TransportMaps.Distributions.Examples.InertialNavigationSystem.INSDistributions
¶
Module Contents¶
Classes¶
Multivariate Gaussian distribution \(\mathcal{N}(\mu,\Sigma)\) |
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Transition probability distribution \(g({\bf x}_{k-1},{\bf x}_k) = \pi_{{\bf X}_k \vert {\bf X}_{k-1}={\bf x}_{k-1}}({\bf x}_k) = \pi({\bf x}_k - F_k {\bf x}_{k-1} - {\bf c}_k)\) where \(\pi \sim \mathcal{N}(\mu_k,Q_k)\). |
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Define the log-likelihood for the additive linear Gaussian model |
Functions¶
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- class TransportMaps.Distributions.Examples.InertialNavigationSystem.INSDistributions.Prior[source]¶
Bases:
TransportMaps.Distributions.FrozenDistributions.NormalDistribution
Multivariate Gaussian distribution \(\mathcal{N}(\mu,\Sigma)\)
- Parameters:
mu (
ndarray
[\(d\)]) – mean vector \(\mu\)covariance (
ndarray
[\(d,d\)]) – covariance matrix \(\Sigma\)precision (
ndarray
[\(d,d\)]) – precision matrix \(\Sigma^{-1}\)square_root_covariance (
ndarray
[\(d,d\)]) – square root \(\Sigma^{\frac{1}{2}}\)square_root_precision (
ndarray
[\(d,d\)]) – square root \(\Sigma^{-\frac{1}{2}}\)square_root_type (str) – type of square root to be used in case
covariance
orprecision``were provided. For ``square_root_type=='sym'
, \(L=U\Lambda^{\frac{1}{2}}U^T\) where \(\Sigma = U\Lambda U^T\) is the eigenvalue decomposition of \(\Sigma\). Forsquare_root_type=='tri'
orsquare_root_type=='chol'
, :maht:`L=C` where \(\Sigma=CC^T\) is the Cholesky decomposition of \(\Sigma\). Forsquare_root_type=='kl'
, \(L=U\Lambda^{\frac{1}{2}}\) where \(\Sigma = U\Lambda U^T\) is the eigenvalue decomposition of \(\Sigma\) (this corresponds to the Karuenen-Loeve expansion). The eigenvalues and eigenvectors are ordered with \(\lambda_i\geq\lambda_{i+1}\). If the parametersquare_root
is provided, then thesquare_root_type
attribute will be setuser
.
Note
The arguments
covariance
,precision
andsquare_root
are mutually exclusive.
- class TransportMaps.Distributions.Examples.InertialNavigationSystem.INSDistributions.Transition[source]¶
-
Transition probability distribution \(g({\bf x}_{k-1},{\bf x}_k) = \pi_{{\bf X}_k \vert {\bf X}_{k-1}={\bf x}_{k-1}}({\bf x}_k) = \pi({\bf x}_k - F_k {\bf x}_{k-1} - {\bf c}_k)\) where \(\pi \sim \mathcal{N}(\mu_k,Q_k)\).
This represents the following Markov transition model:
\[\begin{split}{\bf x}_k = c_k + F_k {\bf x}_{k-1} + {\bf w}_k \\ {\bf w}_k \sim \mathcal{N}(\mu,Q_k)\end{split}\]where the control \({\bf c}_k := B_k {\bf u}_k\) can be used for control purposes
- Parameters:
ck (
ndarray
[\(d\)] orMap
) – constant part or map returning the constant part given some parametersFk (
ndarray
[\(d,d\)] orMap
) – state transition matrix (dynamics) or map returning the linear part given some parametrsmu (
ndarray
[\(d\)] orMap
) – mean \(\mu_k\) or parametric map for \(\mu_k(\theta)\)covariance (
ndarray
[\(d,d\)] orMap
) – covariance \(Q_k\) or parametric map for \(Q_k(\theta)\)precision (
ndarray
[\(d,d\)] orMap
) – precision \(Q_k^{-1}\) or parametric map for \(Q_k^{-1}(\theta)\)
- class TransportMaps.Distributions.Examples.InertialNavigationSystem.INSDistributions.LogLikelihood(y)[source]¶
Bases:
TransportMaps.Likelihoods.LikelihoodBase.AdditiveLinearNormalLogLikelihood
Define the log-likelihood for the additive linear Gaussian model
The model is
\[{\bf y} = {\bf c} + {\bf T}{\bf x} + \varepsilon \;, \quad \varepsilon \sim \mathcal{N}(\mu, \Sigma)\]where \(T \in \mathbb{R}^{d_y \times d_x}\), \(\mu \in \mathbb{R}^{d_y}\) and \(\Sigma \in \mathbb{R}^{d_y \times d_y}\) is symmetric positve definite
- Parameters:
y (
ndarray
[\(d_y\)]) – datac (
ndarray
[\(d_y\)] orMap
) – system constant or parametric map returning the constantT (
ndarray
[\(d_y,d_x\)] orMap
) – system matrix or parametric map returning the system matrixmu (
ndarray
[\(d_y\)] orMap
) – noise mean or parametric map returning the meancovariance (
ndarray
[\(d_y,d_y\)] orMap
) – noise covariance or parametric map returning the covarianceprecision (
ndarray
[\(d_y,d_y\)] orMap
) – noise precision matrix or parametric map returning the precision matrix