From 22d30d9d39c70f806fe5bcb815cce9c8eb0f8dca Mon Sep 17 00:00:00 2001 From: Zhenwen Dai Date: Mon, 10 Nov 2014 16:24:24 +0000 Subject: [PATCH] new ssrbf implementation --- GPy/kern/_src/psi_comp/ssrbf_psi_comp.py | 561 +++++++++++++++-------- 1 file changed, 380 insertions(+), 181 deletions(-) diff --git a/GPy/kern/_src/psi_comp/ssrbf_psi_comp.py b/GPy/kern/_src/psi_comp/ssrbf_psi_comp.py index 6302a590..f6a24c86 100644 --- a/GPy/kern/_src/psi_comp/ssrbf_psi_comp.py +++ b/GPy/kern/_src/psi_comp/ssrbf_psi_comp.py @@ -7,193 +7,392 @@ The package for the psi statistics computation import numpy as np -def psicomputations(variance, lengthscale, Z, variational_posterior): - """ - Z - MxQ - mu - NxQ - S - NxQ - gamma - NxQ - """ - # here are the "statistics" for psi0, psi1 and psi2 - # Produced intermediate results: - # _psi1 NxM - mu = variational_posterior.mean - S = variational_posterior.variance - gamma = variational_posterior.binary_prob +try: + from scipy import weave + + def _psicomputations(variance, lengthscale, Z, variational_posterior): + """ + Z - MxQ + mu - NxQ + S - NxQ + gamma - NxQ + """ + # here are the "statistics" for psi0, psi1 and psi2 + # Produced intermediate results: + # _psi1 NxM + mu = variational_posterior.mean + S = variational_posterior.variance + gamma = variational_posterior.binary_prob + + N,M,Q = mu.shape[0],Z.shape[0],mu.shape[1] + l2 = np.square(lengthscale) + log_denom1 = np.log(S/l2+1) + log_denom2 = np.log(2*S/l2+1) + log_gamma = np.log(gamma) + log_gamma1 = np.log(1.-gamma) + variance = float(variance) + psi0 = np.empty(N) + psi0[:] = variance + psi1 = np.empty((N,M)) + psi2n = np.empty((N,M,M)) + + from ....util.misc import param_to_array + S = param_to_array(S) + mu = param_to_array(mu) + gamma = param_to_array(gamma) + Z = param_to_array(Z) + + support_code = """ + #include + """ + code = """ + for(int n=0; npsi1_exp2)?psi1_exp1+log1p(exp(psi1_exp2-psi1_exp1)):psi1_exp2+log1p(exp(psi1_exp1-psi1_exp2)); + } + // Compute Psi_2 + double muZhat = mu(n,q) - (Zm1q+Zm2q)/2.; + double Z2 = Zm1q*Zm1q+ Zm2q*Zm2q; + double dZ = Zm1q - Zm2q; + + double psi2_exp1 = dZ*dZ/(-4.*lq)-muZhat*muZhat/(2.*Snq+lq) - log_denom2(n,q)/2. + log_gamma(n,q); + double psi2_exp2 = log_gamma1(n,q) - Z2/(2.*lq); + log_psi2_n += (psi2_exp1>psi2_exp2)?psi2_exp1+log1p(exp(psi2_exp2-psi2_exp1)):psi2_exp2+log1p(exp(psi2_exp1-psi2_exp2)); + } + double exp_psi2_n = exp(log_psi2_n); + psi2n(n,m1,m2) = variance*variance*exp_psi2_n; + if(m1!=m2) { psi2n(n,m2,m1) = variance*variance*exp_psi2_n;} + } + psi1(n,m1) = variance*exp(log_psi1); + } + } + """ + weave.inline(code, support_code=support_code, arg_names=['psi1','psi2n','N','M','Q','variance','l2','Z','mu','S','gamma','log_denom1','log_denom2','log_gamma','log_gamma1'], type_converters=weave.converters.blitz) + + psi2 = psi2n.sum(axis=0) + return psi0,psi1,psi2,psi2n + + from GPy.util.caching import Cacher + psicomputations = Cacher(_psicomputations, limit=1) + + def psiDerivativecomputations(dL_dpsi0, dL_dpsi1, dL_dpsi2, variance, lengthscale, Z, variational_posterior): + ARD = (len(lengthscale)!