def test_dot_2args():
from numpy.core import dot
a = np.array([[1, 2], [3, 4]], dtype=float)
b = np.array([[1, 0], [1, 1]], dtype=float)
c = np.array([[3, 2], [7, 4]], dtype=float)
d = dot(a, b)
assert_allclose(c, d)
def test_dot_2args():
from numpy.core import dot
a = np.array([[1, 2], [3, 4]], dtype=float)
b = np.array([[1, 0], [1, 1]], dtype=float)
c = np.array([[3, 2], [7, 4]], dtype=float)
d = dot(a, b)
assert_allclose(c, d)
def pinv(a, rcond=1e-15):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate the generalized inverse of a matrix using its
singular-value decomposition (SVD) and including all
`large` singular values.
Parameters
----------
a : array_like (M, N)
Matrix to be pseudo-inverted.
rcond : float
Cutoff for `small` singular values.
Singular values smaller than rcond*largest_singular_value are
considered zero.
Returns
-------
B : ndarray (N, M)
The pseudo-inverse of `a`. If `a` is an np.matrix instance, then so
is `B`.
Raises
------
LinAlgError
In case SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = np.linalg.pinv(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a, wrap = _makearray(a)
_assertNonEmpty(a)
a = a.conjugate()
u, s, vt = svd(a, 0)
m = u.shape[0]
n = vt.shape[1]
cutoff = rcond * maximum.reduce(s)
for i in range(min(n, m)):
if s[i] > cutoff:
s[i] = 1. / s[i]
else:
s[i] = 0.
res = dot(transpose(vt), multiply(s[:, newaxis], transpose(u)))
return wrap(res)
def pinv(a, rcond=1e-15 ):
"""
Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate the generalized inverse of a matrix using its
singular-value decomposition (SVD) and including all
`large` singular values.
Parameters
----------
a : array_like (M, N)
Matrix to be pseudo-inverted.
rcond : float
Cutoff for `small` singular values.
Singular values smaller than rcond*largest_singular_value are
considered zero.
Returns
-------
B : ndarray (N, M)
The pseudo-inverse of `a`. If `a` is an np.matrix instance, then so
is `B`.
Raises
------
LinAlgError
In case SVD computation does not converge.
Examples
--------
>>> a = np.random.randn(9, 6)
>>> B = np.linalg.pinv(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True
"""
a, wrap = _makearray(a)
_assertNonEmpty(a)
a = a.conjugate()
u, s, vt = svd(a, 0)
m = u.shape[0]
n = vt.shape[1]
cutoff = rcond*maximum.reduce(s)
for i in range(min(n, m)):
if s[i] > cutoff:
s[i] = 1./s[i]
else:
s[i] = 0.;
res = dot(transpose(vt), multiply(s[:, newaxis],transpose(u)))
return wrap(res)
def projectArray(homography, points):
from numpy.core import dot
from numpy.lib.function_base import append
if points.shape[0] != 2:
raise Exception('points of dimension {0} {1}'.format(points.shape[0], points.shape[1]))
if (homography is not None) and homography.size>0:
augmentedPoints = append(points,[[1]*points.shape[1]], 0)
prod = dot(homography, augmentedPoints)
return prod[0:2]/prod[2]
else:
return points
def pinv(a, rcond=1e-15 ):
"""Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array-like, shape (M, N)
Matrix to be pseudo-inverted
rcond : float
Cutoff for 'small' singular values.
Singular values smaller than rcond*largest_singular_value are
considered zero.
Returns
-------
B : array, shape (N, M)
If a is a matrix, then so is B.
Raises LinAlgError if SVD computation does not converge
Examples
--------
>>> from numpy import *
>>> a = random.randn(9, 6)
>>> B = linalg.pinv(a)
>>> allclose(a, dot(a, dot(B, a)))
True
>>> allclose(B, dot(B, dot(a, B)))
True
"""
a, wrap = _makearray(a)
_assertNonEmpty(a)
a = a.conjugate()
u, s, vt = svd(a, 0)
m = u.shape[0]
n = vt.shape[1]
cutoff = rcond*maximum.reduce(s)
for i in range(min(n, m)):
if s[i] > cutoff:
s[i] = 1./s[i]
else:
s[i] = 0.;
res = dot(transpose(vt), multiply(s[:, newaxis],transpose(u)))
return wrap(res)
def pinv(a, rcond=1e-15):
"""Compute the (Moore-Penrose) pseudo-inverse of a matrix.
Calculate a generalized inverse of a matrix using its
singular-value decomposition and including all 'large' singular
values.
Parameters
----------
a : array-like, shape (M, N)
Matrix to be pseudo-inverted
rcond : float
Cutoff for 'small' singular values.
Singular values smaller than rcond*largest_singular_value are
considered zero.
Returns
-------
B : array, shape (N, M)
If a is a matrix, then so is B.
Raises LinAlgError if SVD computation does not converge
Examples
--------
>>> from numpy import *
>>> a = random.randn(9, 6)
>>> B = linalg.pinv(a)
>>> allclose(a, dot(a, dot(B, a)))
True
>>> allclose(B, dot(B, dot(a, B)))
True
"""
a, wrap = _makearray(a)
_assertNonEmpty(a)
a = a.conjugate()
u, s, vt = svd(a, 0)
m = u.shape[0]
n = vt.shape[1]
cutoff = rcond * maximum.reduce(s)
for i in range(min(n, m)):
if s[i] > cutoff:
s[i] = 1. / s[i]
else:
s[i] = 0.
res = dot(transpose(vt), multiply(s[:, newaxis], transpose(u)))
return wrap(res)
def projectArray(homography, points):
'''Returns the coordinates of the projected points (format 2xN points)
through homography'''
from numpy.core import dot
from numpy.core.multiarray import array
from numpy.lib.function_base import append
if points.shape[0] != 2:
raise Exception('points of dimension {0} {1}'.format(
points.shape[0], points.shape[1]))
if (homography != None) and homography.size > 0:
augmentedPoints = append(points, [[1] * points.shape[1]], 0)
prod = dot(homography, augmentedPoints)
return prod[0:2] / prod[2]
else:
return p
def pinv(a, rcond=1e-15 ):
"""Return the (Moore-Penrose) pseudo-inverse of a 2-d array
This method computes the generalized inverse using the
singular-value decomposition and all singular values larger than
rcond of the largest.
"""
a, wrap = _makearray(a)
a = a.conjugate()
u, s, vt = svd(a, 0)
m = u.shape[0]
n = vt.shape[1]
cutoff = rcond*maximum.reduce(s)
for i in range(min(n, m)):
if s[i] > cutoff:
s[i] = 1./s[i]
else:
s[i] = 0.;
return wrap(dot(transpose(vt),
multiply(s[:, newaxis],transpose(u))))
def pinv(a, rcond=1e-15):
"""Return the (Moore-Penrose) pseudo-inverse of a 2-d array
This method computes the generalized inverse using the
singular-value decomposition and all singular values larger than
rcond of the largest.
"""
a, wrap = _makearray(a)
_assertNonEmpty(a)
a = a.conjugate()
u, s, vt = svd(a, 0)
m = u.shape[0]
n = vt.shape[1]
cutoff = rcond * maximum.reduce(s)
for i in range(min(n, m)):
if s[i] > cutoff:
s[i] = 1. / s[i]
else:
s[i] = 0.
return wrap(dot(transpose(vt), multiply(s[:, newaxis], transpose(u))))
def _inv(a, cf, rcond, epsilon):
"""
modified pseudo inverse
"""
def _assertNoEmpty2d(*arrays):
for a in arrays:
if a.size == 0 and product(a.shape[-2:]) == 0:
raise RuntimeError("Arrays cannot be empty")
def _makearray(a):
new = asarray(a)
wrap = getattr(a, "__array_prepare__", new.__array_wrap__)
return new, wrap
a, wrap = _makearray(a)
_assertNoEmpty2d(a)
if epsilon is not None:
epsilon = numpy.repeat(epsilon, a.shape[0])
epsilon = numpy.diag(epsilon)
a = a + epsilon
a = a.conjugate()
#WARNING! the "s" eigenvalues might not equal the eigenvalues of eigh
u, s, vt = numpy.linalg.svd(a, 0)
m = u.shape[0]
n = vt.shape[1]
eigen = numpy.copy(s)
# cutoff = rcond*maximum.reduce(s)
cutoff = cf(s, rcond)
for i in range(min(n, m)):
# The first Singular Value will always be selected because we want at least one, and the first is the highest
if s[i] >= cutoff or i==0:
s[i] = 1. / s[i]
else:
s[i] = 0.
