2 Copyright (C) 2008-2015 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: UnscentedKalmanFilter
25 .. _section_ref_algorithm_UnscentedKalmanFilter:
27 Calculation algorithm "*UnscentedKalmanFilter*"
28 -----------------------------------------------
33 This algorithm realizes an estimation of the state of a dynamic system by a
34 "unscented" Kalman Filter, avoiding to have to perform the tangent and adjoint
35 operators for the observation and evolution operators, as in the simple or
36 extended Kalman filter.
38 Optional and required commands
39 ++++++++++++++++++++++++++++++
41 .. index:: single: Background
42 .. index:: single: BackgroundError
43 .. index:: single: Observation
44 .. index:: single: ObservationError
45 .. index:: single: ObservationOperator
46 .. index:: single: Bounds
47 .. index:: single: ConstrainedBy
48 .. index:: single: EstimationOf
49 .. index:: single: Alpha
50 .. index:: single: Beta
51 .. index:: single: Kappa
52 .. index:: single: Reconditioner
53 .. index:: single: StoreInternalVariables
54 .. index:: single: StoreSupplementaryCalculations
56 The general required commands, available in the editing user interface, are the
60 *Required command*. This indicates the background or initial vector used,
61 previously noted as :math:`\mathbf{x}^b`. Its value is defined as a
62 "*Vector*" or a *VectorSerie*" type object.
65 *Required command*. This indicates the background error covariance matrix,
66 previously noted as :math:`\mathbf{B}`. Its value is defined as a "*Matrix*"
67 type object, a "*ScalarSparseMatrix*" type object, or a
68 "*DiagonalSparseMatrix*" type object.
71 *Required command*. This indicates the observation vector used for data
72 assimilation or optimization, previously noted as :math:`\mathbf{y}^o`. It
73 is defined as a "*Vector*" or a *VectorSerie* type object.
76 *Required command*. This indicates the observation error covariance matrix,
77 previously noted as :math:`\mathbf{R}`. It is defined as a "*Matrix*" type
78 object, a "*ScalarSparseMatrix*" type object, or a "*DiagonalSparseMatrix*"
82 *Required command*. This indicates the observation operator, previously
83 noted :math:`H`, which transforms the input parameters :math:`\mathbf{x}` to
84 results :math:`\mathbf{y}` to be compared to observations
85 :math:`\mathbf{y}^o`. Its value is defined as a "*Function*" type object or
86 a "*Matrix*" type one. In the case of "*Function*" type, different
87 functional forms can be used, as described in the section
88 :ref:`section_ref_operator_requirements`. If there is some control :math:`U`
89 included in the observation, the operator has to be applied to a pair
92 The general optional commands, available in the editing user interface, are
93 indicated in :ref:`section_ref_assimilation_keywords`. In particular, the
94 optional command "*AlgorithmParameters*" allows to choose the specific options,
95 described hereafter, of the algorithm. See
96 :ref:`section_ref_options_AlgorithmParameters` for the good use of this command.
98 The options of the algorithm are the following:
101 This key allows to define upper and lower bounds for every state variable
102 being optimized. Bounds have to be given by a list of list of pairs of
103 lower/upper bounds for each variable, with extreme values every time there
104 is no bound (``None`` is not allowed when there is no bound).
106 Example : ``{"Bounds":[[2.,5.],[1.e-2,10.],[-30.,1.e99],[-1.e99,1.e99]]}``
109 This key allows to choose the type of estimation to be performed. It can be
110 either state-estimation, with a value of "State", or parameter-estimation,
111 with a value of "Parameters". The default choice is "State".
113 Example : ``{"EstimationOf":"Parameters"}``
115 Alpha, Beta, Kappa, Reconditioner
116 These keys are internal scaling parameters. "Alpha" requires a value between
117 1.e-4 and 1. "Beta" has an optimal value of 2 for Gaussian *a priori*
118 distribution. "Kappa" requires an integer value, and the right default is
119 obtained by setting it to 0. "Reconditioner" requires a value between 1.e-3
120 and 10, it defaults to 1.
122 Example : ``{"Alpha":1,"Beta":2,"Kappa":0,"Reconditioner":1}``
124 StoreInternalVariables
125 This Boolean key allows to store default internal variables, mainly the
126 current state during iterative optimization process. Be careful, this can be
127 a numerically costly choice in certain calculation cases. The default is
130 Example : ``{"StoreInternalVariables":True}``
132 StoreSupplementaryCalculations
133 This list indicates the names of the supplementary variables that can be
134 available at the end of the algorithm. It involves potentially costly
135 calculations or memory consumptions. The default is a void list, none of
136 these variables being calculated and stored by default. The possible names
137 are in the following list: ["APosterioriCovariance", "BMA", "Innovation"].
139 Example : ``{"StoreSupplementaryCalculations":["BMA","Innovation"]}``
141 Information and variables available at the end of the algorithm
142 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
144 At the output, after executing the algorithm, there are variables and
145 information originating from the calculation. The description of
146 :ref:`section_ref_output_variables` show the way to obtain them by the method
147 named ``get`` of the variable "*ADD*" of the post-processing. The input
148 variables, available to the user at the output in order to facilitate the
149 writing of post-processing procedures, are described in the
150 :ref:`subsection_r_o_v_Inventaire`.
152 The unconditional outputs of the algorithm are the following:
155 *List of vectors*. Each element is an optimal state :math:`\mathbf{x}*` in
156 optimization or an analysis :math:`\mathbf{x}^a` in data assimilation.
158 Example : ``Xa = ADD.get("Analysis")[-1]``
160 The conditional outputs of the algorithm are the following:
162 APosterioriCovariance
163 *List of matrices*. Each element is an *a posteriori* error covariance
164 matrix :math:`\mathbf{A}*` of the optimal state.
166 Example : ``A = ADD.get("APosterioriCovariance")[-1]``
169 *List of vectors*. Each element is a vector of difference between the
170 background and the optimal state.
172 Example : ``bma = ADD.get("BMA")[-1]``
175 *List of values*. Each element is a value of the error function :math:`J`.
177 Example : ``J = ADD.get("CostFunctionJ")[:]``
180 *List of values*. Each element is a value of the error function :math:`J^b`,
181 that is of the background difference part.
183 Example : ``Jb = ADD.get("CostFunctionJb")[:]``
186 *List of values*. Each element is a value of the error function :math:`J^o`,
187 that is of the observation difference part.
189 Example : ``Jo = ADD.get("CostFunctionJo")[:]``
192 *List of vectors*. Each element is a usual state vector used during the
193 optimization algorithm procedure.
195 Example : ``Xs = ADD.get("CurrentState")[:]``
198 *List of vectors*. Each element is an innovation vector, which is in static
199 the difference between the optimal and the background, and in dynamic the
202 Example : ``d = ADD.get("Innovation")[-1]``
207 References to other sections:
208 - :ref:`section_ref_algorithm_KalmanFilter`
209 - :ref:`section_ref_algorithm_ExtendedKalmanFilter`
211 Bibliographical references: