AutoregressiveModelSequence

Represents a sequence of multivariate autoregressive (AR) models with increasing order $p$. The AR models should be stored as matrix file in the GROOPS definition of AR models. The required AR models can be computed with CovarianceMatrix2AutoregressiveModel, and passed to this class through inputfileAutoregressiveModel in increasing order.

The main purpose of AutoregressiveModelSequence is to use AR models of the form \[ \label{eq:ar-model} \mathbf{y}_e(t_i) = \sum_{k=1}^p \mathbf{\Phi}^{(p)}_k\mathbf{y}_e(t_{i-k}) + \mathbf{w}(t_i), \hspace{5pt} \mathbf{w}(t_i) \sim \mathcal{N}(0, \mathbf{\Sigma}^{(p)}_\mathbf{w}), \]to create pseudo-observation equations \[ \label{eq:pseudo-observations-transformed} 0 = \bar{\mathbf{\Phi}} \Delta\mathbf{y} + \bar{\mathbf{w}}, \hspace{5pt} \bar{\mathbf{w}} \sim \mathcal{N}(0, \bar{\mathbf{\Sigma}}_{\bar{\mathbf{w}}}), \]with \[ \label{eq:pseudo-observations-ar} \bar{\mathbf{\Phi}} = \begin{bmatrix} \mathbf{I} & & & & & \\ -\mathbf{\Phi}^{(1)}_1 & \mathbf{I} & & & & \\ -\mathbf{\Phi}^{(2)}_2 & -\mathbf{\Phi}^{(2)}_1 & \mathbf{I} & & & \\ -\mathbf{\Phi}^{(3)}_3 & -\mathbf{\Phi}^ {(3)}_2 & -\mathbf{\Phi}^ {(3)}_1 & \mathbf{I} & & \\ & -\mathbf{\Phi}^{(3)}_3 & -\mathbf{\Phi}^ {(3)}_2 & -\mathbf{\Phi}^ {(3)}_1 & \mathbf{I} & \\ & & \ddots & \ddots & \ddots & \ddots \\ \end{bmatrix}, \hspace{15pt} \bar{\mathbf{\Sigma}}_{\bar{\mathbf{w}}} = \bar{\mathbf{\Sigma}}_{\bar{\mathbf{w}}} = \begin{bmatrix} \mathbf{\Sigma}^{(0)}_{\mathbf{w}} & & & & & \\ & \mathbf{\Sigma}^{(1)}_{\mathbf{w}} & & & & \\ & & \mathbf{\Sigma}^{(2)}_{\mathbf{w}} & & & \\ & & & \mathbf{\Sigma}^{(3)}_{\mathbf{w}} & & \\ & & & & \mathbf{\Sigma}^{(3)}_{\mathbf{w}} & \\ & & & & & \ddots \\ \end{bmatrix}. \]used to constrain high-frequency temporal gravity field variations (see KalmanSmootherLeastSquares, NormalsBuildShortTimeStaticLongTime, PreprocessingSst).

The corresponding normal equation coefficient matrix is given by \[ \label{eq:ar-normals} \bar{\mathbf{\Phi}}^T\bar{\mathbf{\Sigma}}^{-1}_{\bar{\mathbf{w}}}\bar{\mathbf{\Phi}} \]and if all AR models are estimated from the same sample its inverse is a block-Toeplitz covariance matrix \[ (\mathbf{\Sigma}_{\mathbf{y}_m})_{ij} = \begin{cases} \mathbf{\Sigma}(|j-i|) & \text{for } i \leq j \\ \mathbf{\Sigma}(|j-i|))^T & \text{otherwise} \end{cases}, \]which can be computed using AutoregressiveModel2CovarianceMatrix.

A detailed description with applications can be found in: Kvas, A., Mayer-Gürr, T. GRACE gravity field recovery with background model uncertainties. J Geod 93, 2543–2552 (2019). https://doi.org/10.1007/s00190-019-01314-1