Merge pull request #1101 from janmayer/fix_invalid_escape_sequence

Fix invalid escape sequence
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Martin Bubel 2024-10-27 13:51:56 +01:00 committed by GitHub
commit 1fcb40845d
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52 changed files with 394 additions and 393 deletions

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@ -1,6 +1,7 @@
# Changelog # Changelog
## Unreleased ## Unreleased
* fix invalid escape sequence #1011 [janmayer]
## v1.13.2 (2024-07-21) ## v1.13.2 (2024-07-21)
* update string checks in initialization method for latent variable and put `empirical_samples` init-method on a deprecation path * update string checks in initialization method for latent variable and put `empirical_samples` init-method on a deprecation path

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@ -194,7 +194,7 @@ class GP(Model):
# Make sure to name this variable and the predict functions will "just work" # Make sure to name this variable and the predict functions will "just work"
# In maths the predictive variable is: # In maths the predictive variable is:
# K_{xx} - K_{xp}W_{pp}^{-1}K_{px} # K_{xx} - K_{xp}W_{pp}^{-1}K_{px}
# W_{pp} := \texttt{Woodbury inv} # W_{pp} := \\texttt{Woodbury inv}
# p := _predictive_variable # p := _predictive_variable
@property @property
@ -283,7 +283,7 @@ class GP(Model):
def log_likelihood(self): def log_likelihood(self):
""" """
The log marginal likelihood of the model, :math:`p(\mathbf{y})`, this is the objective function of the model being optimised The log marginal likelihood of the model, :math:`p(\\mathbf{y})`, this is the objective function of the model being optimised
""" """
return self._log_marginal_likelihood return self._log_marginal_likelihood
@ -296,9 +296,9 @@ class GP(Model):
diagonal of the covariance is returned. diagonal of the covariance is returned.
.. math:: .. math::
p(f*|X*, X, Y) = \int^{\inf}_{\inf} p(f*|f,X*)p(f|X,Y) df p(f*|X*, X, Y) = \\int^{\\inf}_{\\inf} p(f*|f,X*)p(f|X,Y) df
= N(f*| K_{x*x}(K_{xx} + \Sigma)^{-1}Y, K_{x*x*} - K_{xx*}(K_{xx} + \Sigma)^{-1}K_{xx*} = N(f*| K_{x*x}(K_{xx} + \\Sigma)^{-1}Y, K_{x*x*} - K_{xx*}(K_{xx} + \\Sigma)^{-1}K_{xx*}
\Sigma := \texttt{Likelihood.variance / Approximate likelihood covariance} \\Sigma := \\texttt{Likelihood.variance / Approximate likelihood covariance}
""" """
mu, var = self.posterior._raw_predict(kern=self.kern if kern is None else kern, Xnew=Xnew, pred_var=self._predictive_variable, full_cov=full_cov) mu, var = self.posterior._raw_predict(kern=self.kern if kern is None else kern, Xnew=Xnew, pred_var=self._predictive_variable, full_cov=full_cov)
if self.mean_function is not None: if self.mean_function is not None:
@ -702,7 +702,7 @@ class GP(Model):
Calculation of the log predictive density Calculation of the log predictive density
.. math: .. math:
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*}) p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\\mu_{*}\\sigma^{2}_{*})
:param x_test: test locations (x_{*}) :param x_test: test locations (x_{*})
:type x_test: (Nx1) array :type x_test: (Nx1) array
@ -718,7 +718,7 @@ class GP(Model):
Calculation of the log predictive density by sampling Calculation of the log predictive density by sampling
.. math: .. math:
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*}) p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\\mu_{*}\\sigma^{2}_{*})
:param x_test: test locations (x_{*}) :param x_test: test locations (x_{*})
:type x_test: (Nx1) array :type x_test: (Nx1) array

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@ -379,7 +379,7 @@ class Symbolic_core():
def _display_expression(self, keys, user_substitutes={}): def _display_expression(self, keys, user_substitutes={}):
"""Helper function for human friendly display of the symbolic components.""" """Helper function for human friendly display of the symbolic components."""
# Create some pretty maths symbols for the display. # Create some pretty maths symbols for the display.
sigma, alpha, nu, omega, l, variance = sym.var('\sigma, \alpha, \nu, \omega, \ell, \sigma^2') sigma, alpha, nu, omega, l, variance = sym.var(r'\sigma, \alpha, \nu, \omega, \ell, \sigma^2')
substitutes = {'scale': sigma, 'shape': alpha, 'lengthscale': l, 'variance': variance} substitutes = {'scale': sigma, 'shape': alpha, 'lengthscale': l, 'variance': variance}
substitutes.update(user_substitutes) substitutes.update(user_substitutes)

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@ -134,10 +134,10 @@ class posteriorParams(posteriorParamsBase):
B = np.eye(num_data) + Sroot_tilde_K * tau_tilde_root[None,:] B = np.eye(num_data) + Sroot_tilde_K * tau_tilde_root[None,:]
L = jitchol(B) L = jitchol(B)
V, _ = dtrtrs(L, Sroot_tilde_K, lower=1) V, _ = dtrtrs(L, Sroot_tilde_K, lower=1)
Sigma = K - np.dot(V.T,V) #K - KS^(1/2)BS^(1/2)K = (K^(-1) + \Sigma^(-1))^(-1) Sigma = K - np.dot(V.T,V) #K - KS^(1/2)BS^(1/2)K = (K^(-1) + \\Sigma^(-1))^(-1)
aux_alpha , _ = dpotrs(L, tau_tilde_root * (np.dot(K, ga_approx.v) + mean_prior), lower=1) aux_alpha , _ = dpotrs(L, tau_tilde_root * (np.dot(K, ga_approx.v) + mean_prior), lower=1)
alpha = ga_approx.v - tau_tilde_root * aux_alpha #(K + Sigma^(\tilde))^(-1) (/mu^(/tilde) - /mu_p) alpha = ga_approx.v - tau_tilde_root * aux_alpha #(K + Sigma^(\\tilde))^(-1) (/mu^(/tilde) - /mu_p)
mu = np.dot(K, alpha) + mean_prior mu = np.dot(K, alpha) + mean_prior
return posteriorParams(mu=mu, Sigma=Sigma, L=L) return posteriorParams(mu=mu, Sigma=Sigma, L=L)
@ -151,8 +151,8 @@ class posteriorParamsDTC(posteriorParamsBase):
DSYR(LLT,Kmn[:,i].copy(),delta_tau) DSYR(LLT,Kmn[:,i].copy(),delta_tau)
L = jitchol(LLT) L = jitchol(LLT)
V,info = dtrtrs(L,Kmn,lower=1) V,info = dtrtrs(L,Kmn,lower=1)
self.Sigma_diag = np.maximum(np.sum(V*V,-2), np.finfo(float).eps) #diag(K_nm (L L^\top)^(-1)) K_mn self.Sigma_diag = np.maximum(np.sum(V*V,-2), np.finfo(float).eps) #diag(K_nm (L L^\\top)^(-1)) K_mn
si = np.sum(V.T*V[:,i],-1) #(V V^\top)[:,i] si = np.sum(V.T*V[:,i],-1) #(V V^\\top)[:,i]
self.mu += (delta_v-delta_tau*self.mu[i])*si self.mu += (delta_v-delta_tau*self.mu[i])*si
#mu = np.dot(Sigma, v_tilde) #mu = np.dot(Sigma, v_tilde)
@ -391,11 +391,11 @@ class EP(EPBase, ExactGaussianInference):
aux_alpha , _ = dpotrs(post_params.L, tau_tilde_root * (np.dot(K, ga_approx.v) + mean_prior), lower=1) aux_alpha , _ = dpotrs(post_params.L, tau_tilde_root * (np.dot(K, ga_approx.v) + mean_prior), lower=1)
alpha = (ga_approx.v - tau_tilde_root * aux_alpha)[:,None] #(K + Sigma^(\tilde))^(-1) (/mu^(/tilde) - /mu_p) alpha = (ga_approx.v - tau_tilde_root * aux_alpha)[:,None] #(K + Sigma^(\\tilde))^(-1) (/mu^(/tilde) - /mu_p)
LWi, _ = dtrtrs(post_params.L, np.diag(tau_tilde_root), lower=1) LWi, _ = dtrtrs(post_params.L, np.diag(tau_tilde_root), lower=1)
Wi = np.dot(LWi.T,LWi) Wi = np.dot(LWi.T,LWi)
symmetrify(Wi) #(K + Sigma^(\tilde))^(-1) symmetrify(Wi) #(K + Sigma^(\\tilde))^(-1)
dL_dK = 0.5 * (tdot(alpha) - Wi) dL_dK = 0.5 * (tdot(alpha) - Wi)
dL_dthetaL = likelihood.ep_gradients(Y, cav_params.tau, cav_params.v, np.diag(dL_dK), Y_metadata=Y_metadata, quad_mode='gh') dL_dthetaL = likelihood.ep_gradients(Y, cav_params.tau, cav_params.v, np.diag(dL_dK), Y_metadata=Y_metadata, quad_mode='gh')
@ -530,7 +530,7 @@ class EPDTC(EPBase, VarDTC):
#initial values - Gaussian factors #initial values - Gaussian factors
#Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma) #Initial values - Posterior distribution parameters: q(f|X,Y) = N(f|mu,Sigma)
LLT0 = Kmm.copy() LLT0 = Kmm.copy()
Lm = jitchol(LLT0) #K_m = L_m L_m^\top Lm = jitchol(LLT0) #K_m = L_m L_m^\\top
Vm,info = dtrtrs(Lm, Kmn,lower=1) Vm,info = dtrtrs(Lm, Kmn,lower=1)
# Lmi = dtrtri(Lm) # Lmi = dtrtri(Lm)
# Kmmi = np.dot(Lmi.T,Lmi) # Kmmi = np.dot(Lmi.T,Lmi)

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@ -27,7 +27,7 @@ class Laplace(LatentFunctionInference):
""" """
Laplace Approximation Laplace Approximation
Find the moments \hat{f} and the hessian at this point Find the moments \\hat{f} and the hessian at this point
(using Newton-Raphson) of the unnormalised posterior (using Newton-Raphson) of the unnormalised posterior
""" """

