Unobserved State Variables Research Articles

Realized Random Graphs, with an Application to the Interbank Network

Abstract We introduce a new inferential methodology for dynamic network models driven by latent state variables. The main idea is to obtain a noisy representation of the state variables dynamics by computing a sequence of cross-sectional estimates of the network model at each point in time. The dynamic modeling of these cross-sectional estimates, that we name realized random graphs, transforms the original nonlinear state-space network model into a linear time-series model that can be easily estimated. Under the assumption of a mixed-membership blockmodel structure, the model parameters and the unobservable state variables can be consistently estimated when both the size of the network and the time-series length are large. By allowing for an extremely rich parameterization of the model in high dimensions, the proposed methodology describes the heterogeneous topology of real-world networks. We corroborate our findings by using this novel framework to estimate and forecast the dynamic common factors driving the evolution of the Italian electronic market of interbank deposits, and we show that the interbank lending rate is a key factor determining the network topology.

The Improvement of Short-Term Quantitative Precipitation Forecast in Mountainous Areas by the Assimilation of Meteorological State Variables Retrieved by Multiple Doppler Radar Data

Abstract This study presents a sequential procedure formulated by combining a multiple-Doppler radar wind synthesis technique with a thermodynamic retrieval method, which can be applied to retrieve the three-dimensional wind, pressure, temperature, rainwater mixing ratio, and moisture over complex terrain. The retrieved meteorological state variables are utilized to reinitialize a high-resolution numerical model, which then carries out time integration using four different microphysical (MP) schemes, including the Goddard Cumulus Ensemble (GCE), Morrison (MOR), WRF single-moment 6-class (WSM6), and WRF double-moment 6-class (WDM6) schemes. It is found that through this procedure, the short-term quantitative precipitation forecast (QPF) skill of a numerical model over mountainous areas can be significantly improved up to 6 h. The moisture field plays a crucial role in producing the correct rainfall forecast. Since no specific microphysical scheme outperforms the others, a combination of various rainfall scenarios forecasted by different MP schemes is suggested in order to provide a stable and reliable rainfall forecast. This work also demonstrates that, with the proposed approach, radar data from only two volume scans are sufficient to improve the rainfall forecasts. This is because the unobserved meteorological state variables are instantaneously retrieved and directly used to reinitialize the model, thereby the model spinup time can be effectively shortened.

The Biases in Applying Static Demand Models Under Dynamic Demand

This article investigates the mechanisms that underlie the biases in price elasticities of demand in applying static demand models under dynamic demand, which have been pointed out by previous empirical studies. It studies three sources of biases: disregard of state variables (affecting short-run elasticity); inconsistent utility parameter estimates; and changing expectations of consumers (affecting long-run elasticity). Disregard of state variables, such as durable product holdings, which is not negligible but not paid much attention to in the literature, leads to an overestimate of short-run own elasticities. Inconsistent utility parameter estimates arises due to the failure to account for consumers’ future expectations and unobserved state variables. Changing expectations of consumers are not explicitly specified in the static model, and this also leads to biased results when applying static models. Regarding the magnitude of the biases, the first and the third sources of biases might induce large biases in price elasticities, especially when the focus is on the large conditional choice probability products. Possible remedies for the use of static demand models are also discussed.

System identifiability in a time-evolving agent-based model.

Mathematical models are a valuable tool for studying and predicting the spread of infectious agents. The accuracy of model simulations and predictions invariably depends on the specification of model parameters. Estimation of these parameters is therefore extremely important; however, while some parameters can be derived from observational studies, the values of others are difficult to measure. Instead, models can be coupled with inference algorithms (i.e., data assimilation methods, or statistical filters), which fit model simulations to existing observations and estimate unobserved model state variables and parameters. Ideally, these inference algorithms should find the best fitting solution for a given model and set of observations; however, as those estimated quantities are unobserved, it is typically uncertain whether the correct parameters have been identified. Further, it is unclear what 'correct' really means for abstract parameters defined based on specific model forms. In this work, we explored the problem of non-identifiability in a stochastic system which, when overlooked, can significantly impede model prediction. We used a network, agent-based model to simulate the transmission of Methicillin-resistant staphylococcus aureus (MRSA) within hospital settings and attempted to infer key model parameters using the Ensemble Adjustment Kalman Filter, an efficient Bayesian inference algorithm. We show that even though the inference method converged and that simulations using the estimated parameters produced an agreement with observations, the true parameters are not fully identifiable. While the model-inference system can exclude a substantial area of parameter space that is unlikely to contain the true parameters, the estimated parameter range still included multiple parameter combinations that can fit observations equally well. We show that analyzing synthetic trajectories can support or contradict claims of identifiability. While we perform this on a specific model system, this approach can be generalized for a variety of stochastic representations of partially observable systems. We also suggest data manipulations intended to improve identifiability that might be applicable in many systems of interest.

