Top PDF Parameter Estimation of a Cardiac Model Using the Local Ensemble Transform Kalman Filter

Parameter Estimation of a Cardiac Model Using the Local Ensemble Transform Kalman Filter

Parameter Estimation of a Cardiac Model Using the Local Ensemble Transform Kalman Filter

In this section we further explore why this method of parameter estimation works better on some parameter choices than others, with the primary motivation being the importance of the magnitude of the parameter we are estimating. We also show experimental evidence that parameters are more likely to make larger relative changes of their values over the course of assimilation if their magnitudes are small in comparison to the rest of the parameters of the model. The explanation behind this phenomenon is that for large parameter values, the perturbations that get added during the algorithm are simply not large enough to notice bigger errors in the parameters. If the parameter is small, the perturbations added are relatively bigger, which means that the parameter estimation algorithm is more likely to change the value of the parameter significantly during the estimation process, in contrast to remaining stagnant, as is the case with many large, insensitive parameters, such as in Figure 5. These larger perturbations in the parameter estimation may increase the likelihood that the algorithm estimates the parameter successfully.
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Comparison between Local Ensemble Transform Kalman Filter and PSAS in the NASA finite volume GCM – perfect model experiments

Comparison between Local Ensemble Transform Kalman Filter and PSAS in the NASA finite volume GCM – perfect model experiments

Although in this study we compared the LETKF with the 3D-Var analysis scheme used in the NASA GEOS-4 opera- tional system, some caveats about the results should be men- tioned. Our experiments are based on a perfect model sce- nario, in which we have avoided additional challenges asso- ciated with the presence of unknown observation and model errors. Also, the observational network only includes rawin- sondes, which is much sparser than the operational observa- tion network. Previous research shows that EnKF has more advantage in data sparse region (Whitaker et al., 2004) so that the advantages of the LETKF should be smaller for currently available operational observations than our results indicate. In addition, the error statistics of PSAS has not been well tuned, and the model resolution used is lower than that cur- rently used in operations. Therefore our very encouraging results could be interpreted as an upper bound for the poten- tial operational advantage of EnKF over 3D-Var. GEOS-4 has been replaced by the GEOS-5 system, which in the fu- ture should be also compared with the LETKF.
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Towards predictive data-driven simulations of wildfire spread – Part I: Reduced-cost Ensemble Kalman Filter based on a Polynomial Chaos surrogate model for parameter estimation

Towards predictive data-driven simulations of wildfire spread – Part I: Reduced-cost Ensemble Kalman Filter based on a Polynomial Chaos surrogate model for parameter estimation

Relevant insight into wildfire dynamics has been obtained in recent years via detailed numerical simulations performed at flame scales (i.e., with a spatial resolution of the or- der of 1 m). For instance, FIRETEC (Linn et al., 2002) or WFDS (Mell et al., 2007) combine advanced physical mod- eling and classical methods of computational fluid dynam- ics (CFD) to accurately describe the combustion-related pro- cesses that control the fire behavior (e.g., thermal degradation of biomass fuel, buoyancy-induced flow, combustion, radia- tion and convection heat transfer). Note that because of the high computational cost, flame-scale CFD is currently re- stricted to research projects (Linn et al., 2002; Mell et al., 2007; Rochoux, 2014) and is not compatible with opera- tional applications. In contrast, a regional-scale viewpoint (i.e., a viewpoint that considers scales ranging from a few tens of meters up to several kilometers) is adopted in the following: the fire is described as a two-dimensional front that self-propagates normal to itself into unburnt vegeta- tion; the local propagation speed is called the rate of spread (ROS). This viewpoint is the dominant approach used in current operational wildfire spread simulators, see for in- stance FARSITE (Finney, 1998), FOREFIRE (Filippi et al., 2009, 2013), PROMETHEUS (Tymstra et al., 2010) and PHOENIX RapidFire (Chong et al., 2013). In particular, FARSITE uses a model due to Rothermel (1972) that treats the ROS as a semi-empirical function of biomass fuel proper- ties associated with a pre-defined fuel category (i.e., the ver- tical thickness of the fuel layer, the fuel moisture content, the fuel particle surface-to-volume ratio, the fuel loading and the fuel particle mass density), topographical properties (i.e., the terrain slope) and meteorological properties (i.e., the wind velocity at mid-flame height). This approach is limited in scope because of the large uncertainties associated with the accuracy of computer models since they do not account for the interaction between the fire and the atmosphere, and since they have a limited domain of validity resulting from a cali- bration procedure based on experiments (Perry, 1998; Sulli- van, 2009; Viegas, 2011; Cruz and Alexander, 2013; Finney et al., 2013). This approach is also limited because of the large uncertainties associated with many of the input param-
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The local ensemble transform Kalman filter and the running-in-place algorithm applied to a global ocean general circulation model