=1) + + _,psi1,_,psi2n = psicomputations(variance, lengthscale, Z, variational_posterior) + + mu = variational_posterior.mean + S = variational_posterior.variance + gamma = variational_posterior.binary_prob + N,M,Q = mu.shape[0],Z.shape[0],mu.shape[1] + l2 = np.square(lengthscale) + log_denom1 = np.log(S/l2+1) + log_denom2 = np.log(2*S/l2+1) + log_gamma = np.log(gamma) + log_gamma1 = np.log(1.-gamma) + variance = float(variance) + + dvar = np.zeros(1) + dmu = np.zeros((N,Q)) + dS = np.zeros((N,Q)) + dgamma = np.zeros((N,Q)) + dl = np.zeros(Q) + dZ = np.zeros((M,Q)) + dvar += np.sum(dL_dpsi0) + + from ....util.misc import param_to_array + S = param_to_array(S) + mu = param_to_array(mu) + gamma = param_to_array(gamma) + Z = param_to_array(Z) + + support_code = """ + #include + """ + code = """ + for(int n=0; nexp2) { + d_exp1 = 1.; + d_exp2 = exp(exp2-exp1); + } else { + d_exp1 = exp(exp1-exp2); + d_exp2 = 1.; + } + double exp_sum = d_exp1+d_exp2; + + dmu(n,q) += lpsi1*Zmu*d_exp1/(denom*exp_sum); + dS(n,q) += lpsi1*(Zmu2_denom-1.)*d_exp1/(denom*exp_sum)/2.; + dgamma(n,q) += lpsi1*(d_exp1/gnq-d_exp2/(1.-gnq))/exp_sum; + dl(q) += lpsi1*((Zmu2_denom+Snq/lq)/denom*d_exp1+Zm1q*Zm1q/(lq*lq)*d_exp2)/(2.*exp_sum); + dZ(m1,q) += lpsi1*(-Zmu/denom*d_exp1-Zm1q/lq*d_exp2)/exp_sum; + } + // Compute Psi_2 + double lpsi2 = psi2n(n,m1,m2)*dL_dpsi2(m1,m2); + if(q==0) {dvar(0) += lpsi2*2/variance;} + + double dZm1m2 = Zm1q - Zm2q; + double Z2 = Zm1q*Zm1q+Zm2q*Zm2q; + double muZhat = mu_nq - (Zm1q + Zm2q)/2.; + double denom = 2.*Snq+lq; + double muZhat2_denom = muZhat*muZhat/denom; + + double exp1 = dZm1m2*dZm1m2/(-4.*lq)-muZhat*muZhat/(2.*Snq+lq) - log_denom2(n,q)/2. + log_gamma(n,q); + double exp2 = log_gamma1(n,q) - Z2/(2.*lq); + double d_exp1,d_exp2; + if(exp1>exp2) { + d_exp1 = 1.; + d_exp2 = exp(exp2-exp1); + } else { + d_exp1 = exp(exp1-exp2); + d_exp2 = 1.; + } + double exp_sum = d_exp1+d_exp2; + + dmu(n,q) += -2.*lpsi2*muZhat/denom*d_exp1/exp_sum; + dS(n,q) += lpsi2*(2.*muZhat2_denom-1.)/denom*d_exp1/exp_sum; + dgamma(n,q) += lpsi2*(d_exp1/gnq-d_exp2/(1.-gnq))/exp_sum; + dl(q) += lpsi2*(((Snq/lq+muZhat2_denom)/denom+dZm1m2*dZm1m2/(4.*lq*lq))*d_exp1+Z2/(2.*lq*lq)*d_exp2)/exp_sum; + dZ(m1,q) += 2.*lpsi2*((muZhat/denom-dZm1m2/(2*lq))*d_exp1-Zm1q/lq*d_exp2)/exp_sum; + } + } + } + } + """ + weave.inline(code, support_code=support_code, arg_names=['dL_dpsi1','dL_dpsi2','psi1','psi2n','N','M','Q','variance','l2','Z','mu','S','gamma','log_denom1','log_denom2','log_gamma','log_gamma1','dvar','dl','dmu','dS','dgamma','dZ'], type_converters=weave.converters.blitz) + + dl *= 2.