Optimal investment and benefit payment adjustment strategies for the target benefit plan under partial information

This article investigates the optimal asset allocation and benefit adjustment problem for the target benefit plan (TBP) with predictable returns in an environment of partial information. We assume that the return rate of the stock depends on an observable and an unobservable state variables, and the pension manager estimates the unobservable component from known information through Bayesian learning. Under the criterion of expected exponential utility maximization, we obtain the optimal strategy in closed-form through the dynamic programming principle approach. Furthermore, numerical simulations are conducted by the Monte Carlo method to discuss the impacts of some parameters on the derived optimal strategy and to compare the optimal strategy with the suboptimal strategy when ignoring learning. It turns out that neglecting learning affects the optimal strategy seriously, thereby leading to significant utility losses.

Combining Stochastic Parameterized Reduced‐Order Models With Machine Learning for Data Assimilation and Uncertainty Quantification With Partial Observations

AbstractA hybrid data assimilation algorithm is developed for complex dynamical systems with partial observations. The method starts with applying a spectral decomposition to the entire spatiotemporal fields, followed by creating a machine learning model that builds a nonlinear map between the coefficients of observed and unobserved state variables for each spectral mode. A cheap low‐order nonlinear stochastic parameterized extended Kalman filter (SPEKF) model is employed as the forecast model in the ensemble Kalman filter to deal with each mode associated with the observed variables. The resulting ensemble members are then fed into the machine learning model to create an ensemble of the corresponding unobserved variables. In addition to the ensemble spread, the training residual in the machine learning‐induced nonlinear map is further incorporated into the state estimation, advancing the diagnostic quantification of the posterior uncertainty. The hybrid data assimilation algorithm is applied to a precipitating quasi‐geostrophic (PQG) model, which includes the effects of water vapor, clouds, and rainfall beyond the classical two‐level QG model. The complicated nonlinearities in the PQG equations prevent traditional methods from building simple and accurate reduced‐order forecast models. In contrast, the SPEKF forecast model is skillful in recovering the intermittent observed states, and the machine learning model effectively estimates the chaotic unobserved signals. Utilizing the calibrated SPEKF and machine learning models under a moderate cloud fraction, the resulting hybrid data assimilation remains reasonably accurate when applied to other geophysical scenarios with nearly clear skies or relatively heavy rainfall, implying the robustness of the algorithm for extrapolation.

Reconstructing action potential and ionic current dynamics of a mammalian ventricular cell using data assimilation.

The dynamics of the cardiac ventricular myocyte action potential (AP) is regulated by intricate and nonlinear interactions between the cell transmembrane potential, and ionic currents and concentrations. Present technology limits the ability to simultaneously measuring multiple ionic currents and concentration during an AP in in vitro single cell experiments, which limits the scope of experimental data to provide a complete snapshot of the cardiac myocyte system state. This limitation presents an obstacle for understanding how cellular perturbations can trigger arrhythmogenic conditions. In this study, we applied a data assimilation approach, which combines limited noisy observations with predictions from a computational model, paired with a parameter estimation to minimize the effects of parameter uncertainty and model error. We performed a series of in silico experiments, in which a forecasting system was tasked with reconstructing AP dynamics of a virtual ventricular myocyte in the presence varying degrees of parameter and model error. With appropriate protocol design, we find that a data assimilation approach can successfully reconstruct and predict the dynamics of not only the directly observed state variables (such as the transmembrane potential and a selected ionic current), but also unobserved state variables, such as unmeasured ionic currents and concentrations. As an extension of the results of these series of in silico experiments, we generated simulations to assess which ionic currents, when measured, offer greater predictive ability using the data assimilation algorithm. This can offer valuable information for the design of single cardiac in vitro experiments, as it is not feasible to measure all ionic currents and concentrations across time within the same cell during an AP. Future work will investigate the longer-term accuracy of this approach predicting potentially pro-arrhythmic perturbations for arrhythmia risk assessment.

