Takens Embedding Research Articles

Model free data assimilation with Takens embedding

In many practical scenarios, the dynamical system is not available and standard data assimilation methods are not applicable. Our objective is to construct a data-driven model for state estimation without the underlying dynamics. Instead of directly modeling the observation operator with noisy observation, we establish the state space model of the denoised observation. Through data assimilation techniques, the denoised observation information could be used to recover the original model state. Takens’ theorem shows that an embedding of the partial and denoised observation is diffeomorphic to the attractor. This gives a theoretical base for estimating the model state using the reconstruction map. To realize the idea, the procedure consists of offline stage and online stage. In the offline stage, we construct the surrogate dynamics using dynamic mode decomposition with noisy snapshots to learn the transition operator for the denoised observation. The filtering distribution of the denoised observation can be estimated using adaptive ensemble Kalman filter, without knowledge of the model error and observation noise covariances. Then the reconstruction map can be established using the posterior mean of the embedding and its corresponding state. In the online stage, the observation is filtered with the surrogate dynamics. Then the online state estimation can be performed utilizing the reconstruction map and the filtered observation. Furthermore, the idea can be generalized to the nonparametric framework with nonparametric time series prediction methods for chaotic problems. The numerical results show the proposed method can estimate the state distribution without the physical dynamical system.

Data-driven discovery of chemotactic migration of bacteria via coordinate-invariant machine learning

BackgroundE. coli chemotactic motion in the presence of a chemonutrient field can be studied using wet laboratory experiments or macroscale-level partial differential equations (PDEs) (among others). Bridging experimental measurements and chemotactic Partial Differential Equations requires knowledge of the evolution of all underlying fields, initial and boundary conditions, and often necessitates strong assumptions. In this work, we propose machine learning approaches, along with ideas from the Whitney and Takens embedding theorems, to circumvent these challenges.ResultsMachine learning approaches for identifying underlying PDEs were (a) validated through the use of simulation data from established continuum models and (b) used to infer chemotactic PDEs from experimental data. Such data-driven models were surrogates either for the entire chemotactic PDE right-hand-side (black box models), or, in a more targeted fashion, just for the chemotactic term (gray box models). Furthermore, it was demonstrated that a short history of bacterial density may compensate for the missing measurements of the field of chemonutrient concentration. In fact, given reasonable conditions, such a short history of bacterial density measurements could even be used to infer chemonutrient concentration.ConclusionData-driven PDEs are an important modeling tool when studying Chemotaxis at the macroscale, as they can learn bacterial motility from various data sources, fidelities (here, computational models, experiments) or coordinate systems. The resulting data-driven PDEs can then be simulated to reproduce/predict computational or experimental bacterial density profile data independent of the coordinate system, approximate meaningful parameters or functional terms, and even possibly estimate the underlying (unmeasured) chemonutrient field evolution.

Extending discrete geometric singular perturbation theory to non-hyperbolic points

Abstract We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-1 map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high-dimensional) local center manifold in n-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.

Chaotic index analysis of ethanol-based graphene nanofluid pulsating heat pipe

Pulsating heat pipes utilizing ethanol-based graphene nanofluids hold significant promise for cooling large electronic devices. However, traditional methods fall short in fully characterizing their properties due to internal phase transitions, latent heat, multiphase flow, and multi-field coupling. To address this, nonlinear chaotic analysis methods are introduced. This study constructs a vertical closed-loop pulsating heat pipe platform and applies Takens embedding theorem, phase space reconstruction theory, mutual information recursive selection method, and associative dimensionality algorithm to analyze ethanol-based graphene nanofluids with various concentrations. The investigation focuses on correlation dimensions and Kolmogorov entropy to provide numerical feedback on nanofluid concentration. Notably, ethanol-based graphene nanofluids concentrations between 0.04 wt% and 0.07 wt% exhibit an average heat transfer thermal resistance of approximately 0.74 K/W, corresponding to correlation dimension and Kolmogorov entropy values of 1.36 and 0.035, respectively. Additionally, reconstructed three-dimensional Singular Attractor phase diagrams offer morphological feedback on the pulsating heat pipe's working conditions. As the ethanol-based graphene nanofluids concentration reaches 0.35 wt%, an increase in the number of free vector points within the three-dimensional coordinate Attractor indicates localized unstable conditions in the heat pipe. Employing a wavelet noise reduction method with a soft threshold of 80 preserves the macroscopic vector flow pattern while eliminating irrelevant local information. Overall, this paper introduces novel approaches for analyzing heat transfer characteristics of graphene nanofluids in pulsating heat pipes.