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@ -8,7 +8,7 @@ log_2_pi = np.log(2*np.pi)
class PEP(LatentFunctionInference): class PEP(LatentFunctionInference):
''' '''
Sparse Gaussian processes using Power-Expectation Propagation Sparse Gaussian processes using Power-Expectation Propagation
for regression: alpha \approx 0 gives VarDTC and alpha = 1 gives FITC for regression: alpha \\approx 0 gives VarDTC and alpha = 1 gives FITC
Reference: A Unifying Framework for Sparse Gaussian Process Approximation using Reference: A Unifying Framework for Sparse Gaussian Process Approximation using
Power Expectation Propagation, https://arxiv.org/abs/1605.07066 Power Expectation Propagation, https://arxiv.org/abs/1605.07066

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@ -82,7 +82,7 @@ class Posterior(object):
Posterior mean Posterior mean
$$ $$
K_{xx}v K_{xx}v
v := \texttt{Woodbury vector} v := \\texttt{Woodbury vector}
$$ $$
""" """
if self._mean is None: if self._mean is None:
@ -95,7 +95,7 @@ class Posterior(object):
Posterior covariance Posterior covariance
$$ $$
K_{xx} - K_{xx}W_{xx}^{-1}K_{xx} K_{xx} - K_{xx}W_{xx}^{-1}K_{xx}
W_{xx} := \texttt{Woodbury inv} W_{xx} := \\texttt{Woodbury inv}
$$ $$
""" """
if self._covariance is None: if self._covariance is None:
@ -146,8 +146,8 @@ class Posterior(object):
""" """
return $L_{W}$ where L is the lower triangular Cholesky decomposition of the Woodbury matrix return $L_{W}$ where L is the lower triangular Cholesky decomposition of the Woodbury matrix
$$ $$
L_{W}L_{W}^{\top} = W^{-1} L_{W}L_{W}^{\\top} = W^{-1}
W^{-1} := \texttt{Woodbury inv} W^{-1} := \\texttt{Woodbury inv}
$$ $$
""" """
if self._woodbury_chol is None: if self._woodbury_chol is None:
@ -178,8 +178,8 @@ class Posterior(object):
""" """
The inverse of the woodbury matrix, in the gaussian likelihood case it is defined as The inverse of the woodbury matrix, in the gaussian likelihood case it is defined as
$$ $$
(K_{xx} + \Sigma_{xx})^{-1} (K_{xx} + \\Sigma_{xx})^{-1}
\Sigma_{xx} := \texttt{Likelihood.variance / Approximate likelihood covariance} \\Sigma_{xx} := \\texttt{Likelihood.variance / Approximate likelihood covariance}
$$ $$
""" """
if self._woodbury_inv is None: if self._woodbury_inv is None:
@ -200,8 +200,8 @@ class Posterior(object):
""" """
Woodbury vector in the gaussian likelihood case only is defined as Woodbury vector in the gaussian likelihood case only is defined as
$$ $$
(K_{xx} + \Sigma)^{-1}Y (K_{xx} + \\Sigma)^{-1}Y
\Sigma := \texttt{Likelihood.variance / Approximate likelihood covariance} \\Sigma := \\texttt{Likelihood.variance / Approximate likelihood covariance}
$$ $$
""" """
if self._woodbury_vector is None: if self._woodbury_vector is None:

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@ -25,12 +25,12 @@ class Coregionalize(Kern):
This covariance has the form: This covariance has the form:
.. math:: .. math::
\mathbf{B} = \mathbf{W}\mathbf{W}^\intercal + \mathrm{diag}(kappa) \\mathbf{B} = \\mathbf{W}\\mathbf{W}^\\intercal + \\mathrm{diag}(kappa)
An intrinsic/linear coregionalization covariance function of the form: An intrinsic/linear coregionalization covariance function of the form:
.. math:: .. math::
k_2(x, y)=\mathbf{B} k(x, y) k_2(x, y)=\\mathbf{B} k(x, y)
it is obtained as the tensor product between a covariance function it is obtained as the tensor product between a covariance function
k(x, y) and B. k(x, y) and B.

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@ -15,7 +15,7 @@ class EQ_ODE1(Kern):
This outputs of this kernel have the form This outputs of this kernel have the form
.. math:: .. math::
\frac{\text{d}y_j}{\text{d}t} = \sum_{i=1}^R w_{j,i} u_i(t-\delta_j) - d_jy_j(t) \\frac{\\text{d}y_j}{\\text{d}t} = \\sum_{i=1}^R w_{j,i} u_i(t-\\delta_j) - d_jy_j(t)
where :math:`R` is the rank of the system, :math:`w_{j,i}` is the sensitivity of the :math:`j`th output to the :math:`i`th latent function, :math:`d_j` is the decay rate of the :math:`j`th output and :math:`u_i(t)` are independent latent Gaussian processes goverened by an exponentiated quadratic covariance. where :math:`R` is the rank of the system, :math:`w_{j,i}` is the sensitivity of the :math:`j`th output to the :math:`i`th latent function, :math:`d_j` is the decay rate of the :math:`j`th output and :math:`u_i(t)` are independent latent Gaussian processes goverened by an exponentiated quadratic covariance.

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@ -15,7 +15,7 @@ class EQ_ODE2(Kern):
This outputs of this kernel have the form This outputs of this kernel have the form
.. math:: .. math::
\frac{\text{d}^2y_j(t)}{\text{d}^2t} + C_j\frac{\text{d}y_j(t)}{\text{d}t} + B_jy_j(t) = \sum_{i=1}^R w_{j,i} u_i(t) \\frac{\\text{d}^2y_j(t)}{\\text{d}^2t} + C_j\\frac{\\text{d}y_j(t)}{\\text{d}t} + B_jy_j(t) = \\sum_{i=1}^R w_{j,i} u_i(t)
where :math:`R` is the rank of the system, :math:`w_{j,i}` is the sensitivity of the :math:`j`th output to the :math:`i`th latent function, :math:`d_j` is the decay rate of the :math:`j`th output and :math:`f_i(t)` and :math:`g_i(t)` are independent latent Gaussian processes goverened by an exponentiated quadratic covariance. where :math:`R` is the rank of the system, :math:`w_{j,i}` is the sensitivity of the :math:`j`th output to the :math:`i`th latent function, :math:`d_j` is the decay rate of the :math:`j`th output and :math:`f_i(t)` and :math:`g_i(t)` are independent latent Gaussian processes goverened by an exponentiated quadratic covariance.

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@ -45,7 +45,7 @@ class GridRBF(GridKern):
.. math:: .. math::
k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg) k(r) = \\sigma^2 \\exp \\bigg(- \\frac{1}{2} r^2 \\bigg)
""" """
_support_GPU = True _support_GPU = True

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@ -146,25 +146,25 @@ class Kern(Parameterized):
def psi0(self, Z, variational_posterior): def psi0(self, Z, variational_posterior):
""" """
.. math:: .. math::
\psi_0 = \sum_{i=0}^{n}E_{q(X)}[k(X_i, X_i)] \\psi_0 = \\sum_{i=0}^{n}E_{q(X)}[k(X_i, X_i)]
""" """
return self.psicomp.psicomputations(self, Z, variational_posterior)[0] return self.psicomp.psicomputations(self, Z, variational_posterior)[0]
def psi1(self, Z, variational_posterior): def psi1(self, Z, variational_posterior):
""" """
.. math:: .. math::
\psi_1^{n,m} = E_{q(X)}[k(X_n, Z_m)] \\psi_1^{n,m} = E_{q(X)}[k(X_n, Z_m)]
""" """
return self.psicomp.psicomputations(self, Z, variational_posterior)[1] return self.psicomp.psicomputations(self, Z, variational_posterior)[1]
def psi2(self, Z, variational_posterior): def psi2(self, Z, variational_posterior):
""" """
.. math:: .. math::
\psi_2^{m,m'} = \sum_{i=0}^{n}E_{q(X)}[ k(Z_m, X_i) k(X_i, Z_{m'})] \\psi_2^{m,m'} = \\sum_{i=0}^{n}E_{q(X)}[ k(Z_m, X_i) k(X_i, Z_{m'})]
""" """
return self.psicomp.psicomputations(self, Z, variational_posterior, return_psi2_n=False)[2] return self.psicomp.psicomputations(self, Z, variational_posterior, return_psi2_n=False)[2]
def psi2n(self, Z, variational_posterior): def psi2n(self, Z, variational_posterior):
""" """
.. math:: .. math::
\psi_2^{n,m,m'} = E_{q(X)}[ k(Z_m, X_n) k(X_n, Z_{m'})] \\psi_2^{n,m,m'} = E_{q(X)}[ k(Z_m, X_n) k(X_n, Z_{m'})]
Thus, we do not sum out n, compared to psi2 Thus, we do not sum out n, compared to psi2
""" """
@ -173,7 +173,7 @@ class Kern(Parameterized):
""" """
.. math:: .. math::
\\frac{\partial L}{\partial X} = \\frac{\partial L}{\partial K}\\frac{\partial K}{\partial X} \\frac{\\partial L}{\\partial X} = \\frac{\\partial L}{\\partial K}\\frac{\\partial K}{\\partial X}
""" """
raise NotImplementedError raise NotImplementedError
def gradients_X_X2(self, dL_dK, X, X2): def gradients_X_X2(self, dL_dK, X, X2):
@ -182,7 +182,7 @@ class Kern(Parameterized):
""" """
.. math:: .. math::
\\frac{\partial^2 L}{\partial X\partial X_2} = \\frac{\partial L}{\partial K}\\frac{\partial^2 K}{\partial X\partial X_2} \\frac{\\partial^2 L}{\\partial X\\partial X_2} = \\frac{\\partial L}{\\partial K}\\frac{\\partial^2 K}{\\partial X\\partial X_2}
""" """
raise NotImplementedError("This is the second derivative of K wrt X and X2, and not implemented for this kernel") raise NotImplementedError("This is the second derivative of K wrt X and X2, and not implemented for this kernel")
def gradients_XX_diag(self, dL_dKdiag, X, cov=True): def gradients_XX_diag(self, dL_dKdiag, X, cov=True):
@ -216,9 +216,9 @@ class Kern(Parameterized):
.. math:: .. math::
\\frac{\partial L}{\partial \\theta_i} & = \\frac{\partial L}{\partial \psi_0}\\frac{\partial \psi_0}{\partial \\theta_i}\\ \\frac{\\partial L}{\\partial \\theta_i} & = \\frac{\\partial L}{\\partial \\psi_0}\\frac{\\partial \\psi_0}{\\partial \\theta_i}\\
& \quad + \\frac{\partial L}{\partial \psi_1}\\frac{\partial \psi_1}{\partial \\theta_i}\\ & \\quad + \\frac{\\partial L}{\\partial \\psi_1}\\frac{\\partial \\psi_1}{\\partial \\theta_i}\\
& \quad + \\frac{\partial L}{\partial \psi_2}\\frac{\partial \psi_2}{\partial \\theta_i} & \\quad + \\frac{\\partial L}{\\partial \\psi_2}\\frac{\\partial \\psi_2}{\\partial \\theta_i}
Thus, we push the different derivatives through the gradients of the psi Thus, we push the different derivatives through the gradients of the psi
statistics. Be sure to set the gradients for all kernel statistics. Be sure to set the gradients for all kernel

