Article

# Iterated Posterior Linearisation Smoother

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## Abstract

This paper considers the problem of Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Sigma-point approximations to the general Gaussian Rauch–Tung–Striebel smoother are widely used methods to tackle this problem. These algorithms perform statistical linear regression (SLR) of the nonlinear functions considering only the previous measurements. We argue that SLR should be done taking all measurements into account. We propose the iterated posterior linearisation smoother (IPLS), which is an iterated algorithm that performs SLR of the nonlinear functions with respect to the current posterior approximation. The algorithm is demonstrated to outperform conventional Gaussian nonlinear smoothers in two numerical examples.

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... In this paper, we propose a different use of sigma-point methods in BP inspired by the posterior linearisation filter (PLF) [14] and smoother [15]. In these papers, it is pointed out that the most commonly used Gaussian filters and smoothers approximate nonlinear functions as affine functions with additive Gaussian noise. ...
... Due to the nonlinear measurements, the beliefs cannot generally be computed in closed-form, so we need approximations. As in popular nonlinear Gaussian filters and smoothers [14], [15], nonlinear functions, see (1), are dealt with by performing an enabling approximation in which they are approximated as affine functions with additive Gaussian noise ...
... Gaussian, which implies that the marginal PDFs are Gaussian. As in many Gaussian filters/smoothers, the accuracy of the marginal PDF approximations only depends on the choice of (3) so it is of utmost importance to select it properly [14], [15]. Under approximation (3), calculating the marginal PDFs directly from p (x|z) is theoretically simple as we can just integrate out the other states. ...
Article
This paper presents the posterior linearisation belief propagation (PLBP) algorithm for cooperative localisation in wireless sensor networks with nonlinear measurements. PLBP performs two steps iteratively: linearisation and belief propagation. At the linearisation step, the nonlinear functions are linearised using statistical linear regression with respect to the current beliefs. This SLR is performed in practice by using sigma-points drawn from the beliefs. In the second step, belief propagation is run on the linearised model. We show by numerical simulations how PLBP can outperform other algorithms in the literature.
... Taylor series expansion based iterated extended Kalman smoother (IEKS) methods [13][14][15] and sigma-point based methods [5] are well-established techniques in literature. Iterated sigma-point methods have been proposed, for example, in [16,17]. Despite the capabilities of the aforementioned methods in state estimation in nonlinear Gaussian models, they lack a framework which enables the computations in a more efficient way when using parallelization. ...
... There are different strategies to effectively select the parameters of (5). In this paper, we will consider two such strategies widely-used in the Gaussian filtering literature, namely iterated sigma-point and extended Kalman smoothers [14][15][16]. In these approaches, the linearized-filtersmoother method is repeated M times, with the linearization parameters leveraging the results of the previous smoothing pass instead of the previous step. ...
... Iterated sigma-point method. In this approach, we select the parameters (F k−1 , c k−1 , Λ k−1 ) and (H k , d k , Ω k ) using sigma-point-based statistical linear regression (SLR) method [16] as follows. First, we select m sigma points X ...
Conference Paper
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... Due to this, contrary to alternatives such as [29,14,16], the proposed approach has no tuning parameters. Instead, the optimisation required is run until convergence to a local maximum. ...
... where v k ∼ N (0, 1), e k ∼ N (0, 1) and x 0 ∼ N (5,4). This is identical to the setup that was used in [16] where the IPLS idea was profiled. It is well-known to be a challenging state estimation problem possessing multi-modal state distributions. ...
... For the IPLS one filter and 50 smoothing iterations, denoted IPLS-1-50, were used. As per [16], using one filter iteration with many smoothing iterations performs best on this system. ...
Preprint
Full-text available
This paper considers parameter estimation for nonlinear state-space models, which is an important but challenging problem. We address this challenge by employing a variational inference (VI) approach, which is a principled method that has deep connections to maximum likelihood estimation. This VI approach ultimately provides estimates of the model as solutions to an optimisation problem, which is deterministic, tractable and can be solved using standard optimisation tools. A specialisation of this approach for systems with additive Gaussian noise is also detailed. The proposed method is examined numerically on a range of simulation and real examples with a focus on robustness to parameter initialisations; we additionally perform favourable comparisons against state-of-the-art alternatives.
... Due to this, contrary to alternatives such as [29,14,16], the proposed approach has no tuning parameters. Instead, the optimisation required is run until convergence to a local maximum. ...
... where v k ∼ N (0, 1), e k ∼ N (0, 1) and x 0 ∼ N (5,4). This is identical to the setup that was used in [16] where the IPLS idea was profiled. It is well-known to be a challenging state estimation problem possessing multi-modal state distributions. ...
... For the IPLS one filter and 50 smoothing iterations, denoted IPLS-1-50, were used. As per [16], using one filter iteration with many smoothing iterations performs best on this system. ...
Preprint
This paper considers the problem of computing Bayesian estimates of both states and model parameters for nonlinear state-space models. Generally, this problem does not have a tractable solution and approximations must be utilised. In this work, a variational approach is used to provide an assumed density which approximates the desired, intractable, distribution. The approach is deterministic and results in an optimisation problem of a standard form. Due to the parametrisation of the assumed density selected first- and second-order derivatives are readily available which allows for efficient solutions. The proposed method is compared against state-of-the-art Hamiltonian Monte Carlo in two numerical examples.
... Taylor series expansion based iterated extended Kalman smoother (IEKS) methods [13][14][15] and sigma-point based methods [5] are well-established techniques in literature. Iterated sigma-point methods have been proposed, for example, in [16,17]. Despite the capabilities of the aforementioned methods in state estimation in nonlinear Gaussian models, they lack a framework which enables the computations in a more efficient way when using parallelization. ...
... There are different strategies to effectively select the parameters of (5). In this paper, we will consider two such strategies widely-used in the Gaussian filtering literature, namely iterated sigma-point and extended Kalman smoothers [14][15][16]. In these approaches, the linearized-filtersmoother method is repeated M times, with the linearization parameters leveraging the results of the previous smoothing pass instead of the previous step. ...
