__author__ = "Jean-Philippe ARGAUD"
import os, time, copy, types, sys, logging
-import math, numpy, scipy, scipy.optimize
+import math, numpy, scipy, scipy.optimize, scipy.version
from daCore.BasicObjects import Operator
from daCore.PlatformInfo import PlatformInfo
mpr = PlatformInfo().MachinePrecision()
# logging.getLogger().setLevel(logging.DEBUG)
# ==============================================================================
-def ExecuteFunction( paire ):
- assert len(paire) == 2, "Incorrect number of arguments"
- X, funcrepr = paire
+def ExecuteFunction( triplet ):
+ assert len(triplet) == 3, "Incorrect number of arguments"
+ X, xArgs, funcrepr = triplet
__X = numpy.asmatrix(numpy.ravel( X )).T
__sys_path_tmp = sys.path ; sys.path.insert(0,funcrepr["__userFunction__path"])
__module = __import__(funcrepr["__userFunction__modl"], globals(), locals(), [])
__fonction = getattr(__module,funcrepr["__userFunction__name"])
sys.path = __sys_path_tmp ; del __sys_path_tmp
- __HX = __fonction( __X )
+ if isinstance(xArgs, dict):
+ __HX = __fonction( __X, **xArgs )
+ else:
+ __HX = __fonction( __X )
return numpy.ravel( __HX )
# ==============================================================================
centeredDF = False,
increment = 0.01,
dX = None,
+ extraArguments = None,
avoidingRedundancy = True,
toleranceInRedundancy = 1.e-18,
lenghtOfRedundancy = -1,
mfEnabled = False,
):
self.__name = str(name)
+ self.__extraArgs = extraArguments
if mpEnabled:
try:
import multiprocessing
self.__userFunction__modl = os.path.basename(mod).replace('.pyc','').replace('.pyo','').replace('.py','')
self.__userFunction__path = os.path.dirname(mod)
del mod
- self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled )
+ self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled, extraArguments = self.__extraArgs )
self.__userFunction = self.__userOperator.appliedTo # Pour le calcul Direct
elif isinstance(Function,types.MethodType):
logging.debug("FDA Calculs en multiprocessing : MethodType")
self.__userFunction__modl = os.path.basename(mod).replace('.pyc','').replace('.pyo','').replace('.py','')
self.__userFunction__path = os.path.dirname(mod)
del mod
- self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled )
+ self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled, extraArguments = self.__extraArgs )
self.__userFunction = self.__userOperator.appliedTo # Pour le calcul Direct
else:
raise TypeError("User defined function or method has to be provided for finite differences approximation.")
else:
- self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled )
+ self.__userOperator = Operator( name = self.__name, fromMethod = Function, avoidingRedundancy = self.__avoidRC, inputAsMultiFunction = self.__mfEnabled, extraArguments = self.__extraArgs )
self.__userFunction = self.__userOperator.appliedTo
#
self.__centeredDF = bool(centeredDF)
return __ac, __iac
# ---------------------------------------------------------
- def DirectOperator(self, X ):
+ def DirectOperator(self, X, **extraArgs ):
"""
Calcul du direct à l'aide de la fonction fournie.
+
+ NB : les extraArgs sont là pour assurer la compatibilité d'appel, mais
+ ne doivent pas être données ici à la fonction utilisateur.
"""
logging.debug("FDA Calcul DirectOperator (explicite)")
if self.__mfEnabled:
_X_moins_dXi = numpy.array( _X.A1, dtype=float )
_X_moins_dXi[i] = _X[i] - _dXi
#
- _jobs.append( (_X_plus_dXi, funcrepr) )
- _jobs.append( (_X_moins_dXi, funcrepr) )
+ _jobs.append( (_X_plus_dXi, self.__extraArgs, funcrepr) )
+ _jobs.append( (_X_moins_dXi, self.__extraArgs, funcrepr) )
#
import multiprocessing
self.__pool = multiprocessing.Pool(self.__mpWorkers)
"__userFunction__name" : self.__userFunction__name,
}
_jobs = []
- _jobs.append( (_X.A1, funcrepr) )
+ _jobs.append( (_X.A1, self.__extraArgs, funcrepr) )
for i in range( len(_dX) ):
_X_plus_dXi = numpy.array( _X.A1, dtype=float )
_X_plus_dXi[i] = _X[i] + _dX[i]
#
- _jobs.append( (_X_plus_dXi, funcrepr) )
+ _jobs.append( (_X_plus_dXi, self.__extraArgs, funcrepr) )
#
import multiprocessing
self.__pool = multiprocessing.Pool(self.__mpWorkers)
return _Jacobienne
# ---------------------------------------------------------
- def TangentOperator(self, paire ):
+ def TangentOperator(self, paire, **extraArgs ):
"""
Calcul du tangent à l'aide de la Jacobienne.
+
+ NB : les extraArgs sont là pour assurer la compatibilité d'appel, mais
+ ne doivent pas être données ici à la fonction utilisateur.
"""
if self.__mfEnabled:
assert len(paire) == 1, "Incorrect lenght of arguments"
else: return _HtX.A1
# ---------------------------------------------------------
- def AdjointOperator(self, paire ):
+ def AdjointOperator(self, paire, **extraArgs ):
"""
Calcul de l'adjoint à l'aide de la Jacobienne.
+
+ NB : les extraArgs sont là pour assurer la compatibilité d'appel, mais
+ ne doivent pas être données ici à la fonction utilisateur.
"""
if self.__mfEnabled:
assert len(paire) == 1, "Incorrect lenght of arguments"
if self.__mfEnabled: return [_HaY.A1,]
else: return _HaY.A1
-# ==============================================================================
-def mmqr(
- func = None,
- x0 = None,
- fprime = None,
- bounds = None,
- quantile = 0.5,
- maxfun = 15000,
- toler = 1.e-06,
- y = None,
- ):
- """
- Implémentation informatique de l'algorithme MMQR, basée sur la publication :
- David R. Hunter, Kenneth Lange, "Quantile Regression via an MM Algorithm",
- Journal of Computational and Graphical Statistics, 9, 1, pp.60-77, 2000.
- """
- #
- # Recuperation des donnees et informations initiales
- # --------------------------------------------------
- variables = numpy.ravel( x0 )
- mesures = numpy.ravel( y )
- increment = sys.float_info[0]
- p = variables.size
- n = mesures.size
- quantile = float(quantile)
- #
- # Calcul des parametres du MM
- # ---------------------------
- tn = float(toler) / n
- e0 = -tn / math.log(tn)
- epsilon = (e0-tn)/(1+math.log(e0))
- #
- # Calculs d'initialisation
- # ------------------------
- residus = mesures - numpy.ravel( func( variables ) )
- poids = 1./(epsilon+numpy.abs(residus))
- veps = 1. - 2. * quantile - residus * poids
- lastsurrogate = -numpy.sum(residus*veps) - (1.-2.*quantile)*numpy.sum(residus)
- iteration = 0
- #
- # Recherche iterative
- # -------------------
- while (increment > toler) and (iteration < maxfun) :
- iteration += 1
- #
- Derivees = numpy.array(fprime(variables))
- Derivees = Derivees.reshape(n,p) # Necessaire pour remettre en place la matrice si elle passe par des tuyaux YACS
- DeriveesT = Derivees.transpose()
- M = numpy.dot( DeriveesT , (numpy.array(numpy.matrix(p*[poids,]).T)*Derivees) )
- SM = numpy.transpose(numpy.dot( DeriveesT , veps ))
- step = - numpy.linalg.lstsq( M, SM, rcond=-1 )[0]
- #
- variables = variables + step
- if bounds is not None:
- # Attention : boucle infinie à éviter si un intervalle est trop petit
- while( (variables < numpy.ravel(numpy.asmatrix(bounds)[:,0])).any() or (variables > numpy.ravel(numpy.asmatrix(bounds)[:,1])).any() ):
- step = step/2.
- variables = variables - step
- residus = mesures - numpy.ravel( func(variables) )
- surrogate = numpy.sum(residus**2 * poids) + (4.*quantile-2.) * numpy.sum(residus)
- #
- while ( (surrogate > lastsurrogate) and ( max(list(numpy.abs(step))) > 1.e-16 ) ) :
- step = step/2.
- variables = variables - step
- residus = mesures - numpy.ravel( func(variables) )
- surrogate = numpy.sum(residus**2 * poids) + (4.*quantile-2.) * numpy.sum(residus)
- #
- increment = lastsurrogate-surrogate
- poids = 1./(epsilon+numpy.abs(residus))
- veps = 1. - 2. * quantile - residus * poids
- lastsurrogate = -numpy.sum(residus * veps) - (1.-2.*quantile)*numpy.sum(residus)
- #
- # Mesure d'écart
- # --------------
- Ecart = quantile * numpy.sum(residus) - numpy.sum( residus[residus<0] )
- #
- return variables, Ecart, [n,p,iteration,increment,0]
-
# ==============================================================================
def EnsembleOfCenteredPerturbations( _bgcenter, _bgcovariance, _nbmembers ):
"Génération d'un ensemble de taille _nbmembers-1 d'états aléatoires centrés"
notes manuscrites de MB et conforme au code de PS avec eps = -1
"""
eps = -1
- Q = numpy.eye(N-1)-numpy.ones((N-1,N-1))/numpy.sqrt(N)/(numpy.sqrt(N)-eps)
+ Q = numpy.identity(N-1)-numpy.ones((N-1,N-1))/numpy.sqrt(N)/(numpy.sqrt(N)-eps)
Q = numpy.concatenate((Q, [eps*numpy.ones(N-1)/numpy.sqrt(N)]), axis=0)
R, _ = numpy.linalg.qr(numpy.random.normal(size = (N-1,N-1)))
Q = numpy.dot(Q,R)
Zr = numpy.dot(Q,Zr)
return Zr.T
#
- _bgcenter = numpy.ravel(_bgcenter)[:,None]
+ _bgcenter = numpy.ravel(_bgcenter).reshape((-1,1))
if _nbmembers < 1:
raise ValueError("Number of members has to be strictly more than 1 (given number: %s)."%(str(_nbmembers),))
if _bgcovariance is None:
return BackgroundEnsemble
# ==============================================================================
-def EnsembleOfAnomalies( _ensemble, _optmean = None):
+def EnsembleOfAnomalies( Ensemble, OptMean = None, Normalisation = 1.):
"Renvoie les anomalies centrées à partir d'un ensemble TailleEtat*NbMembres"
- if _optmean is None:
- Em = numpy.asarray(_ensemble).mean(axis=1, dtype=mfp).astype('float')[:,numpy.newaxis]
+ if OptMean is None:
+ __Em = numpy.asarray(Ensemble).mean(axis=1, dtype=mfp).astype('float').reshape((-1,1))
else:
- Em = numpy.ravel(_optmean)[:,numpy.newaxis]
+ __Em = numpy.ravel(OptMean).reshape((-1,1))
+ #
+ return Normalisation * (numpy.asarray(Ensemble) - __Em)
+
+# ==============================================================================
+def EnsembleErrorCovariance( Ensemble ):
+ "Renvoie la covariance d'ensemble"
+ __Anomalies = EnsembleOfAnomalies( Ensemble )
+ __n, __m = numpy.asarray(__Anomalies).shape
+ # Estimation empirique
+ __Covariance = (__Anomalies @ __Anomalies.T) / (__m-1)
+ # Assure la symétrie
+ __Covariance = (__Covariance + __Covariance.T) * 0.5
+ # Assure la positivité
+ __epsilon = mpr*numpy.trace(__Covariance)
+ __Covariance = __Covariance + __epsilon * numpy.identity(__n)
#
- return numpy.asarray(_ensemble) - Em
+ return __Covariance
# ==============================================================================
def CovarianceInflation(
__n, __m = numpy.asarray(InputCovOrEns).shape
if __n != __m:
raise ValueError("Additive inflation can only be applied to squared (covariance) matrix.")
- OutputCovOrEns = (1. - InflationFactor) * InputCovOrEns + InflationFactor * numpy.eye(__n)
+ OutputCovOrEns = (1. - InflationFactor) * InputCovOrEns + InflationFactor * numpy.identity(__n)
#
elif InflationType == "HybridOnBackgroundCovariance":
if InflationFactor < 0.:
return OutputCovOrEns
# ==============================================================================
-def senkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="KalmanFilterFormula"):
+def enks(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="EnKS16-KalmanFilterFormula"):
+ """
+ EnKS
"""
- Stochastic EnKF (Envensen 1994, Burgers 1998)
+ #
+ # Initialisations
+ # ---------------
+ #
+ # Opérateurs
+ H = HO["Direct"].appliedControledFormTo
+ #
+ if selfA._parameters["EstimationOf"] == "State":
+ M = EM["Direct"].appliedControledFormTo
+ #
+ if CM is not None and "Tangent" in CM and U is not None:
+ Cm = CM["Tangent"].asMatrix(Xb)
+ else:
+ Cm = None
+ #
+ # Précalcul des inversions de B et R
+ RIdemi = R.sqrtmI()
+ #
+ LagL = selfA._parameters["SmootherLagL"]
+ if (not hasattr(Y,"store")) or (not hasattr(Y,"stepnumber")):
+ raise ValueError("Fixed-lag smoother requires a series of observation")
+ if Y.stepnumber() < LagL:
+ raise ValueError("Fixed-lag smoother requires a series of observation greater then the lag L")
+ duration = Y.stepnumber()
+ __p = numpy.cumprod(Y.shape())[-1]
+ __n = Xb.size
+ __m = selfA._parameters["NumberOfMembers"]
+ #
+ if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(__n)
+ else: Pn = B
+ if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
+ else: Qn = Q
+ if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
+ selfA.StoredVariables["Analysis"].store( Xb )
+ if selfA._toStore("APosterioriCovariance"):
+ selfA.StoredVariables["APosterioriCovariance"].store( Pn )
+ covarianceXa = Pn
+ #
+ # Calcul direct initial (on privilégie la mémorisation au recalcul)
+ __seed = numpy.random.get_state()
+ selfB = copy.deepcopy(selfA)
+ selfB._parameters["StoreSupplementaryCalculations"] = ["CurrentEnsembleState"]
+ if VariantM == "EnKS16-KalmanFilterFormula":
+ etkf(selfB, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM = "KalmanFilterFormula")
+ else:
+ raise ValueError("VariantM has to be chosen in the authorized methods list.")