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@ -16,15 +16,15 @@ class Linear(Kern):
.. math:: .. math::
k(x,y) = \sum_{i=1}^{\\text{input_dim}} \sigma^2_i x_iy_i k(x,y) = \\sum_{i=1}^{\\text{input_dim}} \\sigma^2_i x_iy_i
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
:param variances: the vector of variances :math:`\sigma^2_i` :param variances: the vector of variances :math:`\\sigma^2_i`
:type variances: array or list of the appropriate size (or float if there :type variances: array or list of the appropriate size (or float if there
is only one variance parameter) is only one variance parameter)
:param ARD: Auto Relevance Determination. If False, the kernel has only one :param ARD: Auto Relevance Determination. If False, the kernel has only one
variance parameter \sigma^2, otherwise there is one variance variance parameter \\sigma^2, otherwise there is one variance
parameter per dimension. parameter per dimension.
:type ARD: Boolean :type ARD: Boolean
:rtype: kernel object :rtype: kernel object
@ -121,7 +121,7 @@ class Linear(Kern):
the returned array is of shape [NxNxQxQ]. the returned array is of shape [NxNxQxQ].
..math: ..math:
\frac{\partial^2 K}{\partial X2 ^2} = - \frac{\partial^2 K}{\partial X\partial X2} \\frac{\\partial^2 K}{\\partial X2 ^2} = - \\frac{\\partial^2 K}{\\partial X\\partial X2}
..returns: ..returns:
dL2_dXdX2: [NxMxQxQ] for X [NxQ] and X2[MxQ] (X2 is X if, X2 is None) dL2_dXdX2: [NxMxQxQ] for X [NxQ] and X2[MxQ] (X2 is X if, X2 is None)

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@ -20,12 +20,12 @@ class MLP(Kern):
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
:param variance: the variance :math:`\sigma^2` :param variance: the variance :math:`\\sigma^2`
:type variance: float :type variance: float
:param weight_variance: the vector of the variances of the prior over input weights in the neural network :math:`\sigma^2_w` :param weight_variance: the vector of the variances of the prior over input weights in the neural network :math:`\\sigma^2_w`
:type weight_variance: array or list of the appropriate size (or float if there is only one weight variance parameter) :type weight_variance: array or list of the appropriate size (or float if there is only one weight variance parameter)
:param bias_variance: the variance of the prior over bias parameters :math:`\sigma^2_b` :param bias_variance: the variance of the prior over bias parameters :math:`\\sigma^2_b`
:param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one weight variance parameter \sigma^2_w), otherwise there is one weight variance parameter per dimension. :param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one weight variance parameter \\sigma^2_w), otherwise there is one weight variance parameter per dimension.
:type ARD: Boolean :type ARD: Boolean
:rtype: Kernpart object :rtype: Kernpart object

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@ -16,7 +16,7 @@ class RBF(Stationary):
.. math:: .. math::
k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg) k(r) = \\sigma^2 \\exp \\bigg(- \\frac{1}{2} r^2 \\bigg)
""" """
_support_GPU = True _support_GPU = True

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@ -20,7 +20,7 @@ class sde_Brownian(Brownian):
.. math:: .. math::
k(x,y) = \sigma^2 min(x,y) k(x,y) = \\sigma^2 min(x,y)
""" """

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@ -19,7 +19,7 @@ class sde_Linear(Linear):
.. math:: .. math::
k(x,y) = \sum_{i=1}^{input dim} \sigma^2_i x_iy_i k(x,y) = \\sum_{i=1}^{input dim} \\sigma^2_i x_iy_i
""" """
def __init__(self, input_dim, X, variances=None, ARD=False, active_dims=None, name='linear'): def __init__(self, input_dim, X, variances=None, ARD=False, active_dims=None, name='linear'):

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@ -19,7 +19,7 @@ class sde_Matern32(Matern32):
.. math:: .. math::
k(r) = \sigma^2 (1 + \sqrt{3} r) \exp(- \sqrt{3} r) \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} } k(r) = \\sigma^2 (1 + \\sqrt{3} r) \\exp(- \\sqrt{3} r) \\ \\ \\ \\ \\text{ where } r = \\sqrt{\\sum_{i=1}^{input dim} \\frac{(x_i-y_i)^2}{\\ell_i^2} }
""" """
def sde_update_gradient_full(self, gradients): def sde_update_gradient_full(self, gradients):
@ -79,7 +79,7 @@ class sde_Matern52(Matern52):
.. math:: .. math::
k(r) = \sigma^2 (1 + \sqrt{5} r + \frac{5}{3}r^2) \exp(- \sqrt{5} r) \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} } k(r) = \\sigma^2 (1 + \\sqrt{5} r + \\frac{5}{3}r^2) \\exp(- \\sqrt{5} r) \\ \\ \\ \\ \\text{ where } r = \\sqrt{\\sum_{i=1}^{input dim} \\frac{(x_i-y_i)^2}{\\ell_i^2} }
""" """
def sde_update_gradient_full(self, gradients): def sde_update_gradient_full(self, gradients):

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@ -24,8 +24,8 @@ class sde_StdPeriodic(StdPeriodic):
.. math:: .. math::
k(x,y) = \theta_1 \exp \left[ - \frac{1}{2} {}\sum_{i=1}^{input\_dim} k(x,y) = \\theta_1 \\exp \\left[ - \\frac{1}{2} {}\\sum_{i=1}^{input\\_dim}
\left( \frac{\sin(\frac{\pi}{\lambda_i} (x_i - y_i) )}{l_i} \right)^2 \right] } \\left( \\frac{\\sin(\\frac{\\pi}{\\lambda_i} (x_i - y_i) )}{l_i} \\right)^2 \\right] }
""" """
@ -177,7 +177,7 @@ def seriescoeff(m=6, lengthScale=1.0, magnSigma2=1.0, true_covariance=False):
Calculate the coefficients q_j^2 for the covariance function Calculate the coefficients q_j^2 for the covariance function
approximation: approximation:
k(\tau) = \sum_{j=0}^{+\infty} q_j^2 \cos(j\omega_0 \tau) k(\\tau) = \\sum_{j=0}^{+\\infty} q_j^2 \\cos(j\\omega_0 \\tau)
Reference is: Reference is:

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@ -20,7 +20,7 @@ class sde_White(White):
.. math:: .. math::
k(x,y) = \alpha*\delta(x-y) k(x,y) = \\alpha*\\delta(x-y)
""" """
@ -68,7 +68,7 @@ class sde_Bias(Bias):
.. math:: .. math::
k(x,y) = \alpha k(x,y) = \\alpha
""" """
def sde_update_gradient_full(self, gradients): def sde_update_gradient_full(self, gradients):

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@ -29,7 +29,7 @@ class sde_RBF(RBF):
.. math:: .. math::
k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg) \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} } k(r) = \\sigma^2 \\exp \\bigg(- \\frac{1}{2} r^2 \\bigg) \\ \\ \\ \\ \\text{ where } r = \\sqrt{\\sum_{i=1}^{input dim} \\frac{(x_i-y_i)^2}{\\ell_i^2} }
""" """
@ -204,7 +204,7 @@ class sde_Exponential(Exponential):
.. math:: .. math::
k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r \\bigg) \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} } k(r) = \\sigma^2 \\exp \\bigg(- \\frac{1}{2} r \\bigg) \\ \\ \\ \\ \\text{ where } r = \\sqrt{\\sum_{i=1}^{input dim} \\frac{(x_i-y_i)^2}{\\ell_i^2} }
""" """
@ -259,7 +259,7 @@ class sde_RatQuad(RatQuad):
.. math:: .. math::
k(r) = \sigma^2 \\bigg( 1 + \\frac{r^2}{2} \\bigg)^{- \alpha} \\ \\ \\ \\ \text{ where } r = \sqrt{\sum_{i=1}^{input dim} \frac{(x_i-y_i)^2}{\ell_i^2} } k(r) = \\sigma^2 \\bigg( 1 + \\frac{r^2}{2} \\bigg)^{- \\alpha} \\ \\ \\ \\ \\text{ where } r = \\sqrt{\\sum_{i=1}^{input dim} \\frac{(x_i-y_i)^2}{\\ell_i^2} }
""" """

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@ -24,19 +24,19 @@ class StdPeriodic(Kern):
.. math:: .. math::
k(x,y) = \theta_1 \exp \left[ - \frac{1}{2} \sum_{i=1}^{input\_dim} k(x,y) = \\theta_1 \\exp \\left[ - \\frac{1}{2} \\sum_{i=1}^{input\\_dim}
\left( \frac{\sin(\frac{\pi}{T_i} (x_i - y_i) )}{l_i} \right)^2 \right] } \\left( \\frac{\\sin(\\frac{\\pi}{T_i} (x_i - y_i) )}{l_i} \\right)^2 \\right] }
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
:param variance: the variance :math:`\theta_1` in the formula above :param variance: the variance :math:`\\theta_1` in the formula above
:type variance: float :type variance: float
:param period: the vector of periods :math:`\T_i`. If None then 1.0 is assumed. :param period: the vector of periods :math:`\\T_i`. If None then 1.0 is assumed.
:type period: array or list of the appropriate size (or float if there is only one period parameter) :type period: array or list of the appropriate size (or float if there is only one period parameter)
:param lengthscale: the vector of lengthscale :math:`\l_i`. If None then 1.0 is assumed. :param lengthscale: the vector of lengthscale :math:`\\l_i`. If None then 1.0 is assumed.
:type lengthscale: array or list of the appropriate size (or float if there is only one lengthscale parameter) :type lengthscale: array or list of the appropriate size (or float if there is only one lengthscale parameter)
:param ARD1: Auto Relevance Determination with respect to period. :param ARD1: Auto Relevance Determination with respect to period.
If equal to "False" one single period parameter :math:`\T_i` for If equal to "False" one single period parameter :math:`\\T_i` for
each dimension is assumed, otherwise there is one lengthscale each dimension is assumed, otherwise there is one lengthscale
parameter per dimension. parameter per dimension.
:type ARD1: Boolean :type ARD1: Boolean

