# TransportMaps.Distributions.Examples.InertialNavigationSystem.INSDistributions¶

## Module Contents¶

### Classes¶

 Prior Multivariate Gaussian distribution $$\mathcal{N}(\mu,\Sigma)$$ Transition 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 $$\pi \sim \mathcal{N}(\mu_k,Q_k)$$. LogLikelihood Define the log-likelihood for the additive linear Gaussian model

### Functions¶

 generate_data(nsteps)

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 or precisionwere 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 $$\Sigma$$. For square_root_type=='tri' or square_root_type=='chol', :maht:L=C where $$\Sigma=CC^T$$ is the Cholesky decomposition of $$\Sigma$$. For square_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 parameter square_root is provided, then the square_root_type attribute will be set user.

Note

The arguments covariance, precision and square_root are mutually exclusive.

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 [$$d$$] or Map) – constant part or map returning the constant part given some parameters

• Fk (ndarray [$$d,d$$] or Map) – state transition matrix (dynamics) or map returning the linear part given some parametrs

• mu (ndarray [$$d$$] or Map) – mean $$\mu_k$$ or parametric map for $$\mu_k(\theta)$$

• covariance (ndarray [$$d,d$$] or Map) – covariance $$Q_k$$ or parametric map for $$Q_k(\theta)$$

• precision (ndarray [$$d,d$$] or Map) – precision $$Q_k^{-1}$$ or parametric map for $$Q_k^{-1}(\theta)$$

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 [$$d_y$$]) – data

• c (ndarray [$$d_y$$] or Map) – system constant or parametric map returning the constant

• T (ndarray [$$d_y,d_x$$] or Map) – system matrix or parametric map returning the system matrix

• mu (ndarray [$$d_y$$] or Map) – noise mean or parametric map returning the mean

• covariance (ndarray [$$d_y,d_y$$] or Map) – noise covariance or parametric map returning the covariance

• precision (ndarray [$$d_y,d_y$$] or Map) – noise precision matrix or parametric map returning the precision matrix