def deform(self, mesh, anchors):
#t_start = time.time()
delta = np.array(self.L.dot(mesh.points()))
#t_end = time.time()
#print("delta computation time is %.5f seconds." % (t_end - t_start))
#t_start = time.time()
# augment delta solution matrix with weighted anchors
for i in range(self.k):
delta[self.n + i, :] = self.weight * anchors[i, :]
#t_end = time.time()
#print("give anchor value computation time is %.5f seconds." % (t_end - t_start))
#t_start = time.time()
# update mesh vertices with least-squares solution
for i in range(3):
mesh.points()[:, i] = sparseqr.solve(self.L,
delta[:, i],
tolerance=1e-8)
#mesh.points()[:, i] = lsqr(self.L, delta[:, i])[0]
#t_end = time.time()
#print("sparse lsqr time is %.5f seconds." % (t_end - t_start))
return mesh
def deform(self, verts, achr_verts):
self.L = sparse.coo_matrix((self.V, (self.I, self.J)),
shape=(self.n + self.k, self.n)).tocsr()
delta = np.array(self.L.dot(verts))
# augment delta solution matrix with weighted anchors
for i in range(self.k):
delta[self.n + i, :] = self.weight * achr_verts[i, :]
# update mesh vertices with least-squares solution
deformed_verts = np.zeros(verts.shape)
for i in range(3):
deformed_verts[:, i] = sparseqr.solve(self.L,
delta[:, i],
tolerance=1e-8)
return deformed_verts
def solveLaplacianMesh(mesh, anchors, anchorsIdx, cotangent=True):
n = mesh.n_vertices()
k = anchorsIdx.shape[0]
operator = (getLaplacianMatrixUmbrella, getLaplacianMatrixCotangent)
L = operator[1](mesh, anchorsIdx) if cotangent else operator[0](mesh,
anchorsIdx)
delta = np.array(L.dot(mesh.points()))
# augment delta solution matrix with weighted anchors
for i in range(k):
delta[n + i, :] = WEIGHT * anchors[i, :]
# update mesh vertices with least-squares solution
for i in range(3):
#mesh.points()[:, i] = lsqr(L, delta[:, i])[0]
mesh.points()[:, i] = sparseqr.solve(L, delta[:, i], tolerance=1e-8)
return mesh
def run(self):
self.get_patches()
self.poisson_b = []
cols_all = []
vals_all = []
rows_all = []
row_global = 0
for i in self.valid_index:
[colidx, colvals,
bvals] = self.build_patch_for_poisson(self.mask_patches[i],
self.data_patches[i],
self.index_1d_patches[i])
self.poisson_b.append(bvals)
cols_all.append(colidx)
vals_all.append(colvals)
rows_all.append(np.ones_like(colidx) * row_global)
row_global += 1
rows_all_flat = list(itertools.chain.from_iterable(rows_all))
cols_all_flat = list(itertools.chain.from_iterable(cols_all))
vals_all_flat = list(itertools.chain.from_iterable(vals_all))
self.poisson_A = sparse.coo_matrix(
(vals_all_flat, (rows_all_flat, cols_all_flat)),
shape=(row_global, self.valid_num))
self.poisson_b = np.array(self.poisson_b)
# depth fusion
if self.depth_A is not None:
self.poisson_A = sparse.vstack((self.poisson_A, self.depth_A))
self.poisson_b = np.hstack((self.poisson_b, self.depth_b))
depth = sparseqr.solve(self.poisson_A, self.poisson_b, tolerance=0)
self.depth.reshape(-1)[self.valid_index] = depth
return self.depth
print "Average time:", total_time / float(n_trials), "sec."
print "Average error:", total_error / float(n_trials)
# PySPQR
if PySPQR == 1:
total_time = 0
total_error = 0
for i in range(n_trials):
A = scipy.sparse.rand(matrix_size[0],
matrix_size[1],
density=matrix_density)
x_true = np.random.random(matrix_size[1])
b = A * x_true
start = time.clock()
A = A.tocsr()
x = sparseqr.solve(A, b, tolerance=1e-4)
end = time.clock()
total_time += end - start
total_error += np.linalg.norm(x - x_true, ord=2)
print "Average time:", total_time / float(n_trials), "sec."
print "Average error:", total_error / float(n_trials)
# CVXPY
if CVXPY_SCS == 1:
total_time = 0
total_error = 0
for i in range(n_trials):
A = scipy.sparse.rand(matrix_size[0],
matrix_size[1],
density=matrix_density)
x_true = np.random.random(matrix_size[1])
def smooth_xyt_fit(**kwargs):
required_fields = ('data', 'W', 'ctr', 'spacing', 'E_RMS')
args = {
'reference_epoch': 0,
'W_ctr': 1e4,
'mask_file': None,
'mask_scale': None,
'compute_E': False,
'max_iterations': 10,
'srs_WKT': None,
'N_subset': None,
'bias_params': None,
'repeat_res': None,
'repeat_dt': 1,
'Edit_only': False,
'dzdt_lags': [1, 4],
'VERBOSE': True
}
args.update(kwargs)
for field in required_fields:
if field not in kwargs:
raise ValueError("%s must be defined", field)
valid_data = np.ones_like(args['data'].x, dtype=bool)
timing = dict()
if args['N_subset'] is not None:
tic = time()
valid_data = edit_data_by_subset_fit(args['N_subset'], args)
timing['edit_by_subset'] = time() - tic
if args['Edit_only']:
return {
'timing': timing,
'data': args['data'].copy().subset(valid_data)
}
m = dict()
E = dict()
# define the grids
tic = time()
bds = {
coord: args['ctr'][coord] + np.array([-0.5, 0.5]) * args['W'][coord]
for coord in ('x', 'y', 't')
}
grids = dict()
grids['z0'] = fd_grid([bds['y'], bds['x']],
args['spacing']['z0'] * np.ones(2),
name='z0',
srs_WKT=args['srs_WKT'],
mask_file=args['mask_file'])
grids['dz']=fd_grid( [bds['y'], bds['x'], bds['t']], \
[args['spacing']['dz'], args['spacing']['dz'], args['spacing']['dt']], col_0=grids['z0'].N_nodes, name='dz', srs_WKT=args['srs_WKT'], mask_file=args['mask_file'])
grids['z0'].col_N = grids['dz'].col_N
grids['t'] = fd_grid([bds['t']], [args['spacing']['dt']], name='t')
# select only the data points that are within the grid bounds
valid_z0 = grids['z0'].validate_pts((args['data'].coords()[0:2]))
valid_dz = grids['dz'].validate_pts((args['data'].coords()))
valid_data = valid_data & valid_dz & valid_z0
# if repeat_res is given, resample the data to include only repeat data (to within a spatial tolerance of repeat_res)
if args['repeat_res'] is not None:
valid_data[valid_data]=valid_data[valid_data] & \
select_repeat_data(args['data'].copy().subset(valid_data), grids, args['repeat_dt'], args['repeat_res'])
# subset the data based on the valid mask
data = args['data'].copy().subset(valid_data)