... Iterated sigma-point method. In this approach, we select the parameters (F k−1 , c k−1 , Λ k−1 ) and (H k , d k , Ω k ) using sigma-point-based statistical linear regression (SLR) method [16] as follows. First, we select m sigma points X ...
Preprint
Full-text available
The problem of Bayesian filtering and smoothing in nonlinear models with additive noise is an active area of research. Classical Taylor series as well as more recent sigma-point based methods are two well-known strategies to deal with these problems. However, these methods are inherently sequential and do not in their standard formulation allow for parallelization in the time domain. In this paper, we present a set of parallel formulas that replace the existing sequential ones in order to achieve lower time (span) complexity. Our experimental results done with a graphics processing unit (GPU) illustrate the efficiency of the proposed methods over their sequential counterparts.
... As we all know, in terms of estimation accuracy, the smoother is superior to its corresponding filter in most applications [12,13]. Enlightened by the development of CMIEKF as well as the superiority of CKF, a new kind of continuous-discrete distributed smoother can be obtained by using Rauch-Tung-Striebel (RTS) smoothing technology [14], where the filtering results are used to calculate the smoothing solutions. ...
... Applying the newly presented smoother to addressing the DSE of continuous-discrete nonlinear systems, an improved performance may be expected. In order to obtain the continuous-discrete distributed smoother, the third-order cubature rule used in the continuous-discrete CKF [15] is firstly embedded into CMIEKF architecture by using the statistical linear regression (SLR) method [12]. The resulting filtering method is called continuous-discrete consensus on measurements and information CKF (CMICKF). ...
... Notice that the linearized output matrices H i k are needed to calculate the information pair ( i k|k−1 , q i k|k−1 ). If the function h is not continuously differentiable, it is necessary to calculate the pseudo-Jacobian matrices H i k , which is often calculated by SLR method [12]. When the nonlinear function is linearized by SLR, an affine function can be obtained, and the linearization error has the minimum mean-square value. ...
Article
In order to estimate the state of general distributed continuous-discrete nonlinear system, a new consensus on the measurement and information cubature smoothing method is proposed in this article. The distributed continuous-discrete nonlinear dynamic system refers to a system whose process is described as a stochastic differential equation (SDE) and whose measurements are provided by a wireless sensor network (WSN) at discrete times. First, the cubature rule is embedded into the consensus on measurements and information extended Kalman filter (CMIEKF) architecture by using the statistical linear regression (SLR) method, and then the presented consensus-based method is obtained according to the Rauch–Tung–Striebel (RTS) smoothing methodology. In a numerical simulation, the performances of the presented smoothing method and the conventional nonlinear estimating methods are compared and analyzed. The numerical results show that, compared with other distributed estimation methods, the newly proposed smoothing algorithm enjoys not only higher accuracy but also stronger robustness.
... A recent advance, in discrete time inference, is the iterated posterior linearisation smoother (IPLS) [18] (see also [19]) which generalises the iterated extended Kalman smoother [20] to sigma-point methods. This was done on the basis of statistical linear regression [21], where a given smoothing solution is improved upon by re-linearising the system using the current Gaussian smoother approximation and then running the smoother again [18,19]. ...
... A recent advance, in discrete time inference, is the iterated posterior linearisation smoother (IPLS) [18] (see also [19]) which generalises the iterated extended Kalman smoother [20] to sigma-point methods. This was done on the basis of statistical linear regression [21], where a given smoothing solution is improved upon by re-linearising the system using the current Gaussian smoother approximation and then running the smoother again [18,19]. However, an analogue for continuous-time smoothing has yet to appear. ...
... The purpose of this paper is thus to generalise the discrete-time smoother of [18,19] to the continuous-time case. In order to accomplish this the statistical linear regression method [21] is generalised to the setting of stochastic differential equations. ...
Preprint
This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on 1) taking the limit of statistical linear regression of the discretised process and 2) minimising an upper bound to a cost functional. Their difference is manifested in the diffusion of the approximate processes. This in turn gives novel derivations of pre-existing Gaussian smoothers when Method 1 is used and a new class of Gaussian smoothers when Method 2 is used. Furthermore, based on the aforementioned development the iterative Gaussian smoothers in discrete-time are generalised to the continuous-time setting by iteratively re-linearising the stochastic differential equation with respect to the current Gaussian process approximation to the smoothed process. The method is verified in two challenging tracking problems, a reentry problem and a radar tracked coordinated turn model with state dependent diffusion. The results show that the method has better estimation accuracy than state-of-the-art smoothers.
... This approach was recently extended on the basis that it ought to be better to perform the SLR with respect to the posterior distribution rather than the prior distribution. While this is intractable, it leads to a scheme where the SLR is iteratively computed using the current best approximation to the posterior, why it is called iterated posterior linearisation filtering/smoothing (IPLF/IPLS) [25], [26]. However, these methods assume additive Gaussian noise. ...
... The IEKF/IEKS iteratively linearise the system by Taylor series around the current mean of the approximate filtering/smoothing distribution [22], [23]. On the other hand, IPLF/IPLS iteratively linearise the system using SLR with respect to the current approximate filtering/smoothing density [25], [26]. The RTS smoother is given in Alg. 1 for future reference. ...
... In a similar manner to [25], [26] a local convergence analysis can be carried out, resulting in Thm. 2. ...
Article
This letter presents the development of novel iterated filters and smoothers that only require specification of the conditional moments of the dynamic and measurement models. This leads to generalisations of the iterated extended Kalman filter, the iterated extended Kalman smoother, the iterated posterior linearisation filter, and the iterated posterior linearisation smoother. The connections to the previous algorithms are clarified and a convergence analysis is provided. Furthermore, the merits of the proposed algorithms are demonstrated in simulations of the stochastic Ricker map where they are shown to have similar or superior performance to competing algorithms.
... Typically these have been derived from different viewpoints, and from various communities of researchers with different priorities. For example, linearisationbased methods have been intensively studied in the signal processing literature due to their intuitive nature when applied to nonlinear dynamical systems (Bell, 1994;Särkkä, 2013;García-Fernández et al., 2016). Attempts to generalise these methods beyond signal processing motivated the invention of expectation propagation (EP, Minka, 2001) in the machine learning community as an alternative to variational inference (VI, Sato, 2001;Blei et al., 2017). ...
