Cointegrated continuous-time linear state-space and MCARMA models

We investigate the model for an industrial isothermal HDPE slurry reactor. The model, consisting of several nonlinear equations, can be linearized to give sets of linear time invariant state space model. The effectiveness of the linearized model is verified by the numerical simulations. A simple model predictive control scheme is constructed based on the linear state space model. The value of the melt index is obtained from the values of the manipulated and controlled variables generated from the control scheme. The control performance can be evaluated from the comparison between the computed melt index values and measured melt index values. The control scheme shows good tracking performance in the numerical simulations. We believe that the model developed in the present study can be effectively used to predict process variables as well as to control the melt index.

Linear parametric state-space models are a ubiquitous tool for analyzing neural time series data, providing a way to characterize the underlying brain dynamics with much greater statistical efficiency than non-parametric data analysis approaches. However, neural time series data are frequently time-varying, exhibiting rapid changes in dynamics, with transient activity that is often the key feature of interest in the data. Stationary methods can be adapted to time-varying scenarios by employing fixed-duration windows under an assumption of quasi-stationarity. But time-varying dynamics can be explicitly modeled by switching state-space models, i.e., by using a pool of state-space models with different dynamics selected by a probabilistic switching process. Unfortunately, exact solutions for state inference and parameter learning with switching state-space models are intractable. Here we revisit a switching state-space model inference approach first proposed by Ghahramani and Hinton. We provide explicit derivations for solving the inference problem iteratively after applying a variational approximation on the joint posterior of the hidden states and the switching process. We introduce a novel initialization procedure using an efficient leave-one-out strategy to compare among candidate models, which significantly improves performance compared to the existing method that relies on deterministic annealing. We then utilize this state inference solution within a generalized expectation-maximization algorithm to estimate model parameters of the switching process and the linear state-space models with dynamics potentially shared among candidate models. We perform extensive simulations under different settings to benchmark performance against existing switching inference methods and further validate the robustness of our switching inference solution outside the generative switching model class. Finally, we demonstrate the utility of our method for sleep spindle detection in real recordings, showing how switching state-space models can be used to detect and extract transient spindles from human sleep electroencephalograms in an unsupervised manner.

Application of a Linear Gaussian State Space Model and the Kalman Filter in Estimating Expectations of the Natural Logarithmic Monthly Nav Returns of the Open – End Funds of Major Investment Banks.

The framework for the following analysis is the linear state space model. Therefore this chapter treats the formulation, interpretation and estimation of state space models. As this is a rather general class covering many different models it will not be presented in its full generality. Actually, the presentation will be restricted to those models which may be interpreted as linear regression models with time varying coefficients. In particular, only single equation models will be discussed. Hence, VAR (vector autoregressive) or SUR (seemingly unrelated regression) models are excluded. Even though the extension to the multivariate case is sometimes rather obvious, it will not be treated here, because the main part of this study considers single regression equations only.

Deep learning has been used to train neural networks to estimate the Remaining Useful Life (RUL) of a machine given sensor signals from that machine. This has resulted in some accurate state-of-the-art RUL estimation models. Here we present a novel way to construct a generative model to estimate the RUL and quantify its uncertainty. It takes the form of a Linear Gaussian State Space Model (LGSSM) and is trained using the Kalman Filter; hence, uncertainty is quantified using this LGSSM instead of by other popular methods like Monte Carlo Dropout or Deep Ensembles. This means we can train by directly using the marginal log-likelihood loss and don’t require multiple samples to represent the uncertainty. However, this method is limited to using Gaussian distributions to quantify uncertainty. We avoid needing to use non-linear variants of the filter by processing the sensors using a neural network to represent noncausal sensor sequences as a “control variable” in the LGSSM. This noncausal representation is shown to be important for achieving state-of-the-art performance. The model is tested on a turbofan engine and dust filter dataset. The code can be found on GitHub. 1 1 https://github.com/StarMarco/Kalman_DVAE • Deep Neural Networks encode time window of sensor signals. • These are used as control/forcing variables of a Linear Gaussian State Space Model (LGSSM). • The “observables" of the LGSSM are the Remaining Useful Life (RUL) values of the machine. • The Kalman filtering algorithm is used to estimate the marginal log-likelihood of these RUL estimates. • After training the LGSSM can be evaluated over time to estimate the RUL trajectories.