The local ensemble transform Kalman filter and the running-in-place algorithm applied to a global ocean general circulation model

The separation of ensemble members in the state space should occur because of a combination of internal fluid in- stabilities and variations in surface forcing. The impacts of internal instabilities are greatly reduced because of our choice of non-eddy resolving resolution and associated phys- ical parameterization errors. Additionally, limitations on the size of the ensemble further reduce the ensemble variance. We adjust for this missing error variance (1) in the model by adding perturbations to the wind field (as described above), and (2) in the data assimilation procedure through an adaptive error covariance inflation scheme developed by Miyoshi (2011). Miyoshi’s scheme has been applied to adjust inflation to automatically balance the background error with the estimated observation error. Occasionally, the ensemble spread becomes under-dispersive, meaning that the back- ground error covariance estimate is small, while the mean state is “far” from the observations (with respect to the ob- servation error). In these cases the inflation is automatically increased and the ensemble spread increases as a direct re- sult. The ensemble spread continues to increase over multi- ple analysis cycles until the background error covariance is large enough that the observations begin to impact the anal- ysis again. A parameter for adaptive inflation σ b allows the
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Joint state and parameter estimation with an iterative ensemble Kalman smoother

Joint state and parameter estimation with an iterative ensemble Kalman smoother

In this study, an augmented state IEnKS is tested on its estimation of the forcing parameter of the Lorenz-95 model. Since joint state and parameter estimation is especially use- ful in applications where the forcings are uncertain but never- theless determining, typically in atmospheric chemistry, the augmented state IEnKS is tested on a new low-order model that takes its meteorological part from the Lorenz-95 model, and its chemical part from the advection diffusion of a tracer. In these experiments, the IEnKS is compared to the ensemble Kalman filter, the ensemble Kalman smoother, and a 4D-Var, which are considered the methods of choice to solve these joint estimation problems. In this low-order model context, the IEnKS is shown to significantly outperform the other methods regardless of the length of the data assimilation win- dow, and for present time analysis as well as retrospective analysis. Besides which, the performance of the IEnKS is even more striking on parameter estimation; getting close to the same performance with 4D-Var is likely to require both a long data assimilation window and a complex modeling of the background statistics.
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An implementation of the Local Ensemble Kalman Filter in a quasi geostrophic model and comparison with 3D-Var

An implementation of the Local Ensemble Kalman Filter in a quasi geostrophic model and comparison with 3D-Var

