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122-437: The papers establishing the mathematical foundations of Kalman type filters were published between 1959 and 1961. The Kalman filter is the optimal linear estimator for linear system models with additive independent white noise in both the transition and the measurement systems. Unfortunately, in engineering, most systems are nonlinear , so attempts were made to apply this filtering method to nonlinear systems; most of this work

244-429: A Markov chain built on linear operators perturbed by errors that may include Gaussian noise . The state of the target system refers to the ground truth (yet hidden) system configuration of interest, which is represented as a vector of real numbers . At each discrete time increment, a linear operator is applied to the state to generate the new state, with some noise mixed in, and optionally some information from

366-463: A Kalman Filter is often difficult due to the difficulty of getting a good estimate of the noise covariance matrices Q k and R k . Extensive research has been done to estimate these covariances from data. One practical method of doing this is the autocovariance least-squares (ALS) technique that uses the time-lagged autocovariances of routine operating data to estimate the covariances. The GNU Octave and Matlab code used to calculate

488-1143: A chosen θ 0 {\displaystyle \theta _{0}} . By substituting the definition of the operators F and G we obtain the fully explicit projection filter equation in direct metric: + ∑ k = 1 d [ ∑ j = 1 m γ i j ( θ t ) ∫ [ b k ( x , t ) − ∫ b k ( z , t ) p ( z , θ t ) d z ] p ( x , θ t ) ∂ p ( x , θ t ) ∂ θ j d x ] ∘ d Y t k   . {\displaystyle +\sum _{k=1}^{d}\;\left[\sum _{j=1}^{m}\gamma ^{ij}(\theta _{t})\;\int \left[b_{k}(x,t)-\int b_{k}(z,t)p(z,\theta _{t})dz\right]\;p(x,\theta _{t})\;{\frac {\partial p(x,\theta _{t})}{\partial \theta _{j}}}\;dx\right]\circ dY_{t}^{k}\ .} If one uses

610-683: A chosen exponential family. The exponential family can be chosen so as to make the prediction step of the filtering algorithm exact. A different type of projection filters, based on an alternative projection metric, the direct L 2 {\displaystyle L^{2}} metric, was introduced in Armstrong and Brigo (2016). With this metric, the projection filters on families of mixture distributions coincide with filters based on Galerkin methods . Later on, Armstrong, Brigo and Rossi Ferrucci (2021) derive optimal projection filters that satisfy specific optimality criteria in approximating

732-465: A density described informally as where σ ( Y s , s ≤ t ) {\displaystyle \sigma (Y_{s},s\leq t)} is the sigma-field generated by the history of noisy observations Y {\displaystyle Y} up to time t {\displaystyle t} , under suitable technical conditions the density p t {\displaystyle p_{t}} satisfies

854-536: A faux algebraic Riccati equation for the gain design. Another way of improving extended Kalman filter performance is to employ the H-infinity results from robust control. Robust filters are obtained by adding a positive definite term to the design Riccati equation. The additional term is parametrized by a scalar which the designer may tweak to achieve a trade-off between mean-square-error and peak error performance criteria. The invariant extended Kalman filter (IEKF)

976-582: A finite dimensional density p ( x , θ t ) {\displaystyle p(x,\theta _{t})} (or p ( x , θ t ) {\displaystyle {\sqrt {p(x,\theta _{t})}}} ). The fact that the filter SPDE is in Stratonovich form allows for the following. As Stratonovich SPDEs satisfy the chain rule, F {\displaystyle F} and G {\displaystyle G} behave as vector fields. Thus,

1098-498: A join-tree or Markov tree . Additional methods include belief filtering which use Bayes or evidential updates to the state equations. A wide variety of Kalman filters exists by now: Kalman's original formulation - now termed the "simple" Kalman filter, the Kalman–Bucy filter , Schmidt's "extended" filter, the information filter , and a variety of "square-root" filters that were developed by Bierman, Thornton, and many others. Perhaps

1220-618: A linear interpolation, x = ( 1 − t ) ( a ) + t ( b ) {\displaystyle x=(1-t)(a)+t(b)} for t {\displaystyle t} between [0,1]. In our case: This expression also resembles the alpha beta filter update step. If the model is accurate, and the values for x ^ 0 ∣ 0 {\displaystyle {\hat {\mathbf {x} }}_{0\mid 0}} and P 0 ∣ 0 {\displaystyle \mathbf {P} _{0\mid 0}} accurately reflect

1342-463: A linear state error, but from an invariant state error. The main benefit is that the gain and covariance equations converge to constant values on a much bigger set of trajectories than equilibrium points as it is the case for the EKF, which results in a better convergence of the estimation. A nonlinear Kalman filter which shows promise as an improvement over the EKF is the unscented Kalman filter (UKF). In

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1464-419: A locally optimal filter, however, it is not necessarily stable because the solutions of the underlying Riccati equation are not guaranteed to be positive definite. One way of improving performance is the faux algebraic Riccati technique which trades off optimality for stability. The familiar structure of the extended Kalman filter is retained but stability is achieved by selecting a positive definite solution to

1586-480: A manifold S Θ {\displaystyle S_{\Theta }} of mixture families, leads to equivalence with a Galerkin method. The projection filter in Hellinger/Fisher metric when implemented on a manifold S Θ 1 / 2 {\displaystyle S_{\Theta }^{1/2}} of square roots of an exponential family of densities is equivalent to

1708-496: A random dynamical system from partial noisy observations of the signal. The objective is computing the probability distribution of the signal conditional on the history of the noise-perturbed observations. This distribution allows for calculations of all statistics of the signal given the history of observations. If this distribution has a density, the density satisfies specific stochastic partial differential equations (SPDEs) called Kushner-Stratonovich equation, or Zakai equation. It