*lengthscale + if not ARD: + dl = dl.sum() + + return dvar, dl, dZ, dmu, dS, dgamma + +except: + + def psicomputations(variance, lengthscale, Z, variational_posterior): + """ + Z - MxQ + mu - NxQ + S - NxQ + gamma - NxQ + """ + # here are the "statistics" for psi0, psi1 and psi2 + # Produced intermediate results: + # _psi1 NxM + mu = variational_posterior.mean + S = variational_posterior.variance + gamma = variational_posterior.binary_prob + + psi0 = np.empty(mu.shape[0]) + psi0[:] = variance + psi1 = _psi1computations(variance, lengthscale, Z, mu, S, gamma) + psi2 = _psi2computations(variance, lengthscale, Z, mu, S, gamma) + return psi0, psi1, psi2 - psi0 = np.empty(mu.shape[0]) - psi0[:] = variance - psi1 = _psi1computations(variance, lengthscale, Z, mu, S, gamma) - psi2 = _psi2computations(variance, lengthscale, Z, mu, S, gamma) - return psi0, psi1, psi2 - -def _psi1computations(variance, lengthscale, Z, mu, S, gamma): - """ - Z - MxQ - mu - NxQ - S - NxQ - gamma - NxQ - """ - # here are the "statistics" for psi1 - # Produced intermediate results: - # _psi1 NxM - - lengthscale2 = np.square(lengthscale) - - # psi1 - _psi1_denom = S[:, None, :] / lengthscale2 + 1. # Nx1xQ - _psi1_denom_sqrt = np.sqrt(_psi1_denom) #Nx1xQ - _psi1_dist = Z[None, :, :] - mu[:, None, :] # NxMxQ - _psi1_dist_sq = np.square(_psi1_dist) / (lengthscale2 * _psi1_denom) # NxMxQ - _psi1_common = gamma[:,None,:] / (lengthscale2*_psi1_denom*_psi1_denom_sqrt) #Nx1xQ - _psi1_exponent1 = np.log(gamma[:,None,:]) - (_psi1_dist_sq + np.log(_psi1_denom))/2. # NxMxQ - _psi1_exponent2 = np.log(1.-gamma[:,None,:]) - (np.square(Z[None,:,:])/lengthscale2)/2. # NxMxQ - _psi1_exponent_max = np.maximum(_psi1_exponent1,_psi1_exponent2) - _psi1_exponent = _psi1_exponent_max+np.log(np.exp(_psi1_exponent1-_psi1_exponent_max) + np.exp(_psi1_exponent2-_psi1_exponent_max)) #NxMxQ - _psi1_exp_sum = _psi1_exponent.sum(axis=-1) #NxM - _psi1 = variance * np.exp(_psi1_exp_sum) # NxM - - return _psi1 - -def _psi2computations(variance, lengthscale, Z, mu, S, gamma): - """ - Z - MxQ - mu - NxQ - S - NxQ - gamma - NxQ - """ - # here are the "statistics" for psi2 - # Produced intermediate results: - # _psi2 MxM + def _psi1computations(variance, lengthscale, Z, mu, S, gamma): + """ + Z - MxQ + mu - NxQ + S - NxQ + gamma - NxQ + """ + # here are the "statistics" for psi1 + # Produced intermediate results: + # _psi1 NxM - lengthscale2 = np.square(lengthscale) + lengthscale2 = np.square(lengthscale) - _psi2_Zhat = 0.5 * (Z[:, None, :] + Z[None, :, :]) # M,M,Q - _psi2_Zdist = 0.5 * (Z[:, None, :] - Z[None, :, :]) # M,M,Q - _psi2_Zdist_sq = np.square(_psi2_Zdist / lengthscale) # M,M,Q - _psi2_Z_sq_sum = (np.square(Z[:,None,:])+np.square(Z[None,:,:]))/lengthscale2 # MxMxQ - - # psi2 - _psi2_denom = 2.*S[:, None, None, :] / lengthscale2 + 1. # Nx1x1xQ - _psi2_denom_sqrt = np.sqrt(_psi2_denom) - _psi2_mudist = mu[:,None,None,:]-_psi2_Zhat #N,M,M,Q - _psi2_mudist_sq = np.square(_psi2_mudist)/(lengthscale2*_psi2_denom) - _psi2_common = gamma[:,None,None,:]/(lengthscale2 * _psi2_denom * _psi2_denom_sqrt) # Nx1x1xQ - _psi2_exponent1 = -_psi2_Zdist_sq -_psi2_mudist_sq -0.5*np.log(_psi2_denom)+np.log(gamma[:,None,None,:]) #N,M,M,Q - _psi2_exponent2 = np.log(1.