+ if LagL > 0:
+ EL = selfB.StoredVariables["CurrentEnsembleState"][LagL-1]
+ else:
+ EL = EnsembleOfBackgroundPerturbations( Xb, None, __m ) # Cf. etkf
+ selfA._parameters["SetSeed"] = numpy.random.set_state(__seed)
+ #
+ for step in range(LagL,duration-1):
+ #
+ sEL = selfB.StoredVariables["CurrentEnsembleState"][step+1-LagL:step+1]
+ sEL.append(None)
+ #
+ if hasattr(Y,"store"):
+ Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,1))
+ else:
+ Ynpu = numpy.ravel( Y ).reshape((__p,1))
+ #
+ if U is not None:
+ if hasattr(U,"store") and len(U)>1:
+ Un = numpy.asmatrix(numpy.ravel( U[step] )).T
+ elif hasattr(U,"store") and len(U)==1:
+ Un = numpy.asmatrix(numpy.ravel( U[0] )).T
+ else:
+ Un = numpy.asmatrix(numpy.ravel( U )).T
+ else:
+ Un = None
+ #
+ #--------------------------
+ if VariantM == "EnKS16-KalmanFilterFormula":
+ if selfA._parameters["EstimationOf"] == "State": # Forecast
+ EL = M( [(EL[:,i], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ EL = EL + numpy.random.multivariate_normal(numpy.zeros(__n), Qn, size=__m).T
+ EZ = H( [(EL[:,i], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
+ Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
+ EZ = EZ + Cm * Un
+ elif selfA._parameters["EstimationOf"] == "Parameters":
+ # --- > Par principe, M = Id, Q = 0
+ EZ = H( [(EL[:,i], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ #
+ vEm = EL.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ vZm = EZ.mean(axis=1, dtype=mfp).astype('float').reshape((__p,1))
+ #
+ mS = RIdemi @ EnsembleOfAnomalies( EZ, vZm, 1./math.sqrt(__m-1) )
+ delta = RIdemi @ ( Ynpu - vZm )
+ mT = numpy.linalg.inv( numpy.identity(__m) + mS.T @ mS )
+ vw = mT @ mS.T @ delta
+ #
+ Tdemi = numpy.real(scipy.linalg.sqrtm(mT))
+ mU = numpy.identity(__m)
+ wTU = (vw.reshape((__m,1)) + math.sqrt(__m-1) * Tdemi @ mU)
+ #
+ EX = EnsembleOfAnomalies( EL, vEm, 1./math.sqrt(__m-1) )
+ EL = vEm + EX @ wTU
+ #
+ sEL[LagL] = EL
+ for irl in range(LagL): # Lissage des L précédentes analysis
+ vEm = sEL[irl].mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ EX = EnsembleOfAnomalies( sEL[irl], vEm, 1./math.sqrt(__m-1) )
+ sEL[irl] = vEm + EX @ wTU
+ #
+ # Conservation de l'analyse retrospective d'ordre 0 avant rotation
+ Xa = sEL[0].mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ if selfA._toStore("APosterioriCovariance"):
+ EXn = sEL[0]
+ #
+ for irl in range(LagL):
+ sEL[irl] = sEL[irl+1]
+ sEL[LagL] = None
+ #--------------------------
+ else:
+ raise ValueError("VariantM has to be chosen in the authorized methods list.")
+ #
+ selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
+ selfA.StoredVariables["Analysis"].store( Xa )
+ if selfA._toStore("APosterioriCovariance"):
+ selfA.StoredVariables["APosterioriCovariance"].store( EnsembleErrorCovariance(EXn) )
+ #
+ # Stockage des dernières analyses incomplètement remises à jour
+ for irl in range(LagL):
+ selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["Analysis"]) )
+ Xa = sEL[irl].mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ selfA.StoredVariables["Analysis"].store( Xa )
+ #
+ return 0
- selfA est identique au "self" d'algorithme appelant et contient les
- valeurs.
+# ==============================================================================
+def etkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="KalmanFilterFormula"):
+ """
+ Ensemble-Transform EnKF
"""
if selfA._parameters["EstimationOf"] == "Parameters":
selfA._parameters["StoreInternalVariables"] = True
or selfA._toStore("APosterioriCovariance"):
BI = B.getI()
RI = R.getI()
+ elif VariantM != "KalmanFilterFormula":
+ RI = R.getI()
+ if VariantM == "KalmanFilterFormula":
+ RIdemi = R.sqrtmI()
#
# Initialisation
# --------------
__m = selfA._parameters["NumberOfMembers"]
if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(__n)
else: Pn = B
- if hasattr(R,"asfullmatrix"): Rn = R.asfullmatrix(__p)
- else: Rn = R
if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
else: Qn = Q
Xn = EnsembleOfBackgroundPerturbations( Xb, None, __m )
+ #~ Xn = EnsembleOfBackgroundPerturbations( Xb, Pn, __m )
#
if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
- selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
+ selfA.StoredVariables["Analysis"].store( Xb )
if selfA._toStore("APosterioriCovariance"):
selfA.StoredVariables["APosterioriCovariance"].store( Pn )
covarianceXa = Pn
#
for step in range(duration-1):
if hasattr(Y,"store"):
- Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,1))
else:
- Ynpu = numpy.ravel( Y ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y ).reshape((__p,1))
#
if U is not None:
if hasattr(U,"store") and len(U)>1:
returnSerieAsArrayMatrix = True )
#
# Mean of forecast and observation of forecast
- Xfm = Xn_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
- Hfm = HX_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__p,-1))
+ Xfm = Xn_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ Hfm = HX_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__p,1))
+ #
+ # Anomalies
+ EaX = EnsembleOfAnomalies( Xn_predicted, Xfm )
+ EaHX = EnsembleOfAnomalies( HX_predicted, Hfm)
#
#--------------------------
- if VariantM == "KalmanFilterFormula05":
- PfHT, HPfHT = 0., 0.
- for i in range(__m):
- Exfi = Xn_predicted[:,i].reshape((__n,-1)) - Xfm
- Eyfi = HX_predicted[:,i].reshape((__p,-1)) - Hfm
- PfHT += Exfi * Eyfi.T
- HPfHT += Eyfi * Eyfi.T
- PfHT = (1./(__m-1)) * PfHT
- HPfHT = (1./(__m-1)) * HPfHT
- Kn = PfHT * ( R + HPfHT ).I
- del PfHT, HPfHT
+ if VariantM == "KalmanFilterFormula":
+ mS = RIdemi * EaHX / math.sqrt(__m-1)
+ delta = RIdemi * ( Ynpu - Hfm )
+ mT = numpy.linalg.inv( numpy.identity(__m) + mS.T @ mS )
+ vw = mT @ mS.T @ delta
#
- for i in range(__m):
- ri = numpy.random.multivariate_normal(numpy.zeros(__p), Rn)
- Xn[:,i] = numpy.ravel(Xn_predicted[:,i]) + Kn @ (numpy.ravel(Ynpu) + ri - HX_predicted[:,i])
+ Tdemi = numpy.real(scipy.linalg.sqrtm(mT))
+ mU = numpy.identity(__m)
+ #
+ EaX = EaX / math.sqrt(__m-1)
+ Xn = Xfm + EaX @ ( vw.reshape((__m,1)) + math.sqrt(__m-1) * Tdemi @ mU )
#--------------------------
- elif VariantM == "KalmanFilterFormula16":
- EpY = EnsembleOfCenteredPerturbations(Ynpu, Rn, __m)
- EpYm = EpY.mean(axis=1, dtype=mfp).astype('float').reshape((__p,-1))
+ elif VariantM == "Variational":
+ HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
+ def CostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _Jo = 0.5 * _A.T @ (RI * _A)
+ _Jb = 0.5 * (__m-1) * w.T @ w
+ _J = _Jo + _Jb
+ return float(_J)
+ def GradientOfCostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _GardJo = - EaHX.T @ (RI * _A)
+ _GradJb = (__m-1) * w.reshape((__m,1))
+ _GradJ = _GardJo + _GradJb
+ return numpy.ravel(_GradJ)
+ vw = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = numpy.zeros(__m),
+ fprime = GradientOfCostFunction,
+ args = (),
+ disp = False,
+ )
#
- EaX = EnsembleOfAnomalies( Xn_predicted ) / numpy.sqrt(__m-1)
- EaY = (HX_predicted - Hfm - EpY + EpYm) / numpy.sqrt(__m-1)
+ Hto = EaHX.T @ (RI * EaHX)
+ Htb = (__m-1) * numpy.identity(__m)
+ Hta = Hto + Htb
#
- Kn = EaX @ EaY.T @ numpy.linalg.inv( EaY @ EaY.T)
+ Pta = numpy.linalg.inv( Hta )
+ EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
#
- for i in range(__m):
- Xn[:,i] = numpy.ravel(Xn_predicted[:,i]) + Kn @ (numpy.ravel(EpY[:,i]) - HX_predicted[:,i])
+ Xn = Xfm + EaX @ (vw[:,None] + EWa)
#--------------------------
- else:
- raise ValueError("VariantM has to be chosen in the authorized methods list.")
- #
- if selfA._parameters["InflationType"] == "MultiplicativeOnAnalysisAnomalies":
- Xn = CovarianceInflation( Xn,
- selfA._parameters["InflationType"],
- selfA._parameters["InflationFactor"],
+ elif VariantM == "FiniteSize11": # Jauge Boc2011
+ HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
+ def CostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _Jo = 0.5 * _A.T @ (RI * _A)
+ _Jb = 0.5 * __m * math.log(1 + 1/__m + w.T @ w)
+ _J = _Jo + _Jb
+ return float(_J)
+ def GradientOfCostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _GardJo = - EaHX.T @ (RI * _A)
+ _GradJb = __m * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
+ _GradJ = _GardJo + _GradJb
+ return numpy.ravel(_GradJ)
+ vw = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = numpy.zeros(__m),
+ fprime = GradientOfCostFunction,
+ args = (),
+ disp = False,
)
- #
- Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
+ #
+ Hto = EaHX.T @ (RI * EaHX)
+ Htb = __m * \
+ ( (1 + 1/__m + vw.T @ vw) * numpy.identity(__m) - 2 * vw @ vw.T ) \
+ / (1 + 1/__m + vw.T @ vw)**2
+ Hta = Hto + Htb
+ #
+ Pta = numpy.linalg.inv( Hta )
+ EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
+ #
+ Xn = Xfm + EaX @ (vw.reshape((__m,1)) + EWa)
#--------------------------
- #
- if selfA._parameters["StoreInternalVariables"] \
- or selfA._toStore("CostFunctionJ") \
- or selfA._toStore("CostFunctionJb") \
+ elif VariantM == "FiniteSize15": # Jauge Boc2015
+ HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
+ def CostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _Jo = 0.5 * _A.T * RI * _A
+ _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w)
+ _J = _Jo + _Jb
+ return float(_J)
+ def GradientOfCostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _GardJo = - EaHX.T @ (RI * _A)
+ _GradJb = (__m+1) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
+ _GradJ = _GardJo + _GradJb
+ return numpy.ravel(_GradJ)
+ vw = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = numpy.zeros(__m),
+ fprime = GradientOfCostFunction,
+ args = (),
+ disp = False,
+ )
+ #
+ Hto = EaHX.T @ (RI * EaHX)
+ Htb = (__m+1) * \
+ ( (1 + 1/__m + vw.T @ vw) * numpy.identity(__m) - 2 * vw @ vw.T ) \
+ / (1 + 1/__m + vw.T @ vw)**2
+ Hta = Hto + Htb
+ #
+ Pta = numpy.linalg.inv( Hta )
+ EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
+ #
+ Xn = Xfm + EaX @ (vw.reshape((__m,1)) + EWa)
+ #--------------------------
+ elif VariantM == "FiniteSize16": # Jauge Boc2016
+ HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
+ def CostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _Jo = 0.5 * _A.T @ (RI * _A)
+ _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w / (__m-1))
+ _J = _Jo + _Jb
+ return float(_J)
+ def GradientOfCostFunction(w):
+ _A = Ynpu - HXfm.reshape((__p,1)) - (EaHX @ w).reshape((__p,1))
+ _GardJo = - EaHX.T @ (RI * _A)
+ _GradJb = ((__m+1) / (__m-1)) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w / (__m-1))
+ _GradJ = _GardJo + _GradJb
+ return numpy.ravel(_GradJ)
+ vw = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = numpy.zeros(__m),
+ fprime = GradientOfCostFunction,
+ args = (),
+ disp = False,
+ )
+ #
+ Hto = EaHX.T @ (RI * EaHX)
+ Htb = ((__m+1) / (__m-1)) * \
+ ( (1 + 1/__m + vw.T @ vw / (__m-1)) * numpy.identity(__m) - 2 * vw @ vw.T / (__m-1) ) \
+ / (1 + 1/__m + vw.T @ vw / (__m-1))**2
+ Hta = Hto + Htb
+ #
+ Pta = numpy.linalg.inv( Hta )
+ EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
+ #
+ Xn = Xfm + EaX @ (vw[:,None] + EWa)
+ #--------------------------
+ else:
+ raise ValueError("VariantM has to be chosen in the authorized methods list.")
+ #
+ if selfA._parameters["InflationType"] == "MultiplicativeOnAnalysisAnomalies":
+ Xn = CovarianceInflation( Xn,
+ selfA._parameters["InflationType"],
+ selfA._parameters["InflationFactor"],
+ )
+ #
+ Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ #--------------------------
+ #
+ if selfA._parameters["StoreInternalVariables"] \
+ or selfA._toStore("CostFunctionJ") \
+ or selfA._toStore("CostFunctionJb") \
or selfA._toStore("CostFunctionJo") \
or selfA._toStore("APosterioriCovariance") \
or selfA._toStore("InnovationAtCurrentAnalysis") \
if selfA._toStore("CostFunctionJAtCurrentOptimum"):
selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
if selfA._toStore("APosterioriCovariance"):
- Eai = EnsembleOfAnomalies( Xn ) / numpy.sqrt(__m-1) # Anomalies
- Pn = Eai @ Eai.T
- Pn = 0.5 * (Pn + Pn.T)
- selfA.StoredVariables["APosterioriCovariance"].store( Pn )
+ selfA.StoredVariables["APosterioriCovariance"].store( EnsembleErrorCovariance(Xn) )
if selfA._parameters["EstimationOf"] == "Parameters" \
and J < previousJMinimum:
previousJMinimum = J
XaMin = Xa
if selfA._toStore("APosterioriCovariance"):
covarianceXaMin = Pn
+ # ---> Pour les smoothers
+ if selfA._toStore("CurrentEnsembleState"):
+ selfA.StoredVariables["CurrentEnsembleState"].store( Xn )
#
# Stockage final supplémentaire de l'optimum en estimation de paramètres
# ----------------------------------------------------------------------
return 0
# ==============================================================================
-def etkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="KalmanFilterFormula"):
+def ienkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="IEnKF12",
+ BnotT=False, _epsilon=1.e-3, _e=1.e-7, _jmax=15000):
"""
- Ensemble-Transform EnKF (ETKF or Deterministic EnKF: Bishop 2001, Hunt 2007)
-
- selfA est identique au "self" d'algorithme appelant et contient les
- valeurs.
