def log_likelihood(self, p):
r"""Returns the log of the likelihood for the stored data given
parameters ``p``. The likelihood is computed in time
proportional to :math:`\mathcal{O}(N)`, where :math:`N` is the
number of data samples.
"""
p = self.to_params(p)
alphas, betas = self._alphas_betas(p)
bc = self._banded_covariance(p)
ys = self.ys.copy() - p['mu']
ys[1:] = ys[1:] - alphas*ys[0:-1]
dts = self.ts.reshape((-1, 1)) - self.ts.reshape((1, -1))
tau = np.exp(p['lntau'])
nu = self._inv_logit(p['logitnu'])
sigma = np.exp(p['lnsigma'])
full_cov = sigma*sigma*((1-nu)*np.exp(-np.abs(dts)/tau) + nu)
llow = sl.cholesky_banded(bc, lower=True)
logdet = np.sum(np.log(llow[0, :]))
return -0.5*self.ts.shape[0]*np.log(2.0*np.pi) - logdet - 0.5*np.dot(ys, sl.cho_solve_banded((llow, True), ys))
def posterior_cov(self):
r"""
Posterior covariance of the states conditional on the data
Notes
-----
**Warning**: the matrix computed when accessing this property can be
extremely large: it is shaped `(nobs * k_states, nobs * k_states)`. In
most cases, it is better to use the `posterior_cov_inv_chol_sparse`
property if possible, which holds in sparse diagonal banded storage
the Cholesky factor of the inverse of the posterior covariance matrix.
.. math::
Var[\alpha \mid Y^n ]
This posterior covariance matrix is *not* identical to the
`smoothed_state_cov` attribute produced by the Kalman smoother, because
it additionally contains all cross-covariance terms. Instead,
`smoothed_state_cov` contains the `(k_states, k_states)` block
diagonal entries of this posterior covariance matrix.
"""
if self._posterior_cov is None:
from scipy.linalg import cho_solve_banded
inv_chol = self.posterior_cov_inv_chol_sparse
self._posterior_cov = cho_solve_banded((inv_chol, True),
np.eye(inv_chol.shape[1]))
return self._posterior_cov
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
def _step(self):
self.x = cho_solve_banded(
(self.c, False),
self.y + sum(self.ρ[i] * self.D[i].T @ (self.α[i] + self.u[i])
for i in range(self.r)),
check_finite=False,
)
for i in range(self.r):
Dx = self.D[i] @ self.x
for j in range(self.x.shape[1]):
self.α[i][:, j] = ptv.tv1_1d(Dx[:, j] - self.u[i][:, j],
self.λ[i] / self.ρ[i])
self.u[i] += self.α[i] - Dx
def test_lower_real(self):
# Symmetric positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0], [1.0, 4.0, 0.5, 0.0], [0.0, 0.5, 4.0, 0.2], [0.0, 0.0, 0.2, 4.0]])
# Banded storage form of `a`.
ab = array([[4.0, 4.0, 4.0, 4.0], [1.0, 0.5, 0.2, -1.0]])
c = cholesky_banded(ab, lower=True)
lfac = zeros_like(a)
lfac[list(range(4)), list(range(4))] = c[0]
lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
assert_array_almost_equal(a, dot(lfac, lfac.T))
b = array([0.0, 0.5, 4.2, 4.2])
x = cho_solve_banded((c, True), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_lower_complex(self):
# Hermitian positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0], [1.0, 4.0, 0.5, 0.0], [0.0, 0.5, 4.0, -0.2j], [0.0, 0.0, 0.2j, 4.0]])
# Banded storage form of `a`.
ab = array([[4.0, 4.0, 4.0, 4.0], [1.0, 0.5, 0.2j, -1.0]])
c = cholesky_banded(ab, lower=True)
lfac = zeros_like(a)
lfac[list(range(4)), list(range(4))] = c[0]
lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
assert_array_almost_equal(a, dot(lfac, lfac.conj().T))
b = array([0.0, 0.5j, 3.8j, 3.8])
x = cho_solve_banded((c, True), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0j, 1.0])
def Binvprod(self, x):
"""Computes the product B^(-1) * x via the Cholesky factor.