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@ -35,7 +35,7 @@ class Stationary(Kern):
.. math:: .. math::
r(x, x') = \\sqrt{ \\sum_{q=1}^Q \\frac{(x_q - x'_q)^2}{\ell_q^2} }. r(x, x') = \\sqrt{ \\sum_{q=1}^Q \\frac{(x_q - x'_q)^2}{\\ell_q^2} }.
By default, there's only one lengthscale: seaprate lengthscales for each By default, there's only one lengthscale: seaprate lengthscales for each
dimension can be enables by setting ARD=True. dimension can be enables by setting ARD=True.
@ -153,7 +153,7 @@ class Stationary(Kern):
Efficiently compute the scaled distance, r. Efficiently compute the scaled distance, r.
..math:: ..math::
r = \sqrt( \sum_{q=1}^Q (x_q - x'q)^2/l_q^2 ) r = \\sqrt( \\sum_{q=1}^Q (x_q - x'q)^2/l_q^2 )
Note that if thre is only one lengthscale, l comes outside the sum. In Note that if thre is only one lengthscale, l comes outside the sum. In
this case we compute the unscaled distance first (in a separate this case we compute the unscaled distance first (in a separate
@ -259,7 +259,7 @@ class Stationary(Kern):
the returned array is of shape [NxNxQxQ]. the returned array is of shape [NxNxQxQ].
..math: ..math:
\frac{\partial^2 K}{\partial X2 ^2} = - \frac{\partial^2 K}{\partial X\partial X2} \\frac{\\partial^2 K}{\\partial X2 ^2} = - \\frac{\\partial^2 K}{\\partial X\\partial X2}
..returns: ..returns:
dL2_dXdX2: [NxMxQxQ] in the cov=True case, or [NxMxQ] in the cov=False case, dL2_dXdX2: [NxMxQxQ] in the cov=True case, or [NxMxQ] in the cov=False case,
@ -295,7 +295,7 @@ class Stationary(Kern):
Given the derivative of the objective dL_dK, compute the second derivative of K wrt X: Given the derivative of the objective dL_dK, compute the second derivative of K wrt X:
..math: ..math:
\frac{\partial^2 K}{\partial X\partial X} \\frac{\\partial^2 K}{\\partial X\\partial X}
..returns: ..returns:
dL2_dXdX: [NxQxQ] dL2_dXdX: [NxQxQ]
@ -423,7 +423,7 @@ class OU(Stationary):
.. math:: .. math::
k(r) = \\sigma^2 \exp(- r) \\ \\ \\ \\ \\text{ where } r = \sqrt{\sum_{i=1}^{\text{input_dim}} \\frac{(x_i-y_i)^2}{\ell_i^2} } k(r) = \\sigma^2 \\exp(- r) \\ \\ \\ \\ \\text{ where } r = \\sqrt{\\sum_{i=1}^{\\text{input_dim}} \\frac{(x_i-y_i)^2}{\\ell_i^2} }
""" """
@ -460,7 +460,7 @@ class Matern32(Stationary):
.. math:: .. math::
k(r) = \\sigma^2 (1 + \\sqrt{3} r) \exp(- \sqrt{3} r) \\ \\ \\ \\ \\text{ where } r = \sqrt{\sum_{i=1}^{\\text{input_dim}} \\frac{(x_i-y_i)^2}{\ell_i^2} } k(r) = \\sigma^2 (1 + \\sqrt{3} r) \\exp(- \\sqrt{3} r) \\ \\ \\ \\ \\text{ where } r = \\sqrt{\\sum_{i=1}^{\\text{input_dim}} \\frac{(x_i-y_i)^2}{\\ell_i^2} }
""" """
@ -559,7 +559,7 @@ class Matern52(Stationary):
.. math:: .. math::
k(r) = \sigma^2 (1 + \sqrt{5} r + \\frac53 r^2) \exp(- \sqrt{5} r) k(r) = \\sigma^2 (1 + \\sqrt{5} r + \\frac53 r^2) \\exp(- \\sqrt{5} r)
""" """
def __init__(self, input_dim, variance=1., lengthscale=None, ARD=False, active_dims=None, name='Mat52'): def __init__(self, input_dim, variance=1., lengthscale=None, ARD=False, active_dims=None, name='Mat52'):
super(Matern52, self).__init__(input_dim, variance, lengthscale, ARD, active_dims, name) super(Matern52, self).__init__(input_dim, variance, lengthscale, ARD, active_dims, name)
@ -626,7 +626,7 @@ class ExpQuad(Stationary):
.. math:: .. math::
k(r) = \sigma^2 \exp(- 0.5 r^2) k(r) = \\sigma^2 \\exp(- 0.5 r^2)
notes:: notes::
- This is exactly the same as the RBF covariance function, but the - This is exactly the same as the RBF covariance function, but the
@ -667,7 +667,7 @@ class Cosine(Stationary):
.. math:: .. math::
k(r) = \sigma^2 \cos(r) k(r) = \\sigma^2 \\cos(r)
""" """
def __init__(self, input_dim, variance=1., lengthscale=None, ARD=False, active_dims=None, name='Cosine'): def __init__(self, input_dim, variance=1., lengthscale=None, ARD=False, active_dims=None, name='Cosine'):
@ -685,7 +685,7 @@ class ExpQuadCosine(Stationary):
.. math:: .. math::
k(r) = \sigma^2 \exp(-2\pi^2r^2)\cos(2\pi r/T) k(r) = \\sigma^2 \\exp(-2\\pi^2r^2)\\cos(2\\pi r/T)
""" """
@ -720,7 +720,7 @@ class Sinc(Stationary):
.. math:: .. math::
k(r) = \sigma^2 \sinc(\pi r) k(r) = \\sigma^2 \\sinc(\\pi r)
""" """
@ -742,7 +742,7 @@ class RatQuad(Stationary):
.. math:: .. math::
k(r) = \sigma^2 \\bigg( 1 + \\frac{r^2}{2} \\bigg)^{- \\alpha} k(r) = \\sigma^2 \\bigg( 1 + \\frac{r^2}{2} \\bigg)^{- \\alpha}
""" """

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@ -14,7 +14,7 @@ class Eq_ode1(Kernpart):
This outputs of this kernel have the form This outputs of this kernel have the form
.. math:: .. math::
\frac{\text{d}y_j}{\text{d}t} = \sum_{i=1}^R w_{j,i} f_i(t-\delta_j) +\sqrt{\kappa_j}g_j(t) - d_jy_j(t) \\frac{\\text{d}y_j}{\\text{d}t} = \\sum_{i=1}^R w_{j,i} f_i(t-\\delta_j) +\\sqrt{\\kappa_j}g_j(t) - d_jy_j(t)
where :math:`R` is the rank of the system, :math:`w_{j,i}` is the sensitivity of the :math:`j`th output to the :math:`i`th latent function, :math:`d_j` is the decay rate of the :math:`j`th output and :math:`f_i(t)` and :math:`g_i(t)` are independent latent Gaussian processes goverened by an exponentiated quadratic covariance. where :math:`R` is the rank of the system, :math:`w_{j,i}` is the sensitivity of the :math:`j`th output to the :math:`i`th latent function, :math:`d_j` is the decay rate of the :math:`j`th output and :math:`f_i(t)` and :math:`g_i(t)` are independent latent Gaussian processes goverened by an exponentiated quadratic covariance.

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@ -15,7 +15,7 @@ class Gibbs(Kernpart):
r = sqrt((x_i - x_j)'*(x_i - x_j)) r = sqrt((x_i - x_j)'*(x_i - x_j))
k(x_i, x_j) = \sigma^2*Z*exp(-r^2/(l(x)*l(x) + l(x')*l(x'))) k(x_i, x_j) = \\sigma^2*Z*exp(-r^2/(l(x)*l(x) + l(x')*l(x')))
Z = (2*l(x)*l(x')/(l(x)*l(x) + l(x')*l(x')^{q/2} Z = (2*l(x)*l(x')/(l(x)*l(x) + l(x')*l(x')^{q/2}
@ -25,18 +25,18 @@ class Gibbs(Kernpart):
with input location. This leads to an additional term in front of with input location. This leads to an additional term in front of
the kernel. the kernel.
The parameters are :math:`\sigma^2`, the process variance, and The parameters are :math:`\\sigma^2`, the process variance, and
the parameters of l(x) which is a function that can be the parameters of l(x) which is a function that can be
specified by the user, by default an multi-layer peceptron is specified by the user, by default an multi-layer peceptron is
used. used.
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
:param variance: the variance :math:`\sigma^2` :param variance: the variance :math:`\\sigma^2`
:type variance: float :type variance: float
:param mapping: the mapping that gives the lengthscale across the input space (by default GPy.mappings.MLP is used with 20 hidden nodes). :param mapping: the mapping that gives the lengthscale across the input space (by default GPy.mappings.MLP is used with 20 hidden nodes).
:type mapping: GPy.core.Mapping :type mapping: GPy.core.Mapping
:param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one weight variance parameter \sigma^2_w), otherwise there is one weight variance parameter per dimension. :param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one weight variance parameter \\sigma^2_w), otherwise there is one weight variance parameter per dimension.
:type ARD: Boolean :type ARD: Boolean
:rtype: Kernpart object :rtype: Kernpart object

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@ -19,11 +19,11 @@ class Hetero(Kernpart):
.. math:: .. math::
k(x_i, x_j) = \delta_{i,j} \sigma^2(x_i) k(x_i, x_j) = \\delta_{i,j} \\sigma^2(x_i)
where :math:`\sigma^2(x)` is a function giving the variance as a function of input space and :math:`\delta_{i,j}` is the Kronecker delta function. where :math:`\\sigma^2(x)` is a function giving the variance as a function of input space and :math:`\\delta_{i,j}` is the Kronecker delta function.
The parameters are the parameters of \sigma^2(x) which is a The parameters are the parameters of \\sigma^2(x) which is a
function that can be specified by the user, by default an function that can be specified by the user, by default an
multi-layer peceptron is used. multi-layer peceptron is used.

