×

zbMATH — the first resource for mathematics

On the second-order random walk model for irregular locations. (English) Zbl 1199.60276
A new model of second-order random walk \(x_1,\dots,x_n\) is proposed for the case when \(x_i\) should represent a smooth curve at locations \(s_i\), \(s_1<\dots<s_n\). The model is based on the idea to interpret \(x_i\) as observations of a continuous process \(x(t)\) at points \(s_i\) where \(x(t)\) is a weak solution to \( {d^2\over dt^2}x(t)={d\over dt}W(t) \), \(W(t)\) being the standard Wienner process. In fact, a Galerkin approximation is used with some simplification to obtain \(x_i\) as a Gaussian Markov random field with the properties \[ E(x_i\;| \;x_{i-1},x_{i-2})= \left( {d_{i-1}\over d_{i-2}}+1 \right)x_{i-1} -{d_{i-1}\over d_{i-2}} x_{i-2}, \]
\[ \text{Var}(x_i\;| \;x_{i-1},x_{i-2})= {d_{i-1}^2(d_{i-2}-d_{i-1})\over 2} \] The random walk is used as a prior in a semiparametric model for Bayesian analysis of an econometric time series.

MSC:
60J25 Continuous-time Markov processes on general state spaces
62F15 Bayesian inference
91B84 Economic time series analysis
Software:
GMRFLib; Fahrmeir
PDF BibTeX Cite
Full Text: DOI
References:
[1] Arnold, Stochastic differential equations; theory and applications (1974) · Zbl 0278.60039
[2] Chib, Inference in semiparametric dynamic models for binary longitudinal data, J. Amer. Statist. Assoc. 101 pp 685– (2006) · Zbl 1119.62337
[3] Fahrmeir, Smoothing and regression: approaches, computation, and application pp 513– (2000)
[4] Fahrmeir, Bayesian inference for generalized additive mixed models based on Markov random field priors, J. Roy. Statist. Soc. Ser. C 50 pp 201– (2001)
[5] Fahrmeir, Multivariate statistical modelling based on generalized linear models (2001) · Zbl 0980.62052
[6] Gradshteyn, Table of intergrals, series, and products (1994) · Zbl 0918.65002
[7] Green, Nonparametric regression and generalized linear models: a roughness penalty approach (1994) · Zbl 0832.62032
[8] Jones, Applied time series analysis II pp 651– (1981)
[9] Kitagawa, Smoothness priors analysis of time series (1996) · Zbl 0853.62069
[10] Lindgren , F. Rue , H. 2004 Intrinsic Gaussian Markov random fields on triangulated spheres Preprints in Mathematical Sciences
[11] Lindgren , F. Rue , H. 2007 Explicit construction of GMRF approximations to generalised Matérn fields on irregular grids Preprints in Mathematical Sciences
[12] Rue, Fast sampling of Gaussian Markov random fields, J. Roy. Statist. Soc. Ser. B 63 pp 325– (2001) · Zbl 0979.62075
[13] Rue, Gaussian Markov random fields: theory and applications (2005) · Zbl 1093.60003
[14] Rue , H. Martino , S. Chopin , N. 2007 Approximate Bayesian inference for latent Gaussian models using integrated nested Laplace approximations Department of mathematical sciences · Zbl 1248.62156
[15] Thomée, Galerkin finite element methods for parabolic problems (1984)
[16] Wahba, Improper priors, spline smoothing and the problem of guarding against model errors in regression, J. Roy. Statist. Soc. Ser. B 40 pp 364– (1978) · Zbl 0407.62048
[17] Wecker, The signal extraction approach to nonlinear regression and spline smoothing, J. Amer. Statist. Assoc. 78 pp 81– (1983) · Zbl 0536.62071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.