... For example, we show that natural gradient VI is a limiting case of power EP, we discuss the connection between variational inference and Newton's method, and we show that when approximations are applied to Newton's method the extended Kalman smoother is recovered. We also derive an improved version of the posterior linearisation algorithm (García-Fernández et al., 2016) based on our insights. ...
... However, it was shown by Bell (1994) that the iterated extended Kalman smoother (EKS) is equivalent to applying the Gauss-Newton algorithm to a nonlinear state space model. García-Fernández et al. (2016) further discuss how posterior linearisation, which is a generalisation of all nonlinear Kalman smoothers, can also be seen as a Gauss-Newton type of method. ...
Preprint
We formulate natural gradient variational inference (VI), expectation propagation (EP), and posterior linearisation (PL) as extensions of Newton's method for optimising the parameters of a Bayesian posterior distribution. This viewpoint explicitly casts inference algorithms under the framework of numerical optimisation. We show that common approximations to Newton's method from the optimisation literature, namely Gauss-Newton and quasi-Newton methods (e.g., the BFGS algorithm), are still valid under this `Bayes-Newton' framework. This leads to a suite of novel algorithms which are guaranteed to result in positive semi-definite covariance matrices, unlike standard VI and EP. Our unifying viewpoint provides new insights into the connections between various inference schemes. All the presented methods apply to any model with a Gaussian prior and non-conjugate likelihood, which we demonstrate with (sparse) Gaussian processes and state space models.
... The point, around which linearisation is done, can greatly impact the resulting estimated trajectory. In general, we want to perform linearisation around points close to the posterior estimates, motivating iterative extensions of aforementioned smoothers [10]. A full smoothed trajectory is repeatedly computed, with linearisations done around the most recent estimates. ...
... The increased computational complexity can lead to significantly better performance. Examples of iterative methods are the Iterated EKS (IEKS) and the Iterated posterior linearisation smoother (IPLS) [3], [10]. ...
... The IPLS is a more recent iterative smoother, first introduced in [10]. In IPLS, Θ (i) 1:K is selected by performing SLR as in (4a) to (4c) and (5) ...
Preprint
Full-text available
This paper considers the problem of robust iterative Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Iterative methods are known to improve smoothed estimates but are not guaranteed to converge, motivating the development of more robust versions of the algorithms. The aim of this article is to present Levenberg-Marquardt (LM) and line-search extensions of the classical iterated extended Kalman smoother (IEKS) as well as the iterated posterior linearisation smoother (IPLS). The IEKS has previously been shown to be equivalent to the Gauss-Newton (GN) method. We derive a similar GN interpretation for the IPLS. Furthermore, we show that an LM extension for both iterative methods can be achieved with a simple modification of the smoothing iterations, enabling algorithms with efficient implementations. Our numerical experiments show the importance of robust methods, in particular for the IEKS-based smoothers. The computationally expensive IPLS-based smoothers are naturally robust but can still benefit from further regularisation.
... We derive four novel approximate inference algorithms based on this approach. These amount to doubly sparse extensions to conjugatecomputation variational inference (CVI, Khan and Lin, 2017), power-expectation propagation (PEP, Minka, 2004), posterior linearisation (PL, García-Fernández et al., 2016), and the Extended Kalman smoother (EKS, Bell, 1994). We present these algorithms alongside the existing S 2 VGP approach, providing an overview of methods for inference in non-conjugate GP time series. ...
... Wilkinson et al. (2020) showed that classical nonlinear Kalman smoothers, such as the Extended, Unscented and Gauss-Hermite smoothers, can also be formulated as site-based algorithms. These algorithms are based on various forms of linearisation of the likelihood model, and their approach is generalised and improved upon in a method called posterior linearisation (PL,García-Fernández et al., 2016). We derived a sparse extension to the posterior linearisation algorithms presented inWilkinson et al. (2020), including a sparse version of the extended Kalman smoother (S 2 EKS). ...
Preprint
Approximate Bayesian inference methods that scale to very large datasets are crucial in leveraging probabilistic models for real-world time series. Sparse Markovian Gaussian processes combine the use of inducing variables with efficient Kalman filter-like recursions, resulting in algorithms whose computational and memory requirements scale linearly in the number of inducing points, whilst also enabling parallel parameter updates and stochastic optimisation. Under this paradigm, we derive a general site-based approach to approximate inference, whereby we approximate the non-Gaussian likelihood with local Gaussian terms, called sites. Our approach results in a suite of novel sparse extensions to algorithms from both the machine learning and signal processing literature, including variational inference, expectation propagation, and the classical nonlinear Kalman smoothers. The derived methods are suited to large time series, and we also demonstrate their applicability to spatio-temporal data, where the model has separate inducing points in both time and space.
... This results in a VI based assumed density state smoother which at a high level is conceptually similar to the state estimation approaches in Ala-Luhtala et al. (2015), Gultekin and Paisley (2017), and Darling and DeMars (2017). The resultant VI based smoother offers an alternative to the typical Unscented Rauch-Tung-Striebel smoother (URTSS) (Särkkä, 2008) approach and its iterative variants such as the Iterated Posterior Linearization Smoother (IPLS) described by Garcia-Fernandez et al. (2017). The VI smoothing approach, however, has several key differences to these alternatives. ...
... Lastly, the URTSS approach performs function approximations about the prediction densities. However, as discussed by Garcia-Fernandez et al. (2017) better function approximations, and hence state estimation, can be performed utilising the smoothed densities. This is an inherit feature of the VI smoother given here as all integral approximations naturally occur about the estimated smoothed distribution. ...
Preprint
We consider the problem of maximum likelihood parameter estimation for nonlinear state-space models. This is an important, but challenging problem. This challenge stems from the intractable multidimensional integrals that must be solved in order to compute, and maximise, the likelihood. Here we present a new variational family where variational inference is used in combination with tractable approximations of these integrals resulting in a deterministic optimisation problem. Our developments also include a novel means for approximating the smoothed state distributions. We demonstrate our construction on several examples and show that they perform well compared to state of the art methods on real data-sets.
