Set Of Admissible Models Research Articles

Model Error (or Ambiguity) and Its Estimation, with Particular Application to Loss Reserving

This paper is concerned with the estimation of forecast error, particularly in relation to insurance loss reserving. Forecast error is generally regarded as consisting of three components, namely parameter, process and model errors. The first two of these components, and their estimation, are well understood, but less so model error. Model error itself is considered in two parts: one part that is capable of estimation from past data (internal model error), and another part that is not (external model error). Attention is focused here on internal model error. Estimation of this error component is approached by means of Bayesian model averaging, using the Bayesian interpretation of the LASSO. This is used to generate a set of admissible models, each with its prior probability and likelihood of observed data. A posterior on the model set, conditional on the data, may then be calculated. An estimate of model error (for a loss reserve estimate) is obtained as the variance of the loss reserve according to this posterior. The population of models entering materially into the support of the posterior may turn out to be “thinner” than desired, and bootstrapping of the LASSO is used to increase this population. This also provides the bonus of an estimate of parameter error. It turns out that the estimates of parameter and model errors are entangled, and dissociation of them is at least difficult, and possibly not even meaningful. These matters are discussed. The majority of the discussion applies to forecasting generally, but numerical illustration of the concepts is given in relation to insurance data and the problem of insurance loss reserving.

Identification of SVAR Models by Combining Sign Restrictions With External Instruments

ABSTRACT We discuss combining sign restrictions with information in external instruments (proxy variables) to identify structural vector autoregressive (SVAR) models. In one setting, we assume the availability of valid external instruments. Sign restrictions may then be used to identify further orthogonal shocks, or as an additional piece of information to pin down the shocks identified by the external instruments more precisely. In a second setting, we assume that proxy variables are only “plausibly exogenous” and suggest various types of inequality restrictions to bound the relation between structural shocks and the external variable. This can be combined with conventional sign restrictions to further narrow down the set of admissible models. Within a proxy-augmented SVAR, we conduct Bayesian inference and discuss computation of Bayes factors. They can be useful to test either the sign- or IV restrictions as overidentifying. We illustrate the usefulness of our methodology in estimating the effects of oil supply and monetary policy shocks.

Data-Driven Computation of Robust Control Invariant Sets With Concurrent Model Selection

Set invariance in the presence of uncertainty and disturbance is of central importance for the safety of control systems. This article proposes a data-driven method to compute an approximation of a minimal robust control invariant set (mRCI) from experimental data. For a given dynamical model with additive and multiplicative uncertainty, the proposed method is able to compute a polytopic mRCI with fixed complexity via linear programming (LP). Moreover, the method can be combined with model selection to enable mRCI computation directly from experiment data when the system dynamics are unknown. Specifically, given a model structure, our algorithm begins by identifying the set of admissible models with constraints extracted from the experimental data. Each model in the set of admissible models contains information about the nominal model and the characterization of the model uncertainties. Then, two iterative algorithms based on robust optimization are proposed to compute an mRCI while simultaneously searching for a model “optimal” with regard to the mRCI computation and the corresponding invariance-inducing controller. Finally, the method is demonstrated in an experiment with an autonomous vehicle lane-keeping control example.

Adaptive model predictive control for linear time varying MIMO systems

A robust, adaptive Model Predictive Control (MPC) approach for asymptotically stable, constrained linear time-varying (LTV) systems with multiple inputs and outputs is proposed. The approach consists of two-steps, carried out on-line with a receding horizon strategy. In the first one, a real-time Set Membership identification algorithm exploits the measured input–output data and the available prior knowledge to build and refine a set of admissible models of the plant (Feasible Parameter Set, FPS). This set is guaranteed to contain also the true system dynamics under the considered working assumptions. In the second step, a robust finite-horizon optimal control problem is formulated and solved. The variation of system dynamics is taken into account by inflating the FPS over the prediction horizon, according to worst-case bounds, assumed a priori, on the parameters’ rate of change. The resulting optimal control sequence guarantees that the outputs of all possible plants inside the FPS satisfy the operational constraints, also considering all possible future parameter changes. The main theoretical properties of the proposed approach are demonstrated and the method is showcased in numerical simulations, highlighting the fundamental improvement over previous approaches not designed for LTV systems.

