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 CovarianceInflation(
 512         InputCovOrEns,
 513         InflationType   = None,
 514         InflationFactor = None,
 515         BackgroundCov   = None,
 516         ):
 517     """
 518     Inflation applicable soit sur Pb ou Pa, soit sur les ensembles EXb ou EXa
 519
 520     Synthèse : Hunt 2007, section 2.3.5
 521     """
 522     if InflationFactor is None:
 523         return InputCovOrEns
 524     else:
 525         InflationFactor = float(InflationFactor)
 526     #
 527     if InflationType in ["MultiplicativeOnAnalysisCovariance", "MultiplicativeOnBackgroundCovariance"]:
 528         if InflationFactor < 1.:
 529             raise ValueError("Inflation factor for multiplicative inflation has to be greater or equal than 1.")
 530         if InflationFactor < 1.+mpr:
 531             return InputCovOrEns
 532         OutputCovOrEns = InflationFactor**2 * InputCovOrEns
 533     #
 534     elif InflationType in ["MultiplicativeOnAnalysisAnomalies", "MultiplicativeOnBackgroundAnomalies"]:
 535         if InflationFactor < 1.:
 536             raise ValueError("Inflation factor for multiplicative inflation has to be greater or equal than 1.")
 537         if InflationFactor < 1.+mpr:
 538             return InputCovOrEns
 539         InputCovOrEnsMean = InputCovOrEns.mean(axis=1, dtype=mfp).astype('float')
 540         OutputCovOrEns = InputCovOrEnsMean[:,numpy.newaxis] \
 541             + InflationFactor * (InputCovOrEns - InputCovOrEnsMean[:,numpy.newaxis])
 542     #
 543     elif InflationType in ["AdditiveOnBackgroundCovariance", "AdditiveOnAnalysisCovariance"]:
 544         if InflationFactor < 0.:
 545             raise ValueError("Inflation factor for additive inflation has to be greater or equal than 0.")
 546         if InflationFactor < mpr:
 547             return InputCovOrEns
 548         __n, __m = numpy.asarray(InputCovOrEns).shape
 549         if __n != __m:
 550             raise ValueError("Additive inflation can only be applied to squared (covariance) matrix.")
 551         OutputCovOrEns = (1. - InflationFactor) * InputCovOrEns + InflationFactor * numpy.eye(__n)
 552     #
 553     elif InflationType == "HybridOnBackgroundCovariance":
 554         if InflationFactor < 0.:
 555             raise ValueError("Inflation factor for hybrid inflation has to be greater or equal than 0.")
 556         if InflationFactor < mpr:
 557             return InputCovOrEns
 558         __n, __m = numpy.asarray(InputCovOrEns).shape
 559         if __n != __m:
 560             raise ValueError("Additive inflation can only be applied to squared (covariance) matrix.")
 561         if BackgroundCov is None:
 562             raise ValueError("Background covariance matrix B has to be given for hybrid inflation.")
 563         if InputCovOrEns.shape != BackgroundCov.shape:
 564             raise ValueError("Ensemble covariance matrix has to be of same size than background covariance matrix B.")
 565         OutputCovOrEns = (1. - InflationFactor) * InputCovOrEns + InflationFactor * BackgroundCov
 566     #
 567     elif InflationType == "Relaxation":
 568         raise NotImplementedError("InflationType Relaxation")
 569     #
 570     else:
 571         raise ValueError("Error in inflation type, '%s' is not a valid keyword."%InflationType)
 572     #
 573     return OutputCovOrEns
 574
 575 # ==============================================================================
 576 def senkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q):
 577     """
 578     Stochastic EnKF (Envensen 1994, Burgers 1998)
 579
 580     selfA est identique au "self" d'algorithme appelant et contient les
 581     valeurs.
