Non-Linear dynamic models

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Bayesian dynamic linear models for structural health monitoring

Bayesian dynamic linear models for structural health monitoring

A key challenge remaining for enabling large-scale applications of SHM is to identify from time-series, the baseline response of structures without the effect of external actions such as temperature and traffic. To succeed at this task, a methodology must be able to operate seamlessly in conditions with frequent outliers and missing data. This paper proposes a framework for modeling the time-dependent responses of structures by breaking it into generic components, each having its own specific mathematical formulation. This new framework is a generalization of Bayesian Dynamic Linear Models (BDLM) for the field of structural health monitoring (SHM).
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Modelling Environmental Effect Dependencies with Principal Component Analysis and Bayesian Dynamic Linear Models

Modelling Environmental Effect Dependencies with Principal Component Analysis and Bayesian Dynamic Linear Models

CHAPTER 5 CONCLUSION AND RECOMMENDATIONS 5.1 Summary of work This master thesis presents a solution to tackle the issue of indeterminate systems caused by correlated environmental effect observations using Principal Component Analysis and adapted to Bayesian Dynamic Linear Models. For the purpose of illustrating the method’s capacity, an example is built using hourly data of displacement and four temperature obser- vations from a highway concrete bridge located in Canada. Four models are built using all the temperature observations and 1 to 4 PCs, and four models are built using one temper- ature observation at the time. With this dataset, using 2 to 4 principal components with the PCA method produces models with better prediction capacities than the models without the proposed method. The proposed method also increases the robustness and reliability as redundant sensors are necessary, and a simultaneous failure of multiple sensors is unlikely. Thus, in general, if one is confronted to a dataset with multiple environmental effect sensors, for example temperature, humidity, and solar radiation, the recommendation is to include all the available data in the model. The method enables the user to avoid discarding datasets that include information relevant for explaining the structural response.
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Empirical validation of bayesian dynamic linear models in the context of structural health monitoring

Empirical validation of bayesian dynamic linear models in the context of structural health monitoring

Bayesian Dynamic Linear Models (BDLM) are traditionally employed in the fields of applied statis- tics and Machine Learning. This paper performs an empirical validation of BDLM in the context of Structural Health Monitoring (SHM) for separating the observed response of a structure into sub- components. These sub-components describe the baseline response of the structure, the effect of traffic, and the effect of temperature. This utilization of BDLM for SHM is validated with data recorded on the Tamar Bridge (UK). This study is performed in the context of large-scale civil structures where missing data, outliers and non-uniform time steps are present. The study shows that the BDLM is able to sepa- rate observations into generic sub-components allowing to isolate the baseline behavior of the structure. Keywords: Structural Health Monitoring (SHM), Bayesian, Dynamic Lineal Models, Kalman Filter, Bridge, infrastructure, Tamar Bridge
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Bayesian Nonparametric Inference of Switching Dynamic Linear Models

Bayesian Nonparametric Inference of Switching Dynamic Linear Models

VI. C ONCLUSION In this paper, we have addressed the problem of learning switching linear dynamical models with an unknown number of modes for describing complex dynamical phenomena. We presented a Bayesian nonparametric approach and demonstrated both the utility and versatility of the developed HDP-SLDS and HDP-AR-HMM on real applications. Using the same parameter settings, although different model choices, in one case we are able to learn changes in the volatility of the IBOVESPA stock exchange while in another case we learn segmentations of data into waggle, turn-right, and turn-left honey bee dances. We also described a method of applying automatic relevance determination (ARD) as a sparsity-inducing prior, leading to flexible and scalable dynamical models that allow for identification of variable order structure. We concluded by considering adaptations of the HDP-SLDS to specific forms often examined in the literature such as the Markov switching stochastic volatility model and a standard multiple model target tracking formulation.
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Prediction of non-linear time-variant dynamic crop model using bayesian methods

Prediction of non-linear time-variant dynamic crop model using bayesian methods

In the current work, we propose to extend the variational filter to better handle non-linear and non- Gaussian processes (where no a priori information about the states and parameters is available), by assuming time-varying statistical parameters. The objectives of the paper are to compare three methods (EKF, PF and VF) for estimating important state variables of crop models. The LAI (Leaf Area Index) determines the photosynthetic primary production and the plant evapotranspiration and is thus a key state to characterize the plant growth. The moisture content of two soil layers (20 and 50 cm) was considered as it affects the capacity of plants to extract water and soil nutrients. The comparison will rely on the computation of the RMSE and the number of parameters that can be accurately predicted. The model used in this study is mini STICS (Makowski et al., 2006). Using mini STICS model has several advantages since: (1) it can reduce the computing and execution times of the STICS model; and (2) it has the nice property to be a good dynamic model, ensuring the robustness of data processing and estimation.
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Dynamic linear economies with social interactions