n_indep = numpy.count_nonzero(s)
res = dot(transpose(vt), multiply(s[:, newaxis], transpose(u)))
return wrap(res), n_indep, eigen
__all__ = ['matrix_power', 'solve', 'tensorsolve', 'tensorinv', 'inv',
'cholesky', 'eigvals', 'eigvalsh', 'pinv', 'slogdet', 'det',
'svd', 'eig', 'eigh', 'lstsq', 'norm', 'qr', 'cond', 'matrix_rank',
'LinAlgError', 'multi_dot']; import functools
import operator
import warnings; from numpy.core import (
array, asarray, zeros, empty, empty_like, intc, single, double,
csingle, cdouble, inexact, complexfloating, newaxis, all, Inf, dot,
add, multiply, sqrt, fastCopyAndTranspose, sum, isfinite,
finfo, errstate, geterrobj, moveaxis, amin, amax, product, abs,
atleast_2d, intp, asanyarray, object_, matmul,
swapaxes, divide, count_nonzero, isnan, sign
); from numpy.core.multiarray import normalize_axis_index; from numpy.core.overrides import set_module; from numpy.core import overrides; from numpy.lib.twodim_base import triu, eye; from numpy.linalg import lapack_lite, _umath_linalg; array_function_dispatch = functools.partial(
overrides.array_function_dispatch, module='numpy.linalg'); _N = b'N'; _V = b'V'; _A = b'A'; _S = b'S'; _L = b'L' fortran_int = intc @set_module('numpy.linalg') class LinAlgError(Exception): def _determine_error_states(): errobj = geterrobj() bufsize = errobj[0] with errstate(invalid='call', over='ignore', divide='ignore', under='ignore'): invalid_call_errmask = geterrobj()[1] return [bufsize, invalid_call_errmask, None]; _linalg_error_extobj = _determine_error_states(); del _determine_error_states; def _raise_linalgerror_singular(err, flag): raise LinAlgError("Singular matrix"); def _raise_linalgerror_nonposdef(err, flag): raise LinAlgError("Matrix is not positive definite"); def _raise_linalgerror_eigenvalues_nonconvergence(err, flag): raise LinAlgError("Eigenvalues did not converge"); def _raise_linalgerror_svd_nonconvergence(err, flag): raise LinAlgError("SVD did not converge"); def _raise_linalgerror_lstsq(err, flag): raise LinAlgError("SVD did not converge in Linear Least Squares"); def get_linalg_error_extobj(callback): extobj = list(_linalg_error_extobj); extobj[2] = callback; return extobj; def _makearray(a): new = asarray(a); wrap = getattr(a, "__array_prepare__", new.__array_wrap__); return new, wrap; def isComplexType(t): return issubclass(t, complexfloating); _real_types_map = {single: single,; double: double,; csingle: single,; cdouble: double}; _complex_types_map = {single: csingle,; double: cdouble,; csingle: csingle,; cdouble: cdouble}; def _realType(t, default=double): return _real_types_map.get(t, default); def _complexType(t, default=cdouble): return _complex_types_map.get(t, default); def _linalgRealType(t): """Cast the type t to either double or cdouble."""; return double; def _commonType(*arrays): result_type = single; is_complex = False; for a in arrays: if issubclass(a.dtype.type, inexact): if isComplexType(a.dtype.type): is_complex = True; rt = _realType(a.dtype.type, default=None); if rt is None: raise TypeError("array type %s is unsupported in linalg" %; (a.dtype.name,)); else: rt = double; if rt is double: result_type = double; if is_complex: t = cdouble; result_type = _complex_types_map[result_type]; else: t = double; return t, result_type; _fastCT = fastCopyAndTranspose; def _to_native_byte_order(*arrays): ret = []; for arr in arrays: if arr.dtype.byteorder not in ('=', '|'): ret.append(asarray(arr, dtype=arr.dtype.newbyteorder('='))); else: ret.append(arr); if len(ret) == 1: return ret[0]; else: return ret; def _fastCopyAndTranspose(type, *arrays): cast_arrays = (); for a in arrays: if a.dtype.type is type: cast_arrays = cast_arrays + (_fastCT(a),); else: cast_arrays = cast_arrays + (_fastCT(a.astype(type)),); if len(cast_arrays) == 1: return cast_arrays[0]; else: return cast_arrays; def _assert_2d(*arrays): for a in arrays: if a.ndim != 2: raise LinAlgError('%d-dimensional array given. Array must be '; 'two-dimensional' % a.ndim); def _assert_stacked_2d(*arrays): for a in arrays: if a.ndim < 2: raise LinAlgError('%d-dimensional array given. Array must be '; 'at least two-dimensional' % a.ndim); def _assert_stacked_square(*arrays): for a in arrays: m, n = a.shape[-2:]; if m != n: raise LinAlgError('Last 2 dimensions of the array must be square'); def _assert_finite(*arrays): for a in arrays: if not isfinite(a).all(): raise LinAlgError("Array must not contain infs or NaNs"); def _is_empty_2d(arr): return arr.size == 0 and product(arr.shape[-2:]) == 0; def transpose(a): a, wrap = _makearray(a); b = asarray(b); an = a.ndim; if axes is not None: allaxes = list(range(0, an)); for k in axes: allaxes.remove(k); allaxes.insert(an, k); a = a.transpose(allaxes); oldshape = a.shape[-(an-b.ndim):]; prod = 1; for k in oldshape: prod *= k; a = a.reshape(-1, prod); b = b.ravel(); res = wrap(solve(a, b)); res.shape = oldshape; return res; def _solve_dispatcher(a, b): return (a, b); @array_function_dispatch(_solve_dispatcher); def solve(a, b): a, _ = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); b, wrap = _makearray(b); t, result_t = _commonType(a, b); if b.ndim == a.ndim - 1: gufunc = _umath_linalg.solve1; else: gufunc = _umath_linalg.solve; signature = 'DD->D' if isComplexType(t) else 'dd->d'; extobj = get_linalg_error_extobj(_raise_linalgerror_singular); r = gufunc(a, b, signature=signature, extobj=extobj); return wrap(r.astype(result_t, copy=False)); def _tensorinv_dispatcher(a, ind=None): return (a,); @array_function_dispatch(_tensorinv_dispatcher); def tensorinv(a, ind=2): a = asarray(a); oldshape = a.shape; prod = 1; if ind > 0: invshape = oldshape[ind:] + oldshape[:ind]; for k in oldshape[ind:]: prod *= k; else: raise ValueError("Invalid ind argument."); a = a.reshape(prod, -1); ia = inv(a); return ia.reshape(*invshape); def _unary_dispatcher(a): return (a,); @array_function_dispatch(_unary_dispatcher); def inv(a): a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->D' if isComplexType(t) else 'd->d'; extobj = get_linalg_error_extobj(_raise_linalgerror_singular); ainv = _umath_linalg.inv(a, signature=signature, extobj=extobj); return wrap(ainv.astype(result_t, copy=False)); def _matrix_power_dispatcher(a, n): return (a,); @array_function_dispatch(_matrix_power_dispatcher); def matrix_power(a, n): a = asanyarray(a); _assert_stacked_2d(a); _assert_stacked_square(a); try: n = operator.index(n); except TypeError: raise TypeError("exponent must be an integer"); if a.dtype != object: fmatmul = matmul; elif a.ndim == 2: fmatmul = dot; else: raise NotImplementedError(; "matrix_power not supported for stacks of object arrays"); if n == 0: a = empty_like(a); a[...] = eye(a.shape[-2], dtype=a.dtype); return a; elif n < 0: a = inv(a); n = abs(n); if n == 1: return a; elif n == 2: return fmatmul(a, a); elif n == 3: return fmatmul(fmatmul(a, a), a); z = result = None; while n > 0: z = a if z is None else fmatmul(z, z); n, bit = divmod(n, 2); if bit: result = z if result is None else fmatmul(result, z); return result; @array_function_dispatch(_unary_dispatcher); def cholesky(a): extobj = get_linalg_error_extobj(_raise_linalgerror_nonposdef); gufunc = _umath_linalg.cholesky_lo; a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->D' if isComplexType(t) else 'd->d'; r = gufunc(a, signature=signature, extobj=extobj); return wrap(r.astype(result_t, copy=False)); def _qr_dispatcher(a, mode=None): return (a,); @array_function_dispatch(_qr_dispatcher); def qr(a, mode='reduced'): if mode not in ('reduced', 'complete',x 'r', 'raw'): if mode in ('f', 'full'): msg = "".join((; "The 'full' option is deprecated in favor of 'reduced'.