 582     """
 583     if selfA._parameters["EstimationOf"] == "Parameters":
 584         selfA._parameters["StoreInternalVariables"] = True
 585     #
 586     # Opérateurs
 587     # ----------
 588     H = HO["Direct"].appliedControledFormTo
 589     #
 590     if selfA._parameters["EstimationOf"] == "State":
 591         M = EM["Direct"].appliedControledFormTo
 592     #
 593     if CM is not None and "Tangent" in CM and U is not None:
 594         Cm = CM["Tangent"].asMatrix(Xb)
 595     else:
 596         Cm = None
 597     #
 598     # Nombre de pas identique au nombre de pas d'observations
 599     # -------------------------------------------------------
 600     if hasattr(Y,"stepnumber"):
 601         duration = Y.stepnumber()
 602         __p = numpy.cumprod(Y.shape())[-1]
 603     else:
 604         duration = 2
 605         __p = numpy.array(Y).size
 606     #
 607     # Précalcul des inversions de B et R
 608     # ----------------------------------
 609     if selfA._parameters["StoreInternalVariables"] \
 610         or selfA._toStore("CostFunctionJ") \
 611         or selfA._toStore("CostFunctionJb") \
 612         or selfA._toStore("CostFunctionJo") \
 613         or selfA._toStore("CurrentOptimum") \
 614         or selfA._toStore("APosterioriCovariance"):
 615         BI = B.getI()
 616         RI = R.getI()
 617     #
 618     # Initialisation
 619     # --------------
 620     __n = Xb.size
 621     __m = selfA._parameters["NumberOfMembers"]
 622     Xn = numpy.asmatrix(numpy.dot( Xb.reshape(__n,1), numpy.ones((1,__m)) ))
 623     if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(__n)
 624     else:                         Pn = B
 625     if hasattr(R,"asfullmatrix"): Rn = R.asfullmatrix(__p)
 626     else:                         Rn = R
 627     if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
 628     else:                         Qn = Q
 629     #
 630     if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
 631         selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
 632         if selfA._toStore("APosterioriCovariance"):
 633             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
 634             covarianceXa = Pn
 635     #
 636     previousJMinimum = numpy.finfo(float).max
 637     #
 638     # Predimensionnement
 639     Xn_predicted = numpy.asmatrix(numpy.zeros((__n,__m)))
 640     HX_predicted = numpy.asmatrix(numpy.zeros((__p,__m)))
 641     #
 642     for step in range(duration-1):
 643         if hasattr(Y,"store"):
 644             Ynpu = numpy.asmatrix(numpy.ravel( Y[step+1] )).T
 645         else:
 646             Ynpu = numpy.asmatrix(numpy.ravel( Y )).T
 647         #
 648         if U is not None:
 649             if hasattr(U,"store") and len(U)>1:
 650                 Un = numpy.asmatrix(numpy.ravel( U[step] )).T
 651             elif hasattr(U,"store") and len(U)==1:
 652                 Un = numpy.asmatrix(numpy.ravel( U[0] )).T
 653             else:
 654                 Un = numpy.asmatrix(numpy.ravel( U )).T
 655         else:
 656             Un = None
 657         #
 658         if selfA._parameters["InflationType"] == "MultiplicativeOnBackgroundAnomalies":
 659             Xn = CovarianceInflation( Xn,
 660                 selfA._parameters["InflationType"],
 661                 selfA._parameters["InflationFactor"],
 662                 )
 663         #
 664         if selfA._parameters["EstimationOf"] == "State": # Forecast + Q and observation of forecast
 665             for i in range(__m):
 666                 qi = numpy.asmatrix(numpy.random.multivariate_normal(numpy.zeros(__n), Qn, (1,1,1))).T
 667                 Xn_predicted[:,i] = numpy.asmatrix(numpy.ravel( M((Xn[:,i], Un)) )).T + qi
 668                 HX_predicted[:,i] = numpy.asmatrix(numpy.ravel( H((Xn_predicted[:,i], Un)) )).T
 669             if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
 670                 Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
 671                 Xn_predicted = Xn_predicted + Cm * Un
 672         elif selfA._parameters["EstimationOf"] == "Parameters": # Observation of forecast
 673             # --- > Par principe, M = Id, Q = 0
 674             Xn_predicted = Xn
 675             for i in range(__m):
