# Licensed under a 3-clause BSD style license - see LICENSE.rst
# -*- coding: utf-8 -*-
"""This module corresponds to the bspline directory in idlutils.
"""
from warnings import warn
import numpy as np
from numpy.linalg.linalg import LinAlgError
from scipy.linalg import cholesky_banded, cho_solve_banded
from . import PydlutilsUserWarning
from .math import djs_reject
from .trace import fchebyshev
from .. import uniq
from ..goddard.math import flegendre
[docs]
class bspline(object):
"""B-spline class.
Functions in the idlutils bspline library are implemented as methods on this
class.
Parameters
----------
x : :class:`numpy.ndarray`
The data.
nord : :class:`int`, optional
The order of the B-spline. Default is 4, which is cubic.
npoly : :class:`int`, optional
Polynomial order to fit over 2nd variable, if supplied. If not
supplied the order is 1.
bkpt : :class:`numpy.ndarray`, optional
An initial breakpoint vector. If omitted, breakpoints will be
computed from other options.
bkspread : :class:`float`, optional
Applies a scale factor to the difference between successive value of `bkpt`.
placed : :class:`numpy.ndarray`, optional
Precalculated breakpoint positions.
bkspace : :class:`float`, optional
Spacing of breakpoints in units of `x`.
nbkpts : :class:`int`, optional
Number of breakpoints to span `x` range; minimum is 2 (the endpoints).
everyn : :class:`int`, optional
Spacing of breakpoints in good pixels.
Attributes
----------
breakpoints
The processed breakpoints for the fit, based on the initialization options.
nord
The order of the B-spline. Default is 4, which is cubic.
npoly
Polynomial order to fit over 2nd variable, if supplied. If not
supplied the order is 1.
mask
A mask for `breakpoints`.
coeff
Internal attribute used by :meth:`~pydl.pydlutils.bspline.bspline.fit`.
icoeff
Internal attribute used by :meth:`~pydl.pydlutils.bspline.bspline.fit`.
xmin
Normalization minimum for the second variable in 2-dimensional fits.
xmax
Normalization maximum for the second variable in 2-dimensional fits.
funcname
For 2-dimensional fits, this is the function for the second variable.
The default is 'legendre'.
Notes
-----
Although this code has been tested (in the sense that it reproduces the
original idlutils code) and appears to work for a subset 1-dimensional cases,
it should not be used for 2 dimensions or more exotic 1-dimensional cases.
"""
def __init__(self, x, nord=4, npoly=1, bkpt=None, bkspread=1.0,
placed=None, bkspace=None, nbkpts=None, everyn=None):
"""Init creates an object whose attributes are similar to the
structure returned by the ``create_bsplineset()`` function.
"""
#
# Set the breakpoints.
#
if bkpt is None:
startx = x.min()
rangex = x.max() - startx
if placed is not None:
w = ((placed >= startx) &
(placed <= startx+rangex))
if w.sum() < 2:
bkpt = np.arange(2, dtype='f') * rangex + startx
else:
bkpt = placed[w]
elif bkspace is not None:
nbkpts = int(rangex/bkspace) + 1
if nbkpts < 2:
nbkpts = 2
tempbkspace = rangex/float(nbkpts-1)
bkpt = np.arange(nbkpts, dtype='f')*tempbkspace + startx
elif nbkpts is not None:
if nbkpts < 2:
nbkpts = 2
tempbkspace = rangex/float(nbkpts-1)
bkpt = np.arange(nbkpts, dtype='f') * tempbkspace + startx
elif everyn is not None:
nx = x.size
nbkpts = max(nx//everyn, 1)
if nbkpts == 1:
xspot = [0]
else:
xspot = int(nx/(nbkpts-1)) * np.arange(nbkpts, dtype='i4')
bkpt = x[xspot].astype('f')
else:
raise ValueError('No information for bkpts.')
