src/daComposant/daCore/NumericObjects.py

   1 # -*- coding: utf-8 -*-
   2 #
   3 # Copyright (C) 2008-2021 EDF R&D
   4 #
   5 # This library is free software; you can redistribute it and/or
   6 # modify it under the terms of the GNU Lesser General Public
   7 # License as published by the Free Software Foundation; either
   8 # version 2.1 of the License.
   9 #
  10 # This library is distributed in the hope that it will be useful,
  11 # but WITHOUT ANY WARRANTY; without even the implied warranty of
  12 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  13 # Lesser General Public License for more details.
  14 #
  15 # You should have received a copy of the GNU Lesser General Public
  16 # License along with this library; if not, write to the Free Software
  17 # Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307 USA
  18 #
  19 # See http://www.salome-platform.org/ or email : webmaster.salome@opencascade.com
  20 #
  21 # Author: Jean-Philippe Argaud, jean-philippe.argaud@edf.fr, EDF R&D
  22
  23 __doc__ = """
  24     Définit les objets numériques génériques.
  25 """
  26 __author__ = "Jean-Philippe ARGAUD"
  27
  28 import os, time, copy, types, sys, logging
  29 import math, numpy, scipy, scipy.optimize
  30 from daCore.BasicObjects import Operator
  31 from daCore.PlatformInfo import PlatformInfo
  32 mpr = PlatformInfo().MachinePrecision()
  33 mfp = PlatformInfo().MaximumPrecision()
  34 # logging.getLogger().setLevel(logging.DEBUG)
  35
  36 # ==============================================================================
  37 def ExecuteFunction( paire ):
  38     assert len(paire) == 2, "Incorrect number of arguments"
  39     X, funcrepr = paire
  40     __X = numpy.asmatrix(numpy.ravel( X )).T
  41     __sys_path_tmp = sys.path ; sys.path.insert(0,funcrepr["__userFunction__path"])
  42     __module = __import__(funcrepr["__userFunction__modl"], globals(), locals(), [])
  43     __fonction = getattr(__module,funcrepr["__userFunction__name"])
  44     sys.path = __sys_path_tmp ; del __sys_path_tmp
  45     __HX  = __fonction( __X )
  46     return numpy.ravel( __HX )
  47
  48 # ==============================================================================
  49 class FDApproximation(object):
  50     """
  51     Cette classe sert d'interface pour définir les opérateurs approximés. A la
  52     création d'un objet, en fournissant une fonction "Function", on obtient un
  53     objet qui dispose de 3 méthodes "DirectOperator", "TangentOperator" et
  54     "AdjointOperator". On contrôle l'approximation DF avec l'incrément
  55     multiplicatif "increment" valant par défaut 1%, ou avec l'incrément fixe
  56     "dX" qui sera multiplié par "increment" (donc en %), et on effectue de DF
  57     centrées si le booléen "centeredDF" est vrai.
  58     """
  59     def __init__(self,
  60             name                  = "FDApproximation",
  61             Function              = None,
  62             centeredDF            = False,
  63             increment             = 0.01,
  64             dX                    = None,
  65             avoidingRedundancy    = True,
  66             toleranceInRedundancy = 1.e-18,
  67             lenghtOfRedundancy    = -1,
  68             mpEnabled             = False,
  69             mpWorkers             = None,
  70             mfEnabled             = False,
  71             ):
  72         self.__name = str(name)
  73         if mpEnabled:
  74             try:
  75                 import multiprocessing
  76                 self.__mpEnabled = True
  77             except ImportError:
  78                 self.__mpEnabled = False
  79         else:
  80             self.__mpEnabled = False
  81         self.__mpWorkers = mpWorkers
  82         if self.__mpWorkers is not None and self.__mpWorkers < 1:
  83             self.__mpWorkers = None
  84         logging.debug("FDA Calculs en multiprocessing : %s (nombre de processus : %s)"%(self.__mpEnabled,self.__mpWorkers))
  85         #
  86         if mfEnabled:
  87             self.__mfEnabled = True
  88         else:
  89             self.__mfEnabled = False
  90         logging.debug("FDA Calculs en multifonctions : %s"%(self.__mfEnabled,))
  91         #
  92         if avoidingRedundancy:
  93             self.__avoidRC = True
  94             self.__tolerBP = float(toleranceInRedundancy)
  95             self.__lenghtRJ = int(lenghtOfRedundancy)
  96             self.__listJPCP = [] # Jacobian Previous Calculated Points
  97             self.__listJPCI = [] # Jacobian Previous Calculated Increment
  98             self.__listJPCR = [] # Jacobian Previous Calculated Results
  99             self.__listJPPN = [] # Jacobian Previous Calculated Point Norms
 100             self.__listJPIN = [] # Jacobian Previous Calculated Increment Norms
 101         else:
 102             self.__avoidRC = False
 103         #
 104         if self.__mpEnabled:
 105             if isinstance(Function,types.FunctionType):
 106                 logging.debug("FDA Calculs en multiprocessing : FunctionType")
 107                 self.__userFunction__name = Function.__name__
 108                 try:
 109                     mod = os.path.join(Function.__globals__['filepath'],Function.__globals__['filename'])
 110                 except:
 111                     mod = os.path.abspath(Function.__globals__['__file__'])
 112                 if not os.path.isfile(mod):
 113                     raise ImportError("No user defined function or method found with the name %s"%(mod,))
 114                 self.__userFunction__modl = os.path.basename(mod).replace('.pyc','').replace('.pyo','').replace('.py','')
 115                 self.__userFunction__path = os.path.dirname(mod)
 116                 del mod
 117                 self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled )
 118                 self.__userFunction = self.__userOperator.appliedTo # Pour le calcul Direct
 119             elif isinstance(Function,types.MethodType):
 120                 logging.debug("FDA Calculs en multiprocessing : MethodType")
 121                 self.__userFunction__name = Function.__name__
 122                 try:
 123                     mod = os.path.join(Function.__globals__['filepath'],Function.__globals__['filename'])
 124                 except:
 125                     mod = os.path.abspath(Function.__func__.__globals__['__file__'])
 126                 if not os.path.isfile(mod):
 127                     raise ImportError("No user defined function or method found with the name %s"%(mod,))
 128                 self.__userFunction__modl = os.path.basename(mod).replace('.pyc','').replace('.pyo','').replace('.py','')
 129                 self.__userFunction__path = os.path.dirname(mod)
 130                 del mod
 131                 self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled )
 132                 self.__userFunction = self.__userOperator.appliedTo # Pour le calcul Direct
 133             else:
 134                 raise TypeError("User defined function or method has to be provided for finite differences approximation.")
 135         else:
 136             self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled )
 137             self.__userFunction = self.__userOperator.appliedTo
 138         #
 139         self.__centeredDF = bool(centeredDF)
 140         if abs(float(increment)) > 1.e-15:
 141             self.__increment  = float(increment)
 142         else:
 143             self.__increment  = 0.01
 144         if dX is None:
 145             self.__dX     = None
 146         else:
 147             self.__dX     = numpy.asmatrix(numpy.ravel( dX )).T
 148         logging.debug("FDA Reduction des doublons de calcul : %s"%self.__avoidRC)
 149         if self.__avoidRC:
 150             logging.debug("FDA Tolerance de determination des doublons : %.2e"%self.__tolerBP)
 151
 152     # ---------------------------------------------------------
 153     def __doublon__(self, e, l, n, v=None):
 154         __ac, __iac = False, -1
 155         for i in range(len(l)-1,-1,-1):
 156             if numpy.linalg.norm(e - l[i]) < self.__tolerBP * n[i]:
 157                 __ac, __iac = True, i
 158                 if v is not None: logging.debug("FDA Cas%s déja calculé, récupération du doublon %i"%(v,__iac))
 159                 break
 160         return __ac, __iac
 161
 162     # ---------------------------------------------------------
 163     def DirectOperator(self, X ):
 164         """
 165         Calcul du direct à l'aide de la fonction fournie.
 166         """
 167         logging.debug("FDA Calcul DirectOperator (explicite)")
 168         if self.__mfEnabled:
 169             _HX = self.__userFunction( X, argsAsSerie = True )
 170         else:
 171             _X = numpy.asmatrix(numpy.ravel( X )).T
 172             _HX = numpy.ravel(self.__userFunction( _X ))
 173         #
 174         return _HX
 175
 176     # ---------------------------------------------------------
 177     def TangentMatrix(self, X ):
 178         """
 179         Calcul de l'opérateur tangent comme la Jacobienne par différences finies,
 180         c'est-à-dire le gradient de H en X. On utilise des différences finies
 181         directionnelles autour du point X. X est un numpy.matrix.