+ Iterative EnKF
"""
if selfA._parameters["EstimationOf"] == "Parameters":
selfA._parameters["StoreInternalVariables"] = True
or selfA._toStore("CurrentOptimum") \
or selfA._toStore("APosterioriCovariance"):
BI = B.getI()
- RI = R.getI()
- elif VariantM != "KalmanFilterFormula":
- RI = R.getI()
- if VariantM == "KalmanFilterFormula":
- RIdemi = R.choleskyI()
+ RI = R.getI()
#
# Initialisation
# --------------
else: Rn = R
if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
else: Qn = Q
- Xn = EnsembleOfBackgroundPerturbations( Xb, None, __m )
+ Xn = EnsembleOfBackgroundPerturbations( Xb, Pn, __m )
#
if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
- selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
+ selfA.StoredVariables["Analysis"].store( Xb )
if selfA._toStore("APosterioriCovariance"):
selfA.StoredVariables["APosterioriCovariance"].store( Pn )
covarianceXa = Pn
#
for step in range(duration-1):
if hasattr(Y,"store"):
- Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,1))
else:
- Ynpu = numpy.ravel( Y ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y ).reshape((__p,1))
#
if U is not None:
if hasattr(U,"store") and len(U)>1:
selfA._parameters["InflationFactor"],
)
#
- if selfA._parameters["EstimationOf"] == "State": # Forecast + Q and observation of forecast
- EMX = M( [(Xn[:,i], Un) for i in range(__m)],
- argsAsSerie = True,
- returnSerieAsArrayMatrix = True )
- qi = numpy.random.multivariate_normal(numpy.zeros(__n), Qn, size=__m).T
- Xn_predicted = EMX + qi
- HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
- argsAsSerie = True,
- returnSerieAsArrayMatrix = True )
- if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
- Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
- Xn_predicted = Xn_predicted + Cm * Un
- elif selfA._parameters["EstimationOf"] == "Parameters": # Observation of forecast
- # --- > Par principe, M = Id, Q = 0
- Xn_predicted = Xn
- HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
- argsAsSerie = True,
- returnSerieAsArrayMatrix = True )
- #
- # Mean of forecast and observation of forecast
- Xfm = Xn_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
- Hfm = HX_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__p,1))
- #
- # Anomalies
- EaX = EnsembleOfAnomalies( Xn_predicted )
- EaHX = numpy.array(HX_predicted - Hfm)
- #
- #--------------------------
- if VariantM == "KalmanFilterFormula":
- mS = RIdemi * EaHX / numpy.sqrt(__m-1)
- delta = RIdemi * ( Ynpu - Hfm )
- mT = numpy.linalg.inv( numpy.eye(__m) + mS.T @ mS )
- vw = mT @ mS.transpose() @ delta
- #
- Tdemi = numpy.real(scipy.linalg.sqrtm(mT))
- mU = numpy.eye(__m)
- #
- EaX = EaX / numpy.sqrt(__m-1)
- Xn = Xfm + EaX @ ( vw.reshape((__m,-1)) + numpy.sqrt(__m-1) * Tdemi @ mU )
- #--------------------------
- elif VariantM == "Variational":
- HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
- def CostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _Jo = 0.5 * _A.T @ (RI * _A)
- _Jb = 0.5 * (__m-1) * w.T @ w
- _J = _Jo + _Jb
- return float(_J)
- def GradientOfCostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _GardJo = - EaHX.T @ (RI * _A)
- _GradJb = (__m-1) * w.reshape((__m,1))
- _GradJ = _GardJo + _GradJb
- return numpy.ravel(_GradJ)
- vw = scipy.optimize.fmin_cg(
- f = CostFunction,
- x0 = numpy.zeros(__m),
- fprime = GradientOfCostFunction,
- args = (),
- disp = False,
- )
- #
- Hto = EaHX.T @ (RI * EaHX)
- Htb = (__m-1) * numpy.eye(__m)
- Hta = Hto + Htb
- #
- Pta = numpy.linalg.inv( Hta )
- EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
- #
- Xn = Xfm + EaX @ (vw[:,None] + EWa)
#--------------------------
- elif VariantM == "FiniteSize11": # Jauge Boc2011
- HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
- def CostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _Jo = 0.5 * _A.T @ (RI * _A)
- _Jb = 0.5 * __m * math.log(1 + 1/__m + w.T @ w)
- _J = _Jo + _Jb
- return float(_J)
- def GradientOfCostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _GardJo = - EaHX.T @ (RI * _A)
- _GradJb = __m * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
- _GradJ = _GardJo + _GradJb
- return numpy.ravel(_GradJ)
- vw = scipy.optimize.fmin_cg(
- f = CostFunction,
- x0 = numpy.zeros(__m),
- fprime = GradientOfCostFunction,
- args = (),
- disp = False,
- )
- #
- Hto = EaHX.T @ (RI * EaHX)
- Htb = __m * \
- ( (1 + 1/__m + vw.T @ vw) * numpy.eye(__m) - 2 * vw @ vw.T ) \
- / (1 + 1/__m + vw.T @ vw)**2
- Hta = Hto + Htb
- #
- Pta = numpy.linalg.inv( Hta )
- EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
- #
- Xn = Xfm + EaX @ (vw.reshape((__m,-1)) + EWa)
- #--------------------------
- elif VariantM == "FiniteSize15": # Jauge Boc2015
- HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
- def CostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _Jo = 0.5 * _A.T * RI * _A
- _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w)
- _J = _Jo + _Jb
- return float(_J)
- def GradientOfCostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _GardJo = - EaHX.T @ (RI * _A)
- _GradJb = (__m+1) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w)
- _GradJ = _GardJo + _GradJb
- return numpy.ravel(_GradJ)
- vw = scipy.optimize.fmin_cg(
- f = CostFunction,
- x0 = numpy.zeros(__m),
- fprime = GradientOfCostFunction,
- args = (),
- disp = False,
- )
- #
- Hto = EaHX.T @ (RI * EaHX)
- Htb = (__m+1) * \
- ( (1 + 1/__m + vw.T @ vw) * numpy.eye(__m) - 2 * vw @ vw.T ) \
- / (1 + 1/__m + vw.T @ vw)**2
- Hta = Hto + Htb
- #
- Pta = numpy.linalg.inv( Hta )
- EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
- #
- Xn = Xfm + EaX @ (vw.reshape((__m,-1)) + EWa)
- #--------------------------
- elif VariantM == "FiniteSize16": # Jauge Boc2016
- HXfm = H((Xfm[:,None], Un)) # Eventuellement Hfm
- def CostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _Jo = 0.5 * _A.T @ (RI * _A)
- _Jb = 0.5 * (__m+1) * math.log(1 + 1/__m + w.T @ w / (__m-1))
- _J = _Jo + _Jb
- return float(_J)
- def GradientOfCostFunction(w):
- _A = Ynpu - HXfm.reshape((__p,-1)) - (EaHX @ w).reshape((__p,-1))
- _GardJo = - EaHX.T @ (RI * _A)
- _GradJb = ((__m+1) / (__m-1)) * w.reshape((__m,1)) / (1 + 1/__m + w.T @ w / (__m-1))
- _GradJ = _GardJo + _GradJb
- return numpy.ravel(_GradJ)
- vw = scipy.optimize.fmin_cg(
- f = CostFunction,
- x0 = numpy.zeros(__m),
- fprime = GradientOfCostFunction,
- args = (),
- disp = False,
- )
+ if VariantM == "IEnKF12":
+ Xfm = numpy.ravel(Xn.mean(axis=1, dtype=mfp).astype('float'))
+ EaX = EnsembleOfAnomalies( Xn ) / math.sqrt(__m-1)
+ __j = 0
+ Deltaw = 1
+ if not BnotT:
+ Ta = numpy.identity(__m)
+ vw = numpy.zeros(__m)
+ while numpy.linalg.norm(Deltaw) >= _e and __j <= _jmax:
+ vx1 = (Xfm + EaX @ vw).reshape((__n,1))
+ #
+ if BnotT:
+ E1 = vx1 + _epsilon * EaX
+ else:
+ E1 = vx1 + math.sqrt(__m-1) * EaX @ Ta
+ #
+ if selfA._parameters["EstimationOf"] == "State": # Forecast + Q
+ E2 = M( [(E1[:,i,numpy.newaxis], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ elif selfA._parameters["EstimationOf"] == "Parameters":
+ # --- > Par principe, M = Id
+ E2 = Xn
+ vx2 = E2.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ vy1 = H((vx2, Un)).reshape((__p,1))
+ #
+ HE2 = H( [(E2[:,i,numpy.newaxis], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ vy2 = HE2.mean(axis=1, dtype=mfp).astype('float').reshape((__p,1))
+ #
+ if BnotT:
+ EaY = (HE2 - vy2) / _epsilon
+ else:
+ EaY = ( (HE2 - vy2) @ numpy.linalg.inv(Ta) ) / math.sqrt(__m-1)
+ #
+ GradJ = numpy.ravel(vw[:,None] - EaY.transpose() @ (RI * ( Ynpu - vy1 )))
+ mH = numpy.identity(__m) + EaY.transpose() @ (RI * EaY)
+ Deltaw = - numpy.linalg.solve(mH,GradJ)
+ #
+ vw = vw + Deltaw
+ #
+ if not BnotT:
+ Ta = numpy.real(scipy.linalg.sqrtm(numpy.linalg.inv( mH )))
+ #
+ __j = __j + 1
#
- Hto = EaHX.T @ (RI * EaHX)
- Htb = ((__m+1) / (__m-1)) * \
- ( (1 + 1/__m + vw.T @ vw / (__m-1)) * numpy.eye(__m) - 2 * vw @ vw.T / (__m-1) ) \
- / (1 + 1/__m + vw.T @ vw / (__m-1))**2
- Hta = Hto + Htb
+ A2 = EnsembleOfAnomalies( E2 )
#
- Pta = numpy.linalg.inv( Hta )
- EWa = numpy.real(scipy.linalg.sqrtm((__m-1)*Pta)) # Partie imaginaire ~= 10^-18
+ if BnotT:
+ Ta = numpy.real(scipy.linalg.sqrtm(numpy.linalg.inv( mH )))
+ A2 = math.sqrt(__m-1) * A2 @ Ta / _epsilon
#
- Xn = Xfm + EaX @ (vw[:,None] + EWa)
+ Xn = vx2 + A2
#--------------------------
else:
raise ValueError("VariantM has to be chosen in the authorized methods list.")
