def weights_rbf(self, unit_sp, hypers):
# BQ weights for RBF kernel with given hypers, computations adopted from the GP-ADF code [Deisenroth] with
# the following assumptions:
# (A1) the uncertain input is zero-mean with unit covariance
# (A2) one set of hyper-parameters is used for all output dimensions (one GP models all outputs)
d, n = unit_sp.shape
# GP kernel hyper-parameters
alpha, el, jitter = hypers['sig_var'], hypers['lengthscale'], hypers['noise_var']
assert len(el) == d
# pre-allocation for convenience
eye_d, eye_n = np.eye(d), np.eye(n)
iLam1 = np.atleast_2d(np.diag(el ** -1)) # sqrt(Lambda^-1)
iLam2 = np.atleast_2d(np.diag(el ** -2))
inp = unit_sp.T.dot(iLam1) # sigmas / el[:, na] (x - m)^T*sqrt(Lambda^-1) # (numSP, xdim)
K = np.exp(2 * np.log(alpha) - 0.5 * maha(inp, inp))
iK = cho_solve(cho_factor(K + jitter * eye_n), eye_n)
B = iLam2 + eye_d # (D, D)
c = alpha ** 2 / np.sqrt(det(B))
t = inp.dot(inv(B)) # inn*(P + Lambda)^-1
l = np.exp(-0.5 * np.sum(inp * t, 1)) # (N, 1)
zet = 2 * np.log(alpha) - 0.5 * np.sum(inp * inp, 1)
inp = inp.dot(iLam1)
R = 2 * iLam2 + eye_d
t = 1 / np.sqrt(det(R))
L = np.exp((zet[:, na] + zet[:, na].T) + maha(inp, -inp, V=0.5 * inv(R)))
q = c * l # evaluations of the kernel mean map (from the viewpoint of RHKS methods)
# mean weights
wm_f = q.dot(iK)
iKQ = iK.dot(t * L)
# covariance weights
wc_f = iKQ.dot(iK)
# cross-covariance "weights"
wc_fx = np.diag(q).dot(iK)
# used for self.D.dot(x - mean).dot(wc_fx).dot(fx)
self.D = inv(eye_d + np.diag(el ** 2)) # S(S+Lam)^-1; for S=I, (I+Lam)^-1
# model variance; to be added to the covariance
# this diagonal form assumes independent GP outputs (cov(f^a, f^b) = 0 for all a, b: a neq b)
self.model_var = np.diag((alpha ** 2 - np.trace(iKQ)) * np.ones((d, 1)))
return wm_f, wc_f, wc_fx
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