... For large-scale indoor engineering projects, the inspection route determines the level of success of equipment management and route optimization. Unlike the EKF, which can only utilize past and current navigation information, a smoothing algorithm can utilize future information to increase the current position estimation accuracy, and consequently significantly increase the accuracy of the inspection route positioning data [30], [37]. Thus, positioning algorithms for large-scale patrol inspections should include a smoothing function. ...
... Thus, positioning algorithms for large-scale patrol inspections should include a smoothing function. The RTS smoother has been proven to be the best smoothing algorithm for implementation within the EKF framework [37]. Moreover, RTS smoother has been widely applied in navigationrelated fields because of their reliability and availability. ...
Article
A reliable and practical personnel positioning system is necessary for large-scale indoor patrol inspection. In this study, we designed a personnel positioning system that entails the implementation of a foot-mounted inertial module, a smartphone, and pre-determined sparse QR code points. Foot-mounted inertial navigation system (Foot-INS) is self-contained system that send real-time dead-reckoning positions to a smartphone via a built-in Bluetooth module. When the inspector arrives at a planned control point area marked by the QR code, the smartphone actively scans the QR code and obtains the three-dimensional (3D) coordinates of the control point. Then, a newly developed data fusion algorithm performs real-time and post-processing fusion of the relative foot position and 3D QR code control point coordinate data. The proposed algorithm was implemented on an Android smartphone and was evaluated by extensive experiments in an indoor office building and an outdoor open-sky area. The experimental results confirmed the effectiveness and feasibility of the proposed personnel positioning system. The designed system was proven to be decent, robust, and not susceptible to environmental changes. It is also low-cost, has low power consumption, is easy to operate, and is compatible with common smartphones. Moreover, the proposed personnel positioning system provides a practical application pattern of the Foot-INS.
... This has motivated the development of various methods based on approximating smoothing densities in different ways. For instance, the use of Gaussian approximations for the smoothing densities and of sigma points techniques for solving moment matching integrals has been investigated in [8][9][10]. Another class of methods (usually known as particle smoothers) is based on the exploitation of sequential Monte Carlo techniques, i.e. on approximating smoothing densities through a set of weighted particles (e.g., see [3,5,[11][12][13][14] and references therein). ...
... Moreover, the following values have been selected for the parameters of the considered SSM: Some numerical results showing the dependence of RM SE L and RM SE N on the number of particles (N p ) for some of the considered smoothing algorithms are illustrated in Figs. 5 and 6, respectively (simulation results are indicated by markers, whereas continuous lines are drawn to fit them, so facilitating the interpretation of the available data). In this case, n i = 1 has been selected for all the derived particle smoothers, M = N p has been chosen for all the smoothing algorithms generating multiple trajectories and the range [10,150] has been considered for N p (since no real improvement is found for N p 150). Moreover, RM SE L and RM SE N results are also provided for MPF and DBF, since these filtering techniques are employed in the forward pass of Alg-L, the RBSS algorithm and the SPS algorithm, and the DBSA and the SDBSA, respectively; this allows us to assess the improvement in estimation accuracy provided by the backward pass with respect to the forward pass for each smoothing algorithm. ...
Preprint
Full-text available
Recently, a novel method for developing filtering algorithms, based on the interconnection of two Bayesian filters and called double Bayesian filtering, has been proposed. In this manuscript we show that the same conceptual approach can be exploited to devise a new smoothing method, called double Bayesian smoothing. A double Bayesian smoother combines a double Bayesian filter, employed in its forward pass, with the interconnection of two backward information filters used in its backward pass. As a specific application of our general method, a detailed derivation of double Bayesian smoothing algorithms for conditionally linear Gaussian systems is illustrated. Numerical results for two specific dynamic systems evidence that these algorithms can achieve a better complexity-accuracy tradeoff and tracking capability than other smoothing techniques recently appeared in the literature.
... This has motivated the development of various methods based on approximating smoothing densities in different ways. For instance, the use of Gaussian approximations for the smoothing densities and of sigma points techniques for solving moment matching integrals has been investigated in [8]- [10]. Another class of methods (usually known as particle smoothers) is based on the exploitation of sequential Monte Carlo techniques, i.e. on approximating smoothing densities through a set of weighted Manuscript particles (e.g., see [3], [5], [11]- [14] and references therein). ...
... A single realization (i.e., a single smoothed state trajectory) is computed in each backward pass; consequently, generating the whole output of the DBSA requires running a single forward pass and M distinct backward passes. Moreover, the evaluation of the smoothed information is based on the factorisation (10) or (11). In fact, these formulas are exploited to merge the statistical information emerging from the forward pass with that computed in any of the M backward passes. ...
Article
Recently, a novel method for developing filtering algorithms, based on the interconnection of two Bayesian filters and called double Bayesian filtering, has been proposed. In this manuscript we show that the same conceptual approach can be exploited to devise a new smoothing method, called double Bayesian smoothing. A double Bayesian smoother combines a double Bayesian filter, employed in its forward pass, with the interconnection of two backward information filters used in its backward pass. As a specific application of our general method, a detailed derivation of double Bayesian smoothing algorithms for conditionally linear Gaussian systems is illustrated. Numerical results for two specific dynamic systems evidence that these algorithms can achieve a better complexity-accuracy tradeoff and tracking capability than other smoothing techniques recently appeared in the literature.
... Furthermore, the method for reconstructing the trajectory between measurements, used by the basis expansion approaches, appear to suffer from edge effects at the measurement points, inviting further research into the topic. Another line of future research is to generalise the discrete time iterative smoothers to the current frame work [20], [21], [19]. ...
... We can see that a ijk in (27) is equal to a ijk in (28), which proves the associative property of the operator in Definition 5. ...
Preprint
This paper presents algorithms for the temporal parallelization of Bayesian filters and smoothers. We define the elements and the operators to pose these problems as the solutions to all-prefix-sums operations for which efficient parallel scan-algorithms are available. We present the temporal parallelization of the general Bayesian filtering and smoothing equations, and the specific linear/Gaussian models, and discrete hidden Markov models. The advantage of the proposed algorithms is that they reduce the linear complexity of standard filtering and smoothing algorithms with respect to time to logarithmic.