Full‐information estimation of heterogeneous agent models using macro and micro data

We develop a generally applicable full‐information inference method for heterogeneous agent models, combining aggregate time series data and repeated cross‐sections of micro data. To handle unobserved aggregate state variables that affect cross‐sectional distributions, we compute a numerically unbiased estimate of the model‐implied likelihood function. Employing the likelihood estimate in a Markov Chain Monte Carlo algorithm, we obtain fully efficient and valid Bayesian inference. Evaluation of the micro part of the likelihood lends itself naturally to parallel computing. Numerical illustrations in models with heterogeneous households or firms demonstrate that the proposed full‐information method substantially sharpens inference relative to using only macro data, and for some parameters micro data is essential for identification.

Harmonic Instability of LCL-Type Grid-Connected Inverter Caused by the Pole-Zero Cancellation: A Case Study

Pole-zero cancellation technique has drawn much attention to tackle the filter resonance of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> -type grid-connected inverter for simplifying the stability-oriented design. However, its potential stability risk was rarely concerned. As one representative of this technique, the weighted average current (WAC) control is assessed. It is found that the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">LCL</i> resonance in the state variable WAC is cancelled out by pole-zero cancellation, but the ones in some uncontrollable or unobservable state variables such as the grid current are still inherited, resulting in harmonic instability. From this perspective, this article appeals that this technique should be cautiously used or even avoided. An extended WAC control which avoids pole-zero cancellation is proposed accordingly. The theoretical expectations are finally validated by simulation and experimental results.

Structural identification via the inference of the stochastic volatility model conditioned on the time-dependent bridge deflection

The time-dependent deflection data collected by a structural health monitoring (SHM) system contains information indicating the damage accumulation of prestressed concrete bridges (PSCBs). However, for this information to be used, the results must be fully translated into reliable metrics. Model-based structural identification (SI) targeting the time-dependent deflection of PSCBs is affected by time-dependent material behavior such as creep and shrinkage as well as the unknown path of structural deterioration, which should be considered in a stochastic volatility model (SVM). This paper presents an inference framework for an SVM conditioned on the time-dependent deflection of PSCBs. In this framework, the augmented state space includes the full ranges of the unobserved state variables affecting the time-dependent deflection of PSCBs, namely, structural rigidity, creep, shrinkage, prestress level and dead load level, along with the volatility parameters associated with the deterioration path of structural rigidity, which is modeled via the Wiener process. Exploiting the latent structure of the SVM, a cyclic Markov chain Monte Carlo (MCMC) sampler is proposed to draw samples from the joint posterior distribution of the unobserved state variables and volatility parameters. In an illustrative example, two-year continuous deflection monitoring data of an existing bridge are utilized as the target information for inference. The updated model can indicate the accumulated damage of the case bridge over the monitoring period. The proposed SI framework is able to support accurate probabilistic analysis of the time-dependent deflection of PSCB.

Monte Carlo investigation of structural minimality for structures of uncontrolled linear switching systems with Maple

One path to understanding a physical system is to represent it by a model structure (collection of related models). Suppose our system is not subject to external influences, and depends on unobservable state variables (x), and observables (y). Then, a suitable uncontrolled, state-space model structure S is defined by relationships between x and y, involving parameters θ ∈ Θ. That is, each parameter vector in parameter space Θ is associated with a particular model in S.&#x0D; Before using S for prediction, we require system observations for parameter estimation. This process aims to determine θ values for which predictions “best” approximate the data (according to some objective function). The result is some number of estimates of the true parameter vector, θ*. Multiple parameter estimates are problematic when these cause S to produce dissimilar predictions beyond our data's range. This can render us unable to confidently make predictions, resulting in an uninformative study.&#x0D; Non-uniqueness of parameter estimates follows when S lacks the property of structural global identifiability (SGI). Fortunately, we may test S for SGI prior to data collection. The absence of SGI encourages us to rethink our experimental design or model structure.&#x0D; Before testing S for SGI we should check that it is structurally minimal. If so, we cannot replace S by a structure of fewer state variables which produces the same output. Most testing methodology is applicable to structures which employ the same equations for all time. These methods are not appropriate when, for example, a process has an abrupt change in its dynamics. For such a situation, a structure of linear switching systems (LSSs) may be suitable. Any system in the structure has a collection of linear time-invariant state-space systems, and a switching function which determines the system in effect at each instant. As such, we face a novel challenge in testing an LSS structure for SGI.&#x0D; We will consider the case of an uncontrolled LSS structure of one switching event (a ULSS-1 structure). In this setting, we may approach the structural minimality problem via the Laplace transform of the output function on each time interval. Each rational function yields conditions for pole-zero cancellation. If these conditions are not satisfied for almost all θ ∈ Θ, then S is structurally minimal.&#x0D; Analytical approaches can be quite laborious. However, we may expect a numerical approach to provide useful insights quickly. For example, if pole-zero cancellation occurs for almost all of a sufficiently large number of parameter values sampled from Θ, then structural minimality is possible. This result may encourage us to prove the existence of structural minimality.&#x0D; We shall use Maple 2020-2 to conduct a numerical investigation of structural minimality for a test case ULSS-1 structure applicable to flow-cell biosensor experiments used to monitor biochemical interactions, which include the popular Biacore-branded units.

