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Probability Distributions

Summary

The DynaML dynaml.probability.distributions package leverages and extends the breeze.stats.distributions package. Below is a list of distributions implemented.

Specifying Distributions

Every probability density function \rho(x) defined over some domain x \in \mathcal{X} can be represented as \rho(x) = \frac{1}{Z} f(x), where f(x) is the un-normalized probability weight and Z is the normalization constant. The normalization constant ensures that the density function sums to 1 over the whole domain \mathcal{X}.

Describing Skewness

An important analytical way to create skewed distributions was described by Azzalani et. al. It consists of four components.

  • A symmetric probability density \varphi(.)
  • An odd function w(.)
  • A cumulative distribution function G(.) of some symmetric density
  • A cut-off parameter \tau
\rho(x) = \frac{1}{G(\tau)} \times \varphi(x)\times G(w(x) + \tau)

Distributions API

The Density[T] and Rand[T] traits form the API entry points for implementing probability distributions in breeze. In the dynaml.probability.distributions package, these two traits are inherited by GenericDistribution[T] which is extended by AbstractContinuousDistr[T] and AbstractDiscreteDistr[T] classes.

Distributions which can produce confidence intervals

The trait HasErrorBars[T] can be used as a mix in to provide the ability of producing error bars to distributions. To extend it, one has to implement the confidenceInterval(s: Double): (T, T) method.

Skewness

The SkewSymmDistribution[T] class is the generic base implementations for skew symmetric family of distributions in DynaML.

Distributions Library

Apart from the distributions defined in the breeze.stats.distributions, users have access to the following distributions implemented in the dynaml.probability.distributions.

Multivariate Students T

Defines a Students' T distribution over the domain of finite dimensional vectors.

\mathcal{X} \equiv \mathbb{R}^{n}

f(x) = \left[1+{\frac {1}{\nu }}({\mathbf {x} }-{\boldsymbol {\mu }})^{\rm {T}}{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right]^{-(\nu +p)/2}

Z = \frac{\Gamma \left[(\nu +p)/2\right]}{\Gamma (\nu /2)\nu ^{p/2}\pi ^{p/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}

Usage:

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val mu = 2.5
val mean = DenseVector(1.0, 0.0)
val cov = DenseMatrix((1.5, 0.5), (0.5, 2.5))
val d = MultivariateStudentsT(mu, mean, cov)

Matrix T

Defines a Students' T distribution over the domain of matrices.

\mathcal{X} \equiv \mathbb{R}^{n \times p}

f(x) = \left|{\mathbf {I}}_{n}+{\boldsymbol \Sigma }^{{-1}}({\mathbf {X}}-{\mathbf {M}}){\boldsymbol \Omega }^{{-1}}({\mathbf {X}}-{\mathbf {M}})^{{{\rm {T}}}}\right|^{{-{\frac {\nu +n+p-1}{2}}}}

Z = {\frac {\Gamma_{p}\left({\frac {\nu +n+p-1}{2}}\right)}{(\pi )^{{\frac {np}{2}}}\Gamma _{p}\left({\frac {\nu +p-1}{2}}\right)}}|{\boldsymbol \Omega }|^{{-{\frac {n}{2}}}}|{\boldsymbol \Sigma }|^{{-{\frac {p}{2}}}}

Usage:

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val mu = 2.5
val mean = DenseMatrix((-1.5, -0.5), (3.5, -2.5))
val cov_rows = DenseMatrix((1.5, 0.5), (0.5, 2.5))
val cov_cols = DenseMatrix((0.5, 0.1), (0.1, 1.5))
val d = MatrixT(mu, mean, cov_rows, cov_cols)

Matrix Normal

Defines a Gaussian distribution over the domain of matrices.

\mathcal{X} \equiv \mathbb{R}^{n \times p}

f(x) = \exp\left( -\frac{1}{2} \, \mathrm{tr}\left[ \mathbf{V}^{-1} (\mathbf{X} - \mathbf{M})^{T} \mathbf{U}^{-1} (\mathbf{X} - \mathbf{M}) \right] \right)

Z = (2\pi)^{np/2} |\mathbf{V}|^{n/2} |\mathbf{U}|^{p/2}

Usage:

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val mean = DenseMatrix((-1.5, -0.5), (3.5, -2.5))
val cov_rows = DenseMatrix((1.5, 0.5), (0.5, 2.5))
val cov_cols = DenseMatrix((0.5, 0.1), (0.1, 1.5))
val d = MatrixNormal(mean, cov_rows, cov_cols)

Truncated Normal

Defines a univariate Gaussian distribution that is defined in a finite domain.