This paper proposes a linear stochastic state space model for electrocardiogram signal processing and analysis. The model is obtained as a discretized version of Wiener process acceleration model. The model is combined with a fixed-lag Rauch-Tung-Striebel smoother to perform on-line signal denoising, feature extraction, and beat classification. The results indicate that the proposed approach outperforms a conventional FIR filter in terms of improved signal-to-noise ratio, and that the approach can be used for highly accurate online classification of normal beats and premature ventricular contractions. The benefits of the model include the possibility to use closed-form solutions to the optimal filtering and smoothing problems, quick adaptation to sudden changes in beat morphology and heart rate, simple and fast initialization, preprocessing-free operation, intuitive interpretation of the system state, and more.

This paper investigates change point detection in state space models, in which the pre-change distribution f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ0</sup> is given, while the post-distribution f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</sup> after change is unknown. The problem is to raise an alarm as soon as possible after the distribution changes from f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ0</sup> to f <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">θ</sup> , under a restriction on the false alarms. We investigate theoretical properties of a weighted Shiryayev-Roberts-Pollak (SRP) change point detection rule in state space models. By making use of a Markov chain representation for the likelihood function, exponential embedding of the induced Markovian transition operator, nonlinear Markov renewal theory, and sequential hypothesis testing theory for Markov random walks, we show that the weighted SRP procedure is second-order asymptotically optimal. To this end, we derive an asymptotic approximation for the expected stopping time of such a stopping scheme when the change time w = 1. To illustrate our method we apply the results to two types of state space models: general state Markov chains and linear state space models. Simulation study is given to illustrate the method.

Cointegration Between Trends and Their Estimators in State Space Models and CVAR Models

A method is presented for classification of trend curves based on the linear state space model. In this approach information about the smoothness of the trend curves is incorporated into the classification model by a nonstationary stochastic trend model and can thereby be used to obtain a better classification. In the case of small data sets the performance of the classification is significantly improved in comparison with the usual cluster analysis. Maximum likelihood estimation can be used to calculate the parameters of this model and to determine the classification. The classification algorithm is described in detail and the results are compared to those of the usual cluster analysis by simulation studies and by an application to tree ring data.

A method for obtaining linear state space models of the drawing process is developed. Traditionally, computational fluid dynamics methods have been used to model the drawing process. Although these models have the potential to provide very accurate details of the flow field, they incorporate thousands of dynamic states which make them unsuitable for both real-time simulations and control design. Experimental data have also been used to construct frequency response models which are suitable for control design. However, they heavily rely on the particular operating conditions for which they were obtained. Furthermore, they are constructed using large lumping techniques based on the information at the boundary and cannot predict large perturbations in the flow field. The objective of this paper is to bridge the gap between the system theoretic modeling techniques of the control engineer and the more physically motivated modeling methods of computational fluid dynamics. The method presented here consists of using the basic conservation laws (mass, momentum, and energy) to describe the mean flow of glass along the axial direction. Then, a linear state space model is obtained by spatially discretizing and linearizing the nonlinear partial differential equations. The resulting state space model incorporates all the relevant inputs and outputs of the system in a multiple-input/multiple-output framework. Furthermore, it lends itself to the application of modern control design techniques. The method is simple to implement since all that is needed is the steady state operating points, which can be computed from computational fluid dynamics simulations.