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@ -11,28 +11,28 @@ class POLY(Kernpart):
Polynomial kernel parameter initialisation. Included for completeness, but generally not recommended, is the polynomial kernel: Polynomial kernel parameter initialisation. Included for completeness, but generally not recommended, is the polynomial kernel:
.. math:: .. math::
k(x, y) = \sigma^2\*(\sigma_w^2 x'y+\sigma_b^b)^d k(x, y) = \\sigma^2\\*(\\sigma_w^2 x'y+\\sigma_b^b)^d
The kernel parameters are :math:`\sigma^2` (variance), :math:`\sigma^2_w` The kernel parameters are :math:`\\sigma^2` (variance), :math:`\\sigma^2_w`
(weight_variance), :math:`\sigma^2_b` (bias_variance) and d (weight_variance), :math:`\\sigma^2_b` (bias_variance) and d
(degree). Only gradients of the first three are provided for (degree). Only gradients of the first three are provided for
kernel optimisation, it is assumed that polynomial degree would kernel optimisation, it is assumed that polynomial degree would
be set by hand. be set by hand.
The kernel is not recommended as it is badly behaved when the The kernel is not recommended as it is badly behaved when the
:math:`\sigma^2_w\*x'\*y + \sigma^2_b` has a magnitude greater than one. For completeness :math:`\\sigma^2_w\\*x'\\*y + \\sigma^2_b` has a magnitude greater than one. For completeness
there is an automatic relevance determination version of this there is an automatic relevance determination version of this
kernel provided (NOTE YET IMPLEMENTED!). kernel provided (NOTE YET IMPLEMENTED!).
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
:param variance: the variance :math:`\sigma^2` :param variance: the variance :math:`\\sigma^2`
:type variance: float :type variance: float
:param weight_variance: the vector of the variances of the prior over input weights in the neural network :math:`\sigma^2_w` :param weight_variance: the vector of the variances of the prior over input weights in the neural network :math:`\\sigma^2_w`
:type weight_variance: array or list of the appropriate size (or float if there is only one weight variance parameter) :type weight_variance: array or list of the appropriate size (or float if there is only one weight variance parameter)
:param bias_variance: the variance of the prior over bias parameters :math:`\sigma^2_b` :param bias_variance: the variance of the prior over bias parameters :math:`\\sigma^2_b`
:param degree: the degree of the polynomial. :param degree: the degree of the polynomial.
:type degree: int :type degree: int
:param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one weight variance parameter :math:`\sigma^2_w`), otherwise there is one weight variance parameter per dimension. :param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one weight variance parameter :math:`\\sigma^2_w`), otherwise there is one weight variance parameter per dimension.
:type ARD: Boolean :type ARD: Boolean
:rtype: Kernpart object :rtype: Kernpart object

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@ -15,9 +15,9 @@ class RBFInv(RBF):
.. math:: .. math::
k(r) = \sigma^2 \exp \\bigg(- \\frac{1}{2} r^2 \\bigg) \ \ \ \ \ \\text{ where } r^2 = \sum_{i=1}^d \\frac{ (x_i-x^\prime_i)^2}{\ell_i^2} k(r) = \\sigma^2 \\exp \\bigg(- \\frac{1}{2} r^2 \\bigg) \\ \\ \\ \\ \\ \\text{ where } r^2 = \\sum_{i=1}^d \\frac{ (x_i-x^\\prime_i)^2}{\\ell_i^2}
where \ell_i is the lengthscale, \sigma^2 the variance and d the dimensionality of the input. where \\ell_i is the lengthscale, \\sigma^2 the variance and d the dimensionality of the input.
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
@ -25,7 +25,7 @@ class RBFInv(RBF):
:type variance: float :type variance: float
:param lengthscale: the vector of lengthscale of the kernel :param lengthscale: the vector of lengthscale of the kernel
:type lengthscale: array or list of the appropriate size (or float if there is only one lengthscale parameter) :type lengthscale: array or list of the appropriate size (or float if there is only one lengthscale parameter)
:param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one single lengthscale parameter \ell), otherwise there is one lengthscale parameter per dimension. :param ARD: Auto Relevance Determination. If equal to "False", the kernel is isotropic (ie. one single lengthscale parameter \\ell), otherwise there is one lengthscale parameter per dimension.
:type ARD: Boolean :type ARD: Boolean
:rtype: kernel object :rtype: kernel object

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@ -14,15 +14,15 @@ class TruncLinear(Kern):
.. math:: .. math::
k(x,y) = \sum_{i=1}^input_dim \sigma^2_i \max(0, x_iy_i - \sigma_q) k(x,y) = \\sum_{i=1}^input_dim \\sigma^2_i \\max(0, x_iy_i - \\sigma_q)
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
:param variances: the vector of variances :math:`\sigma^2_i` :param variances: the vector of variances :math:`\\sigma^2_i`
:type variances: array or list of the appropriate size (or float if there :type variances: array or list of the appropriate size (or float if there
is only one variance parameter) is only one variance parameter)
:param ARD: Auto Relevance Determination. If False, the kernel has only one :param ARD: Auto Relevance Determination. If False, the kernel has only one
variance parameter \sigma^2, otherwise there is one variance variance parameter \\sigma^2, otherwise there is one variance
parameter per dimension. parameter per dimension.
:type ARD: Boolean :type ARD: Boolean
:rtype: kernel object :rtype: kernel object
@ -113,15 +113,15 @@ class TruncLinear_inf(Kern):
.. math:: .. math::
k(x,y) = \sum_{i=1}^input_dim \sigma^2_i \max(0, x_iy_i - \sigma_q) k(x,y) = \\sum_{i=1}^input_dim \\sigma^2_i \\max(0, x_iy_i - \\sigma_q)
:param input_dim: the number of input dimensions :param input_dim: the number of input dimensions
:type input_dim: int :type input_dim: int
:param variances: the vector of variances :math:`\sigma^2_i` :param variances: the vector of variances :math:`\\sigma^2_i`
:type variances: array or list of the appropriate size (or float if there :type variances: array or list of the appropriate size (or float if there
is only one variance parameter) is only one variance parameter)
:param ARD: Auto Relevance Determination. If False, the kernel has only one :param ARD: Auto Relevance Determination. If False, the kernel has only one
variance parameter \sigma^2, otherwise there is one variance variance parameter \\sigma^2, otherwise there is one variance
parameter per dimension. parameter per dimension.
:type ARD: Boolean :type ARD: Boolean
:rtype: kernel object :rtype: kernel object

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@ -243,7 +243,7 @@ class Bernoulli(Likelihood):
assert np.atleast_1d(inv_link_f).shape == np.atleast_1d(y).shape assert np.atleast_1d(inv_link_f).shape == np.atleast_1d(y).shape
#d3logpdf_dlink3 = 2*(y/(inv_link_f**3) - (1-y)/((1-inv_link_f)**3)) #d3logpdf_dlink3 = 2*(y/(inv_link_f**3) - (1-y)/((1-inv_link_f)**3))
state = np.seterr(divide='ignore') state = np.seterr(divide='ignore')
# TODO check y \in {0, 1} or {-1, 1} # TODO check y \\in {0, 1} or {-1, 1}
d3logpdf_dlink3 = np.where(y==1, 2./(inv_link_f**3), -2./((1.-inv_link_f)**3)) d3logpdf_dlink3 = np.where(y==1, 2./(inv_link_f**3), -2./((1.-inv_link_f)**3))
np.seterr(**state) np.seterr(**state)
return d3logpdf_dlink3 return d3logpdf_dlink3

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@ -14,7 +14,7 @@ class Exponential(Likelihood):
Y is expected to take values in {0,1,2,...} Y is expected to take values in {0,1,2,...}
----- -----
$$ $$
L(x) = \exp(\lambda) * \lambda**Y_i / Y_i! L(x) = \\exp(\\lambda) * \\lambda**Y_i / Y_i!
$$ $$
""" """
def __init__(self,gp_link=None): def __init__(self,gp_link=None):
@ -46,7 +46,7 @@ class Exponential(Likelihood):
Log Likelihood Function given link(f) Log Likelihood Function given link(f)
.. math:: .. math::
\\ln p(y_{i}|\lambda(f_{i})) = \\ln \\lambda(f_{i}) - y_{i}\\lambda(f_{i}) \\ln p(y_{i}|\\lambda(f_{i})) = \\ln \\lambda(f_{i}) - y_{i}\\lambda(f_{i})
:param link_f: latent variables (link(f)) :param link_f: latent variables (link(f))
:type link_f: Nx1 array :type link_f: Nx1 array
@ -65,7 +65,7 @@ class Exponential(Likelihood):
Gradient of the log likelihood function at y, given link(f) w.r.t link(f) Gradient of the log likelihood function at y, given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{d\\lambda(f)} = \\frac{1}{\\lambda(f)} - y_{i} \\frac{d \\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)} = \\frac{1}{\\lambda(f)} - y_{i}
:param link_f: latent variables (f) :param link_f: latent variables (f)
:type link_f: Nx1 array :type link_f: Nx1 array
@ -87,7 +87,7 @@ class Exponential(Likelihood):
The hessian will be 0 unless i == j The hessian will be 0 unless i == j
.. math:: .. math::
\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}\\lambda(f)} = -\\frac{1}{\\lambda(f_{i})^{2}} \\frac{d^{2} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{2}\\lambda(f)} = -\\frac{1}{\\lambda(f_{i})^{2}}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)
:type link_f: Nx1 array :type link_f: Nx1 array
@ -110,7 +110,7 @@ class Exponential(Likelihood):
Third order derivative log-likelihood function at y given link(f) w.r.t link(f) Third order derivative log-likelihood function at y given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d^{3} \\ln p(y_{i}|\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{2}{\\lambda(f_{i})^{3}} \\frac{d^{3} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{2}{\\lambda(f_{i})^{3}}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)
:type link_f: Nx1 array :type link_f: Nx1 array

