def _sort_slices(self, X, X2):
slices = [index_to_slices(x) for x in X[:, -self.levels:].T]
X = X[:, :-self.levels]
if X2 is None:
slices2 = slices
X2 = X
else:
slices2 = [index_to_slices(x) for x in X2[:, -self.levels:].T]
X2 = X2[:, :-self.levels]
return X, X2, slices, slices2
def _sort_slices(self,X,X2):
slices = [index_to_slices(x) for x in X[:,-self.levels:].T]
X = X[:,:-self.levels]
if X2 is None:
slices2 = slices
X2 = X
else:
slices2 = [index_to_slices(x) for x in X2[:,-self.levels:].T]
X2 = X2[:,:-self.levels]
return X, X2, slices, slices2
def K(self, X, X2=None):
"""Compute the covariance matrix between X and X2."""
X, slices = X[:, :-1], index_to_slices(X[:, -1])
if X2 is None:
X2, slices2 = X, slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2, slices2 = X2[:, :-1], index_to_slices(X2[:, -1])
K = np.zeros((X.shape[0], X2.shape[0]))
tdist = (X[:, 0][:, None] - X2[:, 0][None, :])**2
ttdist = (X[:, 0][:, None] - X2[:, 0][None, :])
vyt = self.variance_Yt
lyt = 1 / (2 * self.lengthscale_Yt)
a = -self.a
c = self.c
kyy = lambda tdist: np.exp(-lyt * (tdist))
k1 = lambda tdist: (2 * lyt - 4 * lyt**2 * (tdist))
k4 = lambda tdist: 2 * lyt * (tdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i == 0 and j == 0:
K[ss1, ss2] = vyt * kyy(tdist[ss1, ss2])
elif i == 0 and j == 1:
K[ss1,
ss2] = (k4(ttdist[ss1, ss2]) + 1) * vyt * kyy(
tdist[ss1, ss2])
#K[ss1,ss2] = (2*lyt*(ttdist[ss1,ss2])+1)*vyt*kyy(tdist[ss1,ss2])
elif i == 1 and j == 1:
K[ss1,
ss2] = (k1(tdist[ss1, ss2]) + 1.) * vyt * kyy(
tdist[ss1, ss2]) + self.ubias
else:
K[ss1,
ss2] = (-k4(ttdist[ss1, ss2]) + 1) * vyt * kyy(
tdist[ss1, ss2])
#K[ss1,ss2] = (-2*lyt*(ttdist[ss1,ss2])+1)*vyt*kyy(tdist[ss1,ss2])
#stop
return K
def Kdiag(self, X):
"""Compute the diagonal of the covariance matrix associated to X."""
Kdiag = np.zeros(X.shape[0])
ly = 1 / self.lengthscale_Y
lu = np.sqrt(3) / self.lengthscale_U
Vu = self.variance_U
Vy = self.variance_Y
k1 = (2 * lu + ly) / (lu + ly)**2
k2 = (ly - 2 * lu + 2 * lu - ly) / (ly - lu)**2
k3 = 1 / (lu + ly) + (lu) / (lu + ly)**2
slices = index_to_slices(X[:, -1])
for i, ss1 in enumerate(slices):
for s1 in ss1:
if i == 0:
Kdiag[s1] += self.variance_U
elif i == 1:
Kdiag[s1] += Vu * Vy * (k1 + k2 + k3)
else:
raise ValueError, "invalid input/output index"
#Kdiag[slices[0][0]]+= self.variance_U #matern32 diag
#Kdiag[slices[1][0]]+= self.variance_U*self.variance_Y*(k1+k2+k3) # diag
return Kdiag
def Kdiag(self, X):
"""Compute the diagonal of the covariance matrix associated to X."""
Kdiag = np.zeros(X.shape[0])
ly=1/self.lengthscale_Y
lu=np.sqrt(3)/self.lengthscale_U
Vu = self.variance_U
Vy=self.variance_Y
k1 = (2*lu+ly)/(lu+ly)**2
k2 = (ly-2*lu + 2*lu-ly ) / (ly-lu)**2
k3 = 1/(lu+ly) + (lu)/(lu+ly)**2
slices = index_to_slices(X[:,-1])
for i, ss1 in enumerate(slices):
for s1 in ss1:
if i==0:
Kdiag[s1]+= self.variance_U
elif i==1:
Kdiag[s1]+= Vu*Vy*(k1+k2+k3)
else:
raise ValueError, "invalid input/output index"
#Kdiag[slices[0][0]]+= self.variance_U #matern32 diag
#Kdiag[slices[1][0]]+= self.variance_U*self.variance_Y*(k1+k2+k3) # diag
return Kdiag
def Kdiag(self, X):
"""Compute the diagonal of the covariance matrix associated to X."""
vyt = self.variance_Yt
vyx = self.variance_Yx
lyt = 1./(2*self.lengthscale_Yt)
lyx = 1./(2*self.lengthscale_Yx)
a = self.a
b = self.b
c = self.c
## dk^2/dtdt'
k1 = (2*lyt )*vyt*vyx
## dk^2/dx^2
k2 = ( - 2*lyx )*vyt*vyx
## dk^4/dx^2dx'^2
k3 = ( 4*3*lyx**2 )*vyt*vyx
Kdiag = np.zeros(X.shape[0])
slices = index_to_slices(X[:,-1])
for i, ss1 in enumerate(slices):
for s1 in ss1:
if i==0:
Kdiag[s1]+= vyt*vyx
elif i==1:
#i=1
Kdiag[s1]+= b**2*k1 - 2*a*c*k2 + a**2*k3 + c**2*vyt*vyx
#Kdiag[s1]+= Vu*Vy*(k1+k2+k3)
else:
raise ValueError, "invalid input/output index"
return Kdiag
def Kdiag(self, X):
vyt = self.variance_Yt
lyt = 1./(2*self.lengthscale_Yt)
a = -self.a
c = self.c
k1 = (2*lyt )*vyt
Kdiag = np.zeros(X.shape[0])
slices = index_to_slices(X[:,-1])
for i, ss1 in enumerate(slices):
for s1 in ss1:
if i==0:
Kdiag[s1]+= vyt
elif i==1:
#i=1
Kdiag[s1]+= k1 + vyt+self.ubias
#Kdiag[s1]+= Vu*Vy*(k1+k2+k3)
else:
raise ValueError, "invalid input/output index"
return Kdiag
def Kdiag(self, X):
vyt = self.variance_Yt
lyt = 1. / (2 * self.lengthscale_Yt)
a = -self.a
c = self.c
k1 = (2 * lyt) * vyt
Kdiag = np.zeros(X.shape[0])
slices = index_to_slices(X[:, -1])
for i, ss1 in enumerate(slices):
for s1 in ss1:
if i == 0:
Kdiag[s1] += vyt
elif i == 1:
#i=1
Kdiag[s1] += k1 + vyt + self.ubias
#Kdiag[s1]+= Vu*Vy*(k1+k2+k3)
else:
raise ValueError, "invalid input/output index"
return Kdiag
def K(self, X, X2=None):
"""Compute the covariance matrix between X and X2."""
