13.9. Calculation algorithm “ExtendedKalmanFilter”¶

13.9.1. Description¶

This algorithm realizes an estimation of the state of a dynamic system by a extended Kalman Filter, using a non-linear calculation of the state observation and incremental evolution (process). Technically, the estimation of the state is performed by the classical Kalman filter equations, using at each step the Jacobian obtained by linearization of the observation and the evolution to evaluate the state error covariance. This algorithm is therefore more expensive than the linear Kalman Filter, but it is by nature better adapted as soon as the operators are non-linear, being by principle universally recommended in this case.

Conceptually, we can represent the temporal pattern of action of the evolution and observation operators in this algorithm in the following way, with x the state, P the state error covariance, t the discrete iterative time :

Fig. 13.1 Timeline of steps in extended Kalman filter data assimilation¶

In this scheme, the analysis (x,P) is obtained by means of the “correction” by observing the “prediction” of the previous state. We notice that there is no analysis performed at the initial time step (numbered 0 in the time indexing) because there is no forecast at this time (the background is stored as a pseudo analysis at the initial time step). If the observations are provided in series by the user, the first one is therefore not used.

This filter can also be used to estimate (jointly or solely) parameters and not the state, in which case neither the time nor the evolution have any meaning. The iteration steps are then linked to the insertion of a new observation in the recursive estimation. One should consult the section Going further in data assimilation for dynamics for the implementation concepts.

In case of more pronounced non-linear operators, one can easily use a Calculation algorithm “EnsembleKalmanFilter” or a Calculation algorithm “UnscentedKalmanFilter”, which are often far more adapted to non-linear behavior but sometimes costly. One can verify the linearity of the operators with the help of a Checking algorithm “LinearityTest”.

The extended Kalman filter can take into account bounds on the states (the variant is named “CEKF”, it is recommended and is used by default), or conducted without any constraint (the variant is named “EKF”, and it is not recommended).

13.9.2. Some noteworthy properties of the implemented methods¶

To complete the description, we summarize here a few notable properties of the algorithm methods or of their implementations. These properties may have an influence on how it is used or on its computational performance. For further information, please refer to the more comprehensive references given at the end of this algorithm description.

The optimization methods proposed by this algorithm perform a local search for the minimum, theoretically enabling a locally optimal state (as opposed to a “globally optimal” state) to be reached.

The methods proposed by this algorithm require the derivation of the objective function or of one of the operators. It requires that at least one or both of the observation or evolution operators are differentiable, and this implies an additional calculation time in the case where the derivatives are calculated numerically by multiple evaluations.

The methods proposed by this algorithm have no internal parallelism, but use the numerical derivation of operator(s), which can be parallelized. The potential interaction, between the parallelism of the numerical derivation, and the parallelism that may be present in the observation or evolution operators embedding user codes, must therefore be carefully tuned.

The methods proposed by this algorithm achieve their convergence on one or more static criteria, fixed by some particular algorithmic properties. In practice, there may be several convergence criteria active simultaneously.

The more frequent algorithmic property is the one of direct calculations, which evaluate the converged solution without any controllable iteration. There is no convergence threshold to be adjusted in this case.

13.9.3. Optional and required commands¶

The general required commands, available in the editing user graphical or textual interface, are the following:

Background: Vector. The variable indicates the background or initial vector used, previously noted as $\mathbf{x}^b$ . Its value is defined as a “Vector” or “VectorSerie” type object. Its availability in output is conditioned by the boolean “Stored” associated with input.

BackgroundError: Matrix. This indicates the background error covariance matrix, previously noted as $\mathbf{B}$ . Its value is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

EvolutionError: Matrix. The variable indicates the evolution error covariance matrix, usually noted as $\mathbf{Q}$ . It is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

EvolutionModel: Operator. The variable indicates the evolution model operator, usually noted $M$ , which describes an elementary step of evolution. Its value is defined as a “Function” type object or a “Matrix” type one. In the case of “Function” type, different functional forms can be used, as described in the section Requirements for functions describing an operator. If there is some control $U$ included in the evolution model, the operator has to be applied to a pair $(X,U)$ .