\n",; "For backward compatibility let mode default.")); warnings.warn(msg, DeprecationWarning, stacklevel=3); mode = 'reduced'; elif mode in ('e', 'economic'): msg = "The 'economic' option is deprecated."; warnings.warn(msg, DeprecationWarning, stacklevel=3); mode = 'economic'; else: raise ValueError("Unrecognized mode '%s'" % mode); a, wrap = _makearray(a); _assert_2d(a); m, n = a.shape; t, result_t = _commonType(a); a = _fastCopyAndTranspose(t, a); a = _to_native_byte_order(a); mn = min(m, n); tau = zeros((mn,), t); if isComplexType(t): lapack_routine = lapack_lite.zgeqrf; routine_name = 'zgeqrf'; else: lapack_routine = lapack_lite.dgeqrf; routine_name = 'dgeqrf'; lwork = 1; work = zeros((lwork,), t); results = lapack_routine(m, n, a, max(1, m), tau, work, -1, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); lwork = max(1, n, int(abs(work[0]))); work = zeros((lwork,), t); results = lapack_routine(m, n, a, max(1, m), tau, work, lwork, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); if mode == 'r': r = _fastCopyAndTranspose(result_t, a[:, :mn]); return wrap(triu(r)); if mode == 'raw': return a, tau; if mode == 'economic': if t != result_t : a = a.astype(result_t, copy=False); return wrap(a.T); if mode == 'complete' and m > n: mc = m; q = empty((m, m), t); else: mc = mn; q = empty((n, m), t); q[:n] = a; if isComplexType(t): lapack_routine = lapack_lite.zungqr; routine_name = 'zungqr'; else: lapack_routine = lapack_lite.dorgqr; routine_name = 'dorgqr'; lwork = 1; work = zeros((lwork,), t); results = lapack_routine(m, mc, mn, q, max(1, m), tau, work, -1, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); lwork = max(1, n, int(abs(work[0]))); work = zeros((lwork,), t); results = lapack_routine(m, mc, mn, q, max(1, m), tau, work, lwork, 0); if results['info'] != 0: raise LinAlgError('%s returns %d' % (routine_name, results['info'])); q = _fastCopyAndTranspose(result_t, q[:mc]); r = _fastCopyAndTranspose(result_t, a[:, :mc]); return wrap(q), wrap(triu(r)); @array_function_dispatch(_unary_dispatcher); def eigvals(a): a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); _assert_finite(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); signature = 'D->D' if isComplexType(t) else 'd->D'; w = _umath_linalg.eigvals(a, signature=signature, extobj=extobj); if not isComplexType(t): if all(w.imag == 0): w = w.real; result_t = _realType(result_t); else: result_t = _complexType(result_t); return w.astype(result_t, copy=False); def _eigvalsh_dispatcher(a, UPLO=None): return (a,); @array_function_dispatch(_eigvalsh_dispatcher); def eigvalsh(a, UPLO='L'): UPLO = UPLO.upper(); if UPLO not in ('L', 'U'): raise ValueError("UPLO argument must be 'L' or 'U'"); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); if UPLO == 'L': gufunc = _umath_linalg.eigvalsh_lo; else: gufunc = _umath_linalg.eigvalsh_up; a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->d' if isComplexType(t) else 'd->d'; w = gufunc(a, signature=signature, extobj=extobj); return w.astype(_realType(result_t), copy=False); def _convertarray(a): t, result_t = _commonType(a); a = _fastCT(a.astype(t)); return a, t, result_t; def eig(a): a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); _assert_finite(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); signature = 'D->DD' if isComplexType(t) else 'd->DD'; w, vt = _umath_linalg.eig(a, signature=signature, extobj=extobj); if not isComplexType(t) and all(w.imag == 0.0): w = w.real; vt = vt.real; result_t = _realType(result_t); else: result_t = _complexType(result_t); vt = vt.astype(result_t, copy=False); return w.astype(result_t, copy=False), wrap(vt); @array_function_dispatch(_eigvalsh_dispatcher); def eigh(a, UPLO='L'): UPLO = UPLO.upper(); if UPLO not in ('L', 'U'): raise ValueError("UPLO argument must be 'L' or 'U'"); a, wrap = _makearray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(; _raise_linalgerror_eigenvalues_nonconvergence); if UPLO == 'L': gufunc = _umath_linalg.eigh_lo; else: gufunc = _umath_linalg.eigh_up; signature = 'D->dD' if isComplexType(t) else 'd->dd'; w, vt = gufunc(a, signature=signature, extobj=extobj); w = w.astype(_realType(result_t), copy=False); vt = vt.astype(result_t, copy=False); return w, wrap(vt); def _svd_dispatcher(a, full_matrices=None, compute_uv=None, hermitian=None): return (a,); @array_function_dispatch(_svd_dispatcher); def svd(a, full_matrices=True, compute_uv=True, hermitian=False): a, wrap = _makearray(a); if hermitian: if compute_uv: s, u = eigh(a); s = s[..., ::-1]; u = u[..., ::-1]; vt = transpose(u * sign(s)[..., None, :]).conjugate(); s = abs(s); return wrap(u), s, wrap(vt); else: s = eigvalsh(a); s = s[..., ::-1]; s = abs(s); return s; _assert_stacked_2d(a); t, result_t = _commonType(a); extobj = get_linalg_error_extobj(_raise_linalgerror_svd_nonconvergence); m, n = a.shape[-2:]; if compute_uv: if full_matrices: if m < n: gufunc = _umath_linalg.svd_m_f; else: gufunc = _umath_linalg.svd_n_f; else: if m < n: gufunc = _umath_linalg.svd_m_s; else: gufunc = _umath_linalg.svd_n_s; signature = 'D->DdD' if isComplexType(t) else 'd->ddd'; u, s, vh = gufunc(a, signature=signature, extobj=extobj); u = u.astype(result_t, copy=False); s = s.astype(_realType(result_t), copy=False); vh = vh.astype(result_t, copy=False); return wrap(u), s, wrap(vh); else: if m < n: gufunc = _umath_linalg.svd_m; else: gufunc = _umath_linalg.svd_n; signature = 'D->d' if isComplexType(t) else 'd->d'; s = gufunc(a, signature=signature, extobj=extobj); s = s.astype(_realType(result_t), copy=False); return s; def _cond_dispatcher(x, p=None): return (x,); @array_function_dispatch(_cond_dispatcher); def cond(x, p=None): x = asarray(x); if _is_empty_2d(x): raise LinAlgError("cond is not defined on empty arrays"); if p is None or p == 2 or p == -2: s = svd(x, compute_uv=False); with errstate(all='ignore'): if p == -2: r = s[..., -1] / s[..., 0]; else: r = s[..., 0] / s[..., -1]; else: _assert_stacked_2d(x); _assert_stacked_square(x); t, result_t = _commonType(x); signature = 'D->D' if isComplexType(t) else 'd->d'; with errstate(all='ignore'): invx = _umath_linalg.inv(x, signature=signature); r = norm(x, p, axis=(-2, -1)) * norm(invx, p, axis=(-2, -1)); r = r.astype(result_t, copy=False); r = asarray(r); nan_mask = isnan(r); if nan_mask.any(): nan_mask &= ~isnan(x).any(axis=(-2, -1)); if r.ndim > 0: r[nan_mask] = Inf; elif nan_mask: r[()] = Inf; if r.ndim == 0: r = r[()]; return r; def _matrix_rank_dispatcher(M, tol=None, hermitian=None): return (M,); @array_function_dispatch(_matrix_rank_dispatcher); def matrix_rank(M, tol=None, hermitian=False): M = asarray(M); if M.ndim < 2: return int(not all(M==0)); S = svd(M, compute_uv=False, hermitian=hermitian); if tol is None: tol = S.max(axis=-1, keepdims=True) * max(M.shape[-2:]) * finfo(S.dtype).eps; else: tol = asarray(tol)[..., newaxis]; return count_nonzero(S > tol, axis=-1); def pinv(a, rcond=1e-15, hermitian=False): a, wrap = _makearray(a); rcond = asarray(rcond); if _is_empty_2d(a): m, n = a.shape[-2:]; res = empty(a.shape[:-2] + (n, m), dtype=a.dtype); return wrap(res); a = a.conjugate(); u, s, vt = svd(a, full_matrices=False, hermitian=hermitian); cutoff = rcond[..., newaxis] * amax(s, axis=-1, keepdims=True); large = s > cutoff; s = divide(1, s, where=large, out=s); s[~large] = 0; res = matmul(transpose(vt), multiply(s[..., newaxis], transpose(u))); return wrap(res); def slogdet(a): a = asarray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); real_t = _realType(result_t); signature = 'D->Dd' if isComplexType(t) else 'd->dd'; sign, logdet = _umath_linalg.slogdet(a, signature=signature); sign = sign.astype(result_t, copy=False); logdet = logdet.astype(real_t, copy=False); return sign, logdet; @array_function_dispatch(_unary_dispatcher); def det(a): a = asarray(a); _assert_stacked_2d(a); _assert_stacked_square(a); t, result_t = _commonType(a); signature = 'D->D' if isComplexType(t) else 'd->d'; r = _umath_linalg.det(a, signature=signature); r = r.astype(result_t, copy=False); return r; def lstsq(a, b, rcond="warn"): a, _ = _makearray(a); b, wrap = _makearray(b); is_1d = b.ndim == 1; if is_1d: b = b[:, newaxis]; _assert_2d(a, b); m, n = a.shape[-2:]; m2, n_rhs = b.shape[-2:]; if m != m2: raise LinAlgError('Incompatible dimensions'); t, result_t = _commonType(a, b); real_t = _linalgRealType(t); result_real_t = _realType(result_t); if rcond == "warn": warnings.warn("`rcond` parameter will change to the default of "; "machine precision times ``max(M, N)`` where M and N "; "are the input matrix dimensions.