imin = bkpt.argmin()
imax = bkpt.argmax()
if x.min() < bkpt[imin]:
warn('Lowest breakpoint does not cover lowest x value: changing.',
PydlutilsUserWarning)
bkpt[imin] = x.min()
if x.max() > bkpt[imax]:
warn('Highest breakpoint does not cover highest x value: changing.',
PydlutilsUserWarning)
bkpt[imax] = x.max()
nshortbkpt = bkpt.size
fullbkpt = bkpt.copy()
if nshortbkpt == 1:
bkspace = np.float32(bkspread)
else:
bkspace = (bkpt[1] - bkpt[0]) * np.float32(bkspread)
for i in np.arange(1, nord, dtype=np.float32):
fullbkpt = np.insert(fullbkpt, 0, bkpt[0]-bkspace*i)
fullbkpt = np.insert(fullbkpt, fullbkpt.shape[0],
bkpt[nshortbkpt-1] + bkspace*i)
#
# Set the attributes
#
nc = fullbkpt.size - nord
self.breakpoints = fullbkpt
self.nord = nord
self.npoly = npoly
self.mask = np.ones((fullbkpt.size,), dtype='bool')
if npoly > 1:
self.coeff = np.zeros((npoly, nc), dtype='d')
self.icoeff = np.zeros((npoly, nc), dtype='d')
else:
self.coeff = np.zeros((nc,), dtype='d')
self.icoeff = np.zeros((nc,), dtype='d')
self.xmin = 0.0
self.xmax = 1.0
self.funcname = 'legendre'
return
[docs]
def fit(self, xdata, ydata, invvar, x2=None):
"""Calculate a B-spline in the least-squares sense.
Fit is based on two variables: `xdata` which is sorted and spans a large range
where breakpoints are required `ydata` which can be described with a low order
polynomial.
Parameters
----------
xdata : :class:`numpy.ndarray`
Independent variable.
ydata : :class:`numpy.ndarray`
Dependent variable.
invvar : :class:`numpy.ndarray`
Inverse variance of `ydata`.
x2 : :class:`numpy.ndarray`, optional
Orthogonal dependent variable for 2d fits.
Returns
-------
:class:`tuple`
A tuple containing an integer error code, and the evaluation of the
b-spline at the input values. An error code of -2 is a failure,
-1 indicates dropped breakpoints, 0 is success, and positive
integers indicate ill-conditioned breakpoints.
"""
goodbk = self.mask[self.nord:]
nn = goodbk.sum()
if nn < self.nord:
yfit = np.zeros(ydata.shape, dtype='f')
return (-2, yfit)
nfull = nn * self.npoly
bw = self.npoly * self.nord
a1, lower, upper = self.action(xdata, x2=x2)
foo = np.tile(invvar, bw).reshape(bw, invvar.size).transpose()
a2 = a1 * foo
alpha = np.zeros((bw, nfull+bw), dtype='d')
beta = np.zeros((nfull+bw,), dtype='d')
bi = np.arange(bw, dtype='i4')
bo = np.arange(bw, dtype='i4')
for k in range(1, bw):
bi = np.append(bi, np.arange(bw-k, dtype='i4')+(bw+1)*k)
bo = np.append(bo, np.arange(bw-k, dtype='i4')+bw*k)
for k in range(nn-self.nord+1):
itop = k*self.npoly
ibottom = min(itop, nfull) + bw - 1
ict = upper[k] - lower[k] + 1
if ict > 0:
work = np.dot(a1[lower[k]:upper[k]+1, :].T, a2[lower[k]:upper[k]+1, :])
wb = np.dot(ydata[lower[k]:upper[k]+1], a2[lower[k]:upper[k]+1, :])
alpha.T.flat[bo+itop*bw] += work.flat[bi]
beta[itop:ibottom+1] += wb
min_influence = 1.0e-10 * invvar.sum() / nfull
errb = cholesky_band(alpha, mininf=min_influence)
if isinstance(errb[0], int) and errb[0] == -1:
a = errb[1]
else:
yfit, foo = self.value(xdata, x2=x2, action=a1, upper=upper, lower=lower)
return (self.maskpoints(errb[0]), yfit)
sol = cholesky_solve(a, beta)
if self.npoly > 1:
self.icoeff[:, goodbk] = np.array(a[0, 0:nfull].reshape(self.npoly, nn), dtype=a.dtype)
self.coeff[:, goodbk] = np.array(sol[0:nfull].reshape(self.npoly, nn), dtype=sol.dtype)
else:
self.icoeff[goodbk] = np.array(a[0, 0:nfull], dtype=a.dtype)
self.coeff[goodbk] = np.array(sol[0:nfull], dtype=sol.dtype)
yfit, foo = self.value(xdata, x2=x2, action=a1, upper=upper, lower=lower)
return (0, yfit)
[docs]
def action(self, x, x2=None):
"""Construct banded B-spline matrix, with dimensions [ndata, bandwidth].