Quantum process tomography of unitary maps from time-delayed measurements

Quantum process tomography conventionally uses a multitude of initial quantum states and then performs state tomography on the process output. Here we propose and study an alternative approach which requires only a single (or few) known initial states together with time-delayed measurements for reconstructing the unitary map and corresponding Hamiltonian of the time dynamics. The overarching mathematical framework and feasibility guarantee of our method is provided by the Takens embedding theorem. We explain in detail how the reconstruction of a single-qubit Hamiltonian works in this setting and provide numerical methods and experiments for general few-qubit and lattice systems with local interactions. In particular, the method allows to find the Hamiltonian of a two qubit system by observing only one of the qubits.

Использование топологического анализа данных для визуализации выходной информации приборов

The article discusses Takens embedding for 2D and 3D visualization of 1D time series data. The paper considers the use of topological data analysis in conjunction with Takens embedding to analyze the output information of a dynamic system - a rigid body. 3D images of curves constructed from three components of the angular velocity vector of a rigid body for various values of the main moments of inertia are constructed. Three-dimensional images of the curves constructed on the basis of the Takens embedding for the component of the angular velocity of a rigid body for various values of the principal moments of inertia are constructed. Distances between 3D images of curves are determined by constructing persistent landscape functions. Using the method of topological data analysis in conjunction with Takens embedding for image comparison allows you to classify and identify images and output information of the instrumental composition.

Severe slugging flow identification from topological indicators

In this work, topological data analysis is used to identify the onset of severe slug flow in offshore petroleum production systems. Severe slugging is a multiphase flow regime known to be very inefficient and potentially harmful to process equipment and it is characterized by large oscillations in the production fluid pressure. Time series from pressure sensors in subsea oil wells are processed by means of Takens embedding to produce point clouds of data. Embedded sensor data is then analyzed using persistent homology to obtain topological indicators capable of revealing the occurrence of severe slugging in a condition-based monitoring approach. A large dataset of well events consisting of both real and simulated data is used to demonstrate the possibilty of authomatizing severe slugging detection from live data via topological data analysis. Methods based on persistence diagrams are shown to accurately identify severe slugging and to classify different flow regimes from pressure signals of producing wells with supervised machine learning.

Embedding information onto a dynamical system

The celebrated Takens’ embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In this paper, we show how an arbitrary sequence can be mapped into another space as an attractive solution of a nonautonomous dynamical system. Such mapping also entails a topological conjugacy and an embedding between the sequence and the attractive solution spaces. This result is not a generalisation of Takens embedding theorem but helps us understand what exactly is required by discrete-time state space models widely used in applications to embed an external stimulus onto its solution space. Our results settle another basic problem concerning the perturbation of an autonomous dynamical system. We describe what exactly happens to the dynamics when exogenous noise perturbs continuously a local irreducible attracting set (such as a stable fixed point) of a discrete-time autonomous dynamical system.

Topological feature vectors for chatter detection in turning processes

Machining processes are most accurately described using complex dynamical systems that include nonlinearities, time delays and stochastic effects. Due to the nature of these models as well as the practical challenges which include time-varying parameters, the transition from numerical/analytical modeling of machining to the analysis of real cutting signals remains challenging. Some studies have focused on studying the signals of cutting processes using machine learning algorithms with the goal of identifying and predicting undesirable vibrations during machining referred to as chatter. These tools typically decompose the signal using Wavelet Packet Transforms (WPT) or Ensemble Empirical Mode Decomposition (EEMD). However, these methods require a significant overhead in identifying the feature vectors before a classifier can be trained. In this study, we present an alternative approach based on featurizing the time series of the cutting process using its topological features. We first embed the time series as a point cloud using Takens embedding. We then utilize Support Vector Machine, Logistic Regression, Random Forest and Gradient Boosting classifier combined with feature vectors derived from persistence diagrams, a tool from persistent homology, to encode chatter’s distinguishing characteristics. We present the results for several choices of the topological feature vectors, and we compare our results to the WPT and EEMD methods using experimental turning data. Our results show that in two out of four cutting configurations the Topological Data Analysis (TDA)-based features yield accuracies as high as 97%. We also show that combining Bézier curve approximation method and parallel computing can reduce runtime for persistence diagram computation of a single time series to less than a second thus making our approach suitable for online chatter detection.