Using the band Cholesky factorization of B, the matrix-vector
product y is computed using scipy.linalg.cho_solve_banded to
solve B * y = x.
Argument:
x: Vector to multiply.
Returns:
Matrix-vector product B^(-1) * x.
"""
return la.cho_solve_banded((self.sqrtBdata, True), x)
def test_upper_real(self):
# Symmetric positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0], [1.0, 4.0, 0.5, 0.0], [0.0, 0.5, 4.0, 0.2], [0.0, 0.0, 0.2, 4.0]])
# Banded storage form of `a`.
ab = array([[-1.0, 1.0, 0.5, 0.2], [4.0, 4.0, 4.0, 4.0]])
c = cholesky_banded(ab, lower=False)
ufac = zeros_like(a)
ufac[list(range(4)), list(range(4))] = c[-1]
ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
assert_array_almost_equal(a, dot(ufac.T, ufac))
b = array([0.0, 0.5, 4.2, 4.2])
x = cho_solve_banded((c, False), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_lower_complex(self):
# Hermitian positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0], [1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, -0.2j], [0.0, 0.0, 0.2j, 4.0]])
# Banded storage form of `a`.
ab = array([[4.0, 4.0, 4.0, 4.0], [1.0, 0.5, 0.2j, -1.0]])
c = cholesky_banded(ab, lower=True)
lfac = zeros_like(a)
lfac[list(range(4)), list(range(4))] = c[0]
lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
assert_array_almost_equal(a, dot(lfac, lfac.conj().T))
b = array([0.0, 0.5j, 3.8j, 3.8])
x = cho_solve_banded((c, True), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0j, 1.0])
def test_upper_complex(self):
# Hermitian positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0], [1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, -0.2j], [0.0, 0.0, 0.2j, 4.0]])
# Banded storage form of `a`.
ab = array([[-1.0, 1.0, 0.5, -0.2j], [4.0, 4.0, 4.0, 4.0]])
c = cholesky_banded(ab, lower=False)
ufac = zeros_like(a)
ufac[range(4), range(4)] = c[-1]
ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
assert_array_almost_equal(a, dot(ufac.conj().T, ufac))
b = array([0.0, 0.5, 4.0 - 0.2j, 0.2j + 4.0])
x = cho_solve_banded((c, False), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def Qinvprod(self, x, i):
"""Product of the inverse of Q[i] with a vector x.
The model error covariance matrix Q[i] is assumed to be a
band matrix accessible from sqrtQdata[i], which is stored in
lower form.
Arguments:
x: The vector to multiply.
i: Index indicating which model error covariance to use.
Returns:
Product of the inverse of Q[i] with x.
"""
return la.cho_solve_banded((self.sqrtQdata[i], True), x)
def test_upper_real(self):
# Symmetric positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0], [1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, 0.2], [0.0, 0.0, 0.2, 4.0]])
# Banded storage form of `a`.
ab = array([[-1.0, 1.0, 0.5, 0.2], [4.0, 4.0, 4.0, 4.0]])
c = cholesky_banded(ab, lower=False)
ufac = zeros_like(a)
ufac[list(range(4)), list(range(4))] = c[-1]
ufac[(0, 1, 2), (1, 2, 3)] = c[0, 1:]
assert_array_almost_equal(a, dot(ufac.T, ufac))
b = array([0.0, 0.5, 4.2, 4.2])
x = cho_solve_banded((c, False), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_lower_real(self):
# Symmetric positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0], [1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, 0.2], [0.0, 0.0, 0.2, 4.0]])
# Banded storage form of `a`.
ab = array([[4.0, 4.0, 4.0, 4.0], [1.0, 0.5, 0.2, -1.0]])
c = cholesky_banded(ab, lower=True)
lfac = zeros_like(a)
lfac[list(range(4)), list(range(4))] = c[0]
lfac[(1, 2, 3), (0, 1, 2)] = c[1, :3]
assert_array_almost_equal(a, dot(lfac, lfac.T))
b = array([0.0, 0.5, 4.2, 4.2])
x = cho_solve_banded((c, True), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_upper_complex(self):
# Hermitian positive definite banded matrix `a`
a = array([[4.0, 1.0, 0.0, 0.0],
[1.0, 4.0, 0.5, 0.0],
[0.0, 0.5, 4.0, -0.2j],
[0.0, 0.0, 0.2j, 4.0]])
