# 13.9. Calculation algorithm “*ExtendedKalmanFilter*”¶

## 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 :

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).

## 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 . 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 . 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 . 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 , 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 included in the evolution model, the operator has to be applied to a pair .

- Observation
*List of vectors*. The variable indicates the observation vector used for data assimilation or optimization, and usually noted . 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 . 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 , which transforms the input parameters to results to be compared to observations . 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 included in the observation, the operator has to be applied to a pair .

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 optimisation 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 possibly`None`

every time there is no bound. The bounds can always be specified, but they are taken into account only by the constrained optimizers.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 avalability 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 unconditionnal 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”, “CurrentIterationNumber”, “CurrentOptimum”, “CurrentState”, “ForecastCovariance”, “ForecastState”, “IndexOfOptimum”, “InnovationAtCurrentAnalysis”, “InnovationAtCurrentState”, “SimulatedObservationAtCurrentAnalysis”, “SimulatedObservationAtCurrentOptimum”, “SimulatedObservationAtCurrentState”, ].Example :

`{"StoreSupplementaryCalculations":["BMA", "CurrentState"]}`

- 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"}`

## 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 informations 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 the 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 in optimization or an analysis 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 in optimization or an analysis 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 covariance matrix. In order to get them, this*a posteriori*error covariances calculation has to be requested at the same time.Example:

`C = ADD.get("APosterioriCorrelations")[-1]`

- APosterioriCovariance
*List of matrices*. Each element is an*a posteriori*error covariance matrix of the optimal state.Example:

`A = 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 covariance matrix. In order to get them, this*a posteriori*error covariances calculation has to be requested at the same time.Example:

`S = 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 covariance matrix. In order to get them, this*a posteriori*error covariances calculation has to be requested at the same time.Example:

`V = 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 .Example:

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

- CostFunctionJAtCurrentOptimum
*List of values*. Each element is a value of the error function . 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 , 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 . 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 , 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 , 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")[:]`

- CurrentIterationNumber
*List of integers*. Each element is the iteration index at the current step during the iterative algorithm procedure. There is one iteration index value per assimilation step corresponding to an observed state.Example:

`i = ADD.get("CurrentIterationNumber")[-1]`

- 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")[:]`

- 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:

`Xp = 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:

`i = 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:

`ds = 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]`

## See also¶

References to other sections:

- Calculation algorithm “KalmanFilter”
- Calculation algorithm “EnsembleKalmanFilter”
- Calculation algorithm “UnscentedKalmanFilter”

Bibliographical references: