MultivariateGaussianEstimator object for estimating multivariate normal distribution from sufficient stats.

dim

Dimension of multivariate normal.

Type:: int

pseudo_count

Regularize mean and/or covariance.

Type:: Optional[Tuple[Optional[float], Optional[float]]]

prior_mu

Mean from prior data or used to regularize.

Type:: Optional[np.ndarray]

prior_covar

Covariance matrix from prior data or used to regularize.

Type:: Optional[np.ndarray]

name

Set name to object.

Type:: Optional[str]

keys

Keys for merging sufficient statistics.

Type:: Optional[str]

__init__(dim=None, pseudo_count=(None, None), suff_stat=(None, None), name=None, keys=None)

MultivariateGaussianEstimator object.

Parameters:

dim (Optional[int]) – Dimension of multivariate normal. Inferred from ‘suff_stat’ if None.
pseudo_count (Optional[Tuple[Optional[float], Optional[float]]]) – Regularize mean and/or covariance.
suff_stat (Optional[Tuple[Optional[np.ndarray], Optional[np.ndarray]]]) – Mean and covariance estimated from previous data or used to regularize.
name (Optional[str]) – Set name for object instance.
keys (Optional[str]) – Set keys for estimator.

accumulator_factory()

Create SequenceEncodableStatisticAccumulator object.

Return type:: MultivariateGaussianAccumulatorFactory

estimate(nobs, suff_stat)

Estimate SequenceEncodableProbabilityDistribution for sufficient statistics.

Parameters:

nobs (Optional[float]) – Weighted number of observations.
suff_stat (Tuple[int, np.ndarray, np.ndarray, np.ndarray]) – Sufficient statistics for dirichlet distribution.

Return type:

MultivariateGaussianDistribution

Returns:

SequenceEncodableProbabilityDistribution

MultivariateGaussianSampler

class dml.stats.mvn.MultivariateGaussianSampler(dist, seed=None)

MultivariateGaussianSampler object for sampling from MultivariateGaussianDistribution.

rng

Sets seed for generating samples.

Type:: RandomState

dist

MultivariateGaussianDistribution to sample from.

Type:: MultivariateGaussianDistribution

sample(size=None)

Generate samples from MultivariateGaussianDistribution.

Parameters:: size (Optional[int]) – Number of samples to generate.
Returns:: Size by dim number of samples.
Return type:: np.ndarray