 182
 183         Différences finies centrées (approximation d'ordre 2):
 184         1/ Pour chaque composante i de X, on ajoute et on enlève la perturbation
 185            dX[i] à la  composante X[i], pour composer X_plus_dXi et X_moins_dXi, et
 186            on calcule les réponses HX_plus_dXi = H( X_plus_dXi ) et HX_moins_dXi =
 187            H( X_moins_dXi )
 188         2/ On effectue les différences (HX_plus_dXi-HX_moins_dXi) et on divise par
 189            le pas 2*dXi
 190         3/ Chaque résultat, par composante, devient une colonne de la Jacobienne
 191
 192         Différences finies non centrées (approximation d'ordre 1):
 193         1/ Pour chaque composante i de X, on ajoute la perturbation dX[i] à la
 194            composante X[i] pour composer X_plus_dXi, et on calcule la réponse
 195            HX_plus_dXi = H( X_plus_dXi )
 196         2/ On calcule la valeur centrale HX = H(X)
 197         3/ On effectue les différences (HX_plus_dXi-HX) et on divise par
 198            le pas dXi
 199         4/ Chaque résultat, par composante, devient une colonne de la Jacobienne
 200
 201         """
 202         logging.debug("FDA Début du calcul de la Jacobienne")
 203         logging.debug("FDA   Incrément de............: %s*X"%float(self.__increment))
 204         logging.debug("FDA   Approximation centrée...: %s"%(self.__centeredDF))
 205         #
 206         if X is None or len(X)==0:
 207             raise ValueError("Nominal point X for approximate derivatives can not be None or void (given X: %s)."%(str(X),))
 208         #
 209         _X = numpy.asmatrix(numpy.ravel( X )).T
 210         #
 211         if self.__dX is None:
 212             _dX  = self.__increment * _X
 213         else:
 214             _dX = numpy.asmatrix(numpy.ravel( self.__dX )).T
 215         #
 216         if (_dX == 0.).any():
 217             moyenne = _dX.mean()
 218             if moyenne == 0.:
 219                 _dX = numpy.where( _dX == 0., float(self.__increment), _dX )
 220             else:
 221                 _dX = numpy.where( _dX == 0., moyenne, _dX )
 222         #
 223         __alreadyCalculated  = False
 224         if self.__avoidRC:
 225             __bidon, __alreadyCalculatedP = self.__doublon__(_X,  self.__listJPCP, self.__listJPPN, None)
 226             __bidon, __alreadyCalculatedI = self.__doublon__(_dX, self.__listJPCI, self.__listJPIN, None)
 227             if __alreadyCalculatedP == __alreadyCalculatedI > -1:
 228                 __alreadyCalculated, __i = True, __alreadyCalculatedP
 229                 logging.debug("FDA Cas J déja calculé, récupération du doublon %i"%__i)
 230         #
 231         if __alreadyCalculated:
 232             logging.debug("FDA   Calcul Jacobienne (par récupération du doublon %i)"%__i)
 233             _Jacobienne = self.__listJPCR[__i]
 234         else:
 235             logging.debug("FDA   Calcul Jacobienne (explicite)")
 236             if self.__centeredDF:
 237                 #
 238                 if self.__mpEnabled and not self.__mfEnabled:
 239                     funcrepr = {
 240                         "__userFunction__path" : self.__userFunction__path,
 241                         "__userFunction__modl" : self.__userFunction__modl,
 242                         "__userFunction__name" : self.__userFunction__name,
 243                     }
 244                     _jobs = []
 245                     for i in range( len(_dX) ):
 246                         _dXi            = _dX[i]
 247                         _X_plus_dXi     = numpy.array( _X.A1, dtype=float )
 248                         _X_plus_dXi[i]  = _X[i] + _dXi
 249                         _X_moins_dXi    = numpy.array( _X.A1, dtype=float )
 250                         _X_moins_dXi[i] = _X[i] - _dXi
 251                         #
 252                         _jobs.append( (_X_plus_dXi,  funcrepr) )
 253                         _jobs.append( (_X_moins_dXi, funcrepr) )
 254                     #
 255                     import multiprocessing
 256                     self.__pool = multiprocessing.Pool(self.__mpWorkers)
 257                     _HX_plusmoins_dX = self.__pool.map( ExecuteFunction, _jobs )
 258                     self.__pool.close()
 259                     self.__pool.join()
 260                     #
 261                     _Jacobienne  = []
 262                     for i in range( len(_dX) ):
 263                         _Jacobienne.append( numpy.ravel( _HX_plusmoins_dX[2*i] - _HX_plusmoins_dX[2*i+1] ) / (2.*_dX[i]) )
 264                     #
 265                 elif self.__mfEnabled:
 266                     _xserie = []
 267                     for i in range( len(_dX) ):
 268                         _dXi            = _dX[i]
 269                         _X_plus_dXi     = numpy.array( _X.A1, dtype=float )
 270                         _X_plus_dXi[i]  = _X[i] + _dXi
 271                         _X_moins_dXi    = numpy.array( _X.A1, dtype=float )
 272                         _X_moins_dXi[i] = _X[i] - _dXi
 273                         #
 274                         _xserie.append( _X_plus_dXi )
 275                         _xserie.append( _X_moins_dXi )
 276                     #
 277                     _HX_plusmoins_dX = self.DirectOperator( _xserie )
 278                      #
 279                     _Jacobienne  = []
 280                     for i in range( len(_dX) ):
 281                         _Jacobienne.append( numpy.ravel( _HX_plusmoins_dX[2*i] - _HX_plusmoins_dX[2*i+1] ) / (2.*_dX[i]) )
 282                     #
 283                 else:
 284                     _Jacobienne  = []
 285                     for i in range( _dX.size ):
 286                         _dXi            = _dX[i]
 287                         _X_plus_dXi     = numpy.array( _X.A1, dtype=float )
 288                         _X_plus_dXi[i]  = _X[i] + _dXi
 289                         _X_moins_dXi    = numpy.array( _X.A1, dtype=float )
 290                         _X_moins_dXi[i] = _X[i] - _dXi
 291                         #
 292                         _HX_plus_dXi    = self.DirectOperator( _X_plus_dXi )
 293                         _HX_moins_dXi   = self.DirectOperator( _X_moins_dXi )
 294                         #
 295                         _Jacobienne.append( numpy.ravel( _HX_plus_dXi - _HX_moins_dXi ) / (2.*_dXi) )
 296                 #
 297             else:
 298                 #
 299                 if self.__mpEnabled and not self.__mfEnabled:
 300                     funcrepr = {
 301                         "__userFunction__path" : self.__userFunction__path,
 302                         "__userFunction__modl" : self.__userFunction__modl,
 303                         "__userFunction__name" : self.__userFunction__name,
 304                     }
 305                     _jobs = []
 306                     _jobs.append( (_X.A1, funcrepr) )
 307                     for i in range( len(_dX) ):
 308                         _X_plus_dXi    = numpy.array( _X.A1, dtype=float )
 309                         _X_plus_dXi[i] = _X[i] + _dX[i]
 310                         #
 311                         _jobs.append( (_X_plus_dXi, funcrepr) )
 312                     #
 313                     import multiprocessing
 314                     self.__pool = multiprocessing.Pool(self.__mpWorkers)
 315                     _HX_plus_dX = self.__pool.map( ExecuteFunction, _jobs )
 316                     self.__pool.close()
 317                     self.__pool.join()
 318                     #
 319                     _HX = _HX_plus_dX.pop(0)
 320                     #
 321                     _Jacobienne = []
 322                     for i in range( len(_dX) ):
 323                         _Jacobienne.append( numpy.ravel(( _HX_plus_dX[i] - _HX ) / _dX[i]) )
 324                     #
 325                 elif self.__mfEnabled:
 326                     _xserie = []
 327                     _xserie.append( _X.A1 )
 328                     for i in range( len(_dX) ):
 329                         _X_plus_dXi    = numpy.array( _X.A1, dtype=float )
 330                         _X_plus_dXi[i] = _X[i] + _dX[i]
 331                         #
 332                         _xserie.append( _X_plus_dXi )
 333                     #
 334                     _HX_plus_dX = self.DirectOperator( _xserie )
 335                     #
 336                     _HX = _HX_plus_dX.pop(0)
 337                     #
 338                     _Jacobienne = []
 339                     for i in range( len(_dX) ):
 340                         _Jacobienne.append( numpy.ravel(( _HX_plus_dX[i] - _HX ) / _dX[i]) )
 341                    #
 342                 else:
 343                     _Jacobienne  = []
 344                     _HX = self.DirectOperator( _X )
 345                     for i in range( _dX.size ):
 346                         _dXi            = _dX[i]
 347                         _X_plus_dXi     = numpy.array( _X.A1, dtype=float )
 348                         _X_plus_dXi[i]  = _X[i] + _dXi
 349                         #
 350                         _HX_plus_dXi = self.DirectOperator( _X_plus_dXi )
 351                         #
 352                         _Jacobienne.append( numpy.ravel(( _HX_plus_dXi - _HX ) / _dXi) )
 353                 #
 354             #
 355             _Jacobienne = numpy.asmatrix( numpy.vstack( _Jacobienne ) ).T
 356             if self.__avoidRC:
 357                 if self.__lenghtRJ < 0: self.__lenghtRJ = 2 * _X.size
 358                 while len(self.__listJPCP) > self.__lenghtRJ:
 359                     self.__listJPCP.pop(0)
 360                     self.__listJPCI.pop(0)
 361                     self.__listJPCR.pop(0)
 362                     self.__listJPPN.pop(0)
 363                     self.__listJPIN.pop(0)
 364                 self.__listJPCP.append( copy.copy(_X) )
 365                 self.__listJPCI.append( copy.copy(_dX) )
 366                 self.__listJPCR.append( copy.copy(_Jacobienne) )
 367                 self.__listJPPN.append( numpy.linalg.norm(_X) )
 368                 self.__listJPIN.append( numpy.linalg.norm(_Jacobienne) )
 369         #
 370         logging.debug("FDA Fin du calcul de la Jacobienne")
 371         #
 372         return _Jacobienne
 373
 374     # ---------------------------------------------------------
 375     def TangentOperator(self, paire ):
 376         """