1830-481: A rich structure, or similarly for the quadratic sensor. In such cases the projection filters have been studied as an alternative, having been applied also to navigation. Other general nonlinear filtering methods like full particle filters may be considered in this case. Having stated this, the extended Kalman filter can give reasonable performance, and is arguably the de facto standard in navigation systems and GPS. Model Initialize Predict-Update Unlike

1952-437: A single equation; however, it is most often conceptualized as two distinct phases: "Predict" and "Update". The predict phase uses the state estimate from the previous timestep to produce an estimate of the state at the current timestep. This predicted state estimate is also known as the a priori state estimate because, although it is an estimate of the state at the current timestep, it does not include observation information from

2074-469: A system's state to calculate a new state. The measurements' certainty-grading and current-state estimate are important considerations. It is common to discuss the filter's response in terms of the Kalman filter's gain . The Kalman gain is the weight given to the measurements and current-state estimate, and can be "tuned" to achieve a particular performance. With a high gain, the filter places more weight on

2196-445: A truck. The truck can be equipped with a GPS unit that provides an estimate of the position within a few meters. The GPS estimate is likely to be noisy; readings 'jump around' rapidly, though remaining within a few meters of the real position. In addition, since the truck is expected to follow the laws of physics, its position can also be estimated by integrating its velocity over time, determined by keeping track of wheel revolutions and

2318-435: A two-phase process: a prediction phase and an update phase. In the prediction phase, the Kalman filter produces estimates of the current state variables , including their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some error, including random noise) is observed, these estimates are updated using a weighted average , with more weight given to estimates with greater certainty. The algorithm

2440-620: A worthwhile alternative to the Autocovariance Least-Squares methods. Another approach is the Optimized Kalman Filter ( OKF ), which considers the covariance matrices not as representatives of the noise, but rather, as parameters aimed to achieve the most accurate state estimation. These two views coincide under the KF assumptions, but often contradict each other in real systems. Thus, OKF's state estimation

2562-955: Is d p = 1 2 p [ F ( p ) d t + G T ( p ) ∘ d Y ]   . {\displaystyle d{\sqrt {p}}={\frac {1}{2{\sqrt {p}}}}[F(p)\,dt+G^{T}(p)\circ dY]\ .} These are Stratonovich SPDEs whose solutions evolve in infinite dimensional function spaces. For example p t {\displaystyle p_{t}} may evolve in L 2 {\displaystyle L^{2}} (direct metric d 2 {\displaystyle d_{2}} ) or p t {\displaystyle {\sqrt {p_{t}}}} may evolve in L 2 {\displaystyle L^{2}} (Hellinger metric d H {\displaystyle d_{H}} ) where ‖ ⋅ ‖ {\displaystyle \Vert \cdot \Vert }

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2684-1230: Is d θ i = [ ∑ j = 1 n γ i j ( θ t ) ∫ F ( p ( x , θ t ) ) ∂ p ( x , θ t ) ∂ θ j d x ] d t + ∑ k = 1 d [ ∑ j = 1 n γ i j ( θ t ) ∫ G k ( p ( x , θ t ) ) ∂ p ( x , θ t ) ∂ θ j d x ] ∘ d Y k {\displaystyle d\theta _{i}=\left[\sum _{j=1}^{n}\gamma ^{ij}(\theta _{t})\;\int F(p(x,\theta _{t}))\;{\frac {\partial p(x,\theta _{t})}{\partial \theta _{j}}}dx\right]dt+\sum _{k=1}^{d}\;\left[\sum _{j=1}^{n}\gamma ^{ij}(\theta _{t})\;\int G_{k}(p(x,\theta _{t}))\;{\frac {\partial p(x,\theta _{t})}{\partial \theta _{j}}}\;dx\right]\circ dY_{k}} with initial condition

2806-426: Is where a = σ σ T {\displaystyle a=\sigma \sigma ^{T}} and T {\displaystyle T} denotes transposition. To derive the first version of the projection filters, one needs to put the p t {\displaystyle p_{t}} SPDE in Stratonovich form. One obtains Through the chain rule, it's immediate to derive

2928-581: Is recursive . It can operate in real time , using only the present input measurements and the state calculated previously and its uncertainty matrix; no additional past information is required. Optimality of Kalman filtering assumes that errors have a normal (Gaussian) distribution. In the words of Rudolf E. Kálmán : "The following assumptions are made about random processes: Physical random phenomena may be thought of as due to primary random sources exciting dynamic systems. The primary sources are assumed to be independent gaussian random processes with zero mean;

3050-527: Is a Brownian motion . Validity of all regularity conditions necessary for the results to hold will be assumed, with details given in the references. The associated noisy observation process Y t ∈ R d {\displaystyle Y_{t}\in \mathbb {R} ^{d}} is modelled by where b {\displaystyle b} is R d {\displaystyle \mathbb {R} ^{d}} valued and V t {\displaystyle V_{t}}

3172-459: Is a Brownian motion independent of W t {\displaystyle W_{t}} . As hinted above, the full filter is the conditional distribution of X t {\displaystyle X_{t}} given a prior for X 0 {\displaystyle X_{0}} and the history of Y {\displaystyle Y} up to time t {\displaystyle t} . If this distribution has

3294-411: Is a difficult one and is treated as a problem of control theory using robust control . The Kalman filter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state. In contrast to batch estimation techniques, no history of observations and/or estimates is required. In what follows,

3416-539: Is a first-order extended Kalman filter (EKF). Higher order EKFs may be obtained by retaining more terms of the Taylor series expansions. For example, second and third order EKFs have been described. However, higher order EKFs tend to only provide performance benefits when the measurement noise is small. The typical formulation of the EKF involves the assumption of additive process and measurement noise. This assumption, however,

3538-465: Is a modified version of the EKF for nonlinear systems possessing symmetries (or invariances ). It combines the advantages of both the EKF and the recently introduced symmetry-preserving filters . Instead of using a linear correction term based on a linear output error, the IEKF uses a geometrically adapted correction term based on an invariant output error; in the same way the gain matrix is not updated from

