$L^2$ Optimization in Discrete FIR Estimation: Exploiting State-Space Structure

2013 ◽  
Vol 51 (1) ◽  
pp. 419-441 ◽  
Author(s):  
Yaron Levinson ◽  
Leonid Mirkin
Keyword(s):  
State Space ◽  
Space Structure

scholarly journals Assessment of Local Dynamic Stability in Gait Based on Univariate and Multivariate Time Series

2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
Henryk Josiński ◽  
Adam Świtoński ◽  
Agnieszka Michalczuk ◽  
Piotr Grabiec ◽  
Magdalena Pawlyta ◽  
...  
Keyword(s):  
Time Series ◽  
State Space ◽  
Dynamic Stability ◽  
Multivariate Time Series ◽  
Assessment Method ◽  
Space Structure ◽  
Largest Lyapunov Exponent ◽  
Local Dynamic ◽  
Short Term ◽  
Local Dynamic Stability

The ability of the locomotor system to maintain continuous walking despite very small external or internal disturbances is called local dynamic stability (LDS). The importance of the LDS requires constantly working on different aspects of its assessment method which is based on the short-term largest Lyapunov exponent (LLE). A state space structure is a vital aspect of the LDS assessment because the algorithm of the LLE computation for experimental data requires a reconstruction of a state space trajectory. The gait kinematic data are usually one- or three-dimensional, which enables to construct a state space based on a uni- or multivariate time series. Furthermore, two variants of the short-term LLE are present in the literature which differ in length of a time span, over which the short-term LLE is computed. Both a state space structure and the consistency of the observations based on values of both short-term LLE variants were analyzed using time series representing the joint angles at ankle, knee, and hip joints. The short-term LLE was computed for individual joints in three state spaces constructed on the basis of either univariate or multivariate time series. Each state space revealed walkers’ locally unstable behavior as well as its attenuation in the current stride. The corresponding conclusions made on the basis of both short-term LLE variants were consistent in ca. 59% of cases determined by a joint and a state space. Moreover, the authors present an algorithm for estimation of the embedding dimension in the case of a multivariate gait time series.

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