-gamma[:,None,None,:]) - 0.5*(_psi2_Z_sq_sum) # NxMxMxQ - _psi2_exponent_max = np.maximum(_psi2_exponent1, _psi2_exponent2) - _psi2_exponent = _psi2_exponent_max+np.log(np.exp(_psi2_exponent1-_psi2_exponent_max) + np.exp(_psi2_exponent2-_psi2_exponent_max)) - _psi2_exp_sum = _psi2_exponent.sum(axis=-1) #NxM - _psi2 = variance*variance * (np.exp(_psi2_exp_sum).sum(axis=0)) # MxM - - return _psi2 - -def psiDerivativecomputations(dL_dpsi0, dL_dpsi1, dL_dpsi2, variance, lengthscale, Z, variational_posterior): - ARD = (len(lengthscale)!=1) + # psi1 + _psi1_denom = S[:, None, :] / lengthscale2 + 1. # Nx1xQ + _psi1_denom_sqrt = np.sqrt(_psi1_denom) #Nx1xQ + _psi1_dist = Z[None, :, :] - mu[:, None, :] # NxMxQ + _psi1_dist_sq = np.square(_psi1_dist) / (lengthscale2 * _psi1_denom) # NxMxQ + _psi1_common = gamma[:,None,:] / (lengthscale2*_psi1_denom*_psi1_denom_sqrt) #Nx1xQ + _psi1_exponent1 = np.log(gamma[:,None,:]) - (_psi1_dist_sq + np.log(_psi1_denom))/2. # NxMxQ + _psi1_exponent2 = np.log(1.-gamma[:,None,:]) - (np.square(Z[None,:,:])/lengthscale2)/2. # NxMxQ + _psi1_exponent_max = np.maximum(_psi1_exponent1,_psi1_exponent2) + _psi1_exponent = _psi1_exponent_max+np.log(np.exp(_psi1_exponent1-_psi1_exponent_max) + np.exp(_psi1_exponent2-_psi1_exponent_max)) #NxMxQ + _psi1_exp_sum = _psi1_exponent.sum(axis=-1) #NxM + _psi1 = variance * np.exp(_psi1_exp_sum) # NxM - dvar_psi1, dl_psi1, dZ_psi1, dmu_psi1, dS_psi1, dgamma_psi1 = _psi1compDer(dL_dpsi1, variance, lengthscale, Z, variational_posterior.mean, variational_posterior.variance, variational_posterior.binary_prob) - dvar_psi2, dl_psi2, dZ_psi2, dmu_psi2, dS_psi2, dgamma_psi2 = _psi2compDer(dL_dpsi2, variance, lengthscale, Z, variational_posterior.mean, variational_posterior.variance, variational_posterior.binary_prob) - - dL_dvar = np.sum(dL_dpsi0) + dvar_psi1 + dvar_psi2 + return _psi1 - dL_dlengscale = dl_psi1 + dl_psi2 - if not ARD: - dL_dlengscale = dL_dlengscale.sum() - - dL_dgamma = dgamma_psi1 + dgamma_psi2 - dL_dmu = dmu_psi1 + dmu_psi2 - dL_dS = dS_psi1 + dS_psi2 - dL_dZ = dZ_psi1 + dZ_psi2 + def _psi2computations(variance, lengthscale, Z, mu, S, gamma): + """ + Z - MxQ + mu - NxQ + S - NxQ + gamma - NxQ + """ + # here are the "statistics" for psi2 + # Produced intermediate results: + # _psi2 MxM + + lengthscale2 = np.square(lengthscale) + + _psi2_Zhat = 0.5 * (Z[:, None, :] + Z[None, :, :]) # M,M,Q + _psi2_Zdist = 0.5 * (Z[:, None, :] - Z[None, :, :]) # M,M,Q + _psi2_Zdist_sq = np.square(_psi2_Zdist / lengthscale) # M,M,Q + _psi2_Z_sq_sum = (np.square(Z[:,None,:])+np.square(Z[None,:,:]))/lengthscale2 # MxMxQ - return dL_dvar, dL_dlengscale, dL_dZ, dL_dmu, dL_dS, dL_dgamma - -def _psi1compDer(dL_dpsi1, variance, lengthscale, Z, mu, S, gamma): - """ - dL_dpsi1 - NxM - Z - MxQ - mu - NxQ - S - NxQ - gamma - NxQ - """ - # here are the "statistics" for psi1 - # Produced intermediate results: dL_dparams w.r.t. psi1 - # _dL_dvariance 1 - # _dL_dlengthscale Q - # _dL_dZ MxQ - # _dL_dgamma NxQ - # _dL_dmu NxQ - # _dL_dS NxQ + # psi2 + _psi2_denom = 2.