text: # Copyright (c) 2013, GPy authors (see AUTHORS.txt).
  # Licensed under the BSD 3-clause license (see LICENSE.txt)
  
  from .kern import Kern
  from ...core.parameterization import Param
  from paramz.transformations import Logexp
  import numpy as np
  from .independent_outputs import index_to_slices
  
  class ODE_UYC(Kern):
      def __init__(self, input_dim, variance_U=3., variance_Y=1., lengthscale_U=1., lengthscale_Y=1., ubias =1. ,active_dims=None, name='ode_uyc'):
          assert input_dim ==2, "only defined for 2 input dims"
          super(ODE_UYC, self).__init__(input_dim, active_dims, name)
  
          self.variance_Y = Param('variance_Y', variance_Y, Logexp())
          self.variance_U = Param('variance_U', variance_U, Logexp())
          self.lengthscale_Y = Param('lengthscale_Y', lengthscale_Y, Logexp())
          self.lengthscale_U = Param('lengthscale_U', lengthscale_U, Logexp())
          self.ubias = Param('ubias', ubias, Logexp())
  
          self.link_parameters(self.variance_Y, self.variance_U, self.lengthscale_Y, self.lengthscale_U, self.ubias)
  
      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]))
  
          #stop
          #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)) +self.ubias
  
          # 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 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 + self.ubias
                  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 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[0]
          dktheta1 = np.zeros([rd,rd])
          dktheta2 = np.zeros([rd,rd])
          dkUdvar = np.zeros([rd,rd])
          dkYdvar = np.zeros([rd,rd])
  
          dkdubias = np.zeros([rd,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
                              dkdubias[ss1,ss2] = 1
                          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])  )
                              dkdubias[ss1,ss2] = 0
                          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])) )
                              dkdubias[ss1,ss2] = 0
                          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]))
                              dkdubias[ss1,ss2] = 0
          #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
  
          self.ubias.gradient = np.sum(dkdubias * dL_dK) 
  
  