# if we have a mask file, use it to subset the data
# needs to be done after the valid subset because otherwise the interp_mtx for the mask file fails.
if args['mask_file'] is not None:
temp = fd_grid([bds['y'], bds['x']],
[args['spacing']['z0'], args['spacing']['z0']],
name='z0',
srs_WKT=args['srs_WKT'],
mask_file=args['mask_file'])
data_mask = lin_op(temp, name='interp_z').interp_mtx(
data.coords()[0:2]).toCSR().dot(grids['z0'].mask.ravel())
data_mask[~np.isfinite(data_mask)] = 0
if np.any(data_mask == 0):
data.subset(~(data_mask == 0))
valid_data[valid_data] = ~(data_mask == 0)
# define the interpolation operator, equal to the sum of the dz and z0 operators
G_data = lin_op(grids['z0'],
name='interp_z').interp_mtx(data.coords()[0:2])
G_data.add(lin_op(grids['dz'], name='interp_dz').interp_mtx(data.coords()))
# define the smoothness constraints
grad2_z0 = lin_op(grids['z0'], name='grad2_z0').grad2(DOF='z0')
grad2_dz = lin_op(grids['dz'], name='grad2_dzdt').grad2_dzdt(DOF='z',
t_lag=1)
grad_dzdt = lin_op(grids['dz'], name='grad_dzdt').grad_dzdt(DOF='z',
t_lag=1)
constraint_op_list = [grad2_z0, grad2_dz, grad_dzdt]
if 'd2z_dt2' in args['E_RMS'] and args['E_RMS']['d2z_dt2'] is not None:
d2z_dt2 = lin_op(grids['dz'], name='d2z_dt2').d2z_dt2(DOF='z')
constraint_op_list.append(d2z_dt2)
# if bias params are given, create a set of parameters to estimate them
if args['bias_params'] is not None:
data, bias_model = assign_bias_ID(data, args['bias_params'])
G_bias, Gc_bias, Cvals_bias, bias_model = param_bias_matrix(
data,
bias_model,
bias_param_name='bias_ID',
col_0=grids['dz'].col_N)
G_data.add(G_bias)
constraint_op_list.append(Gc_bias)
# put the equations together
Gc = lin_op(None, name='constraints').vstack(constraint_op_list)
N_eq = G_data.N_eq + Gc.N_eq
# put together all the errors
Ec = np.zeros(Gc.N_eq)
root_delta_V_dz = np.sqrt(np.prod(grids['dz'].delta))
root_delta_A_z0 = np.sqrt(np.prod(grids['z0'].delta))
Ec[Gc.TOC['rows']['grad2_z0']] = args['E_RMS'][
'd2z0_dx2'] / root_delta_A_z0 * grad2_z0.mask_for_ind0(
args['mask_scale'])
Ec[Gc.TOC['rows']['grad2_dzdt']] = args['E_RMS'][
'd3z_dx2dt'] / root_delta_V_dz * grad2_dz.mask_for_ind0(
args['mask_scale'])
Ec[Gc.TOC['rows']['grad_dzdt']] = args['E_RMS'][
'd2z_dxdt'] / root_delta_V_dz * grad_dzdt.mask_for_ind0(
args['mask_scale'])
if 'd2z_dt2' in args['E_RMS'] and args['E_RMS']['d2z_dt2'] is not None:
Ec[Gc.TOC['rows']
['d2z_dt2']] = args['E_RMS']['d2z_dt2'] / root_delta_V_dz
if args['bias_params'] is not None:
Ec[Gc.TOC['rows'][Gc_bias.name]] = Cvals_bias
Ed = data.sigma.ravel()
# calculate the inverse square root of the data covariance matrix
TCinv = sp.dia_matrix((1. / np.concatenate((Ed, Ec)), 0),
shape=(N_eq, N_eq))
# define the right hand side of the equation
rhs = np.zeros([N_eq])
rhs[0:data.size] = data.z.ravel()
# put the fit and constraint matrices together
Gcoo = sp.vstack([G_data.toCSR(), Gc.toCSR()]).tocoo()
cov_rows = G_data.N_eq + np.arange(Gc.N_eq)
# define the matrix that sets dz[reference_epoch]=0 by removing columns from the solution:
# Find the identify the rows and columns that match the reference epoch
temp_r, temp_c = np.meshgrid(np.arange(0, grids['dz'].shape[0]),
np.arange(0, grids['dz'].shape[1]))
z02_mask = grids['dz'].global_ind([
temp_r.transpose().ravel(),
temp_c.transpose().ravel(),
args['reference_epoch'] + np.zeros_like(temp_r).ravel()
])
# Identify all of the DOFs that do not include the reference epoch
cols = np.arange(G_data.col_N, dtype='int')
include_cols = np.setdiff1d(cols, z02_mask)
# Generate a matrix that has diagonal elements corresponding to all DOFs except the reference epoch.
# Multiplying this by a matrix with columns for all model parameters yeilds a matrix with no columns
# corresponding to the reference epoch.
Ip_c = sp.coo_matrix((np.ones_like(include_cols),
(include_cols, np.arange(include_cols.size))),
shape=(Gc.col_N, include_cols.size)).tocsc()
# eliminate the columns for the model variables that are set to zero
Gcoo = Gcoo.dot(Ip_c)
timing['setup'] = time() - tic
if np.any(data.z > 2500):
print('outlier!')
# initialize the book-keeping matrices for the inversion
m0 = np.zeros(Ip_c.shape[0])
if "three_sigma_edit" in data.list_of_fields:
inTSE = np.where(data.three_sigma_edit)[0]
else:
inTSE = np.arange(G_data.N_eq, dtype=int)
if args['VERBOSE']:
print("initial: %d:" % G_data.r.max())
tic_iteration = time()
for iteration in range(args['max_iterations']):
# build the parsing matrix that removes invalid rows
Ip_r = sp.coo_matrix(
(np.ones(Gc.N_eq + inTSE.size),
(np.arange(Gc.N_eq + inTSE.size), np.concatenate(
(inTSE, cov_rows)))),
shape=(Gc.N_eq + inTSE.size, Gcoo.shape[0])).tocsc()
m0_last = m0
if args['VERBOSE']:
print("starting qr solve for iteration %d" % iteration)
# solve the equations
tic = time()
m0 = Ip_c.dot(
sparseqr.solve(Ip_r.dot(TCinv.dot(Gcoo)),
Ip_r.dot(TCinv.dot(rhs))))
timing['sparseqr_solve'] = time() - tic
# quit if the solution is too similar to the previous solution
if (np.max(np.abs(
(m0_last - m0)[Gc.TOC['cols']['dz']])) < 0.05) and (iteration > 2):
break
# calculate the full data residual
rs_data = (data.z - G_data.toCSR().dot(m0)) / data.sigma
# calculate the robust standard deviation of the scaled residuals for the selected data
sigma_hat = RDE(rs_data[inTSE])
inTSE_last = inTSE
# select the data that are within 3*sigma of the solution
inTSE = np.where(np.abs(rs_data) < 3.0 * np.maximum(1, sigma_hat))[0]
if args['VERBOSE']:
print('found %d in TSE, sigma_hat=%3.3f' % (inTSE.size, sigma_hat))
if (sigma_hat <= 1 or
(inTSE.size == inTSE_last.size
and np.all(inTSE_last == inTSE))) and (iteration > 2):
if args['VERBOSE']:
print("sigma_hat LT 1, exiting")
break
timing['iteration'] = time() - tic_iteration
inTSE = inTSE_last
valid_data[valid_data] = (np.abs(rs_data) < 3.0 * np.maximum(1, sigma_hat))
data.assign(
{'three_sigma_edit': np.abs(rs_data) < 3.0 * np.maximum(1, sigma_hat)})
# report the model-based estimate of the data points
data.assign({'z_est': np.reshape(G_data.toCSR().dot(m0), data.shape)})
# reshape the components of m to the grid shapes
m['z0'] = np.reshape(m0[Gc.TOC['cols']['z0']], grids['z0'].shape)
m['dz'] = np.reshape(m0[Gc.TOC['cols']['dz']], grids['dz'].shape)
# calculate height rates
for lag in args['dzdt_lags']:
this_name = 'dzdt_lag%d' % lag
m[this_name] = lin_op(grids['dz'], name='dzdt',
col_N=G_data.col_N).dzdt(lag=lag).grid_prod(m0)
# build a matrix that takes the average of the central 20 km of the delta-z grid
XR = np.mean(grids['z0'].bds[0]) + np.array([-1., 1.]) * args['W_ctr'] / 2.