Adaptive control with optimal tracking performance

ABSTRACTThis paper provides a way to optimise the steady-state tracking performance of the adaptive control system in the presence of unknown external disturbances. A-priori knowledge of the dynamic model of the reference signal to be tracked is not completely known. Especially, the updatable non-empty admissible model set, which is consistent to the a-priori knowledge of the plant parameter and the online measurements, is computed. Treating the overall system performance as the criteria, the nominal model is optimally chosen within the admissible model set. The optimal nominal model is subsequently used to synthesise the optimal closed-loop controller that minimises the steady-state absolute value of the tracking error. Combining the above two aspects, an optimal adaptive control scheme is proposed. Because of the consistency of the identification criteria and control object, the adaptive control scheme proposed in this paper can achieve the overall optimal steady-state tracking performance, and the effect of the interplay between the identification and control of the adaptive system can be handled effectively. In addition, the computable optimal performance is also provided.

The differential effects of oil demand and supply shocks on the global economy

We employ a set of sign restrictions on the impulse responses of a Global VAR model, estimated for 38 countries/regions over the period 1979Q2–2011Q2, as well as bounds on impact price elasticities of oil supply and oil demand to discriminate between supply-driven and demand-driven oil-price shocks, and to study the time profile of their macroeconomic effects across a wide range of countries and real/financial variables. We show that the above identification scheme can greatly benefit from the cross-sectional dimension of the GVAR—by providing a large number of additional cross-country sign restrictions and hence reducing the set of admissible models. The results indicate that the economic consequences of a supply-driven oil-price shock are very different from those of an oil-demand shock driven by global economic activity, and vary for oil-importing countries compared to energy exporters. While oil importers typically face a long-lived fall in economic activity in response to a supply-driven surge in oil prices, the impact is positive for energy-exporting countries that possess large proven oil/gas reserves. However, in response to an oil-demand disturbance, almost all countries in our sample experience long-run inflationary pressures, an increase in real output, a rise in interest rates, and a fall in equity prices.

Oil Price Volatility and the Role of Speculation

How much does speculation contribute to oil price volatility? We revisit this contentious question by estimating a sign-restricted structural vector autoregression (SVAR). First, using a simple storage model, we show that revisions to expectations regarding oil market fundamentals and the effect of mispricing in oil derivative markets can be observationally equivalent in a SVAR model of the world oil market a la Kilian and Murphy (2013), since both imply a positive co-movement of oil prices and inventories. Second, we impose additional restrictions on the set of admissible models embodying the assumption that the impact from noise trading shocks in oil derivative markets is temporary. Our additional restrictions effectively put a bound on the contribution of speculation to short-term oil price volatility (lying between 3 and 22 percent). This estimated short-run impact is smaller than that of flow demand shocks but possibly larger than that of flow supply shocks.

A pricing kernel approach to valuing options on interest rate futures

This paper builds on existing asset pricing models in an intertemporal capital asset pricing model framework to investigate the pricing of options on interest rate futures. It addresses the issues of selecting the preferred pricing kernel model by employing the second Hansen–Jagannathan distance criterion. This criterion restricts the set of admissible models to those with a positive stochastic discount factor that ensures the model is arbitrage-free. The results indicate that the three-term polynomial pricing kernel with three non-wealth-related state variables, namely the real interest rate, maximum Sharpe ratio, and implied volatility, clearly dominates the other candidates. This pricing kernel is always strictly positive and everywhere monotonically decreasing in market returns in conformity with economic theory.

Model selection in regression under structural constraints

The paper considers model selection in regression under the additional structural constraints on admissible models where the number of potential predictors miht be even larger than the available sample size. We develop a Bayesian formalism which is used as a natural tool for generating a wide class of model selection criteria based on penalized least squares estimation with various complexity penalties associated with a prior on a model size. The resulting criteria are adaptive to structural constraints. We establish the upper bound for the quadratic risk of the resulting MAP estimator and the corresponding lower bound for the minimax risk over a set of admissible models of a given size. We then specify the class of priors (and, therefore, the class of complexity penalties) where for the “nearly-orthogonal” design the MAP estimator is asymptotically at least nearly-minimax (up to a log-factor) simultaneously over an entire range of sparse and dense setups. Moreover, when the numbers of admissible models are “small” (e.g., ordered variable selection) or, on the opposite, for the case of complete variable selection, the proposed estimator achieves the exact minimax rates.