# Banded storage form of `a`.
ab = array([[-1.0, 1.0, 0.5, -0.2j],
[4.0, 4.0, 4.0, 4.0]])
c = cholesky_banded(ab, lower=False)
ufac = zeros_like(a)
ufac[range(4),range(4)] = c[-1]
ufac[(0,1,2),(1,2,3)] = c[0,1:]
assert_array_almost_equal(a, dot(ufac.conj().T, ufac))
b = array([0.0, 0.5, 4.0-0.2j, 0.2j + 4.0])
x = cho_solve_banded((c, False), b)
assert_array_almost_equal(x, [0.0, 0.0, 1.0, 1.0])
def test_posterior_cov(self):
# Test the values from the Cython results
inv_chol = np.array(self._sim_cfa.posterior_cov_inv_chol, copy=True)
actual = cho_solve_banded((inv_chol, True), np.eye(inv_chol.shape[1]))
for t in range(self.mod.nobs):
tm = t * self.mod.k_states
t1m = tm + self.mod.k_states
assert_allclose(actual[tm:t1m, tm:t1m],
self.res.smoothed_state_cov[..., t],
atol=self.cov_atol)
# Test the values from the CFASimulationSmoother wrapper results
actual = self.sim_cfa.posterior_cov
for t in range(self.mod.nobs):
tm = t * self.mod.k_states
t1m = tm + self.mod.k_states
assert_allclose(actual[tm:t1m, tm:t1m],
self.res.smoothed_state_cov[..., t],
atol=self.cov_atol)
def drawFrame(i):
#print ''
print 'i:',i
gridT = np.reshape(T[0], (Nx, Ny))
#print 'gridT:', shape(gridT)
#print gridT
#print 'x:',shape(x)
#print 'y:',shape(y)
z = im.zoom(gridT, szmul, order=3)
extent = (xi, xf, yf, yi)
ax.imshow(z, extent=extent, cmap=paleta, interpolation="nearest", vmin=0.0, vmax=vmax)
c1 = ax.contour(x,y,z, levels=np.arange(0.0, vmax, vmax/50.0), colors='k', vmin=0.0, vmax=vmax)
c2 = ax.contourf(x,y,z, levels=np.arange(0.0, vmax, vmax/50.0), cmap=paleta, vmin=0.0, vmax=vmax)
# calculando os valores para o próximo frame da animação
vecB = calcNextTimeResultVec(i)
#print vecB
#vecR = la.solve(matA, vecB)
#print 'choFacA:', choFacA
vecR = la.cho_solve_banded((choA, False), vecB)
#vecR = la.cho_solve(choFacA, vecB)
vecR = list(vecR.flat)
#print 'vecR:', vecR
T[0] = vecR
def test_cho_solve_banded(self):
x = array([[0, -1, -1], [2, 2, 2]])
xcho = cholesky_banded(x)
assert_no_overwrite(lambda b: cho_solve_banded((xcho, False), b),
[(3,)])
def make_lsq_spline(x, y, t, k=3, w=None, axis=0, check_finite=True):
r"""Compute the (coefficients of) an LSQ B-spline.
The result is a linear combination
.. math::
S(x) = \sum_j c_j B_j(x; t)
of the B-spline basis elements, :math:`B_j(x; t)`, which minimizes
.. math::
\sum_{j} \left( w_j \times (S(x_j) - y_j) \right)^2
Parameters
----------
x : array_like, shape (m,)
Abscissas.
y : array_like, shape (m, ...)
Ordinates.
t : array_like, shape (n + k + 1,).
Knots.
Knots and data points must satisfy Schoenberg-Whitney conditions.
k : int, optional
B-spline degree. Default is cubic, k=3.
w : array_like, shape (n,), optional
Weights for spline fitting. Must be positive. If ``None``,
then weights are all equal.
Default is ``None``.
axis : int, optional
Interpolation axis. Default is zero.
check_finite : bool, optional
Whether to check that the input arrays contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default is True.
Returns
-------
b : a BSpline object of the degree `k` with knots `t`.