YR = np.mean(grids['z0'].bds[1]) + np.array([-1., 1.]) * args['W_ctr'] / 2.
center_dzbar = lin_op(grids['dz'], name='center_dzbar',
col_N=G_data.col_N).vstack([
lin_op(grids['dz']).mean_of_bounds(
(XR, YR, [season, season]))
for season in grids['dz'].ctrs[2]
])
G_dzbar = center_dzbar.toCSR()
# calculate the grid mean of dz
m['dz_bar'] = G_dzbar.dot(m0)
# build a matrix that takes the lagged temporal derivative of dzbar (e.g. quarterly dzdt, annual dzdt)
for lag in args['dzdt_lags']:
this_name = 'dzdt_bar_lag%d' % lag
this_op = lin_op(grids['t'], name=this_name).diff(lag=lag).toCSR()
# calculate the grid mean of dz/dt
m[this_name] = this_op.dot(m['dz_bar'].ravel())
# report the parameter biases. Sorted in order of the parameter bias arguments
#???
if args['bias_params'] is not None:
m['bias'] = parse_biases(m0, bias_model['bias_ID_dict'],
args['bias_params'])
# report the entire model vector, just in case we want it.
m['all'] = m0
# report the geolocation of the output map
m['extent'] = np.concatenate((grids['z0'].bds[1], grids['z0'].bds[0]))
# parse the resduals to assess the contributions of the total error:
# Make the C matrix for the constraints
TCinv_cov = sp.dia_matrix((1. / Ec, 0), shape=(Gc.N_eq, Gc.N_eq))
rc = TCinv_cov.dot(Gc.toCSR().dot(m0))
ru = Gc.toCSR().dot(m0)
R = dict()
RMS = dict()
for eq_type in ['d2z_dt2', 'grad2_z0', 'grad2_dzdt']:
if eq_type in Gc.TOC['rows']:
R[eq_type] = np.sum(rc[Gc.TOC['rows'][eq_type]]**2)
RMS[eq_type] = np.sqrt(np.mean(ru[Gc.TOC['rows'][eq_type]]**2))
R['data'] = np.sum(((data.z_est - data.z) / data.sigma)**2)
RMS['data'] = np.sqrt(np.mean((data.z_est - data.z)**2))
# if we need to compute the errors in the solution, continue
if args['compute_E']:
tic = time()
# take the QZ transform of Gcoo
z, R, perm, rank = sparseqr.qz(Ip_r.dot(TCinv.dot(Gcoo)),
Ip_r.dot(TCinv.dot(rhs)))
z = z.ravel()
R = R.tocsr()
R.sort_indices()
R.eliminate_zeros()
timing['decompose_qz'] = time() - tic
E0 = np.zeros(R.shape[0])
# compute Rinv for use in propagating errors.
# what should the tolerance be? We will eventually square Rinv and take its
# row-wise sum. We care about errors at the cm level, so
# size(Rinv)*tol^2 = 0.01 -> tol=sqrt(0.01/size(Rinv))~ 1E-4
tic = time()
RR, CC, VV, status = inv_tr_upper(R, np.int(np.prod(R.shape) / 4),
1.e-5)
# save Rinv as a sparse array. The syntax perm[RR] undoes the permutation from QZ
Rinv = sp.coo_matrix((VV, (perm[RR], CC)), shape=R.shape).tocsr()
timing['Rinv_cython'] = time() - tic
tic = time()
E0 = np.sqrt(Rinv.power(2).sum(axis=1))
timing['propagate_errors'] = time() - tic
# generate the full E vector. E0 appears to be an ndarray,
E0 = np.array(Ip_c.dot(E0)).ravel()
E['z0'] = np.reshape(E0[Gc.TOC['cols']['z0']], grids['z0'].shape)
E['dz'] = np.reshape(E0[Gc.TOC['cols']['dz']], grids['dz'].shape)
# generate the lagged dz errors:
for lag in args['dzdt_lags']:
this_name = 'dzdt_lag%d' % lag
E[this_name] = lin_op(grids['dz'],
name=this_name,
col_N=G_data.col_N).dzdt(lag=lag).grid_error(
Ip_c.dot(Rinv))
this_name = 'dzdt_bar_lag%d' % lag
this_op = lin_op(grids['t'], name=this_name).diff(lag=lag).toCSR()
E[this_name] = np.sqrt(
(this_op.dot(Ip_c).dot(Rinv)).power(2).sum(axis=1))
# calculate the grid mean of dz/dt
# generate the season-to-season errors
#E['dzdt_qyr']=lin_op(grids['dz'], name='dzdt_1yr', col_N=G_data.col_N).dzdt().grid_error(Ip_c.dot(Rinv))
# generate the annual errors
#E['dzdt_1yr']=lin_op(grids['dz'], name='dzdt_1yr', col_N=G_data.col_N).dzdt(lag=4).grid_error(Ip_c.dot(Rinv))
# generate the grid-mean error
E['dz_bar'] = np.sqrt(
(G_dzbar.dot(Ip_c).dot(Rinv)).power(2).sum(axis=1))
# generate the grid-mean quarterly dzdt error
#E['dzdt_bar_qyr']=np.sqrt((ddt_qyr.dot(G_dzbar).dot(Ip_c).dot(Rinv)).power(2).sum(axis=1))
# generate the grid-mean annual dzdt error
#E['dzdt_bar_1yr']=np.sqrt((ddt_1yr.dot(G_dzbar).dot(Ip_c).dot(Rinv)).power(2).sum(axis=1))
# report the rgt bias errors. Sorted by RGT, then by cycle
if args['bias_params'] is not None:
E['bias'] = parse_biases(E0, bias_model['bias_ID_dict'],
args['bias_params'])
TOC = Gc.TOC
return {
'm': m,
'E': E,
'data': data,
'grids': grids,
'valid_data': valid_data,
'TOC': TOC,
'R': R,
'RMS': RMS,
'timing': timing,
'E_RMS': args['E_RMS']
}
def sparsesolve(A, b, eps_w=0, eps_laplace=0):
A_mod, b_mod = modify_system(A, b, eps_w, eps_laplace)
return sparseqr.solve(A_mod, b_mod)
def sparsesolve_qr(A, b, eps=0, R=None):
if eps > 0:
A, b = modify_system(A, b, eps, R)
return scipy.sparse.csc_matrix(sparseqr.solve(A, b))
def main():
######################
# Config
######################
# Choose least-squares solver to use:
#
# lsq_solver = "dense" # LAPACK DGELSD, direct, good for small problems
# lsq_solver = "sparse" # SciPy LSQR, iterative, asymptotically faster, good for large problems
# lsq_solver = "optimize" # general nonlinear optimizer using Trust Region Reflective (trf) algorithm
# lsq_solver = "qr"
# lsq_solver = "cholesky"
# lsq_solver = "sparse_qr"
lsq_solver = "sparse_qr_solve"
######################
# Load multiscale data
######################
print("Loading measurement data...")