selfA._parameters["InflationFactor"],
)
#
- Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
+ Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
#--------------------------
#
if selfA._parameters["StoreInternalVariables"] \
or selfA._toStore("CurrentState"):
selfA.StoredVariables["CurrentState"].store( Xn )
if selfA._toStore("ForecastState"):
- selfA.StoredVariables["ForecastState"].store( EMX )
+ selfA.StoredVariables["ForecastState"].store( E2 )
if selfA._toStore("BMA"):
- selfA.StoredVariables["BMA"].store( EMX - Xa.reshape((__n,1)) )
+ selfA.StoredVariables["BMA"].store( E2 - Xa )
if selfA._toStore("InnovationAtCurrentState"):
- selfA.StoredVariables["InnovationAtCurrentState"].store( - HX_predicted + Ynpu.reshape((__p,1)) )
+ selfA.StoredVariables["InnovationAtCurrentState"].store( - HE2 + Ynpu )
if selfA._toStore("SimulatedObservationAtCurrentState") \
or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
- selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HX_predicted )
+ selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HE2 )
# ---> autres
if selfA._parameters["StoreInternalVariables"] \
or selfA._toStore("CostFunctionJ") \
if selfA._toStore("CostFunctionJAtCurrentOptimum"):
selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
if selfA._toStore("APosterioriCovariance"):
- Eai = EnsembleOfAnomalies( Xn ) / numpy.sqrt(__m-1) # Anomalies
- Pn = Eai @ Eai.T
- Pn = 0.5 * (Pn + Pn.T)
- selfA.StoredVariables["APosterioriCovariance"].store( Pn )
+ selfA.StoredVariables["APosterioriCovariance"].store( EnsembleErrorCovariance(Xn) )
if selfA._parameters["EstimationOf"] == "Parameters" \
and J < previousJMinimum:
previousJMinimum = J
#
return 0
+# ==============================================================================
+def incr3dvar(selfA, Xb, Y, U, HO, EM, CM, R, B, Q):
+ """
+ 3DVAR incrémental
+ """
+ #
+ # Initialisations
+ # ---------------
+ #
+ # Opérateur non-linéaire pour la boucle externe
+ Hm = HO["Direct"].appliedTo
+ #
+ # Précalcul des inversions de B et R
+ BI = B.getI()
+ RI = R.getI()
+ #
+ # Point de démarrage de l'optimisation
+ Xini = selfA._parameters["InitializationPoint"]
+ #
+ HXb = numpy.asmatrix(numpy.ravel( Hm( Xb ) )).T
+ Innovation = Y - HXb
+ #
+ # Outer Loop
+ # ----------
+ iOuter = 0
+ J = 1./mpr
+ DeltaJ = 1./mpr
+ Xr = Xini.reshape((-1,1))
+ while abs(DeltaJ) >= selfA._parameters["CostDecrementTolerance"] and iOuter <= selfA._parameters["MaximumNumberOfSteps"]:
+ #
+ # Inner Loop
+ # ----------
+ Ht = HO["Tangent"].asMatrix(Xr)
+ Ht = Ht.reshape(Y.size,Xr.size) # ADAO & check shape
+ #
+ # Définition de la fonction-coût
+ # ------------------------------
+ def CostFunction(dx):
+ _dX = numpy.asmatrix(numpy.ravel( dx )).T
+ if selfA._parameters["StoreInternalVariables"] or \
+ selfA._toStore("CurrentState") or \
+ selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentState"].store( Xb + _dX )
+ _HdX = Ht * _dX
+ _HdX = numpy.asmatrix(numpy.ravel( _HdX )).T
+ _dInnovation = Innovation - _HdX
+ if selfA._toStore("SimulatedObservationAtCurrentState") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HXb + _HdX )
+ if selfA._toStore("InnovationAtCurrentState"):
+ selfA.StoredVariables["InnovationAtCurrentState"].store( _dInnovation )
+ #
+ Jb = float( 0.5 * _dX.T * BI * _dX )
+ Jo = float( 0.5 * _dInnovation.T * RI * _dInnovation )
+ J = Jb + Jo
+ #
+ selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["CostFunctionJ"]) )
+ selfA.StoredVariables["CostFunctionJb"].store( Jb )
+ selfA.StoredVariables["CostFunctionJo"].store( Jo )
+ selfA.StoredVariables["CostFunctionJ" ].store( J )
+ if selfA._toStore("IndexOfOptimum") or \
+ selfA._toStore("CurrentOptimum") or \
+ selfA._toStore("CostFunctionJAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJbAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJoAtCurrentOptimum") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ if selfA._toStore("IndexOfOptimum"):
+ selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
+ if selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["CurrentState"][IndexMin] )
+ if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin] )
+ if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
+ if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
+ if selfA._toStore("CostFunctionJAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
+ return J
+ #
+ def GradientOfCostFunction(dx):
+ _dX = numpy.asmatrix(numpy.ravel( dx )).T
+ _HdX = Ht * _dX
+ _HdX = numpy.asmatrix(numpy.ravel( _HdX )).T
+ _dInnovation = Innovation - _HdX
+ GradJb = BI * _dX
+ GradJo = - Ht.T @ (RI * _dInnovation)
+ GradJ = numpy.ravel( GradJb ) + numpy.ravel( GradJo )
+ return GradJ
+ #
+ # Minimisation de la fonctionnelle
+ # --------------------------------
+ nbPreviousSteps = selfA.StoredVariables["CostFunctionJ"].stepnumber()
+ #
+ if selfA._parameters["Minimizer"] == "LBFGSB":
+ # Minimum, J_optimal, Informations = scipy.optimize.fmin_l_bfgs_b(
+ if "0.19" <= scipy.version.version <= "1.1.0":
+ import lbfgsbhlt as optimiseur
+ else:
+ import scipy.optimize as optimiseur
+ Minimum, J_optimal, Informations = optimiseur.fmin_l_bfgs_b(
+ func = CostFunction,
+ x0 = numpy.zeros(Xini.size),
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"]-1,
+ factr = selfA._parameters["CostDecrementTolerance"]*1.e14,
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ iprint = selfA._parameters["optiprint"],
+ )
+ nfeval = Informations['funcalls']
+ rc = Informations['warnflag']
+ elif selfA._parameters["Minimizer"] == "TNC":
+ Minimum, nfeval, rc = scipy.optimize.fmin_tnc(
+ func = CostFunction,
+ x0 = numpy.zeros(Xini.size),
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"],
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ ftol = selfA._parameters["CostDecrementTolerance"],
+ messages = selfA._parameters["optmessages"],
+ )
+ elif selfA._parameters["Minimizer"] == "CG":
+ Minimum, fopt, nfeval, grad_calls, rc = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = numpy.zeros(Xini.size),
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "NCG":
+ Minimum, fopt, nfeval, grad_calls, hcalls, rc = scipy.optimize.fmin_ncg(
+ f = CostFunction,
+ x0 = numpy.zeros(Xini.size),
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ avextol = selfA._parameters["CostDecrementTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "BFGS":
+ Minimum, fopt, gopt, Hopt, nfeval, grad_calls, rc = scipy.optimize.fmin_bfgs(
+ f = CostFunction,
+ x0 = numpy.zeros(Xini.size),
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ else:
+ raise ValueError("Error in Minimizer name: %s"%selfA._parameters["Minimizer"])
+ #
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ MinJ = selfA.StoredVariables["CostFunctionJ"][IndexMin]
+ #
+ if selfA._parameters["StoreInternalVariables"] or selfA._toStore("CurrentState"):
+ Minimum = selfA.StoredVariables["CurrentState"][IndexMin]
+ Minimum = numpy.asmatrix(numpy.ravel( Minimum )).T
+ else:
+ Minimum = Xb + numpy.asmatrix(numpy.ravel( Minimum )).T
+ #
+ Xr = Minimum
+ DeltaJ = selfA.StoredVariables["CostFunctionJ" ][-1] - J
+ iOuter = selfA.StoredVariables["CurrentIterationNumber"][-1]
+ #
+ # Obtention de l'analyse
+ # ----------------------
+ Xa = Xr
+ #
+ selfA.StoredVariables["Analysis"].store( Xa )
+ #
+ if selfA._toStore("OMA") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("SimulatedObservationAtOptimum"):
+ if selfA._toStore("SimulatedObservationAtCurrentState"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin]
+ elif selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"][-1]
+ else:
+ HXa = Hm( Xa )
+ #
+ # Calcul de la covariance d'analyse
+ # ---------------------------------
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("JacobianMatrixAtOptimum") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HtM = HO["Tangent"].asMatrix(ValueForMethodForm = Xa)
+ HtM = HtM.reshape(Y.size,Xa.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HaM = HO["Adjoint"].asMatrix(ValueForMethodForm = Xa)
+ HaM = HaM.reshape(Xa.size,Y.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles"):
+ HessienneI = []
+ nb = Xa.size
+ for i in range(nb):
+ _ee = numpy.matrix(numpy.zeros(nb)).T
+ _ee[i] = 1.
+ _HtEE = numpy.dot(HtM,_ee)
+ _HtEE = numpy.asmatrix(numpy.ravel( _HtEE )).T
+ HessienneI.append( numpy.ravel( BI*_ee + HaM * (RI * _HtEE) ) )
+ HessienneI = numpy.matrix( HessienneI )
+ A = HessienneI.I
+ if min(A.shape) != max(A.shape):
+ raise ValueError("The %s a posteriori covariance matrix A is of shape %s, despites it has to be a squared matrix. There is an error in the observation operator, please check it."%(selfA._name,str(A.shape)))
+ if (numpy.diag(A) < 0).any():
+ raise ValueError("The %s a posteriori covariance matrix A has at least one negative value on its diagonal. There is an error in the observation operator, please check it."%(selfA._name,))
+ if logging.getLogger().level < logging.WARNING: # La verification n'a lieu qu'en debug
+ try:
+ L = numpy.linalg.cholesky( A )
+ except:
+ raise ValueError("The %s a posteriori covariance matrix A is not symmetric positive-definite. Please check your a priori covariances and your observation operator."%(selfA._name,))
+ if selfA._toStore("APosterioriCovariance"):
+ selfA.StoredVariables["APosterioriCovariance"].store( A )
+ if selfA._toStore("JacobianMatrixAtOptimum"):
+ selfA.StoredVariables["JacobianMatrixAtOptimum"].store( HtM )
+ if selfA._toStore("KalmanGainAtOptimum"):
+ if (Y.size <= Xb.size): KG = B * HaM * (R + numpy.dot(HtM, B * HaM)).I
+ elif (Y.size > Xb.size): KG = (BI + numpy.dot(HaM, RI * HtM)).I * HaM * RI
+ selfA.StoredVariables["KalmanGainAtOptimum"].store( KG )
+ #
+ # Calculs et/ou stockages supplémentaires
+ # ---------------------------------------
+ if selfA._toStore("Innovation") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("MahalanobisConsistency") or \
+ selfA._toStore("OMB"):
+ d = Y - HXb
+ if selfA._toStore("Innovation"):
+ selfA.StoredVariables["Innovation"].store( numpy.ravel(d) )
+ if selfA._toStore("BMA"):
+ selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(Xa) )
+ if selfA._toStore("OMA"):
+ selfA.StoredVariables["OMA"].store( numpy.ravel(Y) - numpy.ravel(HXa) )
+ if selfA._toStore("OMB"):
+ selfA.StoredVariables["OMB"].store( numpy.ravel(d) )
+ if selfA._toStore("SigmaObs2"):
+ TraceR = R.trace(Y.size)
+ selfA.StoredVariables["SigmaObs2"].store( float( (d.T * (numpy.asmatrix(numpy.ravel(Y)).T-numpy.asmatrix(numpy.ravel(HXa)).T)) ) / TraceR )
+ if selfA._toStore("MahalanobisConsistency"):
+ selfA.StoredVariables["MahalanobisConsistency"].store( float( 2.*MinJ/d.size ) )
+ if selfA._toStore("SimulationQuantiles"):
+ nech = selfA._parameters["NumberOfSamplesForQuantiles"]
+ HXa = numpy.matrix(numpy.ravel( HXa )).T
+ YfQ = None
+ for i in range(nech):
+ if selfA._parameters["SimulationForQuantiles"] == "Linear":
+ dXr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A) - Xa.A1).T
+ dYr = numpy.matrix(numpy.ravel( HtM * dXr )).T
+ Yr = HXa + dYr
+ elif selfA._parameters["SimulationForQuantiles"] == "NonLinear":
+ Xr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A)).T
+ Yr = numpy.matrix(numpy.ravel( Hm( Xr ) )).T
+ if YfQ is None:
+ YfQ = Yr
+ else:
+ YfQ = numpy.hstack((YfQ,Yr))
+ YfQ.sort(axis=-1)
+ YQ = None
+ for quantile in selfA._parameters["Quantiles"]:
+ if not (0. <= float(quantile) <= 1.): continue
+ indice = int(nech * float(quantile) - 1./nech)
+ if YQ is None: YQ = YfQ[:,indice]
+ else: YQ = numpy.hstack((YQ,YfQ[:,indice]))
+ selfA.StoredVariables["SimulationQuantiles"].store( YQ )
+ if selfA._toStore("SimulatedObservationAtBackground"):
+ selfA.StoredVariables["SimulatedObservationAtBackground"].store( numpy.ravel(HXb) )
+ if selfA._toStore("SimulatedObservationAtOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtOptimum"].store( numpy.ravel(HXa) )
+ #
+ return 0
+
# ==============================================================================
def mlef(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="MLEF13",
BnotT=False, _epsilon=1.e-3, _e=1.e-7, _jmax=15000):
"""
- Maximum Likelihood Ensemble Filter (EnKF/MLEF Zupanski 2005, Bocquet 2013)
-
- selfA est identique au "self" d'algorithme appelant et contient les
- valeurs.
+ Maximum Likelihood Ensemble Filter
"""
if selfA._parameters["EstimationOf"] == "Parameters":
selfA._parameters["StoreInternalVariables"] = True
Xn = EnsembleOfBackgroundPerturbations( Xb, None, __m )
#
if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
- selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
+ selfA.StoredVariables["Analysis"].store( Xb )
if selfA._toStore("APosterioriCovariance"):
selfA.StoredVariables["APosterioriCovariance"].store( Pn )
covarianceXa = Pn
#
for step in range(duration-1):
if hasattr(Y,"store"):
- Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,1))
else:
- Ynpu = numpy.ravel( Y ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y ).reshape((__p,1))
#
if U is not None:
if hasattr(U,"store") and len(U)>1:
#--------------------------
if VariantM == "MLEF13":
Xfm = numpy.ravel(Xn_predicted.mean(axis=1, dtype=mfp).astype('float'))
- EaX = EnsembleOfAnomalies( Xn_predicted ) / numpy.sqrt(__m-1)
- Ua = numpy.eye(__m)
+ EaX = EnsembleOfAnomalies( Xn_predicted, Xfm, 1./math.sqrt(__m-1) )
+ Ua = numpy.identity(__m)
__j = 0
Deltaw = 1
if not BnotT:
- Ta = numpy.eye(__m)
+ Ta = numpy.identity(__m)
vw = numpy.zeros(__m)
while numpy.linalg.norm(Deltaw) >= _e and __j <= _jmax:
- vx1 = (Xfm + EaX @ vw).reshape((__n,-1))
+ vx1 = (Xfm + EaX @ vw).reshape((__n,1))
#
if BnotT:
E1 = vx1 + _epsilon * EaX
else:
- E1 = vx1 + numpy.sqrt(__m-1) * EaX @ Ta
+ E1 = vx1 + math.sqrt(__m-1) * EaX @ Ta
#
HE2 = H( [(E1[:,i,numpy.newaxis], Un) for i in range(__m)],
argsAsSerie = True,
returnSerieAsArrayMatrix = True )
- vy2 = HE2.mean(axis=1, dtype=mfp).astype('float').reshape((__p,-1))
+ vy2 = HE2.mean(axis=1, dtype=mfp).astype('float').reshape((__p,1))
#
if BnotT:
EaY = (HE2 - vy2) / _epsilon
else:
- EaY = ( (HE2 - vy2) @ numpy.linalg.inv(Ta) ) / numpy.sqrt(__m-1)
+ EaY = ( (HE2 - vy2) @ numpy.linalg.inv(Ta) ) / math.sqrt(__m-1)
#
GradJ = numpy.ravel(vw[:,None] - EaY.transpose() @ (RI * ( Ynpu - vy2 )))
- mH = numpy.eye(__m) + EaY.transpose() @ (RI * EaY)
+ mH = numpy.identity(__m) + EaY.transpose() @ (RI * EaY)
Deltaw = - numpy.linalg.solve(mH,GradJ)
#
vw = vw + Deltaw
if BnotT:
Ta = numpy.real(scipy.linalg.sqrtm(numpy.linalg.inv( mH )))
#
- Xn = vx1 + numpy.sqrt(__m-1) * EaX @ Ta @ Ua
+ Xn = vx1 + math.sqrt(__m-1) * EaX @ Ta @ Ua
#--------------------------
else:
raise ValueError("VariantM has to be chosen in the authorized methods list.")
selfA._parameters["InflationFactor"],
)
#
- Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
+ Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
#--------------------------
#
if selfA._parameters["StoreInternalVariables"] \
if selfA._toStore("ForecastState"):
selfA.StoredVariables["ForecastState"].store( EMX )
if selfA._toStore("BMA"):
- selfA.StoredVariables["BMA"].store( EMX - Xa.reshape((__n,1)) )
+ selfA.StoredVariables["BMA"].store( EMX - Xa )
if selfA._toStore("InnovationAtCurrentState"):
- selfA.StoredVariables["InnovationAtCurrentState"].store( - HE2 + Ynpu.reshape((__p,-1)) )
+ selfA.StoredVariables["InnovationAtCurrentState"].store( - HE2 + Ynpu )
if selfA._toStore("SimulatedObservationAtCurrentState") \
or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HE2 )
if selfA._toStore("CostFunctionJAtCurrentOptimum"):
selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
if selfA._toStore("APosterioriCovariance"):
- Eai = EnsembleOfAnomalies( Xn ) / numpy.sqrt(__m-1) # Anomalies
- Pn = Eai @ Eai.T
- Pn = 0.5 * (Pn + Pn.T)
- selfA.StoredVariables["APosterioriCovariance"].store( Pn )
+ selfA.StoredVariables["APosterioriCovariance"].store( EnsembleErrorCovariance(Xn) )
if selfA._parameters["EstimationOf"] == "Parameters" \
and J < previousJMinimum:
previousJMinimum = J
return 0
# ==============================================================================
-def ienkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="IEnKF12",
- BnotT=False, _epsilon=1.e-3, _e=1.e-7, _jmax=15000):
+def mmqr(
+ func = None,
+ x0 = None,
+ fprime = None,
+ bounds = None,
+ quantile = 0.5,
+ maxfun = 15000,
+ toler = 1.e-06,
+ y = None,
+ ):
+ """
+ Implémentation informatique de l'algorithme MMQR, basée sur la publication :
+ David R. Hunter, Kenneth Lange, "Quantile Regression via an MM Algorithm",
+ Journal of Computational and Graphical Statistics, 9, 1, pp.60-77, 2000.