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
K = np.zeros((X.shape[0], X2.shape[0]))
tdist = (X[:,0][:,None] - X2[:,0][None,:])**2
ttdist = (X[:,0][:,None] - X2[:,0][None,:])
vyt = self.variance_Yt
lyt=1/(2*self.lengthscale_Yt)
a = -self.a
c = self.c
kyy = lambda tdist: np.exp(-lyt*(tdist))
k1 = lambda tdist: (2*lyt - 4*lyt**2 *(tdist) )
k4 = lambda tdist: 2*lyt*(tdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
K[ss1,ss2] = vyt*kyy(tdist[ss1,ss2])
elif i==0 and j==1:
K[ss1,ss2] = (k4(ttdist[ss1,ss2])+1)*vyt*kyy(tdist[ss1,ss2])
#K[ss1,ss2] = (2*lyt*(ttdist[ss1,ss2])+1)*vyt*kyy(tdist[ss1,ss2])
elif i==1 and j==1:
K[ss1,ss2] = ( k1(tdist[ss1,ss2]) + 1. )*vyt* kyy(tdist[ss1,ss2])+self.ubias
else:
K[ss1,ss2] = (-k4(ttdist[ss1,ss2])+1)*vyt*kyy(tdist[ss1,ss2])
#K[ss1,ss2] = (-2*lyt*(ttdist[ss1,ss2])+1)*vyt*kyy(tdist[ss1,ss2])
#stop
return K
def K(self, X, X2=None):
# model : a * dy/dt + b * y = U
#lu=sqrt(3)/theta1 ly=1/theta2 theta2= a/b :thetay sigma2=1/(2ab) :sigmay
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
K = np.zeros((X.shape[0], X2.shape[0]))
#rdist = X[:,0][:,None] - X2[:,0][:,None].T
rdist = X - X2.T
ly=1/self.lengthscale_Y
lu=np.sqrt(3)/self.lengthscale_U
#iu=self.input_lengthU #dimention of U
Vu=self.variance_U
Vy=self.variance_Y
#Vy=ly/2
#stop
# kernel for kuu matern3/2
kuu = lambda dist:Vu * (1 + lu* np.abs(dist)) * np.exp(-lu * np.abs(dist))
# kernel for kyy
k1 = lambda dist:np.exp(-ly*np.abs(dist))*(2*lu+ly)/(lu+ly)**2
k2 = lambda dist:(np.exp(-lu*dist)*(ly-2*lu+lu*ly*dist-lu**2*dist) + np.exp(-ly*dist)*(2*lu-ly) ) / (ly-lu)**2
k3 = lambda dist:np.exp(-lu*dist) * ( (1+lu*dist)/(lu+ly) + (lu)/(lu+ly)**2 )
kyy = lambda dist:Vu*Vy*(k1(dist) + k2(dist) + k3(dist))
# cross covariance function
kyu3 = lambda dist:np.exp(-lu*dist)/(lu+ly)*(1+lu*(dist+1/(lu+ly)))
#kyu3 = lambda dist: 0
k1cros = lambda dist:np.exp(ly*dist)/(lu-ly) * ( 1- np.exp( (lu-ly)*dist) + lu* ( dist*np.exp( (lu-ly)*dist ) + (1- np.exp( (lu-ly)*dist ) ) /(lu-ly) ) )
#k1cros = lambda dist:0
k2cros = lambda dist:np.exp(ly*dist)*( 1/(lu+ly) + lu/(lu+ly)**2 )
#k2cros = lambda dist:0
Vyu=np.sqrt(Vy*ly*2)
# cross covariance kuy
kuyp = lambda dist:Vu*Vyu*(kyu3(dist)) #t>0 kuy
kuyn = lambda dist:Vu*Vyu*(k1cros(dist)+k2cros(dist)) #t<0 kuy
# cross covariance kyu
kyup = lambda dist:Vu*Vyu*(k1cros(-dist)+k2cros(-dist)) #t>0 kyu
kyun = lambda dist:Vu*Vyu*(kyu3(-dist)) #t<0 kyu
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
K[ss1,ss2] = kuu(np.abs(rdist[ss1,ss2]))
elif i==0 and j==1:
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[ss1,ss2]) ) )
K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kuyp(rdist[ss1,ss2]), kuyn(rdist[ss1,ss2] ) )
elif i==1 and j==1:
K[ss1,ss2] = kyy(np.abs(rdist[ss1,ss2]))
else:
#K[ss1,ss2]= 0
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kyup(np.abs(rdist[ss1,ss2])), kyun(np.abs(rdist[ss1,ss2]) ) )
K[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , kyup(rdist[ss1,ss2]), kyun(rdist[ss1,ss2] ) )
return K
def update_gradients_full(self, dL_dK, X, X2=None):
"""derivative of the covariance matrix with respect to the parameters."""
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
#rdist = X[:,0][:,None] - X2[:,0][:,None].T
rdist = X - X2.T
ly=1/self.lengthscale_Y
lu=np.sqrt(3)/self.lengthscale_U
Vu=self.variance_U
Vy=self.variance_Y
Vyu = np.sqrt(Vy*ly*2)
dVdly = 0.5/np.sqrt(ly)*np.sqrt(2*Vy)
dVdVy = 0.5/np.sqrt(Vy)*np.sqrt(2*ly)
rd=rdist.shape
dktheta1 = np.zeros(rd)
dktheta2 = np.zeros(rd)
dkUdvar = np.zeros(rd)
dkYdvar = np.zeros(rd)
# dk dtheta for UU
UUdtheta1 = lambda dist: np.exp(-lu* dist)*dist + (-dist)*np.exp(-lu* dist)*(1+lu*dist)
UUdtheta2 = lambda dist: 0
#UUdvar = lambda dist: (1 + lu*dist)*np.exp(-lu*dist)
UUdvar = lambda dist: (1 + lu* np.abs(dist)) * np.exp(-lu * np.abs(dist))
# dk dtheta for YY
dk1theta1 = lambda dist: np.exp(-ly*dist)*2*(-lu)/(lu+ly)**3
dk2theta1 = lambda dist: (1.0)*(
np.exp(-lu*dist)*dist*(-ly+2*lu-lu*ly*dist+dist*lu**2)*(ly-lu)**(-2) + np.exp(-lu*dist)*(-2+ly*dist-2*dist*lu)*(ly-lu)**(-2)
+np.exp(-dist*lu)*(ly-2*lu+ly*lu*dist-dist*lu**2)*2*(ly-lu)**(-3)
+np.exp(-dist*ly)*2*(ly-lu)**(-2)
+np.exp(-dist*ly)*2*(2*lu-ly)*(ly-lu)**(-3)
)
dk3theta1 = lambda dist: np.exp(-dist*lu)*(lu+ly)**(-2)*((2*lu+ly+dist*lu**2+lu*ly*dist)*(-dist-2/(lu+ly))+2+2*lu*dist+ly*dist)
#dktheta1 = lambda dist: self.variance_U*self.variance_Y*(dk1theta1+dk2theta1+dk3theta1)