# measurements are provided on a meshgrid over (Hx, sigxx)
# data2.mat contains virtual measurements, generated from a multiscale model.
# data2 = scipy.io.loadmat("data2.mat")
# Hx = np.squeeze(data2["Hx"]) # 1D array, (M,)
# sigxx = np.squeeze(data2["sigxx"]) # 1D array, (N,)
# Bx = data2["Bx"] # 2D array, (M, N)
# lamxx = data2["lamxx"] # --"--
## lamyy = data2["lamyy"] # --"--
## lamzz = data2["lamzz"] # --"--
data2 = scipy.io.loadmat("umair_gal_denoised.mat")
sigxx = -1e6 * np.array([
0, 1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80
][::-1],
dtype=np.float64)
assert sigxx.shape[0] == 18
Hx = data2["Hval"][0, :] # same for all sigma, just take the first row
Bx = data2["Bval"].T
lamxx = data2["LHval"].T * 1e-6
Bx = Bx[::-1, :]
lamxx = lamxx[::-1, :]
# HACK, fix later (must decouple number of knots from number of data sites)
ii = np.arange(Hx.shape[0])
n_newi = 401
newii = np.linspace(0, ii[-1], n_newi)
nsigma = sigxx.shape[0]
fH = scipy.interpolate.interp1d(ii, Hx)
newB = np.empty((n_newi, Bx.shape[1]), dtype=np.float64)
newlam = np.empty((n_newi, lamxx.shape[1]), dtype=np.float64)
for j in range(nsigma):
fB = scipy.interpolate.interp1d(ii, Bx[:, j])
newB[:, j] = fB(newii)
flam = scipy.interpolate.interp1d(ii, lamxx[:, j])
newlam[:, j] = flam(newii)
Hx = fH(newii)
Bx = newB
lamxx = newlam
# Order of spline (as-is! 3 = cubic)
ordr = 3
# Auxiliary variables (H, sig_xx, sig_xy)
Hscale = np.max(Hx)
sscale = np.max(np.abs(sigxx))
x = Hx / Hscale
y = sigxx / sscale
nx = x.shape[0] # number of grid points, x axis
ny = y.shape[0] # number of grid points, y axis
# Partial derivatives (B, lam_xx, lam_xy) from multiscale model
#
# In the magnetostriction components, the multiscale model produces nonzero lamxx at zero stress.
# We normalize this away for purposes of performing the curve fit.
#
dpsi_dx = Bx * Hscale
dpsi_dy = (lamxx - lamxx[0, :]) * sscale
######################
# Set up splines
######################
print("Setting up splines...")
# The evaluation algorithm used in bspline.py uses half-open intervals t_i <= x < t_{i+1}.
#
# This causes havoc for evaluation at the end of each interval, because it is actually the start
# of the next interval.
#
# Especially, the end of the last interval is the start of the next (non-existent) interval.
#
# We work around this by using a small epsilon to avoid evaluation exactly at t_{i+1} (for the last interval).
#
def marginize_end(x):
out = x.copy()
out[-1] += 1e-10 * (x[-1] - x[0])
return out
# create knots and spline basis
xknots = splinelab.aptknt(marginize_end(x), ordr)
yknots = splinelab.aptknt(marginize_end(y), ordr)
splx = bspline.Bspline(xknots, ordr)
sply = bspline.Bspline(yknots, ordr)
# get number of basis functions (perform dummy evaluation and count)
nxb = len(splx(0.))
nyb = len(sply(0.))
# TODO Check if we need to convert input Bx and sigxx to u,v (what is actually stored in the data files?)
# Create collocation matrices:
#
# A[i,j] = d**deriv_order B_j(tau[i])
#
# where d denotes differentiation and B_j is the jth basis function.
#
# We place the collocation sites at the points where we have measurements.
#
Au = splx.collmat(x)
Av = sply.collmat(y)
Du = splx.collmat(x, deriv_order=1)
Dv = sply.collmat(y, deriv_order=1)
######################
# Assemble system
######################
print("Assembling system...")
# Assemble the equation system for fitting against data on the partial derivatives of psi.
#
# By writing psi in the spline basis,
#
# psi_{ij} = A^{u}_{ik} A^{v}_{jl} c_{kl}
#
# the quantities to be fitted, which are the partial derivatives of psi, become
#
# B_{ij} = D^{u}_{ik} A^{v}_{jl} c_{kl}
# lambda_{xx,ij} = A^{u}_{ik} D^{v}_{jl} c_{kl}
#
# Repeated indices are summed over.
#
# Column: kl converted to linear index (k = 0,1,...,nxb-1, l = 0,1,...,nyb-1)
# Row: ij converted to linear index (i = 0,1,...,nx-1, j = 0,1,...,ny-1)
#
# (Paavo's notes, Stresses4.pdf)
nf = 2 # number of unknown fields
nr = nx * ny # equation system rows per unknown field
A = np.empty((nf * nr, nxb * nyb), dtype=np.float64) # global matrix
b = np.empty((nf * nr), dtype=np.float64) # global RHS
# zero array element detection tolerance
tol = 1e-6
I, J, IJ = util.index.genidx((nx, ny))
K, L, KL = util.index.genidx((nxb, nyb))
# loop only over rows of the equation system
for i, j, ij in zip(I, J, IJ):
A[nf * ij, KL] = Du[i, K] * Av[j, L]
A[nf * ij + 1, KL] = Au[i, K] * Dv[j, L]
b[nf * IJ] = dpsi_dx[I, J] # RHS for B_x
b[nf * IJ + 1] = dpsi_dy[I, J] # RHS for lambda_xx
# # the above is equivalent to this much slower version:
# #
# # equation system row
# for j in range(ny):
# for i in range(nx):
# ij = np.ravel_multi_index( (i,j), (nx,ny) )
#
# # equation system column
# for l in range(nyb):
# for k in range(nxb):
# kl = np.ravel_multi_index( (k,l), (nxb,nyb) )
# A[nf*ij, kl] = Du[i,k] * Av[j,l]
# A[nf*ij+1,kl] = Au[i,k] * Dv[j,l]
#
# b[nf*ij] = dpsi_dx[i,j] if abs(dpsi_dx[i,j]) > tol else 0. # RHS for B_x
# b[nf*ij+1] = dpsi_dy[i,j] if abs(dpsi_dy[i,j]) > tol else 0. # RHS for lambda_xx
######################
# Solve
######################
# Solve the optimal coefficients.
# Note that we are constructing a potential function from partial derivatives only,
# so the solution is unique only up to a global additive shift term.
#
# Under the hood, numpy.linalg.lstsq uses LAPACK DGELSD:
#
# http://stackoverflow.com/questions/29372559/what-is-the-difference-between-numpy-linalg-lstsq-and-scipy-linalg-lstsq
#
# DGELSD accepts also rank-deficient input (rank(A) < min(nrows,ncols)), returning arg min( ||x||_2 ) ,
# so we don't need to do anything special to account for this.
#
# Same goes for the sparse LSQR.
# equilibrate row and column norms
#
# See documentation of scipy.sparse.linalg.lsqr, it requires this to work properly.
#
# https://github.com/Technologicat/python-wlsqm
#
print("Equilibrating...")
S = A.copy(order='F') # the rescaler requires Fortran memory layout
A = scipy.sparse.csr_matrix(A) # save memory (dense "A" no longer needed)
# eps = 7./3. - 4./3. - 1 # http://stackoverflow.com/questions/19141432/python-numpy-machine-epsilon
# print( S.max() * max(S.shape) * eps ) # default zero singular value detection tolerance in np.linalg.matrix_rank()
# import wlsqm.utils.lapackdrivers as wul
# rs,cs = wul.do_rescale( S, wul.ScalingAlgo.ALGO_DGEEQU )
# # row scaling only (for weighting)
# with np.errstate(divide='ignore', invalid='ignore'):
# rs = np.where( np.abs(b) > tol, 1./b, 1. )
# for i in range(S.shape[0]):
# S[i,:] *= rs[i]
# cs = 1.
# scale rows corresponding to Bx
#
rs = np.ones_like(b)
rs[nf * IJ] = 2
for i in range(S.shape[0]):
S[i, :] *= rs[i]
cs = 1.