Notes
-----
The number of data points must be larger than the spline degree `k`.
Knots `t` must satisfy the Schoenberg-Whitney conditions,
i.e., there must be a subset of data points ``x[j]`` such that
``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``.
Examples
--------
Generate some noisy data:
>>> x = np.linspace(-3, 3, 50)
>>> y = np.exp(-x**2) + 0.1 * np.random.randn(50)
Now fit a smoothing cubic spline with a pre-defined internal knots.
Here we make the knot vector (k+1)-regular by adding boundary knots:
>>> from scipy.interpolate import make_lsq_spline, BSpline
>>> t = [-1, 0, 1]
>>> k = 3
>>> t = np.r_[(x[0],)*(k+1),
... t,
... (x[-1],)*(k+1)]
>>> spl = make_lsq_spline(x, y, t, k)
For comparison, we also construct an interpolating spline for the same
set of data:
>>> from scipy.interpolate import make_interp_spline
>>> spl_i = make_interp_spline(x, y)
Plot both:
>>> import matplotlib.pyplot as plt
>>> xs = np.linspace(-3, 3, 100)
>>> plt.plot(x, y, 'ro', ms=5)
>>> plt.plot(xs, spl(xs), 'g-', lw=3, label='LSQ spline')
>>> plt.plot(xs, spl_i(xs), 'b-', lw=3, alpha=0.7, label='interp spline')
>>> plt.legend(loc='best')
>>> plt.show()
**NaN handling**: If the input arrays contain ``nan`` values, the result is
not useful since the underlying spline fitting routines cannot deal with
``nan``. A workaround is to use zero weights for not-a-number data points:
>>> y[8] = np.nan
>>> w = np.isnan(y)
>>> y[w] = 0.
>>> tck = make_lsq_spline(x, y, t, w=~w)
Notice the need to replace a ``nan`` by a numerical value (precise value
does not matter as long as the corresponding weight is zero.)
See Also
--------
BSpline : base class representing the B-spline objects
make_interp_spline : a similar factory function for interpolating splines
LSQUnivariateSpline : a FITPACK-based spline fitting routine
splrep : a FITPACK-based fitting routine
"""
x = _as_float_array(x, check_finite)
y = _as_float_array(y, check_finite)
t = _as_float_array(t, check_finite)
if w is not None:
w = _as_float_array(w, check_finite)
else:
w = np.ones_like(x)
k = operator.index(k)
if not -y.ndim <= axis < y.ndim:
raise ValueError("axis {} is out of bounds".format(axis))
if axis < 0:
axis += y.ndim
y = np.rollaxis(y, axis) # now internally interp axis is zero
if x.ndim != 1 or np.any(x[1:] - x[:-1] <= 0):
raise ValueError("Expect x to be a 1-D sorted array_like.")
if x.shape[0] < k+1:
raise ValueError("Need more x points.")
if k < 0:
raise ValueError("Expect non-negative k.")
if t.ndim != 1 or np.any(t[1:] - t[:-1] < 0):
raise ValueError("Expect t to be a 1-D sorted array_like.")
if x.size != y.shape[0]:
raise ValueError('x & y are incompatible.')
if k > 0 and np.any((x < t[k]) | (x > t[-k])):
raise ValueError('Out of bounds w/ x = %s.' % x)
if x.size != w.size:
raise ValueError('Incompatible weights.')
# number of coefficients
n = t.size - k - 1
# construct A.T @ A and rhs with A the collocation matrix, and
# rhs = A.T @ y for solving the LSQ problem ``A.T @ A @ c = A.T @ y``
lower = True
extradim = prod(y.shape[1:])
ab = np.zeros((k+1, n), dtype=np.float_, order='F')
rhs = np.zeros((n, extradim), dtype=y.dtype, order='F')
_bspl._norm_eq_lsq(x, t, k,
y.reshape(-1, extradim),
w,
ab, rhs)
rhs = rhs.reshape((n,) + y.shape[1:])
# have observation matrix & rhs, can solve the LSQ problem
cho_decomp = cholesky_banded(ab, overwrite_ab=True, lower=lower,
check_finite=check_finite)
c = cho_solve_banded((cho_decomp, lower), rhs, overwrite_b=True,
check_finite=check_finite)
c = np.ascontiguousarray(c)
return BSpline.construct_fast(t, c, k, axis=axis)
def l2appr(n_dim, K, vec_timestamps, vec_obs, numObs, knots, weight):