dk1theta2 = lambda dist: np.exp(-ly*dist) * ((lu+ly)**(-2)) * ( (-dist)*(2*lu+ly) + 1 + (-2)*(2*lu+ly)/(lu+ly) )
dk2theta2 =lambda dist: 1*(
np.exp(-dist*lu)*(ly-lu)**(-2) * ( 1+lu*dist+(-2)*(ly-2*lu+lu*ly*dist-dist*lu**2)*(ly-lu)**(-1) )
+np.exp(-dist*ly)*(ly-lu)**(-2) * ( (-dist)*(2*lu-ly) -1+(2*lu-ly)*(-2)*(ly-lu)**(-1) )
)
dk3theta2 = lambda dist: np.exp(-dist*lu) * (-3*lu-ly-dist*lu**2-lu*ly*dist)/(lu+ly)**3
#dktheta2 = lambda dist: self.variance_U*self.variance_Y*(dk1theta2 + dk2theta2 +dk3theta2)
# kyy kernel
k1 = lambda dist: np.exp(-ly*dist)*(2*lu+ly)/(lu+ly)**2
k2 = lambda dist: (np.exp(-lu*dist)*(ly-2*lu+lu*ly*dist-lu**2*dist) + np.exp(-ly*dist)*(2*lu-ly) ) / (ly-lu)**2
k3 = lambda dist: np.exp(-lu*dist) * ( (1+lu*dist)/(lu+ly) + (lu)/(lu+ly)**2 )
#dkdvar = k1+k2+k3
# cross covariance function
kyu3 = lambda dist:np.exp(-lu*dist)/(lu+ly)*(1+lu*(dist+1/(lu+ly)))
k1cros = lambda dist:np.exp(ly*dist)/(lu-ly) * ( 1- np.exp( (lu-ly)*dist) + lu* ( dist*np.exp( (lu-ly)*dist ) + (1- np.exp( (lu-ly)*dist ) ) /(lu-ly) ) )
k2cros = lambda dist:np.exp(ly*dist)*( 1/(lu+ly) + lu/(lu+ly)**2 )
# cross covariance kuy
kuyp = lambda dist:(kyu3(dist)) #t>0 kuy
kuyn = lambda dist:(k1cros(dist)+k2cros(dist)) #t<0 kuy
# cross covariance kyu
kyup = lambda dist:(k1cros(-dist)+k2cros(-dist)) #t>0 kyu
kyun = lambda dist:(kyu3(-dist)) #t<0 kyu
# dk dtheta for UY
dkyu3dtheta2 = lambda dist: np.exp(-lu*dist) * ( (-1)*(lu+ly)**(-2)*(1+lu*dist+lu*(lu+ly)**(-1)) + (lu+ly)**(-1)*(-lu)*(lu+ly)**(-2) )
dkyu3dtheta1 = lambda dist: np.exp(-lu*dist)*(lu+ly)**(-1)* ( (-dist)*(1+dist*lu+lu*(lu+ly)**(-1)) -\
(lu+ly)**(-1)*(1+dist*lu+lu*(lu+ly)**(-1)) +dist+(lu+ly)**(-1)-lu*(lu+ly)**(-2) )
dkcros2dtheta1 = lambda dist: np.exp(ly*dist)* ( -(ly+lu)**(-2) + (ly+lu)**(-2) + (-2)*lu*(lu+ly)**(-3) )
dkcros2dtheta2 = lambda dist: np.exp(ly*dist)*dist* ( (ly+lu)**(-1) + lu*(lu+ly)**(-2) ) + \
np.exp(ly*dist)*( -(lu+ly)**(-2) + lu*(-2)*(lu+ly)**(-3) )
dkcros1dtheta1 = lambda dist: np.exp(ly*dist)*( -(lu-ly)**(-2)*( 1-np.exp((lu-ly)*dist) + lu*dist*np.exp((lu-ly)*dist)+ \
lu*(1-np.exp((lu-ly)*dist))/(lu-ly) ) + (lu-ly)**(-1)*( -np.exp( (lu-ly)*dist )*dist + dist*np.exp( (lu-ly)*dist)+\
lu*dist**2*np.exp((lu-ly)*dist)+(1-np.exp((lu-ly)*dist))/(lu-ly) - lu*np.exp((lu-ly)*dist)*dist/(lu-ly) -\
lu*(1-np.exp((lu-ly)*dist))/(lu-ly)**2 ) )
dkcros1dtheta2 = lambda t: np.exp(ly*t)*t/(lu-ly)*( 1-np.exp((lu-ly)*t) +lu*t*np.exp((lu-ly)*t)+\
lu*(1-np.exp((lu-ly)*t))/(lu-ly) )+\
np.exp(ly*t)/(lu-ly)**2* ( 1-np.exp((lu-ly)*t) +lu*t*np.exp((lu-ly)*t) + lu*( 1-np.exp((lu-ly)*t) )/(lu-ly) )+\
np.exp(ly*t)/(lu-ly)*( np.exp((lu-ly)*t)*t -lu*t*t*np.exp((lu-ly)*t) +lu*t*np.exp((lu-ly)*t)/(lu-ly)+\
lu*( 1-np.exp((lu-ly)*t) )/(lu-ly)**2 )
dkuypdtheta1 = lambda dist:(dkyu3dtheta1(dist)) #t>0 kuy
dkuyndtheta1 = lambda dist:(dkcros1dtheta1(dist)+dkcros2dtheta1(dist)) #t<0 kuy
# cross covariance kyu
dkyupdtheta1 = lambda dist:(dkcros1dtheta1(-dist)+dkcros2dtheta1(-dist)) #t>0 kyu
dkyundtheta1 = lambda dist:(dkyu3dtheta1(-dist)) #t<0 kyu
dkuypdtheta2 = lambda dist:(dkyu3dtheta2(dist)) #t>0 kuy
dkuyndtheta2 = lambda dist:(dkcros1dtheta2(dist)+dkcros2dtheta2(dist)) #t<0 kuy
# cross covariance kyu
dkyupdtheta2 = lambda dist:(dkcros1dtheta2(-dist)+dkcros2dtheta2(-dist)) #t>0 kyu
dkyundtheta2 = lambda dist:(dkyu3dtheta2(-dist)) #t<0 kyu
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
#target[ss1,ss2] = kuu(np.abs(rdist[ss1,ss2]))
dktheta1[ss1,ss2] = Vu*UUdtheta1(np.abs(rdist[ss1,ss2]))
dktheta2[ss1,ss2] = 0
dkUdvar[ss1,ss2] = UUdvar(np.abs(rdist[ss1,ss2]))
dkYdvar[ss1,ss2] = 0
elif i==0 and j==1:
########target[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[s1[0],s2[0]]) ) )
#np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[s1[0],s2[0]]) ) )
#dktheta1[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , self.variance_U*self.variance_Y*dkcrtheta1(np.abs(rdist[ss1,ss2])) ,self.variance_U*self.variance_Y*(dk1theta1(np.abs(rdist[ss1,ss2]))+dk2theta1(np.abs(rdist[ss1,ss2]))) )
#dktheta2[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , self.variance_U*self.variance_Y*dkcrtheta2(np.abs(rdist[ss1,ss2])) ,self.variance_U*self.variance_Y*(dk1theta2(np.abs(rdist[ss1,ss2]))+dk2theta2(np.abs(rdist[ss1,ss2]))) )
dktheta1[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vu*Vyu*dkuypdtheta1(rdist[ss1,ss2]),Vu*Vyu*dkuyndtheta1(rdist[ss1,ss2]) )
dkUdvar[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vyu*kuyp(rdist[ss1,ss2]), Vyu* kuyn(rdist[ss1,ss2]) )
dktheta2[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vu*Vyu*dkuypdtheta2(rdist[ss1,ss2])+Vu*dVdly*kuyp(rdist[ss1,ss2]),Vu*Vyu*dkuyndtheta2(rdist[ss1,ss2])+Vu*dVdly*kuyn(rdist[ss1,ss2]) )
dkYdvar[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vu*dVdVy*kuyp(rdist[ss1,ss2]), Vu*dVdVy* kuyn(rdist[ss1,ss2]) )
elif i==1 and j==1:
#target[ss1,ss2] = kyy(np.abs(rdist[ss1,ss2]))
dktheta1[ss1,ss2] = self.variance_U*self.variance_Y*(dk1theta1(np.abs(rdist[ss1,ss2]))+dk2theta1(np.abs(rdist[ss1,ss2]))+dk3theta1(np.abs(rdist[ss1,ss2])))
dktheta2[ss1,ss2] = self.variance_U*self.variance_Y*(dk1theta2(np.abs(rdist[ss1,ss2])) + dk2theta2(np.abs(rdist[ss1,ss2])) +dk3theta2(np.abs(rdist[ss1,ss2])))
dkUdvar[ss1,ss2] = self.variance_Y*(k1(np.abs(rdist[ss1,ss2]))+k2(np.abs(rdist[ss1,ss2]))+k3(np.abs(rdist[ss1,ss2])) )
dkYdvar[ss1,ss2] = self.variance_U*(k1(np.abs(rdist[ss1,ss2]))+k2(np.abs(rdist[ss1,ss2]))+k3(np.abs(rdist[ss1,ss2])) )
else:
#######target[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , kyup(np.abs(rdist[ss1,ss2])), kyun(np.abs(rdist[s1[0],s2[0]]) ) )
#dktheta1[ss1,ss2] = np.where( rdist[ss1,ss2]>0 ,self.variance_U*self.variance_Y*(dk1theta1(np.abs(rdist[ss1,ss2]))+dk2theta1(np.abs(rdist[ss1,ss2]))) , self.variance_U*self.variance_Y*dkcrtheta1(np.abs(rdist[ss1,ss2])) )
#dktheta2[ss1,ss2] = np.where( rdist[ss1,ss2]>0 ,self.variance_U*self.variance_Y*(dk1theta2(np.abs(rdist[ss1,ss2]))+dk2theta2(np.abs(rdist[ss1,ss2]))) , self.variance_U*self.variance_Y*dkcrtheta2(np.abs(rdist[ss1,ss2])) )
dktheta1[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vu*Vyu*dkyupdtheta1(rdist[ss1,ss2]),Vu*Vyu*dkyundtheta1(rdist[ss1,ss2]) )
dkUdvar[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vyu*kyup(rdist[ss1,ss2]),Vyu*kyun(rdist[ss1,ss2]))
dktheta2[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vu*Vyu*dkyupdtheta2(rdist[ss1,ss2])+Vu*dVdly*kyup(rdist[ss1,ss2]),Vu*Vyu*dkyundtheta2(rdist[ss1,ss2])+Vu*dVdly*kyun(rdist[ss1,ss2]) )
dkYdvar[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , Vu*dVdVy*kyup(rdist[ss1,ss2]), Vu*dVdVy*kyun(rdist[ss1,ss2]))
#stop
self.variance_U.gradient = np.sum(dkUdvar * dL_dK) # Vu
self.variance_Y.gradient = np.sum(dkYdvar * dL_dK) # Vy
self.lengthscale_U.gradient = np.sum(dktheta1*(-np.sqrt(3)*self.lengthscale_U**(-2))* dL_dK) #lu
self.lengthscale_Y.gradient = np.sum(dktheta2*(-self.lengthscale_Y**(-2)) * dL_dK) #ly
def K(self, X, X2=None):
# model : -a d^2y/dx^2 + b dy/dt + c * y = U
# kernel Kyy rbf spatiol temporal
# vyt Y temporal variance vyx Y spatiol variance lyt Y temporal lengthscale lyx Y spatiol lengthscale
# kernel Kuu doper( doper(Kyy))
# a b c lyt lyx vyx*vyt
"""Compute the covariance matrix between X and X2."""
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
K = np.zeros((X.shape[0], X2.shape[0]))
tdist = (X[:,0][:,None] - X2[:,0][None,:])**2
xdist = (X[:,1][:,None] - X2[:,1][None,:])**2
ttdist = (X[:,0][:,None] - X2[:,0][None,:])
#rdist = [tdist,xdist]
#dist = np.abs(X - X2.T)
vyt = self.variance_Yt
vyx = self.variance_Yx
lyt=1/(2*self.lengthscale_Yt)
lyx=1/(2*self.lengthscale_Yx)
a = self.a ## -a is used in the model, negtive diffusion
b = self.b
c = self.c
kyy = lambda tdist,xdist: np.exp(-lyt*(tdist) -lyx*(xdist))
k1 = lambda tdist: (2*lyt - 4*lyt**2 * (tdist) )
k2 = lambda xdist: ( 4*lyx**2 * (xdist) - 2*lyx )
k3 = lambda xdist: ( 3*4*lyx**2 - 6*8*xdist*lyx**3 + 16*xdist**2*lyx**4 )
k4 = lambda ttdist: 2*lyt*(ttdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
K[ss1,ss2] = vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
elif i==0 and j==1:
K[ss1,ss2] = (-a*k2(xdist[ss1,ss2]) + b*k4(ttdist[ss1,ss2]) + c)*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[ss1,ss2]) ) )
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kuyp(rdist[ss1,ss2]), kuyn(rdist[ss1,ss2] ) )
elif i==1 and j==1:
K[ss1,ss2] = ( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 )* vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
else:
K[ss1,ss2] = (-a*k2(xdist[ss1,ss2]) - b*k4(ttdist[ss1,ss2]) + c)*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kyup(np.abs(rdist[ss1,ss2])), kyun(np.abs(rdist[ss1,ss2]) ) )
#K[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , kyup(rdist[ss1,ss2]), kyun(rdist[ss1,ss2] ) )
#stop
return K
def update_gradients_full(self, dL_dK, X, X2=None):
"""derivative of the covariance matrix with respect to the parameters."""
