2 Copyright (C) 2008-2015 EDF R&D
4 This file is part of SALOME ADAO module.
6 This library is free software; you can redistribute it and/or
7 modify it under the terms of the GNU Lesser General Public
8 License as published by the Free Software Foundation; either
9 version 2.1 of the License, or (at your option) any later version.
11 This library is distributed in the hope that it will be useful,
12 but WITHOUT ANY WARRANTY; without even the implied warranty of
13 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
14 Lesser General Public License for more details.
16 You should have received a copy of the GNU Lesser General Public
17 License along with this library; if not, write to the Free Software
18 Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
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 type of estimation to be performed. It can be
105 either state-estimation, with a value of "State", or parameter-estimation,
106 with a value of "Parameters". The default choice is "State".
108 Example : ``{"EstimationOf":"Parameters"}``
110 StoreSupplementaryCalculations
111 This list indicates the names of the supplementary variables that can be
112 available at the end of the algorithm. It involves potentially costly
113 calculations or memory consumptions. The default is a void list, none of
114 these variables being calculated and stored by default. The possible names
115 are in the following list: ["APosterioriCorrelations",
116 "APosterioriCovariance", "APosterioriStandardDeviations",
117 "APosterioriVariances", "BMA", "CostFunctionJ", "CurrentState",
120 Example : ``{"StoreSupplementaryCalculations":["BMA","Innovation"]}``
122 Information and variables available at the end of the algorithm
123 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
125 At the output, after executing the algorithm, there are variables and
126 information originating from the calculation. The description of
127 :ref:`section_ref_output_variables` show the way to obtain them by the method
128 named ``get`` of the variable "*ADD*" of the post-processing. The input
129 variables, available to the user at the output in order to facilitate the
130 writing of post-processing procedures, are described in the
131 :ref:`subsection_r_o_v_Inventaire`.
133 The unconditional outputs of the algorithm are the following:
136 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
137 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
139 Example : ``Xa = ADD.get("Analysis")[-1]``
141 The conditional outputs of the algorithm are the following:
143 APosterioriCorrelations
144 *List of matrices*. Each element is an *a posteriori* error correlation
145 matrix of the optimal state.
147 Example : ``C = ADD.get("APosterioriCorrelations")[-1]``
149 APosterioriCovariance
150 *List of matrices*. Each element is an *a posteriori* error covariance
151 matrix :math:`\mathbf{A}*` of the optimal state.
153 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
155 APosterioriStandardDeviations
156 *List of matrices*. Each element is an *a posteriori* error standard
157 deviation matrix of the optimal state.
159 Example : ``E = ADD.get("APosterioriStandardDeviations")[-1]``
162 *List of matrices*. Each element is an *a posteriori* error variance matrix
163 of the optimal state.
165 Example : ``V = ADD.get("APosterioriVariances")[-1]``
168 *List of vectors*. Each element is a vector of difference between the
169 background and the optimal state.
171 Example : ``bma = ADD.get("BMA")[-1]``
174 *List of values*. Each element is a value of the error function :math:`J`.
176 Example : ``J = ADD.get("CostFunctionJ")[:]``
179 *List of values*. Each element is a value of the error function :math:`J^b`,
180 that is of the background difference part.
182 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
185 *List of values*. Each element is a value of the error function :math:`J^o`,
186 that is of the observation difference part.
188 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
191 *List of vectors*. Each element is a usual state vector used during the
192 optimization algorithm procedure.
194 Example : ``Xs = ADD.get("CurrentState")[:]``
197 *List of vectors*. Each element is an innovation vector, which is in static
198 the difference between the optimal and the background, and in dynamic the
201 Exemple : ``d = ADD.get("Innovation")[-1]``
206 References to other sections:
207 - :ref:`section_ref_algorithm_KalmanFilter`
208 - :ref:`section_ref_algorithm_UnscentedKalmanFilter`