 377         Calcul du tangent à l'aide de la Jacobienne.
 378         """
 379         if self.__mfEnabled:
 380             assert len(paire) == 1, "Incorrect lenght of arguments"
 381             _paire = paire[0]
 382             assert len(_paire) == 2, "Incorrect number of arguments"
 383         else:
 384             assert len(paire) == 2, "Incorrect number of arguments"
 385             _paire = paire
 386         X, dX = _paire
 387         _Jacobienne = self.TangentMatrix( X )
 388         if dX is None or len(dX) == 0:
 389             #
 390             # Calcul de la forme matricielle si le second argument est None
 391             # -------------------------------------------------------------
 392             if self.__mfEnabled: return [_Jacobienne,]
 393             else:                return _Jacobienne
 394         else:
 395             #
 396             # Calcul de la valeur linéarisée de H en X appliqué à dX
 397             # ------------------------------------------------------
 398             _dX = numpy.asmatrix(numpy.ravel( dX )).T
 399             _HtX = numpy.dot(_Jacobienne, _dX)
 400             if self.__mfEnabled: return [_HtX.A1,]
 401             else:                return _HtX.A1
 402
 403     # ---------------------------------------------------------
 404     def AdjointOperator(self, paire ):
 405         """
 406         Calcul de l'adjoint à l'aide de la Jacobienne.
 407         """
 408         if self.__mfEnabled:
 409             assert len(paire) == 1, "Incorrect lenght of arguments"
 410             _paire = paire[0]
 411             assert len(_paire) == 2, "Incorrect number of arguments"
 412         else:
 413             assert len(paire) == 2, "Incorrect number of arguments"
 414             _paire = paire
 415         X, Y = _paire
 416         _JacobienneT = self.TangentMatrix( X ).T
 417         if Y is None or len(Y) == 0:
 418             #
 419             # Calcul de la forme matricielle si le second argument est None
 420             # -------------------------------------------------------------
 421             if self.__mfEnabled: return [_JacobienneT,]
 422             else:                return _JacobienneT
 423         else:
 424             #
 425             # Calcul de la valeur de l'adjoint en X appliqué à Y
 426             # --------------------------------------------------
 427             _Y = numpy.asmatrix(numpy.ravel( Y )).T
 428             _HaY = numpy.dot(_JacobienneT, _Y)
 429             if self.__mfEnabled: return [_HaY.A1,]
 430             else:                return _HaY.A1
 431
 432 # ==============================================================================
 433 def mmqr(
 434         func     = None,
 435         x0       = None,
 436         fprime   = None,
 437         bounds   = None,
 438         quantile = 0.5,
 439         maxfun   = 15000,
 440         toler    = 1.e-06,
 441         y        = None,
 442         ):
 443     """
 444     Implémentation informatique de l'algorithme MMQR, basée sur la publication :
 445     David R. Hunter, Kenneth Lange, "Quantile Regression via an MM Algorithm",
 446     Journal of Computational and Graphical Statistics, 9, 1, pp.60-77, 2000.
 447     """
 448     #
 449     # Recuperation des donnees et informations initiales
 450     # --------------------------------------------------
 451     variables = numpy.ravel( x0 )
 452     mesures   = numpy.ravel( y )
 453     increment = sys.float_info[0]
 454     p         = variables.size
 455     n         = mesures.size
 456     quantile  = float(quantile)
 457     #
 458     # Calcul des parametres du MM
 459     # ---------------------------
 460     tn      = float(toler) / n
 461     e0      = -tn / math.log(tn)
 462     epsilon = (e0-tn)/(1+math.log(e0))
 463     #
 464     # Calculs d'initialisation
 465     # ------------------------
 466     residus  = mesures - numpy.ravel( func( variables ) )
 467     poids    = 1./(epsilon+numpy.abs(residus))
 468     veps     = 1. - 2. * quantile - residus * poids
 469     lastsurrogate = -numpy.sum(residus*veps) - (1.-2.*quantile)*numpy.sum(residus)
 470     iteration = 0
 471     #
 472     # Recherche iterative
 473     # -------------------
 474     while (increment > toler) and (iteration < maxfun) :
 475         iteration += 1
 476         #
 477         Derivees  = numpy.array(fprime(variables))
 478         Derivees  = Derivees.reshape(n,p) # Necessaire pour remettre en place la matrice si elle passe par des tuyaux YACS
 479         DeriveesT = Derivees.transpose()
 480         M         =   numpy.dot( DeriveesT , (numpy.array(numpy.matrix(p*[poids,]).T)*Derivees) )
 481         SM        =   numpy.transpose(numpy.dot( DeriveesT , veps ))
 482         step      = - numpy.linalg.lstsq( M, SM, rcond=-1 )[0]
 483         #
 484         variables = variables + step
 485         if bounds is not None:
 486             # Attention : boucle infinie à éviter si un intervalle est trop petit
 487             while( (variables < numpy.ravel(numpy.asmatrix(bounds)[:,0])).any() or (variables > numpy.ravel(numpy.asmatrix(bounds)[:,1])).any() ):
 488                 step      = step/2.
 489                 variables = variables - step
 490         residus   = mesures - numpy.ravel( func(variables) )
 491         surrogate = numpy.sum(residus**2 * poids) + (4.*quantile-2.) * numpy.sum(residus)
 492         #
 493         while ( (surrogate > lastsurrogate) and ( max(list(numpy.abs(step))) > 1.e-16 ) ) :
 494             step      = step/2.