Observation: List of vectors. The variable indicates the observation vector used for data assimilation or optimization, and usually noted $\mathbf{y}^o$ . Its value is defined as an object of type “Vector” if it is a single observation (temporal or not) or “VectorSeries” if it is a succession of observations. Its availability in output is conditioned by the boolean “Stored” associated in input.

ObservationError: Matrix. The variable indicates the observation error covariance matrix, usually noted as $\mathbf{R}$ . It is defined as a “Matrix” type object, a “ScalarSparseMatrix” type object, or a “DiagonalSparseMatrix” type object, as described in detail in the section Requirements to describe covariance matrices. Its availability in output is conditioned by the boolean “Stored” associated with input.

ObservationOperator: Operator. The variable indicates the observation operator, usually noted as $H$ , which transforms the input parameters $\mathbf{x}$ to results $\mathbf{y}$ to be compared to observations $\mathbf{y}^o$ . Its value is defined as a “Function” type object or a “Matrix” type one. In the case of “Function” type, different functional forms can be used, as described in the section Requirements for functions describing an operator. If there is some control $U$ included in the observation, the operator has to be applied to a pair $(X,U)$ .

The general optional commands, available in the editing user graphical or textual interface, are indicated in List of commands and keywords for data assimilation or optimization case. Moreover, the parameters of the command “AlgorithmParameters” allows to choose the specific options, described hereafter, of the algorithm. See Description of options of an algorithm by “AlgorithmParameters” for the good use of this command.

The options are the following:

Bounds

List of pairs of real values. This key allows to define pairs of upper and lower bounds for every state variable being optimized. Bounds have to be given by a list of list of pairs of lower/upper bounds for each variable, with a value of None each time there is no bound. The bounds can always be specified, but they are taken into account only by the constrained optimizers. If the list is empty, there are no bounds.

Example: {"Bounds":[[2.,5.],[1.e-2,10.],[-30.,None],[None,None]]}

ConstrainedBy

Predefined name. This key allows to choose the method to take into account the bounds constraints. The only one available is the “EstimateProjection”, which projects the current state estimate on the bounds constraints.

Example: {"ConstrainedBy":"EstimateProjection"}

EstimationOf

Predefined name. This key allows to choose the type of estimation to be performed. It can be either state-estimation, with a value of “State”, or parameter-estimation, with a value of “Parameters”. The default choice is “State”.

Example: {"EstimationOf":"Parameters"}

StoreSupplementaryCalculations

List of names. This list indicates the names of the supplementary variables, that can be available during or at the end of the algorithm, if they are initially required by the user. Their availability involves, potentially, costly calculations or memory consumptions. The default is then a void list, none of these variables being calculated and stored by default (excepted the unconditional variables). The possible names are in the following list (the detailed description of each named variable is given in the following part of this specific algorithmic documentation, in the sub-section “Information and variables available at the end of the algorithm”): [ “Analysis”, “APosterioriCorrelations”, “APosterioriCovariance”, “APosterioriStandardDeviations”, “APosterioriVariances”, “BMA”, “CostFunctionJ”, “CostFunctionJAtCurrentOptimum”, “CostFunctionJb”, “CostFunctionJbAtCurrentOptimum”, “CostFunctionJo”, “CostFunctionJoAtCurrentOptimum”, “CurrentOptimum”, “CurrentState”, “CurrentStepNumber”, “EnsembleOfSimulations”, “EnsembleOfStates”, “ForecastCovariance”, “ForecastState”, “IndexOfOptimum”, “InnovationAtCurrentAnalysis”, “InnovationAtCurrentState”, “SimulatedObservationAtCurrentAnalysis”, “SimulatedObservationAtCurrentOptimum”, “SimulatedObservationAtCurrentState”, ].