 676                 HX_predicted[:,i] = numpy.asmatrix(numpy.ravel( H((Xn_predicted[:,i], Un)) )).T
 677         #
 678         # Mean of forecast and observation of forecast
 679         Xfm  = numpy.asmatrix(numpy.ravel(Xn_predicted.mean(axis=1, dtype=mfp).astype('float'))).T
 680         Hfm  = numpy.asmatrix(numpy.ravel(HX_predicted.mean(axis=1, dtype=mfp).astype('float'))).T
 681         #
 682         PfHT, HPfHT = 0., 0.
 683         for i in range(__m):
 684             Exfi = Xn_predicted[:,i] - Xfm
 685             Eyfi = HX_predicted[:,i] - Hfm
 686             PfHT  += Exfi * Eyfi.T
 687             HPfHT += Eyfi * Eyfi.T
 688         PfHT  = (1./(__m-1)) * PfHT
 689         HPfHT = (1./(__m-1)) * HPfHT
 690         K     = PfHT * ( R + HPfHT ).I
 691         del PfHT, HPfHT
 692         #
 693         for i in range(__m):
 694             ri = numpy.asmatrix(numpy.random.multivariate_normal(numpy.zeros(__p), Rn, (1,1,1))).T
 695             Xn[:,i] = Xn_predicted[:,i] + K * (Ynpu + ri - HX_predicted[:,i])
 696         #
 697         if selfA._parameters["InflationType"] == "MultiplicativeOnAnalysisAnomalies":
 698             Xn = CovarianceInflation( Xn,
 699                 selfA._parameters["InflationType"],
 700                 selfA._parameters["InflationFactor"],
 701                 )
 702         #
 703         Xa = Xn.mean(axis=1, dtype=mfp).astype('float')
 704         #
 705         if selfA._parameters["StoreInternalVariables"] \
 706             or selfA._toStore("CostFunctionJ") \
 707             or selfA._toStore("CostFunctionJb") \
 708             or selfA._toStore("CostFunctionJo") \
 709             or selfA._toStore("APosterioriCovariance") \
 710             or selfA._toStore("InnovationAtCurrentAnalysis") \
 711             or selfA._toStore("SimulatedObservationAtCurrentAnalysis") \
 712             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 713             _HXa = numpy.asmatrix(numpy.ravel( H((Xa, Un)) )).T
 714             _Innovation = Ynpu - _HXa
 715         #
 716         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
 717         # ---> avec analysis
 718         selfA.StoredVariables["Analysis"].store( Xa )
 719         if selfA._toStore("SimulatedObservationAtCurrentAnalysis"):
 720             selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"].store( _HXa )
 721         if selfA._toStore("InnovationAtCurrentAnalysis"):
 722             selfA.StoredVariables["InnovationAtCurrentAnalysis"].store( _Innovation )
 723         # ---> avec current state
 724         if selfA._parameters["StoreInternalVariables"] \
 725             or selfA._toStore("CurrentState"):
 726             selfA.StoredVariables["CurrentState"].store( Xn )
 727         if selfA._toStore("ForecastState"):
 728             selfA.StoredVariables["ForecastState"].store( Xn_predicted )
 729         if selfA._toStore("BMA"):
 730             selfA.StoredVariables["BMA"].store( Xn_predicted - Xa )
 731         if selfA._toStore("InnovationAtCurrentState"):
 732             selfA.StoredVariables["InnovationAtCurrentState"].store( - HX_predicted + Ynpu )
 733         if selfA._toStore("SimulatedObservationAtCurrentState") \
 734             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 735             selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HX_predicted )
 736         # ---> autres
 737         if selfA._parameters["StoreInternalVariables"] \
 738             or selfA._toStore("CostFunctionJ") \
 739             or selfA._toStore("CostFunctionJb") \
 740             or selfA._toStore("CostFunctionJo") \
 741             or selfA._toStore("CurrentOptimum") \
 742             or selfA._toStore("APosterioriCovariance"):
 743             Jb  = float( 0.5 * (Xa - Xb).T * BI * (Xa - Xb) )
 744             Jo  = float( 0.5 * _Innovation.T * RI * _Innovation )
 745             J   = Jb + Jo
 746             selfA.StoredVariables["CostFunctionJb"].store( Jb )
 747             selfA.StoredVariables["CostFunctionJo"].store( Jo )
 748             selfA.StoredVariables["CostFunctionJ" ].store( J )
 749             #
 750             if selfA._toStore("IndexOfOptimum") \
 751                 or selfA._toStore("CurrentOptimum") \
 752                 or selfA._toStore("CostFunctionJAtCurrentOptimum") \
 753                 or selfA._toStore("CostFunctionJbAtCurrentOptimum") \
 754                 or selfA._toStore("CostFunctionJoAtCurrentOptimum") \
 755                 or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 756                 IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
 757             if selfA._toStore("IndexOfOptimum"):
 758                 selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
 759             if selfA._toStore("CurrentOptimum"):
 760                 selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["Analysis"][IndexMin] )
 761             if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
 762                 selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"][IndexMin] )
 763             if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
 764                 selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
 765             if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
 766                 selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
 767             if selfA._toStore("CostFunctionJAtCurrentOptimum"):
 768                 selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
 769         if selfA._toStore("APosterioriCovariance"):
 770             Eai = (1/numpy.sqrt(__m-1)) * (Xn - Xa.reshape((__n,-1))) # Anomalies
 771             Pn = Eai @ Eai.T
 772             Pn = 0.5 * (Pn + Pn.T)
 773             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
 774         if selfA._parameters["EstimationOf"] == "Parameters" \
 775             and J < previousJMinimum:
 776             previousJMinimum    = J
 777             XaMin               = Xa
 778             if selfA._toStore("APosterioriCovariance"):
 779                 covarianceXaMin = Pn
 780     #
 781     # Stockage final supplémentaire de l'optimum en estimation de paramètres
 782     # ----------------------------------------------------------------------
 783     if selfA._parameters["EstimationOf"] == "Parameters":
 784         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
 785         selfA.StoredVariables["Analysis"].store( XaMin )
 786         if selfA._toStore("APosterioriCovariance"):
 787             selfA.StoredVariables["APosterioriCovariance"].store( covarianceXaMin )
 788         if selfA._toStore("BMA"):
 789             selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(XaMin) )
 790     #
 791     return 0
 792
 793 # ==============================================================================
 794 def etkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, KorV="KalmanFilterFormula"):
 795     """
 796     Ensemble-Transform EnKF (ETKF or Deterministic EnKF: Bishop 2001, Hunt 2007)
 797
 798     selfA est identique au "self" d'algorithme appelant et contient les
 799     valeurs.
 800     """
 801     if selfA._parameters["EstimationOf"] == "Parameters":
 802         selfA._parameters["StoreInternalVariables"] = True
 803     #
 804     # Opérateurs
 805     # ----------
 806     H = HO["Direct"].appliedControledFormTo
 807     #
 808     if selfA._parameters["EstimationOf"] == "State":
 809         M = EM["Direct"].appliedControledFormTo
 810     #
 811     if CM is not None and "Tangent" in CM and U is not None:
 812         Cm = CM["Tangent"].asMatrix(Xb)
 813     else:
 814         Cm = None
 815     #
 816     # Nombre de pas identique au nombre de pas d'observations
 817     # -------------------------------------------------------
 818     if hasattr(Y,"stepnumber"):
 819         duration = Y.stepnumber()
 820         __p = numpy.cumprod(Y.shape())[-1]
 821     else:
 822         duration = 2
 823         __p = numpy.array(Y).size
 824     #
 825     # Précalcul des inversions de B et R
 826     # ----------------------------------
 827     if selfA._parameters["StoreInternalVariables"] \
 828         or selfA._toStore("CostFunctionJ") \
 829         or selfA._toStore("CostFunctionJb") \
 830         or selfA._toStore("CostFunctionJo") \
 831         or selfA._toStore("CurrentOptimum") \
 832         or selfA._toStore("APosterioriCovariance"):
 833         BI = B.getI()
 834         RI = R.getI()
 835     elif KorV != "KalmanFilterFormula":
 836         RI = R.getI()
 837     if KorV == "KalmanFilterFormula":
 838         RIdemi = R.choleskyI()
 839     #
 840     # Initialisation
 841     # --------------
 842     __n = Xb.size
 843     __m = selfA._parameters["NumberOfMembers"]
 844     Xn = numpy.asmatrix(numpy.dot( Xb.reshape(__n,1), numpy.ones((1,__m)) ))
 845     if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(__n)
 846     else:                         Pn = B
 847     if hasattr(R,"asfullmatrix"): Rn = R.asfullmatrix(__p)
 848     else:                         Rn = R
 849     if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
 850     else:                         Qn = Q
 851     #
 852     if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
 853         selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
 854         if selfA._toStore("APosterioriCovariance"):
 855             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
 856             covarianceXa = Pn
 857     #
 858     previousJMinimum = numpy.finfo(float).max
 859     #
 860     # Predimensionnement
 861     Xn_predicted = numpy.asmatrix(numpy.zeros((__n,__m)))
 862     HX_predicted = numpy.asmatrix(numpy.zeros((__p,__m)))
 863     #
 864     for step in range(duration-1):
 865         if hasattr(Y,"store"):
 866             Ynpu = numpy.asmatrix(numpy.ravel( Y[step+1] )).T
 867         else:
 868             Ynpu = numpy.asmatrix(numpy.ravel( Y )).T
 869         #
 870         if U is not None:
 871             if hasattr(U,"store") and len(U)>1:
 872                 Un = numpy.asmatrix(numpy.ravel( U[step] )).T
 873             elif hasattr(U,"store") and len(U)==1:
 874                 Un = numpy.asmatrix(numpy.ravel( U[0] )).T
 875             else:
 876                 Un = numpy.asmatrix(numpy.ravel( U )).T
 877         else:
 878             Un = None
 879         #
 880         if selfA._parameters["InflationType"] == "MultiplicativeOnBackgroundAnomalies":
 881             Xn = CovarianceInflation( Xn,
 882                 selfA._parameters["InflationType"],
 883                 selfA._parameters["InflationFactor"],
 884                 )
 885         #
 886         if selfA._parameters["EstimationOf"] == "State": # Forecast + Q and observation of forecast
 887             EMX = M( [(Xn[:,i], Un) for i in range(__m)], argsAsSerie = True )
 888             for i in range(__m):
 889                 qi = numpy.asmatrix(numpy.random.multivariate_normal(numpy.zeros(__n), Qn, (1,1,1))).T
 890                 Xn_predicted[:,i] = numpy.asmatrix(numpy.ravel( EMX[i] )).T + qi
 891             EHX = H( [(Xn_predicted[:,i], Un) for i in range(__m)], argsAsSerie = True )
 892             for i in range(__m):
 893                 HX_predicted[:,i] = numpy.asmatrix(numpy.ravel( EHX[i] )).T
 894             if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
 895                 Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
 896                 Xn_predicted = Xn_predicted + Cm * Un
 897         elif selfA._parameters["EstimationOf"] == "Parameters": # Observation of forecast
 898             # --- > Par principe, M = Id, Q = 0
 899             Xn_predicted = Xn
 900             EHX = H( [(Xn_predicted[:,i], Un) for i in range(__m)], argsAsSerie = True )
 901             for i in range(__m):
 902                 HX_predicted[:,i] = numpy.asmatrix(numpy.ravel( EHX[i] )).T
 903         #
 904         # Mean of forecast and observation of forecast
 905         Xfm  = Xn_predicted.mean(axis=1, dtype=mfp).astype('float')
 906         Hfm  = HX_predicted.mean(axis=1, dtype=mfp).astype('float')
 907         #
 908         EaX   = (Xn_predicted - Xfm.reshape((__n,-1)))
 909         EaHX  = (HX_predicted - Hfm.reshape((__p,-1)))
 910         #
 911         #--------------------------
 912         if KorV == "KalmanFilterFormula":
 913             EaX    = EaX / numpy.sqrt(__m-1)
 914             mS    = RIdemi * EaHX / numpy.sqrt(__m-1)
 915             delta = RIdemi * ( Ynpu.reshape((__p,-1)) - Hfm.reshape((__p,-1)) )
 916             mT    = numpy.linalg.inv( numpy.eye(__m) + mS.T @ mS )
 917             vw    = mT @ mS.transpose() @ delta
 918             #