X, slices = X[:, :-1], index_to_slices(X[:, -1])
if X2 is None:
X2, slices2 = X, slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2, slices2 = X2[:, :-1], index_to_slices(X2[:, -1])
vyt = self.variance_Yt
lyt = 1. / (2 * self.lengthscale_Yt)
tdist = (X[:, 0][:, None] - X2[:, 0][None, :])**2
ttdist = (X[:, 0][:, None] - X2[:, 0][None, :])
#rdist = [tdist,xdist]
rd = tdist.shape[0]
dka = np.zeros([rd, rd])
dkc = np.zeros([rd, rd])
dkYdvart = np.zeros([rd, rd])
dkYdlent = np.zeros([rd, rd])
dkdubias = np.zeros([rd, rd])
kyy = lambda tdist: np.exp(-lyt * (tdist))
dkyydlyt = lambda tdist: kyy(tdist) * (-tdist)
k1 = lambda tdist: (2 * lyt - 4 * lyt**2 * (tdist))
k4 = lambda ttdist: 2 * lyt * (ttdist)
dk1dlyt = lambda tdist: 2. - 4 * 2. * lyt * tdist
dk4dlyt = lambda ttdist: 2 * (ttdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i == 0 and j == 0:
dkYdvart[ss1, ss2] = kyy(tdist[ss1, ss2])
dkYdlent[ss1,
ss2] = vyt * dkyydlyt(tdist[ss1, ss2])
dkdubias[ss1, ss2] = 0
elif i == 0 and j == 1:
dkYdvart[ss1,
ss2] = (k4(ttdist[ss1, ss2]) + 1) * kyy(
tdist[ss1, ss2])
#dkYdvart[ss1,ss2] = ((2*lyt*ttdist[ss1,ss2])+1)*kyy(tdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (k4(ttdist[ss1,ss2])+1.)+\
vyt*kyy(tdist[ss1,ss2])*(dk4dlyt(ttdist[ss1,ss2]))
#dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (2*lyt*(ttdist[ss1,ss2])+1.)+\
#vyt*kyy(tdist[ss1,ss2])*(2*ttdist[ss1,ss2])
dkdubias[ss1, ss2] = 0
elif i == 1 and j == 1:
dkYdvart[ss1,
ss2] = (k1(tdist[ss1, ss2]) + 1.) * kyy(
tdist[ss1, ss2])
dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])*( k1(tdist[ss1,ss2]) + 1. ) +\
vyt*kyy(tdist[ss1,ss2])*dk1dlyt(tdist[ss1,ss2])
dkdubias[ss1, ss2] = 1
else:
dkYdvart[ss1,
ss2] = (-k4(ttdist[ss1, ss2]) + 1) * kyy(
tdist[ss1, ss2])
#dkYdvart[ss1,ss2] = (-2*lyt*(ttdist[ss1,ss2])+1)*kyy(tdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (-k4(ttdist[ss1,ss2])+1.)+\
vyt*kyy(tdist[ss1,ss2])*(-dk4dlyt(ttdist[ss1,ss2]) )
dkdubias[ss1, ss2] = 0
#dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (-2*lyt*(ttdist[ss1,ss2])+1.)+\
#vyt*kyy(tdist[ss1,ss2])*(-2)*(ttdist[ss1,ss2])
self.variance_Yt.gradient = np.sum(dkYdvart * dL_dK)
self.lengthscale_Yt.gradient = np.sum(
dkYdlent * (-0.5 * self.lengthscale_Yt**(-2)) * dL_dK)
self.ubias.gradient = np.sum(dkdubias * dL_dK)
def update_gradients_full(self, dL_dK, X, X2=None):
"""derivative of the covariance matrix with respect to the parameters."""
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
vyt = self.variance_Yt
lyt = 1./(2*self.lengthscale_Yt)
tdist = (X[:,0][:,None] - X2[:,0][None,:])**2
ttdist = (X[:,0][:,None] - X2[:,0][None,:])
#rdist = [tdist,xdist]
rd=tdist.shape[0]
dka = np.zeros([rd,rd])
dkc = np.zeros([rd,rd])
dkYdvart = np.zeros([rd,rd])
dkYdlent = np.zeros([rd,rd])
dkdubias = np.zeros([rd,rd])
kyy = lambda tdist: np.exp(-lyt*(tdist))
dkyydlyt = lambda tdist: kyy(tdist)*(-tdist)
k1 = lambda tdist: (2*lyt - 4*lyt**2 * (tdist) )
k4 = lambda ttdist: 2*lyt*(ttdist)
dk1dlyt = lambda tdist: 2. - 4*2.*lyt*tdist
dk4dlyt = lambda ttdist: 2*(ttdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
dkYdvart[ss1,ss2] = kyy(tdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])
dkdubias[ss1,ss2] = 0
elif i==0 and j==1:
dkYdvart[ss1,ss2] = (k4(ttdist[ss1,ss2])+1)*kyy(tdist[ss1,ss2])
#dkYdvart[ss1,ss2] = ((2*lyt*ttdist[ss1,ss2])+1)*kyy(tdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (k4(ttdist[ss1,ss2])+1.)+\
vyt*kyy(tdist[ss1,ss2])*(dk4dlyt(ttdist[ss1,ss2]))
#dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (2*lyt*(ttdist[ss1,ss2])+1.)+\
#vyt*kyy(tdist[ss1,ss2])*(2*ttdist[ss1,ss2])
dkdubias[ss1,ss2] = 0
elif i==1 and j==1:
dkYdvart[ss1,ss2] = (k1(tdist[ss1,ss2]) + 1. )* kyy(tdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])*( k1(tdist[ss1,ss2]) + 1. ) +\
vyt*kyy(tdist[ss1,ss2])*dk1dlyt(tdist[ss1,ss2])
dkdubias[ss1,ss2] = 1
else:
dkYdvart[ss1,ss2] = (-k4(ttdist[ss1,ss2])+1)*kyy(tdist[ss1,ss2])
#dkYdvart[ss1,ss2] = (-2*lyt*(ttdist[ss1,ss2])+1)*kyy(tdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (-k4(ttdist[ss1,ss2])+1.)+\
vyt*kyy(tdist[ss1,ss2])*(-dk4dlyt(ttdist[ss1,ss2]) )
dkdubias[ss1,ss2] = 0
#dkYdlent[ss1,ss2] = vyt*dkyydlyt(tdist[ss1,ss2])* (-2*lyt*(ttdist[ss1,ss2])+1.)+\
#vyt*kyy(tdist[ss1,ss2])*(-2)*(ttdist[ss1,ss2])
self.variance_Yt.gradient = np.sum(dkYdvart * dL_dK)
self.lengthscale_Yt.gradient = np.sum(dkYdlent*(-0.5*self.lengthscale_Yt**(-2)) * dL_dK)
self.ubias.gradient = np.sum(dkdubias * dL_dK)
def update_gradients_full(self, dL_dK, X, X2=None):
"""derivative of the covariance matrix with respect to the parameters."""