 495             variables = variables - step
 496             residus   = mesures - numpy.ravel( func(variables) )
 497             surrogate = numpy.sum(residus**2 * poids) + (4.*quantile-2.) * numpy.sum(residus)
 498         #
 499         increment     = lastsurrogate-surrogate
 500         poids         = 1./(epsilon+numpy.abs(residus))
 501         veps          = 1. - 2. * quantile - residus * poids
 502         lastsurrogate = -numpy.sum(residus * veps) - (1.-2.*quantile)*numpy.sum(residus)
 503     #
 504     # Mesure d'écart
 505     # --------------
 506     Ecart = quantile * numpy.sum(residus) - numpy.sum( residus[residus<0] )
 507     #
 508     return variables, Ecart, [n,p,iteration,increment,0]
 509
 510 # ==============================================================================
 511 def BackgroundEnsembleGeneration( _bgcenter, _bgcovariance, _nbmembers, _withSVD = True):
 512     "Génération d'un ensemble d'ébauches aléatoires de taille _nbmembers-1"
 513     def __CenteredRandomAnomalies(Zr, N):
 514         """
 515         Génère une matrice de N anomalies aléatoires centrées sur Zr selon les
 516         notes manuscrites de MB et conforme au code de PS avec eps = -1
 517         """
 518         eps = -1
 519         Q = numpy.eye(N-1)-numpy.ones((N-1,N-1))/numpy.sqrt(N)/(numpy.sqrt(N)-eps)
 520         Q = numpy.concatenate((Q, [eps*numpy.ones(N-1)/numpy.sqrt(N)]), axis=0)
 521         R, _ = numpy.linalg.qr(numpy.random.normal(size = (N-1,N-1)))
 522         Q = numpy.dot(Q,R)
 523         Zr = numpy.dot(Q,Zr)
 524         return Zr.T
 525     #
 526     if _nbmembers < 1:
 527         raise ValueError("Number of members has to be strictly more than 1 (given number: %s)."%(str(_nbmembers),))
 528     if _bgcovariance is None:
 529         BackgroundEnsemble = numpy.tile( numpy.ravel(_bgcenter)[:,None], _nbmembers)
 530     else:
 531         if _withSVD:
 532             U, s, V = numpy.linalg.svd(_bgcovariance, full_matrices=False)
 533             _nbctl = numpy.ravel(_bgcenter).size
 534             if _nbmembers > _nbctl:
 535                 _Z = numpy.concatenate((numpy.dot(
 536                     numpy.diag(numpy.sqrt(s[:_nbctl])), V[:_nbctl]),
 537                     numpy.random.multivariate_normal(numpy.zeros(_nbctl),_bgcovariance,_nbmembers-1-_nbctl)), axis = 0)
 538             else:
 539                 _Z = numpy.dot(numpy.diag(numpy.sqrt(s[:_nbmembers-1])), V[:_nbmembers-1])
 540             _Zca = __CenteredRandomAnomalies(_Z, _nbmembers)
 541             BackgroundEnsemble = numpy.ravel(_bgcenter)[:,None] + _Zca
 542         else:
 543             if max(abs(_bgcovariance.flatten())) > 0:
 544                 _nbctl = numpy.ravel(_bgcenter).size
 545                 _Z = numpy.random.multivariate_normal(numpy.zeros(_nbctl),_bgcovariance,_nbmembers-1)
 546                 _Zca = __CenteredRandomAnomalies(_Z, _nbmembers)
 547                 BackgroundEnsemble = numpy.ravel(_bgcenter)[:,None] + _Zca
 548             else:
 549                 BackgroundEnsemble = numpy.tile( numpy.ravel(_bgcenter)[:,None], _nbmembers)
 550     #
 551     return BackgroundEnsemble
 552
 553 # ==============================================================================
 554 def EnsembleCenteredAnomalies( _ensemble, _optmean = None):
 555     "Renvoie les anomalies centrées à partir d'un ensemble TailleEtat*NbMembres"
 556     if _optmean is None:
 557         Em = numpy.asarray(_ensemble).mean(axis=1, dtype=mfp).astype('float')[:,numpy.newaxis]
 558     else:
 559         Em = numpy.ravel(_optmean)[:,numpy.newaxis]
 560     #
 561     return numpy.asarray(_ensemble) - Em
 562
 563 # ==============================================================================
 564 def CovarianceInflation(
 565         InputCovOrEns,
 566         InflationType   = None,
 567         InflationFactor = None,
 568         BackgroundCov   = None,
 569         ):
 570     """
 571     Inflation applicable soit sur Pb ou Pa, soit sur les ensembles EXb ou EXa
 572
 573     Synthèse : Hunt 2007, section 2.3.5
 574     """
 575     if InflationFactor is None:
 576         return InputCovOrEns
 577     else:
 578         InflationFactor = float(InflationFactor)
 579     #
 580     if InflationType in ["MultiplicativeOnAnalysisCovariance", "MultiplicativeOnBackgroundCovariance"]:
 581         if InflationFactor < 1.:
 582             raise ValueError("Inflation factor for multiplicative inflation has to be greater or equal than 1.")
 583         if InflationFactor < 1.+mpr:
 584             return InputCovOrEns
 585         OutputCovOrEns = InflationFactor**2 * InputCovOrEns
 586     #
 587     elif InflationType in ["MultiplicativeOnAnalysisAnomalies", "MultiplicativeOnBackgroundAnomalies"]:
 588         if InflationFactor < 1.:
 589             raise ValueError("Inflation factor for multiplicative inflation has to be greater or equal than 1.")
 590         if InflationFactor < 1.+mpr:
 591             return InputCovOrEns
 592         InputCovOrEnsMean = InputCovOrEns.mean(axis=1, dtype=mfp).astype('float')
 593         OutputCovOrEns = InputCovOrEnsMean[:,numpy.newaxis] \
 594             + InflationFactor * (InputCovOrEns - InputCovOrEnsMean[:,numpy.newaxis])
 595     #
 596     elif InflationType in ["AdditiveOnBackgroundCovariance", "AdditiveOnAnalysisCovariance"]:
 597         if InflationFactor < 0.:
 598             raise ValueError("Inflation factor for additive inflation has to be greater or equal than 0.")
 599         if InflationFactor < mpr:
 600             return InputCovOrEns
 601         __n, __m = numpy.asarray(InputCovOrEns).shape
 602         if __n != __m:
 603             raise ValueError("Additive inflation can only be applied to squared (covariance) matrix.")
 604         OutputCovOrEns = (1. - InflationFactor) * InputCovOrEns + InflationFactor * numpy.eye(__n)
 605     #
 606     elif InflationType == "HybridOnBackgroundCovariance":
 607         if InflationFactor < 0.:
 608             raise ValueError("Inflation factor for hybrid inflation has to be greater or equal than 0.")
 609         if InflationFactor < mpr:
 610             return InputCovOrEns
 611         __n, __m = numpy.asarray(InputCovOrEns).shape
 612         if __n != __m:
 613             raise ValueError("Additive inflation can only be applied to squared (covariance) matrix.")
 614         if BackgroundCov is None:
 615             raise ValueError("Background covariance matrix B has to be given for hybrid inflation.")
 616         if InputCovOrEns.shape != BackgroundCov.shape:
 617             raise ValueError("Ensemble covariance matrix has to be of same size than background covariance matrix B.")
 618         OutputCovOrEns = (1. - InflationFactor) * InputCovOrEns + InflationFactor * BackgroundCov
 619     #
 620     elif InflationType == "Relaxation":
 621         raise NotImplementedError("InflationType Relaxation")
 622     #
 623     else:
 624         raise ValueError("Error in inflation type, '%s' is not a valid keyword."%InflationType)
 625     #
 626     return OutputCovOrEns
 627
 628 # ==============================================================================
 629 def senkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="KalmanFilterFormula"):
 630     """
 631     Stochastic EnKF (Envensen 1994, Burgers 1998)
 632
 633     selfA est identique au "self" d'algorithme appelant et contient les
 634     valeurs.
 635     """
 636     if selfA._parameters["EstimationOf"] == "Parameters":
 637         selfA._parameters["StoreInternalVariables"] = True
 638     #
 639     # Opérateurs
 640     # ----------
 641     H = HO["Direct"].appliedControledFormTo
 642     #
 643     if selfA._parameters["EstimationOf"] == "State":
 644         M = EM["Direct"].appliedControledFormTo
 645     #
 646     if CM is not None and "Tangent" in CM and U is not None:
 647         Cm = CM["Tangent"].asMatrix(Xb)
 648     else:
 649         Cm = None
 650     #
 651     # Nombre de pas identique au nombre de pas d'observations
 652     # -------------------------------------------------------
 653     if hasattr(Y,"stepnumber"):
 654         duration = Y.stepnumber()
 655         __p = numpy.cumprod(Y.shape())[-1]
 656     else:
 657         duration = 2
 658         __p = numpy.array(Y).size
 659     #
 660     # Précalcul des inversions de B et R
 661     # ----------------------------------
 662     if selfA._parameters["StoreInternalVariables"] \
 663         or selfA._toStore("CostFunctionJ") \
 664         or selfA._toStore("CostFunctionJb") \
 665         or selfA._toStore("CostFunctionJo") \
 666         or selfA._toStore("CurrentOptimum") \
 667         or selfA._toStore("APosterioriCovariance"):
 668         BI = B.getI()
 669         RI = R.getI()
 670     #
 671     # Initialisation
 672     # --------------
 673     __n = Xb.size
 674     __m = selfA._parameters["NumberOfMembers"]
 675     if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(__n)
 676     else:                         Pn = B
 677     if hasattr(R,"asfullmatrix"): Rn = R.asfullmatrix(__p)
 678     else:                         Rn = R
 679     if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
 680     else:                         Qn = Q
 681     Xn = numpy.asmatrix(numpy.dot( Xb.reshape((__n,1)), numpy.ones((1,__m)) ))
 682     #
 683     if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
 684         selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
 685         if selfA._toStore("APosterioriCovariance"):
 686             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
 687             covarianceXa = Pn
 688     #
 689     previousJMinimum = numpy.finfo(float).max
 690     #
 691     # Predimensionnement
 692     Xn_predicted = numpy.asmatrix(numpy.zeros((__n,__m)))
 693     HX_predicted = numpy.asmatrix(numpy.zeros((__p,__m)))
 694     #
 695     for step in range(duration-1):
 696         if hasattr(Y,"store"):
 697             Ynpu = numpy.asmatrix(numpy.ravel( Y[step+1] )).T
 698         else:
 699             Ynpu = numpy.asmatrix(numpy.ravel( Y )).T
 700         #
 701         if U is not None:
 702             if hasattr(U,"store") and len(U)>1:
 703                 Un = numpy.asmatrix(numpy.ravel( U[step] )).T
 704             elif hasattr(U,"store") and len(U)==1:
 705                 Un = numpy.asmatrix(numpy.ravel( U[0] )).T
 706             else:
 707                 Un = numpy.asmatrix(numpy.ravel( U )).T
 708         else:
 709             Un = None
 710         #
 711         if selfA._parameters["InflationType"] == "MultiplicativeOnBackgroundAnomalies":
 712             Xn = CovarianceInflation( Xn,
 713                 selfA._parameters["InflationType"],
 714                 selfA._parameters["InflationFactor"],
 715                 )
 716         #
 717         if selfA._parameters["EstimationOf"] == "State": # Forecast + Q and observation of forecast
 718             EMX = M( [(Xn[:,i], Un) for i in range(__m)], argsAsSerie = True )
 719             for i in range(__m):
 720                 qi = numpy.random.multivariate_normal(numpy.zeros(__n), Qn)
 721                 Xn_predicted[:,i] = (numpy.ravel( EMX[i] ) + qi).reshape((__n,-1))
 722             HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
 723                 argsAsSerie = True,
 724                 returnSerieAsArrayMatrix = True )
 725             if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
 726                 Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
 727                 Xn_predicted = Xn_predicted + Cm * Un
 728         elif selfA._parameters["EstimationOf"] == "Parameters": # Observation of forecast
 729             # --- > Par principe, M = Id, Q = 0
 730             Xn_predicted = Xn
 731             HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
 732                 argsAsSerie = True,
 733                 returnSerieAsArrayMatrix = True )
 734         #
 735         # Mean of forecast and observation of forecast
 736         Xfm  = Xn_predicted.mean(axis=1, dtype=mfp).astype('float')
 737         Hfm  = HX_predicted.mean(axis=1, dtype=mfp).astype('float')
 738         #
 739         #--------------------------
 740         if VariantM == "KalmanFilterFormula":