RESEARCH ON WIND PRESSURE FIELD OF LARGE-SPAN FLAT ROOF BASED ON DYNAMIC MODE DECOMPOSITION

The dynamic mode decomposition (DMD) method improved by embedded dimension and the proper orthogonal decomposition (POD) method are used to analyze the random wind pressure field of a large-span flat roof. The results show that the fuzzy dynamic features hidden in the wind tunnel test data can be discovered by using the Takens embedding theorem to increase the spatial dimension of the original snapshot matrix. The modal distributions of POD and DMD can reflect the main characteristics of typical vortices in a fluctuating pressure field. However, each mode of POD contains the information of multiple frequencies, which to a certain extent makes the mode of POD become the fluctuating coupling of multiple frequency bands. The mode decomposed by DMD method is a single-frequency mode and the stability of each mode can be obtained. At the same proportion, POD modes contain more energy than DMD modes. But when the same proportion of modes are used to reconstruct the fluctuating wind pressure field, the reconstruction results of DMD method can better describe and fit the local characteristics of the original fluctuating pressure field than those of POD method. This is because DMD method reconstructs the pressure field directly, while POD method mainly reconstructs the energy field. Therefore, the DMD method has more advantages in revealing the spatial-temporal evolution characteristics of a random wind pressure field.

Dynamic Mode Decomposition of Random Pressure Fields over Bluff Bodies

Aerodynamic pressure field over bluff bodies immersed in boundary layer flows is correlated both in space and time. Conventional approaches for the analysis of distributed aerodynamic pressures, e.g., the proper orthogonal decomposition (POD), can only offer relevant spatial patterns in a set of coherent structures. This study provides an operator-theoretic approach that describes dynamic pressure fields in a functional space rather than conventional phase space via the Koopman operator. Subsequently, spectral analysis of the Koopman operator provides a spatiotemporal characterization of the pressure field. An augmented dynamic mode decomposition (DMD) method is proposed to perform the spectral decomposition. The augmentation is achieved by the use of the Takens's embedding theorem, where time delay coordinates are considered. Consequently, the identified eigen-tuples (eigenvalues, eigenvectors, and time evolution) can capture not only dominant spatial structures but also identify each structure with a specific frequency and a corresponding temporal growth/decay. This study encompasses learning the evolution dynamics of the random aerodynamic pressure field over a scaled model of a finite height prism using limited wind tunnel data. The POD analysis of the experimental data was also carried out. To demonstrate the unique feature of the proposed approach, the DMD and POD based learning results including algorithm convergence, data sufficiency, and modal analysis are examined. The ensuing observations offer a glimpse of the complex dynamics of the surface pressure field over bluff bodies that lends insights to features previously masked by conventional analysis approaches.

Manifold learning for organizing unstructured sets of process observations.

Data mining is routinely used to organize ensembles of short temporal observations so as to reconstruct useful, low-dimensional realizations of an underlying dynamical system. In this paper, we use manifold learning to organize unstructured ensembles of observations ("trials") of a system's response surface. We have no control over where every trial starts, and during each trial, operating conditions are varied by turning "agnostic" knobs, which change system parameters in a systematic, but unknown way. As one (or more) knobs "turn," we record (possibly partial) observations of the system response. We demonstrate how such partial and disorganized observation ensembles can be integrated into coherent response surfaces whose dimension and parametrization can be systematically recovered in a data-driven fashion. The approach can be justified through the Whitney and Takens embedding theorems, allowing reconstruction of manifolds/attractors through different types of observations. We demonstrate our approach by organizing unstructured observations of response surfaces, including the reconstruction of a cusp bifurcation surface for hydrogen combustion in a continuous stirred tank reactor. Finally, we demonstrate how this observation-based reconstruction naturally leads to informative transport maps between the input parameter space and output/state variable spaces.