... When considering smoothing problems, it is even better to iteratively linearize with respect to the smoothing trajectory as is done in the iterated extended Kalman smoother (Bell, 1994). A general framework of iterated posterior linearization smoothers using this idea was developed in García-Fernández et al. (2017) and this was further generalized to more general state-space models in Tronarp et al. (2018). These methods result in Gaussian approximations to the marginals p(x t | y 1:T ) ≈ N (x t ; m l t , P l t ) which are optimal in a Kullback-Leibler sense (García-Fernández et al., 2015). ...
Preprint
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Particle smoothers are SMC (Sequential Monte Carlo) algorithms designed to approximate the joint distribution of the states given observations from a state-space model. We propose dSMC (de-Sequentialized Monte Carlo), a new particle smoother that is able to process $T$ observations in $\mathcal{O}(\log T)$ time on parallel architecture. This compares favourably with standard particle smoothers, the complexity of which is linear in $T$. We derive $\mathcal{L}_p$ convergence results for dSMC, with an explicit upper bound, polynomial in $T$. We then discuss how to reduce the variance of the smoothing estimates computed by dSMC by (i) designing good proposal distributions for sampling the particles at the initialization of the algorithm, as well as by (ii) using lazy resampling to increase the number of particles used in dSMC. Finally, we design a particle Gibbs sampler based on dSMC, which is able to perform parameter inference in a state-space model at a $\mathcal{O}(\log(T))$ cost on parallel hardware.
... Instead, we can resort to general Gaussian filtering and smoothing approaches for nonlinear systems such as statistically linearized and sigma-point Kalman filters (e.g., the cubature or unscented Kalman filters) [21,22]. A particularly useful state-of-the-art tool for this purpose is the iterated posterior linearization filter (IPLF) and smoother (IPLS) [23]. ...
... Their corresponding RTS smoothers include the extended RTS (ERTS) [1], the unscented RTS (URTS) [10], the cubature Kalman smoother (CKS) [11], and the Gauss-Hermite RTS (GHRTS) [1]. Moreover, the general nonlinear Gaussian smoother [12]- [14] and the iterated nonlinear smoothers [15], [16] have been proposed. ...
... The RTS algorithm is also applied to optimal smooth the position and attitude in backpack INS. RTS smoother recursively updates the smoothed state vector and its covariance matrix in a backward-forward manner using the following equation [39], [40]: ...
Article
A reliable and accurate position and orientation system (POS) is essential for indoor mobile mapping system (MMS). We propose a novel pedestrian POS solution for indoor MMS using backpack INS, foot mounted pedestrian dead reckoning (Foot-PDR) and sparse control points with known coordinates. The Foot-PDR can provide continuous high precision positioning with the corrections of the control points to maintain its absolute accuracy. Such precise positioning is integrated with the backpack INS to estimate the accurate position and orientation, similar to the GNSS/INS integration for the outdoor environment. The lever-arm variation issue between the backpack INS and the Foot-PDR is solved by catching a specific phase of the gait cycle that has consistent relative pose between the back and the heel of pedestrians. The feasibility of the proposed POS was verified by an outdoor large-scale open-sky area and an indoor office environment tests. The outdoor experimental results demonstrate that the estimated horizontal position and yaw angle has an accuracy of 0.07 m and 0.6 °, respectively (when the control points are as sparse as 50 meters). The indoor experimental results demonstrate that the position accuracy is 0.07 m, 0.08 m, and 0.04 m, along the north, east, and height directions. Furthermore, the proposed POS does not require expensive infrastructure support and is immune to indoor environment changes.
... In this paper, to further improve the estimation performance and enhance the flexibility, we propose the novel robust recursive filter and smoother based on the cost function induced by the maximum mixture correntropy criterion. In the maximum mixture correntropy based outlier-robust nonlinear filter (MMC-ORNF), a weighted sum of the Gaussian kernel functions of the prediction error and residual error is chosen as the cost function, and the nonlinear measurement model function is linearized by using the statistical linear regression (SLR) method [36]. Meanwhile, an additional weight is introduced into the proposed robust recursive filter to modify the filtering gain. ...
Article
In this paper, we are dedicated to studying the robust filtering and smoothing problem for a nonlinear non-Gaussian system. Considering the advantage of mixture correntropy with two kernel bandwidths in dealing with non-Gaussian noise, we propose the novel robust recursive filter and smoother based on the cost functions induced by the maximum mixture correntropy criterion, where the nonlinear dynamic model function and measurement model function are linearized by using the statistical linear regression (SLR) method. In the proposed robust recursive filter and smoother, we apply a third-order spherical cubature rule to obtain the prior estimation of the state and covariance matrix, and approximate the multi-dimensional Gaussian integrals encountered in the SLR method. Furthermore, two additional weights are introduced into the proposed robust recursive filter and smoother to modify the filtering and smoothing gains, respectively. The simulation results of manoeuvring target tracking under different non-Gaussian noise scenarios illustrate the effectiveness of the proposed robust recursive filter and smoother.
... and observation models are linear and Gaussian, the Kalman smoother (KS) [1], [7] provides the optimal Bayesian solution, which coincides with the optimal minimum mean square error estimator in that case. In the case of nonlinear dynamic systems, the iterated extended KS (IEKS) [8]- [10] makes use of local affine approximations by means of a Taylor series for the nonlinear functions, and then iteratively carries out KS. Sigma-point-based smoothing methods [11], [12] employ sigma-points to approximate the probability density of the states, which can preserve higher order accuracy than IEKS. Random sampling-based filters, such as particle filters [1], [13]- [15], can be used to deal with nonlinear tracking situations involving potentially arbitrary nonlinearities, noise, and constraints. ...
