Premium
Wiener–Kolmogorov Filtering and Smoothing for Multivariate Series With State–Space Structure
Author(s) -
Gómez Víctor
Publication year - 2007
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/j.1467-9892.2006.00514.x
Subject(s) - mathematics , smoothing , covariance , series (stratigraphy) , kolmogorov equations (markov jump process) , kalman filter , state space , wiener filter , projection (relational algebra) , equivalence (formal languages) , multivariate statistics , mathematical analysis , algorithm , discrete mathematics , statistics , differential algebraic equation , paleontology , ordinary differential equation , biology , differential equation
. Wiener–Kolmogorov filtering and smoothing usually deal with projection problems for stochastic processes that are observed over semi‐infinite and doubly infinite intervals. For multivariate stationary series, there exist closed formulae based on covariance generating functions that were first given independently by N. Wiener and A.N. Kolmogorov around 1940. In this article, we consider multivariate series with a state–space structure and, using a new purely algebraic approach to the problem, we prove the equivalence between Wiener–Kolmogorov filtering and Kalman filtering. Up to now, this equivalence has only been partially shown. In addition, we get some new recursions for smoothing and some new recursions to compute the filter weights and the covariance generating functions of the errors. The results are extended to nonstationary series.