Topological data analysis in investment decisions

This article explores the applications of Topological Data Analysis (TDA) in the finance field, especially addressing the primordial problem of asset allocation. Firstly, we build a rationale on why TDA can be a better alternative to traditional risk indicators such as standard deviation using real data sets. We apply Takens embedding theorem to reconstruct the time series of returns in a high dimensional space. We adopt the sliding window approach to draw the time-dependent point cloud data sets and associate a topological space with them. We then apply the persistent homology to discover the topological patterns that appear in the multidimensional time series. The temporal changes in the persistence landscapes, which are the real-valued functions that encode the persistence of topological patterns, are captured via Lp norm. The time series of the Lp norms shows that it is better at measuring the dynamics of returns than the standard deviation.Inspired by our findings, we explore an application of TDA in Enhanced Indexing (EI) that aims to build a portfolio of fewer assets than that in the index to outperform the latter. We propose a two-step procedure to accomplish this task. In step one, we utilize the Lp norms of the assets to propose a filtration technique of selecting a few assets from a larger pool of assets. In step two, we propose an optimization model to construct an optimal portfolio from the class of filtered assets for EI. To test the efficiency of this enhanced algorithm, experiments are carried out on ten data sets from financial markets across the globe. Our extensive empirical analysis exhibits that the proposed strategy delivers superior performance on several measures, including excess mean returns from the benchmark index and tail reward-risk ratios than some of the existing models of EI in the literature. The proposed filtering strategy is also noted to be beneficial for both risk-seeking and risk-averse investors.

Control of Continuous Time Chaotic Systems With Unknown Dynamics and Limitation on State Measurement

This paper has dedicated to study the control of chaos when the system dynamics is unknown and there are some limitations on measuring states. There are many chaotic systems with these features occurring in many biological, economical and mechanical systems. The usual chaos control methods do not have the ability to present a systematic control method for these kinds of systems. To fulfill these strict conditions, we have employed Takens embedding theorem which guarantees the preservation of topological characteristics of the chaotic attractor under an embedding named “Takens transformation.” Takens transformation just needs time series of one of the measurable states. This transformation reconstructs a new chaotic attractor which is topologically similar to the unknown original attractor. After reconstructing a new attractor its governing dynamics has been identified. The measurable state of the original system which is one of the states of the reconstructed system has been controlled by delayed feedback method. Then the controlled measurable state induced a stable response to all of the states of the original system.

Time series network induced subgraph distance as a metonym for dynamical invariants

The complex behaviour of dynamical systems can be discovered through the study of observed scalar time series via reconstruction of a suitable state space. Reconstruction can readily be achieved thanks to Takens embedding theorem and further insights about a system can be found by constructing complex networks based on the proximity to each other of these reconstructed states. An example of such a proximity network is the so-called phase space network which is a type of recurrence network. The structure and properties of these networks, including small induced subgraphs within them, describe the spatial distribution of reconstructed states in a way different from, but related to, correlation dimension. Beyond the makeup of the reconstructed states the time structure inherent in the time series is typically ignored in any subsequent analysis of proximity networks. Here, we construct proximity networks but retain information about the time structure of the reconstructed state space trajectory and hence assign time as an attribute of the nodes of the network. We show that metrics capturing the change in structure of induced subgraphs along the trajectory can qualitatively track behavioural changes of the system dynamics subject to a bifurcation parameter in a manner similar to time-averaging estimation methods of dynamical invariants. Moreover, we demonstrate that these new methods retain the capability of monitoring system change in the case of irregularly sampled data.

Damage Detection Using Dissimilarity in Phase Space Topology of Dynamic Response of Structure Subjected to Shock Wave Loading

Shockwave is a high pressure and short duration pulse that induce damage and lead to progressive collapse of the structure. The shock load excites high-frequency vibrational modes and causes failure due to large deformation in the structure. Shockwave experiments were conducted by imparting repetitive localized shock loads to create progressive damage states in the structure. Two-phase novel damage detection algorithm is proposed, that quantify and segregate perturbative damage from microscale damage. The first phase performs dimension reduction and damage state segregation using principal component analysis (PCA). In the second phase, the embedding dimension was reduced through empirical mode decomposition (EMD). The embedding parameters were derived using singular system analysis (SSA) and average mutual information function (AMIF). Based, on Takens theorem and embedding parameters, the response was represented in a multidimensional phase space trajectory (PST). The dissimilarity in the multidimensional PST was used to derive the damage sensitive features (DSFs). The DSFs namely: (i) change in phase space topology (CPST) and (ii) Mahalanobis distance between phase space topology (MDPST) are evaluated to quantify progressive damage states. The DSFs are able to quantify the occurrence, magnitude, and localization of progressive damage state in the structure. The proposed algorithm is robust and efficient to detect and quantify the evolution of damage state for extreme loading scenarios.