Article
Full-text available
We address the problem of autonomous tracking and state estimation for marine vessels, autonomous vehicles, and other dynamic signals under a (structured) sparsity assumption. The aim is to improve the tracking and estimation accuracy with respect to the classical Bayesian filters and smoothers. We formulate the estimation problem as a dynamic generalized group Lasso problem and develop a class of smoothing-and-splitting methods to solve it. The Levenberg-Marquardt iterated extended Kalman smoother-based multiblock alternating direction method of multipliers (LM-IEKS-mADMMs) algorithms are based on the alternating direction method of multipliers (ADMMs) framework. This leads to minimization subproblems with an inherent structure to which three new augmented recursive smoothers are applied. Our methods can deal with large-scale problems without preprocessing for dimensionality reduction. Moreover, the methods allow one to solve nonsmooth nonconvex optimization problems. We then prove that under mild conditions, the proposed methods converge to a stationary point of the optimization problem. By simulated and real-data experiments, including multisensor range measurement problems, marine vessel tracking, autonomous vehicle tracking, and audio signal restoration, we show the practical effectiveness of the proposed methods.
Article
This paper presents algorithms for temporal parallelization of Bayesian smoothers. We define the elements and the operators to pose these problems as the solutions to all-prefix-sums operations for which efficient parallel scan-algorithms are available. We present the temporal parallelization of the general Bayesian filtering and smoothing equations and specialize them to linear/Gaussian models. The advantage of the proposed algorithms is that they reduce the linear complexity of standard smoothing algorithms with respect to time to logarithmic.
Article
In the article, several concise and efficient iterated posterior linearization filtering and smoothing methodologies are proposed for nonlinear systems with cross-correlated noises. Based on the Gaussian approximation (GA), the presented methods are derived via performing statistical linear regressions (SLRs) of the nonlinear state-space models w.r.t the current posterior distribution in an iterated way. Various posterior linearization methods can be developed by employing different approximation computation approaches for the Gaussian-weighted integrals encountered in SLRs. These new estimation methods enjoy not only the accuracy and robustness of the GA filter but also the lower computational complexity. Estimation performances of the designed methods are illustrated and compared with conventional estimation schemes by two common numerical examples.
Conference Paper
We propose the algorithm to determine the power supply to the printed circuit board electronic components from real-time analysis of trends in the readings of built-in temperature sensors. The algorithm is based on the consistent application of two linear regressions and contains two parameters. The details of the impact of these parameters on the accuracy of power determination is discussed. The algorithm is verified on data obtained from a multifunctional laboratory test bench. The test bench simulates the operational conditions of printed circuit boards that are close to the onboard spacecraft conditions. The central part of the laboratory test bench is a thermal simulator of a single printed circuit board with a built-in temperature sensor system. The efficiency of the algorithm is shown so it can be applied in the the conditional monitoring.
Book
Stochastic differential equations are differential equations whose solutions are stochastic processes. They exhibit appealing mathematical properties that are useful in modeling uncertainties and noisy phenomena in many disciplines. This book is motivated by applications of stochastic differential equations in target tracking and medical technology and, in particular, their use in methodologies such as filtering, smoothing, parameter estimation, and machine learning. It builds an intuitive hands-on understanding of what stochastic differential equations are all about, but also covers the essentials of Itô calculus, the central theorems in the field, and such approximation schemes as stochastic Runge-Kutta. Greater emphasis is given to solution methods than to analysis of theoretical properties of the equations. The book's practical approach assumes only prior understanding of ordinary differential equations. The numerous worked examples and end-of-chapter exercises include application-driven derivations and computational assignments. MATLAB/Octave source code is available for download, promoting hands-on work with the methods.
Article
This paper proposes the posterior linearisation backward simultaneous localisation and mapping (PLB-SLAM) algorithm for batch SLAM problems. Based on motion and landmark measurements, we aim to estimate the trajectory of the mobile agent and the landmark positions using an approximate Rao-Blackwellised Monte Carlo solution, as in FastSLAM. PLB-SLAM improves the accuracy of current FastSLAM solutions due to two key aspects: smoothing of the trajectory distribution via backward trajectory simulation and the use of iterated posterior linearisation to obtain Gaussian approximations of the distribution of the landmarks. PLB-SLAM is assessed via numerical simulations and real experiments for indoor localisation and mapping of radio beacons using a smartphone, Bluetooth beacons, and Wi-Fi access points.
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This paper considers approximate smoothing for discretely observed non-linear stochastic differential equations. The problem is tackled by developing methods for linearising stochastic differential equations with respect to an arbitrary Gaussian process. Two methods are developed based on (1) taking the limit of statistical linear regression of the discretised process and (2) minimising an upper bound to a cost functional. Their difference is manifested in the diffusion of the approximate processes. This in turn gives novel derivations of pre-existing Gaussian smoothers when Method 1 is used and a new class of Gaussian smoothers when Method 2 is used. Furthermore, based on the aforementioned development the iterative Gaussian smoothers in discrete-time are generalised to the continuous-time setting by iteratively re-linearising the stochastic differential equation with respect to the current Gaussian process approximation to the smoothed process. The method is verified in two challenging tracking problems, a reentry problem and a radar tracked coordinated turn model with state dependent diffusion. The results show that the method has competitive estimation accuracy with state-of-the-art smoothers.
Article
In this note, we extend the accurate continuousdiscrete extended-cubature Kalman filter (ACD-ECKF) and continuous-discrete cubature Kalman filter (CD-CKF) to deal with the problem of Bayesian optimal smoothing in nonlinear dynamic systems. The dynamics can be modeled with nonlinear stochastic differential equations (SDEs) and the noise corrupted measurements are obtained at discrete time instants. To be consistent with the literature, the resulting nonlinear smoothers are referred to as the accurate continuous-discrete extendedcubature Kalman smoother (ACD-ECKS) and the continuousdiscrete cubature Kalman smoother (CD-CKS), respectively. We first present two approximation methodologies to solve the SDE encountered in the prediction step of continuous-discrete filter. Then two types of novel Gaussian approximation smoothing methods are derived based on the fixed-interval Rauch-Tung- Striebel (RTS) smoother, which computes the smoothing solution according to the stored filtering results. The performances of the proposed smoothing methods are demonstrated in a simulated application and the numerical results show that the newly presented approaches are more flexible and robust than other smoothing algorithms with lower computational cost.
