Statistical Inference for Curved Fibrous Objects in 3D – Based on Multiple Short Observations of Multivariate Autoregressive Processes

Abstract

SummaryThis paper deals with statistical inference on the parameters of a stochastic model, describing curved fibrous objects in three dimensions, that is based on multivariate autoregressive processes. The model is fitted to experimental data consisting of a large number of short independently sampled trajectories of multivariate autoregressive processes. We discuss relevant statistical properties (e.g. asymptotic behaviour as the number of trajectories tends to infinity) of the maximum likelihood (ML) estimators for such processes. Numerical studies are also performed to analyse some of the more intractable properties of the ML estimators. Finally the whole methodology, i.e., the fibre model and its statistical inference, is applied to appropriately describe the tracking of fibres in real materials.

Stochastic 3D modeling of non-woven materials with wet-proofing agent

“…Therefore, we first transform the polygonal tracks to the incremental representation, see Section 3.1. Then, the parameters of {Y i , i ∈ Z} are estimated by the maximumlikelihood technique, introduced in [10] and further analyzed in [8]. Following the AIC criterion, which is widely applied to estimate the order q of autoregres-sive processes, see also [8], we obtain q = 2 which yields Note that the significant degree of cross-correlations expressed by the nondiagonal entries of A 1 , A 2 and Σ justifies the necessity to use multivariate (i.e., multi-dimensional) time series instead of univariate (i.e., one-dimensional) ones.…”

“…Then, the parameters of {Y i , i ∈ Z} are estimated by the maximumlikelihood technique, introduced in [10] and further analyzed in [8]. Following the AIC criterion, which is widely applied to estimate the order q of autoregres-sive processes, see also [8], we obtain q = 2 which yields Note that the significant degree of cross-correlations expressed by the nondiagonal entries of A 1 , A 2 and Σ justifies the necessity to use multivariate (i.e., multi-dimensional) time series instead of univariate (i.e., one-dimensional) ones. We also remark that the diagonal entries in the matrices A 1 and A 2 describe the dependency between the individual components (length, azimuth angle, polar angle) and the corresponding components of the previous two line segments, whereas the non-diagonal entries indicate dependencies between the individual components of the current line segment and other components of previous line segments, e.g.…”