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@ -54,7 +54,7 @@ class Gamma(Likelihood):
Log Likelihood Function given link(f) Log Likelihood Function given link(f)
.. math:: .. math::
\\ln p(y_{i}|\lambda(f_{i})) = \\alpha_{i}\\log \\beta - \\log \\Gamma(\\alpha_{i}) + (\\alpha_{i} - 1)\\log y_{i} - \\beta y_{i}\\\\ \\ln p(y_{i}|\\lambda(f_{i})) = \\alpha_{i}\\log \\beta - \\log \\Gamma(\\alpha_{i}) + (\\alpha_{i} - 1)\\log y_{i} - \\beta y_{i}\\\\
\\alpha_{i} = \\beta y_{i} \\alpha_{i} = \\beta y_{i}
:param link_f: latent variables (link(f)) :param link_f: latent variables (link(f))
@ -101,7 +101,7 @@ class Gamma(Likelihood):
The hessian will be 0 unless i == j The hessian will be 0 unless i == j
.. math:: .. math::
\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}\\lambda(f)} = -\\beta^{2}\\frac{d\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\ \\frac{d^{2} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{2}\\lambda(f)} = -\\beta^{2}\\frac{d\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\
\\alpha_{i} = \\beta y_{i} \\alpha_{i} = \\beta y_{i}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)
@ -126,7 +126,7 @@ class Gamma(Likelihood):
Third order derivative log-likelihood function at y given link(f) w.r.t link(f) Third order derivative log-likelihood function at y given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d^{3} \\ln p(y_{i}|\lambda(f_{i}))}{d^{3}\\lambda(f)} = -\\beta^{3}\\frac{d^{2}\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\ \\frac{d^{3} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{3}\\lambda(f)} = -\\beta^{3}\\frac{d^{2}\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\
\\alpha_{i} = \\beta y_{i} \\alpha_{i} = \\beta y_{i}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)

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@ -130,7 +130,7 @@ class Likelihood(Parameterized):
Calculation of the log predictive density Calculation of the log predictive density
.. math: .. math:
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*}) p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\\mu_{*}\\sigma^{2}_{*})
:param y_test: test observations (y_{*}) :param y_test: test observations (y_{*})
:type y_test: (Nx1) array :type y_test: (Nx1) array
@ -199,7 +199,7 @@ class Likelihood(Parameterized):
.. math: .. math:
log p(y_{*}|D) = log 1/num_samples prod^{S}_{s=1} p(y_{*}|f_{*s}) log p(y_{*}|D) = log 1/num_samples prod^{S}_{s=1} p(y_{*}|f_{*s})
f_{*s} ~ p(f_{*}|\mu_{*}\\sigma^{2}_{*}) f_{*s} ~ p(f_{*}|\\mu_{*}\\sigma^{2}_{*})
:param y_test: test observations (y_{*}) :param y_test: test observations (y_{*})
:type y_test: (Nx1) array :type y_test: (Nx1) array

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@ -180,7 +180,7 @@ class Cloglog(GPTransformation):
or or
f = \log (-\log(1-p)) f = \\log (-\\log(1-p))
""" """
def transf(self,f): def transf(self,f):

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@ -54,7 +54,7 @@ class Poisson(Likelihood):
Log Likelihood Function given link(f) Log Likelihood Function given link(f)
.. math:: .. math::
\\ln p(y_{i}|\lambda(f_{i})) = -\\lambda(f_{i}) + y_{i}\\log \\lambda(f_{i}) - \\log y_{i}! \\ln p(y_{i}|\\lambda(f_{i})) = -\\lambda(f_{i}) + y_{i}\\log \\lambda(f_{i}) - \\log y_{i}!
:param link_f: latent variables (link(f)) :param link_f: latent variables (link(f))
:type link_f: Nx1 array :type link_f: Nx1 array
@ -72,7 +72,7 @@ class Poisson(Likelihood):
Gradient of the log likelihood function at y, given link(f) w.r.t link(f) Gradient of the log likelihood function at y, given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{d\\lambda(f)} = \\frac{y_{i}}{\\lambda(f_{i})} - 1 \\frac{d \\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)} = \\frac{y_{i}}{\\lambda(f_{i})} - 1
:param link_f: latent variables (f) :param link_f: latent variables (f)
:type link_f: Nx1 array :type link_f: Nx1 array
@ -92,7 +92,7 @@ class Poisson(Likelihood):
The hessian will be 0 unless i == j The hessian will be 0 unless i == j
.. math:: .. math::
\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}\\lambda(f)} = \\frac{-y_{i}}{\\lambda(f_{i})^{2}} \\frac{d^{2} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{2}\\lambda(f)} = \\frac{-y_{i}}{\\lambda(f_{i})^{2}}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)
:type link_f: Nx1 array :type link_f: Nx1 array
@ -113,7 +113,7 @@ class Poisson(Likelihood):
Third order derivative log-likelihood function at y given link(f) w.r.t link(f) Third order derivative log-likelihood function at y given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d^{3} \\ln p(y_{i}|\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{2y_{i}}{\\lambda(f_{i})^{3}} \\frac{d^{3} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{2y_{i}}{\\lambda(f_{i})^{3}}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)
:type link_f: Nx1 array :type link_f: Nx1 array

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@ -78,7 +78,7 @@ class StudentT(Likelihood):
Log Likelihood Function given link(f) Log Likelihood Function given link(f)
.. math:: .. math::
\\ln p(y_{i}|\lambda(f_{i})) = \\ln \\Gamma\\left(\\frac{v+1}{2}\\right) - \\ln \\Gamma\\left(\\frac{v}{2}\\right) - \\ln \\sqrt{v \\pi\\sigma^{2}} - \\frac{v+1}{2}\\ln \\left(1 + \\frac{1}{v}\\left(\\frac{(y_{i} - \lambda(f_{i}))^{2}}{\\sigma^{2}}\\right)\\right) \\ln p(y_{i}|\\lambda(f_{i})) = \\ln \\Gamma\\left(\\frac{v+1}{2}\\right) - \\ln \\Gamma\\left(\\frac{v}{2}\\right) - \\ln \\sqrt{v \\pi\\sigma^{2}} - \\frac{v+1}{2}\\ln \\left(1 + \\frac{1}{v}\\left(\\frac{(y_{i} - \\lambda(f_{i}))^{2}}{\\sigma^{2}}\\right)\\right)
:param inv_link_f: latent variables (link(f)) :param inv_link_f: latent variables (link(f))
:type inv_link_f: Nx1 array :type inv_link_f: Nx1 array
@ -107,7 +107,7 @@ class StudentT(Likelihood):
Gradient of the log likelihood function at y, given link(f) w.r.t link(f) Gradient of the log likelihood function at y, given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{d\\lambda(f)} = \\frac{(v+1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^{2} + \\sigma^{2}v} \\frac{d \\ln p(y_{i}|\\lambda(f_{i}))}{d\\lambda(f)} = \\frac{(v+1)(y_{i}-\\lambda(f_{i}))}{(y_{i}-\\lambda(f_{i}))^{2} + \\sigma^{2}v}
:param inv_link_f: latent variables (f) :param inv_link_f: latent variables (f)
:type inv_link_f: Nx1 array :type inv_link_f: Nx1 array
@ -129,7 +129,7 @@ class StudentT(Likelihood):
The hessian will be 0 unless i == j The hessian will be 0 unless i == j
.. math:: .. math::
\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}\\lambda(f)} = \\frac{(v+1)((y_{i}-\lambda(f_{i}))^{2} - \\sigma^{2}v)}{((y_{i}-\lambda(f_{i}))^{2} + \\sigma^{2}v)^{2}} \\frac{d^{2} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{2}\\lambda(f)} = \\frac{(v+1)((y_{i}-\\lambda(f_{i}))^{2} - \\sigma^{2}v)}{((y_{i}-\\lambda(f_{i}))^{2} + \\sigma^{2}v)^{2}}
:param inv_link_f: latent variables inv_link(f) :param inv_link_f: latent variables inv_link(f)
:type inv_link_f: Nx1 array :type inv_link_f: Nx1 array
@ -154,7 +154,7 @@ class StudentT(Likelihood):
Third order derivative log-likelihood function at y given link(f) w.r.t link(f) Third order derivative log-likelihood function at y given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d^{3} \\ln p(y_{i}|\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{-2(v+1)((y_{i} - \lambda(f_{i}))^3 - 3(y_{i} - \lambda(f_{i})) \\sigma^{2} v))}{((y_{i} - \lambda(f_{i})) + \\sigma^{2} v)^3} \\frac{d^{3} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{3}\\lambda(f)} = \\frac{-2(v+1)((y_{i} - \\lambda(f_{i}))^3 - 3(y_{i} - \\lambda(f_{i})) \\sigma^{2} v))}{((y_{i} - \\lambda(f_{i})) + \\sigma^{2} v)^3}
:param inv_link_f: latent variables link(f) :param inv_link_f: latent variables link(f)
:type inv_link_f: Nx1 array :type inv_link_f: Nx1 array
@ -175,7 +175,7 @@ class StudentT(Likelihood):
Gradient of the log-likelihood function at y given f, w.r.t variance parameter (t_noise) Gradient of the log-likelihood function at y given f, w.r.t variance parameter (t_noise)
.. math:: .. math::
\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{d\\sigma^{2}} = \\frac{v((y_{i} - \lambda(f_{i}))^{2} - \\sigma^{2})}{2\\sigma^{2}(\\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})} \\frac{d \\ln p(y_{i}|\\lambda(f_{i}))}{d\\sigma^{2}} = \\frac{v((y_{i} - \\lambda(f_{i}))^{2} - \\sigma^{2})}{2\\sigma^{2}(\\sigma^{2}v + (y_{i} - \\lambda(f_{i}))^{2})}
:param inv_link_f: latent variables link(f) :param inv_link_f: latent variables link(f)
:type inv_link_f: Nx1 array :type inv_link_f: Nx1 array
@ -199,7 +199,7 @@ class StudentT(Likelihood):
Derivative of the dlogpdf_dlink w.r.t variance parameter (t_noise) Derivative of the dlogpdf_dlink w.r.t variance parameter (t_noise)
.. math:: .. math::
\\frac{d}{d\\sigma^{2}}(\\frac{d \\ln p(y_{i}|\lambda(f_{i}))}{df}) = \\frac{-2\\sigma v(v + 1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^2 + \\sigma^2 v)^2} \\frac{d}{d\\sigma^{2}}(\\frac{d \\ln p(y_{i}|\\lambda(f_{i}))}{df}) = \\frac{-2\\sigma v(v + 1)(y_{i}-\\lambda(f_{i}))}{(y_{i}-\\lambda(f_{i}))^2 + \\sigma^2 v)^2}
:param inv_link_f: latent variables inv_link_f :param inv_link_f: latent variables inv_link_f
:type inv_link_f: Nx1 array :type inv_link_f: Nx1 array
@ -220,7 +220,7 @@ class StudentT(Likelihood):
Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (t_noise) Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (t_noise)
.. math:: .. math::
\\frac{d}{d\\sigma^{2}}(\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}f}) = \\frac{v(v+1)(\\sigma^{2}v - 3(y_{i} - \lambda(f_{i}))^{2})}{(\\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})^{3}} \\frac{d}{d\\sigma^{2}}(\\frac{d^{2} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{2}f}) = \\frac{v(v+1)(\\sigma^{2}v - 3(y_{i} - \\lambda(f_{i}))^{2})}{(\\sigma^{2}v + (y_{i} - \\lambda(f_{i}))^{2})^{3}}
:param inv_link_f: latent variables link(f) :param inv_link_f: latent variables link(f)
:type inv_link_f: Nx1 array :type inv_link_f: Nx1 array