 919             Tdemi = numpy.real(scipy.linalg.sqrtm(mT))
 920             mU    = numpy.eye(__m)
 921             #
 922             Xn = Xfm.reshape((__n,-1)) + EaX @ ( vw.reshape((__m,-1)) + numpy.sqrt(__m-1) * Tdemi @ mU )
 923         #--------------------------
 924         elif KorV == "Variational":
 925             HXfm = H((Xfm, Un)) # Eventuellement Hfm
 926             def CostFunction(w):
 927                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 928                 _Jo = 0.5 * _A.T * RI * _A
 929                 _Jb = 0.5 * (__m-1) * w.T @ w
 930                 _J  = _Jo + _Jb
 931                 return float(_J)
 932             def GradientOfCostFunction(w):
 933                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 934                 _GardJo = - EaHX.T * RI * _A
 935                 _GradJb = (__m-1) * w.reshape((__m,1))
 936                 _GradJ  = _GardJo + _GradJb
 937                 return numpy.ravel(_GradJ)
 938             vw = scipy.optimize.fmin_cg(
 939                 f           = CostFunction,
 940                 x0          = numpy.zeros(__m),
 941                 fprime      = GradientOfCostFunction,
 942                 args        = (),
 943                 disp        = False,
 944                 )
 945             #
 946             Hto = EaHX.T * RI * EaHX
 947             Htb = (__m-1) * numpy.eye(__m)
 948             Hta = Hto + Htb
 949             #
 950             Pta = numpy.linalg.inv( Hta )
 951             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
 952             #
 953             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
 954         #--------------------------
 955         elif KorV == "FiniteSize11": # Jauge Boc2011
 956             HXfm = H((Xfm, Un)) # Eventuellement Hfm
 957             def CostFunction(w):
 958                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 959                 _Jo = 0.5 * _A.T * RI * _A
 960                 _Jb = 0.5 * __m * math.log(1 + 1/__m + w.T @ w)
 961                 _J  = _Jo + _Jb
 962                 return float(_J)
 963             def GradientOfCostFunction(w):
 964                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 965                 _GardJo = - EaHX.T * RI * _A
 966                 _GradJb = __m * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
 967                 _GradJ  = _GardJo + _GradJb
 968                 return numpy.ravel(_GradJ)
 969             vw = scipy.optimize.fmin_cg(
 970                 f           = CostFunction,
 971                 x0          = numpy.zeros(__m),
 972                 fprime      = GradientOfCostFunction,
 973                 args        = (),
 974                 disp        = False,
 975                 )
 976             #
 977             Hto = EaHX.T * RI * EaHX
 978             Htb = __m * \
 979                 ( (1 + 1/__m + vw.T @ vw) * numpy.eye(__m) - 2 * vw @ vw.T ) \
 980                 / (1 + 1/__m + vw.T @ vw)**2
 981             Hta = Hto + Htb
 982             #
 983             Pta = numpy.linalg.inv( Hta )
 984             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
 985             #
 986             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
 987         #--------------------------
 988         elif KorV == "FiniteSize15": # Jauge Boc2015
 989             HXfm = H((Xfm, Un)) # Eventuellement Hfm
 990             def CostFunction(w):
 991                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 992                 _Jo = 0.5 * _A.T * RI * _A
 993                 _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w)
 994                 _J  = _Jo + _Jb
 995                 return float(_J)
 996             def GradientOfCostFunction(w):
 997                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
 998                 _GardJo = - EaHX.T * RI * _A
 999                 _GradJb = (__m+1) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
1000                 _GradJ  = _GardJo + _GradJb
1001                 return numpy.ravel(_GradJ)
1002             vw = scipy.optimize.fmin_cg(
1003                 f           = CostFunction,
1004                 x0          = numpy.zeros(__m),