*S[:, None, None, :] / lengthscale2 + 1. # Nx1x1xQ + _psi2_denom_sqrt = np.sqrt(_psi2_denom) + _psi2_mudist = mu[:,None,None,:]-_psi2_Zhat #N,M,M,Q + _psi2_mudist_sq = np.square(_psi2_mudist)/(lengthscale2*_psi2_denom) + _psi2_common = gamma[:,None,None,:]/(lengthscale2 * _psi2_denom * _psi2_denom_sqrt) # Nx1x1xQ + _psi2_exponent1 = -_psi2_Zdist_sq -_psi2_mudist_sq -0.5*np.log(_psi2_denom)+np.log(gamma[:,None,None,:]) #N,M,M,Q + _psi2_exponent2 = np.log(1.-gamma[:,None,None,:]) - 0.5*(_psi2_Z_sq_sum) # NxMxMxQ + _psi2_exponent_max = np.maximum(_psi2_exponent1, _psi2_exponent2) + _psi2_exponent = _psi2_exponent_max+np.log(np.exp(_psi2_exponent1-_psi2_exponent_max) + np.exp(_psi2_exponent2-_psi2_exponent_max)) + _psi2_exp_sum = _psi2_exponent.sum(axis=-1) #NxM + _psi2 = variance*variance * (np.exp(_psi2_exp_sum).sum(axis=0)) # MxM - lengthscale2 = np.square(lengthscale) - - # psi1 - _psi1_denom = S / lengthscale2 + 1. # NxQ - _psi1_denom_sqrt = np.sqrt(_psi1_denom) #NxQ - _psi1_dist = Z[None, :, :] - mu[:, None, :] # NxMxQ - _psi1_dist_sq = np.square(_psi1_dist) / (lengthscale2 * _psi1_denom[:,None,:]) # NxMxQ - _psi1_common = gamma / (lengthscale2*_psi1_denom*_psi1_denom_sqrt) #NxQ - _psi1_exponent1 = np.log(gamma[:,None,:]) -0.5 * (_psi1_dist_sq + np.log(_psi1_denom[:, None,:])) # NxMxQ - _psi1_exponent2 = np.log(1.-gamma[:,None,:]) -0.5 * (np.square(Z[None,:,:])/lengthscale2) # NxMxQ - _psi1_exponent_max = np.maximum(_psi1_exponent1,_psi1_exponent2) - _psi1_exponent = _psi1_exponent_max+np.log(np.exp(_psi1_exponent1-_psi1_exponent_max) + np.exp(_psi1_exponent2-_psi1_exponent_max)) #NxMxQ - _psi1_exp_sum = _psi1_exponent.sum(axis=-1) #NxM - _psi1_exp_dist_sq = np.exp(-0.5*_psi1_dist_sq) # NxMxQ - _psi1_exp_Z = np.exp(-0.5*np.square(Z[None,:,:])/lengthscale2) # 1xMxQ - _psi1_q = variance * np.exp(_psi1_exp_sum[:,:,None] - _psi1_exponent) # NxMxQ - _psi1 = variance * np.exp(_psi1_exp_sum) # NxM - _dL_dvariance = np.einsum('nm,nm->',dL_dpsi1, _psi1)/variance # 1 - _dL_dgamma = np.einsum('nm,nmq,nmq->nq',dL_dpsi1, _psi1_q, (_psi1_exp_dist_sq/_psi1_denom_sqrt[:,None,:]-_psi1_exp_Z)) # NxQ - _dL_dmu = np.einsum('nm, nmq, nmq, nmq, nq->nq',dL_dpsi1,_psi1_q,_psi1_exp_dist_sq,_psi1_dist,_psi1_common) # NxQ - _dL_dS = np.einsum('nm,nmq,nmq,nq,nmq->nq',dL_dpsi1,_psi1_q,_psi1_exp_dist_sq,_psi1_common,(_psi1_dist_sq-1.))