\n"; "To use the future default and silence this warning "; "we advise to pass `rcond=None`, to keep using the old, "; "explicitly pass `rcond=-1`.",; FutureWarning, stacklevel=3); rcond = -1; if rcond is None: rcond = finfo(t).eps * max(n, m); if m <= n: gufunc = _umath_linalg.lstsq_m; else: gufunc = _umath_linalg.lstsq_n; signature = 'DDd->Ddid' if isComplexType(t) else 'ddd->ddid'; extobj = get_linalg_error_extobj(_raise_linalgerror_lstsq); if n_rhs == 0: b = zeros(b.shape[:-2] + (m, n_rhs + 1), dtype=b.dtype); x, resids, rank, s = gufunc(a, b, rcond, signature=signature, extobj=extobj); if m == 0: x[...] = 0; if n_rhs == 0: x = x[..., :n_rhs]; resids = resids[..., :n_rhs]; if is_1d: x = x.squeeze(axis=-1); if rank != n or m <= n: resids = array([], result_real_t); s = s.astype(result_real_t, copy=False); resids = resids.astype(result_real_t, copy=False); x = x.astype(result_t, copy=True); return wrap(x), wrap(resids), rank, s; def _multi_svd_norm(x, row_axis, col_axis, op): y = moveaxis(x, (row_axis, col_axis), (-2, -1)); result = op(svd(y, compute_uv=False), axis=-1); return result; def _norm_dispatcher(x, ord=None, axis=None, keepdims=None): return (x,); @array_function_dispatch(_norm_dispatcher); def norm(x, ord=None, axis=None, keepdims=False): x = asarray(x); if not issubclass(x.dtype.type, (inexact, object_)): x = x.astype(float); if axis is None: ndim = x.ndim; if ((ord is None) or; (ord in ('f', 'fro') and ndim == 2) or; (ord == 2 and ndim == 1)): x = x.ravel(order='K'); if isComplexType(x.dtype.type): sqnorm = dot(x.real, x.real) + dot(x.imag, x.imag); else: sqnorm = dot(x, x); ret = sqrt(sqnorm); if keepdims: ret = ret.reshape(ndim*[1]); return ret; nd = x.ndim; if axis is None: axis = tuple(range(nd)); elif not isinstance(axis, tuple): try: axis = int(axis); except Exception: raise TypeError("'axis' must be None, an integer or a tuple of integers"); axis = (axis,); if len(axis) == 1: if ord == Inf: return abs(x).max(axis=axis, keepdims=keepdims); elif ord == -Inf: return abs(x).min(axis=axis, keepdims=keepdims); elif ord == 0: return (x != 0).astype(x.real.dtype).sum(axis=axis, keepdims=keepdims); elif ord == 1: return add.reduce(abs(x), axis=axis, keepdims=keepdims); elif ord is None or ord == 2: s = (x.conj() * x).real; return sqrt(add.reduce(s, axis=axis, keepdims=keepdims)); else: try: ord + 1; except TypeError: raise ValueError("Invalid norm order for vectors."); absx = abs(x); absx **= ord; ret = add.reduce(absx, axis=axis, keepdims=keepdims); ret **= (1 / ord); return ret; elif len(axis) == 2: row_axis, col_axis = axis; row_axis = normalize_axis_index(row_axis, nd); col_axis = normalize_axis_index(col_axis, nd); if row_axis == col_axis: raise ValueError('Duplicate axes given.'); if ord == 2: ret =_multi_svd_norm(x, row_axis, col_axis, amax); elif ord == -2: ret = _multi_svd_norm(x, row_axis, col_axis, amin); elif ord == 1: if col_axis > row_axis: col_axis -= 1; ret = add.reduce(abs(x), axis=row_axis).max(axis=col_axis); elif ord == Inf: if row_axis > col_axis: row_axis -= 1; ret = add.reduce(abs(x), axis=col_axis).max(axis=row_axis); elif ord == -1: if col_axis > row_axis: col_axis -= 1; ret = add.reduce(abs(x), axis=row_axis).min(axis=col_axis); elif ord == -Inf: if row_axis > col_axis: row_axis -= 1; ret = add.reduce(abs(x), axis=col_axis).min(axis=row_axis); elif ord in [None, 'fro', 'f']: ret = sqrt(add.reduce((x.conj() * x).real, axis=axis)); elif ord == 'nuc': ret = _multi_svd_norm(x, row_axis, col_axis, sum); else: raise ValueError("Invalid norm order for matrices."); if keepdims: ret_shape = list(x.shape); ret_shape[axis[0]] = 1; ret_shape[axis[1]] = 1; ret = ret.reshape(ret_shape); return ret; else: raise ValueError("Improper number of dimensions to norm."); def multi_dot(arrays): n = len(arrays); if n < 2: raise ValueError("Expecting at least two arrays."); elif n == 2: return dot(arrays[0], arrays[1]); arrays = [asanyarray(a) for a in arrays]; ndim_first, ndim_last = arrays[0].ndim, arrays[-1].ndim; if arrays[0].ndim == 1: arrays[0] = atleast_2d(arrays[0]); if arrays[-1].ndim == 1: arrays[-1] = atleast_2d(arrays[-1]).T; _assert_2d(*arrays)
if n == 3: result = _multi_dot_three(arrays[0], arrays[1], arrays[2])
else: order = _multi_dot_matrix_chain_order(arrays)
result = _multi_dot(arrays, order, 0, n - 1)
if ndim_first == 1 and ndim_last == 1: return result[0, 0]
elif ndim_first == 1 or ndim_last == 1: return result.ravel()
else: return result; def _multi_dot_three(A, B, C): a0, a1b0 = A.shape
b1c0, c1 = C.shape; cost1 = a0 * b1c0 * (a1b0 + c1); cost2 = a1b0 * c1 * (a0 + b1c0); if cost1 < cost2: return dot(dot(A, B), C); else: return dot(A, dot(B, C)); def _multi_dot_matrix_chain_order(arrays, return_costs=False): n = len(arrays); p = [a.shape[0] for a in arrays] + [arrays[-1].shape[1]]; m = zeros((n, n), dtype=double); s = empty((n, n), dtype=intp); for l in range(1, n): for i in range(n - l): j = i + l; m[i, j] = Inf; for k in range(i, j): q = m[i, k] + m[k+1, j] + p[i]*p[k+1]*p[j+1]; if q < m[i, j]: m[i, j] = q; s[i, j] = k; if i == j: return arrays[i]; else: return dot(_multi_dot(arrays, order, i, order[i, j]),; _multi_dot(arrays, order, order[i, j] + 1, j))
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False):
"""
Least squares polynomial fit.
Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg`
to points `(x, y)`. Returns a vector of coefficients `p` that minimises
the squared error.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int
Degree of the fitting polynomial
rcond : float, optional
Relative condition number of the fit. Singular values smaller than this
relative to the largest singular value will be ignored. The default
value is len(x)*eps, where eps is the relative precision of the float
type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is
False (the default) just the coefficients are returned, when True
diagnostic information from the singular value decomposition is also
returned.
w : array_like, shape (M,), optional
weights to apply to the y-coordinates of the sample points.
cov : bool, optional
Return the estimate and the covariance matrix of the estimate
If full is True, then cov is not returned.
Returns
-------
p : ndarray, shape (M,) or (M, K)
Polynomial coefficients, highest power first.
If `y` was 2-D, the coefficients for `k`-th data set are in ``p[:,k]``.
residuals, rank, singular_values, rcond : present only if `full` = True
Residuals of the least-squares fit, the effective rank of the scaled
Vandermonde coefficient matrix, its singular values, and the specified
value of `rcond`. For more details, see `linalg.lstsq`.