 741             PfHT, HPfHT = 0., 0.
 742             for i in range(__m):
 743                 Exfi = Xn_predicted[:,i] - Xfm.reshape((__n,-1))
 744                 Eyfi = (HX_predicted[:,i] - Hfm).reshape((__p,1))
 745                 PfHT  += Exfi * Eyfi.T
 746                 HPfHT += Eyfi * Eyfi.T
 747             PfHT  = (1./(__m-1)) * PfHT
 748             HPfHT = (1./(__m-1)) * HPfHT
 749             K     = PfHT * ( R + HPfHT ).I
 750             del PfHT, HPfHT
 751             #
 752             for i in range(__m):
 753                 ri = numpy.random.multivariate_normal(numpy.zeros(__p), Rn)
 754                 Xn[:,i] = Xn_predicted[:,i] + K @ (numpy.ravel(Ynpu) + ri - HX_predicted[:,i]).reshape((__p,1))
 755         #--------------------------
 756         else:
 757             raise ValueError("VariantM has to be chosen in the authorized methods list.")
 758         #
 759         if selfA._parameters["InflationType"] == "MultiplicativeOnAnalysisAnomalies":
 760             Xn = CovarianceInflation( Xn,
 761                 selfA._parameters["InflationType"],
 762                 selfA._parameters["InflationFactor"],
 763                 )
 764         #
 765         Xa = Xn.mean(axis=1, dtype=mfp).astype('float')
 766         #--------------------------
 767         #
 768         if selfA._parameters["StoreInternalVariables"] \
 769             or selfA._toStore("CostFunctionJ") \
 770             or selfA._toStore("CostFunctionJb") \
 771             or selfA._toStore("CostFunctionJo") \
 772             or selfA._toStore("APosterioriCovariance") \
 773             or selfA._toStore("InnovationAtCurrentAnalysis") \
 774             or selfA._toStore("SimulatedObservationAtCurrentAnalysis") \
 775             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 776             _HXa = numpy.asmatrix(numpy.ravel( H((Xa, Un)) )).T
 777             _Innovation = Ynpu - _HXa
 778         #
 779         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
 780         # ---> avec analysis
 781         selfA.StoredVariables["Analysis"].store( Xa )
 782         if selfA._toStore("SimulatedObservationAtCurrentAnalysis"):
 783             selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"].store( _HXa )
 784         if selfA._toStore("InnovationAtCurrentAnalysis"):
 785             selfA.StoredVariables["InnovationAtCurrentAnalysis"].store( _Innovation )
 786         # ---> avec current state
 787         if selfA._parameters["StoreInternalVariables"] \
 788             or selfA._toStore("CurrentState"):
 789             selfA.StoredVariables["CurrentState"].store( Xn )
 790         if selfA._toStore("ForecastState"):
 791             selfA.StoredVariables["ForecastState"].store( Xn_predicted )
 792         if selfA._toStore("BMA"):
 793             selfA.StoredVariables["BMA"].store( Xn_predicted - Xa )
 794         if selfA._toStore("InnovationAtCurrentState"):
 795             selfA.StoredVariables["InnovationAtCurrentState"].store( - HX_predicted + Ynpu )
 796         if selfA._toStore("SimulatedObservationAtCurrentState") \
 797             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 798             selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HX_predicted )
 799         # ---> autres
 800         if selfA._parameters["StoreInternalVariables"] \
 801             or selfA._toStore("CostFunctionJ") \
 802             or selfA._toStore("CostFunctionJb") \
 803             or selfA._toStore("CostFunctionJo") \
 804             or selfA._toStore("CurrentOptimum") \
 805             or selfA._toStore("APosterioriCovariance"):
 806             Jb  = float( 0.5 * (Xa - Xb).T * BI * (Xa - Xb) )
 807             Jo  = float( 0.5 * _Innovation.T * RI * _Innovation )
 808             J   = Jb + Jo
 809             selfA.StoredVariables["CostFunctionJb"].store( Jb )
 810             selfA.StoredVariables["CostFunctionJo"].store( Jo )
 811             selfA.StoredVariables["CostFunctionJ" ].store( J )
 812             #
 813             if selfA._toStore("IndexOfOptimum") \
 814                 or selfA._toStore("CurrentOptimum") \
 815                 or selfA._toStore("CostFunctionJAtCurrentOptimum") \
 816                 or selfA._toStore("CostFunctionJbAtCurrentOptimum") \
 817                 or selfA._toStore("CostFunctionJoAtCurrentOptimum") \
 818                 or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 819                 IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
 820             if selfA._toStore("IndexOfOptimum"):
 821                 selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
 822             if selfA._toStore("CurrentOptimum"):
 823                 selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["Analysis"][IndexMin] )
 824             if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 825                 selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"][IndexMin] )
 826             if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
 827                 selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
 828             if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
 829                 selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
 830             if selfA._toStore("CostFunctionJAtCurrentOptimum"):
 831                 selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
 832         if selfA._toStore("APosterioriCovariance"):
 833             Eai = (1/numpy.sqrt(__m-1)) * (Xn - Xa.reshape((__n,-1))) # Anomalies
 834             Pn = Eai @ Eai.T
 835             Pn = 0.5 * (Pn + Pn.T)
 836             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
 837         if selfA._parameters["EstimationOf"] == "Parameters" \
 838             and J < previousJMinimum:
 839             previousJMinimum    = J
 840             XaMin               = Xa
 841             if selfA._toStore("APosterioriCovariance"):
 842                 covarianceXaMin = Pn
 843     #
 844     # Stockage final supplémentaire de l'optimum en estimation de paramètres
 845     # ----------------------------------------------------------------------
 846     if selfA._parameters["EstimationOf"] == "Parameters":
 847         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
 848         selfA.StoredVariables["Analysis"].store( XaMin )
 849         if selfA._toStore("APosterioriCovariance"):
 850             selfA.StoredVariables["APosterioriCovariance"].store( covarianceXaMin )
 851         if selfA._toStore("BMA"):
 852             selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(XaMin) )
 853     #
 854     return 0
 855
 856 # ==============================================================================
 857 def etkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="KalmanFilterFormula"):
 858     """
 859     Ensemble-Transform EnKF (ETKF or Deterministic EnKF: Bishop 2001, Hunt 2007)
 860
 861     selfA est identique au "self" d'algorithme appelant et contient les
 862     valeurs.