/2. # NxQ - _dL_dZ = np.einsum('nm,nmq,nmq->mq',dL_dpsi1,_psi1_q, (- _psi1_common[:,None,:] * _psi1_dist * _psi1_exp_dist_sq - (1-gamma[:,None,:])/lengthscale2*Z[None,:,:]*_psi1_exp_Z)) - _dL_dlengthscale = lengthscale* np.einsum('nm,nmq,nmq->q',dL_dpsi1,_psi1_q,(_psi1_common[:,None,:]*(S[:,None,:]/lengthscale2+_psi1_dist_sq)*_psi1_exp_dist_sq + (1-gamma[:,None,:])*np.square(Z[None,:,:]/lengthscale2)*_psi1_exp_Z)) - - return _dL_dvariance, _dL_dlengthscale, _dL_dZ, _dL_dmu, _dL_dS, _dL_dgamma - -def _psi2compDer(dL_dpsi2, variance, lengthscale, Z, mu, S, gamma): - """ - Z - MxQ - mu - NxQ - S - NxQ - gamma - NxQ - dL_dpsi2 - MxM - """ - # here are the "statistics" for psi2 - # Produced the derivatives w.r.t. psi2: - # _dL_dvariance 1 - # _dL_dlengthscale Q - # _dL_dZ MxQ - # _dL_dgamma NxQ - # _dL_dmu NxQ - # _dL_dS NxQ + return _psi2 - lengthscale2 = np.square(lengthscale) + def psiDerivativecomputations(dL_dpsi0, dL_dpsi1, dL_dpsi2, variance, lengthscale, Z, variational_posterior): + ARD = (len(lengthscale)!=1) + + dvar_psi1, dl_psi1, dZ_psi1, dmu_psi1, dS_psi1, dgamma_psi1 = _psi1compDer(dL_dpsi1, variance, lengthscale, Z, variational_posterior.mean, variational_posterior.variance, variational_posterior.binary_prob) + dvar_psi2, dl_psi2, dZ_psi2, dmu_psi2, dS_psi2, dgamma_psi2 = _psi2compDer(dL_dpsi2, variance, lengthscale, Z, variational_posterior.mean, variational_posterior.variance, variational_posterior.binary_prob) + + dL_dvar = np.sum(dL_dpsi0) + dvar_psi1 + dvar_psi2 + + dL_dlengscale = dl_psi1 + dl_psi2 + if not ARD: + dL_dlengscale = dL_dlengscale.sum() + + dL_dgamma = dgamma_psi1 + dgamma_psi2 + dL_dmu = dmu_psi1 + dmu_psi2 + dL_dS = dS_psi1 + dS_psi2 + dL_dZ = dZ_psi1 + dZ_psi2 + + return dL_dvar, dL_dlengscale, dL_dZ, dL_dmu, dL_dS, dL_dgamma - _psi2_Zhat = 0.5 * (Z[:, None, :] + Z[None, :, :]) # M,M,Q - _psi2_Zdist = 0.5 * (Z[:, None, :] - Z[None, :, :]) # M,M,Q - _psi2_Zdist_sq = np.square(_psi2_Zdist / lengthscale) # M,M,Q - _psi2_Z_sq_sum = (np.square(Z[:,None,:])+np.square(Z[None,:,:]))/lengthscale2 # MxMxQ - - # psi2 - _psi2_denom = 2.*S / lengthscale2 + 1. # NxQ - _psi2_denom_sqrt = np.sqrt(_psi2_denom) - _psi2_mudist = mu[:,None,None,:]-_psi2_Zhat #N,M,M,Q - _psi2_mudist_sq = np.square(_psi2_mudist)/(lengthscale2*_psi2_denom[:,None,None,:]) - _psi2_common = gamma/(lengthscale2 * _psi2_denom * _psi2_denom_sqrt) # NxQ - _psi2_exponent1 = -_psi2_Zdist_sq -_psi2_mudist_sq -0.5*np.log(_psi2_denom[:,None,None,:])+np.log(gamma[:,None,None,:]) #N,M,M,Q - _psi2_exponent2 = np.log(1.-gamma[:,None,None,:]) - 0.5*(_psi2_Z_sq_sum) # NxMxMxQ - _psi2_exponent_max = np.maximum(_psi2_exponent1, _psi2_exponent2) - _psi2_exponent = _psi2_exponent_max+np.log(np.exp(_psi2_exponent1-_psi2_exponent_max) + np.exp(_psi2_exponent2-_psi2_exponent_max)) - _psi2_exp_sum = _psi2_exponent.sum(axis=-1) #NxM - _psi2_q = variance*variance * np.exp(_psi2_exp_sum[:,:,:,None]-_psi2_exponent) # NxMxMxQ - _psi2_exp_dist_sq = np.exp(-_psi2_Zdist_sq -_psi2_mudist_sq) # NxMxMxQ - _psi2_exp_Z = np.exp(-0.5*_psi2_Z_sq_sum) # MxMxQ - _psi2 = variance*variance * (np.exp(_psi2_exp_sum).sum(axis=0)) # MxM - _dL_dvariance = np.einsum('mo,mo->',dL_dpsi2,_psi2)*2./variance - _dL_dgamma = np.einsum('mo,nmoq,nmoq->nq',dL_dpsi2,_psi2_q,(_psi2_exp_dist_sq/_psi2_denom_sqrt[:,None,None,:] - _psi2_exp_Z)) - _dL_dmu = -2.