2 Copyright (C) 2008-2017 EDF R&D
4 This file is part of SALOME ADAO module.
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20 See http://www.salome-platform.org/ or email : webmaster.salome@opencascade.com
22 Author: Jean-Philippe Argaud, jean-philippe.argaud@edf.fr, EDF R&D
24 .. index:: single: ExtendedKalmanFilter
25 .. _section_ref_algorithm_ExtendedKalmanFilter:
27 Calculation algorithm "*ExtendedKalmanFilter*"
28 ----------------------------------------------
33 This algorithm realizes an estimation of the state of a dynamic system by a
34 extended Kalman Filter, using a non-linear calculation of the state.
36 Optional and required commands
37 ++++++++++++++++++++++++++++++
39 .. index:: single: AlgorithmParameters
40 .. index:: single: Background
41 .. index:: single: BackgroundError
42 .. index:: single: Observation
43 .. index:: single: ObservationError
44 .. index:: single: ObservationOperator
45 .. index:: single: Bounds
46 .. index:: single: ConstrainedBy
47 .. index:: single: EstimationOf
48 .. index:: single: StoreSupplementaryCalculations
50 The general required commands, available in the editing user interface, are the
54 *Required command*. This indicates the background or initial vector used,
55 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
56 "*Vector*" or a *VectorSerie*" type object.
59 *Required command*. This indicates the background error covariance matrix,
60 previously noted as :math:`\mathbf{B}`. Its value is defined as a "*Matrix*"
61 type object, a "*ScalarSparseMatrix*" type object, or a
62 "*DiagonalSparseMatrix*" type object.
65 *Required command*. This indicates the observation vector used for data
66 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
67 is defined as a "*Vector*" or a *VectorSerie* type object.
70 *Required command*. This indicates the observation error covariance matrix,
71 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
72 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
76 *Required command*. This indicates the observation operator, previously
77 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
78 results :math:`\mathbf{y}` to be compared to observations
79 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
80 a "*Matrix*" type one. In the case of "*Function*" type, different
81 functional forms can be used, as described in the section
82 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
83 included in the observation, the operator has to be applied to a pair
86 The general optional commands, available in the editing user interface, are
87 indicated in :ref:`section_ref_assimilation_keywords`. Moreover, the parameters
88 of the command "*AlgorithmParameters*" allows to choose the specific options,
89 described hereafter, of the algorithm. See
90 :ref:`section_ref_options_Algorithm_Parameters` for the good use of this
93 The options of the algorithm are the following:
96 This key allows to define upper and lower bounds for every state variable
97 being optimized. Bounds have to be given by a list of list of pairs of
98 lower/upper bounds for each variable, with extreme values every time there
99 is no bound (``None`` is not allowed when there is no bound).
101 Example : ``{"Bounds":[[2.,5.],[1.e-2,10.],[-30.,1.e99],[-1.e99,1.e99]]}``
104 This key allows to choose the method to take into account the bounds
105 constraints. The only one available is the "EstimateProjection", which
106 projects the current state estimate on the bounds constraints.
108 Example : ``{"ConstrainedBy":"EstimateProjection"}``
111 This key allows to choose the type of estimation to be performed. It can be
112 either state-estimation, with a value of "State", or parameter-estimation,
113 with a value of "Parameters". The default choice is "State".
115 Example : ``{"EstimationOf":"Parameters"}``
117 StoreSupplementaryCalculations
118 This list indicates the names of the supplementary variables that can be
119 available at the end of the algorithm. It involves potentially costly
120 calculations or memory consumptions. The default is a void list, none of
121 these variables being calculated and stored by default. The possible names
122 are in the following list: ["APosterioriCorrelations",
123 "APosterioriCovariance", "APosterioriStandardDeviations",
124 "APosterioriVariances", "BMA", "CostFunctionJ", "CostFunctionJb",
125 "CostFunctionJo", "CurrentState", "Innovation"].
127 Example : ``{"StoreSupplementaryCalculations":["BMA", "Innovation"]}``
129 Information and variables available at the end of the algorithm
130 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
132 At the output, after executing the algorithm, there are variables and
133 information originating from the calculation. The description of
134 :ref:`section_ref_output_variables` show the way to obtain them by the method
135 named ``get`` of the variable "*ADD*" of the post-processing. The input
136 variables, available to the user at the output in order to facilitate the
137 writing of post-processing procedures, are described in the
138 :ref:`subsection_r_o_v_Inventaire`.
140 The unconditional outputs of the algorithm are the following:
143 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
144 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
146 Example : ``Xa = ADD.get("Analysis")[-1]``
148 The conditional outputs of the algorithm are the following:
150 APosterioriCorrelations
151 *List of matrices*. Each element is an *a posteriori* error correlation
152 matrix of the optimal state.
154 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
156 APosterioriCovariance
157 *List of matrices*. Each element is an *a posteriori* error covariance
158 matrix :math:`\mathbf{A}*` of the optimal state.
160 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
162 APosterioriStandardDeviations
163 *List of matrices*. Each element is an *a posteriori* error standard
164 deviation matrix of the optimal state.
166 Example : ``E = ADD.get("APosterioriStandardDeviations")[-1]``
169 *List of matrices*. Each element is an *a posteriori* error variance matrix
170 of the optimal state.
172 Example : ``V = ADD.get("APosterioriVariances")[-1]``
175 *List of vectors*. Each element is a vector of difference between the
176 background and the optimal state.
178 Example : ``bma = ADD.get("BMA")[-1]``
181 *List of values*. Each element is a value of the error function :math:`J`.
183 Example : ``J = ADD.get("CostFunctionJ")[:]``
186 *List of values*. Each element is a value of the error function :math:`J^b`,
187 that is of the background difference part.
189 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
192 *List of values*. Each element is a value of the error function :math:`J^o`,
193 that is of the observation difference part.
195 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
198 *List of vectors*. Each element is a usual state vector used during the
199 optimization algorithm procedure.
201 Example : ``Xs = ADD.get("CurrentState")[:]``
204 *List of vectors*. Each element is an innovation vector, which is in static
205 the difference between the optimal and the background, and in dynamic the
208 Exemple : ``d = ADD.get("Innovation")[-1]``
213 References to other sections:
214 - :ref:`section_ref_algorithm_KalmanFilter`
215 - :ref:`section_ref_algorithm_UnscentedKalmanFilter`