X, slices = X[:, :-1], index_to_slices(X[:, -1])
if X2 is None:
X2, slices2 = X, slices
else:
X2, slices2 = X2[:, :-1], index_to_slices(X2[:, -1])
#rdist = X[:,0][:,None] - X2[:,0][:,None].T
rdist = X - X2.T
ly = 1 / self.lengthscale_Y
lu = np.sqrt(3) / self.lengthscale_U
Vu = self.variance_U
Vy = self.variance_Y
Vyu = np.sqrt(Vy * ly * 2)
dVdly = 0.5 / np.sqrt(ly) * np.sqrt(2 * Vy)
dVdVy = 0.5 / np.sqrt(Vy) * np.sqrt(2 * ly)
rd = rdist.shape
dktheta1 = np.zeros(rd)
dktheta2 = np.zeros(rd)
dkUdvar = np.zeros(rd)
dkYdvar = np.zeros(rd)
# dk dtheta for UU
UUdtheta1 = lambda dist: np.exp(-lu * dist) * dist + (-dist) * np.exp(
-lu * dist) * (1 + lu * dist)
UUdtheta2 = lambda dist: 0
#UUdvar = lambda dist: (1 + lu*dist)*np.exp(-lu*dist)
UUdvar = lambda dist: (1 + lu * np.abs(dist)) * np.exp(-lu * np.abs(
dist))
# dk dtheta for YY
dk1theta1 = lambda dist: np.exp(-ly * dist) * 2 * (-lu) / (lu + ly)**3
dk2theta1 = lambda dist: (
1.0) * (np.exp(-lu * dist) * dist *
(-ly + 2 * lu - lu * ly * dist + dist * lu**2) *
(ly - lu)**(-2) + np.exp(-lu * dist) *
(-2 + ly * dist - 2 * dist * lu) *
(ly - lu)**(-2) + np.exp(-dist * lu) *
(ly - 2 * lu + ly * lu * dist - dist * lu**2) * 2 *
(ly - lu)**(-3) + np.exp(-dist * ly) * 2 *
(ly - lu)**(-2) + np.exp(-dist * ly) * 2 * (2 * lu - ly) *
(ly - lu)**(-3))
dk3theta1 = lambda dist: np.exp(-dist * lu) * (lu + ly)**(-2) * (
(2 * lu + ly + dist * lu**2 + lu * ly * dist) *
(-dist - 2 / (lu + ly)) + 2 + 2 * lu * dist + ly * dist)
#dktheta1 = lambda dist: self.variance_U*self.variance_Y*(dk1theta1+dk2theta1+dk3theta1)
dk1theta2 = lambda dist: np.exp(-ly * dist) * ((lu + ly)**(-2)) * (
(-dist) * (2 * lu + ly) + 1 + (-2) * (2 * lu + ly) / (lu + ly))
dk2theta2 = lambda dist: 1 * (np.exp(-dist * lu) * (ly - lu)**(
-2) * (1 + lu * dist + (-2) *
(ly - 2 * lu + lu * ly * dist - dist * lu**2) *
(ly - lu)**(-1)) + np.exp(-dist * ly) * (ly - lu)**(-2) *
((-dist) * (2 * lu - ly) - 1 +
(2 * lu - ly) * (-2) * (ly - lu)**(-1)))
dk3theta2 = lambda dist: np.exp(-dist * lu) * (
-3 * lu - ly - dist * lu**2 - lu * ly * dist) / (lu + ly)**3
#dktheta2 = lambda dist: self.variance_U*self.variance_Y*(dk1theta2 + dk2theta2 +dk3theta2)
# kyy kernel
k1 = lambda dist: np.exp(-ly * dist) * (2 * lu + ly) / (lu + ly)**2
k2 = lambda dist: (np.exp(-lu * dist) * (
ly - 2 * lu + lu * ly * dist - lu**2 * dist) + np.exp(-ly * dist) *
(2 * lu - ly)) / (ly - lu)**2
k3 = lambda dist: np.exp(-lu * dist) * ((1 + lu * dist) / (lu + ly) +
(lu) / (lu + ly)**2)
#dkdvar = k1+k2+k3
# cross covariance function
kyu3 = lambda dist: np.exp(-lu * dist) / (lu + ly) * (1 + lu *
(dist + 1 /
(lu + ly)))
k1cros = lambda dist: np.exp(ly * dist) / (lu - ly) * (1 - np.exp(
(lu - ly) * dist) + lu * (dist * np.exp(
(lu - ly) * dist) + (1 - np.exp((lu - ly) * dist)) /
(lu - ly)))
k2cros = lambda dist: np.exp(ly * dist) * (1 / (lu + ly) + lu /
(lu + ly)**2)
# cross covariance kuy
kuyp = lambda dist: (kyu3(dist)) #t>0 kuy
kuyn = lambda dist: (k1cros(dist) + k2cros(dist)) #t<0 kuy
# cross covariance kyu
kyup = lambda dist: (k1cros(-dist) + k2cros(-dist)) #t>0 kyu
kyun = lambda dist: (kyu3(-dist)) #t<0 kyu
# dk dtheta for UY
dkyu3dtheta2 = lambda dist: np.exp(-lu * dist) * (
(-1) * (lu + ly)**(-2) * (1 + lu * dist + lu * (lu + ly)**(-1)) +
(lu + ly)**(-1) * (-lu) * (lu + ly)**(-2))
dkyu3dtheta1 = lambda dist: np.exp(-lu*dist)*(lu+ly)**(-1)* ( (-dist)*(1+dist*lu+lu*(lu+ly)**(-1)) -\
(lu+ly)**(-1)*(1+dist*lu+lu*(lu+ly)**(-1)) +dist+(lu+ly)**(-1)-lu*(lu+ly)**(-2) )
dkcros2dtheta1 = lambda dist: np.exp(ly * dist) * (-(ly + lu)**(-2) +
(ly + lu)**(-2) +
(-2) * lu *
(lu + ly)**(-3))
dkcros2dtheta2 = lambda dist: np.exp(ly*dist)*dist* ( (ly+lu)**(-1) + lu*(lu+ly)**(-2) ) + \
np.exp(ly*dist)*( -(lu+ly)**(-2) + lu*(-2)*(lu+ly)**(-3) )
dkcros1dtheta1 = lambda dist: np.exp(ly*dist)*( -(lu-ly)**(-2)*( 1-np.exp((lu-ly)*dist) + lu*dist*np.exp((lu-ly)*dist)+ \
lu*(1-np.exp((lu-ly)*dist))/(lu-ly) ) + (lu-ly)**(-1)*( -np.exp( (lu-ly)*dist )*dist + dist*np.exp( (lu-ly)*dist)+\
lu*dist**2*np.exp((lu-ly)*dist)+(1-np.exp((lu-ly)*dist))/(lu-ly) - lu*np.exp((lu-ly)*dist)*dist/(lu-ly) -\
lu*(1-np.exp((lu-ly)*dist))/(lu-ly)**2 ) )
dkcros1dtheta2 = lambda t: np.exp(ly*t)*t/(lu-ly)*( 1-np.exp((lu-ly)*t) +lu*t*np.exp((lu-ly)*t)+\
lu*(1-np.exp((lu-ly)*t))/(lu-ly) )+\
np.exp(ly*t)/(lu-ly)**2* ( 1-np.exp((lu-ly)*t) +lu*t*np.exp((lu-ly)*t) + lu*( 1-np.exp((lu-ly)*t) )/(lu-ly) )+\
np.exp(ly*t)/(lu-ly)*( np.exp((lu-ly)*t)*t -lu*t*t*np.exp((lu-ly)*t) +lu*t*np.exp((lu-ly)*t)/(lu-ly)+\
lu*( 1-np.exp((lu-ly)*t) )/(lu-ly)**2 )
dkuypdtheta1 = lambda dist: (dkyu3dtheta1(dist)) #t>0 kuy
dkuyndtheta1 = lambda dist: (dkcros1dtheta1(dist) + dkcros2dtheta1(
dist)) #t<0 kuy
# cross covariance kyu
dkyupdtheta1 = lambda dist: (dkcros1dtheta1(-dist) + dkcros2dtheta1(
-dist)) #t>0 kyu
dkyundtheta1 = lambda dist: (dkyu3dtheta1(-dist)) #t<0 kyu
dkuypdtheta2 = lambda dist: (dkyu3dtheta2(dist)) #t>0 kuy
dkuyndtheta2 = lambda dist: (dkcros1dtheta2(dist) + dkcros2dtheta2(
dist)) #t<0 kuy
# cross covariance kyu
dkyupdtheta2 = lambda dist: (dkcros1dtheta2(-dist) + dkcros2dtheta2(
-dist)) #t>0 kyu
dkyundtheta2 = lambda dist: (dkyu3dtheta2(-dist)) #t<0 kyu
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i == 0 and j == 0:
#target[ss1,ss2] = kuu(np.abs(rdist[ss1,ss2]))
dktheta1[ss1, ss2] = Vu * UUdtheta1(
np.abs(rdist[ss1, ss2]))
dktheta2[ss1, ss2] = 0
dkUdvar[ss1, ss2] = UUdvar(np.abs(rdist[ss1, ss2]))
dkYdvar[ss1, ss2] = 0
elif i == 0 and j == 1:
########target[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[s1[0],s2[0]]) ) )
#np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[s1[0],s2[0]]) ) )
#dktheta1[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , self.variance_U*self.variance_Y*dkcrtheta1(np.abs(rdist[ss1,ss2])) ,self.variance_U*self.variance_Y*(dk1theta1(np.abs(rdist[ss1,ss2]))+dk2theta1(np.abs(rdist[ss1,ss2]))) )