+ """
+ #
+ # Recuperation des donnees et informations initiales
+ # --------------------------------------------------
+ variables = numpy.ravel( x0 )
+ mesures = numpy.ravel( y )
+ increment = sys.float_info[0]
+ p = variables.size
+ n = mesures.size
+ quantile = float(quantile)
+ #
+ # Calcul des parametres du MM
+ # ---------------------------
+ tn = float(toler) / n
+ e0 = -tn / math.log(tn)
+ epsilon = (e0-tn)/(1+math.log(e0))
+ #
+ # Calculs d'initialisation
+ # ------------------------
+ residus = mesures - numpy.ravel( func( variables ) )
+ poids = 1./(epsilon+numpy.abs(residus))
+ veps = 1. - 2. * quantile - residus * poids
+ lastsurrogate = -numpy.sum(residus*veps) - (1.-2.*quantile)*numpy.sum(residus)
+ iteration = 0
+ #
+ # Recherche iterative
+ # -------------------
+ while (increment > toler) and (iteration < maxfun) :
+ iteration += 1
+ #
+ Derivees = numpy.array(fprime(variables))
+ Derivees = Derivees.reshape(n,p) # Necessaire pour remettre en place la matrice si elle passe par des tuyaux YACS
+ DeriveesT = Derivees.transpose()
+ M = numpy.dot( DeriveesT , (numpy.array(numpy.matrix(p*[poids,]).T)*Derivees) )
+ SM = numpy.transpose(numpy.dot( DeriveesT , veps ))
+ step = - numpy.linalg.lstsq( M, SM, rcond=-1 )[0]
+ #
+ variables = variables + step
+ if bounds is not None:
+ # Attention : boucle infinie à éviter si un intervalle est trop petit
+ while( (variables < numpy.ravel(numpy.asmatrix(bounds)[:,0])).any() or (variables > numpy.ravel(numpy.asmatrix(bounds)[:,1])).any() ):
+ step = step/2.
+ variables = variables - step
+ residus = mesures - numpy.ravel( func(variables) )
+ surrogate = numpy.sum(residus**2 * poids) + (4.*quantile-2.) * numpy.sum(residus)
+ #
+ while ( (surrogate > lastsurrogate) and ( max(list(numpy.abs(step))) > 1.e-16 ) ) :
+ step = step/2.
+ variables = variables - step
+ residus = mesures - numpy.ravel( func(variables) )
+ surrogate = numpy.sum(residus**2 * poids) + (4.*quantile-2.) * numpy.sum(residus)
+ #
+ increment = lastsurrogate-surrogate
+ poids = 1./(epsilon+numpy.abs(residus))
+ veps = 1. - 2. * quantile - residus * poids
+ lastsurrogate = -numpy.sum(residus * veps) - (1.-2.*quantile)*numpy.sum(residus)
+ #
+ # Mesure d'écart
+ # --------------
+ Ecart = quantile * numpy.sum(residus) - numpy.sum( residus[residus<0] )
+ #
+ return variables, Ecart, [n,p,iteration,increment,0]
+
+# ==============================================================================
+def multi3dvar(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, oneCycle):
+ """
+ 3DVAR multi-pas et multi-méthodes
+ """
+ #
+ # Initialisation
+ # --------------
+ Xn = numpy.ravel(Xb).reshape((-1,1))
+ #
+ if selfA._parameters["EstimationOf"] == "State":
+ M = EM["Direct"].appliedTo
+ #
+ if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
+ selfA.StoredVariables["Analysis"].store( Xn )
+ if selfA._toStore("APosterioriCovariance"):
+ if hasattr(B,"asfullmatrix"): Pn = B.asfullmatrix(Xn.size)
+ else: Pn = B
+ selfA.StoredVariables["APosterioriCovariance"].store( Pn )
+ if selfA._toStore("ForecastState"):
+ selfA.StoredVariables["ForecastState"].store( Xn )
+ #
+ if hasattr(Y,"stepnumber"):
+ duration = Y.stepnumber()
+ else:
+ duration = 2
+ #
+ # Multi-pas
+ # ---------
+ for step in range(duration-1):
+ if hasattr(Y,"store"):
+ Ynpu = numpy.ravel( Y[step+1] ).reshape((-1,1))
+ else:
+ Ynpu = numpy.ravel( Y ).reshape((-1,1))
+ #
+ if selfA._parameters["EstimationOf"] == "State": # Forecast
+ Xn = selfA.StoredVariables["Analysis"][-1]
+ Xn_predicted = M( Xn )
+ if selfA._toStore("ForecastState"):
+ selfA.StoredVariables["ForecastState"].store( Xn_predicted )
+ elif selfA._parameters["EstimationOf"] == "Parameters": # No forecast
+ # --- > Par principe, M = Id, Q = 0
+ Xn_predicted = Xn
+ Xn_predicted = numpy.ravel(Xn_predicted).reshape((-1,1))
+ #
+ oneCycle(selfA, Xn_predicted, Ynpu, U, HO, None, None, R, B, None)
+ #
+ return 0
+
+# ==============================================================================
+def psas3dvar(selfA, Xb, Y, U, HO, EM, CM, R, B, Q):
+ """
+ 3DVAR PSAS
"""
- Iterative EnKF (Sakov 2012, Sakov 2018)
+ #
+ # Initialisations
+ # ---------------
+ #
+ # Opérateurs
+ Hm = HO["Direct"].appliedTo
+ #
+ # Utilisation éventuelle d'un vecteur H(Xb) précalculé
+ if HO["AppliedInX"] is not None and "HXb" in HO["AppliedInX"]:
+ HXb = Hm( Xb, HO["AppliedInX"]["HXb"] )
+ else:
+ HXb = Hm( Xb )
+ HXb = numpy.asmatrix(numpy.ravel( HXb )).T
+ if Y.size != HXb.size:
+ raise ValueError("The size %i of observations Y and %i of observed calculation H(X) are different, they have to be identical."%(Y.size,HXb.size))
+ if max(Y.shape) != max(HXb.shape):
+ raise ValueError("The shapes %s of observations Y and %s of observed calculation H(X) are different, they have to be identical."%(Y.shape,HXb.shape))
+ #
+ if selfA._toStore("JacobianMatrixAtBackground"):
+ HtMb = HO["Tangent"].asMatrix(ValueForMethodForm = Xb)
+ HtMb = HtMb.reshape(Y.size,Xb.size) # ADAO & check shape
+ selfA.StoredVariables["JacobianMatrixAtBackground"].store( HtMb )
+ #
+ Ht = HO["Tangent"].asMatrix(Xb)
+ BHT = B * Ht.T
+ HBHTpR = R + Ht * BHT
+ Innovation = Y - HXb
+ #
+ # Point de démarrage de l'optimisation
+ Xini = numpy.zeros(Xb.shape)
+ #
+ # Définition de la fonction-coût
+ # ------------------------------
+ def CostFunction(w):
+ _W = numpy.asmatrix(numpy.ravel( w )).T
+ if selfA._parameters["StoreInternalVariables"] or \
+ selfA._toStore("CurrentState") or \
+ selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentState"].store( Xb + BHT * _W )
+ if selfA._toStore("SimulatedObservationAtCurrentState") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( Hm( Xb + BHT * _W ) )
+ if selfA._toStore("InnovationAtCurrentState"):
+ selfA.StoredVariables["InnovationAtCurrentState"].store( Innovation )
+ #
+ Jb = float( 0.5 * _W.T * HBHTpR * _W )
+ Jo = float( - _W.T * Innovation )
+ J = Jb + Jo
+ #
+ selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["CostFunctionJ"]) )
+ selfA.StoredVariables["CostFunctionJb"].store( Jb )
+ selfA.StoredVariables["CostFunctionJo"].store( Jo )
+ selfA.StoredVariables["CostFunctionJ" ].store( J )
+ if selfA._toStore("IndexOfOptimum") or \
+ selfA._toStore("CurrentOptimum") or \
+ selfA._toStore("CostFunctionJAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJbAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJoAtCurrentOptimum") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ if selfA._toStore("IndexOfOptimum"):
+ selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
+ if selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["CurrentState"][IndexMin] )
+ if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin] )
+ if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
+ if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
+ if selfA._toStore("CostFunctionJAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
+ return J
+ #
+ def GradientOfCostFunction(w):
+ _W = numpy.asmatrix(numpy.ravel( w )).T
+ GradJb = HBHTpR * _W
+ GradJo = - Innovation
+ GradJ = numpy.ravel( GradJb ) + numpy.ravel( GradJo )
+ return GradJ
+ #
+ # Minimisation de la fonctionnelle
+ # --------------------------------
+ nbPreviousSteps = selfA.StoredVariables["CostFunctionJ"].stepnumber()
+ #
+ if selfA._parameters["Minimizer"] == "LBFGSB":
+ if "0.19" <= scipy.version.version <= "1.1.0":
+ import lbfgsbhlt as optimiseur
+ else:
+ import scipy.optimize as optimiseur
+ Minimum, J_optimal, Informations = optimiseur.fmin_l_bfgs_b(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"]-1,
+ factr = selfA._parameters["CostDecrementTolerance"]*1.e14,
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ iprint = selfA._parameters["optiprint"],
+ )
+ nfeval = Informations['funcalls']
+ rc = Informations['warnflag']
+ elif selfA._parameters["Minimizer"] == "TNC":
+ Minimum, nfeval, rc = scipy.optimize.fmin_tnc(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"],
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ ftol = selfA._parameters["CostDecrementTolerance"],
+ messages = selfA._parameters["optmessages"],
+ )
+ elif selfA._parameters["Minimizer"] == "CG":
+ Minimum, fopt, nfeval, grad_calls, rc = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "NCG":
+ Minimum, fopt, nfeval, grad_calls, hcalls, rc = scipy.optimize.fmin_ncg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ avextol = selfA._parameters["CostDecrementTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "BFGS":
+ Minimum, fopt, gopt, Hopt, nfeval, grad_calls, rc = scipy.optimize.fmin_bfgs(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ else:
+ raise ValueError("Error in Minimizer name: %s"%selfA._parameters["Minimizer"])
+ #
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ MinJ = selfA.StoredVariables["CostFunctionJ"][IndexMin]
+ #
+ # Correction pour pallier a un bug de TNC sur le retour du Minimum
+ # ----------------------------------------------------------------
+ if selfA._parameters["StoreInternalVariables"] or selfA._toStore("CurrentState"):
+ Minimum = selfA.StoredVariables["CurrentState"][IndexMin]
+ Minimum = numpy.asmatrix(numpy.ravel( Minimum )).T
+ else:
+ Minimum = Xb + BHT * numpy.asmatrix(numpy.ravel( Minimum )).T
+ #
+ # Obtention de l'analyse
+ # ----------------------
+ Xa = Minimum
+ #
+ selfA.StoredVariables["Analysis"].store( Xa )
+ #
+ if selfA._toStore("OMA") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("SimulatedObservationAtOptimum"):
+ if selfA._toStore("SimulatedObservationAtCurrentState"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin]
+ elif selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"][-1]
+ else:
+ HXa = Hm( Xa )
+ #
+ # Calcul de la covariance d'analyse
+ # ---------------------------------
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("JacobianMatrixAtOptimum") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HtM = HO["Tangent"].asMatrix(ValueForMethodForm = Xa)
+ HtM = HtM.reshape(Y.size,Xa.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HaM = HO["Adjoint"].asMatrix(ValueForMethodForm = Xa)
+ HaM = HaM.reshape(Xa.size,Y.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles"):
+ BI = B.getI()
+ RI = R.getI()
+ HessienneI = []
+ nb = Xa.size
+ for i in range(nb):
+ _ee = numpy.matrix(numpy.zeros(nb)).T
+ _ee[i] = 1.
+ _HtEE = numpy.dot(HtM,_ee)
+ _HtEE = numpy.asmatrix(numpy.ravel( _HtEE )).T
+ HessienneI.append( numpy.ravel( BI*_ee + HaM * (RI * _HtEE) ) )
+ HessienneI = numpy.matrix( HessienneI )
+ A = HessienneI.I
+ if min(A.shape) != max(A.shape):
+ raise ValueError("The %s a posteriori covariance matrix A is of shape %s, despites it has to be a squared matrix. There is an error in the observation operator, please check it."%(selfA._name,str(A.shape)))
+ if (numpy.diag(A) < 0).any():
+ raise ValueError("The %s a posteriori covariance matrix A has at least one negative value on its diagonal. There is an error in the observation operator, please check it."%(selfA._name,))
+ if logging.getLogger().level < logging.WARNING: # La verification n'a lieu qu'en debug
+ try:
+ L = numpy.linalg.cholesky( A )
+ except:
+ raise ValueError("The %s a posteriori covariance matrix A is not symmetric positive-definite. Please check your a priori covariances and your observation operator."%(selfA._name,))
+ if selfA._toStore("APosterioriCovariance"):
+ selfA.StoredVariables["APosterioriCovariance"].store( A )
+ if selfA._toStore("JacobianMatrixAtOptimum"):
+ selfA.StoredVariables["JacobianMatrixAtOptimum"].store( HtM )
+ if selfA._toStore("KalmanGainAtOptimum"):
+ if (Y.size <= Xb.size): KG = B * HaM * (R + numpy.dot(HtM, B * HaM)).I
+ elif (Y.size > Xb.size): KG = (BI + numpy.dot(HaM, RI * HtM)).I * HaM * RI
+ selfA.StoredVariables["KalmanGainAtOptimum"].store( KG )
+ #
+ # Calculs et/ou stockages supplémentaires
+ # ---------------------------------------
+ if selfA._toStore("Innovation") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("MahalanobisConsistency") or \
+ selfA._toStore("OMB"):
+ d = Y - HXb
+ if selfA._toStore("Innovation"):
+ selfA.StoredVariables["Innovation"].store( numpy.ravel(d) )
+ if selfA._toStore("BMA"):
+ selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(Xa) )
+ if selfA._toStore("OMA"):
+ selfA.StoredVariables["OMA"].store( numpy.ravel(Y) - numpy.ravel(HXa) )
+ if selfA._toStore("OMB"):
+ selfA.StoredVariables["OMB"].store( numpy.ravel(d) )
+ if selfA._toStore("SigmaObs2"):
+ TraceR = R.trace(Y.size)
+ selfA.StoredVariables["SigmaObs2"].store( float( (d.T * (numpy.asmatrix(numpy.ravel(Y)).T-numpy.asmatrix(numpy.ravel(HXa)).T)) ) / TraceR )
+ if selfA._toStore("MahalanobisConsistency"):
+ selfA.StoredVariables["MahalanobisConsistency"].store( float( 2.*MinJ/d.size ) )
+ if selfA._toStore("SimulationQuantiles"):
+ nech = selfA._parameters["NumberOfSamplesForQuantiles"]
+ HXa = numpy.matrix(numpy.ravel( HXa )).T
+ YfQ = None
+ for i in range(nech):
+ if selfA._parameters["SimulationForQuantiles"] == "Linear":
+ dXr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A) - Xa.A1).T
+ dYr = numpy.matrix(numpy.ravel( HtM * dXr )).T
+ Yr = HXa + dYr
+ elif selfA._parameters["SimulationForQuantiles"] == "NonLinear":
+ Xr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A)).T
+ Yr = numpy.matrix(numpy.ravel( Hm( Xr ) )).T
+ if YfQ is None:
+ YfQ = Yr
+ else:
+ YfQ = numpy.hstack((YfQ,Yr))
+ YfQ.sort(axis=-1)
+ YQ = None
+ for quantile in selfA._parameters["Quantiles"]:
+ if not (0. <= float(quantile) <= 1.): continue
+ indice = int(nech * float(quantile) - 1./nech)
+ if YQ is None: YQ = YfQ[:,indice]
+ else: YQ = numpy.hstack((YQ,YfQ[:,indice]))
+ selfA.StoredVariables["SimulationQuantiles"].store( YQ )
+ if selfA._toStore("SimulatedObservationAtBackground"):
+ selfA.StoredVariables["SimulatedObservationAtBackground"].store( numpy.ravel(HXb) )
+ if selfA._toStore("SimulatedObservationAtOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtOptimum"].store( numpy.ravel(HXa) )
+ #
+ return 0
- selfA est identique au "self" d'algorithme appelant et contient les
- valeurs.