Semi-supervised time series classification on positive and unlabeled problems using cross-recurrence quantification analysis

When dealing with semi-supervised scenarios, the Positive and Unlabeled (PU) problem is a special case in which few labeled examples from a single class of interest are received to proceed with the classification of unseen instances, according to their similarities with the known class. In the scope of time series, most of the current studies propose to address this subject using a self-training approach based on the 1-Nearest Neighbor algorithm. In order to compute the most similar instance, they compare features along the time domain using the Euclidean Distance and the Dynamic Time Warping-Delta. Despite time-domain measurements permit the analysis of local series shapes, they disconsider temporal recurrences commonly found in natural phenomena (e.g. population growth, climate studies) and are more sensitive to local noise and fluctuations, leading to poor classification performances as confirmed in this paper. This drawback motivated us to propose the use of the Maximum Diagonal Line of the Cross-Recurrence Quantification Analysis (MDL-CRQA), applied on the time series phase space, as similarity measurement. The phase space is obtained after applying Takens embedding theorem on the series, unfolding temporal relationships and dependencies among data observations. As consequence, by comparing phase spaces rather than the series themselves, we can assess how their trajectories evolve along time, including their periodicities and temporal cycles, as well as decreasing noise influences. Experimental results confirm MDL-CRQA improves classification results for PU time series when compared against the mostly used time-domain similarity measurements.

On the control of unknown continuous time chaotic systems by applying Takens embedding theory

In this paper, a new approach to control continuous time chaotic systems with an unknown governing equation and limitation on the measurement of states, has been investigated. In many chaotic systems, disability to measure all of the states is a usual limitation, like in some economical, biological and many other engineering systems. Takens showed that a chaotic attractor has an astonishing feature in which it can embed to a mathematically similar attractor by using time series of one of the states. The new embedded attractor saves much information from the original attractor. This phenomenon has been deployed to present a new way to control continuous time chaotic systems, when only one of the states of the system is measurable and the system model is not also available.

Chatter Classification in Turning using Machine Learning and Topological Data Analysis

Chatter identification and detection in machining processes has been an active area of research in the past two decades. Part of the challenge in studying chatter is that machining equations that describe its occurrence are often nonlinear delay differential equations. The majority of the available tools for chatter identification rely on defining a metric that captures the characteristics of chatter, and a threshold that signals its occurrence. The difficulty in choosing these parameters can be somewhat alleviated by utilizing machine learning techniques. However, even with a successful classification algorithm, the transferability of typical machine learning methods from one data set to another remains very limited. In this paper we combine supervised machine learning with Topological Data Analysis (TDA) to obtain a descriptor of the process which can detect chatter. The features we use are derived from the persistence diagram of an attractor reconstructed from the time series via Takens embedding. We test the approach using deterministic and stochastic turning models, where the stochasticity is introduced via the cutting coefficient term. Our results show a 97% successful classification rate on the deterministic model labeled by the stability diagram obtained using the spectral element method. The features gleaned from the deterministic model are then utilized for characterization of chatter in a stochastic turning model where there are very limited analysis methods.

Chaos control in delayed phase space constructed by the Takens embedding theory

In this paper, the problem of chaos control in discrete-time chaotic systems with unknown governing equations and limited measurable states is investigated. Using the time-series of only one measurable state, an algorithm is proposed to stabilize unstable fixed points. The approach consists of three steps: first, using Takens embedding theory, a delayed phase space preserving the topological characteristics of the unknown system is reconstructed. Second, a dynamic model is identified by recursive least squares method to estimate the time-series data in the delayed phase space. Finally, based on the reconstructed model, an appropriate linear delayed feedback controller is obtained for stabilizing unstable fixed points of the system. Controller gains are computed using a systematic approach. The effectiveness of the proposed algorithm is examined by applying it to the generalized hyperchaotic Henon system, prey-predator population map, and the discrete-time Lorenz system.