#dktheta2[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , self.variance_U*self.variance_Y*dkcrtheta2(np.abs(rdist[ss1,ss2])) ,self.variance_U*self.variance_Y*(dk1theta2(np.abs(rdist[ss1,ss2]))+dk2theta2(np.abs(rdist[ss1,ss2]))) )
dktheta1[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vu * Vyu * dkuypdtheta1(rdist[ss1, ss2]),
Vu * Vyu * dkuyndtheta1(rdist[ss1, ss2]))
dkUdvar[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vyu * kuyp(rdist[ss1, ss2]),
Vyu * kuyn(rdist[ss1, ss2]))
dktheta2[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vu * Vyu * dkuypdtheta2(rdist[ss1, ss2]) +
Vu * dVdly * kuyp(rdist[ss1, ss2]),
Vu * Vyu * dkuyndtheta2(rdist[ss1, ss2]) +
Vu * dVdly * kuyn(rdist[ss1, ss2]))
dkYdvar[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vu * dVdVy * kuyp(rdist[ss1, ss2]),
Vu * dVdVy * kuyn(rdist[ss1, ss2]))
elif i == 1 and j == 1:
#target[ss1,ss2] = kyy(np.abs(rdist[ss1,ss2]))
dktheta1[
ss1,
ss2] = self.variance_U * self.variance_Y * (
dk1theta1(np.abs(rdist[ss1, ss2])) +
dk2theta1(np.abs(rdist[ss1, ss2])) +
dk3theta1(np.abs(rdist[ss1, ss2])))
dktheta2[
ss1,
ss2] = self.variance_U * self.variance_Y * (
dk1theta2(np.abs(rdist[ss1, ss2])) +
dk2theta2(np.abs(rdist[ss1, ss2])) +
dk3theta2(np.abs(rdist[ss1, ss2])))
dkUdvar[ss1, ss2] = self.variance_Y * (
k1(np.abs(rdist[ss1, ss2])) +
k2(np.abs(rdist[ss1, ss2])) +
k3(np.abs(rdist[ss1, ss2])))
dkYdvar[ss1, ss2] = self.variance_U * (
k1(np.abs(rdist[ss1, ss2])) +
k2(np.abs(rdist[ss1, ss2])) +
k3(np.abs(rdist[ss1, ss2])))
else:
#######target[ss1,ss2] = np.where( rdist[ss1,ss2]>0 , kyup(np.abs(rdist[ss1,ss2])), kyun(np.abs(rdist[s1[0],s2[0]]) ) )
#dktheta1[ss1,ss2] = np.where( rdist[ss1,ss2]>0 ,self.variance_U*self.variance_Y*(dk1theta1(np.abs(rdist[ss1,ss2]))+dk2theta1(np.abs(rdist[ss1,ss2]))) , self.variance_U*self.variance_Y*dkcrtheta1(np.abs(rdist[ss1,ss2])) )
#dktheta2[ss1,ss2] = np.where( rdist[ss1,ss2]>0 ,self.variance_U*self.variance_Y*(dk1theta2(np.abs(rdist[ss1,ss2]))+dk2theta2(np.abs(rdist[ss1,ss2]))) , self.variance_U*self.variance_Y*dkcrtheta2(np.abs(rdist[ss1,ss2])) )
dktheta1[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vu * Vyu * dkyupdtheta1(rdist[ss1, ss2]),
Vu * Vyu * dkyundtheta1(rdist[ss1, ss2]))
dkUdvar[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vyu * kyup(rdist[ss1, ss2]),
Vyu * kyun(rdist[ss1, ss2]))
dktheta2[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vu * Vyu * dkyupdtheta2(rdist[ss1, ss2]) +
Vu * dVdly * kyup(rdist[ss1, ss2]),
Vu * Vyu * dkyundtheta2(rdist[ss1, ss2]) +
Vu * dVdly * kyun(rdist[ss1, ss2]))
dkYdvar[ss1, ss2] = np.where(
rdist[ss1, ss2] > 0,
Vu * dVdVy * kyup(rdist[ss1, ss2]),
Vu * dVdVy * kyun(rdist[ss1, ss2]))
#stop
self.variance_U.gradient = np.sum(dkUdvar * dL_dK) # Vu
self.variance_Y.gradient = np.sum(dkYdvar * dL_dK) # Vy
self.lengthscale_U.gradient = np.sum(
dktheta1 * (-np.sqrt(3) * self.lengthscale_U**(-2)) * dL_dK) #lu
self.lengthscale_Y.gradient = np.sum(
dktheta2 * (-self.lengthscale_Y**(-2)) * dL_dK) #ly
def K(self, X, X2=None):
# model : a * dy/dt + b * y = U
#lu=sqrt(3)/theta1 ly=1/theta2 theta2= a/b :thetay sigma2=1/(2ab) :sigmay
X, slices = X[:, :-1], index_to_slices(X[:, -1])
if X2 is None:
X2, slices2 = X, slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2, slices2 = X2[:, :-1], index_to_slices(X2[:, -1])
K = np.zeros((X.shape[0], X2.shape[0]))
#rdist = X[:,0][:,None] - X2[:,0][:,None].T
rdist = X - X2.T
ly = 1 / self.lengthscale_Y
lu = np.sqrt(3) / self.lengthscale_U
#iu=self.input_lengthU #dimention of U
Vu = self.variance_U
Vy = self.variance_Y
#Vy=ly/2
#stop
# kernel for kuu matern3/2
kuu = lambda dist: Vu * (1 + lu * np.abs(dist)) * np.exp(-lu * np.abs(
dist))
# kernel for kyy
k1 = lambda dist: np.exp(-ly * np.abs(dist)) * (2 * lu + ly) / (lu + ly
)**2
k2 = lambda dist: (np.exp(-lu * dist) * (
ly - 2 * lu + lu * ly * dist - lu**2 * dist) + np.exp(-ly * dist) *
(2 * lu - ly)) / (ly - lu)**2
k3 = lambda dist: np.exp(-lu * dist) * ((1 + lu * dist) / (lu + ly) +
(lu) / (lu + ly)**2)
kyy = lambda dist: Vu * Vy * (k1(dist) + k2(dist) + k3(dist))
# cross covariance function
kyu3 = lambda dist: np.exp(-lu * dist) / (lu + ly) * (1 + lu *
(dist + 1 /
(lu + ly)))
#kyu3 = lambda dist: 0
k1cros = lambda dist: np.exp(ly * dist) / (lu - ly) * (1 - np.exp(
(lu - ly) * dist) + lu * (dist * np.exp(
(lu - ly) * dist) + (1 - np.exp((lu - ly) * dist)) /
(lu - ly)))
#k1cros = lambda dist:0
k2cros = lambda dist: np.exp(ly * dist) * (1 / (lu + ly) + lu /
(lu + ly)**2)
#k2cros = lambda dist:0
Vyu = np.sqrt(Vy * ly * 2)
# cross covariance kuy
kuyp = lambda dist: Vu * Vyu * (kyu3(dist)) #t>0 kuy
kuyn = lambda dist: Vu * Vyu * (k1cros(dist) + k2cros(dist)) #t<0 kuy
# cross covariance kyu
kyup = lambda dist: Vu * Vyu * (k1cros(-dist) + k2cros(-dist)
) #t>0 kyu
kyun = lambda dist: Vu * Vyu * (kyu3(-dist)) #t<0 kyu
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i == 0 and j == 0:
K[ss1, ss2] = kuu(np.abs(rdist[ss1, ss2]))
elif i == 0 and j == 1:
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kuyp(np.abs(rdist[ss1,ss2])), kuyn(np.abs(rdist[ss1,ss2]) ) )
K[ss1, ss2] = np.where(rdist[ss1, ss2] > 0,
kuyp(rdist[ss1, ss2]),
kuyn(rdist[ss1, ss2]))
elif i == 1 and j == 1:
K[ss1, ss2] = kyy(np.abs(rdist[ss1, ss2]))
else:
#K[ss1,ss2]= 0
#K[ss1,ss2]= np.where( rdist[ss1,ss2]>0 , kyup(np.abs(rdist[ss1,ss2])), kyun(np.abs(rdist[ss1,ss2]) ) )
K[ss1, ss2] = np.where(rdist[ss1, ss2] > 0,
kyup(rdist[ss1, ss2]),
kyun(rdist[ss1, ss2]))
return K
def update_gradients_full(self, dL_dK, X, X2=None):
#def dK_dtheta(self, dL_dK, X, X2, target):
"""derivative of the covariance matrix with respect to the parameters."""