4.2 Sensitivity of the LEKF to choice of parameters As could be expected, LEKF results are sensitively depen- dent on the number of vectors forming the ensemble. In Fig. 7a the error averaged in space and time for potential vorticity at midlevel is shown for the LEKF system running with 10, 15, 20, 25, 30 and 40 vectors, using a fixed size of the local domain (l equal to 3). The results with 3D-Var are also shown for reference. It can be seen that for this system 10 vectors are not sufficient to improve upon 3D-Var. The performance of the LEKF improves with the number of en- semble members but only up to 30 (chosen as reference in the previous results). The reduction of errors with ensem- ble size converges and no further significant improvements are observed with 40 members. An ensemble of 30 mem- bers has much fewer degrees of freedom than the model, and supports the hypothesis that the local subspace of the most unstable modes is small and that the LEKF could be used for operational purposes with a reasonable computational effort. Another important parameter in the LEKF is the size of the local volumes (“patches”) used in the localization. Fig- ure 7b shows the analysis and forecast errors for different patch sizes (l=2, 3, 4, 5, corresponding to squares of 5 × 5, 7 × 7, 9 × 9 and 11 × 11 grid points respectively), using an en- semble of 30 members. The analysis errors improve with size up to l=4 (9×9 grid points), and then they become slightly worse. This can be explained by the fact that when the num- ber of points of each local domain increases, the local dimen- sion of the system also increases, and the number of vectors required to describe well the local instabilities of the system eventually needs to be larger. In other words, the larger the local dimension, the larger the sampling errors in the repre- sentation of the background errors with a limited number of ensemble perturbations.
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Inflation method for ensemble Kalman filter in soil hydrology

Inflation method for ensemble Kalman filter in soil hydrology

In this paper, we only show the inflation related to the closed-eye period (Fig. B1), which presents the major chal- lenge to the inflation in that particular application. During this time, preferential flow occurs and the underlying local equilibrium assumption of the Richards equation is violated. With a standard approach, parameters become biased to com- pensate these errors. To avoid this, Bauser et al. (2016) in- troduced the closed-eye period, which pauses the parameter estimation and only guides the water content states through times, when assumptions are violated. Compared to the stan- dard approach, this leads to a reduced bias in the parameters, but effectively increases the model errors during the closed- eye period. A strong inflation is required to compensate this error. The inflation method used in Bauser et al. (2016) was just able to accomplish this and the authors were concerned that a too slow adjustment speed of the inflation limits the applicability of the closed-eye period for cases with larger model errors.
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A Bayesian consistent dual ensemble Kalman filter for state parameter estimation in subsurface hydrology

A Bayesian consistent dual ensemble Kalman filter for state parameter estimation in subsurface hydrology

The EnKF is widely used in surface and subsurface hydro- logical studies to tackle state-parameter estimation problems (Zhou et al., 2014; Panzeri et al., 2014). Two approaches are usually considered based on the joint and the dual estima- tion strategies. The standard joint approach concurrently es- timates the state and the parameters by augmenting (in the same vector) the state variables with the unknown parame- ters, that do not vary in time. The parameters could also be set to follow an artificial evolution (random walk) before they get updated with incoming observations (Wan et al., 1999). One of the early applications of the joint EnKF to subsurface groundwater flow models was presented by Chen and Zhang (2006). In their study, a conceptual subsurface flow system was considered and ensemble filtering was performed to es- timate the transient pressure field alongside the hydraulic conductivity. In a reservoir engineering application, Næv- dal et al. (2005) considered a two-dimensional (2-D) North Sea field model and considered the joint estimation prob- lem of the dynamic pressure and saturation on top of the
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Stator Fault Detection in Induction Machines by Parameter Estimation Using Adaptive Kalman Filter

Stator Fault Detection in Induction Machines by Parameter Estimation Using Adaptive Kalman Filter

In this paper some useful modification has been applied on IM modeling in [5]. The electrical part of the model depends on electrical angle in [5] but it depends on rotor velocity in this paper. This is preferable to obtain a description of the IM where the model depends on rotor velocity. It is clear that angle measurement is more complicated than velocity measurement. The other modification is that the states and parameters of the model are estimated recursively by an adaptive Kalman filter. The approach to this problem has been pointed out by Ljung [6]. In spite of many nonlinear optimization methods like Gauss-Newton that is used in [5], adaptive Kalman filter eliminates two drawbacks. Firstly, the identification process can be formulated in a recursive manner. Secondly, there is no possibility of getting stuck at local minima and of the algorithm being unstable.
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Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling

Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling

For reservoir models the terms “data assimilation” and “history matching” are used interchangeably, as the goal of data assimilation is the same as that of history match- ing, where observations are used to improve a solution of a model. Ensemble data assimilation methods such as en- semble Kalman filters (Evensen, 2009) were originally de- veloped in meteorology and oceanography for the state es- timation. Now it is one of the frequently employed ap- proaches for parameter estimation in subsurface flow models as well (e.g., Oliver et al., 2008). A detailed review of en- semble Kalman filter developments in reservoir engineering is written by Aanonsen et al. (2009). An ensemble Kalman filter efficiently approximates a true posterior distribution if the distribution is not far from Gaussian, as it corrects only the mean and the variance. For nonlinear models with multi- modal distributions, however, an ensemble Kalman filter fails to correctly estimate the posterior, as shown by Dovera and Della Rossa (2011).
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Volcanic ash forecast using ensemble-based data assimilation: an ensemble transform Kalman filter coupled with the FALL3D-7.2 model (ETKF–FALL3D version 1.0)

Volcanic ash forecast using ensemble-based data assimilation: an ensemble transform Kalman filter coupled with the FALL3D-7.2 model (ETKF–FALL3D version 1.0)

Inverse modeling and, in particular, data assimilation methods are techniques that can be used to estimate the state of dynamical systems based on partial and noisy observa- tions. In a broad sense, these techniques build on continu- ous or quasi-continuous observations to produce model ini- tial conditions (analyses) that can be used to better predict the future state, taking into account uncertainties in observa- tions and model formulation. Data assimilation methods have been successfully applied to the estimation of the state of the ocean or the atmosphere (e.g., Kalnay, 2003; Carrassi et al., 2018) as well as for the optimization of uncertain model pa- rameters (e.g., Ruiz et al., 2013). More recently, applications have been extended to atmospheric constituents (e.g., Boc- quet et al., 2010; Hutchinson et al., 2017), including ash dis- persion models with the purpose of estimating the 3-D dis- tribution of ash concentrations to be used as initial condi- tions for forecasts. Surprisingly, examples of the application of data assimilation techniques to volcanic ash dispersion are scarce and still mainly limited to a research level. For ex- ample, Wilkins et al. (2015) implemented a data insertion methodology to improve the initial conditions of ash concen- trations based on satellite estimations of ash mass loadings in a Lagrangian dispersion model. Fu et al. (2015, 2017a) applied an ensemble Kalman filter technique to the estima- tion of ash concentrations in an Eulerian dispersion model based on flight concentration measurements and satellite es- timations using idealized experiments and real observations. Their results showed that both observational sets (flight mea- surements and satellite mass loads) reduced forecast errors, which in their particular case were attributed to a wrong model representation of ash sedimentation processes. One important issue when using satellite estimates of ash mass loadings is that observations only provide a 2-D distribution of ash mass, while models usually require the vertical profile
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Operational hydrological data assimilation with the recursive ensemble Kalman filter

Operational hydrological data assimilation with the recursive ensemble Kalman filter

TopNet uses seven calibrated parameters for each sub- catchment. To reduce the dimensionality of the parameter estimation problem, initial values for the parameters were estimated from the sources described. The spatial distribu- tion of the parameters was then preserved, and the values were adjusted uniformly using a spatially constant set of pa- rameter multipliers. Calibration used precipitation and cli- mate data from Tait et al. (2006) who interpolated data from over 500 climate stations in New Zealand across a regular 0.05 ◦ latitude/longitude grid (approximately 5 km × 5 km). The precipitation data were bias-corrected using a water bal- ance approach (Woods et al., 2006). These data are pro- vided at daily time steps, and are disaggregated to hourly data before use in the model. The parameter values used here are those in current use for the operational forecasting sys- tem. The calibration methods varied by catchment accord- ing to the responsible hydrologist, and consisted of a semi- automatic method using either Monte Carlo simulation (two catchments), or the ROPE (RObust Parameter Estimation) calibration method (Bardossy and Singh, 2008) (five catch- ments) to obtain a small ensemble of possible parameter sets. This was followed by review by a hydrologist to determine a single preferred set based on visual inspection of the model simulation results. Note that parameter values are not per- turbed during the assimilation. Instead, we allow for model error by perturbing the state variables.
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State and parameter estimation of two land surface models using the ensemble Kalman filter and the particle filter