Optimal Secondary Control to Suppress Voltage Fluctuations in an HVDC-Linked Wind Farm Grid

High-voltage direct-current (HVDC) systems using voltage source converters (VSCs) have been developed to suppress voltage fluctuations in offshore grids due to the intermittent power generation of offshore wind farms (OWFs). This paper proposes a new strategy for the optimal secondary voltage control of HVDC-linked OWFs. In the proposed strategy, optimal coordinated control of an HVDC system and OWFs is achieved to minimize real-time voltage fluctuations throughout an offshore grid. A full-order dynamic model of HVDC-linked OWFs is implemented, and its unobservable and uncontrollable state variables are then eliminated using a model reduction algorithm. Given the dynamic model, a linear quadratic Gaussian regulator is designed for optimal coordinated control, and an eigenvalue analysis is conducted focusing on the effects of communication time delays on the stability of the proposed strategy. Simulation case studies are also carried out to verify that the proposed strategy more effectively improves real-time regulation of offshore grid voltages, compared to conventional strategies, while ensuring voltage stability even for relatively large delays in communications systems.

On correlated measurement errors in the Schwartz–Smith two-factor model

Abstract The Schwartz–Smith two-factor model is commonly used for pricing of derivatives in commodity markets. For estimating and forecasting the term structures of futures prices, the logarithm of commodity spot price is represented as the sum of short- and long-term factors being the unobservable state variables. The futures prices derived as functions of the spot price lead to the simultaneous set of measurement equations, which is used for joint estimation of unobservable state variables and the model parameters through a filtering procedure. We propose a modified model where the error terms in the measurement equations are assumed to be serially correlated. In addition, for comparative analysis, the modelling of the logarithmic returns of futures prices is also considered. Out-of-sample prediction performances of two proposed models were illustrated using European Unit Allowances (EUA) futures prices from January 2017 to April 2021. Historically, this period corresponds to the second half of Phase III, and the beginning of Phase IV of the European Union Emission Trading System (EU-ETS).

Conditional Gaussian nonlinear system: A fast preconditioner and a cheap surrogate model for complex nonlinear systems.

Developing suitable approximate models for analyzing and simulating complex nonlinear systems is practically important. This paper aims at exploring the skill of a rich class of nonlinear stochastic models, known as the conditional Gaussian nonlinear system (CGNS), as both a cheap surrogate model and a fast preconditioner for facilitating many computationally challenging tasks. The CGNS preserves the underlying physics to a large extent and can reproduce intermittency, extreme events, and other non-Gaussian features in many complex systems arising from practical applications. Three interrelated topics are studied. First, the closed analytic formulas of solving the conditional statistics provide an efficient and accurate data assimilation scheme. It is shown that the data assimilation skill of a suitable CGNS approximate forecast model outweighs that by applying an ensemble method even to the perfect model with strong nonlinearity, where the latter suffers from filter divergence. Second, the CGNS allows the development of a fast algorithm for simultaneously estimating the parameters and the unobserved variables with uncertainty quantification in the presence of only partial observations. Utilizing an appropriate CGNS as a preconditioner significantly reduces the computational cost in accurately estimating the parameters in the original complex system. Finally, the CGNS advances rapid and statistically accurate algorithms for computing the probability density function and sampling the trajectories of the unobserved state variables. These fast algorithms facilitate the development of an efficient and accurate data-driven method for predicting the linear response of the original system with respect to parameter perturbations based on a suitable CGNS preconditioner.

Recurrent neural networks for partially observed dynamical systems.