# # It seems this is not needed in the 2D problem (fitting error is slightly smaller without it).
#
# # Additional row scaling.
# #
# # This equilibrates equation weights, but deteriorates the condition number of the matrix.
# #
# # Note that in a least-squares problem the row weighting *does* matter, because it affects
# # the fitting error contribution from the rows.
# #
# with np.errstate(divide='ignore', invalid='ignore'):
# rs2 = np.where( np.abs(b) > tol, 1./b, 1. )
# for i in range(S.shape[0]):
# S[i,:] *= rs2[i]
# rs *= rs2
# a = np.abs(rs2)
# print( np.min(a), np.mean(a), np.max(a) )
# rs = np.asanyarray(rs)
# cs = np.asanyarray(cs)
# a = np.abs(rs)
# print( np.min(a), np.mean(a), np.max(a) )
b *= rs # scale RHS accordingly
# colnorms = np.linalg.norm(S, ord=np.inf, axis=0) # sum over rows -> column norms
# rownorms = np.linalg.norm(S, ord=np.inf, axis=1) # sum over columns -> row norms
# print( " rescaled column norms min = %g, avg = %g, max = %g" % (np.min(colnorms), np.mean(colnorms), np.max(colnorms)) )
# print( " rescaled row norms min = %g, avg = %g, max = %g" % (np.min(rownorms), np.mean(rownorms), np.max(rownorms)) )
print("Solving with algorithm = '%s'..." % (lsq_solver))
if lsq_solver == "dense":
print(" matrix shape %s = %d elements" %
(S.shape, np.prod(S.shape)))
ret = numpy.linalg.lstsq(S, b) # c,residuals,rank,singvals
c = ret[0]
elif lsq_solver == "sparse":
S = scipy.sparse.coo_matrix(S)
print(" matrix shape %s = %d elements; %d nonzeros (%g%%)" %
(S.shape, np.prod(
S.shape), S.nnz, 100. * S.nnz / np.prod(S.shape)))
ret = scipy.sparse.linalg.lsqr(S, b)
c, exit_reason, iters = ret[:3]
if exit_reason != 2: # 2 = least-squares solution found
print("WARNING: solver did not converge (exit_reason = %d)" %
(exit_reason))
print(" sparse solver iterations taken: %d" % (iters))
elif lsq_solver == "optimize":
# make sparse matrix (faster for dot products)
S = scipy.sparse.coo_matrix(S)
print(" matrix shape %s = %d elements; %d nonzeros (%g%%)" %
(S.shape, np.prod(
S.shape), S.nnz, 100. * S.nnz / np.prod(S.shape)))
def fitting_error(c):
return S.dot(c) - b
ret = scipy.optimize.least_squares(fitting_error,
np.ones(S.shape[1],
dtype=np.float64),
method="trf",
loss="linear")
c = ret.x
if ret.status < 1:
# status codes: https://docs.scipy.org/doc/scipy-0.18.1/reference/generated/scipy.optimize.least_squares.html
print("WARNING: solver did not converge (status = %d)" %
(ret.status))
elif lsq_solver == "qr":
print(" matrix shape %s = %d elements" %
(S.shape, np.prod(S.shape)))
# http://glowingpython.blogspot.fi/2012/03/solving-overdetermined-systems-with-qr.html
Q, R = np.linalg.qr(S) # qr decomposition of A
Qb = (Q.T).dot(b) # computing Q^T*b (project b onto the range of A)
# c = np.linalg.solve(R,Qb) # solving R*x = Q^T*b
c = scipy.linalg.solve_triangular(R, Qb, check_finite=False)
elif lsq_solver == "cholesky":
# S is rank-deficient by one, because we are solving a potential based on data on its partial derivatives.
#
# Before solving, force S to have full rank by fixing one coefficient.
#
S[0, :] = 0.
S[0, 0] = 1.
b[0] = 1.
rs[0] = 1.
S = scipy.sparse.csr_matrix(S)
print(" matrix shape %s = %d elements; %d nonzeros (%g%%)" %
(S.shape, np.prod(
S.shape), S.nnz, 100. * S.nnz / np.prod(S.shape)))
# Be sure to use the new sksparse from
#
# https://github.com/scikit-sparse/scikit-sparse
#
# instead of the old scikits.sparse (which will fail with an error).
#
# Requires libsuitesparse-dev for CHOLMOD headers.
#
from sksparse.cholmod import cholesky_AAt
# Notice that CHOLMOD computes AA' and we want M'M, so we must set A = M'!
factor = cholesky_AAt(S.T)
c = factor.solve_A(S.T * b)
elif lsq_solver == "sparse_qr":
# S is rank-deficient by one, because we are solving a potential based on data on its partial derivatives.
#
# Before solving, force S to have full rank by fixing one coefficient;
# otherwise the linear solve step will fail because R will be exactly singular.
#
S[0, :] = 0.
S[0, 0] = 1.
b[0] = 1.
rs[0] = 1.
S = scipy.sparse.coo_matrix(S)
print(" matrix shape %s = %d elements; %d nonzeros (%g%%)" %
(S.shape, np.prod(
S.shape), S.nnz, 100. * S.nnz / np.prod(S.shape)))
# pip install sparseqr
# or https://github.com/yig/PySPQR
#
# Works like MATLAB's [Q,R,e] = qr(...):
#
# https://se.mathworks.com/help/matlab/ref/qr.html
#
# [Q,R,E] = qr(A) or [Q,R,E] = qr(A,'matrix') produces unitary Q, upper triangular R and a permutation matrix E
# so that A*E = Q*R. The column permutation E is chosen to reduce fill-in in R.
#
# [Q,R,e] = qr(A,'vector') returns the permutation information as a vector instead of a matrix.
# That is, e is a row vector such that A(:,e) = Q*R.
#
import sparseqr
print(" performing sparse QR decomposition...")
Q, R, E, rank = sparseqr.qr(S)
# produce reduced QR (for least-squares fitting)
#
# - cut away bottom part of R (zeros!)
# - cut away the corresponding far-right part of Q
#
# see
# np.linalg.qr
# https://andreask.cs.illinois.edu/cs357-s15/public/demos/06-qr-applications/Solving%20Least-Squares%20Problems.html
#
# # inefficient way:
# k = min(S.shape)
# R = scipy.sparse.csr_matrix( R.A[:k,:] )
# Q = scipy.sparse.csr_matrix( Q.A[:,:k] )
print(" reducing matrices...")
# somewhat more efficient way:
k = min(S.shape)
R = R.tocsr()[:k, :]
Q = Q.tocsc()[:, :k]
# # maybe somewhat efficient way: manipulate data vectors, create new coo matrix
# #
# # (incomplete, needs work; need to shift indices of rows/cols after the removed ones)
# #
# k = min(S.shape)
# mask = np.nonzero( R.row < k )[0]
# R = scipy.sparse.coo_matrix( ( R.data[mask], (R.row[mask], R.col[mask]) ), shape=(k,k) )
# mask = np.nonzero( Q.col < k )[0]
# Q = scipy.sparse.coo_matrix( ( Q.data[mask], (Q.row[mask], Q.col[mask]) ), shape=(k,k) )
print(" solving...")
Qb = (Q.T).dot(b)
x = scipy.sparse.linalg.spsolve(R, Qb)
c = np.empty_like(x)
c[E] = x[:] # apply inverse permutation
elif lsq_solver == "sparse_qr_solve":
S[0, :] = 0.
S[0, 0] = 1.
b[0] = 1.
rs[0] = 1.
S = scipy.sparse.coo_matrix(S)
print(" matrix shape %s = %d elements; %d nonzeros (%g%%)" %
(S.shape, np.prod(
S.shape), S.nnz, 100. * S.nnz / np.prod(S.shape)))
import sparseqr
c = sparseqr.solve(S, b)
else:
raise ValueError("unknown solver '%s'; valid: 'dense', 'sparse'" %
(lsq_solver))
c *= cs # undo column scaling in solution
# now c contains the spline coefficients, c_{kl}, where kl has been raveled into a linear index.
######################
# Save
######################
filename = "tmp_s2d.mat"
L = locals()
data = {
key: L[key]
for key in ["ordr", "xknots", "yknots", "c", "Hscale", "sscale"]
}
scipy.io.savemat(filename, data, format='5', oned_as='row')
######################
# Plot
######################
print("Visualizing...")