+# ==============================================================================
+def senkf(selfA, Xb, Y, U, HO, EM, CM, R, B, Q, VariantM="KalmanFilterFormula"):
+ """
+ Stochastic EnKF
"""
if selfA._parameters["EstimationOf"] == "Parameters":
selfA._parameters["StoreInternalVariables"] = True
or selfA._toStore("CurrentOptimum") \
or selfA._toStore("APosterioriCovariance"):
BI = B.getI()
- RI = R.getI()
+ RI = R.getI()
#
# Initialisation
# --------------
else: Rn = R
if hasattr(Q,"asfullmatrix"): Qn = Q.asfullmatrix(__n)
else: Qn = Q
- Xn = EnsembleOfBackgroundPerturbations( Xb, Pn, __m )
+ Xn = EnsembleOfBackgroundPerturbations( Xb, None, __m )
#
if len(selfA.StoredVariables["Analysis"])==0 or not selfA._parameters["nextStep"]:
- selfA.StoredVariables["Analysis"].store( numpy.ravel(Xb) )
+ selfA.StoredVariables["Analysis"].store( Xb )
if selfA._toStore("APosterioriCovariance"):
selfA.StoredVariables["APosterioriCovariance"].store( Pn )
covarianceXa = Pn
#
for step in range(duration-1):
if hasattr(Y,"store"):
- Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y[step+1] ).reshape((__p,1))
else:
- Ynpu = numpy.ravel( Y ).reshape((__p,-1))
+ Ynpu = numpy.ravel( Y ).reshape((__p,1))
#
if U is not None:
if hasattr(U,"store") and len(U)>1:
selfA._parameters["InflationFactor"],
)
#
+ if selfA._parameters["EstimationOf"] == "State": # Forecast + Q and observation of forecast
+ EMX = M( [(Xn[:,i], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ qi = numpy.random.multivariate_normal(numpy.zeros(__n), Qn, size=__m).T
+ Xn_predicted = EMX + qi
+ HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ if Cm is not None and Un is not None: # Attention : si Cm est aussi dans M, doublon !
+ Cm = Cm.reshape(__n,Un.size) # ADAO & check shape
+ Xn_predicted = Xn_predicted + Cm * Un
+ elif selfA._parameters["EstimationOf"] == "Parameters": # Observation of forecast
+ # --- > Par principe, M = Id, Q = 0
+ Xn_predicted = Xn
+ HX_predicted = H( [(Xn_predicted[:,i], Un) for i in range(__m)],
+ argsAsSerie = True,
+ returnSerieAsArrayMatrix = True )
+ #
+ # Mean of forecast and observation of forecast
+ Xfm = Xn_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
+ Hfm = HX_predicted.mean(axis=1, dtype=mfp).astype('float').reshape((__p,1))
+ #
#--------------------------
- if VariantM == "IEnKF12":
- Xfm = numpy.ravel(Xn.mean(axis=1, dtype=mfp).astype('float'))
- EaX = EnsembleOfAnomalies( Xn ) / numpy.sqrt(__m-1)
- __j = 0
- Deltaw = 1
- if not BnotT:
- Ta = numpy.eye(__m)
- vw = numpy.zeros(__m)
- while numpy.linalg.norm(Deltaw) >= _e and __j <= _jmax:
- vx1 = (Xfm + EaX @ vw).reshape((__n,-1))
- #
- if BnotT:
- E1 = vx1 + _epsilon * EaX
- else:
- E1 = vx1 + numpy.sqrt(__m-1) * EaX @ Ta
- #
- if selfA._parameters["EstimationOf"] == "State": # Forecast + Q
- E2 = M( [(E1[:,i,numpy.newaxis], Un) for i in range(__m)],
- argsAsSerie = True,
- returnSerieAsArrayMatrix = True )
- elif selfA._parameters["EstimationOf"] == "Parameters":
- # --- > Par principe, M = Id
- E2 = Xn
- vx2 = E2.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
- vy1 = H((vx2, Un)).reshape((__p,-1))
- #
- HE2 = H( [(E2[:,i,numpy.newaxis], Un) for i in range(__m)],
- argsAsSerie = True,
- returnSerieAsArrayMatrix = True )
- vy2 = HE2.mean(axis=1, dtype=mfp).astype('float').reshape((__p,-1))
- #
- if BnotT:
- EaY = (HE2 - vy2) / _epsilon
- else:
- EaY = ( (HE2 - vy2) @ numpy.linalg.inv(Ta) ) / numpy.sqrt(__m-1)
- #
- GradJ = numpy.ravel(vw[:,None] - EaY.transpose() @ (RI * ( Ynpu - vy1 )))
- mH = numpy.eye(__m) + EaY.transpose() @ (RI * EaY)
- Deltaw = - numpy.linalg.solve(mH,GradJ)
- #
- vw = vw + Deltaw
- #
- if not BnotT:
- Ta = numpy.real(scipy.linalg.sqrtm(numpy.linalg.inv( mH )))
- #
- __j = __j + 1
+ if VariantM == "KalmanFilterFormula05":
+ PfHT, HPfHT = 0., 0.
+ for i in range(__m):
+ Exfi = Xn_predicted[:,i].reshape((__n,1)) - Xfm
+ Eyfi = HX_predicted[:,i].reshape((__p,1)) - Hfm
+ PfHT += Exfi * Eyfi.T
+ HPfHT += Eyfi * Eyfi.T
+ PfHT = (1./(__m-1)) * PfHT
+ HPfHT = (1./(__m-1)) * HPfHT
+ Kn = PfHT * ( R + HPfHT ).I
+ del PfHT, HPfHT
#
- A2 = EnsembleOfAnomalies( E2 )
+ for i in range(__m):
+ ri = numpy.random.multivariate_normal(numpy.zeros(__p), Rn)
+ Xn[:,i] = numpy.ravel(Xn_predicted[:,i]) + Kn @ (numpy.ravel(Ynpu) + ri - HX_predicted[:,i])
+ #--------------------------
+ elif VariantM == "KalmanFilterFormula16":
+ EpY = EnsembleOfCenteredPerturbations(Ynpu, Rn, __m)
+ EpYm = EpY.mean(axis=1, dtype=mfp).astype('float').reshape((__p,1))
#
- if BnotT:
- Ta = numpy.real(scipy.linalg.sqrtm(numpy.linalg.inv( mH )))
- A2 = numpy.sqrt(__m-1) * A2 @ Ta / _epsilon
+ EaX = EnsembleOfAnomalies( Xn_predicted ) / math.sqrt(__m-1)
+ EaY = (HX_predicted - Hfm - EpY + EpYm) / math.sqrt(__m-1)
#
- Xn = vx2 + A2
+ Kn = EaX @ EaY.T @ numpy.linalg.inv( EaY @ EaY.T)
+ #
+ for i in range(__m):
+ Xn[:,i] = numpy.ravel(Xn_predicted[:,i]) + Kn @ (numpy.ravel(EpY[:,i]) - HX_predicted[:,i])
#--------------------------
else:
raise ValueError("VariantM has to be chosen in the authorized methods list.")
selfA._parameters["InflationFactor"],
)
#
- Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,-1))
+ Xa = Xn.mean(axis=1, dtype=mfp).astype('float').reshape((__n,1))
#--------------------------
#
if selfA._parameters["StoreInternalVariables"] \
or selfA._toStore("CurrentState"):
selfA.StoredVariables["CurrentState"].store( Xn )
if selfA._toStore("ForecastState"):
- selfA.StoredVariables["ForecastState"].store( E2 )
+ selfA.StoredVariables["ForecastState"].store( EMX )
if selfA._toStore("BMA"):
- selfA.StoredVariables["BMA"].store( E2 - Xa )
+ selfA.StoredVariables["BMA"].store( EMX - Xa )
if selfA._toStore("InnovationAtCurrentState"):
- selfA.StoredVariables["InnovationAtCurrentState"].store( - HE2 + Ynpu.reshape((__p,-1)) )
+ selfA.StoredVariables["InnovationAtCurrentState"].store( - HX_predicted + Ynpu )
if selfA._toStore("SimulatedObservationAtCurrentState") \
or selfA._toStore("SimulatedObservationAtCurrentOptimum"):
- selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HE2 )
+ selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( HX_predicted )
# ---> autres
if selfA._parameters["StoreInternalVariables"] \
or selfA._toStore("CostFunctionJ") \
if selfA._toStore("CostFunctionJAtCurrentOptimum"):
selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
if selfA._toStore("APosterioriCovariance"):
- Eai = EnsembleOfAnomalies( Xn ) / numpy.sqrt(__m-1) # Anomalies
- Pn = Eai @ Eai.T
- Pn = 0.5 * (Pn + Pn.T)
- selfA.StoredVariables["APosterioriCovariance"].store( Pn )
+ selfA.StoredVariables["APosterioriCovariance"].store( EnsembleErrorCovariance(Xn) )
if selfA._parameters["EstimationOf"] == "Parameters" \
and J < previousJMinimum:
previousJMinimum = J
#
return 0
+# ==============================================================================
+def std3dvar(selfA, Xb, Y, U, HO, EM, CM, R, B, Q):
+ """
+ 3DVAR
+ """
+ #
+ # Initialisations
+ # ---------------
+ #
+ # Opérateurs
+ Hm = HO["Direct"].appliedTo
+ Ha = HO["Adjoint"].appliedInXTo
+ #
+ # Utilisation éventuelle d'un vecteur H(Xb) précalculé
+ if HO["AppliedInX"] is not None and "HXb" in HO["AppliedInX"]:
+ HXb = Hm( Xb, HO["AppliedInX"]["HXb"] )
+ else:
+ HXb = Hm( Xb )
+ HXb = numpy.asmatrix(numpy.ravel( HXb )).T
+ if Y.size != HXb.size:
+ raise ValueError("The size %i of observations Y and %i of observed calculation H(X) are different, they have to be identical."%(Y.size,HXb.size))
+ if max(Y.shape) != max(HXb.shape):
+ raise ValueError("The shapes %s of observations Y and %s of observed calculation H(X) are different, they have to be identical."%(Y.shape,HXb.shape))
+ #
+ if selfA._toStore("JacobianMatrixAtBackground"):
+ HtMb = HO["Tangent"].asMatrix(ValueForMethodForm = Xb)
+ HtMb = HtMb.reshape(Y.size,Xb.size) # ADAO & check shape
+ selfA.StoredVariables["JacobianMatrixAtBackground"].store( HtMb )
+ #
+ # Précalcul des inversions de B et R
+ BI = B.getI()
+ RI = R.getI()
+ #
+ # Point de démarrage de l'optimisation
+ Xini = selfA._parameters["InitializationPoint"]
+ #
+ # Définition de la fonction-coût
+ # ------------------------------
+ def CostFunction(x):
+ _X = numpy.asmatrix(numpy.ravel( x )).T
+ if selfA._parameters["StoreInternalVariables"] or \
+ selfA._toStore("CurrentState") or \
+ selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentState"].store( _X )
+ _HX = Hm( _X )
+ _HX = numpy.asmatrix(numpy.ravel( _HX )).T
+ _Innovation = Y - _HX
+ if selfA._toStore("SimulatedObservationAtCurrentState") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( _HX )
+ if selfA._toStore("InnovationAtCurrentState"):
+ selfA.StoredVariables["InnovationAtCurrentState"].store( _Innovation )
+ #
+ Jb = float( 0.5 * (_X - Xb).T * BI * (_X - Xb) )
+ Jo = float( 0.5 * _Innovation.T * RI * _Innovation )
+ J = Jb + Jo
+ #
+ selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["CostFunctionJ"]) )
+ selfA.StoredVariables["CostFunctionJb"].store( Jb )
+ selfA.StoredVariables["CostFunctionJo"].store( Jo )
+ selfA.StoredVariables["CostFunctionJ" ].store( J )
+ if selfA._toStore("IndexOfOptimum") or \
+ selfA._toStore("CurrentOptimum") or \
+ selfA._toStore("CostFunctionJAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJbAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJoAtCurrentOptimum") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ if selfA._toStore("IndexOfOptimum"):
+ selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
+ if selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["CurrentState"][IndexMin] )
+ if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin] )
+ if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
+ if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
+ if selfA._toStore("CostFunctionJAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
+ return J
+ #
+ def GradientOfCostFunction(x):
+ _X = numpy.asmatrix(numpy.ravel( x )).T
+ _HX = Hm( _X )
+ _HX = numpy.asmatrix(numpy.ravel( _HX )).T
+ GradJb = BI * (_X - Xb)
+ GradJo = - Ha( (_X, RI * (Y - _HX)) )
+ GradJ = numpy.ravel( GradJb ) + numpy.ravel( GradJo )
+ return GradJ
+ #
+ # Minimisation de la fonctionnelle
+ # --------------------------------
+ nbPreviousSteps = selfA.StoredVariables["CostFunctionJ"].stepnumber()
+ #
+ if selfA._parameters["Minimizer"] == "LBFGSB":
+ if "0.19" <= scipy.version.version <= "1.1.0":
+ import lbfgsbhlt as optimiseur
+ else:
+ import scipy.optimize as optimiseur
+ Minimum, J_optimal, Informations = optimiseur.fmin_l_bfgs_b(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"]-1,
+ factr = selfA._parameters["CostDecrementTolerance"]*1.e14,
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ iprint = selfA._parameters["optiprint"],
+ )
+ nfeval = Informations['funcalls']
+ rc = Informations['warnflag']
+ elif selfA._parameters["Minimizer"] == "TNC":
+ Minimum, nfeval, rc = scipy.optimize.fmin_tnc(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"],
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ ftol = selfA._parameters["CostDecrementTolerance"],
+ messages = selfA._parameters["optmessages"],
+ )
+ elif selfA._parameters["Minimizer"] == "CG":
+ Minimum, fopt, nfeval, grad_calls, rc = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "NCG":
+ Minimum, fopt, nfeval, grad_calls, hcalls, rc = scipy.optimize.fmin_ncg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ avextol = selfA._parameters["CostDecrementTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "BFGS":
+ Minimum, fopt, gopt, Hopt, nfeval, grad_calls, rc = scipy.optimize.fmin_bfgs(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ else:
+ raise ValueError("Error in Minimizer name: %s"%selfA._parameters["Minimizer"])
+ #
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ MinJ = selfA.StoredVariables["CostFunctionJ"][IndexMin]
+ #
+ # Correction pour pallier a un bug de TNC sur le retour du Minimum
+ # ----------------------------------------------------------------
+ if selfA._parameters["StoreInternalVariables"] or selfA._toStore("CurrentState"):
+ Minimum = selfA.StoredVariables["CurrentState"][IndexMin]
+ #
+ # Obtention de l'analyse
+ # ----------------------
+ Xa = numpy.asmatrix(numpy.ravel( Minimum )).T
+ #
+ selfA.StoredVariables["Analysis"].store( Xa )
+ #
+ if selfA._toStore("OMA") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("SimulatedObservationAtOptimum"):
+ if selfA._toStore("SimulatedObservationAtCurrentState"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin]
+ elif selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"][-1]
+ else:
+ HXa = Hm( Xa )
+ #
+ # Calcul de la covariance d'analyse
+ # ---------------------------------
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("JacobianMatrixAtOptimum") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HtM = HO["Tangent"].asMatrix(ValueForMethodForm = Xa)
+ HtM = HtM.reshape(Y.size,Xa.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HaM = HO["Adjoint"].asMatrix(ValueForMethodForm = Xa)
+ HaM = HaM.reshape(Xa.size,Y.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles"):
+ HessienneI = []
+ nb = Xa.size
+ for i in range(nb):
+ _ee = numpy.matrix(numpy.zeros(nb)).T
+ _ee[i] = 1.