An adaptive control with optimal disturbances rejection

This paper provides a way to optimize the overall disturbances rejection performance of the adaptive control system in the presence of unknown external disturbances. Especially, the updatable non-empty admissible model set, which is consistent to the a priori knowledge of the plant parameter and the online measurements, is computed. With the overall system performance as the criteria, the nominal model is optimally chosen within the admissible model set. The optimal nominal model is subsequently used to synthesize the optimal closed-loop controller based on the l1 design methodology. Combining the above two aspects, an optimal adaptive control scheme is proposed. Because of the consistency of the identification criteria and control object, the adaptive control scheme proposed in this paper can achieve the overall optimal disturbances rejection performance, and the effect of the interplay between the identification and control of the adaptive system can be handled effectively. In addition, the computable optimal performance is also provided.

A Pricing Kernel Approach to Valuing Options on Interest Rate Futures

This paper builds on existing asset pricing models in an intertemporal CAPM framework to investigate the pricing of options on interest rate futures. It addresses the issues of selecting the preferred pricing kernel model by employing the second Hansen-Jagannathan distance (HJD) criterion. This criterion restricts the set of admissible models to those with a positive stochastic discount factor that ensures the model is arbitrage free. The results indicate that the 3-term polynomial pricing kernel with three non-wealth related state variables - the real interest rate, maximum Sharpe ratio, and implied volatility - clearly dominates the other candidates. The pricing kernel is always strictly positive and everywhere monotonically decreasing in market returns in conformity with economic theory.

Step Tracking in the Presence of Persistent Plant Changes

In this paper, we consider the use of periodic controllers for simultaneous stabilization and step tracking, including the case when there are occasional, though persistent, plant changes. We consider the case of a finite set of admissible models; we provide a design procedure that yields a controller which stabilizes each such model while providing near optimal LQR step tracking. We then demonstrate that this controller has the facility to tolerate occasional (but persistent) switches between these models without an undue effect on the performance. The controller is periodic with a slight nonlinearity.

Reasoning under minimal upper bounds in propositional logic

Reasoning from the minimal models of a theory, as fostered by circumscription, is in the area of Artificial Intelligence an important method to formalize common sense reasoning. However, as it appears, minimal models may not always be suitable to capture the intuitive semantics of a knowledge base, aiming intuitively at an exclusive interpretation of disjunctions of atoms, i.e., if possible then assign at most one of the disjuncts the value true in a model. In this paper, we consider an approach which is more lenient and also admits non-minimal models, such that inclusive interpretation of disjunction also may be possible in cases where minimal model reasoning adopts an exclusive interpretation. Nonetheless, in the spirit of minimization, the approach aims at including only positive information that is necessary. This is achieved by closing the set of admissible models of a theory under minimal upper bounds in the set of models of the theory, which we refer to as curbing. We demonstrate this method on some examples, and investigate its semantical and computational properties. We establish that curbing is an expressive reasoning method, since the main reasoning tasks are shown to be PSPACE-complete. On the other hand, we also present cases of lower complexity, and in particular cases in which the complexity is located, just as for ordinary minimal model reasoning, at the second level of the Polynomial Hierarchy, or even below.

FAULT TOLERANT CONTROL OF THE BOEING 747 SHORT-PERIOD MODE USING THE ADMISSIBLE MODEL MATCHING TECHNIQUE

The paper analyzes the different model matching approaches (Exact, Pseudo-Inverse Method / PIM, Modified Pseudo-Inverse Method / MPIM) for the fault tolerant control of the short-period mode of the longitudinal model of the Boeing 747 aircraft. To overcome the drawbacks of the PIM and MPIM approaches, the paper proposes a new philosophy for the model matching problem, namely to use a set of admissible models instead of trying to best approximate a single reference one. The application shows the benefits associated with using this approach.