# CONSTRUCTS THE (WEIGHTED DISCRETE) L2-APPROXIMATION BY SPLINES OF ORDER
# K WITH KNOT SEQUENCE T(1), .., T(N+K) TO GIVEN DATA POINTS
# (TAU(I),GTAU(I)), I=1,...,NTAU. THE B-SPLINE COEFFICIENTS
# B C O E F OF THE APPROXIMATING SPLINE ARE DETERMINED FROM THE
# NORMAL EQUATIONS USING CHOLESKY'S METHOD.
# n_dim : dimension of the B-spline space
# K : order
# numObs : number of data points
# number of knots = n_dim + K
bcoeff = np.asarray([0.0 for i in range(n_dim)])
biatx = np.asarray([0.0 for i in range(K)])
Q = np.zeros((K, n_dim))
left_py = K - 1
leftmk_py = -1 # left - K
# Note that left cors. to the index of the knot just left of x, by Dr. Watson's notes indices.
# The knots t run from t_1 to t_{n_dim+K}.
# The python vector t will, however, go from 0 to n_dim+K-1.
# t_1 is stored at location t[0].
# t_2 is stored at location t[1].
# t_K is stored at location t[K-1]
# t[i] is stored at location t[i-1].
# If we identify an index 'left' st. t[left] < x < t[left+1], then
# x is actually in between t_{left} and t_{left+1}
for ll in range(0, numObs):
# print 'll =', ll, 'x_ll =', vec_timestamps[ll],
# print 'knots[left_py]', knots[left_py], 'vec_timestamps[ll] >= knots[left_py]', vec_timestamps[ll] >= knots[left_py]
# corner case
if (left_py == n_dim - 1) or (vec_timestamps[ll] < knots[left_py + 1]):
# we want: vec_timestamps(ll) \in (knots(left_fort) , knots(left_fort+1))
# i.e., vec_timestamps(ll) \in (knots(left_py - 1) , knots(left_py))
# call bsplvb directly
left_fort = left_py + 1
biatx[0:K + 1] = bsplvb(knots, K, 1, vec_timestamps[ll], left_fort)
else:
# Locate left st. vec_timestamps(ll) \in (knots(left_fort) , knots(left_fort+1))
# i.e., vec_timestamps(ll) \in (knots(left_py - 1) , knots(left_py))
while (vec_timestamps[ll] >= knots[left_py + 1]):
left_py += 1
leftmk_py += 1
# now call bsplvb
left_fort = left_py + 1
biatx[0:K + 1] = bsplvb(knots, K, 1, vec_timestamps[ll], left_fort)
# BIATX(MM-1) (originally, BIATX(MM) in Fortran)
# CONTAINS THE VALUE OF B(LEFT-K+MM), MM = 1,2,...,K, AT TAU(LL).
# HENCE, WITH DW := BIATX(MM-1)*WEIGHT(LL), (originally, BIATX(MM)*WEIGHT(LL) in Fortran)
# THE NUMBER DW*GTAU(LL)
# IS A SUMMAND IN THE INNER PRODUCT
# (B(LEFT-K+MM),G) WHICH GOES INTO BCOEF(LEFT-K+MM-1) (originally, BCOEF(LEFT-K+MM) in Fortran)
# AND THE NUMBER BIATX(JJ)*DW IS A SUMMAND IN THE INNER PRODUCT
# (B(LEFT-K+JJ),B(LEFT-K+MM)), INTO Q(JJ-MM+1,LEFT-K+MM)
# SINCE (LEFT-K+JJ) - (LEFT-K+MM) + 1 = JJ - MM + 1 .
# Solving: min || Ax - b ||
# So, we solve: A^tA x = A^t b
#
# Entries of A^t A:
#
# ---
# \
# [AtA]_{ij} = B_i(x_l) * B_j(x_j) * w(l)
# /
# ---
# i,j=
#
# AtA is a symmetric banded matrix due to local support of B-splines.
# No point storing (or even calculating) all entries.
# Only nonzero entries are calculated.
#
# Entries of A^t b:
#
for mm in range(1, K + 1):
dw = biatx[mm - 1] * weight[ll]
j = (leftmk_py + 1) + mm
# The rhs < A^t, b >
bcoeff[j - 1] = dw * vec_obs[ll] + bcoeff[j - 1]
i = 1
for jj in range(mm, K + 1):
Q[i - 1, j - 1] = biatx[jj - 1] * dw + Q[
i - 1, j - 1] # j is fixed for currnt x_ll and current mm.