Complex nonlinear dynamics are ubiquitous in many fields. Moreover, we rarely have access to all of the relevant state variables governing the dynamics. Delay embedding allows us, in principle, to account for unobserved state variables. Here we provide an algebraic approach to delay embedding that permits explicit approximation of error. We also provide the asymptotic dependence of the first-order approximation error on the system size. More importantly, this formulation of delay embedding can be directly implemented using a recurrent neural network (RNN). This observation expands the interpretability of both delay embedding and the RNN and facilitates principled incorporation of structure and other constraints into these approaches.

Decentralized Robust Frequency Regulation of Multi-terminal HVDC-linked Grids

This paper proposes a new strategy for optimal grid frequency regulation (FR) in an interconnected power system where regional ac grids and an offshore wind farm are linked via a multi-terminal high voltage direct-current (MTDC) network. In the proposed strategy, decentralized <i>H</i>_&#x221E; controllers are developed to coordinate the operations of synchronous generators and MTDC converters, thus achieving optimal power sharing of interconnected ac grids and minimizing frequency deviations in each grid. To develop the controllers, robust optimization problems are formulated and solved using a dynamic model of the hybrid MTDC-linked grids with model parameter uncertainty and decentralized control inputs and outputs. The model orders of the controllers are then reduced using a balanced truncation algorithm to eliminate unobservable and uncontrollable state variables while preserving their dominant response characteris- tics. Sensitivity and eigenvalue analyses are conducted focusing on the effects of grid measurements, parameter uncertainty levels, and communication time delays. Comparative case studies are also carried out to verify that the proposed strategy improves the effectiveness, stability, and robustness of real-time FR in MTDC-linked grids under various conditions.

High accuracy estimation technique of radiated sound from structural vibration by virtual sensing

The present study is concerned with the application of radiated sound simulation by the virtual sensing. The virtual sensing is a method to estimate unobservable or unmeasurable state variables by combining exsisting measurement data with model information. When FE model is applied to the virtual sensing, vibration response data estimated at all nodes. Therefore, it enables to simulate sound radiation from mechanical structure with vibration response data. In this paper, it is illustrated how to estimate vibration response data, and simulate sound radiation. It then examines the effect of excitation force on the vibration estimation results. Based on the results of this verification, a method to improve the accuracy of vibration response estimation is proposed. Finally, it tests how the quality of estimation for sound radiation.

An efficient continuous data assimilation algorithm for the Sabra shell model of turbulence.

Complex nonlinear turbulent dynamical systems are ubiquitous in many areas of research. Recovering unobserved state variables is an important topic for the data assimilation of turbulent systems. In this article, an efficient continuous-in-time data assimilation scheme is developed, which exploits closed analytic formulas for updating the unobserved state variables. Therefore, it is computationally efficient and accurate. The new data assimilation scheme is combined with a simple reduced order modeling technique that involves a cheap closure approximation and noise inflation. In such a way, many complicated turbulent dynamical systems can satisfy the requirements of the mathematical structures for the proposed efficient data assimilation scheme. The new data assimilation scheme is then applied to the Sabra shell model, which is a conceptual model for nonlinear turbulence. The goal is to recover the unobserved shell velocities across different spatial scales. It is shown that the new data assimilation scheme is skillful in capturing the nonlinear features of turbulence including the intermittency and extreme events in both the chaotic and the turbulent dynamical regimes. It is also shown that the new data assimilation scheme is more accurate and computationally cheaper than the standard ensemble Kalman filter and nudging data assimilation schemes for assimilating the Sabra shell model with partial observations.

Option Pricing with Markov Switching

In this article, we consider a model of time-varying volatility which generalizes the classical Black-Scholes model to include regime-switching properties. Specically, the unobservable state variables for stock uctu- ations are modeled by a Markov process, and the drift and volatility pa- rameters take dierent values depending on the state of this hidden Markov process. We provide a closed-form formula for the arbitrage-free price of the European call option, when the hidden Markov process has nite number of states. Two simulation methods, the discrete diusion method and the Markovian tree method, for computing the European call option price are presented for comparison.

Nearly Exact Bayesian Estimation of Non-linear No-Arbitrage Term-Structure Models

Abstract We propose a general method for the Bayesian estimation of a very broad class of non-linear no-arbitrage term-structure models. The main innovation we introduce is a computationally efficient method, based on deep learning techniques, for approximating no-arbitrage model-implied bond yields to any desired degree of accuracy. Once the pricing function is approximated, the posterior distribution of model parameters and unobservable state variables can be estimated by standard Markov Chain Monte Carlo methods. As an illustrative example, we apply the proposed techniques to the estimation of a shadow-rate model with a time-varying lower bound and unspanned macroeconomic factors.