Optimal State Space Reconstruction via Monte Carlo Decision Tree Search

A novel idea for an optimal time delay state space reconstruction from uni- and multivariate time series is presented. The entire embedding process is considered as a game, in which each move corresponds to an embedding cycle and is subject to an evaluation through an objective function. This way the embedding procedure can be modeled as a tree, in which each leaf holds a specific value of the objective function. By using a Monte Carlo ansatz the proposed algorithm populates the tree with many leafs by computing different possible embedding paths and the final embedding is chosen as that particular path, which ends at the leaf with the lowest achieved value of the objective function. The method aims to prevent getting stuck in a local minimum of the objective function and can be used in a modular way, enabling practitioners to choose a statistic for possible delays in each embedding cycle as well as a suitable objective function themselves. As a proof of concept, we demonstrate the superiority of the proposed method over the classical time delay embedding methods using a variety of application examples. We compare recurrence plot based statistics inferred from reconstructions of a Lorenz-96 system and highlight an improved forecast accuracy for map-like model data as well as for paleoclimate isotope time series. Finally we utilize state space reconstruction for the detection of causality and its strength between observables of a gas turbine type thermoacoustic combustor.

Slow–fast control of eye movements: an instance of Zeeman’s model for an action

The rapid eye movements (saccades) used to transfer gaze between targets are examples of an action. The behaviour of saccades matches that of the slow–fast model of actions originally proposed by Zeeman. Here, we extend Zeeman’s model by incorporating an accumulator that represents the increase in certainty of the presence of a target, together with an integrator that converts a velocity command to a position command. The saccadic behaviour of several foveate species, including human, rhesus monkey and mouse, is replicated by the augmented model. Predictions of the linear stability of the saccadic system close to equilibrium are made, and it is shown that these could be tested by applying state-space reconstruction techniques to neurophysiological recordings. Moreover, each model equation describes behaviour that can be matched to specific classes of neurons found throughout the oculomotor system, and the implication of the model is that build-up, burst and omnipause neurons are found throughout the oculomotor pathway because they constitute the simplest circuit that can produce the motor commands required to specify the trajectories of motor actions.

Day-to-Day Reliability of Nonlinear Methods to Assess Walking Dynamics

The present study investigated the day-to-day reliability (quantified by the absolute and relative reliability) of nonlinear methods used to assess human locomotion dynamics. Twenty-four participants of whom twelve were diagnosed with knee osteoarthritis completed 5 min of treadmill walking at self-selected preferred speed on two separate days. Lower limb kinematics were recorded at 100 Hz and hip, knee, and ankle joint angles, three-dimensional (3D) sacrum marker displacement and stride time intervals were extracted for 170 consecutive strides. The largest Lyapunov exponent and correlation dimension were calculated for the joint angle and sacrum displacement data using three different state space reconstruction methods (group average, test-retest average, individual time delay and embedding dimension). Sample entropy and detrended fluctuation analysis (DFA) were applied to the stride time interval time series. Relative reliability was assessed using intraclass correlation coefficients and absolute reliability was determined using measurement error (ME). For both joint angles and sacrum displacement, there was a general pattern that the group average state space reconstruction method provided the highest relative reliability and lowest ME compared to the individual and test-retest average methods. The DFA exhibited good reliability, while the sample entropy showed poor reliability. The results comprise a reference material that can inspire and guide future studies of nonlinear gait dynamics.