Hierarchical Dynamical System Research Articles

Turbulent energy dissipation rate modification due to particle insertion: Effect of Reynolds number

A hierarchical spatiotemporal dynamical system model (tree model), modified for the insertion of dispersed phase particle, is utilized to study the turbulence modulation caused by the presence of particle in high Reynolds numbers. The focus in this study is to see how the energy dissipation rate is escalated in the vicinity of particle surface. In the present study, shell model simulations are performed for two different Reynolds numbers, i.e. Re∼1/ν, as 5 × 104 and 106, corresponding to two particle sizes of 256 and 64 times of the Kolmogorov length scale, respectively. The results are compared to the corresponding unladen cases, as well as the lower Reynolds number of 104. The scaling properties of the modified tree model are assessed through the velocity structure functions, which reveals the existence of intermittency.

Hierarchical linear dynamical systems for unsupervised musical note recognition

In this paper we develop a new framework for time series segmentation based on a Hierarchical Linear Dynamical System (HLDS), and test its performance on monophonic and polyphonic musical note recognition. The center piece of our approach is the inclusion of constraints in the filter topology, instead of on the cost function as normally done in machine learning. Just by slowing down the dynamics of the top layer of an augmented (multilayer) state model, which is still compatible with the recursive update equation proposed originally by Kalman, the system learns directly from data all the musical notes, without labels, effectively creating a time series clustering algorithm that does not require segmentation. We analyze the HLDS properties and show that it provides better classification accuracy compared to current state-of-the-art approaches.

A Study of Musical Pitch Distance Using a Self-Organized Hierarchical Linear Dynamical System on Acoustic Signals

The hierarchical linear dynamical system (HLDS) is a self-organizing architecture to cluster acoustic time series. The HLDS architecture is equivalent to a Kalman filter whose top-layer state learns to create subspaces that tessellate the acoustic signal in regions that correspond to different musical pitches. The observation layer of the HLDS is built from a biologically plausible gammatone filter bank that provides the representation space for the state assignments. An important characteristic of the methodology is that it is adaptive and self-organizing, i.e., previous exposure to the acoustic input is the only requirement for learning and recognition. In this article we show that the representation space that the algorithm learns from acoustic signals preserves the organization found in monophonic notes, and exhibits (for isolated pitches and triads) properties suggested in the theory of efficient chromatic voice leading and neo-Riemannian theories.

Clinical time series prediction: Toward a hierarchical dynamical system framework.

Developing machine learning and data mining algorithms for building temporal models of clinical time series is important for understanding of the patient condition, the dynamics of a disease, effect of various patient management interventions and clinical decision making. In this work, we propose and develop a novel hierarchical framework for modeling clinical time series data of varied length and with irregularly sampled observations. Our hierarchical dynamical system framework for modeling clinical time series combines advantages of the two temporal modeling approaches: the linear dynamical system and the Gaussian process. We model the irregularly sampled clinical time series by using multiple Gaussian process sequences in the lower level of our hierarchical framework and capture the transitions between Gaussian processes by utilizing the linear dynamical system. The experiments are conducted on the complete blood count (CBC) panel data of 1000 post-surgical cardiac patients during their hospitalization. Our framework is evaluated and compared to multiple baseline approaches in terms of the mean absolute prediction error and the absolute percentage error. We tested our framework by first learning the time series model from data for the patients in the training set, and then using it to predict future time series values for the patients in the test set. We show that our model outperforms multiple existing models in terms of its predictive accuracy. Our method achieved a 3.13% average prediction accuracy improvement on ten CBC lab time series when it was compared against the best performing baseline. A 5.25% average accuracy improvement was observed when only short-term predictions were considered. A new hierarchical dynamical system framework that lets us model irregularly sampled time series data is a promising new direction for modeling clinical time series and for improving their predictive performance.

Minimax program control for the approach process in a two-level hierarchical discrete dynamical system

We consider a dynamical system consisting of three objects whose dynamics is given by vector linear discrete recurrent equations. The system contains two levels of managerial decision making, basic and secondary, that have different operational criteria and are united by informational and managerial connections defined in advance. For the dynamical system in question, we propose a mathematical formalization in the form of solving a multistage problem of two-level hierarchical minimax program control over the approach process with incomplete information and give a general scheme for its solution.

Identifying damage using local flow variation method

In this paper, a damage identification method called local flow variation is introduced. Itis a practical implementation of a phase space warping concept. A hierarchicaldynamical system is considered where a slow-time damage process causes drifts inthe parameters of a fast-time system describing the measurable response of astructure. The method is based on a hypothesis that the probability distributionfunction of the fast-time trajectory in its phase space is a function of a damagestate. In this method, an ensemble of estimated expectations of a trajectory indifferent locations of the reconstructed phase space is used as a damage featurevector. Using these feature vectors, damage identification is realized by a smoothorthogonal decomposition. An experiment is conducted to validate the method. Atwo-dimensional slow-time damage process is identified from experimental fast-time data.Although damage identification results from the local flow variation are not asaccurate as those from the direct application of phase space warping trackingfunctions, the required computation time is about two orders of magnitude shorter.