# unpack results onto meshgrid
#
fitted = A.dot(
c
) # function values corresponding to each row in the global equation system
X, Y = np.meshgrid(
Hx, sigxx,
indexing='ij') # indexed like X[i,j] (i is x index, j is y index)
Z_Bx = np.empty_like(X)
Z_lamxx = np.empty_like(X)
Z_Bx[I, J] = fitted[nf * IJ]
Z_lamxx[I, J] = fitted[nf * IJ + 1]
# # the above is equivalent to:
# for ij in range(nr):
# i,j = np.unravel_index( ij, (nx,ny) )
# Z_Bx[i,j] = fitted[nf*ij]
# Z_lamxx[i,j] = fitted[nf*ij+1]
data_Bx = {
"x": (X, r"$H_{x}$"),
"y": (Y, r"$\sigma_{xx}$"),
"z": (Z_Bx / Hscale, r"$B_{x}$")
}
data_lamxx = {
"x": (X, r"$H_{x}$"),
"y": (Y, r"$\sigma_{xx}$"),
"z": (Z_lamxx / sscale, r"$\lambda_{xx}$")
}
def relerr(data, refdata):
refdata_linview = refdata.reshape(-1)
return 100. * np.linalg.norm(refdata_linview - data.reshape(-1)
) / np.linalg.norm(refdata_linview)
plt.figure(1)
plt.clf()
ax = util.plot.plot_wireframe(data_Bx, legend_label="Spline", figno=1)
ax.plot_wireframe(X, Y, dpsi_dx / Hscale, label="Multiscale", color="r")
plt.legend(loc="best")
print("B_x relative error %g%%" % (relerr(Z_Bx, dpsi_dx)))
plt.figure(2)
plt.clf()
ax = util.plot.plot_wireframe(data_lamxx, legend_label="Spline", figno=2)
ax.plot_wireframe(X, Y, dpsi_dy / sscale, label="Multiscale", color="r")
plt.legend(loc="best")
print("lambda_{xx} relative error %g%%" % (relerr(Z_lamxx, dpsi_dy)))
# match the grid point numbering used in MATLAB version of this script
#
def t(A):
return np.transpose(A, [1, 0])
dpsi_dx = t(dpsi_dx)
Z_Bx = t(Z_Bx)
dpsi_dy = t(dpsi_dy)
Z_lamxx = t(Z_lamxx)
plt.figure(3)
plt.clf()
ax = plt.subplot(1, 1, 1)
ax.plot(dpsi_dx.reshape(-1) / Hscale,
'ro',
markersize='2',
label="Multiscale")
ax.plot(Z_Bx.reshape(-1) / Hscale, 'ko', markersize='2', label="Spline")
ax.set_xlabel("Grid point number")
ax.set_ylabel(r"$B_{x}$")
plt.legend(loc="best")
plt.figure(4)
plt.clf()
ax = plt.subplot(1, 1, 1)
ax.plot(dpsi_dy.reshape(-1) / sscale,
'ro',
markersize='2',
label="Multiscale")
ax.plot(Z_lamxx.reshape(-1) / sscale, 'ko', markersize='2', label="Spline")
ax.set_xlabel("Grid point number")
ax.set_ylabel(r"$\lambda_{xx}$")
plt.legend(loc="best")
print("All done.")
def solve(self, vector):
return sparseqr.solve(self.matrix, vector)
def iterate_fit(data, Gcoo, rhs, TCinv, G_data, Gc, in_TSE, Ip_c, timing, args,\
bias_model=None):
cov_rows = G_data.N_eq + np.arange(Gc.N_eq)
#print(f"iterate_fit: G.shape={Gcoo.shape}, G.nnz={Gcoo.nnz}, data.shape={data.shape}", flush=True)
in_TSE_original = np.zeros(data.shape, dtype=bool)
in_TSE_original[in_TSE] = True
min_tse_iterations = 2
if args['bias_nsigma_iteration'] is not None:
min_tse_iterations = np.max(
[min_tse_iterations, args['bias_nsigma_iteration'] + 1])
for iteration in range(args['max_iterations']):
# build the parsing matrix that removes invalid rows
Ip_r=sp.coo_matrix((np.ones(Gc.N_eq+in_TSE.size), \
(np.arange(Gc.N_eq+in_TSE.size), \
np.concatenate((in_TSE, cov_rows)))), \
shape=(Gc.N_eq+in_TSE.size, Gcoo.shape[0])).tocsc()
m0_last = np.zeros(Ip_c.shape[0])
if args['VERBOSE']:
print("starting qr solve for iteration %d at %s" %
(iteration, ctime()),
flush=True)
# solve the equations
tic = time()
m0 = Ip_c.dot(
sparseqr.solve(Ip_r.dot(TCinv.dot(Gcoo)),
Ip_r.dot(TCinv.dot(rhs))))
timing['sparseqr_solve'] = time() - tic
# calculate the full data residual
rs_data = (data.z - G_data.toCSR().dot(m0)) / data.sigma
# calculate the robust standard deviation of the scaled residuals for the selected data
sigma_hat = RDE(rs_data[in_TSE])
# select the data that have scaled residuals < 3 *max(1, sigma_hat)
in_TSE_last = in_TSE
###testing
in_TSE = (np.abs(rs_data) < 3.0 * np.maximum(1, sigma_hat))
# if bias_nsigma_edit is specified, check for biases that are more than
# args['bias_nsigma_edit'] times their expected values.
if args['bias_nsigma_edit'] is not None and iteration >= args[
'bias_nsigma_iteration']:
if 'edited' not in bias_model['bias_param_dict']:
bias_model['bias_param_dict']['edited'] = np.zeros_like(
bias_model['bias_param_dict']['ID'], dtype=bool)
bias_dict, slope_bias_dict = parse_biases(m0, bias_model,
args['bias_params'])
bad_bias_IDs=np.array(bias_dict['ID'])\
[(np.abs(bias_dict['val']) > args['bias_nsigma_edit'] * np.array(bias_dict['expected'])\
* np.maximum(1, sigma_hat)) \
| bias_model['bias_param_dict']['edited']]
print(bad_bias_IDs)
for ID in bad_bias_IDs:
mask = np.ones(data.size, dtype=bool)
#Mark the ID as edited (because it will have a bias estimate of zero in subsequent iterations)
bias_model['bias_param_dict']['edited'][
bias_model['bias_param_dict']['ID'].index(ID)] = True
# mark all data associated with the ID as invalid
for field, field_val in bias_model['bias_ID_dict'][ID].items():
if field in data.fields:
mask &= (getattr(data, field).ravel() == field_val)
in_TSE[mask == 1] = 0
if 'editable' in data.fields:
in_TSE[data.editable == 0] = in_TSE_original[data.editable == 0]
in_TSE = np.flatnonzero(in_TSE)
if args['DEM_tol'] is not None:
in_TSE = check_data_against_DEM(in_TSE, data, m0, G_data,
args['DEM_tol'])
# quit if the solution is too similar to the previous solution
if (np.max(np.abs((m0_last - m0)[Gc.TOC['cols']['dz']])) <
args['converge_tol_dz']) and (iteration > 2):
if args['VERBOSE']:
print(
"Solution identical to previous iteration with tolerance %3.1f, exiting after iteration %d"
% (args['converge_tol_dz'], iteration))
break
# select the data that are within 3*sigma of the solution
if args['VERBOSE']:
print('found %d in TSE, sigma_hat=%3.3f, dt=%3.0f' %
(in_TSE.size, sigma_hat, timing['sparseqr_solve']),
flush=True)
if iteration > 0:
if in_TSE.size == in_TSE_last.size and np.all(
in_TSE_last == in_TSE):
if args['VERBOSE']:
print("filtering unchanged, exiting after iteration %d" %
iteration)
break
if iteration >= min_tse_iterations:
if sigma_hat <= 1:
if args['VERBOSE']:
print("sigma_hat LT 1, exiting after iteration %d" %
iteration,
flush=True)
break
return m0, sigma_hat, in_TSE, in_TSE_last, rs_data
def smooth_xyt_fit(**kwargs):
required_fields=('data','W','ctr','spacing','E_RMS')
args={'reference_epoch':0,
'W_ctr':1e4,