3660-403: Is a new state estimate that lies between the predicted and measured state, and has a better estimated uncertainty than either alone. This process is repeated at every time step, with the new estimate and its covariance informing the prediction used in the following iteration. This means that Kalman filter works recursively and requires only the last "best guess", rather than the entire history, of

3782-481: Is a strong analogy between the equations of a Kalman Filter and those of the hidden Markov model. A review of this and other models is given in Roweis and Ghahramani (1999) and Hamilton (1994), Chapter 13. In order to use the Kalman filter to estimate the internal state of a process given only a sequence of noisy observations, one must model the process in accordance with the following framework. This means specifying

Extended Kalman filter - Misplaced Pages Continue

3904-461: Is also important for robotic motion planning and control, and can be used for trajectory optimization . Kalman filtering also works for modeling the central nervous system 's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback , the use of Kalman filters provides a realistic model for making estimates of the current state of a motor system and issuing updated commands. The algorithm works via

4026-422: Is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, to produce estimates of unknown variables that tend to be more accurate than those based on a single measurement, by estimating a joint probability distribution over the variables for each time-step. The filter is constructed as a mean squared error minimiser, but an alternative derivation of

4148-416: Is an important topic in control theory and control systems engineering. Together with the linear-quadratic regulator (LQR), the Kalman filter solves the linear–quadratic–Gaussian control problem (LQG). The Kalman filter, the linear-quadratic regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the most fundamental problems of control theory. In most applications,

4270-522: Is better than the estimate obtained by using only one measurement alone. As such, it is a common sensor fusion and data fusion algorithm. Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not accounted for, all limit how well it is possible to determine the system's state. The Kalman filter deals effectively with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter produces an estimate of

4392-403: Is defined differently. The Jacobian matrix H k {\displaystyle {{\boldsymbol {H}}_{k}}} is defined as before, but determined from the implicit observation model h ( x k , z k ) {\displaystyle h({\boldsymbol {x}}_{k},{\boldsymbol {z}}_{k})} . The iterated extended Kalman filter improves

4514-508: Is generally credited with developing the first implementation of a Kalman filter. He realized that the filter could be divided into two distinct parts, with one part for time periods between sensor outputs and another part for incorporating measurements. It was during a visit by Kálmán to the NASA Ames Research Center that Schmidt saw the applicability of Kálmán's ideas to the nonlinear problem of trajectory estimation for

4636-564: Is independent of time. The initial state, and the noise vectors at each step { x 0 , w 1 , … , w k , v 1 , … , v k } {\displaystyle \{\mathbf {x} _{0},\mathbf {w} _{1},\dots ,\mathbf {w} _{k},\mathbf {v} _{1},\dots ,\mathbf {v} _{k}\}} are all assumed to be mutually independent . Many real-time dynamic systems do not exactly conform to this model. In fact, unmodeled dynamics can seriously degrade

4758-435: Is known that the nonlinear filter density evolves in an infinite dimensional function space. One can choose a finite dimensional family of probability densities, for example Gaussian densities , Gaussian mixtures , or exponential families , on which the infinite-dimensional filter density can be approximated. The basic idea of the projection filter is to use a geometric structure in the chosen spaces of densities to project

4880-672: Is minimized, not zeroed, and one never attains ( δ t ) 2 {\displaystyle (\delta t)^{2}} convergence, only δ t {\displaystyle \delta t} convergence. A further benefit of the Ito vector projection is that it minimizes the order 1 Taylor expansion in δ t {\displaystyle \delta t} of To achieve ( δ t ) 2 {\displaystyle (\delta t)^{2}} convergence, rather than δ t {\displaystyle \delta t} convergence,

5002-473: Is more robust to modeling inaccuracies. It follows from theory that the Kalman filter provides an optimal state estimation in cases where a) the model matches the real system perfectly, b) the entering noise is "white" (uncorrelated), and c) the covariances of the noise are known exactly. Correlated noise can also be treated using Kalman filters. Several methods for the noise covariance estimation have been proposed during past decades, including ALS, mentioned in

Extended Kalman filter - Misplaced Pages Continue

5124-582: Is named for Hungarian émigré Rudolf E. Kálmán , although Thorvald Nicolai Thiele and Peter Swerling developed a similar algorithm earlier. Richard S. Bucy of the Johns Hopkins Applied Physics Laboratory contributed to the theory, causing it to be known sometimes as Kalman–Bucy filtering. Kalman was inspired to derive the Kalman filter by applying state variables to the Wiener filtering problem . Stanley F. Schmidt

5246-669: Is not necessary for EKF implementation. Instead, consider a more general system of the form: Here w k and v k are the process and observation noises which are both assumed to be zero mean multivariate Gaussian noises with covariance Q k and R k respectively. Then the covariance prediction and innovation equations become where the matrices L k − 1 {\displaystyle {\boldsymbol {L}}_{k-1}} and M k {\displaystyle {\boldsymbol {M}}_{k}} are Jacobian matrices: The predicted state estimate and measurement residual are evaluated at

5368-474: Is the closest point on S Θ {\displaystyle S_{\Theta }} (or S Θ 1 / 2 {\displaystyle S_{\Theta }^{1/2}} ) to p {\displaystyle p} (or p {\displaystyle {\sqrt {p}}} ). Denote it by π ( p ) {\displaystyle \pi (p)} . The metric projection is, by definition, according to

5490-561: Is the norm of Hilbert space L 2 {\displaystyle L^{2}} . In any case, p t {\displaystyle p_{t}} (or p t {\displaystyle {\sqrt {p_{t}}}} ) will not evolve inside any finite dimensional family of densitities, The projection filter idea is approximating p t ( x ) {\displaystyle p_{t}(x)} (or p t ( x ) {\displaystyle {\sqrt {p_{t}(x)}}} ) via