Article
In this paper, we consider Rao-Blackwellization of linear substructures in sigma-point-based Gaussian assumed density smoothers. We derive marginalized prediction, smoothing, and update steps for the mixed linear/nonlinear Gaussian state-space model as well as for a hierarchical model for both conventional and iterated posterior linearization Gaussian smoothers. The proposed method is evaluated in a numerical example and it is shown that the computational complexity is reduced considerably compared to non-Rao--Blackwellized Gaussian smoothers for systems with high-dimensional linear subspaces.
Article
Full-text available
In this paper, we present a new nonlinear filter for high-dimensional state estimation, which we have named the cubature Kalman filter (CKF). The heart of the CKF is a spherical-radial cubature rule, which makes it possible to numerically compute multivariate moment integrals encountered in the nonlinear Bayesian filter. Specifically, we derive a third-degree spherical-radial cubature rule that provides a set of cubature points scaling linearly with the state-vector dimension. The CKF may therefore provide a systematic solution for high-dimensional nonlinear filtering problems. The paper also includes the derivation of a square-root version of the CKF for improved numerical stability. The CKF is tested experimentally in two nonlinear state estimation problems. In the first problem, the proposed cubature rule is used to compute the second-order statistics of a nonlinearly transformed Gaussian random variable. The second problem addresses the use of the CKF for tracking a maneuvering aircraft. The results of both experiments demonstrate the improved performance of the CKF over conventional nonlinear filters.
Article
Full-text available
In this note we shall present a new Gaussian approximation based framework for approximate optimal smoothing of non-linear stochastic state space models. The approximation framework can be used for efficiently solving non-linear fixed-interval, fixed-point and fixed-lag optimal smoothing problems. We shall also numerically compare accuracies of approximations, which are based on Taylor series expansion, unscented transformation, central differences and Gauss-Hermite quadrature.
Article
Full-text available
This note considers the application of the unscented transform to optimal smoothing of nonlinear state-space models. In this note, a new Rauch-Tung-Striebel type form of the fixed-interval unscented Kalman smoother is derived. The new smoother differs from the previously proposed two-filter-formulation-based unscented Kalman smoother in the sense that it is not based on running two independent filters forward and backward in time. Instead, a separate backward smoothing pass is used, which recursively computes corrections to the forward filtering result. The smoother equations are derived as approximations to the formal Bayesian optimal smoothing equations. The performance of the new smoother is demonstrated with a simulation.
Article
Full-text available
In this paper, a new version of the quadrature Kalman filter (QKF) is developed theoretically and tested experimentally. We first derive the new QKF for nonlinear systems with additive Gaussian noise by linearizing the process and measurement functions using statistical linear regression (SLR) through a set of Gauss-Hermite quadrature points that parameterize the Gaussian density. Moreover, we discuss how the new QKF can be extended and modified to take into account specific details of a given application. We then go on to extend the use of the new QKF to discrete-time, nonlinear systems with additive, possibly non-Gaussian noise. A bank of parallel QKFs, called the Gaussian sum-quadrature Kalman filter (GS-QKF) approximates the predicted and posterior densities as a finite number of weighted sums of Gaussian densities. The weights are obtained from the residuals of the QKFs. Three different Gaussian mixture reduction techniques are presented to alleviate the growing number of the Gaussian sum terms inherent to the GS-QKFs. Simulation results exhibit a significant improvement of the GS-QKFs over other nonlinear filtering approaches, namely, the basic bootstrap (particle) filters and Gaussian-sum extended Kalman filters, to solve nonlinear non- Gaussian filtering problems.
Article
A new algorithm for the prediction, filtering, and smoothing of non-Gaussian nonlinear state space models is shown. The algorithm is based on a Monte Carlo method in which successive prediction, filtering (and subsequently smoothing), conditional probability density functions are approximated by many of their realizations. The particular contribution of this algorithm is that it can be applied to a broad class of nonlinear non-Gaussian higher dimensional state space models on the provision that the dimensions of the system noise and the observation noise are relatively low. Several numerical examples are shown.
Book
The purpose of this chapter is to present state estimation techniques than can “adapt” themselves to certain types of uncertainties beyond those treated in earlier chapters—adaptive estimation algorithms. One type of uncertainty to be considered is the case of unknown inputs into the system, which typifies maneuvering targets. The other type will be a combination of system parameter uncertainties with unknown inputs where the system parameters (are assumed to) take values in a discrete set. The input estimation with state estimate correction technique is presented. The technique of estimating the input and, when “statistically significant,” augmenting the state with it (which leads to variable state dimension), is detailed. These two algorithms and the noise level switching technique are later compared. The design of an IMM estimator for air traffic control (ATC) is discussed in detail. Guidelines are also developed for when an adaptive estimator is really needed, i.e., when a (single model based) Kalman filter is not adequate. The chapter concludes with a brief presentation of the use of the extended Kalman filter for state and system parameter estimation. A problem solving section appears at the end of the chapter.
Book
Filtering and smoothing methods are used to produce an accurate estimate of the state of a time-varying system based on multiple observational inputs (data). Interest in these methods has exploded in recent years, with numerous applications emerging in fields such as navigation, aerospace engineering, telecommunications and medicine. This compact, informal introduction for graduate students and advanced undergraduates presents the current state-of-the-art filtering and smoothing methods in a unified Bayesian framework. Readers learn what non-linear Kalman filters and particle filters are, how they are related, and their relative advantages and disadvantages. They also discover how state-of-the-art Bayesian parameter estimation methods can be combined with state-of-the-art filtering and smoothing algorithms. The book's practical and algorithmic approach assumes only modest mathematical prerequisites. Examples include Matlab computations, and the numerous end-of-chapter exercises include computational assignments. Matlab code is available for download at www.cambridge.org/sarkka, promoting hands-on work with the methods.
Article
The Kalman smoother is known to be the maximum likelihood estimator when the measurement and transition functions are affine; i.e., a linear function plus a constant. A new proof of this result is presented that shows that the Kalman smoother decomposes a large least squares problem into a sequence of much smaller problems. The iterated Kalman smoother is then presented and shown to be a Gauss-Newton method for maximizing the likelihood function in the nonaffine case. This method takes advantage of the decomposition obtained with the Kalman smoother.