# only i and jj are incrementing.
# Q[*, j-1] is getting filled.
# Then mm increments. j will also increment.
# Q[*, j_old+1] will get filled.
#
# Then next x_ll comes in.
# Suppose x_ll is in the same knot interval.
# mm will still run from 1 to K.
# j, by formula, will again take same values as previous loop.
# So same columns of Q will get updated.
# i will also take same values as previous ll-loop.
# So, same rows of Q will update.
# Essentially, for fixed [left, left+1), same entries
# of Q will update.
#
# Suppose x_ll is in a different interval.
# i.e., left is different, say, left_new = left_old+1
# Then, j takes 1 higher values than prevoius ll-loops.
# Meaning, a new (1 up) set of columns is filled.
# mm and K are always same. So each row is filled again,
# just in a different column.
# That's the story of 'matrix filling' here.
# Figure out the actual working of bandedness etc. -- TODO.
i += 1
# Get the CHOLESKY FACTORIZATION FOR banded matrix Q, THEN USE
# IT TO SOLVE THE NORMAL EQUATIONS
# C*X = BCOEF
# FOR X , AND STORE X IN BCOEF
# print 'In l2appr'
# print 'n_dim =', n_dim
# with open("ltr_python_output.csv", "a") as fh:
# fh.write('writing Q:' + '\n')
# for i in range(K):
# fh.write(' '.join([str(Q[i, col]) for col in range(n_dim)] ) + '\n ')
# fh.close()
cb = la.cholesky_banded(Q,
overwrite_ab=False,
lower=True,
check_finite=True)
bcoeff = la.cho_solve_banded((cb, True),
bcoeff,
overwrite_b=True,
check_finite=True)
return bcoeff
def make_lsq_spline(x, y, t, k=3, w=None, axis=0, check_finite=True):
r"""Compute the (coefficients of) an LSQ B-spline.
The result is a linear combination
.. math::
S(x) = \sum_j c_j B_j(x; t)
of the B-spline basis elements, :math:`B_j(x; t)`, which minimizes
.. math::
\sum_{j} \left( w_j \times (S(x_j) - y_j) \right)^2
Parameters
----------
x : array_like, shape (m,)
Abscissas.
y : array_like, shape (m, ...)
Ordinates.
t : array_like, shape (n + k + 1,).
Knots.
Knots and data points must satisfy Schoenberg-Whitney conditions.
k : int, optional
B-spline degree. Default is cubic, k=3.
w : array_like, shape (n,), optional
Weights for spline fitting. Must be positive. If ``None``,
then weights are all equal.
Default is ``None``.
axis : int, optional
Interpolation axis. Default is zero.
check_finite : bool, optional
Whether to check that the input arrays contain only finite numbers.
Disabling may give a performance gain, but may result in problems
(crashes, non-termination) if the inputs do contain infinities or NaNs.
Default is True.
Returns
-------
b : a BSpline object of the degree `k` with knots `t`.
Notes
-----
The number of data points must be larger than the spline degree `k`.
Knots `t` must satisfy the Schoenberg-Whitney conditions,
i.e., there must be a subset of data points ``x[j]`` such that
``t[j] < x[j] < t[j+k+1]``, for ``j=0, 1,...,n-k-2``.
Examples
--------
Generate some noisy data:
>>> x = np.linspace(-3, 3, 50)
>>> y = np.exp(-x**2) + 0.1 * np.random.randn(50)
Now fit a smoothing cubic spline with a pre-defined internal knots.
