Mixed orthogonality graphs for continuous-time state space models and orthogonal projections
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2023
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909285477253120 |
|---|---|
| author | Fasen-Hartmann, Vicky Schenk, Lea |
| author_facet | Fasen-Hartmann, Vicky Schenk, Lea |
| contents | In this paper, we derive (local) orthogonality graphs for the popular continuous-time state space models, including in particular multivariate continuous-time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen-Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well-defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving Lévy process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_04478 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mixed orthogonality graphs for continuous-time state space models and orthogonal projections Fasen-Hartmann, Vicky Schenk, Lea Probability Primary 62H22, 62M20, Secondary 62M10, 60G25 G.3 In this paper, we derive (local) orthogonality graphs for the popular continuous-time state space models, including in particular multivariate continuous-time ARMA (MCARMA) processes. In these (local) orthogonality graphs, vertices represent the components of the process, directed edges between the vertices indicate causal influences and undirected edges indicate contemporaneous correlations between the component processes. We present sufficient criteria for state space models to satisfy the assumptions of Fasen-Hartmann and Schenk (2024a) so that the (local) orthogonality graphs are well-defined and various Markov properties hold. Both directed and undirected edges in these graphs are characterised by orthogonal projections on well defined linear spaces. To compute these orthogonal projections, we use the unique controller canonical form of a state space model, which exists under mild assumptions, to recover the input process from the output process. We are then able to derive some alternative representations of the output process and its highest derivative. Finally, we apply these representations to calculate the necessary orthogonal projections, which culminate in the characterisations of the edges in the (local) orthogonality graph. These characterisations are given by the parameters of the controller canonical form and the covariance matrix of the driving Lévy process. |
| title | Mixed orthogonality graphs for continuous-time state space models and orthogonal projections |
| topic | Probability Primary 62H22, 62M20, Secondary 62M10, 60G25 G.3 |
| url | https://arxiv.org/abs/2311.04478 |