5612-415: The δ t {\displaystyle \delta t} term of the Taylor expansion for the mean square error finding the drift term in the approximating Ito equation that minimizes the ( δ t ) 2 {\displaystyle (\delta t)^{2}} term of the same difference. Here the δ t {\displaystyle \delta t} order term

5734-596: The Apollo program resulting in its incorporation in the Apollo navigation computer . This digital filter is sometimes termed the Stratonovich–Kalman–Bucy filter because it is a special case of a more general, nonlinear filter developed by the Soviet mathematician Ruslan Stratonovich . In fact, some of the special case linear filter's equations appeared in papers by Stratonovich that were published before

5856-430: The nonlinear filtering problem and the inadequacy of a simple mean and variance-covariance estimator to fully represent the optimal filter. It should also be noted that the extended Kalman filter may give poor performances even for very simple one-dimensional systems such as the cubic sensor, where the optimal filter can be bimodal and as such cannot be effectively represented by a single mean and variance estimator, having

5978-542: The EKF in its estimation of error in all the directions. "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." A 2012 paper includes simulation results which suggest that some published variants of

6100-2507: The Hellinger distance instead, square roots of densities are needed. The tangent space basis is then and one defines the metric The metric g {\displaystyle g} is the Fisher information metric. One follows steps completely analogous to the direct metric case and the filter equation in Hellinger/Fisher metric is again with initial condition a chosen θ 0 {\displaystyle \theta _{0}} . Substituting F and G one obtains d θ i ( t ) = [ ∑ j = 1 m g i j ( θ t ) ∫ L t ∗ p ( x , θ t ) p ( x , θ t ) ∂ p ( x , θ t ) ∂ θ j d x − ∑ j = 1 m g i j ( θ t ) ∫ 1 2 | b t ( x ) | 2 ∂ p ( x , θ t ) ∂ θ j d x ] d t {\displaystyle d\theta _{i}(t)=\left[\sum _{j=1}^{m}g^{ij}(\theta _{t})\;\int {\frac {{\cal {L}}_{t}^{\ast }\,p(x,\theta _{t})}{p(x,\theta _{t})}}\;{\frac {\partial p(x,\theta _{t})}{\partial \theta _{j}}}\;dx-\sum _{j=1}^{m}g^{ij}(\theta _{t})\int {\frac {1}{2}}\vert b_{t}(x)\vert ^{2}{\frac {\partial p(x,\theta _{t})}{\partial \theta _{j}}}dx\right]dt} + ∑ k = 1 d [ ∑ j = 1 m g i j ( θ t ) ∫ b k ( x , t ) ∂ p ( x , θ t ) ∂ θ j d x ] ∘ d Y t k   . {\displaystyle +\sum _{k=1}^{d}\;\left[\sum _{j=1}^{m}g^{ij}(\theta _{t})\;\int b_{k}(x,t)\;{\frac {\partial p(x,\theta _{t})}{\partial \theta _{j}}}\;dx\right]\circ dY_{t}^{k}\ .} The projection filter in direct metric, when implemented on

6222-725: The Ito vector and the Ito jet projection result in final SDEs, driven by the observations d Y {\displaystyle dY} , for the parameter θ t {\displaystyle \theta _{t}} that best approximates the exact filter evolution for small times. Jones and Soatto (2011) mention projection filters as possible algorithms for on-line estimation in visual-inertial navigation , mapping and localization, while again on navigation Azimi-Sadjadi and Krishnaprasad (2005) use projection filters algorithms. The projection filter has been also considered for applications in ocean dynamics by Lermusiaux 2006. Kutschireiter, Rast, and Drugowitsch (2022) refer to

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6344-630: The Ito-jet projection is introduced. It is based on the notion of metric projection. The metric projection of a density p ∈ L 2 {\displaystyle p\in L^{2}} (or p ∈ L 2 {\displaystyle {\sqrt {p}}\in L^{2}} ) onto the manifold S Θ {\displaystyle S_{\Theta }} (or S Θ 1 / 2 {\displaystyle S_{\Theta }^{1/2}} )

6466-489: The Kushner—Stratonovich SPDE: where E p {\displaystyle E_{p}} is the expectation E p [ h ] = ∫ h ( x ) p ( x ) d x , {\displaystyle E_{p}[h]=\int h(x)p(x)dx,} and the forward diffusion operator L t ∗ {\displaystyle {\cal {L}}_{t}^{*}}

6588-611: The SPDE for d p t {\displaystyle d{\sqrt {p_{t}}}} . To shorten notation one may rewrite this last SPDE as d p = F ( p ) d t + G T ( p ) ∘ d Y   , {\displaystyle dp=F(p)\,dt+G^{T}(p)\circ dY\ ,} where the operators F ( p ) {\displaystyle F(p)} and G T ( p ) {\displaystyle G^{T}(p)} are defined as The square root version

6710-410: The Taylor expansion of the mean square distance in L 2 {\displaystyle L^{2}} between π ( p 0 + δ t ) {\displaystyle \pi (p_{0+\delta t})} and p ( ⋅ , θ 0 + δ t ) {\displaystyle p(\cdot ,\theta _{0+\delta t})} . Both

6832-672: The UKF fail to be as accurate as the Second Order Extended Kalman Filter (SOEKF), also known as the augmented Kalman filter. The SOEKF predates the UKF by approximately 35 years with the moment dynamics first described by Bass et al. The difficulty in implementing any Kalman-type filters for nonlinear state transitions stems from the numerical stability issues required for precision, however the UKF does not escape this difficulty in that it uses linearization as well, namely linear regression . The stability issues for