 863     """
 864     if selfA._parameters["EstimationOf"] == "Parameters":
 865         selfA._parameters["StoreInternalVariables"] = True
 866     #
 867     # Opérateurs
 868     # ----------
 869     H = HO["Direct"].appliedControledFormTo
 870     #
 871     if selfA._parameters["EstimationOf"] == "State":
 872         M = EM["Direct"].appliedControledFormTo
 873     #
 874     if CM is not None and "Tangent" in CM and U is not None:
 875         Cm = CM["Tangent"].asMatrix(Xb)
 876     else:
 877         Cm = None
 878     #
 879     # Nombre de pas identique au nombre de pas d'observations
 880     # -------------------------------------------------------
 881     if hasattr(Y,"stepnumber"):
 882         duration = Y.stepnumber()
 883         __p = numpy.cumprod(Y.shape())[-1]
 884     else:
 885         duration = 2
 886         __p = numpy.array(Y).size
 887     #
 888     # Précalcul des inversions de B et R
 889     # ----------------------------------
 890     if selfA._parameters["StoreInternalVariables"] \
 891         or selfA._toStore("CostFunctionJ") \
 892         or selfA._toStore("CostFunctionJb") \
 893         or selfA._toStore("CostFunctionJo") \
 894         or selfA._toStore("CurrentOptimum") \
 895         or selfA._toStore("APosterioriCovariance"):
 896         BI = B.getI()
 897         RI = R.getI()
 898     elif VariantM != "KalmanFilterFormula":
 899         RI = R.getI()
 900     if VariantM == "KalmanFilterFormula":
 901         RIdemi = R.choleskyI()
 902     #
 903     # Initialisation
 904     # --------------
 905     __n = Xb.size
 906     __m = selfA._parameters["NumberOfMembers"]
 907     if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(__n)
 908     else:                         Pn = B
 909     if hasattr(R,"asfullmatrix"): Rn = R.asfullmatrix(__p)
 910     else:                         Rn = R
 911     if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
 912     else:                         Qn = Q
 913     Xn = numpy.asmatrix(numpy.dot( Xb.reshape((__n,1)), numpy.ones((1,__m)) ))
 914     #
 915     if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
 916         selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
 917         if selfA._toStore("APosterioriCovariance"):
 918             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
 919             covarianceXa = Pn
 920     #
 921     previousJMinimum = numpy.finfo(float).max
 922     #
 923     # Predimensionnement
 924     Xn_predicted = numpy.asmatrix(numpy.zeros((__n,__m)))
 925     #
 926     for step in range(duration-1):
 927         if hasattr(Y,"store"):
 928             Ynpu = numpy.asmatrix(numpy.ravel( Y[step+1] )).T
 929         else:
 930             Ynpu = numpy.asmatrix(numpy.ravel( Y )).T
 931         #
 932         if U is not None:
 933             if hasattr(U,"store") and len(U)>1:
 934                 Un = numpy.asmatrix(numpy.ravel( U[step] )).T
 935             elif hasattr(U,"store") and len(U)==1:
 936                 Un = numpy.asmatrix(numpy.ravel( U[0] )).T
 937             else:
 938                 Un = numpy.asmatrix(numpy.ravel( U )).T
 939         else:
 940             Un = None
 941         #
 942         if selfA._parameters["InflationType"] == "MultiplicativeOnBackgroundAnomalies":
 943             Xn = CovarianceInflation( Xn,
 944                 selfA._parameters["InflationType"],
 945                 selfA._parameters["InflationFactor"],
 946                 )
 947         #
 948         if selfA._parameters["EstimationOf"] == "State": # Forecast + Q and observation of forecast
 949             EMX = M( [(Xn[:,i], Un) for i in range(__m)], argsAsSerie = True )
 950             for i in range(__m):
 951                 qi = numpy.random.multivariate_normal(numpy.zeros(__n), Qn)
 952                 Xn_predicted[:,i] = (numpy.ravel( EMX[i] ) + qi).reshape((__n,-1))
 953             HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
 954                 argsAsSerie = True,
 955                 returnSerieAsArrayMatrix = True )
 956             if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
 957                 Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
 958                 Xn_predicted = Xn_predicted + Cm * Un
 959         elif selfA._parameters["EstimationOf"] == "Parameters": # Observation of forecast
 960             # --- > Par principe, M = Id, Q = 0
 961             Xn_predicted = Xn
 962             HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
 963                 argsAsSerie = True,
 964                 returnSerieAsArrayMatrix = True )
 965         #
 966         # Mean of forecast and observation of forecast
 967         Xfm  = Xn_predicted.mean(axis=1, dtype=mfp).astype('float')
 968         Hfm  = HX_predicted.mean(axis=1, dtype=mfp).astype('float')
 969         #
 970         # Anomalies
 971         EaX   = numpy.matrix(Xn_predicted - Xfm.reshape((__n,-1)))
 972         EaHX  = numpy.matrix(HX_predicted - Hfm.reshape((__p,-1)))
 973         #
 974         #--------------------------
 975         if VariantM == "KalmanFilterFormula":
 976             EaX    = EaX / numpy.sqrt(__m-1)
 977             mS    = RIdemi * EaHX / numpy.sqrt(__m-1)
 978             delta = RIdemi * ( Ynpu.reshape((__p,-1)) - Hfm.reshape((__p,-1)) )
 979             mT    = numpy.linalg.inv( numpy.eye(__m) + mS.T @ mS )
 980             vw    = mT @ mS.transpose() @ delta
 981             #
 982             Tdemi = numpy.real(scipy.linalg.sqrtm(mT))
 983             mU    = numpy.eye(__m)
 984             #
 985             Xn = Xfm.reshape((__n,-1)) + EaX @ ( vw.reshape((__m,-1)) + numpy.sqrt(__m-1) * Tdemi @ mU )
 986         #--------------------------
 987         elif VariantM == "Variational":
 988             HXfm = H((Xfm, Un)) # Eventuellement Hfm
 989             def CostFunction(w):
 990                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 991                 _Jo = 0.5 * _A.T * RI * _A
 992                 _Jb = 0.5 * (__m-1) * w.T @ w
 993                 _J  = _Jo + _Jb
 994                 return float(_J)
 995             def GradientOfCostFunction(w):
 996                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 997                 _GardJo = - EaHX.T * RI * _A
 998                 _GradJb = (__m-1) * w.reshape((__m,1))
 999                 _GradJ  = _GardJo + _GradJb
1000                 return numpy.ravel(_GradJ)
1001             vw = scipy.optimize.fmin_cg(
1002                 f           = CostFunction,
1003                 x0          = numpy.zeros(__m),
1004                 fprime      = GradientOfCostFunction,
1005                 args        = (),
1006                 disp        = False,
1007                 )
1008             #
1009             Hto = EaHX.T * RI * EaHX
1010             Htb = (__m-1) * numpy.eye(__m)
1011             Hta = Hto + Htb
1012             #
1013             Pta = numpy.linalg.inv( Hta )
1014             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
1015             #
1016             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
1017         #--------------------------
1018         elif VariantM == "FiniteSize11": # Jauge Boc2011
1019             HXfm = H((Xfm, Un)) # Eventuellement Hfm
1020             def CostFunction(w):
1021                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1022                 _Jo = 0.5 * _A.T * RI * _A
1023                 _Jb = 0.5 * __m * math.log(1 + 1/__m + w.T @ w)
1024                 _J  = _Jo + _Jb
1025                 return float(_J)
1026             def GradientOfCostFunction(w):
1027                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1028                 _GardJo = - EaHX.T * RI * _A
1029                 _GradJb = __m * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
1030                 _GradJ  = _GardJo + _GradJb
1031                 return numpy.ravel(_GradJ)
1032             vw = scipy.optimize.fmin_cg(
1033                 f           = CostFunction,
1034                 x0          = numpy.zeros(__m),
1035                 fprime      = GradientOfCostFunction,
1036                 args        = (),
1037                 disp        = False,
1038                 )
1039             #
1040             Hto = EaHX.T * RI * EaHX
1041             Htb = __m * \
1042                 ( (1 + 1/__m + vw.T @ vw) * numpy.eye(__m) - 2 * vw @ vw.T ) \
1043                 / (1 + 1/__m + vw.T @ vw)**2
1044             Hta = Hto + Htb
1045             #
1046             Pta = numpy.linalg.inv( Hta )
1047             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
1048             #
1049             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
1050         #--------------------------
1051         elif VariantM == "FiniteSize15": # Jauge Boc2015
1052             HXfm = H((Xfm, Un)) # Eventuellement Hfm
1053             def CostFunction(w):
1054                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1055                 _Jo = 0.5 * _A.T * RI * _A
1056                 _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w)
1057                 _J  = _Jo + _Jb
1058                 return float(_J)
1059             def GradientOfCostFunction(w):
1060                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1061                 _GardJo = - EaHX.T * RI * _A
1062                 _GradJb = (__m+1) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
1063                 _GradJ  = _GardJo + _GradJb
1064                 return numpy.ravel(_GradJ)
1065             vw = scipy.optimize.fmin_cg(
1066                 f           = CostFunction,
1067                 x0          = numpy.zeros(__m),
1068                 fprime      = GradientOfCostFunction,
1069                 args        = (),
1070                 disp        = False,
1071                 )
1072             #
1073             Hto = EaHX.T * RI * EaHX
1074             Htb = (__m+1) * \
1075                 ( (1 + 1/__m + vw.T @ vw) * numpy.eye(__m) - 2 * vw @ vw.T ) \
1076                 / (1 + 1/__m + vw.T @ vw)**2
1077             Hta = Hto + Htb
1078             #
1079             Pta = numpy.linalg.inv( Hta )
1080             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
1081             #
1082             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
1083         #--------------------------
1084         elif VariantM == "FiniteSize16": # Jauge Boc2016
1085             HXfm = H((Xfm, Un)) # Eventuellement Hfm
1086             def CostFunction(w):
1087                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1088                 _Jo = 0.5 * _A.T * RI * _A
1089                 _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w / (__m-1))
1090                 _J  = _Jo + _Jb
1091                 return float(_J)
1092             def GradientOfCostFunction(w):
1093                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1094                 _GardJo = - EaHX.T * RI * _A
1095                 _GradJb = ((__m+1) / (__m-1)) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w / (__m-1))
1096                 _GradJ  = _GardJo + _GradJb
1097                 return numpy.ravel(_GradJ)
1098             vw = scipy.optimize.fmin_cg(
1099                 f           = CostFunction,
1100                 x0          = numpy.zeros(__m),
1101                 fprime      = GradientOfCostFunction,
1102                 args        = (),
1103                 disp        = False,
1104                 )
1105             #
1106             Hto = EaHX.T * RI * EaHX
1107             Htb = ((__m+1) / (__m-1)) * \
1108                 ( (1 + 1/__m + vw.T @ vw / (__m-1)) * numpy.eye(__m) - 2 * vw @ vw.T / (__m-1) ) \
1109                 / (1 + 1/__m + vw.T @ vw / (__m-1))**2
1110             Hta = Hto + Htb
1111             #
1112             Pta = numpy.linalg.inv( Hta )
1113             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
1114             #
1115             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
1116         #--------------------------
1117         else:
1118             raise ValueError("VariantM has to be chosen in the authorized methods list.")