'mask_file':None,
'mask_scale':None,
'compute_E':False,
'max_iterations':10,
'srs_proj4': None,
'N_subset': None,
'bias_params': None,
'repeat_res':None,
'converge_tol_dz':0.05,
'repeat_dt': 1,
'Edit_only': False,
'dzdt_lags':[1, 4],
'data_slope_sensors':None,
'E_slope':0.05,
'VERBOSE': True}
args.update(kwargs)
for field in required_fields:
if field not in kwargs:
raise ValueError("%s must be defined", field)
valid_data = np.isfinite(args['data'].z) & np.isfinite(args['data'].sigma)
timing=dict()
if args['N_subset'] is not None:
tic=time()
valid_data &= edit_data_by_subset_fit(args['N_subset'], args)
timing['edit_by_subset']=time()-tic
if args['Edit_only']:
return {'timing':timing, 'data':args['data'].copy()[valid_data]}
m={}
E={}
R={}
RMS={}
# define the grids
tic=time()
bds={coord:args['ctr'][coord]+np.array([-0.5, 0.5])*args['W'][coord] for coord in ('x','y','t')}
grids=dict()
grids['z0']=fd_grid( [bds['y'], bds['x']], args['spacing']['z0']*np.ones(2),\
name='z0', srs_proj4=args['srs_proj4'], mask_file=args['mask_file'])
grids['dz']=fd_grid( [bds['y'], bds['x'], bds['t']], \
[args['spacing']['dz'], args['spacing']['dz'], args['spacing']['dt']], \
name='dz', col_0=grids['z0'].N_nodes, srs_proj4=args['srs_proj4'], \
mask_file=args['mask_file'])
grids['z0'].col_N=grids['dz'].col_N
grids['t']=fd_grid([bds['t']], [args['spacing']['dt']], name='t')
# select only the data points that are within the grid bounds
valid_z0=grids['z0'].validate_pts((args['data'].coords()[0:2]))
valid_dz=grids['dz'].validate_pts((args['data'].coords()))
valid_data=valid_data & valid_dz & valid_z0
if not np.any(valid_data):
return {'m':m, 'E':E, 'data':None, 'grids':grids, 'valid_data': valid_data, 'TOC':{},'R':{}, 'RMS':{}, 'timing':timing,'E_RMS':args['E_RMS']}
# if repeat_res is given, resample the data to include only repeat data (to within a spatial tolerance of repeat_res)
if args['repeat_res'] is not None:
N_before_repeat=np.sum(valid_data)
valid_data[valid_data]=valid_data[valid_data] & \
select_repeat_data(args['data'].copy_subset(valid_data), grids, args['repeat_dt'], args['repeat_res'], reference_time=grids['t'].ctrs[0][args['reference_epoch']])
if args['VERBOSE']:
print("before repeat editing found %d data" % N_before_repeat)
print("after repeat editing found %d data" % valid_data.sum())
# subset the data based on the valid mask
data=args['data'].copy_subset(valid_data)
# if we have a mask file, use it to subset the data
# needs to be done after the valid subset because otherwise the interp_mtx for the mask file fails.
if args['mask_file'] is not None:
temp=fd_grid( [bds['y'], bds['x']], [args['spacing']['z0'], args['spacing']['z0']], name='z0', srs_proj4=args['srs_proj4'], mask_file=args['mask_file'])
data_mask=lin_op(temp, name='interp_z').interp_mtx(data.coords()[0:2]).toCSR().dot(grids['z0'].mask.ravel())
data_mask[~np.isfinite(data_mask)]=0
if np.any(data_mask==0):
data.index(~(data_mask==0))
valid_data[valid_data]= ~(data_mask==0)
# Check if we have any data. If not, quit
if data.size==0:
return {'m':m, 'E':E, 'data':data, 'grids':grids, 'valid_data': valid_data, 'TOC':{},'R':{}, 'RMS':{}, 'timing':timing,'E_RMS':args['E_RMS']}
# define the interpolation operator, equal to the sum of the dz and z0 operators
G_data=lin_op(grids['z0'], name='interp_z').interp_mtx(data.coords()[0:2])
G_data.add(lin_op(grids['dz'], name='interp_dz').interp_mtx(data.coords()))
# define the smoothness constraints
grad2_z0=lin_op(grids['z0'], name='grad2_z0').grad2(DOF='z0')
grad2_dz=lin_op(grids['dz'], name='grad2_dzdt').grad2_dzdt(DOF='z', t_lag=1)
grad_dzdt=lin_op(grids['dz'], name='grad_dzdt').grad_dzdt(DOF='z', t_lag=1)
constraint_op_list=[grad2_z0, grad2_dz, grad_dzdt]
if 'd2z_dt2' in args['E_RMS'] and args['E_RMS']['d2z_dt2'] is not None:
d2z_dt2=lin_op(grids['dz'], name='d2z_dt2').d2z_dt2(DOF='z')
constraint_op_list.append(d2z_dt2)
# if bias params are given, create a set of parameters to estimate them
if args['bias_params'] is not None:
data, bias_model=assign_bias_ID(data, args['bias_params'])
G_bias, Gc_bias, Cvals_bias, bias_model=\
param_bias_matrix(data, bias_model, bias_param_name='bias_ID',
col_0=grids['dz'].col_N)
G_data.add(G_bias)
constraint_op_list.append(Gc_bias)
if args['data_slope_sensors'] is not None:
bias_model['E_slope']=args['E_slope']
G_slope_bias, Gc_slope_bias, Cvals_slope_bias, bias_model= data_slope_bias(data, bias_model, sensors=args['data_slope_sensors'], col_0=G_data.col_N)
G_data.add(G_slope_bias)
constraint_op_list.append(Gc_slope_bias)
# put the equations together
Gc=lin_op(None, name='constraints').vstack(constraint_op_list)
N_eq=G_data.N_eq+Gc.N_eq
# put together all the errors
Ec=np.zeros(Gc.N_eq)
root_delta_V_dz=np.sqrt(np.prod(grids['dz'].delta))
root_delta_A_z0=np.sqrt(np.prod(grids['z0'].delta))
Ec[Gc.TOC['rows']['grad2_z0']]=args['E_RMS']['d2z0_dx2']/root_delta_A_z0*grad2_z0.mask_for_ind0(args['mask_scale'])
Ec[Gc.TOC['rows']['grad2_dzdt']]=args['E_RMS']['d3z_dx2dt']/root_delta_V_dz*grad2_dz.mask_for_ind0(args['mask_scale'])
Ec[Gc.TOC['rows']['grad_dzdt']]=args['E_RMS']['d2z_dxdt']/root_delta_V_dz*grad_dzdt.mask_for_ind0(args['mask_scale'])
if 'd2z_dt2' in args['E_RMS'] and args['E_RMS']['d2z_dt2'] is not None:
Ec[Gc.TOC['rows']['d2z_dt2']]=args['E_RMS']['d2z_dt2']/root_delta_V_dz
if args['bias_params'] is not None:
Ec[Gc.TOC['rows'][Gc_bias.name]] = Cvals_bias
if args['data_slope_sensors'] is not None:
Ec[Gc.TOC['rows'][Gc_slope_bias.name]] = Cvals_slope_bias
Ed=data.sigma.ravel()
# calculate the inverse square root of the data covariance matrix
TCinv=sp.dia_matrix((1./np.concatenate((Ed, Ec)), 0), shape=(N_eq, N_eq))
# define the right hand side of the equation
rhs=np.zeros([N_eq])
rhs[0:data.size]=data.z.ravel()
# put the fit and constraint matrices together
Gcoo=sp.vstack([G_data.toCSR(), Gc.toCSR()]).tocoo()
cov_rows=G_data.N_eq+np.arange(Gc.N_eq)
# build a matrix that takes the average of the center of the delta-z grid
# this gets used both in the averaging and error-calculation codes
XR=np.mean(grids['z0'].bds[0])+np.array([-1., 1.])*args['W_ctr']/2.
