def create_Lambda(self,f,sigma):
N = self.N
m = self.m
sample_points, weights = np.polynomial.hermite.hermgauss(self.n_gausshermite_sample_points)
''' Diagonal matrix '''
Lambda_diagonal = [0]*N
constant = 1/(m*(sigma**2))
for i in range(N):
if not self.is_pseudobs(i):
Lambda_diagonal[i] = -constant*self.sum_Phi(i,2,f,sigma) #Note that sum_Phi used so minus sign needed!
else:
latest_x_index = np.max([ind for ind in self.obs_indices if ind < i])
Delta_i = (f[i]-f[latest_x_index])/sigma
Lambda_diagonal[i] = 0.5*constant*Delta_i*var2_normal_pdf(Delta_i)
Lambda_diagonal = np.diag(Lambda_diagonal)
''' Upper triangular off-diagonal matrix'''
Lambda_uppertri = np.zeros((N,N))
for row_ind in range(N):
if not self.is_pseudobs(row_ind):
for col_ind in range(row_ind+1,row_ind+m+1):
if self.is_pseudobs(col_ind):
Delta = (f[col_ind]-f[row_ind])/sigma
Lambda_uppertri[row_ind,col_ind] = -0.5*constant*Delta*var2_normal_pdf(Delta)
''' Final Lambda is sum of diagonal + upper_tringular + lower_tringular '''
Lambda = Lambda_diagonal + Lambda_uppertri + Lambda_uppertri.T
return Lambda
def sum_Phi(self,i,order_of_derivative,f,sigma,sample_points=None, weights=None):
'''
Auxiliary function that computes summation of Phi-function values
i = index of x observation, i.e. some element of list 'obs_indices'
order_of_derivative = how many this the c.d.f of the standard normal (Phi) is differentiated?
f = vector of f-values
'''
m = self.m
#Delta_i_j = (f[i+j+1]-f[i])/(sigma_)
Delta = f[i+1:i+m+1]
Delta = (Delta - f[i])/sigma
sum_=0
if order_of_derivative==0:
for j in range(0,m):
#sum_ = sum_ + integrate(lambda x: std_normal_cdf(Delta[j]+x)*std_normal_pdf(x), -np.inf, np.inf)[0]
#Do integration by using Gaussian-Hermite quadrature:
sum_ = sum_ + (1/np.sqrt(np.pi))*np.dot(weights,std_normal_cdf(Delta[j]-np.sqrt(2)*sample_points)) #Do to change of variables to get integteragl into the form int exp(x^2)*....dx. This is why np.sqrt(2)
return(sum_)
if order_of_derivative==1:
for j in range(0,m):
sum_ = sum_ + float(var2_normal_pdf(Delta[j]))
return(sum_)
if order_of_derivative==2:
for j in range(0,m):
sum_ = sum_ - 0.5*Delta[j]*float(var2_normal_pdf(Delta[j]))
return(sum_)
else:
print("The derivatives of an order higher than 2 are not needed!")
return None
def sum_Phi(self,
i,
order_of_derivative,
f,
sigma,
sample_points=None,
weights=None):
'''
Auxiliary function that computes summation of Phi-function values
i = index of x observation, i.e. some element of list 'obs_indices'
dimensionality of the output depends on order_of_derivative
order_of_derivative = how many this the c.d.f of the standard normal (Phi) is differentiated?
f = vector of f-values
'''
m = self.m
phi_X = self.phi_X
#Delta_i_j = (f[i+j+1]-f[i])/(sigma_)
Delta = f[i + 1:i + m + 1]
Delta = (Delta - f[i]) / sigma
if order_of_derivative == 0:
sum_ = 0
for j in range(0, m):
sum_ = sum_ + (1 / np.sqrt(np.pi)) * np.dot(
weights,
std_normal_cdf(Delta[j] - np.sqrt(2) * sample_points)
) #Do to change of variables to get integteragl into the form int exp(x^2)*....dx. This is why np.sqrt(2)
return (sum_)
if order_of_derivative == 1:
sum_ = np.zeros(self.nFeatures)
for j in range(0, m):
sum_ = sum_ + (phi_X[:, i + 1 + j] - phi_X[:, i]) * float(
var2_normal_pdf(Delta[j]))
return (sum_)
if order_of_derivative == 2:
sum_ = np.zeros(self.nFeatures)
for j in range(0, m):
sum_ = sum_ - (
(phi_X[:, i + 1 + j] - phi_X[:, i])**
2) * 0.5 * Delta[j] * float(var2_normal_pdf(Delta[j]))
return (sum_)
else:
print("The derivatives of an order higher than 2 are not needed!")
return None
def T_grad(self,f,theta,Sigma_inv_=None):
if Sigma_inv_ is None:
Sigma_inv_=self.Sigma_inv
N = self.N
m = self.m
latest_obs_indices = self.latest_obs_indices
beta = np.zeros((N,1)).reshape(N,)
Phi_der_vec_ = np.array([float(var2_normal_pdf((f[j]-f[latest_obs_indices[j]])/theta[0])) for j in self.pseudobs_indices])
sum_Phi_der_vec_ = self.sum_Phi_vec(1,f,theta[0])
beta[self.obs_indices] = sum_Phi_der_vec_/(theta[0]*m)
beta[self.pseudobs_indices] = -Phi_der_vec_/(theta[0]*m)
T_grad = -Sigma_inv_.dot(f).reshape(N,) + beta.reshape(N,)
return T_grad