Dynamic linear economies with social interactions

The simplicity of linear models allows us to extend our analysis in several directions which are important in applications and empirical work. This is the case, for instance of general (including asymmetric) neighborhood network structures for social interactions. But our analysis extends also to general stochastic processes for preference shocks and to the addition of global interactions. One particular form of global interactions occurs when each agent’s preferences depend on an average of actions of all other agents in the population, e.g. Brock and Durlauf (2001a), and Glaeser and Scheinkman (2003). This is the case, for instance, if agents have preferences for social status. More generally, global interactions could capture preferences to adhere to aggregate norms of behavior, such as specific group cultures, or other externalities as well as price effects. Finally, and perhaps most importantly, we extend our analysis to encompass a richer structure of dynamic dependence of agents’ actions at equilibrium. In particular we study an economy in which agents’ past behavior is aggregated through an accumulated stock variable which carries habit persistence, which can be directly applied e.g., to the issue of teenage substance addiction due to peer pressure at school. With respect to the addiction literature, as e.g., Becker and Murphy (1988), we model the dynamics of addiction considering peer effects not only in a single-person decision problem, but rather as an equilibrium effect allowing for the intertemporal feedback channel between agents across social space and through time. 12 In this context we show that in
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Dynamic Factor Models

Dynamic Factor Models

The one-step methods used in all these papers (most of them following Kim’s (1994) approximation of the likelihood, others relying on Kim and Nelson’s (1998) Gibbs sampling) have been successful in estimating MS-DFMs of very small dimensions. In order to estimate MS-DFM of larger dimensions, it is possible to take advantage of the two-step approach originally applied to a small number of variables by Diebold and Rudebusch (1996). Indeed, it is possible to use a two-step approach similar to the one of Doz et al. (2011) in a standard DFM framework: in the first step, a linear DFM is estimated by principal components, in the second step, a Markov-switching model, as in Hamilton (1989), is specified for the estimated factor(s) and is estimated by maximum likelihood. Camacho et al. (2015) compared this two-step approach to a one-step approach applied to a small dataset of coincident indicators. They concluded that the one-step approach was better at turning point detection when the small dataset contained good quality business cycle indicators, and they also observed a decreasing marginal gain in accuracy when the number of indicators increased. However other authors have obtained satisfying results with the two-step approach. Bessec and Bouabdallah (2015) applied MS-factor MIDAS models 5 to a large dataset of mixed frequency variables. They ran Monte Carlo simulations and applied their model to US data containing a large number of financial series and real GDP growth: in both cases the model properly detected recessions. Doz and Petronevich (2016) also used the two-step approach: using French data, they com- pared business cycle dates obtained from a one factor MS-DFM estimated on a small dataset with Kim’s (1994) method, to the dates obtained using a one factor MS-DFM estimated in two steps from a large database. The two-step approach successfully predicted the turning point dates released by the OECD. As a complement, Doz and Petronevich (2017) conducted a Monte-Carlo experiment, which provided evidence that the two-step method is asymptotically valid for large N and T and provides good turning points prediction. Thus the relative performances of the one-step and two-step methods under the MS-DFM framework are worth exploring further, both from a turning point detection perspective and a forecasting perspective.
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Convex optimization in identification of stable non-linear state space models

Convex optimization in identification of stable non-linear state space models

I. I NTRODUCTION Converting numerical data, originating from either phys- ical measurements or computer simulations, to compact mathematical models is a common challenge in engineering. The case of static system identification, where models y = h(u) defined by “simple” functions h(·) are fitted to data records of u and y, is a major topic of research in statistics and machine learning. This paper is focused on a subset of dynamic system identification tasks, where state space models of the form

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An approximate dynamic programming approach to solving dynamic oligopoly models

An approximate dynamic programming approach to solving dynamic oligopoly models

Theorems 4.1, 4.2 and 4.3 together establish the quality of our approximation to a best response com- puted via a tractable linear program (4.10). In particular, we showed that by sampling a sufficiently large, but tractable, number of constraints via an appropriate sampling distribution, one could compute an approximate best response whose quality is similar to that of an approximate best response computed via the intractable linear program (4.7); the quality of that approximate best response was established in the preceding section. This section established a tractable computational scheme to compute an approximation to the best response operator F (·) in step 4 of Algorithm 1: in particular, we suggest approximating F (µ, λ) by a strategy ˜ µ satisfying T µ,λ µ ˜ Φ˜ r µ,λ = T µ,λ Φ˜ r µ,λ where ˜ r µ,λ is an optimal solution to the tractable LP (4.10). A number of details, such as the choice of basis functions Φ, and the precise way the state relevant weights c and sampling distribution ψ are determined, remain unresolved. These will be addressed in subsequent sections where we describe our algorithm (in the format of a ‘guide to implementation’) to compute an approximation to MPE precisely, and discuss our computational experiments. The implementation issues that we discuss are key to the efficiency of our algorithm and the accuracy of our approximation.
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Toward robust parameterized reduced-order models of non-linear structures using POD

