Conditional independence graphs for multivariate autoregressive models by convex optimization: Efficient algorithms☆
Section snippets
Motivation
In recent years, there is a very serious interest in the signal processing community for novel methods in graphical modeling of time series (see, for example, [1], [2]). The newly proposed algorithms are designed to have a relatively low computational complexity and, at the same time, to guarantee good estimation performance when a particular set of assumptions is satisfied. The use of convex optimization is deemed to be very promising in finding graphical models for multivariate time series [3]
Two-stage approach
Instead of the Full-Search from [3], we propose the two-stage approach, which is described in Algorithm 1. Remark that the stages for our estimation procedure are different from those in [6]. In our case, in Stage1, an is employed to select the best order, say (see Section 3.1). The most important consequence is that, in Stage2, all the estimations are performed only for order and not for all the orders within the set . Algorithm 1 Two-Stage MethodStage1 [Select ]: for all
Main algorithmic steps and their computational complexity
Stage 1. This is a classical problem in signal processing and for solving it we employ the ARFIT algorithm which guarantees that the complexity of computing the estimates for all considered orders is (see [24, Section 3.3]). We note in passing that, in the ARFIT algorithm, VAR is recast in the form of a linear regression model (see also A.1). The estimates of the parameters are obtained by solving downdating least squares problems in which the order of the model is decreased from
Model selection rules
For selecting the VAR-order we do not use only the and criteria, but also [15], [30], – final prediction error [31], – Kullback Information Criterion [32], - “corrected” [33]. Note that in [3] only , and have been considered. We emphasize that the formula of we use in Stage1 is not the same with the one from [3]. We prefer the from [30] because its derivation is tailored to VAR-models.
Another aspect concerns the alteration of
Final remarks
In this paper, we have proposed a family of algorithms for inferring the conditional independence graph of a for K-variate time series. Our theoretical and empirical results demonstrate that the algorithms from this family can be used when , and . Thus far, the methods which rely on convex optimization and do not ask the user to make subjective choice of parameters have been suitable only for much smaller values of p and K. Another important feature of our method is
References (48)
- et al.
Identification of synaptic connections in neural ensembles by graphical models
J. Neurosci. Methods
(1997) - et al.
Model selection by sequentially normalized least squares
J. Multivar. Anal.
(2010) - et al.
Variable selection in linear regressionseveral approaches based on normalized maximum likelihood
Signal Process.
(2011) - et al.
An efficient approach to graphical modeling of time series
IEEE Trans. Signal Process.
(2015) Learning the conditional independence structure of stationary time seriesa multitask learning approach
IEEE Trans. Signal Process.
(2015)- et al.
Graphical models of autoregressive processes
Graphical interaction models for multivariate time series
Metrika
(2000)Remarks concerning graphical models for time series and point processes
Rev. De. Econom.
(1996)- R. Davis, P. Zang, T. Zheng, Sparse vector autoregressive modeling, . Comput. Graphical Stat.,...
- et al.
Learning graphical models for stationary time series
IEEE Trans. Signal Process.
(2004)
Gaussian Markov distributions over finite graphs
Ann. Stat.
A test statistic for graphical modelling of multivariate time series
Biometrika
New Introduction to Multiple Time Series Analysis
Graphical Models
Estimating the dimension of a model
Ann. Stat.
A new look at the statistical model identification
IEEE Trans. Autom. Control
Time Series: Theory and Methods
Topology selection in graphical models of autoregressive processes
J. Mach. Learn. Res.
ARMA identification of graphical models
IEEE Trans. Autom. Control
AR identification of latent-variable graphical models
IEEE Trans. Autom. Control
MDL denoising
IEEE Trans. Inf. Theory
Information and Complexity in Statistical Modeling
Cited by (0)
- ☆
This work was supported by Dept. of Statistics (UOA) Doctoral Scholarship.