State and parameter estimation of two land surface models using the ensemble Kalman filter and the particle filter

For CLM, larger differences were observed in the perfor- mance of the different data assimilation methods. This larger disparity among the methods is explained by the consider- ably larger number of soil layers (10) used by CLM. This increased significantly the dimensionality of the parameter estimation problem. The overall best results at the 5, 20, and 50 cm measurement depths were observed for EnKF-AUG and EnKF-DUAL, with RMSE values that were somewhat smaller than their counterparts derived from PMCMC. This was true for both the calibration and evaluation periods. The RRPF exhibited the worst performance, in part determined by the use of a relatively small ensemble of N = 100 par- ticles. The superiority of the EnKF-AUG and EnKF-DUAL methods for CLM is consistent with our expectations articu- lated previously in Sect. 3.1. The analysis step of the EnKF makes it much easier for EnKF-AUG and EnKF-DUAL to track the measured soil moisture dynamics, thereby promot- ing convergence in high-dimensional state-parameter spaces. PF-based methods, by contrast, deteriorate in robustness and efficiency with larger dimensionality of the state-parameter space as they lack a state-analysis step and approximate the transient state-parameter PDF via the particles’ likeli- hoods. This likelihood is only a low-dimensional summary statistic of the distance between the forecasted and mea- sured values of the states. Resampling with MCMC via the likelihood thus becomes increasingly more difficult in high- dimensional state-parameter spaces. For CLM, the PMCMC method still achieves comparable results to EnKF-AUG and EnKF-DUAL as the dimensionality of the state-parameter PDF of this model is only somewhat larger than its coun- terpart of VIC-3L. Of course, the use of a larger ensem- ble size makes it easier to characterize the transient state- parameter PDF, but at the expense of a significantly increased CPU cost. For PMCMC, multiple different MCMC resam- pling steps can also enhance significantly the particle ensem- ble by allowing each particle trajectory to improve its likeli- hood. Yet, this deteriorates significantly the efficiency of im- plementation as each candidate particle requires a separate model evaluation of VIC-3L or CLM to determine its likeli- hood. Thus, for LSMs with relatively few state variables and model parameters, we expect the EnKF and PF methods to achieve a comparable performance. For larger-dimensional state-parameter spaces we would recommend EnKF-AUG and EnKF-DUAL, unless one can afford a very large num- ber of particles.
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Parameter estimation in an atmospheric GCM using the Ensemble Kalman Filter

Parameter estimation in an atmospheric GCM using the Ensemble Kalman Filter

The ensemble mean temperature matches the data fairly well (Fig. 4), with a typical RMS error of 3K at each grid- point. However, it should be noted that in tuning to reanalysis data, we have chosen a somewhat easier target than if we had used pure observations. Substantial cold biases over the high plateaux (especially Tibet) are present in almost all AGCMs, including that used for the NCEP reanalysis. This compari- son therefore gives an optimistic impression of model skill by masking the same failing in our model. In contrast to the rea- sonable temperature fields, precipitation remains poor in all simulations (Fig. 5), with an RMS error as high as 3mm per day. In fact, even attempting to tune to precipitation alone, completely ignoring the fit to the temperature data, did not improve that result. The model parameters appear to have very little effect on precipitation patterns, despite several of them relating directly to hydrology. Re-examining the re- sults from the univariate sensitivity analysis indicated that the wider range of 29 parameters tested all had a minimal effect on the precipitation. Disabling the convection scheme entirely, changed the model precipitation more substantially (and in fact led to an overall improvement). Clearly this points to a significant structural deficiency and research is
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Resolving structural errors in a spatially distributed hydrologic  model using ensemble Kalman filter state updates

Resolving structural errors in a spatially distributed hydrologic model using ensemble Kalman filter state updates