*np.einsum('mo,nmoq,nq,nmoq,nmoq->nq',dL_dpsi2,_psi2_q,_psi2_common,_psi2_mudist,_psi2_exp_dist_sq) - _dL_dS = np.einsum('mo,nmoq,nq,nmoq,nmoq->nq',dL_dpsi2,_psi2_q, _psi2_common, (2.*_psi2_mudist_sq-1.), _psi2_exp_dist_sq) - _dL_dZ = 2.*np.einsum('mo,nmoq,nmoq->mq',dL_dpsi2,_psi2_q,(_psi2_common[:,None,None,:]*(-_psi2_Zdist*_psi2_denom[:,None,None,:]+_psi2_mudist)*_psi2_exp_dist_sq - (1-gamma[:,None,None,:])*Z[:,None,:]/lengthscale2*_psi2_exp_Z)) - _dL_dlengthscale = 2.*lengthscale* np.einsum('mo,nmoq,nmoq->q',dL_dpsi2,_psi2_q,(_psi2_common[:,None,None,:]*(S[:,None,None,:]/lengthscale2+_psi2_Zdist_sq*_psi2_denom[:,None,None,:]+_psi2_mudist_sq)*_psi2_exp_dist_sq+(1-gamma[:,None,None,:])*_psi2_Z_sq_sum*0.5/lengthscale2*_psi2_exp_Z)) - - return _dL_dvariance, _dL_dlengthscale, _dL_dZ, _dL_dmu, _dL_dS, _dL_dgamma + def _psi1compDer(dL_dpsi1, variance, lengthscale, Z, mu, S, gamma): + """ + dL_dpsi1 - NxM + Z - MxQ + mu - NxQ + S - NxQ + gamma - NxQ + """ + # here are the "statistics" for psi1 + # Produced intermediate results: dL_dparams w.r.t. psi1 + # _dL_dvariance 1 + # _dL_dlengthscale Q + # _dL_dZ MxQ + # _dL_dgamma NxQ + # _dL_dmu NxQ + # _dL_dS NxQ + + lengthscale2 = np.square(lengthscale) + + # psi1 + _psi1_denom = S / lengthscale2 + 1. # NxQ + _psi1_denom_sqrt = np.sqrt(_psi1_denom) #NxQ + _psi1_dist = Z[None, :, :] - mu[:, None, :] # NxMxQ + _psi1_dist_sq = np.square(_psi1_dist) / (lengthscale2 * _psi1_denom[:,None,:]) # NxMxQ + _psi1_common = gamma / (lengthscale2*_psi1_denom*_psi1_denom_sqrt) #NxQ + _psi1_exponent1 = np.log(gamma[:,None,:]) -0.5 * (_psi1_dist_sq + np.log(_psi1_denom[:, None,:])) # NxMxQ + _psi1_exponent2 = np.log(1.-gamma[:,None,:]) -0.5 * (np.square(Z[None,:,:])/lengthscale2) # NxMxQ + _psi1_exponent_max = np.maximum(_psi1_exponent1,_psi1_exponent2) + _psi1_exponent = _psi1_exponent_max+np.log(np.exp(_psi1_exponent1-_psi1_exponent_max) + np.exp(_psi1_exponent2-_psi1_exponent_max)) #NxMxQ + _psi1_exp_sum = _psi1_exponent.sum(axis=-1) #NxM + _psi1_exp_dist_sq = np.exp(-0.5*_psi1_dist_sq) # NxMxQ + _psi1_exp_Z = np.exp(-0.5*np.square(Z[None,:,:])/lengthscale2) # 1xMxQ + _psi1_q = variance * np.exp(_psi1_exp_sum[:,:,None] - _psi1_exponent) # NxMxQ + _psi1 = variance * np.exp(_psi1_exp_sum) # NxM + _dL_dvariance = np.einsum('nm,nm->',dL_dpsi1, _psi1)/variance # 1 + _dL_dgamma = np.einsum('nm,nmq,nmq->nq',dL_dpsi1, _psi1_q, (_psi1_exp_dist_sq/_psi1_denom_sqrt[:,None,:]-_psi1_exp_Z)) # NxQ + _dL_dmu = np.einsum('nm, nmq, nmq, nmq, nq->nq',dL_dpsi1,_psi1_q,_psi1_exp_dist_sq,_psi1_dist,_psi1_common) # NxQ + _dL_dS = np.einsum('nm,nmq,nmq,nq,nmq->nq',dL_dpsi1,_psi1_q,_psi1_exp_dist_sq,_psi1_common,(_psi1_dist_sq-1.))