6954-616: The UKF generally stem from the numerical approximation to the square root of the covariance matrix, whereas the stability issues for both the EKF and the SOEKF stem from possible issues in the Taylor Series approximation along the trajectory. The UKF was in fact predated by the Ensemble Kalman filter , invented by Evensen in 1994. It has the advantage over the UKF that the number of ensemble members used can be much smaller than

7076-447: The UKF, the probability density is approximated by a deterministic sampling of points which represent the underlying distribution as a Gaussian . The nonlinear transformation of these points are intended to be an estimation of the posterior distribution , the moments of which can then be derived from the transformed samples. The transformation is known as the unscented transform . The UKF tends to be more robust and more accurate than

7198-421: The angle of the steering wheel. This is a technique known as dead reckoning . Typically, the dead reckoning will provide a very smooth estimate of the truck's position, but it will drift over time as small errors accumulate. For this example, the Kalman filter can be thought of as operating in two distinct phases: predict and update. In the prediction phase, the truck's old position will be modified according to

7320-695: The assumed density filters. One should note that it is also possible to project the simpler Zakai equation for an unnormalized version of the density p. This would result in the same Hellinger projection filter but in a different direct metric projection filter. Finally, if in the exponential family case one includes among the sufficient statistics of the exponential family the observation function in d Y t {\displaystyle dY_{t}} , namely b ( x ) {\displaystyle b(x)} 's components and | b ( x ) | 2 {\displaystyle |b(x)|^{2}} , then one can see that

7442-799: The chosen metric, the best one can ever do for approximating p {\displaystyle p} in S Θ {\displaystyle S_{\Theta }} . Thus the idea is finding a projection filter that comes as close as possible to the metric projection. In other terms, one considers the criterion θ 0 + δ t ≈ argmin θ   ‖ π ( p 0 + δ t ) − p ( ⋅ , θ ) ‖ . {\displaystyle \theta _{0+\delta t}\approx {\mbox{argmin}}_{\theta }\ \|\pi (p_{0+\delta t})-p(\cdot ,\theta )\|.} The detailed calculations are lengthy and laborious, but

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7564-465: The context of hazard position estimation, while Vellekoop and Clark (2006) generalize the projection filter theory to deal with changepoint detection . Harel, Meir and Opper (2015) apply the projection filters in assumed density form to the filtering of optimal point processes with applications to neural encoding . Broecker and Parlitz (2000) study projection filter methods for noise reduction in chaotic time series . Zhang, Wang, Wu and Xu (2014) apply

7686-418: The controls on the system if they are known. Then, another linear operator mixed with more noise generates the measurable outputs (i.e., observation) from the true ("hidden") state. The Kalman filter may be regarded as analogous to the hidden Markov model, with the difference that the hidden state variables have values in a continuous space as opposed to a discrete state space as for the hidden Markov model. There

7808-431: The correction step at each new observation is exact, as the related Bayes formula entails no approximation. Now rather than considering the exact filter SPDE in Stratonovich calculus form, one keeps it in Ito calculus form In the Stratonovich projection filters above, the vector fields F {\displaystyle F} and G {\displaystyle G} were projected separately. By definition,

7930-443: The correction step in the filtering algorithm becomes exact. In other terms, the projection of the vector field G {\displaystyle G} is exact, resulting in G {\displaystyle G} itself. Writing the filtering algorithm in a setting with continuous state X {\displaystyle X} and discrete time observations Y {\displaystyle Y} , one can see that

8052-403: The covariances are set, it is useful to evaluate the performance of the filter; i.e., whether it is possible to improve the state estimation quality. If the Kalman filter works optimally, the innovation sequence (the output prediction error) is a white noise, therefore the whiteness property of the innovations measures filter performance. Several different methods can be used for this purpose. If

8174-417: The current timestep. In the update phase, the innovation (the pre-fit residual), i.e. the difference between the current a priori prediction and the current observation information, is multiplied by the optimal Kalman gain and combined with the previous state estimate to refine the state estimate. This improved estimate based on the current observation is termed the a posteriori state estimate. Typically,

8296-673: The discrete-time extended Kalman filter, the prediction and update steps are coupled in the continuous-time extended Kalman filter. Most physical systems are represented as continuous-time models while discrete-time measurements are frequently taken for state estimation via a digital processor. Therefore, the system model and measurement model are given by where x k = x ( t k ) {\displaystyle \mathbf {x} _{k}=\mathbf {x} (t_{k})} . Initialize Predict where Update where The update equations are identical to those of discrete-time extended Kalman filter. The above recursion

8418-449: The distribution of the initial state values, then the following invariants are preserved: where E ⁡ [ ξ ] {\displaystyle \operatorname {E} [\xi ]} is the expected value of ξ {\displaystyle \xi } . That is, all estimates have a mean error of zero. Also: so covariance matrices accurately reflect the covariance of estimates. Practical implementation of

8540-493: The dynamic systems will be linear." Regardless of Gaussianity, however, if the process and measurement covariances are known, then the Kalman filter is the best possible linear estimator in the minimum mean-square-error sense , although there may be better nonlinear estimators. It is a common misconception (perpetuated in the literature) that the Kalman filter cannot be rigorously applied unless all noise processes are assumed to be Gaussian. Extensions and generalizations of

8662-508: The equation is characterized by a d t {\displaystyle dt} vector field F {\displaystyle F} and a d Y t {\displaystyle dY_{t}} vector field G {\displaystyle G} . For this version of the projection filter one is satisfied with dealing with the two vector fields separately. One may project F {\displaystyle F} and G {\displaystyle G} on

8784-441: The extended Kalman filter, the state transition and observation models don't need to be linear functions of the state but may instead be differentiable functions. Here w k and v k are the process and observation noises which are both assumed to be zero mean multivariate Gaussian noises with covariance Q k and R k respectively. u k is the control vector. The function f can be used to compute