1005                 fprime      = GradientOfCostFunction,
1006                 args        = (),
1007                 disp        = False,
1008                 )
1009             #
1010             Hto = EaHX.T * RI * EaHX
1011             Htb = (__m+1) * \
1012                 ( (1 + 1/__m + vw.T @ vw) * numpy.eye(__m) - 2 * vw @ vw.T ) \
1013                 / (1 + 1/__m + vw.T @ vw)**2
1014             Hta = Hto + Htb
1015             #
1016             Pta = numpy.linalg.inv( Hta )
1017             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
1018             #
1019             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
1020         #--------------------------
1021         elif KorV == "FiniteSize16": # Jauge Boc2016
1022             HXfm = H((Xfm, Un)) # Eventuellement Hfm
1023             def CostFunction(w):
1024                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1025                 _Jo = 0.5 * _A.T * RI * _A
1026                 _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w / (__m-1))
1027                 _J  = _Jo + _Jb
1028                 return float(_J)
1029             def GradientOfCostFunction(w):
1030                 _A  = Ynpu.reshape((__p,-1)) - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
1031                 _GardJo = - EaHX.T * RI * _A
1032                 _GradJb = ((__m+1) / (__m-1)) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w / (__m-1))
1033                 _GradJ  = _GardJo + _GradJb
1034                 return numpy.ravel(_GradJ)
1035             vw = scipy.optimize.fmin_cg(
1036                 f           = CostFunction,
1037                 x0          = numpy.zeros(__m),
1038                 fprime      = GradientOfCostFunction,
1039                 args        = (),
1040                 disp        = False,
1041                 )
1042             #
1043             Hto = EaHX.T * RI * EaHX
1044             Htb = ((__m+1) / (__m-1)) * \
1045                 ( (1 + 1/__m + vw.T @ vw / (__m-1)) * numpy.eye(__m) - 2 * vw @ vw.T / (__m-1) ) \
1046                 / (1 + 1/__m + vw.T @ vw / (__m-1))**2
1047             Hta = Hto + Htb
1048             #
1049             Pta = numpy.linalg.inv( Hta )
1050             EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
1051             #
1052             Xn = Xfm.reshape((__n,-1)) + EaX @ (vw.reshape((__m,-1)) + EWa)
1053         #--------------------------
1054         else:
1055             raise ValueError("KorV has to be chosen in the authorized methods list.")
1056         #
1057         if selfA._parameters["InflationType"] == "MultiplicativeOnAnalysisAnomalies":
1058             Xn = CovarianceInflation( Xn,
1059                 selfA._parameters["InflationType"],
1060                 selfA._parameters["InflationFactor"],
1061                 )
1062         #
1063         Xa = Xn.mean(axis=1, dtype=mfp).astype('float')
1064         #--------------------------
1065         #
1066         if selfA._parameters["StoreInternalVariables"] \
1067             or selfA._toStore("CostFunctionJ") \
1068             or selfA._toStore("CostFunctionJb") \
1069             or selfA._toStore("CostFunctionJo") \
1070             or selfA._toStore("APosterioriCovariance") \
1071             or selfA._toStore("InnovationAtCurrentAnalysis") \
1072             or selfA._toStore("SimulatedObservationAtCurrentAnalysis") \
1073             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1074             _HXa = numpy.asmatrix(numpy.ravel( H((Xa, Un)) )).T
1075             _Innovation = Ynpu - _HXa
1076         #
1077         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
1078         # ---> avec analysis
1079         selfA.StoredVariables["Analysis"].store( Xa )
1080         if selfA._toStore("SimulatedObservationAtCurrentAnalysis"):
1081             selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"].store( _HXa )
1082         if selfA._toStore("InnovationAtCurrentAnalysis"):
1083             selfA.StoredVariables["InnovationAtCurrentAnalysis"].store( _Innovation )
1084         # ---> avec current state
1085         if selfA._parameters["StoreInternalVariables"] \
1086             or selfA._toStore("CurrentState"):