Parameters
----------
x : :class:`numpy.ndarray`
Independent variable.
x2 : :class:`numpy.ndarray`, optional
Orthogonal dependent variable for 2d fits.
Returns
-------
:class:`tuple`
A tuple containing the B-spline action matrix; the 'lower' parameter,
a list of pixel positions, each corresponding to the first
occurence of position greater than breakpoint indx; and 'upper',
Same as lower, but denotes the upper pixel positions.
"""
nx = x.size
nbkpt = self.mask.sum()
if nbkpt < 2*self.nord:
return (-2, 0, 0)
n = nbkpt - self.nord
# gb = self.breakpoints[self.mask]
bw = self.npoly*self.nord
lower = np.zeros((n - self.nord + 1,), dtype='i4')
upper = np.zeros((n - self.nord + 1,), dtype='i4') - 1
indx = self.intrv(x)
bf1 = self.bsplvn(x, indx)
action = bf1
aa = uniq(indx, np.arange(indx.size, dtype='i4'))
upper[indx[aa]-self.nord+1] = aa
rindx = indx[::-1]
bb = uniq(rindx, np.arange(rindx.size, dtype='i4'))
lower[rindx[bb]-self.nord+1] = nx - bb - 1
if x2 is not None:
if x2.size != nx:
raise ValueError('Dimensions of x and x2 do not match.')
x2norm = 2.0 * (x2 - self.xmin) / (self.xmax - self.xmin) - 1.0
if self.funcname == 'poly':
temppoly = np.ones((nx, self.npoly), dtype='f')
for i in range(1, self.npoly):
temppoly[:, i] = temppoly[:, i-1] * x2norm
elif self.funcname == 'poly1':
temppoly = np.tile(x2norm, self.npoly).reshape(nx, self.npoly)
for i in range(1, self.npoly):
temppoly[:, i] = temppoly[:, i-1] * x2norm
elif self.funcname == 'chebyshev':
temppoly = fchebyshev(x2norm, self.npoly)
elif self.funcname == 'legendre':
temppoly = flegendre(x2norm, self.npoly)
else:
raise ValueError('Unknown value of funcname.')
action = np.zeros((nx, bw), dtype='d')
counter = -1
for ii in range(self.nord):
for jj in range(self.npoly):
counter += 1
action[:, counter] = bf1[:, ii]*temppoly[:, jj]
return (action, lower, upper)
[docs]
def intrv(self, x):
"""Find the segment between breakpoints which contain each value in the array `x`.
The minimum breakpoint is ``nbkptord - 1``, and the maximum
is ``nbkpt - nbkptord - 1``.
Parameters
----------
x : :class:`numpy.ndarray`
Data values, assumed to be monotonically increasing.
Returns
-------
:class:`numpy.ndarray`
Position of array elements with respect to breakpoints.
"""
gb = self.breakpoints[self.mask]
n = gb.size - self.nord
indx = np.zeros((x.size,), dtype='i4')
ileft = self.nord - 1
for i in range(x.size):
while x[i] > gb[ileft+1] and ileft < n - 1:
ileft += 1
indx[i] = ileft
return indx
[docs]
def bsplvn(self, x, ileft):
"""Calculates the value of all possibly nonzero B-splines at `x`
of a certain order.
Parameters
----------
x : :class:`numpy.ndarray`
Independent variable.
ileft : :class:`int`
Breakpoint segements that contain `x`.