A Dynamical Systems Approach to Failure Prognosis

In this paper, a previously published damage tracking method is extended to provide failure prognosis, and applied experimentally to an electromechanical system with a failing supply battery. The method is based on a dynamical systems approach to the problem of damage evolution. In this approach, damage processes are viewed as occurring in a hierarchical dynamical system consisting of a “fast,” directly observable subsystem coupled to a “slow,” hidden subsystem describing damage evolution. Damage tracking is achieved using a two-time-scale modeling strategy based on phase space reconstruction. Using the reconstructed phase space of the reference (undamaged) system, short-time predictive models are constructed. Fast-time data from later stages of damage evolution of a given system are collected and used to estimate a tracking function by calculating the short time reference model prediction error. In this paper, the tracking metric based on these estimates is used as an input to a nonlinear recursive filter, the output of which provides continuous refined estimates of the current damage (or, equivalently, health) state. Estimates of remaining useful life (or, equivalently, time to failure) are obtained recursively using the current damage state estimates under the assumption of a particular damage evolution model. The method is experimentally demonstrated using an electromechanical system, in which mechanical vibrations of a cantilever beam are dynamically coupled to electrical oscillations in an electromagnet circuit. Discharge of a battery powering the electromagnet (the “damage” process in this case) is tracked using strain gauge measurements from the beam. The method is shown to accurately estimate both the battery state and the time to failure throughout virtually the entire experiment.

A Dynamical Systems Approach to Damage Evolution Tracking, Part 1: Description and Experimental Application

AbstractIn this two-part paper we present a novel method for tracking a slowly evolving hidden damage process responsible for nonstationarity in a fast dynamical system. The development of the method and its application to an electromechanical experiment is the core of Part 1. In Part 2, a mathematical model of the experimental system is developed and used to validate the experimental results. In addition, an analytical connection is established between the tracking method and the physics of the system based on the idea of averaging and the slow flow equations for the hidden process. The tracking method developed in this study uses a nonlinear, two-time-scale modeling strategy based on the delay reconstruction of a system’s phase space. The method treats damage-induced nonstationarity as evolving in a hierarchical dynamical system containing a fast, directly observable subsystem coupled to a slow, hidden subsystem. The utility of the method is demonstrated by tracking battery discharge in a vibrating beam system with a battery-powered electromagnetic restoring force. Applications to systems with evolving material damage are also discussed.

Some special restricted four-body problems—I. Modelling the Caledonian problem

A four-body system analogous to the Copenhagen problem (the restricted circular coplanar three-body problem with two finite equal masses) is modelled. The coplanar, restricted, circular equal mass four-body problem, or the Caledonian problem as it is named, involves four point-like bodies of equal mass moving in coplanar orbits about their centre of mass. Initially, the bodies are collinear and move co-rotationally with their velocity vectors perpendicular to the line they form. Models are developed for both a linear hierarchical dynamical system (the smallest binary's centre of mass forms a binary with the third body, while their centre of mass forms a binary with the fourth body) and a double binary hierarchical dynamical system (two binaries whose centres of mass form an additional binary). This simplified four-body model reduces the four-body problem from 24th-order with 28 initial parameters to 16th-order with only four independent initial parameter values, viz the mass m and three distance parameters a, b, c marking the initial separations between the four bodies. The advantages in minimisation of the numbers of variables and boundary condition parameters are obvious, yet, despite its simplifications, the Caledonian problem is shown to be suitable for application to the study of real four-body stellar systems. An exhaustive study of the model utilising ranges of the four parameters should provide: insight into why real four-body stellar systems are always found in either a linear or double binary hierarchy; an understanding of the part played by the parameters in the stability lifetimes of this model; a measure of the reliability of the c 2 H criteria applied to the triple subsets and the Roy-Walker empirical stability criteria in predicting the stability of the four body systems.

Asymptotic Approach to Mirror Conditions as a Trapping Mechanism in N-Body Hierarchical Dynamical Systems

AbstractThe consequences for hierarchical stability of almost circular almost coplanar, low perturbation orbits in an n-body hierarchical dynamical system is discussed. It is shown that frequent close approaches to mirror conditions with subsequent reversing of perturbations is ensured by such properties. The part played by near commensurabilities in mean motion is also discussed, the Sun-Jupiter-Saturn case being taken as an example.