1119         #
1120         if selfA._parameters["InflationType"] == "MultiplicativeOnAnalysisAnomalies":
1121             Xn = CovarianceInflation( Xn,
1122                 selfA._parameters["InflationType"],
1123                 selfA._parameters["InflationFactor"],
1124                 )
1125         #
1126         Xa = Xn.mean(axis=1, dtype=mfp).astype('float')
1127         #--------------------------
1128         #
1129         if selfA._parameters["StoreInternalVariables"] \
1130             or selfA._toStore("CostFunctionJ") \
1131             or selfA._toStore("CostFunctionJb") \
1132             or selfA._toStore("CostFunctionJo") \
1133             or selfA._toStore("APosterioriCovariance") \
1134             or selfA._toStore("InnovationAtCurrentAnalysis") \
1135             or selfA._toStore("SimulatedObservationAtCurrentAnalysis") \
1136             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1137             _HXa = numpy.asmatrix(numpy.ravel( H((Xa, Un)) )).T
1138             _Innovation = Ynpu - _HXa
1139         #
1140         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
1141         # ---> avec analysis
1142         selfA.StoredVariables["Analysis"].store( Xa )
1143         if selfA._toStore("SimulatedObservationAtCurrentAnalysis"):
1144             selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"].store( _HXa )
1145         if selfA._toStore("InnovationAtCurrentAnalysis"):
1146             selfA.StoredVariables["InnovationAtCurrentAnalysis"].store( _Innovation )
1147         # ---> avec current state
1148         if selfA._parameters["StoreInternalVariables"] \
1149             or selfA._toStore("CurrentState"):
1150             selfA.StoredVariables["CurrentState"].store( Xn )
1151         if selfA._toStore("ForecastState"):
1152             selfA.StoredVariables["ForecastState"].store( Xn_predicted )
1153         if selfA._toStore("BMA"):
1154             selfA.StoredVariables["BMA"].store( Xn_predicted - Xa )
1155         if selfA._toStore("InnovationAtCurrentState"):
1156             selfA.StoredVariables["InnovationAtCurrentState"].store( - HX_predicted + Ynpu )
1157         if selfA._toStore("SimulatedObservationAtCurrentState") \
1158             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1159             selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HX_predicted )
1160         # ---> autres
1161         if selfA._parameters["StoreInternalVariables"] \
1162             or selfA._toStore("CostFunctionJ") \
1163             or selfA._toStore("CostFunctionJb") \
1164             or selfA._toStore("CostFunctionJo") \
1165             or selfA._toStore("CurrentOptimum") \
1166             or selfA._toStore("APosterioriCovariance"):
1167             Jb  = float( 0.5 * (Xa - Xb).T * BI * (Xa - Xb) )
1168             Jo  = float( 0.5 * _Innovation.T * RI * _Innovation )
1169             J   = Jb + Jo
1170             selfA.StoredVariables["CostFunctionJb"].store( Jb )
1171             selfA.StoredVariables["CostFunctionJo"].store( Jo )
1172             selfA.StoredVariables["CostFunctionJ" ].store( J )
1173             #
1174             if selfA._toStore("IndexOfOptimum") \
1175                 or selfA._toStore("CurrentOptimum") \
1176                 or selfA._toStore("CostFunctionJAtCurrentOptimum") \
1177                 or selfA._toStore("CostFunctionJbAtCurrentOptimum") \
1178                 or selfA._toStore("CostFunctionJoAtCurrentOptimum") \
1179                 or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1180                 IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
1181             if selfA._toStore("IndexOfOptimum"):
1182                 selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
1183             if selfA._toStore("CurrentOptimum"):
1184                 selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["Analysis"][IndexMin] )
1185             if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1186                 selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"][IndexMin] )
1187             if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
1188                 selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
1189             if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
1190                 selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
1191             if selfA._toStore("CostFunctionJAtCurrentOptimum"):
1192                 selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
1193         if selfA._toStore("APosterioriCovariance"):
1194             Eai = (1/numpy.sqrt(__m-1)) * (Xn - Xa.reshape((__n,-1))) # Anomalies
1195             Pn = Eai @ Eai.T
1196             Pn = 0.5 * (Pn + Pn.T)
1197             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
1198         if selfA._parameters["EstimationOf"] == "Parameters" \
1199             and J < previousJMinimum:
1200             previousJMinimum    = J
1201             XaMin               = Xa
1202             if selfA._toStore("APosterioriCovariance"):
1203                 covarianceXaMin = Pn
1204     #
1205     # Stockage final supplémentaire de l'optimum en estimation de paramètres
1206     # ----------------------------------------------------------------------
1207     if selfA._parameters["EstimationOf"] == "Parameters":
1208         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
1209         selfA.StoredVariables["Analysis"].store( XaMin )
1210         if selfA._toStore("APosterioriCovariance"):
1211             selfA.StoredVariables["APosterioriCovariance"].store( covarianceXaMin )
1212         if selfA._toStore("BMA"):
1213             selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(XaMin) )
1214     #
1215     return 0
1216
1217 # ==============================================================================
1218 def mlef(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="MLEF13",
1219     BnotT=False, _epsilon=1.e-3, _e=1.e-7, _jmax=15000):
1220     """
1221     Maximum Likelihood Ensemble Filter (EnKF/MLEF Zupanski 2005, Bocquet 2013)
1222
1223     selfA est identique au "self" d'algorithme appelant et contient les
1224     valeurs.