Returns
-------
:class:`numpy.ndarray`
B-spline values.
"""
bkpt = self.breakpoints[self.mask]
vnikx = np.zeros((x.size, self.nord), dtype=x.dtype)
deltap = vnikx.copy()
deltam = vnikx.copy()
j = 0
vnikx[:, 0] = 1.0
while j < self.nord - 1:
ipj = ileft+j+1
deltap[:, j] = bkpt[ipj] - x
imj = ileft-j
deltam[:, j] = x - bkpt[imj]
vmprev = 0.0
for l in range(j+1):
vm = vnikx[:, l]/(deltap[:, l] + deltam[:, j-l])
vnikx[:, l] = vm*deltap[:, l] + vmprev
vmprev = vm*deltam[:, j-l]
j += 1
vnikx[:, j] = vmprev
return vnikx
[docs]
def value(self, x, x2=None, action=None, lower=None, upper=None):
"""Evaluate a B-spline at specified values.
Parameters
----------
x : :class:`numpy.ndarray`
Independent variable.
x2 : :class:`numpy.ndarray`, optional
Orthogonal dependent variable for 2d fits.
action : :class:`numpy.ndarray`, optional
Action matrix to use. If not supplied it is calculated.
lower : :class:`numpy.ndarray`, optional
If the action parameter is supplied, this parameter must also
be supplied.
upper : :class:`numpy.ndarray`, optional
If the action parameter is supplied, this parameter must also
be supplied.
Returns
-------
:class:`tuple`
A tuple containing the results of the bspline evaluation and a
mask indicating where the evaluation was good.
"""
xsort = x.argsort()
xwork = x[xsort]
if x2 is not None:
x2work = x2[xsort]
else:
x2work = None
if action is not None:
if lower is None or upper is None:
raise ValueError('Must specify lower and upper if action is set.')
else:
action, lower, upper = self.action(xwork, x2=x2work)
yfit = np.zeros(x.shape, dtype=x.dtype)
bw = self.npoly * self.nord
spot = np.arange(bw, dtype='i4')
goodbk = self.mask.nonzero()[0]
coeffbk = self.mask[self.nord:].nonzero()[0]
n = self.mask.sum() - self.nord
if self.npoly > 1:
goodcoeff = self.coeff[:, coeffbk]
else:
goodcoeff = self.coeff[coeffbk]
# maskthis = np.zeros(xwork.shape,dtype=xwork.dtype)
for i in range(n-self.nord+1):
ict = upper[i] - lower[i] + 1
if ict > 0:
yfit[lower[i]:upper[i]+1] = np.dot(
action[lower[i]:upper[i]+1, :], goodcoeff[i*self.npoly+spot])
yy = yfit.copy()
yy[xsort] = yfit
mask = np.ones(x.shape, dtype='bool')
gb = self.breakpoints[goodbk]
outside = ((x < gb[self.nord-1]) | (x > gb[n]))
if outside.any():
mask[outside] = False
hmm = ((np.diff(goodbk) > 2).nonzero())[0]
for jj in range(hmm.size):
inside = ((x >= self.breakpoints[goodbk[hmm[jj]]]) &
(x <= self.breakpoints[goodbk[hmm[jj]+1]-1]))
if inside.any():
mask[inside] = False
return (yy, mask)
[docs]
def maskpoints(self, err):
"""Perform simple logic of which breakpoints to mask.
Parameters
----------
err : :class:`numpy.ndarray`
The list of indexes returned by the cholesky routines.
Returns
-------
:class:`int`
An integer indicating the results of the masking. -1 indicates
that the error points were successfully masked. -2 indicates
failure; the calculation should be aborted.
Notes
-----
The mask attribute is modified, assuming it is possible to create the
mask.