/2. # NxQ + _dL_dZ = np.einsum('nm,nmq,nmq->mq',dL_dpsi1,_psi1_q, (- _psi1_common[:,None,:] * _psi1_dist * _psi1_exp_dist_sq - (1-gamma[:,None,:])/lengthscale2*Z[None,:,:]*_psi1_exp_Z)) + _dL_dlengthscale = lengthscale* np.einsum('nm,nmq,nmq->q',dL_dpsi1,_psi1_q,(_psi1_common[:,None,:]*(S[:,None,:]/lengthscale2+_psi1_dist_sq)*_psi1_exp_dist_sq + (1-gamma[:,None,:])*np.square(Z[None,:,:]/lengthscale2)*_psi1_exp_Z)) + + return _dL_dvariance, _dL_dlengthscale, _dL_dZ, _dL_dmu, _dL_dS, _dL_dgamma + + def _psi2compDer(dL_dpsi2, variance, lengthscale, Z, mu, S, gamma): + """ + Z - MxQ + mu - NxQ + S - NxQ + gamma - NxQ + dL_dpsi2 - MxM + """ + # here are the "statistics" for psi2 + # Produced the derivatives w.r.t. psi2: + # _dL_dvariance 1 + # _dL_dlengthscale Q + # _dL_dZ MxQ + # _dL_dgamma NxQ + # _dL_dmu NxQ + # _dL_dS NxQ + + lengthscale2 = np.square(lengthscale) + + _psi2_Zhat = 0.5 * (Z[:, None, :] + Z[None, :, :]) # M,M,Q + _psi2_Zdist = 0.5 * (Z[:, None, :] - Z[None, :, :]) # M,M,Q + _psi2_Zdist_sq = np.square(_psi2_Zdist / lengthscale) # M,M,Q + _psi2_Z_sq_sum = (np.square(Z[:,None,:])+np.square(Z[None,:,:]))/lengthscale2 # MxMxQ + + # psi2 + _psi2_denom = 2.*S / lengthscale2 + 1. # NxQ + _psi2_denom_sqrt = np.sqrt(_psi2_denom) + _psi2_mudist = mu[:,None,None,:]-_psi2_Zhat #N,M,M,Q + _psi2_mudist_sq = np.square(_psi2_mudist)/(lengthscale2*_psi2_denom[:,None,None,:]) + _psi2_common = gamma/(lengthscale2 * _psi2_denom * _psi2_denom_sqrt) # NxQ + _psi2_exponent1 = -_psi2_Zdist_sq -_psi2_mudist_sq -0.5*np.log(_psi2_denom[:,None,None,:])+np.log(gamma[:,None,None,:]) #N,M,M,Q + _psi2_exponent2 = np.log(1.-gamma[:,None,None,:]) - 0.5*(_psi2_Z_sq_sum) # NxMxMxQ + _psi2_exponent_max = np.maximum(_psi2_exponent1, _psi2_exponent2) + _psi2_exponent = _psi2_exponent_max+np.log(np.exp(_psi2_exponent1-_psi2_exponent_max) + np.exp(_psi2_exponent2-_psi2_exponent_max)) + _psi2_exp_sum = _psi2_exponent.sum(axis=-1) #NxM + _psi2_q = variance*variance * np.exp(_psi2_exp_sum[:,:,:,None]-_psi2_exponent) # NxMxMxQ + _psi2_exp_dist_sq = np.exp(-_psi2_Zdist_sq -_psi2_mudist_sq) # NxMxMxQ + _psi2_exp_Z = np.exp(-0.5*_psi2_Z_sq_sum) # MxMxQ + _psi2 = variance*variance * (np.exp(_psi2_exp_sum).sum(axis=0)) # MxM + _dL_dvariance = np.einsum('mo,mo->',dL_dpsi2,_psi2)*2./variance + _dL_dgamma = np.einsum('mo,nmoq,nmoq->nq',dL_dpsi2,_psi2_q,(_psi2_exp_dist_sq/_psi2_denom_sqrt[:,None,None,:] - _psi2_exp_Z)) + _dL_dmu = -2.*np.einsum('mo,nmoq,nq,nmoq,nmoq->nq',dL_dpsi2,_psi2_q,_psi2_common,_psi2_mudist,_psi2_exp_dist_sq) + _dL_dS = np.einsum('mo,nmoq,nq,nmoq,nmoq->nq',dL_dpsi2,_psi2_q, _psi2_common, (2.*_psi2_mudist_sq-1.), _psi2_exp_dist_sq) + _dL_dZ = 2.*np.einsum('mo,nmoq,nmoq->mq',dL_dpsi2,_psi2_q,(_psi2_common[:,None,None,:]*(-_psi2_Zdist*_psi2_denom[:,None,None,:]+_psi2_mudist)*_psi2_exp_dist_sq - (1-gamma[:,None,None,:])*Z[:,None,:]/lengthscale2*_psi2_exp_Z)) + _dL_dlengthscale = 2.*lengthscale* np.einsum('mo,nmoq,nmoq->q',dL_dpsi2,_psi2_q,(_psi2_common[:,None,None,:]*(S[:,None,None,:]/lengthscale2+_psi2_Zdist_sq*_psi2_denom[:,None,None,:]+_psi2_mudist_sq)*_psi2_exp_dist_sq+(1-gamma[:,None,None,:])*_psi2_Z_sq_sum*0.5/lengthscale2*_psi2_exp_Z)) + + return _dL_dvariance, _dL_dlengthscale, _dL_dZ, _dL_dmu, _dL_dS, _dL_dgamma