1225     """
1226     if selfA._parameters["EstimationOf"] == "Parameters":
1227         selfA._parameters["StoreInternalVariables"] = True
1228     #
1229     # Opérateurs
1230     # ----------
1231     H = HO["Direct"].appliedControledFormTo
1232     #
1233     if selfA._parameters["EstimationOf"] == "State":
1234         M = EM["Direct"].appliedControledFormTo
1235     #
1236     if CM is not None and "Tangent" in CM and U is not None:
1237         Cm = CM["Tangent"].asMatrix(Xb)
1238     else:
1239         Cm = None
1240     #
1241     # Nombre de pas identique au nombre de pas d'observations
1242     # -------------------------------------------------------
1243     if hasattr(Y,"stepnumber"):
1244         duration = Y.stepnumber()
1245         __p = numpy.cumprod(Y.shape())[-1]
1246     else:
1247         duration = 2
1248         __p = numpy.array(Y).size
1249     #
1250     # Précalcul des inversions de B et R
1251     # ----------------------------------
1252     if selfA._parameters["StoreInternalVariables"] \
1253         or selfA._toStore("CostFunctionJ") \
1254         or selfA._toStore("CostFunctionJb") \
1255         or selfA._toStore("CostFunctionJo") \
1256         or selfA._toStore("CurrentOptimum") \
1257         or selfA._toStore("APosterioriCovariance"):
1258         BI = B.getI()
1259     RI = R.getI()
1260     #
1261     # Initialisation
1262     # --------------
1263     __n = Xb.size
1264     __m = selfA._parameters["NumberOfMembers"]
1265     if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(__n)
1266     else:                         Pn = B
1267     if hasattr(R,"asfullmatrix"): Rn = R.asfullmatrix(__p)
1268     else:                         Rn = R
1269     if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
1270     else:                         Qn = Q
1271     Xn = BackgroundEnsembleGeneration( Xb, None, __m )
1272     #
1273     if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
1274         selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
1275         if selfA._toStore("APosterioriCovariance"):
1276             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
1277             covarianceXa = Pn
1278     #
1279     previousJMinimum = numpy.finfo(float).max
1280     #
1281     Xn_predicted = numpy.zeros((__n,__m))
1282     for step in range(duration-1):
1283         if hasattr(Y,"store"):
1284             Ynpu = numpy.ravel( Y[step+1] )[:,None]
1285         else:
1286             Ynpu = numpy.ravel( Y )[:,None]
1287         #
1288         if U is not None:
1289             if hasattr(U,"store") and len(U)>1:
1290                 Un = numpy.asmatrix(numpy.ravel( U[step] )).T
1291             elif hasattr(U,"store") and len(U)==1:
1292                 Un = numpy.asmatrix(numpy.ravel( U[0] )).T
1293             else:
1294                 Un = numpy.asmatrix(numpy.ravel( U )).T
1295         else:
1296             Un = None
1297         #
1298         if selfA._parameters["InflationType"] == "MultiplicativeOnBackgroundAnomalies":
1299             Xn = CovarianceInflation( Xn,
1300                 selfA._parameters["InflationType"],
1301                 selfA._parameters["InflationFactor"],
1302                 )
1303         #
1304         if selfA._parameters["EstimationOf"] == "State": # Forecast + Q and observation of forecast
1305             EMX = M( [(Xn[:,i,numpy.newaxis], Un) for i in range(__m)], argsAsSerie = True )
1306             for i in range(__m):
1307                 qi = numpy.random.multivariate_normal(numpy.zeros(__n), Qn)
1308                 Xn_predicted[:,i] = numpy.ravel( EMX[i] ) + qi
1309             if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
1310                 Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
1311                 Xn_predicted = Xn_predicted + Cm * Un
1312         elif selfA._parameters["EstimationOf"] == "Parameters": # Observation of forecast
1313             # --- > Par principe, M = Id, Q = 0
1314             Xn_predicted = Xn
1315         #
1316         #--------------------------
1317         if VariantM == "MLEF13":
1318             Xfm = numpy.asarray(Xn_predicted.mean(axis=1, dtype=mfp).astype('float'))
1319             EaX = numpy.asarray((Xn_predicted - Xfm.reshape((__n,-1))) / numpy.sqrt(__m-1))
1320             Ua  = numpy.eye(__m)
1321             __j = 0
1322             Deltaw = 1
1323             if not BnotT:
1324                 Ta  = numpy.eye(__m)
1325             vw  = numpy.zeros(__m)
1326             while numpy.linalg.norm(Deltaw) >= _e and __j <= _jmax:
1327                 vx1 = numpy.ravel(Xfm) + EaX @ vw
1328                 #
1329                 if BnotT:
1330                     E1 = vx1.reshape((__n,-1)) + _epsilon * EaX
1331                 else:
1332                     E1 = vx1.reshape((__n,-1)) + numpy.sqrt(__m-1) * EaX @ Ta
1333                 #
1334                 HE2 = H( [(E1[:,i,numpy.newaxis], Un) for i in range(__m)],
1335                     argsAsSerie = True,
1336                     returnSerieAsArrayMatrix = True )
1337                 vy2 = HE2.mean(axis=1, dtype=mfp).astype('float').reshape((__p,-1))
1338                 #
1339                 if BnotT:
1340                     EaY = (HE2 - vy2) / _epsilon
1341                 else:
1342                     EaY = ( (HE2 - vy2) @ numpy.linalg.inv(Ta) ) / numpy.sqrt(__m-1)
1343                 #
1344                 GradJ = numpy.ravel(vw[:,None] - EaY.transpose() @ (RI * ( Ynpu - vy2 )))
1345                 mH = numpy.eye(__m) + EaY.transpose() @ (RI * EaY)
1346                 Deltaw = - numpy.linalg.solve(mH,GradJ)
1347                 #
1348                 vw = vw + Deltaw
1349                 #
1350                 if not BnotT:
1351                     Ta = numpy.real(scipy.linalg.sqrtm(numpy.linalg.inv( mH )))
1352                 #
1353                 __j = __j + 1
1354             #
1355             if BnotT:
1356                 Ta = numpy.real(scipy.linalg.sqrtm(numpy.linalg.inv( mH )))
1357             #
1358             Xn = vx1.reshape((__n,-1)) + numpy.sqrt(__m-1) * EaX @ Ta @ Ua
1359         #--------------------------
1360         else:
1361             raise ValueError("VariantM has to be chosen in the authorized methods list.")
1362         #
1363         if selfA._parameters["InflationType"] == "MultiplicativeOnAnalysisAnomalies":
1364             Xn = CovarianceInflation( Xn,
1365                 selfA._parameters["InflationType"],
1366                 selfA._parameters["InflationFactor"],
1367                 )
1368         #
1369         Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
1370         #--------------------------
1371         #
1372         if selfA._parameters["StoreInternalVariables"] \
1373             or selfA._toStore("CostFunctionJ") \
1374             or selfA._toStore("CostFunctionJb") \
1375             or selfA._toStore("CostFunctionJo") \
1376             or selfA._toStore("APosterioriCovariance") \
1377             or selfA._toStore("InnovationAtCurrentAnalysis") \
1378             or selfA._toStore("SimulatedObservationAtCurrentAnalysis") \
1379             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1380             _HXa = numpy.asmatrix(numpy.ravel( H((Xa, Un)) )).T
1381             _Innovation = Ynpu - _HXa
1382         #
1383         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
1384         # ---> avec analysis
1385         selfA.StoredVariables["Analysis"].store( Xa )
1386         if selfA._toStore("SimulatedObservationAtCurrentAnalysis"):
1387             selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"].store( _HXa )
1388         if selfA._toStore("InnovationAtCurrentAnalysis"):
1389             selfA.StoredVariables["InnovationAtCurrentAnalysis"].store( _Innovation )
1390         # ---> avec current state
1391         if selfA._parameters["StoreInternalVariables"] \
1392             or selfA._toStore("CurrentState"):
1393             selfA.StoredVariables["CurrentState"].store( Xn )
1394         if selfA._toStore("ForecastState"):
1395             selfA.StoredVariables["ForecastState"].store( Xn_predicted )
1396         if selfA._toStore("BMA"):
1397             selfA.StoredVariables["BMA"].store( Xn_predicted - Xa )
1398         #~ if selfA._toStore("InnovationAtCurrentState"):
1399             #~ selfA.StoredVariables["InnovationAtCurrentState"].store( - HX_predicted + Ynpu )
1400         #~ if selfA._toStore("SimulatedObservationAtCurrentState") \
1401             #~ or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1402             #~ selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HX_predicted )
1403         # ---> autres
1404         if selfA._parameters["StoreInternalVariables"] \
1405             or selfA._toStore("CostFunctionJ") \
1406             or selfA._toStore("CostFunctionJb") \
1407             or selfA._toStore("CostFunctionJo") \
1408             or selfA._toStore("CurrentOptimum") \
1409             or selfA._toStore("APosterioriCovariance"):
1410             Jb  = float( 0.5 * (Xa - Xb).T * BI * (Xa - Xb) )
1411             Jo  = float( 0.5 * _Innovation.T * RI * _Innovation )
1412             J   = Jb + Jo
1413             selfA.StoredVariables["CostFunctionJb"].store( Jb )
1414             selfA.StoredVariables["CostFunctionJo"].store( Jo )
1415             selfA.StoredVariables["CostFunctionJ" ].store( J )
1416             #
1417             if selfA._toStore("IndexOfOptimum") \
1418                 or selfA._toStore("CurrentOptimum") \
1419                 or selfA._toStore("CostFunctionJAtCurrentOptimum") \
1420                 or selfA._toStore("CostFunctionJbAtCurrentOptimum") \
1421                 or selfA._toStore("CostFunctionJoAtCurrentOptimum") \
1422                 or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1423                 IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
1424             if selfA._toStore("IndexOfOptimum"):
1425                 selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
1426             if selfA._toStore("CurrentOptimum"):
1427                 selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["Analysis"][IndexMin] )
1428             if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1429                 selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"][IndexMin] )
1430             if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
1431                 selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
1432             if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
1433                 selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
1434             if selfA._toStore("CostFunctionJAtCurrentOptimum"):
1435                 selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
1436         if selfA._toStore("APosterioriCovariance"):
1437             Eai = numpy.asarray((Xn - Xa.reshape((__n,-1))) / numpy.sqrt(__m-1)) # Anomalies
1438             Pn = Eai @ Eai.T
1439             Pn = 0.5 * (Pn + Pn.T)
1440             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
1441         if selfA._parameters["EstimationOf"] == "Parameters" \
1442             and J < previousJMinimum:
1443             previousJMinimum    = J
1444             XaMin               = Xa
1445             if selfA._toStore("APosterioriCovariance"):
1446                 covarianceXaMin = Pn
1447     #
1448     # Stockage final supplémentaire de l'optimum en estimation de paramètres
1449     # ----------------------------------------------------------------------
1450     if selfA._parameters["EstimationOf"] == "Parameters":
1451         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
1452         selfA.StoredVariables["Analysis"].store( XaMin )
1453         if selfA._toStore("APosterioriCovariance"):
1454             selfA.StoredVariables["APosterioriCovariance"].store( covarianceXaMin )
1455         if selfA._toStore("BMA"):
1456             selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(XaMin) )
1457     #
1458     return 0
1459
1460 # ==============================================================================
1461 if __name__ == "__main__":
1462     print('\n AUTODIAGNOSTIC\n')