+ _HtEE = numpy.dot(HtM,_ee)
+ _HtEE = numpy.asmatrix(numpy.ravel( _HtEE )).T
+ HessienneI.append( numpy.ravel( BI*_ee + HaM * (RI * _HtEE) ) )
+ HessienneI = numpy.matrix( HessienneI )
+ A = HessienneI.I
+ if min(A.shape) != max(A.shape):
+ raise ValueError("The %s a posteriori covariance matrix A is of shape %s, despites it has to be a squared matrix. There is an error in the observation operator, please check it."%(selfA._name,str(A.shape)))
+ if (numpy.diag(A) < 0).any():
+ raise ValueError("The %s a posteriori covariance matrix A has at least one negative value on its diagonal. There is an error in the observation operator, please check it."%(selfA._name,))
+ if logging.getLogger().level < logging.WARNING: # La verification n'a lieu qu'en debug
+ try:
+ L = numpy.linalg.cholesky( A )
+ except:
+ raise ValueError("The %s a posteriori covariance matrix A is not symmetric positive-definite. Please check your a priori covariances and your observation operator."%(selfA._name,))
+ if selfA._toStore("APosterioriCovariance"):
+ selfA.StoredVariables["APosterioriCovariance"].store( A )
+ if selfA._toStore("JacobianMatrixAtOptimum"):
+ selfA.StoredVariables["JacobianMatrixAtOptimum"].store( HtM )
+ if selfA._toStore("KalmanGainAtOptimum"):
+ if (Y.size <= Xb.size): KG = B * HaM * (R + numpy.dot(HtM, B * HaM)).I
+ elif (Y.size > Xb.size): KG = (BI + numpy.dot(HaM, RI * HtM)).I * HaM * RI
+ selfA.StoredVariables["KalmanGainAtOptimum"].store( KG )
+ #
+ # Calculs et/ou stockages supplémentaires
+ # ---------------------------------------
+ if selfA._toStore("Innovation") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("MahalanobisConsistency") or \
+ selfA._toStore("OMB"):
+ d = Y - HXb
+ if selfA._toStore("Innovation"):
+ selfA.StoredVariables["Innovation"].store( numpy.ravel(d) )
+ if selfA._toStore("BMA"):
+ selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(Xa) )
+ if selfA._toStore("OMA"):
+ selfA.StoredVariables["OMA"].store( numpy.ravel(Y) - numpy.ravel(HXa) )
+ if selfA._toStore("OMB"):
+ selfA.StoredVariables["OMB"].store( numpy.ravel(d) )
+ if selfA._toStore("SigmaObs2"):
+ TraceR = R.trace(Y.size)
+ selfA.StoredVariables["SigmaObs2"].store( float( (d.T * (numpy.asmatrix(numpy.ravel(Y)).T-numpy.asmatrix(numpy.ravel(HXa)).T)) ) / TraceR )
+ if selfA._toStore("MahalanobisConsistency"):
+ selfA.StoredVariables["MahalanobisConsistency"].store( float( 2.*MinJ/d.size ) )
+ if selfA._toStore("SimulationQuantiles"):
+ nech = selfA._parameters["NumberOfSamplesForQuantiles"]
+ HXa = numpy.matrix(numpy.ravel( HXa )).T
+ YfQ = None
+ for i in range(nech):
+ if selfA._parameters["SimulationForQuantiles"] == "Linear":
+ dXr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A) - Xa.A1).T
+ dYr = numpy.matrix(numpy.ravel( HtM * dXr )).T
+ Yr = HXa + dYr
+ elif selfA._parameters["SimulationForQuantiles"] == "NonLinear":
+ Xr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A)).T
+ Yr = numpy.matrix(numpy.ravel( Hm( Xr ) )).T
+ if YfQ is None:
+ YfQ = Yr
+ else:
+ YfQ = numpy.hstack((YfQ,Yr))
+ YfQ.sort(axis=-1)
+ YQ = None
+ for quantile in selfA._parameters["Quantiles"]:
+ if not (0. <= float(quantile) <= 1.): continue
+ indice = int(nech * float(quantile) - 1./nech)
+ if YQ is None: YQ = YfQ[:,indice]
+ else: YQ = numpy.hstack((YQ,YfQ[:,indice]))
+ selfA.StoredVariables["SimulationQuantiles"].store( YQ )
+ if selfA._toStore("SimulatedObservationAtBackground"):
+ selfA.StoredVariables["SimulatedObservationAtBackground"].store( numpy.ravel(HXb) )
+ if selfA._toStore("SimulatedObservationAtOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtOptimum"].store( numpy.ravel(HXa) )
+ #
+ return 0
+
+# ==============================================================================
+def std4dvar(selfA, Xb, Y, U, HO, EM, CM, R, B, Q):
+ """
+ 4DVAR
+ """
+ #
+ # Initialisations
+ # ---------------
+ #
+ # Opérateurs
+ Hm = HO["Direct"].appliedControledFormTo
+ Mm = EM["Direct"].appliedControledFormTo
+ #
+ if CM is not None and "Tangent" in CM and U is not None:
+ Cm = CM["Tangent"].asMatrix(Xb)
+ else:
+ Cm = None
+ #
+ def Un(_step):
+ if U is not None:
+ if hasattr(U,"store") and 1<=_step<len(U) :
+ _Un = numpy.asmatrix(numpy.ravel( U[_step] )).T
+ elif hasattr(U,"store") and len(U)==1:
+ _Un = numpy.asmatrix(numpy.ravel( U[0] )).T
+ else:
+ _Un = numpy.asmatrix(numpy.ravel( U )).T
+ else:
+ _Un = None
+ return _Un
+ def CmUn(_xn,_un):
+ if Cm is not None and _un is not None: # Attention : si Cm est aussi dans M, doublon !
+ _Cm = Cm.reshape(_xn.size,_un.size) # ADAO & check shape
+ _CmUn = _Cm * _un
+ else:
+ _CmUn = 0.
+ return _CmUn
+ #
+ # Remarque : les observations sont exploitées à partir du pas de temps
+ # numéro 1, et sont utilisées dans Yo comme rangées selon ces indices.
+ # Donc le pas 0 n'est pas utilisé puisque la première étape commence
+ # avec l'observation du pas 1.
+ #
+ # Nombre de pas identique au nombre de pas d'observations
+ if hasattr(Y,"stepnumber"):
+ duration = Y.stepnumber()
+ else:
+ duration = 2
+ #
+ # Précalcul des inversions de B et R
+ BI = B.getI()
+ RI = R.getI()
+ #
+ # Point de démarrage de l'optimisation
+ Xini = selfA._parameters["InitializationPoint"]
+ #
+ # Définition de la fonction-coût
+ # ------------------------------
+ selfA.DirectCalculation = [None,] # Le pas 0 n'est pas observé
+ selfA.DirectInnovation = [None,] # Le pas 0 n'est pas observé
+ def CostFunction(x):
+ _X = numpy.asmatrix(numpy.ravel( x )).T
+ if selfA._parameters["StoreInternalVariables"] or \
+ selfA._toStore("CurrentState") or \
+ selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentState"].store( _X )
+ Jb = float( 0.5 * (_X - Xb).T * BI * (_X - Xb) )
+ selfA.DirectCalculation = [None,]
+ selfA.DirectInnovation = [None,]
+ Jo = 0.
+ _Xn = _X
+ for step in range(0,duration-1):
+ if hasattr(Y,"store"):
+ _Ynpu = numpy.asmatrix(numpy.ravel( Y[step+1] )).T
+ else:
+ _Ynpu = numpy.asmatrix(numpy.ravel( Y )).T
+ _Un = Un(step)
+ #
+ # Etape d'évolution
+ if selfA._parameters["EstimationOf"] == "State":
+ _Xn = Mm( (_Xn, _Un) ) + CmUn(_Xn, _Un)
+ elif selfA._parameters["EstimationOf"] == "Parameters":
+ pass
+ #
+ if selfA._parameters["Bounds"] is not None and selfA._parameters["ConstrainedBy"] == "EstimateProjection":
+ _Xn = numpy.max(numpy.hstack((_Xn,numpy.asmatrix(selfA._parameters["Bounds"])[:,0])),axis=1)
+ _Xn = numpy.min(numpy.hstack((_Xn,numpy.asmatrix(selfA._parameters["Bounds"])[:,1])),axis=1)
+ #
+ # Etape de différence aux observations
+ if selfA._parameters["EstimationOf"] == "State":
+ _YmHMX = _Ynpu - numpy.asmatrix(numpy.ravel( Hm( (_Xn, None) ) )).T
+ elif selfA._parameters["EstimationOf"] == "Parameters":
+ _YmHMX = _Ynpu - numpy.asmatrix(numpy.ravel( Hm( (_Xn, _Un) ) )).T - CmUn(_Xn, _Un)
+ #
+ # Stockage de l'état
+ selfA.DirectCalculation.append( _Xn )
+ selfA.DirectInnovation.append( _YmHMX )
+ #
+ # Ajout dans la fonctionnelle d'observation
+ Jo = Jo + 0.5 * float( _YmHMX.T * RI * _YmHMX )
+ J = Jb + Jo
+ #
+ selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["CostFunctionJ"]) )
+ selfA.StoredVariables["CostFunctionJb"].store( Jb )
+ selfA.StoredVariables["CostFunctionJo"].store( Jo )
+ selfA.StoredVariables["CostFunctionJ" ].store( J )
+ if selfA._toStore("IndexOfOptimum") or \
+ selfA._toStore("CurrentOptimum") or \
+ selfA._toStore("CostFunctionJAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJbAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJoAtCurrentOptimum"):
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ if selfA._toStore("IndexOfOptimum"):
+ selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
+ if selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["CurrentState"][IndexMin] )
+ if selfA._toStore("CostFunctionJAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
+ if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
+ if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
+ return J
+ #
+ def GradientOfCostFunction(x):
+ _X = numpy.asmatrix(numpy.ravel( x )).T
+ GradJb = BI * (_X - Xb)
+ GradJo = 0.