The closed-loop brain stimulation system can adjust stimulation parameters based on neural activity feedback, so as to provide precise and personalized treatment for neurological and neuropsychiatric diseases. The design of a model-based closed-loop system requires the choice of a suitable system identification framework to learn a dynamic mapping from input and output data to quantify the effect of input stimulation on output neural activity, and design the controller with the identification model. We first designed a control-theory system identification framework to build a dynamic input-output (IO) model of neural activity suitable for closed-loop control design. To achieve effective model-based control, we use a linear state space model (LSSM), which uses low-dimensional hidden neural states to characterize the effect of inputs on neural activity. Secondly, in order to train the model parameters, we choose a pulse sequence with independent modulation of parameters by pseudo-random sequence (PRS), and proved that this is the best choice for collecting informational IO data sets in system identification. The input waveform can be used as a deep brain stimulation (DBS) for treating neuropsychiatric diseases while meeting clinical safety requirements. Thirdly, we take the beta frequency band of the local field potential (LFP) as a biomarker of Parkinson's disease (PD) and system feedback signal to identify the system and design a closed-loop controller. The simulation results show that the closed-loop controller designed based on the LSSM model identified by the PRS modulation waveform can achieve the ideal control effect, and the performance of the model is similar to that of a real physiological IO model describing neural activity. The new PRS modulation waveform and system identification framework can help develop future model-based closed-loop stimulation systems, which is of great significance for the treatment of neuropsychiatric diseases.

A Mini Aerial Vehicle as a Support for Modelling, Control and Estimation Teaching

In a recent paper by Jiang (1988), a sufficient condition is presented for the stability of real interval matrices. This result and another in the paper have appeared previously.

Within the paradigm of population dynamics a central task is to identify environmental factors affecting population change and to estimate the strength of these effects. We here investigate the impact of observation errors in measurements of population densities on estimates of environmental effects. Adding observation errors may change the autocorrelation of a population time series with potential consequences for estimates of effects of autocorrelated environmental covariates. Using Monte Carlo simulations, we compare the performance of maximum likelihood estimates from three stochastic versions of the Gompertz model (log–linear first order autoregressive model), assuming 1) process error only, 2) observation error only, and 3) both process and observation error (the linear state–space model on log‐scale). We also simulated population dynamics using the Ricker model, and evaluated the corresponding maximum likelihood estimates for process error models. When there is observation error in the data and the considered environmental variable is strongly autocorrelated, its estimated effect is likely to be biased when using process error models. The environmental effect is overestimated when the sign of the autocorrelations of the intrinsic dynamics and the environment are the same and underestimated when the signs differ. With non‐autocorrelated environmental covariates, process error models produce fairly exact point estimates as well as reliable confidence intervals for environmental effects. In all scenarios, observation error models produce unbiased estimates with reasonable precision, but confidence intervals derived from the likelihood profiles are far too optimistic if there is process error present. The safest approach is to use state–space models in presence of observation error. These are factors worthwhile to consider when interpreting earlier empirical results on population time series, and in future studies, we recommend choosing carefully the modelling approach with respect to intrinsic population dynamics and covariate autocorrelation.

Voluntary residual motor efforts to the affected limb of patients after stroke have not been involved enough in most rehabilitation robot control strategies. In this paper, a natural integration between human and machine is proposed by using the surface electromyography (EMG) signals from six muscles which mainly contribute to the upper limb movement. A linear state space model is trained, which can estimate the real-time movement intention by using EMG signals, to calculate the movement needed forces and then provided by a cable-based upper limb rehabilitation robot. Ten healthy subjects are recruited to complete the tasks with and without robot assistances. The performances of the subjects with the assistances are compared to that of the subjects without assistances. Results show that the forces from the model were real-time continuously estimated and accurate. Furthermore, there is no significant difference in the group mean root mean square error (RMSE) and muscle activations between the task without assistance and with assistance. These results show that the robot using the state space model could provide physiologically appreciate assistance to the subject, and the robot could conduct the rehabilitation training combined with the voluntary residual motor efforts. Clinical test will be carried out to validate the feasibility of the robot-aided rehabilitation using myoelectrical control.