V : ndaray, shape (M,M) or (M,M,K) : present only if `full` = False and `cov`=True
The covariance matrix of the polynomial coefficient estimates. The diagonal
of this matrix are the variance estimates for each coefficient. If y is a 2-d
array, then the covariance matrix for the `k`-th data set are in ``V[:,:,k]``
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if `full` = False.
The warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
polyval : Computes polynomial values.
linalg.lstsq : Computes a least-squares fit.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution minimizes the squared error
.. math ::
E = \\sum_{j=0}^k |p(x_j) - y_j|^2
in the equations::
x[0]**n * p[n] + ... + x[0] * p[1] + p[0] = y[0]
x[1]**n * p[n] + ... + x[1] * p[1] + p[0] = y[1]
...
x[k]**n * p[n] + ... + x[k] * p[1] + p[0] = y[k]
The coefficient matrix of the coefficients `p` is a Vandermonde matrix.
`polyfit` issues a `RankWarning` when the least-squares fit is badly
conditioned. This implies that the best fit is not well-defined due
to numerical error. The results may be improved by lowering the polynomial
degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter
can also be set to a value smaller than its default, but the resulting
fit may be spurious: including contributions from the small singular
values can add numerical noise to the result.
Note that fitting polynomial coefficients is inherently badly conditioned
when the degree of the polynomial is large or the interval of sample points
is badly centered. The quality of the fit should always be checked in these
cases. When polynomial fits are not satisfactory, splines may be a good
alternative.
References
----------
.. [1] Wikipedia, "Curve fitting",
http://en.wikipedia.org/wiki/Curve_fitting
.. [2] Wikipedia, "Polynomial interpolation",
http://en.wikipedia.org/wiki/Polynomial_interpolation
Examples
--------
>>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0])
>>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
>>> z = np.polyfit(x, y, 3)
>>> z
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
It is convenient to use `poly1d` objects for dealing with polynomials:
>>> p = np.poly1d(z)
>>> p(0.5)
0.6143849206349179
>>> p(3.5)
-0.34732142857143039
>>> p(10)
22.579365079365115
High-order polynomials may oscillate wildly:
>>> p30 = np.poly1d(np.polyfit(x, y, 30))
/... RankWarning: Polyfit may be poorly conditioned...
>>> p30(4)
-0.80000000000000204
>>> p30(5)
-0.99999999999999445
>>> p30(4.5)
-0.10547061179440398
Illustration:
>>> import matplotlib.pyplot as plt
>>> xp = np.linspace(-2, 6, 100)
>>> plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--')
[<matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>, <matplotlib.lines.Line2D object at 0x...>]
>>> plt.ylim(-2,2)
(-2, 2)
>>> plt.show()
"""
order = int(deg) + 1
x = NX.asarray(x) + 0.0
y = NX.asarray(y) + 0.0
# check arguments.
if deg < 0 :
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
if y.ndim < 1 or y.ndim > 2 :
raise TypeError("expected 1D or 2D array for y")
if x.shape[0] != y.shape[0] :
raise TypeError("expected x and y to have same length")
# set rcond
if rcond is None :
rcond = len(x)*finfo(x.dtype).eps
# set up least squares equation for powers of x
lhs = vander(x, order)
rhs = y
# apply weighting
if w is not None:
w = NX.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError("expected a 1-d array for weights")
if w.shape[0] != y.shape[0] :
raise TypeError("expected w and y to have the same length")
lhs *= w[:, NX.newaxis]
if rhs.ndim == 2:
rhs *= w[:, NX.newaxis]
else:
rhs *= w
# scale lhs to improve condition number and solve
scale = NX.sqrt((lhs*lhs).sum(axis=0))
lhs /= scale
c, resids, rank, s = lstsq(lhs, rhs, rcond)
c = (c.T/scale).T # broadcast scale coefficients
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
msg = "Polyfit may be poorly conditioned"
warnings.warn(msg, RankWarning)
if full :
return c, resids, rank, s, rcond
elif cov :
Vbase = inv(dot(lhs.T,lhs))
Vbase /= NX.outer(scale, scale)
# Some literature ignores the extra -2.0 factor in the denominator, but
# it is included here because the covariance of Multivariate Student-T
# (which is implied by a Bayesian uncertainty analysis) includes it.
# Plus, it gives a slightly more conservative estimate of uncertainty.
fac = resids / (len(x) - order - 2.0)
if y.ndim == 1:
return c, Vbase * fac
else:
return c, Vbase[:,:,NX.newaxis] * fac
else :
return c
def norm(x, ord=None, axis=None, keepdims=False):
x = asarray(x)
if not issubclass(x.dtype.type, (inexact, object_)):
x = x.astype(float)
if axis is None:
ndim = x.ndim
if ((ord is None) or (ord in ('f', 'fro') and ndim == 2)
or (ord == 2 and ndim == 1)):
x = x.ravel(order='K')
if isComplexType(x.dtype.type):
sqnorm = dot(x.real, x.real) + dot(x.imag, x.imag)
else:
sqnorm = dot(x, x)
ret = sqrt(sqnorm)
if keepdims:
ret = ret.reshape(ndim * [1])
return ret
nd = x.ndim
if axis is None:
axis = tuple(range(nd))
elif not isinstance(axis, tuple):
try:
axis = int(axis)
except Exception as e:
raise TypeError(
"'axis' debe ser None, un entero o una tupla de enteros"
) from e
axis = (axis, )
if len(axis) == 1:
if ord == Inf:
return abs(x).max(axis=axis, keepdims=keepdims)
elif ord == -Inf:
return abs(x).min(axis=axis, keepdims=keepdims)
elif ord == 0:
return (x != 0).astype(x.real.dtype).sum(axis=axis,
keepdims=keepdims)
elif ord == 1:
return add.reduce(abs(x), axis=axis, keepdims=keepdims)
elif ord is None or ord == 2:
s = (x.conj() * x).real
return sqrt(add.reduce(s, axis=axis, keepdims=keepdims))
elif isinstance(ord, str):
raise ValueError(
f"Orden de norma inválida '{ord}' para los vectores")
else:
absx = abs(x)
absx **= ord
ret = add.reduce(absx, axis=axis, keepdims=keepdims)
ret **= (1 / ord)
return ret
elif len(axis) == 2:
row_axis, col_axis = axis
row_axis = normalize_axis_index(row_axis, nd)
col_axis = normalize_axis_index(col_axis, nd)
if row_axis == col_axis:
raise ValueError('Duplicado de los ejes dados.')
if ord == 2:
ret = _multi_svd_norm(x, row_axis, col_axis, amax)
elif ord == -2:
ret = _multi_svd_norm(x, row_axis, col_axis, amin)
elif ord == 1:
if col_axis > row_axis:
col_axis -= 1
ret = add.reduce(abs(x), axis=row_axis).max(axis=col_axis)
elif ord == Inf:
if row_axis > col_axis:
row_axis -= 1
ret = add.reduce(abs(x), axis=col_axis).max(axis=row_axis)
elif ord == -1:
if col_axis > row_axis:
col_axis -= 1
ret = add.reduce(abs(x), axis=row_axis).min(axis=col_axis)
elif ord == -Inf:
if row_axis > col_axis:
row_axis -= 1
ret = add.reduce(abs(x), axis=col_axis).min(axis=row_axis)
elif ord in [None, 'fro', 'f']:
ret = sqrt(add.reduce((x.conj() * x).real, axis=axis))
elif ord == 'nuc':
ret = _multi_svd_norm(x, row_axis, col_axis, sum)
else:
raise ValueError("Orden normativo inválido para las matrices.")
if keepdims:
ret_shape = list(x.shape)
ret_shape[axis[0]] = 1
ret_shape[axis[1]] = 1
ret = ret.reshape(ret_shape)
return ret
else:
raise ValueError("Número inadecuado de dimensiones a la norma.")
def norm(x, ord=None, axis=None):
"""
Norm of a sparse matrix
This function is able to return one of seven different matrix norms,
depending on the value of the ``ord`` parameter.
Parameters
----------
x : a sparse matrix
Input sparse matrix. If `axis` is None, `x` must be 1-D or 2-D sparse matrix.
ord : {non-zero int, inf, -inf, 'fro'}, optional
Order of the norm (see table under ``Notes``). inf means numpy's
`inf` object.
axis : {int, None}, optional
If `axis` is an integer, it specifies the axis of `x` along which to
compute the vector norms.
Returns
-------
n : float or matrix
Notes
-----
Some of the ord are not implemented because some associated functions like,
_multi_svd_norm, are not yet available for sparse matrix.