+ for step in range(duration-1,0,-1):
+ # Étape de récupération du dernier stockage de l'évolution
+ _Xn = selfA.DirectCalculation.pop()
+ # Étape de récupération du dernier stockage de l'innovation
+ _YmHMX = selfA.DirectInnovation.pop()
+ # Calcul des adjoints
+ Ha = HO["Adjoint"].asMatrix(ValueForMethodForm = _Xn)
+ Ha = Ha.reshape(_Xn.size,_YmHMX.size) # ADAO & check shape
+ Ma = EM["Adjoint"].asMatrix(ValueForMethodForm = _Xn)
+ Ma = Ma.reshape(_Xn.size,_Xn.size) # ADAO & check shape
+ # Calcul du gradient par état adjoint
+ GradJo = GradJo + Ha * RI * _YmHMX # Équivaut pour Ha linéaire à : Ha( (_Xn, RI * _YmHMX) )
+ GradJo = Ma * GradJo # Équivaut pour Ma linéaire à : Ma( (_Xn, GradJo) )
+ GradJ = numpy.ravel( GradJb ) - numpy.ravel( GradJo )
+ return GradJ
+ #
+ # Minimisation de la fonctionnelle
+ # --------------------------------
+ nbPreviousSteps = selfA.StoredVariables["CostFunctionJ"].stepnumber()
+ #
+ if selfA._parameters["Minimizer"] == "LBFGSB":
+ if "0.19" <= scipy.version.version <= "1.1.0":
+ import lbfgsbhlt as optimiseur
+ else:
+ import scipy.optimize as optimiseur
+ Minimum, J_optimal, Informations = optimiseur.fmin_l_bfgs_b(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"]-1,
+ factr = selfA._parameters["CostDecrementTolerance"]*1.e14,
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ iprint = selfA._parameters["optiprint"],
+ )
+ nfeval = Informations['funcalls']
+ rc = Informations['warnflag']
+ elif selfA._parameters["Minimizer"] == "TNC":
+ Minimum, nfeval, rc = scipy.optimize.fmin_tnc(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"],
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ ftol = selfA._parameters["CostDecrementTolerance"],
+ messages = selfA._parameters["optmessages"],
+ )
+ elif selfA._parameters["Minimizer"] == "CG":
+ Minimum, fopt, nfeval, grad_calls, rc = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "NCG":
+ Minimum, fopt, nfeval, grad_calls, hcalls, rc = scipy.optimize.fmin_ncg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ avextol = selfA._parameters["CostDecrementTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "BFGS":
+ Minimum, fopt, gopt, Hopt, nfeval, grad_calls, rc = scipy.optimize.fmin_bfgs(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ else:
+ raise ValueError("Error in Minimizer name: %s"%selfA._parameters["Minimizer"])
+ #
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ MinJ = selfA.StoredVariables["CostFunctionJ"][IndexMin]
+ #
+ # Correction pour pallier a un bug de TNC sur le retour du Minimum
+ # ----------------------------------------------------------------
+ if selfA._parameters["StoreInternalVariables"] or selfA._toStore("CurrentState"):
+ Minimum = selfA.StoredVariables["CurrentState"][IndexMin]
+ #
+ # Obtention de l'analyse
+ # ----------------------
+ Xa = numpy.asmatrix(numpy.ravel( Minimum )).T
+ #
+ selfA.StoredVariables["Analysis"].store( Xa )
+ #
+ # Calculs et/ou stockages supplémentaires
+ # ---------------------------------------
+ if selfA._toStore("BMA"):
+ selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(Xa) )
+ #
+ return 0
+
+# ==============================================================================
+def van3dvar(selfA, Xb, Y, U, HO, EM, CM, R, B, Q):
+ """
+ 3DVAR variational analysis with no inversion of B
+ """
+ #
+ # Initialisations
+ # ---------------
+ #
+ # Opérateurs
+ Hm = HO["Direct"].appliedTo
+ Ha = HO["Adjoint"].appliedInXTo
+ #
+ # Précalcul des inversions de B et R
+ BT = B.getT()
+ RI = R.getI()
+ #
+ # Point de démarrage de l'optimisation
+ Xini = numpy.zeros(Xb.shape)
+ #
+ # Définition de la fonction-coût
+ # ------------------------------
+ def CostFunction(v):
+ _V = numpy.asmatrix(numpy.ravel( v )).T
+ _X = Xb + B * _V
+ if selfA._parameters["StoreInternalVariables"] or \
+ selfA._toStore("CurrentState") or \
+ selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentState"].store( _X )
+ _HX = Hm( _X )
+ _HX = numpy.asmatrix(numpy.ravel( _HX )).T
+ _Innovation = Y - _HX
+ if selfA._toStore("SimulatedObservationAtCurrentState") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentState"].store( _HX )
+ if selfA._toStore("InnovationAtCurrentState"):
+ selfA.StoredVariables["InnovationAtCurrentState"].store( _Innovation )
+ #
+ Jb = float( 0.5 * _V.T * BT * _V )
+ Jo = float( 0.5 * _Innovation.T * RI * _Innovation )
+ J = Jb + Jo
+ #
+ selfA.StoredVariables["CurrentIterationNumber"].store( len(selfA.StoredVariables["CostFunctionJ"]) )
+ selfA.StoredVariables["CostFunctionJb"].store( Jb )
+ selfA.StoredVariables["CostFunctionJo"].store( Jo )
+ selfA.StoredVariables["CostFunctionJ" ].store( J )
+ if selfA._toStore("IndexOfOptimum") or \
+ selfA._toStore("CurrentOptimum") or \
+ selfA._toStore("CostFunctionJAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJbAtCurrentOptimum") or \
+ selfA._toStore("CostFunctionJoAtCurrentOptimum") or \
+ selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ if selfA._toStore("IndexOfOptimum"):
+ selfA.StoredVariables["IndexOfOptimum"].store( IndexMin )
+ if selfA._toStore("CurrentOptimum"):
+ selfA.StoredVariables["CurrentOptimum"].store( selfA.StoredVariables["CurrentState"][IndexMin] )
+ if selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"].store( selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin] )
+ if selfA._toStore("CostFunctionJbAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJbAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJb"][IndexMin] )
+ if selfA._toStore("CostFunctionJoAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJoAtCurrentOptimum"].store( selfA.StoredVariables["CostFunctionJo"][IndexMin] )
+ if selfA._toStore("CostFunctionJAtCurrentOptimum"):
+ selfA.StoredVariables["CostFunctionJAtCurrentOptimum" ].store( selfA.StoredVariables["CostFunctionJ" ][IndexMin] )
+ return J
+ #
+ def GradientOfCostFunction(v):
+ _V = numpy.asmatrix(numpy.ravel( v )).T
+ _X = Xb + B * _V
+ _HX = Hm( _X )
+ _HX = numpy.asmatrix(numpy.ravel( _HX )).T
+ GradJb = BT * _V
+ GradJo = - Ha( (_X, RI * (Y - _HX)) )
+ GradJ = numpy.ravel( GradJb ) + numpy.ravel( GradJo )
+ return GradJ
+ #
+ # Minimisation de la fonctionnelle
+ # --------------------------------
+ nbPreviousSteps = selfA.StoredVariables["CostFunctionJ"].stepnumber()
+ #
+ if selfA._parameters["Minimizer"] == "LBFGSB":
+ if "0.19" <= scipy.version.version <= "1.1.0":
+ import lbfgsbhlt as optimiseur
+ else:
+ import scipy.optimize as optimiseur
+ Minimum, J_optimal, Informations = optimiseur.fmin_l_bfgs_b(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"]-1,
+ factr = selfA._parameters["CostDecrementTolerance"]*1.e14,
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ iprint = selfA._parameters["optiprint"],
+ )
+ nfeval = Informations['funcalls']
+ rc = Informations['warnflag']
+ elif selfA._parameters["Minimizer"] == "TNC":
+ Minimum, nfeval, rc = scipy.optimize.fmin_tnc(
+ func = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ bounds = selfA._parameters["Bounds"],
+ maxfun = selfA._parameters["MaximumNumberOfSteps"],
+ pgtol = selfA._parameters["ProjectedGradientTolerance"],
+ ftol = selfA._parameters["CostDecrementTolerance"],
+ messages = selfA._parameters["optmessages"],
+ )
+ elif selfA._parameters["Minimizer"] == "CG":
+ Minimum, fopt, nfeval, grad_calls, rc = scipy.optimize.fmin_cg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "NCG":
+ Minimum, fopt, nfeval, grad_calls, hcalls, rc = scipy.optimize.fmin_ncg(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ avextol = selfA._parameters["CostDecrementTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ elif selfA._parameters["Minimizer"] == "BFGS":
+ Minimum, fopt, gopt, Hopt, nfeval, grad_calls, rc = scipy.optimize.fmin_bfgs(
+ f = CostFunction,
+ x0 = Xini,
+ fprime = GradientOfCostFunction,
+ args = (),
+ maxiter = selfA._parameters["MaximumNumberOfSteps"],
+ gtol = selfA._parameters["GradientNormTolerance"],
+ disp = selfA._parameters["optdisp"],
+ full_output = True,
+ )
+ else:
+ raise ValueError("Error in Minimizer name: %s"%selfA._parameters["Minimizer"])
+ #
+ IndexMin = numpy.argmin( selfA.StoredVariables["CostFunctionJ"][nbPreviousSteps:] ) + nbPreviousSteps
+ MinJ = selfA.StoredVariables["CostFunctionJ"][IndexMin]
+ #
+ # Correction pour pallier a un bug de TNC sur le retour du Minimum
+ # ----------------------------------------------------------------
+ if selfA._parameters["StoreInternalVariables"] or selfA._toStore("CurrentState"):
+ Minimum = selfA.StoredVariables["CurrentState"][IndexMin]
+ Minimum = numpy.asmatrix(numpy.ravel( Minimum )).T
+ else:
+ Minimum = Xb + B * numpy.asmatrix(numpy.ravel( Minimum )).T
+ #
+ # Obtention de l'analyse
+ # ----------------------
+ Xa = Minimum
+ #
+ selfA.StoredVariables["Analysis"].store( Xa )
+ #
+ if selfA._toStore("OMA") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("SimulatedObservationAtOptimum"):
+ if selfA._toStore("SimulatedObservationAtCurrentState"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentState"][IndexMin]
+ elif selfA._toStore("SimulatedObservationAtCurrentOptimum"):
+ HXa = selfA.StoredVariables["SimulatedObservationAtCurrentOptimum"][-1]
+ else:
+ HXa = Hm( Xa )
+ #
+ # Calcul de la covariance d'analyse
+ # ---------------------------------
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("JacobianMatrixAtOptimum") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HtM = HO["Tangent"].asMatrix(ValueForMethodForm = Xa)
+ HtM = HtM.reshape(Y.size,Xa.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles") or \
+ selfA._toStore("KalmanGainAtOptimum"):
+ HaM = HO["Adjoint"].asMatrix(ValueForMethodForm = Xa)
+ HaM = HaM.reshape(Xa.size,Y.size) # ADAO & check shape
+ if selfA._toStore("APosterioriCovariance") or \
+ selfA._toStore("SimulationQuantiles"):
+ BI = B.getI()
+ HessienneI = []
+ nb = Xa.size
+ for i in range(nb):
+ _ee = numpy.matrix(numpy.zeros(nb)).T
+ _ee[i] = 1.
+ _HtEE = numpy.dot(HtM,_ee)
+ _HtEE = numpy.asmatrix(numpy.ravel( _HtEE )).T
+ HessienneI.append( numpy.ravel( BI*_ee + HaM * (RI * _HtEE) ) )
+ HessienneI = numpy.matrix( HessienneI )
+ A = HessienneI.I
+ if min(A.shape) != max(A.shape):
+ raise ValueError("The %s a posteriori covariance matrix A is of shape %s, despites it has to be a squared matrix. There is an error in the observation operator, please check it."%(selfA._name,str(A.shape)))
+ if (numpy.diag(A) < 0).any():
+ raise ValueError("The %s a posteriori covariance matrix A has at least one negative value on its diagonal. There is an error in the observation operator, please check it."%(selfA._name,))
+ if logging.getLogger().level < logging.WARNING: # La verification n'a lieu qu'en debug
+ try:
+ L = numpy.linalg.cholesky( A )
+ except:
+ raise ValueError("The %s a posteriori covariance matrix A is not symmetric positive-definite. Please check your a priori covariances and your observation operator."%(selfA._name,))
+ if selfA._toStore("APosterioriCovariance"):
+ selfA.StoredVariables["APosterioriCovariance"].store( A )
+ if selfA._toStore("JacobianMatrixAtOptimum"):
+ selfA.StoredVariables["JacobianMatrixAtOptimum"].store( HtM )
+ if selfA._toStore("KalmanGainAtOptimum"):
+ if (Y.size <= Xb.size): KG = B * HaM * (R + numpy.dot(HtM, B * HaM)).I
+ elif (Y.size > Xb.size): KG = (BI + numpy.dot(HaM, RI * HtM)).I * HaM * RI
+ selfA.StoredVariables["KalmanGainAtOptimum"].store( KG )
+ #
+ # Calculs et/ou stockages supplémentaires
+ # ---------------------------------------
+ if selfA._toStore("Innovation") or \
+ selfA._toStore("SigmaObs2") or \
+ selfA._toStore("MahalanobisConsistency") or \
+ selfA._toStore("OMB"):
+ d = Y - HXb
+ if selfA._toStore("Innovation"):
+ selfA.StoredVariables["Innovation"].store( numpy.ravel(d) )
+ if selfA._toStore("BMA"):
+ selfA.StoredVariables["BMA"].store( numpy.ravel(Xb) - numpy.ravel(Xa) )
+ if selfA._toStore("OMA"):
+ selfA.StoredVariables["OMA"].store( numpy.ravel(Y) - numpy.ravel(HXa) )
+ if selfA._toStore("OMB"):
+ selfA.StoredVariables["OMB"].store( numpy.ravel(d) )
+ if selfA._toStore("SigmaObs2"):
+ TraceR = R.trace(Y.size)
+ selfA.StoredVariables["SigmaObs2"].store( float( (d.T * (numpy.asmatrix(numpy.ravel(Y)).T-numpy.asmatrix(numpy.ravel(HXa)).T)) ) / TraceR )
+ if selfA._toStore("MahalanobisConsistency"):
+ selfA.StoredVariables["MahalanobisConsistency"].store( float( 2.*MinJ/d.size ) )
+ if selfA._toStore("SimulationQuantiles"):
+ nech = selfA._parameters["NumberOfSamplesForQuantiles"]
+ HXa = numpy.matrix(numpy.ravel( HXa )).T
+ YfQ = None
+ for i in range(nech):
+ if selfA._parameters["SimulationForQuantiles"] == "Linear":
+ dXr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A) - Xa.A1).T
+ dYr = numpy.matrix(numpy.ravel( HtM * dXr )).T
+ Yr = HXa + dYr
+ elif selfA._parameters["SimulationForQuantiles"] == "NonLinear":
+ Xr = numpy.matrix(numpy.random.multivariate_normal(Xa.A1,A)).T
+ Yr = numpy.matrix(numpy.ravel( Hm( Xr ) )).T
+ if YfQ is None:
+ YfQ = Yr
+ else:
+ YfQ = numpy.hstack((YfQ,Yr))
+ YfQ.sort(axis=-1)
+ YQ = None
+ for quantile in selfA._parameters["Quantiles"]:
+ if not (0. <= float(quantile) <= 1.): continue
+ indice = int(nech * float(quantile) - 1./nech)
+ if YQ is None: YQ = YfQ[:,indice]
+ else: YQ = numpy.hstack((YQ,YfQ[:,indice]))
+ selfA.StoredVariables["SimulationQuantiles"].store( YQ )
+ if selfA._toStore("SimulatedObservationAtBackground"):
+ selfA.StoredVariables["SimulatedObservationAtBackground"].store( numpy.ravel(HXb) )
+ if selfA._toStore("SimulatedObservationAtOptimum"):
+ selfA.StoredVariables["SimulatedObservationAtOptimum"].store( numpy.ravel(HXa) )
+ #
+ return 0
+
# ==============================================================================
if __name__ == "__main__":
print('\n AUTODIAGNOSTIC\n')