FAULT TOLERANT CONTROL : THE PSEUDO-INVERSE METHOD REVISITED

In the pseudo-inverse method (PIM) the Frobenius norm based distance between the closed loop model of the faulty system and some reference model is minimized. Stability issues are considered in the Modified PIM (MPIM). This paper proposes to use a set of admissible models, rather than searching for an optimal one which does not provide any stability / adequation guarantee. The approach allows to characterize the set of accommodable faults, and to quantify the robustness of the fault adaptation scheme.

Knowledge bounded least squares method for the identification of fuzzy systems

This paper presents the knowledge bounded least squares method that uses both linguistic information (i.e., human knowledge and experience) and numerical data to identify fuzzy models. Based on the concept of fuzzy interval systems, the basic idea of this method is: first, to utilize all the available linguistic information to obtain a fuzzy interval system and then use the obtained fuzzy interval system to give the admissible model set (i.e., the set of all fuzzy models which are acceptable and reasonable from the point of view of linguistic information); second, to find a fuzzy model in the admissible fuzzy model set which best fits the available numerical data. It is shown that such a fuzzy model can be obtained by a quadratic programming approach. By comparing this method with the least squares method, it is proved that the fuzzy model obtained by the proposed method fits the real model better than the fuzzy model obtained by the least squares method.

Fuzzy bounded least-squares method for the identification of linear systems

This paper presents the fuzzy bounded least-squares method which uses both linguistic information and numerical data to identify linear systems. This method introduces a new type of fuzzy system, i.e., a fuzzy interval system. The steps in the method are as follows: 1) to utilize all the available linguistic information to obtain a fuzzy interval system and then to use the fuzzy interval system to give the admissible model set (i.e., the set of all models which are acceptable and reasonable from the point of view of linguistic information); 2) to find a model in the admissible model set which best fits the available numerical data. It is shown that such a model can be obtained by a quadratic programming approach. By comparing this method with the least-squares method, it is proved that the model obtained by this method fits a real system better than the model obtained by the least-squares method. In addition, this method also checks the adequacy of linear models for modeling a given system during the identification process and can help one to decide whether it is necessary to use nonlinear models.

Unstable ones in understood algebraic questions of modelling for control design

The use of a necessarily approximate system model for the design of a feedback controller for an actual plant introduces strictures on the quality of the model which are different from those pertaining in open loop. This has been the focus of a raft of recent papers on closed loop modelling, model validation, and iterative control design where appropriate measures of approximation quality have been proposed, analysed and implemented. Our attention here will concentrate on underlying implicit or understood questions of an algebraic nature which need to be resolved prior to any quantitative analysis of approximation. For example, it should be evident that a model P is useful as a design input for a controller for the actual plant P only if the pair ([Pcirc], P) is simultaneously stabilizable. Yet this question has not really been asked. For given P and T we characterize the set of admissible models {[Pcirc]}, where admissible means that a controller designed from [Pcirc] and T yields a stable closed loop. Necessary conditions on (P, T) are derived for this set to be nonempty; they connect the unstable poles and zeros of P on the positive real axis with the unstable zeros and ones of T on that axis.

An Intelligent System For Automated MathematicalModelling And Simulation Of DynamicalEngineering Systems

We describe in this paper a computer program for Automated Mathematical Modelling (AMM) and Automated Simulation (AS) of dynamical engineering systems using Artificial Intelligence (AI) techniques. This computer program is an implementation of a new method for AMM using Fuzzy Logic techniques and Fractal Theory, and a new method for AS using Expert Systems technology. Our new method for AMM consists of three main parts: Time Series Analysis, Developing a set of Admissible Models and Selecting the Best model. Our method for Time Series Analysis consists in the use of the fractal dimension of a set of points as a measure of the geometrical complexity of the time series. Our method for developing a set of admissible models is based on the use of Fuzzy Logic techniques to simulate the reasoning process of the human experts in mathematical modelling of engineering systems. The selection of the best model for the engineering systems is done using heuristics from the experts and statistical calculations. The simulation of the best model can be done by using a new method developed by the authors, that enables automated computer exploration of all the dynamical behaviors of the engineering system. Given a mathematical model for a specific Dynamical Engineering System (DES), this method will automatically select (using a rule base) the best set of parameter values to perform numerical simulations of the system. This simulations will in turn enable the identification of all the dynamical behaviors of the engineering system.