\mathcal{X} \equiv [a, b]

f(x) = \begin{cases} \phi ({\frac {x-\mu }{\sigma }}) & a \leq x \leq b\\0 & else\end{cases}

Z = \sigma \left(\Phi ({\frac {b-\mu }{\sigma }})-\Phi ({\frac {a-\mu }{\sigma }})\right)

\phi() and \Phi() being the gaussian density function and cumulative distribution function respectively

Usage:

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val mean = 1.5
val sigma = 1.5
val (a,b) = (-0.5, 2.5)
val d = TruncatedGaussian(mean, sigma, a, b)

Skew Gaussian

Univariate

\mathcal{X} \equiv \mathbb{R}

f(x) = \phi(\frac{x - \mu}{\sigma}) \Phi(\alpha (\frac{x-\mu}{\sigma}))

Z = \frac{1}{2}

\phi() and \Phi() being the standard gaussian density function and cumulative distribution function respectively

Multivariate

\mathcal{X} \equiv \mathbb{R}^d

f(x) = \phi_{d}(\mathbf{x}; \mathbf{\mu}, {\Sigma}) \Phi(\mathbf{\alpha}^{\intercal} L^{-1}(\mathbf{x} - \mathbf{\mu}))

Z = \frac{1}{2}

\phi_{d}(.; \mathbf{\mu}, {\Sigma}) and \Phi() are the multivariate gaussian density function and standard gaussian univariate cumulative distribution function respectively and L is the lower triangular Cholesky decomposition of \Sigma.

Skewness parameter \alpha

The parameter \alpha determines the skewness of the distribution and its sign tells us in which direction the distribution has a fatter tail. In the univariate case the parameter \alpha is a scalar, while in the multivariate case \alpha \in \mathbb{R}^d, so for the multivariate skew gaussian distribution, there is a skewness value for each dimension.

Usage:

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//Univariate
val mean = 1.5
val sigma = 1.5
val a = -0.5
val d = SkewGaussian(a, mean, sigma)

//Multivariate
val mu = DenseVector.ones[Double](4)
val alpha = DenseVector.fill[Double](4)(1.2)
val cov = DenseMatrix.eye[Double](4)*1.5
val md = MultivariateSkewNormal(alpha, mu, cov)

Extended Skew Gaussian

Univariate

The generalization of the univariate skew Gaussian distribution.

\mathcal{X} \equiv \mathbb{R}

f(x) = \phi(\frac{x - \mu}{\sigma}) \Phi(\alpha (\frac{x-\mu}{\sigma}) + \tau\sqrt{1 + \alpha^{2}})

Z = \Phi(\tau)

\phi() and \Phi() being the standard gaussian density function and cumulative distribution function respectively

Multivariate

\mathcal{X} \equiv \mathbb{R}^d

f(x) = \phi_{d}(\mathbf{x}; \mathbf{\mu}, {\Sigma}) \Phi(\mathbf{\alpha}^{\intercal} L^{-1}(\mathbf{x} - \mathbf{\mu}) + \tau\sqrt{1 + \mathbf{\alpha}^{\intercal}\mathbf{\alpha}})

Z = \Phi(\tau)

\phi_{d}(.; \mathbf{\mu}, {\Sigma}) and \Phi() are the multivariate gaussian density function and standard gaussian univariate cumulative distribution function respectively and L is the lower triangular Cholesky decomposition of \Sigma.

Usage:

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//Univariate
val mean = 1.5
val sigma = 1.5
val a = -0.5
val c = 0.5
val d = ExtendedSkewGaussian(c, a, mean, sigma)

//Multivariate
val mu = DenseVector.ones[Double](4)
val alpha = DenseVector.fill[Double](4)(1.2)
val cov = DenseMatrix.eye[Double](4)*1.5
val tau = 0.2
val md = ExtendedMultivariateSkewNormal(tau, alpha, mu, cov)

Confusing Nomenclature

The following distribution has a very similar form and name to the extended skew gaussian distribution shown above. But despite its deceptively similar formula, it is a very different object.

We use the name MESN to denote the variant below instead of its expanded form.

MESN

The Multivariate Extended Skew Normal or MESN distribution was formulated by Adcock and Schutes. It is given by

\mathcal{X} \equiv \mathbb{R}^d

f(x) = \phi_{d}(\mathbf{x}; \mathbf{\mu} + \mathbf{\alpha}\tau, {\Sigma} + \mathbf{\alpha}\mathbf{\alpha}^\intercal) \Phi\left(\frac{\mathbf{\alpha}^{\intercal} \Sigma^{-1}(\mathbf{x} - \mathbf{\mu}) + \tau}{\sqrt{1 + \mathbf{\alpha}^{\intercal}\Sigma^{-1}\mathbf{\alpha}}}\right)

Z = \Phi(\tau)

\phi_{d}(.; \mathbf{\mu}, {\Sigma}) and \Phi() are the multivariate gaussian density function and standard gaussian univariate cumulative distribution function respectively.

Usage:

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//Univariate
val mean = 1.5
val sigma = 1.5
val a = -0.5
val c = 0.5
val d = UESN(c, a, mean, sigma)

//Multivariate
val mu = DenseVector.ones[Double](4)
val alpha = DenseVector.fill[Double](4)(1.2)
val cov = DenseMatrix.eye[Double](4)*1.5
val tau = 0.2
val md = MESN(tau, alpha, mu, cov)

Extended Skew Gaussian Process ESGP

The MESN distribution is used to define the finite dimensional probabilities for the ESGP process.

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