Toward robust parameterized reduced-order models of non-linear structures using POD

Methods for the reduction of linear structures have been intensively studied in the past and are now well-established (e.g. component mode technique, balanced truncation, moment matching). Some exten- sions of these linear methods have been proposed in order to account for changes in the model parameters. However, these reduction methods failed to give acceptable results when applied to non-linear structures. Several model reduction techniques have been recently developed for non-linear dynamic systems. Most of them involve the selection of modes that capture the dominant behavior of the system to be sim- ulated. The system dynamics is then projected on a low-dimensional space spanned by the modes via a Galerkin process. A popular approach is the Proper Orthogonal Decomposition [1] (a.k.a. the Karhunen- Loève expansion (KLE), principal component analysis (PCA) and empirical orthogonal function (EOF)). In order to construct the POD modes, the full model is first simulated and some relevant snapshots of the numerical results are considered [2]. The POD provides an orthonormal basis that allows to represent the given data in a least-square optimal sense. The POD method is thus a data-driven method. Unfortunately, the POD-based ROM is sensitive to parameter variations. Indeed, any significant parameter change re- quires to rebuild the modal basis, thereby involving a computationally expensive simulation of the full model. The development of more robust ROM with respect to parameter changes is now an important research field.
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Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors

Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors

The resolution of such a sensor is given by the minimum detectable frequency shift which is inversely proportional to the resonator drive amplitude. It is limited by the onset of nonlinearities which occur proportionally sooner with respect to the device size [4, 5]. Moreover, below the bistability limit, it is extremely di fficult to detect the oscillations of such small sensors. Hence, it is important to model the nonlinear dynamics of NEMS-based resonant sensors at large amplitudes. The dynamic behavior of MEMS has been investigated by several authors so far. A survey of the electrostatic force modeling, a comprehensive review on MEMS modeling and an overview of the nonlinear dynamics of MEMS can be found in [6], [7] and [8]. Existing models are of variable complexity, depending on assumptions concerning the mechanical and electrostatic non- linearities. For example a simple lumped spring-mass model was considered in [9, 10] and the continuous linear Euler-Bernoulli beam theory was used in [11]. Nevertheless, several studies pointed out the importance of deformation dependent mid-plane stretching for axially constrained microbeams undergoing large-amplitude vibration [12] and the nonlinear Euler-Bernoulli beam theory is therefore used in most recent studies [13–21]. The electrostatic force is usually modeled using the parallel-plate approximation together with correction terms accounting for the fringing field e ffects due to beam’s finite width [22]. In order to facilitate the use of numerical or analytical methods, the nonlinearity of the electrostatic forcing term was further simplified by means of a high-order Taylor expansion in [10, 13, 21, 23] or a least squares approximation in [15].
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Fast smoothing in switching approximations of non-linear and non-Gaussian models

Fast smoothing in switching approximations of non-linear and non-Gaussian models

Organization of the paper is the following. We show in Section 2 that fast exact smoothing is feasible in the general CMSHLM model. We recall in Section 3 the SCGOMSM model, specifying why it is a particular CMSHLM. SCGOMSM is then used as an approximation of (1), as specified in Section 4. Some experimental results within the context of stochastic volatility [11, 12, 13, 14, 15, 16, 17, 18, 19] and the dynamic beta regression [20] are presented in Section 5. The last Section exposes conclusions and perspectives.

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Dynamic inflation of non-linear elastic and viscoelastic rubberlike membranes

Dynamic inflation of non-linear elastic and viscoelastic rubberlike membranes

Another diculty of such models implementation is that the kinematical variables (elementary local axes for example) change during the time step. As we cannot evaluate this change, we have to approximate it. Therefore, we assume that the principal directions of the deformation tensor remain constant during the time step t between discrete times t − t and t, and that these directions are equal to the principal directions at current time t. Because of large rotations involved in the present problem, this stress update procedure must be used with very small load steps. Here, because of the time-integration procedure, the time steps must be less than the critical time step to ensure convergence. This critical time-step size is a function of elements size and of the constitutive equation. It is ever very small [15]. Thus, the corresponding load steps are suciently small to adopt the stress update method. Rachik et al. [18] use a similar procedure for the implementation of a di erential viscoelastic constitutive law in the context of blow-moulding and thermoforming processes simulation. The previous recurrence formula (29) remains valid only for the principal directions
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Estimating the Number of Regimes of Non-linear Autoregressive Models.

