Using state space models to understand trait evolution in fossil lineages

The paper presents a broad general review of the state space approach to time series analysis. It begins with an introduction to the linear Gaussian state space model. Applications to problems in practical time series analysis are considered. The state space approach is briefly compared with the Box–Jenkins approach. The Kalman filter and smoother and the simulation smoother are described. Missing observations, forecasting and initialisation are considered. A representation of a multivariate series as a univariate series is displayed. The construction and maximisation of the likelihood function are discussed. An application to real data is presented. The treatment is extended to non-Gaussian and nonlinear state space models. A simulation technique based on importance sampling is described for analysing these models. The use of antithetic variables in the simulation is considered. Bayesian analysis of the models is developed based on an extension of the importance sampling technique. Classical and Bayesian methods are applied to a real time series.

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.

Low-order network models, commonly used to assess the thermo-acoustic stability of combustors, can be cast in a linear, time-continuous state-space representation. A standard linear eigenvalue problem for the system modes results, which can be solved in a robust and efficient manner. To represent the linear dynamics of any time-invariant flame in the state-space framework, this study presents an approximation of the distributed time-delayed flame response to acoustic velocity perturbations based on a spatially discretized propagation equation (PE). We derive the rational flame transfer function of a first-order-upwind-PE state-space model and discuss its relation to the Tustin approximation of flame transfer functions. For an exemplary discrete finite impulse response of a flame, a third-order-upwind-PE state-space model is shown to match the discrete flame frequency response with comparable accuracy as a rational approximation found by non-linear optimization. The numerical dissipation introduced by discretization of the PE ensures zero gain above the Nyquist frequency of the underlying discrete flame impulse response. Finally, we apply the PE state-space flame model to a generic Rijke tube and show that the predicted thermo-acoustic modes agree well with results obtained from a classical non-linearly optimized rational approximation of the frequency response function of the flame.

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.

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.

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.

We develop a state space approach to multimodal integration of simultaneously recorded EEG and fMRI. The EEG is represented with a distributed current source model using realistic MRI-based forward models, whose temporal evolution is governed by a linear state space model. The fMRI signal is similarly modeled by a linear state space model describing the hemodynamic response to underlying EEG current activity. We explore the feasibility of high dimensional dynamic estimation of simultaneous EEG/fMRI using simulation studies of the alpha wave.

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.

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.

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.

Previous studies on Sound Event Localization and Detection (SELD) have primarily focused on CNN- and Transformer-based designs. While CNNs possess local receptive fields, making it difficult to capture global dependencies over long sequences, Transformers excel at modeling long-range dependencies but have limited sensitivity to local time–frequency features. Recently, the VMamba architecture, built upon the Visual State Space (VSS) model, has shown great promise in handling long sequences, yet it remains limited in modeling local spatial details. To address this issue, we propose a novel state space model with an attention-enhanced feature fusion mechanism, termed FFMamba, which balances both local spatial modeling and long-range dependency capture. At a fine-grained level, we design two key modules: the Multi-Scale Fusion Visual State Space (MSFVSS) module and the Wavelet Transform-Enhanced Downsampling (WTED) module. Specifically, the MSFVSS module integrates a Multi-Scale Fusion (MSF) component into the VSS framework, enhancing its ability to capture both long-range temporal dependencies and detailed local spatial information. Meanwhile, the WTED module employs a dual-branch design to fuse spatial and frequency domain features, improving the richness of feature representations. Comparative experiments were conducted on the DCASE2021 Task 3 and DCASE2022 Task 3 datasets. The results demonstrate that the proposed FFMamba model outperforms recent approaches in capturing long-range temporal dependencies and effectively integrating multi-scale audio features. In addition, ablation studies confirmed the effectiveness of the MSFVSS and WTED modules.

State space is one of the key concepts of system theory. In general, a state space is introduced into a system description without examining its specific physical meaning. It is known, however, that if we select a suitable state space representation, it becomes easier for us to understand or to manipulate the property of a system. Thus, although it may not be meaningful to try to arrive at a physical meaning of the state space, it is important to find a way where a choice of a state space can be made in accordance with the system of interest, taking into account the theoretical meaning of that choice. In this paper, the choice of a state space is discussed in terms of the concepts that general systems theory have provided and the canonical forms of linear control theory are interpreted using the state space representation. Four realizations of a single-input and single-output discrete basic linear system are constructed, whose state spaces are derived from system properties. such as reachability, past-det...

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

Various analytical techniques are available for modeling multivariate time series data. One such approach is the State Space Model, which can be employed to model this type of data. In this study, the data to be analyzed are data on the Indonesia Rupiah (IDR) exchange rate (ExR) against the US Dollar (USD), oil and gas exports (OGE), money supply (MS) and non-oil and gas exports (non-OGE) from January 2008 to December 2019. The aim of this study is to identify the most suitable state space model for the given data. In this research, the state space method will be applied to multivariate time series data, with the state space represented in the Vector Autoregressive (VAR) model to explore the interrelationships among groups of observed variables. The VAR model is a statistical technique used to analyze the relationships between variables in the dataset, employing the Granger Causality Test. The state space model is utilized to model and forecast multiple interconnected time series, where the variables exhibit dynamic interactions and to examine additional unobserved variables in the time series data. Based on the analysis results and the minimum value of the Akaike Information Criterion (AIC), the optimal VAR model identified is the VAR (6) model. The results of forecasting values using the state space model show that the predicted values and the real values for the state space model are very closed to each other.

Although control algorithms have been conceived for industrial chemical systems, their acceptance by industry has been slow due to a lack of direct experimental evidence of their effectiveness and to volumes of conflicting, or at least incompatible, recommendations on control structure design. This thesis provides the basis for a concerted theoretical and experimental program in multivariable process control structure design for packed bed chemical reactors by presenting an in-depth control analysis of a practical, multivariable, distributed parameter system-the heat conduction problem defined by the simple diffusion equation-using both frequency-domain and time-domain analyses and the formulation, numerical solution, and analysis of a detailed model for packed bed reactors, along with reduction to a low-order state-space representation suitable for on-line process control. The study of the heat conduction system allowed for consideration of various control design techniques and the relation between measurement structure and control system design. This study shows that the choice of measurements and their locations significantly affects the optimal control design and the usefulness of the different design techniques and the importance of an accurate process model and the necessity of model reduction to a low-order state-space representation for control structure design and implementation. The second portion of this study provides a detailed mathematical modeling analysis of packed bed catalytic reactors that significantly extends previous studies in the detail of the model and in the consideration of all aspects of the model development and reduction to a state-space control representation. The general view that modeling simplifications are desired since they lead to a reduction in numerical solution effort is contested, and it is shown that many simplifications are no longer necessary with today's advanced computational capabilities. A unified approach to dynamic reactor modeling is developed and its importance in the accurate description of dynamic and steady state reactor behavior, in the investigation of reactor start-up or the effects of process disturbances, and in the development of an accurate reduced state-space model for the design of control structures to stabilize the reactor under various disturbances or to provide optimal system recovery from input changes is shown.