Book
Hidden Markov models have become a widely used class of statistical models with applications in diverse areas such as communications engineering, bioinformatics, finance and many more. This book is a comprehensive treatment of inference for hidden Markov models, including both algorithms and statistical theory. Topics range from filtering and smoothing of the hidden Markov chain to parameter estimation, Bayesian methods and estimation of the number of states. In a unified way the book covers both models with finite state spaces, which allow for exact algorithms for filtering, estimation etc. and models with continuous state spaces (also called state-space models) requiring approximate simulation-based algorithms that are also described in detail. Simulation in hidden Markov models is addressed in five different chapters that cover both Markov chain Monte Carlo and sequential Monte Carlo approaches. Many examples illustrate the algorithms and theory. The book also carefully treats Gaussian linear state-space models and their extensions and it contains a chapter on general Markov chain theory and probabilistic aspects of hidden Markov models. This volume will suit anybody with an interest in inference for stochastic processes, and it will be useful for researchers and practitioners in areas such as statistics, signal processing, communications engineering, control theory, econometrics, finance and more. The algorithmic parts of the book do not require an advanced mathematical background, while the more theoretical parts require knowledge of probability theory at the measure-theoretical level. Olivier Cappé is Researcher for the French National Center for Scientific Research (CNRS). He received the Ph.D. degree in 1993 from Ecole Nationale Supérieure des Télécommunications, Paris, France, where he is currently a Research Associate. Most of his current research concerns computational statistics and statistical learning. Eric Moulines is Professor at Ecole Nationale Supérieure des Télécommunications (ENST), Paris, France. He graduated from Ecole Polytechnique, France, in 1984 and received the Ph.D. degree from ENST in 1990. He has authored more than 150 papers in applied probability, mathematical statistics and signal processing. Tobias Rydén is Professor of Mathematical Statistics at Lund University, Sweden, where he also received his Ph.D. in 1993. His publications include papers ranging from statistical theory to algorithmic developments for hidden Markov models.
Article
This paper presents a new deterministic approximation technique in Bayesian networks. This method, "Expectation Propagation", unifies two previous techniques: assumed-density filtering, an extension of the Kalman filter, and loopy belief propagation, an extension of belief propagation in Bayesian networks. All three algorithms try to recover an approximate distribution which is close in KL divergence to the true distribution. Loopy belief propagation, because it propagates exact belief states, is useful for a limited class of belief networks, such as those which are purely discrete. Expectation Propagation approximates the belief states by only retaining certain expectations, such as mean and variance, and iterates until these expectations are consistent throughout the network. This makes it applicable to hybrid networks with discrete and continuous nodes. Expectation Propagation also extends belief propagation in the opposite direction - it can propagate richer belief states that incorporate correlations between nodes. Experiments with Gaussian mixture models show Expectation Propagation to be convincingly better than methods with similar computational cost: Laplace's method, variational Bayes, and Monte Carlo. Expectation Propagation also provides an efficient algorithm for training Bayes point machine classifiers.
Book
From the Publisher: "Estimation with Applications to Tracking and Navigation treats the estimation of various quantities from inherently inaccurate remote observations. It explains state estimator design using a balanced combination of linear systems, probability, and statistics." "The authors provide a review of the necessary background mathematical techniques and offer an overview of the basic concepts in estimation. They then provide detailed treatments of all the major issues in estimation with a focus on applying these techniques to real systems." "Suitable for graduate engineering students and engineers working in remote sensors and tracking, Estimation with Applications to Tracking and Navigation provides expert coverage of this important area."--BOOK JACKET.
Article
The cubature Kalman filter (CKF) is a relatively new addition to derivative-free approximate Bayesian filters built under the Gaussian assumption. This paper extends the CKF theory to address nonlinear smoothing problems; the resulting state estimator is named the fixed-interval cubature Kalman smoother (FI-CKS). Moreover, the FI-CKS is reformulated to propagate the square-root error covariances. Although algebraically equivalent to the FI-CKS, the square-root variant ensures reliable implementation when committed to embedded systems with fixed precision or when the inference problem itself is ill-conditioned. Finally, to validate the formulation, the square-root FI-CKS is applied to track a ballistic target on reentry.
Article
The derivative-free nonlinear estimation methods exploiting the Stirling’s interpolation and the unscented transformation for discrete-time nonlinear stochastic systems are treated. The divided difference and unscented filters, smoothers, and predictors based on the methods are introduced in the unified framework. The new relations among the first order Stirling’s interpolation, the second order Stirling’s interpolation, and the unscented transformation are derived and their impact on the covariance matrices of the state estimates of the corresponding filters is analysed. The theoretical results are illustrated and used for the explanation of the unexpected behaviour of the sigma point Gaussian sum filters given as a mixture of the derivative-free filters.
Article
Book on stochastic processes and filtering theory covering probability theory, Markov processes, linear and nonlinear filters, etc
Article
A solution to the optimum linear smoothing problem is presented in which the smoother is interpreted as a combination of two optimum linear filters. This result is obtained from the well-known equation for the maximum likelihood combination of two independent estimates and equivalence to previous formulations is demonstrated. Forms of the solution which are convenient for practical computation are developed.
Article
It is shown that the iterated Kalman filter (IKF) update is an application of the Gauss-Newton method for approximating a maximum likelihood estimate. An example is presented in which the iterated Kalman filter update and maximum likelihood estimate show correct convergence behavior as the observation becomes more accurate, whereas the extended Kalman filter update does not
Article
The extended Kalman filter (EKF) is probably the most widely used estimation algorithm for nonlinear systems. However, more than 35 years of experience in the estimation community has shown that is difficult to implement, difficult to tune, and only reliable for systems that are almost linear on the time scale of the updates. Many of these difficulties arise from its use of linearization. To overcome this limitation, the unscented transformation (UT) was developed as a method to propagate mean and covariance information through nonlinear transformations. It is more accurate, easier to implement, and uses the same order of calculations as linearization. This paper reviews the motivation, development, use, and implications of the UT.
The iterated Kalman filter update as a Gauss-Newton method
• B Bell
• F Cathey