Abstract. In hydrological modeling, model structures are de- veloped in an iterative cycle as more and different types of measurements become available and our understanding of the hillslope or watershed improves. However, with increas- ing complexity of the model, it becomes more and more diffi- cult to detect which parts of the model are deficient, or which processes should also be incorporated into the model during the next development step. In this study, we first compare two methods (the Shuffled Complex Evolution Metropo- lis algorithm (SCEM-UA) and the Simultaneous parame- ter Optimization and Data Assimilation algorithm (SODA)) to calibrate a purposely deficient 3-D hillslope-scale model to error-free, artificially generated measurements. We use a multi-objective approach based on distributed pressure head at the soil–bedrock interface and hillslope-scale dis- charge and water balance. For these idealized circumstances, SODA’s usefulness as a diagnostic methodology is demon- strated by its ability to identify the timing and location of pro- cesses that are missing in the model. We show that SODA’s state updates provide information that could readily be incor- porated into an improved model structure, and that this type of information cannot be gained from parameter estimation methods such as SCEM-UA. We then expand on the SODA result by performing yet another calibration, in which we in- vestigate whether SODA’s state updating patterns are still capable of providing insight into model structure deficien- cies when there are fewer measurements, which are more- over subject to measurement noise. We conclude that SODA can help guide the discussion between experimentalists and modelers by providing accurate and detailed information on how to improve spatially distributed hydrologic models.
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Identification of hydrological model parameter variation using ensemble Kalman filter

Identification of hydrological model parameter variation using ensemble Kalman filter

The data assimilation (DA) actually provides another method to identify the potential temporal variations of model parameters by updating them in real time when observa- tions are available (Liu and Gupta, 2007; Xie and Zhang, 2013). The DA method has been widely applied in hydrol- ogy for soil moisture estimation (Han et al., 2012; Kumar et al., 2012; Yan et al., 2015) and flood forecasting (Y. Li et al., 2013; Liu et al., 2012; Abaza et al., 2014). It has also been successfully used to estimate model parameters (Moradkhani et al., 2005; Kurtz et al., 2012; Montzka et al., 2013; Panzeri et al., 2013; Vrugt et al., 2013; Xie and Zhang, 2013; Shi et al., 2014; Xie et al., 2014). For example, Vrugt et al. (2013) proposed two Particle-DREAM (DiffeR- ential Evolution Adaptive Metropolis) methods, i.e., Particle- DREAM for time-variant and time-invariant parameters, to track the evolving target distribution of HyMOD parame- ters, while both results were approximately similar and sta- tistically coherent since only 3 years of data were used. Xie and Zhang (2013) used a partitioned forecast-update scheme based on the ensemble Kalman filter (EnKF) to retrieve op- timal parameters in a distributed hydrological model. Al- though the DA method has been used to estimate model pa- rameters, these studies are focused on the estimation of con- stant parameters. Little attention has been paid to the iden- tification of time-variant model parameters by using the DA method.
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Simultaneous estimation of land surface scheme states and parameters using the ensemble Kalman filter: identical twin experiments

Simultaneous estimation of land surface scheme states and parameters using the ensemble Kalman filter: identical twin experiments

Abstract. The performance of the ensemble Kalman filter (EnKF) in soil moisture assimilation applications is inves- tigated in the context of simultaneous state-parameter esti- mation in the presence of uncertainties from model parame- ters, soil moisture initial condition and atmospheric forcing. A physically based land surface model is used for this pur- pose. Using a series of identical twin experiments in two kinds of initial parameter distribution (IPD) scenarios, the narrow IPD (NIPD) scenario and the wide IPD (WIPD) sce- nario, model-generated near surface soil moisture observa- tions are assimilated to estimate soil moisture state and three hydraulic parameters (the saturated hydraulic conductivity, the saturated soil moisture suction and a soil texture empiri- cal parameter) in the model. The estimation of single imper- fect parameter is successful with the ensemble mean value of all three estimated parameters converging to their true val- ues respectively in both NIPD and WIPD scenarios. Increas- ing the number of imperfect parameters leads to a decline in the estimation performance. A wide initial distribution of estimated parameters can produce improved simultaneous multi-parameter estimation performances compared to that of the NIPD scenario. However, when the number of esti- mated parameters increased to three, not all parameters were estimated successfully for both NIPD and WIPD scenarios. By introducing constraints between estimated hydraulic pa- rameters, the performance of the constrained three-parameter estimation was successful, even if temporally sparse obser- vations were available for assimilation. The constrained es- timation method can reduce RMSE much more in soil mois- ture forecasting compared to the non-constrained estimation method and traditional non-parameter-estimation assimila-
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Correlated Estimation Problems and the Ensemble Kalman Filter