This docstring is modified based on numpy.linalg.norm.
https://github.com/numpy/numpy/blob/master/numpy/linalg/linalg.py
The following norms can be calculated:
===== ============================
ord norm for sparse matrices
===== ============================
None Frobenius norm
'fro' Frobenius norm
inf max(sum(abs(x), axis=1))
-inf min(sum(abs(x), axis=1))
0 abs(x).sum(axis=axis)
1 max(sum(abs(x), axis=0))
-1 min(sum(abs(x), axis=0))
2 Not implemented
-2 Not implemented
other Not implemented
===== ============================
The Frobenius norm is given by [1]_:
:math:`||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}`
References
----------
.. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15
Examples
--------
>>> from scipy.sparse import *
>>> import numpy as np
>>> from scipy.sparse.linalg import norm
>>> a = np.arange(9) - 4
>>> a
array([-4, -3, -2, -1, 0, 1, 2, 3, 4])
>>> b = a.reshape((3, 3))
>>> b
array([[-4, -3, -2],
[-1, 0, 1],
[ 2, 3, 4]])
>>> b = csr_matrix(b)
>>> norm(b)
7.745966692414834
>>> norm(b, 'fro')
7.745966692414834
>>> norm(b, np.inf)
9
>>> norm(b, -np.inf)
2
>>> norm(b, 1)
7
>>> norm(b, -1)
6
Using the `axis` argument to compute vector norms:
>>> c = np.array([[ 1, 2, 3],
... [-1, 1, 4]])
>>> c = csr_matrix(c)
>>> norm(c, axis=0)
matrix[[ 1.41421356, 2.23606798, 5. ]]
>>> norm(c, axis=1)
matrix[[ 3.74165739, 4.24264069]]
>>> norm(c, ord=1, axis=1)
matrix[[6]
[6]]
"""
if not issparse(x):
raise TypeError("input is not sparse. use numpy.linalg.norm")
# Check the default case first and handle it immediately.
if ord in [None, 'fro', 'f'] and axis is None:
if isComplexType(x.dtype.type):
sqnorm = dot(x.real, x.real) + dot(x.imag, x.imag)
else:
sqnorm = x.power(2).sum()
return sqrt(sqnorm)
# Normalize the `axis` argument to a tuple.
nd = x.ndim
if axis is None:
axis = tuple(range(nd))
if np.isscalar(axis):
if ord == Inf:
return max(abs(x).sum(axis=axis))
elif ord == -Inf:
return min(abs(x).sum(axis=axis))
elif ord == 0:
# Zero norm
return (x != 0).sum(axis=axis)
elif ord == 1:
# special case for speedup
return abs(x).sum(axis=axis)
elif ord == -1:
return min(abs(x).sum(axis=axis))
elif ord is None:
return sqrt(x.power(2).sum(axis=axis))
else:
raise NotImplementedError
elif len(axis) == 2:
row_axis, col_axis = axis
if not (-nd <= row_axis < nd and -nd <= col_axis < nd):
raise ValueError('Invalid axis %r for an array with shape %r' %
(axis, x.shape))
if row_axis % nd == col_axis % nd:
raise ValueError('Duplicate axes given.')
if ord == 2:
raise NotImplementedError
#return _multi_svd_norm(x, row_axis, col_axis, amax)
elif ord == -2:
raise NotImplementedError
#return _multi_svd_norm(x, row_axis, col_axis, amin)
elif ord == 1:
return abs(x).sum(axis=row_axis).max(axis=col_axis)[0,0]
elif ord == Inf:
return abs(x).sum(axis=col_axis).max(axis=row_axis)[0,0]
elif ord == -1:
return abs(x).sum(axis=row_axis).min(axis=col_axis)[0,0]
elif ord == -Inf:
return abs(x).sum(axis=col_axis).min(axis=row_axis)[0,0]
elif ord in [None, 'fro', 'f']:
return sqrt(x.power(2).sum(axis=axis))
else:
raise ValueError("Invalid norm order for matrices.")
else:
raise ValueError("Improper number of dimensions to norm.")
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False):
import numpy.core.numeric as NX
from numpy.core import isscalar, abs, dot
from numpy.lib.twodim_base import diag, vander
from numpy.linalg import eigvals, lstsq, inv
try:
from numpy.core import finfo # 1.7
except:
from numpy.lib.getlimits import finfo # 1.3 support for cluster
order = int(deg) + 1
x = NX.asarray(x) + 0.0
y = NX.asarray(y) + 0.0
# check arguments.
if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if x.shape[0] != y.shape[0]:
raise TypeError("expected x and y to have same length")
# set rcond
if rcond is None:
rcond = len(x) * finfo(x.dtype).eps
# set up least squares equation for powers of x
lhs = vander(x, order)
rhs = y
# apply weighting
if w is not None:
w = NX.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError, "expected a 1-d array for weights"
if w.shape[0] != y.shape[0]:
raise TypeError, "expected w and y to have the same length"
lhs *= w[:, NX.newaxis]
if rhs.ndim == 2:
rhs *= w[:, NX.newaxis]
else:
rhs *= w
# scale lhs to improve condition number and solve
scale = NX.sqrt((lhs * lhs).sum(axis=0))
lhs /= scale
c, resids, rank, s = lstsq(lhs, rhs, rcond)
c = (c.T / scale).T # broadcast scale coefficients
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
msg = "Polyfit may be poorly conditioned"
warnings.warn(msg, RankWarning)
if full:
return c, resids, rank, s, rcond
elif cov:
Vbase = inv(dot(lhs.T, lhs))
Vbase /= NX.outer(scale, scale)
# Some literature ignores the extra -2.0 factor in the denominator, but
# it is included here because the covariance of Multivariate Student-T
# (which is implied by a Bayesian uncertainty analysis) includes it.
# Plus, it gives a slightly more conservative estimate of uncertainty.
fac = resids / (len(x) - order - 2.0)
if y.ndim == 1:
return c, Vbase * fac
else:
return c, Vbase[:, :, NX.newaxis] * fac
else:
return c
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False):
"""
Least squares polynomial fit.
.. note::
This forms part of the old polynomial API. Since version 1.4, the
new polynomial API defined in `numpy.polynomial` is preferred.
A summary of the differences can be found in the
:doc:`transition guide </reference/routines.polynomials>`.
Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg`
to points `(x, y)`. Returns a vector of coefficients `p` that minimises
the squared error in the order `deg`, `deg-1`, ... `0`.
The `Polynomial.fit <numpy.polynomial.polynomial.Polynomial.fit>` class
method is recommended for new code as it is more stable numerically. See
the documentation of the method for more information.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int
Degree of the fitting polynomial
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (M,), optional
Weights. If not None, the weight ``w[i]`` applies to the unsquared
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
chosen so that the errors of the products ``w[i]*y[i]`` all have the
same variance. When using inverse-variance weighting, use
``w[i] = 1/sigma(y[i])``. The default value is None.
cov : bool or str, optional
If given and not `False`, return not just the estimate but also its
covariance matrix. By default, the covariance are scaled by
chi2/dof, where dof = M - (deg + 1), i.e., the weights are presumed
to be unreliable except in a relative sense and everything is scaled
such that the reduced chi2 is unity. This scaling is omitted if
``cov='unscaled'``, as is relevant for the case that the weights are
w = 1/sigma, with sigma known to be a reliable estimate of the
uncertainty.
Returns
-------
p : ndarray, shape (deg + 1,) or (deg + 1, K)
Polynomial coefficients, highest power first. If `y` was 2-D, the
coefficients for `k`-th data set are in ``p[:,k]``.
residuals, rank, singular_values, rcond
These values are only returned if ``full == True``
- residuals -- sum of squared residuals of the least squares fit
- rank -- the effective rank of the scaled Vandermonde
coefficient matrix
- singular_values -- singular values of the scaled Vandermonde
coefficient matrix
- rcond -- value of `rcond`.
For more details, see `numpy.linalg.lstsq`.
V : ndarray, shape (M,M) or (M,M,K)
Present only if ``full == False`` and ``cov == True``. The covariance
matrix of the polynomial coefficient estimates. The diagonal of
this matrix are the variance estimates for each coefficient. If y
is a 2-D array, then the covariance matrix for the `k`-th data set
are in ``V[:,:,k]``
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if ``full == False``.
The warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
polyval : Compute polynomial values.
linalg.lstsq : Computes a least-squares fit.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution minimizes the squared error
.. math::
E = \\sum_{j=0}^k |p(x_j) - y_j|^2
in the equations::
x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0]
x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1]
...
x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k]
The coefficient matrix of the coefficients `p` is a Vandermonde matrix.