8906-458: The filter equation. This is based on research on the geometry of Ito Stochastic differential equations on manifolds based on the jet bundle , the so-called 2-jet interpretation of Ito stochastic differential equations on manifolds. Here the derivation of the different projection filters is sketched. This is a derivation of both the initial filter in Hellinger/Fisher metric sketched by Hanzon and fully developed by Brigo, Hanzon and LeGland, and

9028-491: The filter is also provided showing how the filter relates to maximum likelihood statistics. The filter is named after Rudolf E. Kálmán . Kalman filtering has numerous technological applications. A common application is for guidance, navigation, and control of vehicles, particularly aircraft, spacecraft and ships positioned dynamically . Furthermore, Kalman filtering is much applied in time series analysis tasks such as signal processing and econometrics . Kalman filtering

9150-431: The filter performance, even when it was supposed to work with unknown stochastic signals as inputs. The reason for this is that the effect of unmodeled dynamics depends on the input, and, therefore, can bring the estimation algorithm to instability (it diverges). On the other hand, independent white noise signals will not make the algorithm diverge. The problem of distinguishing between measurement noise and unmodeled dynamics

9272-640: The full infinite-dimensional filter in an optimal way, beyond the optimal approximation of the SPDE coefficients alone, according to precise criteria such as mean square minimization. Projection filters have been studied by the Swedish Defense Research Agency and have also been successfully applied to a variety of fields including navigation , ocean dynamics , quantum optics and quantum systems , estimation of fiber diameters, estimation of chaotic time series , change point detection and other areas. The term "projection filter"

9394-512: The guidance and navigation systems of reusable launch vehicles and the attitude control and navigation systems of spacecraft which dock at the International Space Station . Kalman filtering uses a system's dynamic model (e.g., physical laws of motion), known control inputs to that system, and multiple sequential measurements (such as from sensors) to form an estimate of the system's varying quantities (its state ) that

9516-420: The infinite dimensional SPDE of the optimal filter onto the chosen finite dimensional family, obtaining a finite dimensional stochastic differential equation (SDE) for the parameter of the density in the finite dimensional family that approximates the full filter evolution. To do this, the chosen finite dimensional family is equipped with a manifold structure as in information geometry . The projection filter

9638-476: The infinite dimensional optimal filter. Indeed, the Stratonovich-based projection filters optimized the approximations of the SPDE separate coefficients on the chosen manifold but not the SPDE solution as a whole. This has been dealt with by introducing the optimal projection filters. The innovation here is to work directly with Ito calculus, instead of resorting to the Stratonovich calculus version of

9760-499: The internal state is much larger (has more degrees of freedom ) than the few "observable" parameters which are measured. However, by combining a series of measurements, the Kalman filter can estimate the entire internal state. For the Dempster–Shafer theory , each state equation or observation is considered a special case of a linear belief function and the Kalman filtering is a special case of combining linear belief functions on

9882-547: The later projection filter in direct L2 metric by Armstrong and Brigo (2016). It is assumed that the unobserved random signal X t ∈ R m {\displaystyle X_{t}\in \mathbb {R} ^{m}} is modelled by the Ito stochastic differential equation : where f and σ d W {\displaystyle \sigma \,dW} are R m {\displaystyle \mathbb {R} ^{m}} valued and W t {\displaystyle W_{t}}

10004-404: The linearization of the extended Kalman filter by recursively modifying the centre point of the Taylor expansion. This reduces the linearization error at the cost of increased computational requirements. The robust extended Kalman filter arises by linearizing the signal model about the current state estimate and using the linear Kalman filter to predict the next estimate. This attempts to produce

10126-612: The manifold S Θ {\displaystyle S_{\Theta }} , and where, when applied to a vector such as G T {\displaystyle G^{T}} , it is assumed to act component-wise by projecting each of G T {\displaystyle G^{T}} 's components. As a basis of this tangent space is by denoting the inner product of L 2 {\displaystyle L^{2}} with ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , one defines

10248-622: The matrices, for each time-step k {\displaystyle k} , following: As seen below, it is common in many applications that the matrices F {\displaystyle \mathbf {F} } , H {\displaystyle \mathbf {H} } , Q {\displaystyle \mathbf {Q} } , R {\displaystyle \mathbf {R} } , and B {\displaystyle \mathbf {B} } are constant across time, in which case their k {\displaystyle k} index may be dropped. The Kalman filter model assumes

10370-619: The mean of the process and measurement noise terms, which is assumed to be zero. Otherwise, the non-additive noise formulation is implemented in the same manner as the additive noise EKF . In certain cases, the observation model of a nonlinear system cannot be solved for z k {\displaystyle {\boldsymbol {z}}_{k}} , but can be expressed by the implicit function : where z k = z ′ k + v k {\displaystyle {\boldsymbol {z}}_{k}={\boldsymbol {z'}}_{k}+{\boldsymbol {v}}_{k}} are

10492-501: The method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems . The basis is a hidden Markov model such that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions. Kalman filtering has been used successfully in multi-sensor fusion , and distributed sensor networks to develop distributed or consensus Kalman filtering. The filtering method

10614-428: The metric and the projection is thus where γ i j {\displaystyle \gamma ^{ij}} is the inverse of γ i j {\displaystyle \gamma _{ij}} . The projected equation thus reads which can be written as where it has been crucial that Stratonovich calculus obeys the chain rule. From the above equation, the final projection filter SDE

10736-425: The most commonly used type of very simple Kalman filter is the phase-locked loop , which is now ubiquitous in radios, especially frequency modulation (FM) radios, television sets, satellite communications receivers, outer space communications systems, and nearly any other electronic communications equipment. Kalman filtering is based on linear dynamic systems discretized in the time domain. They are modeled on

10858-411: The most recent measurements, and thus conforms to them more responsively. With a low gain, the filter conforms to the model predictions more closely. At the extremes, a high gain (close to one) will result in a more jumpy estimated trajectory, while a low gain (close to zero) will smooth out noise but decrease the responsiveness. When performing the actual calculations for the filter (as discussed below),