Example : {"StoreSupplementaryCalculations":["CurrentState", "Residu"]}

Variant

Predefined name. This key allows to choose one of the possible variants for the main algorithm. The default variant is the constrained version “CEKF” of the original algorithm “EKF”, and the possible choices are “EKF” (Extended Kalman Filter), “CEKF” (Constrained Extended Kalman Filter). It is highly recommended to keep the default value.

Example : {"Variant":"CEKF"}

13.9.4. Information and variables available at the end of the algorithm¶

At the output, after executing the algorithm, there are information and variables originating from the calculation. The description of Variables and information available at the output show the way to obtain them by the method named get, of the variable “ADD” of the post-processing in graphical interface, or of the case in textual interface. The input variables, available to the user at the output in order to facilitate the writing of post-processing procedures, are described in an Inventory of potentially available information at the output.

Permanent outputs (non conditional)

The unconditional outputs of the algorithm are the following:

Analysis

List of vectors. Each element of this variable is an optimal state $\mathbf{x}^*$ in optimization, an interpolate or an analysis $\mathbf{x}^a$ in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

Set of on-demand outputs (conditional or not)

The whole set of algorithm outputs (conditional or not), sorted by alphabetical order, is the following:

Analysis

List of vectors. Each element of this variable is an optimal state $\mathbf{x}^*$ in optimization, an interpolate or an analysis $\mathbf{x}^a$ in data assimilation.

Example: xa = ADD.get("Analysis")[-1]

APosterioriCorrelations

List of matrices. Each element is an a posteriori error correlations matrix of the optimal state, coming from the $\mathbf{A}$ covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: apc = ADD.get("APosterioriCorrelations")[-1]

APosterioriCovariance

List of matrices. Each element is an a posteriori error covariance matrix $\mathbf{A}$ of the optimal state.

Example: apc = ADD.get("APosterioriCovariance")[-1]

APosterioriStandardDeviations

List of matrices. Each element is an a posteriori error standard errors diagonal matrix of the optimal state, coming from the $\mathbf{A}$ covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: aps = ADD.get("APosterioriStandardDeviations")[-1]

APosterioriVariances

List of matrices. Each element is an a posteriori error variance errors diagonal matrix of the optimal state, coming from the $\mathbf{A}$ covariance matrix. In order to get them, this a posteriori error covariances calculation has to be requested at the same time.

Example: apv = ADD.get("APosterioriVariances")[-1]

BMA

List of vectors. Each element is a vector of difference between the background and the optimal state.

Example: bma = ADD.get("BMA")[-1]

CostFunctionJ

List of values. Each element is a value of the chosen error function $J$ .

Example: J = ADD.get("CostFunctionJ")[:]

CostFunctionJAtCurrentOptimum

List of values. Each element is a value of the error function $J$ . At each step, the value corresponds to the optimal state found from the beginning.

Example: JACO = ADD.get("CostFunctionJAtCurrentOptimum")[:]

CostFunctionJb

List of values. Each element is a value of the error function $J^b$ , that is of the background difference part. If this part does not exist in the error function, its value is zero.

Example: Jb = ADD.get("CostFunctionJb")[:]

CostFunctionJbAtCurrentOptimum

List of values. Each element is a value of the error function $J^b$ . At each step, the value corresponds to the optimal state found from the beginning. If this part does not exist in the error function, its value is zero.

Example: JbACO = ADD.get("CostFunctionJbAtCurrentOptimum")[:]

CostFunctionJo

List of values. Each element is a value of the error function $J^o$ , that is of the observation difference part.

Example: Jo = ADD.get("CostFunctionJo")[:]

CostFunctionJoAtCurrentOptimum

List of values. Each element is a value of the error function $J^o$ , that is of the observation difference part. At each step, the value corresponds to the optimal state found from the beginning.