`polyfit` issues a `RankWarning` when the least-squares fit is badly
conditioned. This implies that the best fit is not well-defined due
to numerical error. The results may be improved by lowering the polynomial
degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter
can also be set to a value smaller than its default, but the resulting
fit may be spurious: including contributions from the small singular
values can add numerical noise to the result.
Note that fitting polynomial coefficients is inherently badly conditioned
when the degree of the polynomial is large or the interval of sample points
is badly centered. The quality of the fit should always be checked in these
cases. When polynomial fits are not satisfactory, splines may be a good
alternative.
References
----------
.. [1] Wikipedia, "Curve fitting",
https://en.wikipedia.org/wiki/Curve_fitting
.. [2] Wikipedia, "Polynomial interpolation",
https://en.wikipedia.org/wiki/Polynomial_interpolation
Examples
--------
>>> import warnings
>>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0])
>>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
>>> z = np.polyfit(x, y, 3)
>>> z
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254]) # may vary
It is convenient to use `poly1d` objects for dealing with polynomials:
>>> p = np.poly1d(z)
>>> p(0.5)
0.6143849206349179 # may vary
>>> p(3.5)
-0.34732142857143039 # may vary
>>> p(10)
22.579365079365115 # may vary
High-order polynomials may oscillate wildly:
>>> with warnings.catch_warnings():
... warnings.simplefilter('ignore', np.RankWarning)
... p30 = np.poly1d(np.polyfit(x, y, 30))
...
>>> p30(4)
-0.80000000000000204 # may vary
>>> p30(5)
-0.99999999999999445 # may vary
>>> p30(4.5)
-0.10547061179440398 # may vary
Illustration:
>>> import matplotlib.pyplot as plt
>>> xp = np.linspace(-2, 6, 100)
>>> _ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--')
>>> plt.ylim(-2,2)
(-2, 2)
>>> plt.show()
"""
order = int(deg) + 1
x = NX.asarray(x) + 0.0
y = NX.asarray(y) + 0.0
# check arguments.
if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if x.shape[0] != y.shape[0]:
raise TypeError("expected x and y to have same length")
# set rcond
if rcond is None:
rcond = len(x)*finfo(x.dtype).eps
# set up least squares equation for powers of x
lhs = vander(x, order)
rhs = y
# apply weighting
if w is not None:
w = NX.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError("expected a 1-d array for weights")
if w.shape[0] != y.shape[0]:
raise TypeError("expected w and y to have the same length")
lhs *= w[:, NX.newaxis]
if rhs.ndim == 2:
rhs *= w[:, NX.newaxis]
else:
rhs *= w
# scale lhs to improve condition number and solve
scale = NX.sqrt((lhs*lhs).sum(axis=0))
lhs /= scale
c, resids, rank, s = lstsq(lhs, rhs, rcond)
c = (c.T/scale).T # broadcast scale coefficients
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
msg = "Polyfit may be poorly conditioned"
warnings.warn(msg, RankWarning, stacklevel=4)
if full:
return c, resids, rank, s, rcond
elif cov:
Vbase = inv(dot(lhs.T, lhs))
Vbase /= NX.outer(scale, scale)
if cov == "unscaled":
fac = 1
else:
if len(x) <= order:
raise ValueError("the number of data points must exceed order "
"to scale the covariance matrix")
# note, this used to be: fac = resids / (len(x) - order - 2.0)
# it was deciced that the "- 2" (originally justified by "Bayesian
# uncertainty analysis") is not what the user expects
# (see gh-11196 and gh-11197)
fac = resids / (len(x) - order)
if y.ndim == 1:
return c, Vbase * fac
else:
return c, Vbase[:,:, NX.newaxis] * fac
else:
return c
def norm(x):
return sqrt(dot(x, x))
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False):
import numpy.core.numeric as NX
from numpy.core import isscalar, abs, dot
from numpy.lib.twodim_base import diag, vander
from numpy.linalg import eigvals, lstsq, inv
try:
from numpy.core import finfo # 1.7
except:
from numpy.lib.getlimits import finfo # 1.3 support for cluster
order = int(deg) + 1
x = NX.asarray(x) + 0.0
y = NX.asarray(y) + 0.0
# check arguments.
if deg < 0 :
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
if y.ndim < 1 or y.ndim > 2 :
raise TypeError("expected 1D or 2D array for y")
if x.shape[0] != y.shape[0] :
raise TypeError("expected x and y to have same length")
# set rcond
if rcond is None :
rcond = len(x)*finfo(x.dtype).eps
# set up least squares equation for powers of x
lhs = vander(x, order)
rhs = y
# apply weighting
if w is not None:
w = NX.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError, "expected a 1-d array for weights"
if w.shape[0] != y.shape[0] :
raise TypeError, "expected w and y to have the same length"
lhs *= w[:, NX.newaxis]
if rhs.ndim == 2:
rhs *= w[:, NX.newaxis]
else:
rhs *= w
# scale lhs to improve condition number and solve
scale = NX.sqrt((lhs*lhs).sum(axis=0))
lhs /= scale
c, resids, rank, s = lstsq(lhs, rhs, rcond)
c = (c.T/scale).T # broadcast scale coefficients
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
msg = "Polyfit may be poorly conditioned"
warnings.warn(msg, RankWarning)
if full :
return c, resids, rank, s, rcond
elif cov :
Vbase = inv(dot(lhs.T,lhs))
Vbase /= NX.outer(scale, scale)
# Some literature ignores the extra -2.0 factor in the denominator, but
# it is included here because the covariance of Multivariate Student-T
# (which is implied by a Bayesian uncertainty analysis) includes it.
# Plus, it gives a slightly more conservative estimate of uncertainty.
fac = resids / (len(x) - order - 2.0)
if y.ndim == 1:
return c, Vbase * fac
else:
return c, Vbase[:,:,NX.newaxis] * fac
else :
return c
def joint_analysis(context, gene):
g, g_n, pvalue, n, n_indep, p_i_best, t_i_best, p_i_worst, t_i_worst, eigen_max, eigen_min, eigen_min_kept, z_min, z_max, z_mean, z_sd, tmi, status \
= None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, None, CalculationStatus.NO_DATA
g = gene.split(".")[0] if context.get_trimmed_ensemble_id() else gene
g_n = context.get_gene_name(g)
####################################################################################################################
zscores, tissue_labels = context.get_metaxcan_zscores(gene)
if not zscores or len(zscores) == 0:
status = CalculationStatus.NO_METAXCAN_RESULTS
return g, g_n, pvalue, n, n_indep, p_i_best, t_i_best, p_i_worst, t_i_worst, eigen_max, eigen_min, eigen_min_kept, z_min, z_max, z_mean, z_sd, tmi, status
n = len(zscores)
z_min = numpy.min(zscores)
z_max = numpy.max(zscores)
z_mean = numpy.mean(zscores)
if (len(zscores)>1):
z_sd = numpy.std(zscores, ddof=1)
####################################################################################################################
labels, matrix = context.get_model_matrix(gene, tissue_labels)
if not labels or len(labels) == 0:
status = CalculationStatus.NO_PRODUCT
return g, g_n, pvalue, n, n_indep, p_i_best, t_i_best, p_i_worst, t_i_worst, eigen_max, eigen_min, eigen_min_kept, z_min, z_max, z_mean, z_sd, tmi, status
# also, check that the matrix actually makes sense. We are currently returning it just in case but matrices with complex covariance are suspicious.
e, v = numpy.linalg.eigh(matrix)
if numpy.imag(e).any():
status = CalculationStatus.COMPLEX_COVARIANCE
e = numpy.real(e)
eigen_max, eigen_min = numpy.max(e), numpy.min(e)
return g, g_n, pvalue, n, n_indep, p_i_best, t_i_best, p_i_worst, t_i_worst, eigen_max, eigen_min, eigen_min_kept, z_min, z_max, z_mean, z_sd, tmi, status
# If no eigenvalue satisfies our cutoff criteria, at least the first component will be used
# Note there is a slight numerical mismatch between the resolution in eigh and the svd
cutoff = context.get_cutoff(matrix)
_d = {tissue_labels[i]:zscores[i] for i in xrange(0, len(tissue_labels))}
zscores = array([_d[l] for l in labels])
inv, n_indep, eigen = Math.capinv(matrix, cutoff, context.epsilon)
eigen_max, eigen_min = numpy.max(eigen), numpy.min(eigen)
eigen_min_kept = numpy.min([x for x in eigen[0:n_indep]])
_absz = numpy.abs(zscores)
_maxzi = numpy.argmax(_absz)
max_z = _absz[_maxzi]
p_i_best = 2*stats.norm.sf(max_z)
t_i_best = labels[_maxzi]
_minzi = numpy.argmin(_absz)
min_z = _absz[_minzi]
p_i_worst = 2*stats.norm.sf(min_z)
t_i_worst = labels[_minzi]
#TODO: implement a better heuristic
try:
eigen_w, eigen_v = numpy.linalg.eigh(inv)
except:
#WTCCC 'ENSG00000204560.5'
logging.log(8, "Problems with inverse for %s, skipping", gene)
status = CalculationStatus.INVERSE_ERROR
return g, g_n, pvalue, n, n_indep, p_i_best, t_i_best, p_i_worst, t_i_worst, eigen_max, eigen_min, eigen_min_kept, z_min, z_max, z_mean, z_sd, tmi, status
####################################################################################################################
w = float(dot(dot(zscores, inv), zscores))
chi2_p = stats.chi2.sf(w, n_indep)
tmi = numpy.trace(numpy.dot(matrix,inv))
# if we got to this point, we are ok-ish. The chi distribution might have been unable to calculate the pvalue because it is too small...