10980-603: The noise covariance matrices using the ALS technique is available online using the GNU General Public License . Field Kalman Filter (FKF), a Bayesian algorithm, which allows simultaneous estimation of the state, parameters and noise covariance has been proposed. The FKF algorithm has a recursive formulation, good observed convergence, and relatively low complexity, thus suggesting that the FKF algorithm may possibly be

11102-701: The noise has no explicit knowledge of time. At time k {\displaystyle k} an observation (or measurement) z k {\displaystyle \mathbf {z} _{k}} of the true state x k {\displaystyle \mathbf {x} _{k}} is made according to where Analogously to the situation for w k {\displaystyle \mathbf {w} _{k}} , one may write v ∙ {\displaystyle \mathbf {v} _{\bullet }} instead of v k {\displaystyle \mathbf {v} _{k}} if R {\displaystyle \mathbf {R} }

11224-402: The noise terms are distributed in a non-Gaussian manner, methods for assessing performance of the filter estimate, which use probability inequalities or large-sample theory, are known in the literature. Consider a truck on frictionless, straight rails. Initially, the truck is stationary at position 0, but it is buffeted this way and that by random uncontrolled forces. We measure the position of

11346-402: The noisy observations. The conventional extended Kalman filter can be applied with the following substitutions: where: Here the original observation covariance matrix R k {\displaystyle {{\boldsymbol {R}}_{k}}} is transformed, and the innovation y ~ k {\displaystyle {\tilde {\boldsymbol {y}}}_{k}}

11468-487: The non-linear function around the current estimate. See the Kalman Filter article for notational remarks. Notation x ^ n ∣ m {\displaystyle {\hat {\mathbf {x} }}_{n\mid m}} represents the estimate of x {\displaystyle \mathbf {x} } at time n given observations up to and including at time m ≤ n . where

11590-489: The notation x ^ n ∣ m {\displaystyle {\hat {\mathbf {x} }}_{n\mid m}} represents the estimate of x {\displaystyle \mathbf {x} } at time n given observations up to and including at time m ≤ n . The state of the filter is represented by two variables: The algorithm structure of the Kalman filter resembles that of Alpha beta filter . The Kalman filter can be written as

11712-438: The parameter that can be implemented efficiently. Projection filters are also flexible, as they allow fine tuning the precision of the approximation by choosing richer approximating families, and some exponential families make the correction step in the projection filtering algorithm exact. Some formulations coincide with heuristic based assumed density filters or with Galerkin methods . Projection filters can also approximate

11834-417: The physical laws of motion (the dynamic or "state transition" model). Not only will a new position estimate be calculated, but also a new covariance will be calculated as well. Perhaps the covariance is proportional to the speed of the truck because we are more uncertain about the accuracy of the dead reckoning position estimate at high speeds but very certain about the position estimate at low speeds. Next, in

11956-444: The position estimate back toward the real position but not disturb it to the point of becoming noisy and rapidly jumping. The Kalman filter is an efficient recursive filter estimating the internal state of a linear dynamic system from a series of noisy measurements. It is used in a wide range of engineering and econometric applications from radar and computer vision to estimation of structural macroeconomic models, and

12078-573: The predicted state from the previous estimate and similarly the function h can be used to compute the predicted measurement from the predicted state. However, f and h cannot be applied to the covariance directly. Instead a matrix of partial derivatives (the Jacobian ) is computed. At each time step, the Jacobian is evaluated with current predicted states. These matrices can be used in the Kalman filter equations. This process essentially linearizes

12200-423: The process is modeled incorrectly, the filter may quickly diverge, owing to its linearization. Another problem with the extended Kalman filter is that the estimated covariance matrix tends to underestimate the true covariance matrix and therefore risks becoming inconsistent in the statistical sense without the addition of "stabilising noise" . More generally one should consider the infinite dimensional nature of

12322-537: The projection filter in the context of continuous time circular filtering. For quantum systems applications, see for example van Handel and Mabuchi (2005), who applied the quantum projection filter to quantum optics , studying a quantum model of optical phase bistability of a strongly coupled two-level atom in an optical cavity. Further applications to quantum systems are considered in Gao, Zhang and Petersen (2019). Ma, Zhao, Chen and Chang (2015) refer to projection filters in

12444-462: The projection is the optimal approximation for F {\displaystyle F} and G {\displaystyle G} separately, although this does not imply it provides the best approximation for the filter SPDE solution as a whole. Indeed, the Stratonovich projection, acting on the two terms F {\displaystyle F} and G {\displaystyle G} separately, does not guarantee optimality of

12566-442: The resulting approximation achieves ( δ t ) 2 {\displaystyle (\delta t)^{2}} convergence. Indeed, the Ito jet projection attains the following optimality criterion. It zeroes the δ t {\displaystyle \delta t} order term and it minimizes the ( δ t ) 2 {\displaystyle (\delta t)^{2}} order term of

12688-400: The section above. More generally, if the model assumptions do not match the real system perfectly, then optimal state estimation is not necessarily obtained by setting Q k and R k to the covariances of the noise. Instead, in that case, the parameters Q k and R k may be set to explicitly optimize the state estimation, e.g., using standard supervised learning . After

12810-503: The solution p ( ⋅ , θ 0 + δ t ) {\displaystyle p(\cdot ,\theta _{0+\delta t})} as an approximation of the exact p 0 + δ t {\displaystyle p_{0+\delta t}} for say small δ t {\displaystyle \delta t} . One may look for a norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} to be applied to

12932-470: The solution, for which The Ito-vector projection is obtained as follows. Let us choose a norm for the space of densities, ‖ ⋅ ‖ {\displaystyle \|\cdot \|} , which might be associated with the direct metric or the Hellinger metric. One chooses the diffusion term in the approximating Ito equation for θ t {\displaystyle \theta _{t}} by minimizing (but not zeroing)