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@ -54,7 +54,7 @@ class Weibull(Likelihood):
Log Likelihood Function given link(f) Log Likelihood Function given link(f)
.. math:: .. math::
\\ln p(y_{i}|\lambda(f_{i})) = \\alpha_{i}\\log \\beta - \\log \\Gamma(\\alpha_{i}) + (\\alpha_{i} - 1)\\log y_{i} - \\beta y_{i}\\\\ \\ln p(y_{i}|\\lambda(f_{i})) = \\alpha_{i}\\log \\beta - \\log \\Gamma(\\alpha_{i}) + (\\alpha_{i} - 1)\\log y_{i} - \\beta y_{i}\\\\
\\alpha_{i} = \\beta y_{i} \\alpha_{i} = \\beta y_{i}
:param link_f: latent variables (link(f)) :param link_f: latent variables (link(f))
@ -117,7 +117,7 @@ class Weibull(Likelihood):
The hessian will be 0 unless i == j The hessian will be 0 unless i == j
.. math:: .. math::
\\frac{d^{2} \\ln p(y_{i}|\lambda(f_{i}))}{d^{2}\\lambda(f)} = -\\beta^{2}\\frac{d\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\ \\frac{d^{2} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{2}\\lambda(f)} = -\\beta^{2}\\frac{d\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\
\\alpha_{i} = \\beta y_{i} \\alpha_{i} = \\beta y_{i}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)
@ -150,7 +150,7 @@ class Weibull(Likelihood):
Third order derivative log-likelihood function at y given link(f) w.r.t link(f) Third order derivative log-likelihood function at y given link(f) w.r.t link(f)
.. math:: .. math::
\\frac{d^{3} \\ln p(y_{i}|\lambda(f_{i}))}{d^{3}\\lambda(f)} = -\\beta^{3}\\frac{d^{2}\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\ \\frac{d^{3} \\ln p(y_{i}|\\lambda(f_{i}))}{d^{3}\\lambda(f)} = -\\beta^{3}\\frac{d^{2}\\Psi(\\alpha_{i})}{d\\alpha_{i}}\\\\
\\alpha_{i} = \\beta y_{i} \\alpha_{i} = \\beta y_{i}
:param link_f: latent variables link(f) :param link_f: latent variables link(f)

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@ -10,7 +10,7 @@ class Additive(Mapping):
.. math:: .. math::
f(\mathbf{x}*) = f_1(\mathbf{x}*) + f_2(\mathbf(x)*) f(\\mathbf{x}*) = f_1(\\mathbf{x}*) + f_2(\\mathbf(x)*)
:param mapping1: first mapping to add together. :param mapping1: first mapping to add together.
:type mapping1: GPy.mappings.Mapping :type mapping1: GPy.mappings.Mapping

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@ -9,7 +9,7 @@ class Compound(Mapping):
.. math:: .. math::
f(\mathbf{x}) = f_2(f_1(\mathbf{x})) f(\\mathbf{x}) = f_2(f_1(\\mathbf{x}))
:param mapping1: first mapping :param mapping1: first mapping
:type mapping1: GPy.mappings.Mapping :type mapping1: GPy.mappings.Mapping

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@ -9,7 +9,7 @@ class Constant(Mapping):
.. math:: .. math::
F(\mathbf{x}) = c F(\\mathbf{x}) = c
:param input_dim: dimension of input. :param input_dim: dimension of input.

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@ -12,20 +12,20 @@ class Kernel(Mapping):
.. math:: .. math::
f(\mathbf{x}) = \sum_i \alpha_i k(\mathbf{z}_i, \mathbf{x}) f(\\mathbf{x}) = \\sum_i \\alpha_i k(\\mathbf{z}_i, \\mathbf{x})
or for multple outputs or for multple outputs
.. math:: .. math::
f_i(\mathbf{x}) = \sum_j \alpha_{i,j} k(\mathbf{z}_i, \mathbf{x}) f_i(\\mathbf{x}) = \\sum_j \\alpha_{i,j} k(\\mathbf{z}_i, \\mathbf{x})
:param input_dim: dimension of input. :param input_dim: dimension of input.
:type input_dim: int :type input_dim: int
:param output_dim: dimension of output. :param output_dim: dimension of output.
:type output_dim: int :type output_dim: int
:param Z: input observations containing :math:`\mathbf{Z}` :param Z: input observations containing :math:`\\mathbf{Z}`
:type Z: ndarray :type Z: ndarray
:param kernel: a GPy kernel, defaults to GPy.kern.RBF :param kernel: a GPy kernel, defaults to GPy.kern.RBF
:type kernel: GPy.kern.kern :type kernel: GPy.kern.kern

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@ -12,7 +12,7 @@ class Linear(Mapping):
.. math:: .. math::
F(\mathbf{x}) = \mathbf{A} \mathbf{x}) F(\\mathbf{x}) = \\mathbf{A} \\mathbf{x})
:param input_dim: dimension of input. :param input_dim: dimension of input.

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@ -22,7 +22,7 @@ class GPKroneckerGaussianRegression(Model):
.. rubric:: References .. rubric:: References
.. [stegle_et_al_2011] Stegle, O.; Lippert, C.; Mooij, J.M.; Lawrence, N.D.; Borgwardt, K.:Efficient inference in matrix-variate Gaussian models with \iid observation noise. In: Advances in Neural Information Processing Systems, 2011, Pages 630-638 .. [stegle_et_al_2011] Stegle, O.; Lippert, C.; Mooij, J.M.; Lawrence, N.D.; Borgwardt, K.:Efficient inference in matrix-variate Gaussian models with \\iid observation noise. In: Advances in Neural Information Processing Systems, 2011, Pages 630-638
""" """
def __init__(self, X1, X2, Y, kern1, kern2, noise_var=1., name='KGPR'): def __init__(self, X1, X2, Y, kern1, kern2, noise_var=1., name='KGPR'):

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@ -4002,10 +4002,10 @@ class ContDescrStateSpace(DescreteStateSpace):
""" """
Linear Time-Invariant Stochastic Differential Equation (LTI SDE): Linear Time-Invariant Stochastic Differential Equation (LTI SDE):
dx(t) = F x(t) dt + L d \beta ,where dx(t) = F x(t) dt + L d \\beta ,where
x(t): (vector) stochastic process x(t): (vector) stochastic process
\beta: (vector) Brownian motion process \\beta: (vector) Brownian motion process
F, L: (time invariant) matrices of corresponding dimensions F, L: (time invariant) matrices of corresponding dimensions
Qc: covariance of noise. Qc: covariance of noise.
@ -4022,7 +4022,7 @@ class ContDescrStateSpace(DescreteStateSpace):
F,L: LTI SDE matrices of corresponding dimensions F,L: LTI SDE matrices of corresponding dimensions
Qc: matrix (n,n) Qc: matrix (n,n)
Covarince between different dimensions of noise \beta. Covarince between different dimensions of noise \\beta.
n is the dimensionality of the noise. n is the dimensionality of the noise.
dt: double or iterable dt: double or iterable

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@ -171,7 +171,7 @@ class TPRegression(Model):
def log_likelihood(self): def log_likelihood(self):
""" """
The log marginal likelihood of the model, :math:`p(\mathbf{y})`, this is the objective function of the model being optimised The log marginal likelihood of the model, :math:`p(\\mathbf{y})`, this is the objective function of the model being optimised
""" """
return self._log_marginal_likelihood or self.inference()[1] return self._log_marginal_likelihood or self.inference()[1]
@ -184,10 +184,10 @@ class TPRegression(Model):
diagonal of the covariance is returned. diagonal of the covariance is returned.
.. math:: .. math::
p(f*|X*, X, Y) = \int^{\inf}_{\inf} p(f*|f,X*)p(f|X,Y) df p(f*|X*, X, Y) = \\int^{\\inf}_{\\inf} p(f*|f,X*)p(f|X,Y) df
= MVN\left(\nu + N,f*| K_{x*x}(K_{xx})^{-1}Y, = MVN\\left(\\nu + N,f*| K_{x*x}(K_{xx})^{-1}Y,
\frac{\nu + \beta - 2}{\nu + N - 2}K_{x*x*} - K_{xx*}(K_{xx})^{-1}K_{xx*}\right) \\frac{\\nu + \\beta - 2}{\\nu + N - 2}K_{x*x*} - K_{xx*}(K_{xx})^{-1}K_{xx*}\\right)
\nu := \texttt{Degrees of freedom} \\nu := \\texttt{Degrees of freedom}
""" """
mu, var = self.posterior._raw_predict(kern=self.kern if kern is None else kern, Xnew=Xnew, mu, var = self.posterior._raw_predict(kern=self.kern if kern is None else kern, Xnew=Xnew,
pred_var=self._predictive_variable, full_cov=full_cov) pred_var=self._predictive_variable, full_cov=full_cov)