Estimating the Number of Regimes of Non-linear Autoregressive Models.

els which allow the series to switch between regimes and we propose to study such models without knowing the form of the density of the noise. The problem we address here is how to select the number of components or number of regimes. One possible method to answer this problem is to consider penalized criteria. The consistency of a modified BIC crite- rion was recently proven in the framework of likelihood criterion for linear switching models (see Olt´eanu and Rynkiewicz 2012). We extend these results to mixtures of nonlinear autoregressive models with mean square error criterion and prove the consistency of a penalized estimate for the number of components under some regularity conditions.
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Analysis of linear structures with non linear dampers

Analysis of linear structures with non linear dampers

Figure 3 : Illustration of the Newton-Raphson method for solv- ing non linear equations This method encounters several problems with Equation 11 which has to be solved. They result mainly from the vertical slope inflexion point in the function F. Indeed, the first term in this function (see Eq. 12) is linear whereas the second one has the same shape as the constitutive law. The function F exhibits therefore a vertical slope inflexion point.

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Non destructive testing using non linear vibroacoustic

Non destructive testing using non linear vibroacoustic

We recall some papers related to the use of the frequency response for non destructive testing; in particular generation of higher harmonics, cross-modulation of a high frequency by a lo[r]

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Limits on Non-Linear Electrodynamics

Limits on Non-Linear Electrodynamics

Laboratoire National des Champs Magn´etiques Intenses (UPR 3228, CNRS-UPS-UJF-INSA), F-31400 Toulouse Cedex, France, EU (Dated: 12 mai 2016) In this paper we set a framework in which experiments whose goal is to test QED predictions can be used in a more general way to test non-linear electrodynamics (NLED) which contains low-energy QED as a special case. We review some of these experiments and we establish limits on the different free parameters by generalizing QED predictions in the framework of NLED. We finally discuss the implications of these limits on bound systems and isolated charged particles for which QED has been widely and successfully tested.
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Stability analysis and gain synthesis for Lipschitz non linear systems under dynamic event triggered sampling

Stability analysis and gain synthesis for Lipschitz non linear systems under dynamic event triggered sampling

In recent year the concept of dynamic event trigger [9] has been proposed in order to further reduce communication (see also [19] and [5]). The main idea of dynamic event trigger is to create a new dynamic including the static event triggering policy acting as a ’filter’ to the static event triggering criterion. The main objective of this paper is to address the problem of dynamic event trigger stabilization of a Lipschitz nonlinear system. Namely we aim at providing numerically tractable sufficient conditions to synthesize a dynamic event triggering policy. The usual methodology to address stability is to consider a gain obtained from the continuous time model and to show that the resulting event triggered system is stable, this emulation (of the continuous model) methodology will be first considered and sufficient conditions for known gain analysis will be considered. Then, sufficient conditions for gain synthesis directly based on the hybrid model will be given, thus avoiding an emulation based approach.
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Development and assessment of non-linear and non-stationary seasonal rainfall forecast models for the Sirba watershed, West Africa

Development and assessment of non-linear and non-stationary seasonal rainfall forecast models for the Sirba watershed, West Africa

4. Conclusion Seasonal forecast models, with either changing parameters or constant parameters, were devel- oped and tested in this study, using three predictors (air temperature, sea level pressure and relative humidity). Normalized Bayes factors, and graphs of the likelihood of forecasted rainfall under each model, were compared. It was found that the best seasonal rainfall forecast model uses air temperature as the predictor and allows parameter changes according to rainfall magnitude. Thus, seasonal fore- cast models with changing parameters could be the best for seasonal rainfall forecasting in the Sirba watershed. Indeed, changes in the predictand–predictor relationship according to rainfall amplitude, combined with the Bayesian model selection procedure, appear to be the best technique for forecasting seasonal rainfall in the Sahel.
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Learning Linear Transformations using models

Learning Linear Transformations using models

were also important in focusing students’ attention in some aspects of the model that they had not previously taken into account. The analysis of students’ productions and discussions made the recognition of those constructions that seem to play a fundamental role in the learning of linear transformations such as making sense of what linearity means in the context of Linear Algebra and the relation of linear transformation with functions, matrices and with the concept of basis, although not all of this has been described in this paper. The concept of transformation emerged quite easily from work with the modeling problem. However, linear transformations had to be introduced in the activities so that students were able to conclude that translations are not linear transformations. It is important to underline, how, while working on the model, students were able to develop on their own powerful conceptual tools, such as a way to determine the difference between rigid and non rigid transformations; and a relation to the concept of matrix. The emergence of these ideas gives evidence that the use of modeling situations in the classroom promotes the construction of knowledge. The complementary use of activities designed with the genetic decomposition played an important role in the development of students’ schema for function and for linearity. REFERENCES
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