YR=np.mean(grids['z0'].bds[1])+np.array([-1., 1.])*args['W_ctr']/2.
center_dzbar=lin_op(grids['dz'], name='center_dzbar', col_N=G_data.col_N).vstack([lin_op(grids['dz']).mean_of_bounds((XR, YR, [season, season] )) for season in grids['dz'].ctrs[2]])
G_dzbar=center_dzbar.toCSR()
# define the matrix that sets dz[reference_epoch]=0 by removing columns from the solution:
# Find the rows and columns that match the reference epoch
temp_r, temp_c=np.meshgrid(np.arange(0, grids['dz'].shape[0]), np.arange(0, grids['dz'].shape[1]))
z02_mask=grids['dz'].global_ind([temp_r.transpose().ravel(), temp_c.transpose().ravel(),\
args['reference_epoch']+np.zeros_like(temp_r).ravel()])
# Identify all of the DOFs that do not include the reference epoch
cols=np.arange(G_data.col_N, dtype='int')
include_cols=np.setdiff1d(cols, z02_mask)
# Generate a matrix that has diagonal elements corresponding to all DOFs except the reference epoch.
# Multiplying this by a matrix with columns for all model parameters yeilds a matrix with no columns
# corresponding to the reference epoch.
Ip_c=sp.coo_matrix((np.ones_like(include_cols), (include_cols, np.arange(include_cols.size))), \
shape=(Gc.col_N, include_cols.size)).tocsc()
# eliminate the columns for the model variables that are set to zero
Gcoo=Gcoo.dot(Ip_c)
timing['setup']=time()-tic
# initialize the book-keeping matrices for the inversion
m0=np.zeros(Ip_c.shape[0])
if "three_sigma_edit" in data.fields:
inTSE=np.flatnonzero(data.three_sigma_edit)
else:
inTSE=np.arange(G_data.N_eq, dtype=int)
inTSE_last = np.zeros([0])
if args['VERBOSE']:
print("initial: %d:" % G_data.r.max())
tic_iteration=time()
for iteration in range(args['max_iterations']):
# build the parsing matrix that removes invalid rows
Ip_r=sp.coo_matrix((np.ones(Gc.N_eq+inTSE.size), (np.arange(Gc.N_eq+inTSE.size), np.concatenate((inTSE, cov_rows)))), \
shape=(Gc.N_eq+inTSE.size, Gcoo.shape[0])).tocsc()
m0_last=m0
if args['VERBOSE']:
print("starting qr solve for iteration %d" % iteration)
# solve the equations
tic=time();
m0=Ip_c.dot(sparseqr.solve(Ip_r.dot(TCinv.dot(Gcoo)), Ip_r.dot(TCinv.dot(rhs))));
timing['sparseqr_solve']=time()-tic
# calculate the full data residual
rs_data=(data.z-G_data.toCSR().dot(m0))/data.sigma
# calculate the robust standard deviation of the scaled residuals for the selected data
sigma_hat=RDE(rs_data[inTSE])
# select the data that have scaled residuals < 3 *max(1, sigma_hat)
inTSE_last=inTSE
inTSE = np.flatnonzero(np.abs(rs_data) < 3.0 * np.maximum(1, sigma_hat))
# quit if the solution is too similar to the previous solution
if (np.max(np.abs((m0_last-m0)[Gc.TOC['cols']['dz']])) < args['converge_tol_dz']) and (iteration > 2):
if args['VERBOSE']:
print("Solution identical to previous iteration with tolerance %3.1f, exiting after iteration %d" % (args['converge_tol_dz'], iteration))
break
# select the data that are within 3*sigma of the solution
if args['VERBOSE']:
print('found %d in TSE, sigma_hat=%3.3f' % ( inTSE.size, sigma_hat ))
if iteration > 0:
if inTSE.size == inTSE_last.size and np.all( inTSE_last == inTSE ):
if args['VERBOSE']:
print("filtering unchanged, exiting after iteration %d" % iteration)
break
if iteration >= 2:
if sigma_hat <= 1:
if args['VERBOSE']:
print("sigma_hat LT 1, exiting after iteration %d" % iteration)
break
# if we've done any iterations, parse the model and the data residuals
if args['max_iterations'] > 0:
timing['iteration']=time()-tic_iteration
inTSE=inTSE_last
valid_data[valid_data]=(np.abs(rs_data)<3.0*np.maximum(1, sigma_hat))
data.assign({'three_sigma_edit':np.abs(rs_data)<3.0*np.maximum(1, sigma_hat)})
# report the model-based estimate of the data points
data.assign({'z_est':np.reshape(G_data.toCSR().dot(m0), data.shape)})
parse_model(m, m0, G_data, G_dzbar, Gc.TOC, grids, args['bias_params'], bias_model, dzdt_lags=args['dzdt_lags'])
# parse the resduals to assess the contributions of the total error:
# Make the C matrix for the constraints
TCinv_cov=sp.dia_matrix((1./Ec, 0), shape=(Gc.N_eq, Gc.N_eq))
rc=TCinv_cov.dot(Gc.toCSR().dot(m0))
ru=Gc.toCSR().dot(m0)
for eq_type in ['d2z_dt2','grad2_z0','grad2_dzdt']:
if eq_type in Gc.TOC['rows']:
R[eq_type]=np.sum(rc[Gc.TOC['rows'][eq_type]]**2)
RMS[eq_type]=np.sqrt(np.mean(ru[Gc.TOC['rows'][eq_type]]**2))
R['data']=np.sum((((data.z_est[data.three_sigma_edit==1]-data.z[data.three_sigma_edit==1])/data.sigma[data.three_sigma_edit==1])**2))
RMS['data']=np.sqrt(np.mean((data.z_est[data.three_sigma_edit==1]-data.z[data.three_sigma_edit==1])**2))
# Compute the error in the solution if requested
if args['compute_E']:
# We have generally not done any iterations at this point, so need to make the Ip_r matrix
Ip_r=sp.coo_matrix((np.ones(Gc.N_eq+inTSE.size), (np.arange(Gc.N_eq+inTSE.size), np.concatenate((inTSE, cov_rows)))), \
shape=(Gc.N_eq+inTSE.size, Gcoo.shape[0])).tocsc()
parse_errors(E, Gcoo, TCinv, rhs, Ip_c, Ip_r, grids, G_data, Gc, G_dzbar, \
bias_model, args['bias_params'], dzdt_lags=args['dzdt_lags'], timing=timing)
TOC=Gc.TOC
return {'m':m, 'E':E, 'data':data, 'grids':grids, 'valid_data': valid_data, 'TOC':TOC,'R':R, 'RMS':RMS, 'timing':timing,'E_RMS':args['E_RMS'], 'dzdt_lags':args['dzdt_lags']}
import scipy.sparse.linalg
import sparseqr
# QR decompose a sparse matrix M such that Q R = M E
#
M = scipy.sparse.rand(10, 10, density=0.1)
Q, R, E, rank = sparseqr.qr(M)
print(abs(Q * R - M * sparseqr.permutation_vector_to_matrix(E)).sum()
) # should be approximately zero
# Solve many linear systems "M x = b for b in columns(B)"
#
B = scipy.sparse.rand(
10, 5, density=0.1
) # many RHS, sparse (could also have just one RHS with shape (10,))
x = sparseqr.solve(M, B, tolerance=0)
# Solve an overdetermined linear system A x = b in the least-squares sense
#
# The same routine also works for the usual non-overdetermined case.
#
A = scipy.sparse.rand(20, 10, density=0.1) # 20 equations, 10 unknowns
b = numpy.random.random(
20) # one RHS, dense, but could also have many (in shape (20,k))
x = sparseqr.solve(A, b, tolerance=0)
# Solve a linear system M x = B via QR decomposition
#
# This approach is slow due to the explicit construction of Q, but may be
# useful if a large number of systems need to be solved with the same M.
#