"""
nbkpt = self.mask.sum()
if nbkpt <= 2*self.nord:
return -2
hmm = err[np.unique(err/self.npoly)]/self.npoly
n = nbkpt - self.nord
if np.any(hmm >= n):
return -2
test = np.zeros(nbkpt, dtype='bool')
for jj in range(-np.ceil(self.nord/2.0), self.nord/2.0):
foo = np.where((hmm+jj) > 0, hmm+jj, np.zeros(hmm.shape, dtype=hmm.dtype))
inside = np.where((foo+self.nord) < n-1, foo+self.nord, np.zeros(hmm.shape, dtype=hmm.dtype)+n-1)
test[inside] = True
if test.any():
reality = self.mask[test]
if self.mask[reality].any():
self.mask[reality] = False
return -1
else:
return -2
else:
return -2
[docs]
def cholesky_band(l, mininf=0.0):
"""Compute *lower* Cholesky decomposition of a banded matrix.
This function provides informative error messages to pass back to the
:class:`~pydl.pydlutils.bspline.bspline` machinery; the actual
computation is delegated to :func:`scipy.linalg.cholesky_banded`.
Parameters
----------
l : :class:`numpy.ndarray`
A matrix on which to perform the Cholesky decomposition. The
matrix must be in a special, *lower* form described in
:func:`scipy.linalg.cholesky_banded`. In addition, the input
must be padded. If the original, square matrix has size
:math:`N \\times N`, and the width of the band is :math:`b`,
`l` must be :math:`b \\times (N + b)`.
mininf : :class:`float`, optional
Entries in the `l` matrix are considered negative if they are less
than this value (default 0.0).
Returns
-------
:class:`tuple`
If problems were detected, the first item will be the index or
indexes where the problem was detected, and the second item will simply
be the input matrix. If no problems were detected, the first item
will be -1, and the second item will be the Cholesky decomposition.
"""
bw, nn = l.shape
n = nn - bw
negative = l[0, 0:n] <= mininf
if negative.any() or not np.all(np.isfinite(l)):
warn('Bad entries: ' + str(negative.nonzero()[0]), PydlutilsUserWarning)
return (negative.nonzero()[0], l)
try:
lower = cholesky_banded(l[:, 0:n], lower=True)
except LinAlgError:
#
# Figure out where the error is.
#
lower = l.copy()
kn = bw - 1
spot = np.arange(kn, dtype='i4') + 1
for j in range(n):
lower[0, j] = np.sqrt(lower[0, j])
lower[spot, j] /= lower[0, j]
x = lower[spot, j]
if not np.all(np.isfinite(x)):
warn('NaN found in cholesky_band.', PydlutilsUserWarning)
return (j, l)
#
# Restore padding.
#
L = np.zeros(l.shape, dtype=l.dtype)
L[:, 0:n] = lower
return (-1, L)
[docs]
def cholesky_solve(a, bb):
"""Solve the equation :math:`A x = b` where `a` is a *lower*
Cholesky-banded matrix.
In the :class:`~pydl.pydlutils.bspline.bspline` machinery, `a` needs to
be padded. This function should only used with the output of
:func:`~pydl.pydlutils.bspline.cholesky_band`, to ensure the proper
padding on `a`. Otherwise the computation is delegated to
:func:`scipy.linalg.cho_solve_banded`.
Parameters
----------
a : :class:`numpy.ndarray`
*Lower* Cholesky decomposition of :math:`A` in :math:`A x = b`.
bb : :class:`numpy.ndarray`
:math:`b` in :math:`A x = b`.
Returns
-------
:class:`numpy.ndarray`
The solution, padded to be the same shape as `bb`.
"""
bw = a.shape[0]
n = bb.shape[0] - bw
x = np.zeros(bb.shape, dtype=bb.dtype)
x[0:n] = cho_solve_banded((a[:, 0:n], True), bb[0:n])
return x
[docs]
def iterfit(xdata, ydata, invvar=None, upper=5, lower=5, x2=None,
maxiter=10, **kwargs):
"""Iteratively fit a B-spline set to data, with rejection.
Additional keyword parameters are passed to
:class:`~pydl.pydlutils.bspline.bspline`.