Correlated Estimation Problems and the Ensemble Kalman Filter

applied in real-time estimation problems. The asymptotic computational complexity of a direct implementation of the Kalman filter based on Definition 2.4 is provided in Table 2.1, where n and m denote the number of dimensions of the state space and the observation, respectively. Note that the most tedious operation - the prediction step with O(n 3 ) time complexity - can be performed more efficiently in problems where only a part of the state vector is altered by the process model. For example, in simultaneous localisation and mapping (SLAM), the robot movement only updates a small portion of the map, and hence the prediction step can be implemented in O(n) time (Smith et al., 1990). Similarly, in certain estimation problems, it is possible to exploit the structure of the state space in order to reduce the computational cost of the update step, in particular to eliminate the O(n 2 ) term. For example, in SLAM it is possible to decompose the state estimate’s vector and covariance matrix (representing a global map) into several smaller components (local sub-maps) and maintain these separately. The component estimates only need to be combined sporadically and thus the quadratic computation cost can be amortised between more steps of the filter (Paz et al., 2007; Huang et al., 2008). However, these approaches are beyond the scope of this thesis, as it deals with general estimation problems. The Kalman filter’s favourable computational properties and general simplicity of implemen- tation, its versatility to various problems and observation schedules, and the fact that it often provides reasonable state estimates even if the model of the dynamical system is not precise, are arguably the main reasons why it reached such popularity and widespread adoption in technol- ogy over the last 50 years. The applications of the Kalman filter span areas such as guidance, navigation and control, robotics, target tracking, or time series analysis in signal processing and econometrics. As Harold W. Sorenson famously put it (Sorenson, 1985),
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Market Risk Beta Estimation using Adaptive Kalman Filter

Market Risk Beta Estimation using Adaptive Kalman Filter

Literature shows that there have been quite a number of techniques for beta estimation: OLS [5, 4, 15], GLS [27], KF [6, 12, 4, 26, 14], Adaptive KF [28]. [27] compared GLS and KF for purpose of estimation of non-stationary beta parameter in time varying extended CAPM. [19, 29, 30] used modified KF for estimating daily betas with high frequency Indian data exhibiting significant non-Gaussianity in the distribution of beta. [31 & 32] approached to estimate time varying beta using KF and Quadratic Filter. [33] integrated realized beta as function of realized volatility [34, 35] in a unified framework. The present work first applied an AKF for the dynamic beta estimation using state space model where measurement noise covariance and state noise covariance are not known and estimated during filtering. A small modification of the AKF is proposed to take care of the situations arises by characterizing AKF for beta estimation.
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Automatic modulation classification using interacting multiple model - Kalman filter for channel estimation

Automatic modulation classification using interacting multiple model - Kalman filter for channel estimation

An efficient AMC paradigm featuring the IMM-KF mixture to elicit a robust and reliable CSI estimate is proposed in this paper. The adaptive IMM-KF estimation is based on decomposing the CSI parameters into a small number of parallel components, by applying the well-known SVD methodology, and then adding up their dominant ETs, which is newly called Frobenius ET (FET) in this paper. This accordingly will maintain the total power of all ETs instead of the waste resulting from relying on individual ETs only. The estimate is applied over Rayleigh MIMO channel and directly admitted to the EM recursion to invoke the Q(HLRT) for AMC core framework. Further to the viable performance, the overall IMM-KF computation is significantly less demanding compared to the optimal MLE especially with large constellation sizes. The IMM-KF only needs one search loop to find the best constellation candidate and is hence more economical than the MLE, which requires two extensive loops to fulfil the EM complete cycle.
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