X,slices = X[:,:-1],index_to_slices(X[:,-1])
if X2 is None:
X2,slices2 = X,slices
K = np.zeros((X.shape[0], X.shape[0]))
else:
X2,slices2 = X2[:,:-1],index_to_slices(X2[:,-1])
vyt = self.variance_Yt
vyx = self.variance_Yx
lyt = 1./(2*self.lengthscale_Yt)
lyx = 1./(2*self.lengthscale_Yx)
a = self.a
b = self.b
c = self.c
tdist = (X[:,0][:,None] - X2[:,0][None,:])**2
xdist = (X[:,1][:,None] - X2[:,1][None,:])**2
#rdist = [tdist,xdist]
ttdist = (X[:,0][:,None] - X2[:,0][None,:])
rd=tdist.shape[0]
dka = np.zeros([rd,rd])
dkb = np.zeros([rd,rd])
dkc = np.zeros([rd,rd])
dkYdvart = np.zeros([rd,rd])
dkYdvarx = np.zeros([rd,rd])
dkYdlent = np.zeros([rd,rd])
dkYdlenx = np.zeros([rd,rd])
kyy = lambda tdist,xdist: np.exp(-lyt*(tdist) -lyx*(xdist))
#k1 = lambda tdist: (lyt - lyt**2 * (tdist) )
#k2 = lambda xdist: ( lyx**2 * (xdist) - lyx )
#k3 = lambda xdist: ( 3*lyx**2 - 6*xdist*lyx**3 + xdist**2*lyx**4 )
#k4 = lambda tdist: -lyt*np.sqrt(tdist)
k1 = lambda tdist: (2*lyt - 4*lyt**2 * (tdist) )
k2 = lambda xdist: ( 4*lyx**2 * (xdist) - 2*lyx )
k3 = lambda xdist: ( 3*4*lyx**2 - 6*8*xdist*lyx**3 + 16*xdist**2*lyx**4 )
k4 = lambda ttdist: 2*lyt*(ttdist)
dkyydlyx = lambda tdist,xdist: kyy(tdist,xdist)*(-xdist)
dkyydlyt = lambda tdist,xdist: kyy(tdist,xdist)*(-tdist)
dk1dlyt = lambda tdist: 2. - 4*2.*lyt*tdist
dk2dlyx = lambda xdist: (4.*2.*lyx*xdist -2.)
dk3dlyx = lambda xdist: (6.*4.*lyx - 18.*8*xdist*lyx**2 + 4*16*xdist**2*lyx**3)
dk4dlyt = lambda ttdist: 2*(ttdist)
for i, s1 in enumerate(slices):
for j, s2 in enumerate(slices2):
for ss1 in s1:
for ss2 in s2:
if i==0 and j==0:
dka[ss1,ss2] = 0
dkb[ss1,ss2] = 0
dkc[ss1,ss2] = 0
dkYdvart[ss1,ss2] = vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])
elif i==0 and j==1:
dka[ss1,ss2] = -k2(xdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkb[ss1,ss2] = k4(ttdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkc[ss1,ss2] = vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#dkYdvart[ss1,ss2] = 0
#dkYdvarx[ss1,ss2] = 0
#dkYdlent[ss1,ss2] = 0
#dkYdlenx[ss1,ss2] = 0
dkYdvart[ss1,ss2] = (-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = (-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])* (-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*b*dk4dlyt(ttdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*k2(xdist[ss1,ss2])+b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*dk2dlyx(xdist[ss1,ss2]))
elif i==1 and j==1:
dka[ss1,ss2] = (2*a*k3(xdist[ss1,ss2]) - 2*c*k2(xdist[ss1,ss2]))*vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkb[ss1,ss2] = 2*b*k1(tdist[ss1,ss2])*vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkc[ss1,ss2] = (-2*a*k2(xdist[ss1,ss2]) + 2*c )*vyt*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvart[ss1,ss2] = ( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 )*vyx* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = ( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 )*vyt* kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])*( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 ) +\
vyx*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*b**2*dk1dlyt(tdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])*( b**2*k1(tdist[ss1,ss2]) - 2*a*c*k2(xdist[ss1,ss2]) + a**2*k3(xdist[ss1,ss2]) + c**2 ) +\
vyx*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])* (-2*a*c*dk2dlyx(xdist[ss1,ss2]) + a**2*dk3dlyx(xdist[ss1,ss2]) )
else:
dka[ss1,ss2] = -k2(xdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkb[ss1,ss2] = -k4(ttdist[ss1,ss2])*vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkc[ss1,ss2] = vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
#dkYdvart[ss1,ss2] = 0
#dkYdvarx[ss1,ss2] = 0
#dkYdlent[ss1,ss2] = 0
#dkYdlenx[ss1,ss2] = 0
dkYdvart[ss1,ss2] = (-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdvarx[ss1,ss2] = (-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)*vyt*kyy(tdist[ss1,ss2],xdist[ss1,ss2])
dkYdlent[ss1,ss2] = vyt*vyx*dkyydlyt(tdist[ss1,ss2],xdist[ss1,ss2])* (-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*(-1)*b*dk4dlyt(ttdist[ss1,ss2])
dkYdlenx[ss1,ss2] = vyt*vyx*dkyydlyx(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*k2(xdist[ss1,ss2])-b*k4(ttdist[ss1,ss2])+c)+\
vyt*vyx*kyy(tdist[ss1,ss2],xdist[ss1,ss2])*(-a*dk2dlyx(xdist[ss1,ss2]))
self.a.gradient = np.sum(dka * dL_dK)
self.b.gradient = np.sum(dkb * dL_dK)
self.c.gradient = np.sum(dkc * dL_dK)
self.variance_Yt.gradient = np.sum(dkYdvart * dL_dK) # Vy
self.variance_Yx.gradient = np.sum(dkYdvarx * dL_dK)
self.lengthscale_Yt.gradient = np.sum(dkYdlent*(-0.5*self.lengthscale_Yt**(-2)) * dL_dK) #ly np.sum(dktheta2*(-self.lengthscale_Y**(-2)) * dL_dK)
self.lengthscale_Yx.gradient = np.sum(dkYdlenx*(-0.5*self.lengthscale_Yx**(-2)) * dL_dK)