Parameters
----------
xdata : :class:`numpy.ndarray`
Independent variable.
ydata : :class:`numpy.ndarray`
Dependent variable.
invvar : :class:`numpy.ndarray`, optional
Inverse variance of `ydata`. If not set, it will be calculated based
on the standard deviation.
upper : :class:`int` or :class:`float`, optional
Upper rejection threshold in units of sigma, defaults to 5 sigma.
lower : :class:`int` or :class:`float`, optional
Lower rejection threshold in units of sigma, defaults to 5 sigma.
x2 : :class:`numpy.ndarray`, optional
Orthogonal dependent variable for 2d fits.
maxiter : :class:`int`, optional
Maximum number of rejection iterations, default 10. Set this to
zero to disable rejection.
Returns
-------
:class:`tuple`
A tuple containing the fitted bspline object and an output mask.
"""
nx = xdata.size
if ydata.size != nx:
raise ValueError('Dimensions of xdata and ydata do not agree.')
if invvar is not None:
if invvar.size != nx:
raise ValueError('Dimensions of xdata and invvar do not agree.')
else:
#
# This correction to the variance makes it the same
# as IDL's variance()
#
var = ydata.var()*(float(nx)/float(nx-1))
if var == 0:
var = 1.0
invvar = np.ones(ydata.shape, dtype=ydata.dtype)/var
if x2 is not None:
if x2.size != nx:
raise ValueError('Dimensions of xdata and x2 do not agree.')
yfit = np.zeros(ydata.shape, dtype=ydata.dtype)
if invvar.size == 1:
outmask = True
else:
outmask = np.ones(invvar.shape, dtype='bool')
xsort = xdata.argsort()
maskwork = (outmask & (invvar > 0))[xsort]
if 'oldset' in kwargs:
sset = kwargs['oldset']
sset.mask = True
sset.coeff = 0
else:
if not maskwork.any():
raise ValueError('No valid data points.')
if 'fullbkpt' in kwargs:
# fullbkpt = kwargs['fullbkpt']
raise ValueError('Input via fullbkpt is not supported!')
else:
sset = bspline(xdata[xsort[maskwork]], **kwargs)
if maskwork.sum() < sset.nord:
warn('Number of good data points fewer than nord.',
PydlutilsUserWarning)
return (sset, outmask)
if x2 is not None:
if 'xmin' in kwargs:
xmin = kwargs['xmin']
else:
xmin = x2.min()
if 'xmax' in kwargs:
xmax = kwargs['xmax']
else:
xmax = x2.max()
if xmin == xmax:
xmax = xmin + 1
sset.xmin = xmin
sset.xmax = xmax
if 'funcname' in kwargs:
sset.funcname = kwargs['funcname']
xwork = xdata[xsort]
ywork = ydata[xsort]
invwork = invvar[xsort]
if x2 is not None:
warn('2D bspline fits may be buggy and will be fully removed in the future.',
DeprecationWarning)
x2work = x2[xsort]
else:
x2work = None
iiter = 0
error = 0
qdone = -1
while (error != 0 or qdone == -1) and iiter <= maxiter:
goodbk = sset.mask.nonzero()[0]
if maskwork.sum() <= 1 or not sset.mask.any():
sset.coeff = 0
iiter = maxiter + 1
else:
if 'requiren' in kwargs:
i = 0
while xwork[i] < sset.breakpoints[goodbk[sset.nord]] and i < nx-1:
i += 1
ct = 0
for ileft in range(sset.nord, sset.mask.sum()-sset.nord+1):
while (xwork[i] >= sset.breakpoints[goodbk[ileft]] and
xwork[i] < sset.breakpoints[goodbk[ileft+1]] and
i < nx-1):
ct += invwork[i]*maskwork[i] > 0
i += 1
if ct >= kwargs['requiren']:
ct = 0
else:
sset.mask[goodbk[ileft]] = False
error, yfit = sset.fit(xwork, ywork, invwork*maskwork,
x2=x2work)
iiter += 1
inmask = maskwork
if error == -2:
return (sset, outmask)
elif error == 0:
maskwork, qdone = djs_reject(ywork, yfit, invvar=invwork,
inmask=inmask, outmask=maskwork,
upper=upper, lower=lower)
else:
pass
outmask[xsort] = maskwork
temp = yfit
yfit[xsort] = temp
return (sset, outmask)