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@ -146,7 +146,7 @@ class WarpedGP(GP):
the jacobian of the warping function here. the jacobian of the warping function here.
.. math: .. math:
p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\mu_{*}\\sigma^{2}_{*}) p(y_{*}|D) = p(y_{*}|f_{*})p(f_{*}|\\mu_{*}\\sigma^{2}_{*})
:param x_test: test locations (x_{*}) :param x_test: test locations (x_{*})
:type x_test: (Nx1) array :type x_test: (Nx1) array

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@ -159,7 +159,7 @@ def generate_brownian_data(
): ):
""" """
Generate brownian data - data from Brownian motion. Generate brownian data - data from Brownian motion.
First point is always 0, and \Beta(0) = 0 - standard conditions for Brownian motion. First point is always 0, and \\Beta(0) = 0 - standard conditions for Brownian motion.
Input: Input:
-------------------------------- --------------------------------

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@ -269,8 +269,8 @@ class TestMisc:
from GPy.core.parameterization.variational import NormalPosterior from GPy.core.parameterization.variational import NormalPosterior
X_pred = NormalPosterior(X_pred_mu, X_pred_var) X_pred = NormalPosterior(X_pred_mu, X_pred_var)
# mu = \int f(x)q(x|mu,S) dx = \int 2x.q(x|mu,S) dx = 2.mu # mu = \\int f(x)q(x|mu,S) dx = \\int 2x.q(x|mu,S) dx = 2.mu
# S = \int (f(x) - m)^2q(x|mu,S) dx = \int f(x)^2 q(x) dx - mu**2 = 4(mu^2 + S) - (2.mu)^2 = 4S # S = \\int (f(x) - m)^2q(x|mu,S) dx = \\int f(x)^2 q(x) dx - mu**2 = 4(mu^2 + S) - (2.mu)^2 = 4S
Y_mu_true = 2 * X_pred_mu Y_mu_true = 2 * X_pred_mu
Y_var_true = 4 * X_pred_var Y_var_true = 4 * X_pred_var
Y_mu_pred, Y_var_pred = m.predict_noiseless(X_pred) Y_mu_pred, Y_var_pred = m.predict_noiseless(X_pred)
@ -684,7 +684,7 @@ class TestMisc:
warp_m = GPy.models.WarpedGP( warp_m = GPy.models.WarpedGP(
X, Y X, Y
) # , kernel=warp_k)#, warping_function=warp_f) ) # , kernel=warp_k)#, warping_function=warp_f)
warp_m[".*\.d"].constrain_fixed(1.0) warp_m[r".*\.d"].constrain_fixed(1.0)
warp_m.optimize_restarts( warp_m.optimize_restarts(
parallel=False, robust=False, num_restarts=5, max_iters=max_iters parallel=False, robust=False, num_restarts=5, max_iters=max_iters
) )

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@ -47,7 +47,7 @@ class TestRVTransformation:
# ax.hist(phi_s, normed=True, bins=100, alpha=0.25, label='Histogram') # ax.hist(phi_s, normed=True, bins=100, alpha=0.25, label='Histogram')
# ax.plot(phi, kde(phi), '--', linewidth=2, label='Kernel Density Estimation') # ax.plot(phi, kde(phi), '--', linewidth=2, label='Kernel Density Estimation')
# ax.plot(phi, pdf_phi, ':', linewidth=2, label='Transformed PDF') # ax.plot(phi, pdf_phi, ':', linewidth=2, label='Transformed PDF')
# ax.set_xlabel(r'transformed $\theta$', fontsize=16) # ax.set_xlabel(r'transformed $\\theta$', fontsize=16)
# ax.set_ylabel('PDF', fontsize=16) # ax.set_ylabel('PDF', fontsize=16)
# plt.legend(loc='best') # plt.legend(loc='best')
# plt.show(block=True) # plt.show(block=True)

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@ -537,7 +537,7 @@ http://nbviewer.ipython.org/github/sahuguet/notebooks/blob/master/GoogleTrends%2
# In the notebook they did some data cleaning: remove Javascript header+footer, and translate new Date(....,..,..) into YYYY-MM-DD. # In the notebook they did some data cleaning: remove Javascript header+footer, and translate new Date(....,..,..) into YYYY-MM-DD.
header = """// Data table response\ngoogle.visualization.Query.setResponse(""" header = """// Data table response\ngoogle.visualization.Query.setResponse("""
data = data[len(header):-2] data = data[len(header):-2]
data = re.sub('new Date\((\d+),(\d+),(\d+)\)', (lambda m: '"%s-%02d-%02d"' % (m.group(1).strip(), 1+int(m.group(2)), int(m.group(3)))), data) data = re.sub(r'new Date\((\d+),(\d+),(\d+)\)', (lambda m: '"%s-%02d-%02d"' % (m.group(1).strip(), 1+int(m.group(2)), int(m.group(3)))), data)
timeseries = json.loads(data) timeseries = json.loads(data)
columns = [k['label'] for k in timeseries['table']['cols']] columns = [k['label'] for k in timeseries['table']['cols']]
rows = map(lambda x: [k['v'] for k in x['c']], timeseries['table']['rows']) rows = map(lambda x: [k['v'] for k in x['c']], timeseries['table']['rows'])
@ -782,7 +782,7 @@ def hapmap3(data_set='hapmap3'):
/ 1, iff SNPij==(B1,B1) / 1, iff SNPij==(B1,B1)
Aij = | 0, iff SNPij==(B1,B2) Aij = | 0, iff SNPij==(B1,B2)
\ -1, iff SNPij==(B2,B2) \\ -1, iff SNPij==(B2,B2)
The SNP data and the meta information (such as iid, sex and phenotype) are The SNP data and the meta information (such as iid, sex and phenotype) are
stored in the dataframe datadf, index is the Individual ID, stored in the dataframe datadf, index is the Individual ID,
@ -1011,7 +1011,7 @@ def singlecell_rna_seq_deng(dataset='singlecell_deng'):
sample_info.columns = c sample_info.columns = c
# get the labels right: # get the labels right:
rep = re.compile('\(.*\)') rep = re.compile(r'\(.*\)')
def filter_dev_stage(row): def filter_dev_stage(row):
if isnull(row): if isnull(row):
row = "2-cell stage embryo" row = "2-cell stage embryo"
@ -1050,7 +1050,7 @@ def singlecell_rna_seq_deng(dataset='singlecell_deng'):
#gene_info[file_info.name[:-18]] = inner.Refseq_IDs #gene_info[file_info.name[:-18]] = inner.Refseq_IDs
# Strip GSM number off data index # Strip GSM number off data index
rep = re.compile('GSM\d+_') rep = re.compile(r'GSM\d+_')
from pandas import MultiIndex from pandas import MultiIndex
columns = MultiIndex.from_tuples([row.split('_', 1) for row in data.columns]) columns = MultiIndex.from_tuples([row.split('_', 1) for row in data.columns])

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@ -180,24 +180,24 @@ class NetpbmFile(object):
"""Read PAM header and initialize instance.""" """Read PAM header and initialize instance."""
regroups = re.search( regroups = re.search(
b"(^P7[\n\r]+(?:(?:[\n\r]+)|(?:#.*)|" b"(^P7[\n\r]+(?:(?:[\n\r]+)|(?:#.*)|"
b"(HEIGHT\s+\d+)|(WIDTH\s+\d+)|(DEPTH\s+\d+)|(MAXVAL\s+\d+)|" rb"(HEIGHT\s+\d+)|(WIDTH\s+\d+)|(DEPTH\s+\d+)|(MAXVAL\s+\d+)|"
b"(?:TUPLTYPE\s+\w+))*ENDHDR\n)", data).groups() rb"(?:TUPLTYPE\s+\w+))*ENDHDR\n)", data).groups()
self.header = regroups[0] self.header = regroups[0]
self.magicnum = b'P7' self.magicnum = b'P7'
for group in regroups[1:]: for group in regroups[1:]:
key, value = group.split() key, value = group.split()
setattr(self, unicode(key).lower(), int(value)) setattr(self, unicode(key).lower(), int(value))
matches = re.findall(b"(TUPLTYPE\s+\w+)", self.header) matches = re.findall(rb"(TUPLTYPE\s+\w+)", self.header)
self.tupltypes = [s.split(None, 1)[1] for s in matches] self.tupltypes = [s.split(None, 1)[1] for s in matches]
def _read_pnm_header(self, data): def _read_pnm_header(self, data):
"""Read PNM header and initialize instance.""" """Read PNM header and initialize instance."""
bpm = data[1:2] in b"14" bpm = data[1:2] in b"14"
regroups = re.search(b"".join(( regroups = re.search(b"".join((
b"(^(P[123456]|P7 332)\s+(?:#.*[\r\n])*", rb"(^(P[123456]|P7 332)\s+(?:#.*[\r\n])*",
b"\s*(\d+)\s+(?:#.*[\r\n])*", rb"\s*(\d+)\s+(?:#.*[\r\n])*",
b"\s*(\d+)\s+(?:#.*[\r\n])*" * (not bpm), rb"\s*(\d+)\s+(?:#.*[\r\n])*" * (not bpm),
b"\s*(\d+)\s(?:\s*#.*[\r\n]\s)*)")), data).groups() + (1, ) * bpm rb"\s*(\d+)\s(?:\s*#.*[\r\n]\s)*)")), data).groups() + (1, ) * bpm
self.header = regroups[0] self.header = regroups[0]
self.magicnum = regroups[1] self.magicnum = regroups[1]
self.width = int(regroups[2]) self.width = int(regroups[2])

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@ -150,7 +150,7 @@ with open('../../GPy/__version__.py', 'r') as f:
version = f.read() version = f.read()
release = version release = version
print version print(version)
# version = '0.8.8' # version = '0.8.8'
# The full version, including alpha/beta/rc tags. # The full version, including alpha/beta/rc tags.