Here we make the knot vector (k+1)-regular by adding boundary knots:
>>> from scipy.interpolate import make_lsq_spline, BSpline
>>> t = [-1, 0, 1]
>>> k = 3
>>> t = np.r_[(x[0],)*(k+1),
... t,
... (x[-1],)*(k+1)]
>>> spl = make_lsq_spline(x, y, t, k)
For comparison, we also construct an interpolating spline for the same
set of data:
>>> from scipy.interpolate import make_interp_spline
>>> spl_i = make_interp_spline(x, y)
Plot both:
>>> import matplotlib.pyplot as plt
>>> xs = np.linspace(-3, 3, 100)
>>> plt.plot(x, y, 'ro', ms=5)
>>> plt.plot(xs, spl(xs), 'g-', lw=3, label='LSQ spline')
>>> plt.plot(xs, spl_i(xs), 'b-', lw=3, alpha=0.7, label='interp spline')
>>> plt.legend(loc='best')
>>> plt.show()
**NaN handling**: If the input arrays contain ``nan`` values, the result is
not useful since the underlying spline fitting routines cannot deal with
``nan``. A workaround is to use zero weights for not-a-number data points:
>>> y[8] = np.nan
>>> w = np.isnan(y)
>>> y[w] = 0.
>>> tck = make_lsq_spline(x, y, t, w=~w)
Notice the need to replace a ``nan`` by a numerical value (precise value
does not matter as long as the corresponding weight is zero.)
See Also
--------
BSpline : base class representing the B-spline objects
make_interp_spline : a similar factory function for interpolating splines
LSQUnivariateSpline : a FITPACK-based spline fitting routine
splrep : a FITPACK-based fitting routine
"""
x = _as_float_array(x, check_finite)
y = _as_float_array(y, check_finite)
t = _as_float_array(t, check_finite)
if w is not None:
w = _as_float_array(w, check_finite)
else:
w = np.ones_like(x)
k = operator.index(k)
axis = normalize_axis_index(axis, y.ndim)
y = np.rollaxis(y, axis) # now internally interp axis is zero
if x.ndim != 1 or np.any(x[1:] - x[:-1] <= 0):
raise ValueError("Expect x to be a 1-D sorted array_like.")
if x.shape[0] < k + 1:
raise ValueError("Need more x points.")
if k < 0:
raise ValueError("Expect non-negative k.")
if t.ndim != 1 or np.any(t[1:] - t[:-1] < 0):
raise ValueError("Expect t to be a 1-D sorted array_like.")
if x.size != y.shape[0]:
raise ValueError('x & y are incompatible.')
if k > 0 and np.any((x < t[k]) | (x > t[-k])):
raise ValueError('Out of bounds w/ x = %s.' % x)
if x.size != w.size:
raise ValueError('Incompatible weights.')
# number of coefficients
n = t.size - k - 1
# construct A.T @ A and rhs with A the collocation matrix, and
# rhs = A.T @ y for solving the LSQ problem ``A.T @ A @ c = A.T @ y``
lower = True
extradim = prod(y.shape[1:])
ab = np.zeros((k + 1, n), dtype=np.float_, order='F')
rhs = np.zeros((n, extradim), dtype=y.dtype, order='F')
_bspl._norm_eq_lsq(x, t, k, y.reshape(-1, extradim), w, ab, rhs)
rhs = rhs.reshape((n, ) + y.shape[1:])
# have observation matrix & rhs, can solve the LSQ problem
cho_decomp = cholesky_banded(ab,
overwrite_ab=True,
lower=lower,
check_finite=check_finite)
c = cho_solve_banded((cho_decomp, lower),
rhs,
overwrite_b=True,
check_finite=check_finite)
c = np.ascontiguousarray(c)
return BSpline.construct_fast(t, c, k, axis=axis)
def test_cho_solve_banded(self):
x = array([[0, -1, -1], [2, 2, 2]])
xcho = cholesky_banded(x)
assert_no_overwrite(lambda b: cho_solve_banded((xcho, False), b),
[(3,)])
def linear_map(self, x):
if not self.banded:
return cho_solve(self._cholesky, x)
else:
return cho_solve_banded(self._cholesky, x)
def linear_map(self, x):
if not self.banded:
return cho_solve(self._cholesky, x)
else:
return cho_solve_banded(self._cholesky, x)