13054-639: The state dimension, allowing for applications in very high-dimensional systems, such as weather prediction, with state-space sizes of a billion or more. Fuzzy Kalman filter with a new method to represent possibility distributions was recently proposed to replace probability distributions by possibility distributions in order to obtain a genuine possibilistic filter, enabling the use of non-symmetric process and observation noises as well as higher inaccuracies in both process and observation models. Kalman filter In statistics and control theory , Kalman filtering (also known as linear quadratic estimation )

13176-412: The state estimate and covariances are coded into matrices because of the multiple dimensions involved in a single set of calculations. This allows for a representation of linear relationships between different state variables (such as position, velocity, and acceleration) in any of the transition models or covariances. As an example application, consider the problem of determining the precise location of

13298-401: The state of the system as an average of the system's predicted state and of the new measurement using a weighted average . The purpose of the weights is that values with better (i.e., smaller) estimated uncertainty are "trusted" more. The weights are calculated from the covariance , a measure of the estimated uncertainty of the prediction of the system's state. The result of the weighted average

13420-413: The state transition and observation matrices are defined to be the following Jacobians Unlike its linear counterpart, the extended Kalman filter in general is not an optimal estimator (it is optimal if the measurement and the state transition model are both linear, as in that case the extended Kalman filter is identical to the regular one). In addition, if the initial estimate of the state is wrong, or if

13542-561: The summer of 1961, when Kalman met with Stratonovich during a conference in Moscow. This Kalman filtering was first described and developed partially in technical papers by Swerling (1958), Kalman (1960) and Kalman and Bucy (1961). The Apollo computer used 2k of magnetic core RAM and 36k wire rope [...]. The CPU was built from ICs [...]. Clock speed was under 100 kHz [...]. The fact that the MIT engineers were able to pack such good software (one of

13664-497: The tangent space of the densities in S Θ {\displaystyle S_{\Theta }} (direct metric) or of their square roots (Hellinger metric). The direct metric case yields where Π p ( ⋅ , θ ) {\displaystyle \Pi _{p(\cdot ,\theta )}} is the tangent space projection at the point p ( ⋅ , θ ) {\displaystyle p(\cdot ,\theta )} for

13786-407: The truck are described by the linear state space Projection filters Projection filters are a set of algorithms based on stochastic analysis and information geometry , or the differential geometric approach to statistics, used to find approximate solutions for filtering problems for nonlinear state-space systems. The filtering problem consists of estimating the unobserved signal of

13908-439: The truck every Δ t seconds, but these measurements are imprecise; we want to maintain a model of the truck's position and velocity . We show here how we derive the model from which we create our Kalman filter. Since F , H , R , Q {\displaystyle \mathbf {F} ,\mathbf {H} ,\mathbf {R} ,\mathbf {Q} } are constant, their time indices are dropped. The position and velocity of

14030-535: The true state at time k {\displaystyle k} is evolved from the state at k − 1 {\displaystyle k-1} according to where If Q {\displaystyle \mathbf {Q} } is independent of time, one may, following Roweis and Ghahramani ( op. cit. ), write w ∙ {\displaystyle \mathbf {w} _{\bullet }} instead of w k {\displaystyle \mathbf {w} _{k}} to emphasize that

14152-508: The two phases alternate, with the prediction advancing the state until the next scheduled observation, and the update incorporating the observation. However, this is not necessary; if an observation is unavailable for some reason, the update may be skipped and multiple prediction procedures performed. Likewise, if multiple independent observations are available at the same time, multiple update procedures may be performed (typically with different observation matrices H k ). The formula for

14274-464: The update phase, a measurement of the truck's position is taken from the GPS unit. Along with this measurement comes some amount of uncertainty, and its covariance relative to that of the prediction from the previous phase determines how much the new measurement will affect the updated prediction. Ideally, as the dead reckoning estimates tend to drift away from the real position, the GPS measurement should pull

14396-536: The updated ( a posteriori ) estimate covariance above is valid for the optimal K k gain that minimizes the residual error, in which form it is most widely used in applications. Proof of the formulae is found in the derivations section, where the formula valid for any K k is also shown. A more intuitive way to express the updated state estimate ( x ^ k ∣ k {\displaystyle {\hat {\mathbf {x} }}_{k\mid k}} ) is: This expression reminds us of

14518-416: The very first applications of the Kalman filter) into such a tiny computer is truly remarkable. Kalman filters have been vital in the implementation of the navigation systems of U.S. Navy nuclear ballistic missile submarines , and in the guidance and navigation systems of cruise missiles such as the U.S. Navy's Tomahawk missile and the U.S. Air Force 's Air Launched Cruise Missile . They are also used in

14640-474: Was done at NASA Ames . The EKF adapted techniques from calculus , namely multivariate Taylor series expansions, to linearize a model about a working point. If the system model (as described below) is not well known or is inaccurate, then Monte Carlo methods , especially particle filters , are employed for estimation. Monte Carlo techniques predate the existence of the EKF but are more computationally expensive for any moderately dimensioned state-space . In

14762-526: Was first coined in 1987 by Bernard Hanzon, and the related theory and numerical examples were fully developed, expanded and made rigorous during the Ph.D. work of Damiano Brigo , in collaboration with Bernard Hanzon and Francois LeGland. These works dealt with the projection filters in Hellinger distance and Fisher information metric , that were used to project the optimal filter infinite-dimensional SPDE on

14884-413: Was tested against the optimal filter for the cubic sensor problem. The projection filter could track effectively bimodal densities of the optimal filter that would have been difficult to approximate with standard algorithms like the extended Kalman filter . Projection filters are ideal for in-line estimation, as they are quick to implement and run efficiently in time, providing a finite dimensional SDE for

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