1087             selfA.StoredVariables["CurrentState"].store( Xn )
1088         if selfA._toStore("ForecastState"):
1089             selfA.StoredVariables["ForecastState"].store( Xn_predicted )
1090         if selfA._toStore("BMA"):
1091             selfA.StoredVariables["BMA"].store( Xn_predicted - Xa )
1092         if selfA._toStore("InnovationAtCurrentState"):
1093             selfA.StoredVariables["InnovationAtCurrentState"].store( - HX_predicted + Ynpu )
1094         if selfA._toStore("SimulatedObservationAtCurrentState") \
1095             or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1096             selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HX_predicted )
1097         # ---> autres
1098         if selfA._parameters["StoreInternalVariables"] \
1099             or selfA._toStore("CostFunctionJ") \
1100             or selfA._toStore("CostFunctionJb") \
1101             or selfA._toStore("CostFunctionJo") \
1102             or selfA._toStore("CurrentOptimum") \
1103             or selfA._toStore("APosterioriCovariance"):
1104             Jb  = float( 0.5 * (Xa - Xb).T * BI * (Xa - Xb) )
1105             Jo  = float( 0.5 * _Innovation.T * RI * _Innovation )
1106             J   = Jb + Jo
1107             selfA.StoredVariables["CostFunctionJb"].store( Jb )
1108             selfA.StoredVariables["CostFunctionJo"].store( Jo )
1109             selfA.StoredVariables["CostFunctionJ" ].store( J )
1110             #
1111             if selfA._toStore("IndexOfOptimum") \
1112                 or selfA._toStore("CurrentOptimum") \
1113                 or selfA._toStore("CostFunctionJAtCurrentOptimum") \
1114                 or selfA._toStore("CostFunctionJbAtCurrentOptimum") \
1115                 or selfA._toStore("CostFunctionJoAtCurrentOptimum") \
1116                 or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1117                 IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
1118             if selfA._toStore("IndexOfOptimum"):
1119                 selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
1120             if selfA._toStore("CurrentOptimum"):
1121                 selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["Analysis"][IndexMin] )
1122             if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
1123                 selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentAnalysis"][IndexMin] )
1124             if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
1125                 selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
1126             if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
1127                 selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
1128             if selfA._toStore("CostFunctionJAtCurrentOptimum"):
1129                 selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
1130         if selfA._toStore("APosterioriCovariance"):
1131             Eai = (1/numpy.sqrt(__m-1)) * (Xn - Xa.reshape((__n,-1))) # Anomalies
1132             Pn = Eai @ Eai.T
1133             Pn = 0.5 * (Pn + Pn.T)
1134             selfA.StoredVariables["APosterioriCovariance"].store( Pn )
1135         if selfA._parameters["EstimationOf"] == "Parameters" \
1136             and J < previousJMinimum:
1137             previousJMinimum    = J
1138             XaMin               = Xa
1139             if selfA._toStore("APosterioriCovariance"):
1140                 covarianceXaMin = Pn
1141     #
1142     # Stockage final supplémentaire de l'optimum en estimation de paramètres
1143     # ----------------------------------------------------------------------
1144     if selfA._parameters["EstimationOf"] == "Parameters":
1145         selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
1146         selfA.StoredVariables["Analysis"].store( XaMin )
1147         if selfA._toStore("APosterioriCovariance"):
1148             selfA.StoredVariables["APosterioriCovariance"].store( covarianceXaMin )
1149         if selfA._toStore("BMA"):
1150             selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(XaMin) )
1151     #
1152     return 0
1153
1154 # ==============================================================================
1155 if __name__ == "__main__":
1156     print('\n AUTODIAGNOSTIC\n')