Example: JoACO = ADD.get("CostFunctionJoAtCurrentOptimum")[:]

CurrentOptimum

List of vectors. Each element is the optimal state obtained at the usual step of the iterative algorithm procedure of the optimization algorithm. It is not necessarily the last state.

Example: xo = ADD.get("CurrentOptimum")[:]

CurrentState

List of vectors. Each element is a usual state vector used during the iterative algorithm procedure.

Example: xs = ADD.get("CurrentState")[:]

CurrentStepNumber

List of integers. Each element is the index of the current step in the iterative process, driven by the series of observations, of the algorithm used. This corresponds to the observation step used. Note: it is not the index of the current iteration of the algorithm even if it coincides for non-iterative algorithms.

Example: csn = ADD.get("CurrentStepNumber")[-1]

EnsembleOfSimulations

List of vectors or matrix. This key contains an ordered collection of physical state vectors or simulated state vectors $\mathbf{y}$ that may be observed. These are $H$ operator outputs, i.e. simulated observation states (called “snapshots” in reduced-base terminology). At each step index, there is 1 state per column if this list is in matrix form, or 1 state per element if it’s actually a list. Caution: the numbering of the support or points, on which or to which a state value is given in each vector, is implicitly that of the natural order of numbering of the state vector, from 0 to the “size minus 1” of this vector.

Example : {"EnsembleOfSimulations":[y1, y2, y3...]}

EnsembleOfStates

List of vectors or matrix. Each element is an ordered collection of physical or parameter state vectors $\mathbf{x}$ . These are $H$ operator entries, i.e. current states before observation. At each step index, there is 1 state per column if this list is in matrix form, or 1 state per element if it’s actually a list. Caution: the numbering of the support or points, on which or to which a state value is given in each vector, is implicitly that of the natural order of numbering of the state vector, from 0 to the “size minus 1” of this vector.

Example : {"EnsembleOfStates":[x1, x2, x3...]}

ForecastCovariance

Liste of matrices. Each element is a forecast state error covariance matrix predicted by the model during the time iteration of the algorithm used.

Example : pf = ADD.get("ForecastCovariance")[-1]

ForecastState

List of vectors. Each element is a state vector forecasted by the model during the iterative algorithm procedure.

Example: xf = ADD.get("ForecastState")[:]

IndexOfOptimum

List of integers. Each element is the iteration index of the optimum obtained at the current step of the iterative algorithm procedure of the optimization algorithm. It is not necessarily the number of the last iteration.

Example: ioo = ADD.get("IndexOfOptimum")[-1]

InnovationAtCurrentAnalysis

List of vectors. Each element is an innovation vector at current analysis. This quantity is identical to the innovation vector at analysed state in the case of a single-state assimilation.

Example: da = ADD.get("InnovationAtCurrentAnalysis")[-1]

InnovationAtCurrentState

List of vectors. Each element is an innovation vector at current state before analysis.

Example: ds = ADD.get("InnovationAtCurrentState")[-1]

SimulatedObservationAtCurrentAnalysis

List of vectors. Each element is an observed vector simulated by the observation operator from the current analysis, that is, in the observation space. This quantity is identical to the observed vector simulated at current state in the case of a single-state assimilation.

Example: hxs = ADD.get("SimulatedObservationAtCurrentAnalysis")[-1]

SimulatedObservationAtCurrentOptimum

List of vectors. Each element is a vector of observation simulated from the optimal state obtained at the current step the optimization algorithm, that is, in the observation space.

Example: hxo = ADD.get("SimulatedObservationAtCurrentOptimum")[-1]

SimulatedObservationAtCurrentState

List of vectors. Each element is an observed vector simulated by the observation operator from the current state, that is, in the observation space.

Example: hxs = ADD.get("SimulatedObservationAtCurrentState")[-1]

13.9.5. See also¶

References to other sections:

Bibliographical references:

[WikipediaEKF]