if chi2_p == 0:
status = CalculationStatus.INSUFFICIENT_NUMERICAL_RESOLUTION
else:
status = CalculationStatus.OK
pvalue = chi2_p
return g, g_n, pvalue, n, n_indep, p_i_best, t_i_best, p_i_worst, t_i_worst, eigen_max, eigen_min, eigen_min_kept, z_min, z_max, z_mean, z_sd, tmi, status
def polyfit(x, y, deg, rcond=None, full=False, w=None, cov=False):
"""
Least squares polynomial fit.
Fit a polynomial ``p(x) = p[0] * x**deg + ... + p[deg]`` of degree `deg`
to points `(x, y)`. Returns a vector of coefficients `p` that minimises
the squared error.
Parameters
----------
x : array_like, shape (M,)
x-coordinates of the M sample points ``(x[i], y[i])``.
y : array_like, shape (M,) or (M, K)
y-coordinates of the sample points. Several data sets of sample
points sharing the same x-coordinates can be fitted at once by
passing in a 2D-array that contains one dataset per column.
deg : int
Degree of the fitting polynomial
rcond : float, optional
Relative condition number of the fit. Singular values smaller than
this relative to the largest singular value will be ignored. The
default value is len(x)*eps, where eps is the relative precision of
the float type, about 2e-16 in most cases.
full : bool, optional
Switch determining nature of return value. When it is False (the
default) just the coefficients are returned, when True diagnostic
information from the singular value decomposition is also returned.
w : array_like, shape (M,), optional
weights to apply to the y-coordinates of the sample points.
cov : bool, optional
Return the estimate and the covariance matrix of the estimate
If full is True, then cov is not returned.
Returns
-------
p : ndarray, shape (M,) or (M, K)
Polynomial coefficients, highest power first. If `y` was 2-D, the
coefficients for `k`-th data set are in ``p[:,k]``.
residuals, rank, singular_values, rcond :
Present only if `full` = True. Residuals of the least-squares fit,
the effective rank of the scaled Vandermonde coefficient matrix,
its singular values, and the specified value of `rcond`. For more
details, see `linalg.lstsq`.
V : ndarray, shape (M,M) or (M,M,K)
Present only if `full` = False and `cov`=True. The covariance
matrix of the polynomial coefficient estimates. The diagonal of
this matrix are the variance estimates for each coefficient. If y
is a 2-D array, then the covariance matrix for the `k`-th data set
are in ``V[:,:,k]``
Warns
-----
RankWarning
The rank of the coefficient matrix in the least-squares fit is
deficient. The warning is only raised if `full` = False.
The warnings can be turned off by
>>> import warnings
>>> warnings.simplefilter('ignore', np.RankWarning)
See Also
--------
polyval : Computes polynomial values.
linalg.lstsq : Computes a least-squares fit.
scipy.interpolate.UnivariateSpline : Computes spline fits.
Notes
-----
The solution minimizes the squared error
.. math ::
E = \\sum_{j=0}^k |p(x_j) - y_j|^2
in the equations::
x[0]**n * p[0] + ... + x[0] * p[n-1] + p[n] = y[0]
x[1]**n * p[0] + ... + x[1] * p[n-1] + p[n] = y[1]
...
x[k]**n * p[0] + ... + x[k] * p[n-1] + p[n] = y[k]
The coefficient matrix of the coefficients `p` is a Vandermonde matrix.
`polyfit` issues a `RankWarning` when the least-squares fit is badly
conditioned. This implies that the best fit is not well-defined due
to numerical error. The results may be improved by lowering the polynomial
degree or by replacing `x` by `x` - `x`.mean(). The `rcond` parameter
can also be set to a value smaller than its default, but the resulting
fit may be spurious: including contributions from the small singular
values can add numerical noise to the result.
Note that fitting polynomial coefficients is inherently badly conditioned
when the degree of the polynomial is large or the interval of sample points
is badly centered. The quality of the fit should always be checked in these
cases. When polynomial fits are not satisfactory, splines may be a good
alternative.
References
----------
.. [1] Wikipedia, "Curve fitting",
http://en.wikipedia.org/wiki/Curve_fitting
.. [2] Wikipedia, "Polynomial interpolation",
http://en.wikipedia.org/wiki/Polynomial_interpolation
Examples
--------
>>> x = np.array([0.0, 1.0, 2.0, 3.0, 4.0, 5.0])
>>> y = np.array([0.0, 0.8, 0.9, 0.1, -0.8, -1.0])
>>> z = np.polyfit(x, y, 3)
>>> z
array([ 0.08703704, -0.81349206, 1.69312169, -0.03968254])
It is convenient to use `poly1d` objects for dealing with polynomials:
>>> p = np.poly1d(z)
>>> p(0.5)
0.6143849206349179
>>> p(3.5)
-0.34732142857143039
>>> p(10)
22.579365079365115
High-order polynomials may oscillate wildly:
>>> p30 = np.poly1d(np.polyfit(x, y, 30))
/... RankWarning: Polyfit may be poorly conditioned...
>>> p30(4)
-0.80000000000000204
>>> p30(5)
-0.99999999999999445
>>> p30(4.5)
-0.10547061179440398
Illustration:
>>> import matplotlib.pyplot as plt
>>> xp = np.linspace(-2, 6, 100)
>>> _ = plt.plot(x, y, '.', xp, p(xp), '-', xp, p30(xp), '--')
>>> plt.ylim(-2,2)
(-2, 2)
>>> plt.show()
"""
order = int(deg) + 1
x = NX.asarray(x) + 0.0
y = NX.asarray(y) + 0.0
# check arguments.
if deg < 0:
raise ValueError("expected deg >= 0")
if x.ndim != 1:
raise TypeError("expected 1D vector for x")
if x.size == 0:
raise TypeError("expected non-empty vector for x")
if y.ndim < 1 or y.ndim > 2:
raise TypeError("expected 1D or 2D array for y")
if x.shape[0] != y.shape[0]:
raise TypeError("expected x and y to have same length")
# set rcond
if rcond is None:
rcond = len(x) * finfo(x.dtype).eps
# set up least squares equation for powers of x
lhs = vander(x, order)
rhs = y
# apply weighting
if w is not None:
w = NX.asarray(w) + 0.0
if w.ndim != 1:
raise TypeError("expected a 1-d array for weights")
if w.shape[0] != y.shape[0]:
raise TypeError("expected w and y to have the same length")
lhs *= w[:, NX.newaxis]
if rhs.ndim == 2:
rhs *= w[:, NX.newaxis]
else:
rhs *= w
# scale lhs to improve condition number and solve
scale = NX.sqrt((lhs * lhs).sum(axis=0))
lhs /= scale
c, resids, rank, s = lstsq(lhs, rhs, rcond)
c = (c.T / scale).T # broadcast scale coefficients
# warn on rank reduction, which indicates an ill conditioned matrix
if rank != order and not full:
msg = "Polyfit may be poorly conditioned"
warnings.warn(msg, RankWarning)
if full:
return c, resids, rank, s, rcond
elif cov:
Vbase = inv(dot(lhs.T, lhs))
Vbase /= NX.outer(scale, scale)
# Some literature ignores the extra -2.0 factor in the denominator, but
# it is included here because the covariance of Multivariate Student-T
# (which is implied by a Bayesian uncertainty analysis) includes it.
# Plus, it gives a slightly more conservative estimate of uncertainty.
fac = resids / (len(x) - order - 2.0)
if y.ndim == 1:
return c, Vbase * fac
else:
return c, Vbase[:, :, NX.newaxis] * fac
else:
return c
def m_norm(x):
x = asanyarray(x).ravel(order='K')
return sqrt(dot(x, x))