Exemplo n.º 1
def preprocess(C, data_locs, grids, x, n_blocks_x, n_blocks_y, tdata, pdata, relp, mean_ondata): 

    xbi = np.asarray(np.linspace(0,grids[0][2],n_blocks_x+1),dtype=int)
    ybi = np.asarray(np.linspace(0,grids[1][2],n_blocks_y+1),dtype=int)

    dev = (tdata-mean_ondata-pdata)

    # Figure out which data locations are relevant to the different prediction blocks
    cutoff = C.params['amp']*relp
    scale = C.params['scale']
    inc = C.params['inc']
    ecc = C.params['ecc']
    eff_spat_scale = scale/np.sqrt(2)
    rel_data_ind = np.empty((n_blocks_x, n_blocks_y), dtype=object)
    C_eval = C(data_locs,data_locs)
    U, n_posdef, pivots = ichol_full(c=C_eval, reltol=relp)
    U = U[:n_posdef, :n_posdef]

    dl_posdef = data_locs[pivots[:n_posdef]]
    dev_posdef = dev[pivots[:n_posdef]]

    # Backsolve data-data covariance against dev
    pm.gp.trisolve(U, dev_posdef, uplo='U', transa='T', inplace=True)
    pm.gp.trisolve(U, dev_posdef, uplo='U', transa='N', inplace=True)
    return dev_posdef, xbi, ybi, dl_posdef
Exemplo n.º 2
def ApplyICDonSymmetricMatrix(K, center=True):
    Apply Incomplete Cholesky Decomposition on Gram matrix.

    A Gram matrix K (n*n) assumes the following approximation:

        K \approx G * G^T
        G = U*S*V

    where G is a n*m matrix (m < n), which can be SVD decomposed. So U^T * U = I.

    So by combining them, we have:

        K \approx U * Lambda * U^T

    where Lambda = S^2, which is the m leading eigen values of K.

        2002, Bach and Jordan, Kernel Independent Component Analysis, Journal of Machine Learning Research

    :param K: numpy matrix, uncentered Gram matrix

    :param center: boolean, True by default.
        If True, K = G * G^T is a centered matrix, i.e. for a un-centered Gram matrix K, we have

            (I - Ones/n) * K * (I - Ones/n) \approx G * G^T

        after permuting K according to pind.

        U: numpy matrix. A n*m lower triangular matrix such that U^T * U = I.
        Lambda: 1-d numpy array. A vector of m leading eigen-values of K.
        pind: 1-d numpy array of int32. A vector of permutation indices, which are the column numbers of K in the same order as ICD retains them.

    # ICD
    reltol = 1e-6 # fixed threshold
    L, m, pind = pyichol.ichol_full(K, reltol)
    G = np.matrix(L[:m].T)
    n = G.shape[0]

    if center:
        I = np.identity(n)
        Ones = np.matrix(np.ones((n, n)))
        G = (I - Ones/n) * G  # (column) centered G

    # SVD
    U, d, V = np.linalg.svd(G, full_matrices=False)
    Lambda = d**2 # eigen values of Gram matrix

    return U, Lambda, pind

# def ApplyICDonSymmetricMatrix(K, center=True, reltol=1e-6):
#     if center:
#         n = K.shape[0]
#         N0 = np.identity(n) - np.ones((n,n))/n
#         K = N0 * K * N0
#     U, s, V = np.linalg.svd(K)
#     ind = np.where(s >= reltol)[0]
#     return U[:,ind], s[ind]

# TODO: add Nystroem method for kernel matrix approximation
# TODO: add function to stack (row, column, diagonal) matrices ???