def _setdiag(self, values, k):
M, N = self.shape
if k < 0:
if values.ndim == 0:
# broadcast
max_index = min(M+k, N)
for i in xrange(max_index):
self[i - k, i] = values
else:
max_index = min(M+k, N, len(values))
if max_index <= 0:
return
for i,v in enumerate(values[:max_index]):
self[i - k, i] = v
else:
if values.ndim == 0:
# broadcast
max_index = min(M, N-k)
for i in xrange(max_index):
self[i, i + k] = values
else:
max_index = min(M, N-k, len(values))
if max_index <= 0:
return
for i,v in enumerate(values[:max_index]):
self[i, i + k] = v
def _evaluate_derivatives(self, x, der=None):
n = self.n
r = self.r
if der is None:
der = self.n
pi = np.zeros((n, len(x)))
w = np.zeros((n, len(x)))
pi[0] = 1
p = np.zeros((len(x), self.r))
p += self.c[0,np.newaxis,:]
for k in xrange(1,n):
w[k-1] = x - self.xi[k-1]
pi[k] = w[k-1]*pi[k-1]
p += pi[k,:,np.newaxis]*self.c[k]
cn = np.zeros((max(der,n+1), len(x), r), dtype=self.dtype)
cn[:n+1,:,:] += self.c[:n+1,np.newaxis,:]
cn[0] = p
for k in xrange(1,n):
for i in xrange(1,n-k+1):
pi[i] = w[k+i-1]*pi[i-1]+pi[i]
cn[k] = cn[k]+pi[i,:,np.newaxis]*cn[k+i]
cn[k] *= factorial(k)
cn[n,:,:] = 0
return cn[:der]
def __init__(self, xi, yi, axis=0):
_Interpolator1DWithDerivatives.__init__(self, xi, yi, axis)
self.xi = np.asarray(xi)
self.yi = self._reshape_yi(yi)
self.n, self.r = self.yi.shape
c = np.zeros((self.n+1, self.r), dtype=self.dtype)
c[0] = self.yi[0]
Vk = np.zeros((self.n, self.r), dtype=self.dtype)
for k in xrange(1,self.n):
s = 0
while s <= k and xi[k-s] == xi[k]:
s += 1
s -= 1
Vk[0] = self.yi[k]/float(factorial(s))
for i in xrange(k-s):
if xi[i] == xi[k]:
raise ValueError("Elements if `xi` can't be equal.")
if s == 0:
Vk[i+1] = (c[i]-Vk[i])/(xi[i]-xi[k])
else:
Vk[i+1] = (Vk[i+1]-Vk[i])/(xi[i]-xi[k])
c[k] = Vk[k-s]
self.c = c
def __add__(self, other):
# First check if argument is a scalar
if isscalarlike(other):
res_dtype = upcast_scalar(self.dtype, other)
new = dok_matrix(self.shape, dtype=res_dtype)
# Add this scalar to every element.
M, N = self.shape
for i in xrange(M):
for j in xrange(N):
aij = self.get((i, j), 0) + other
if aij != 0:
new[i, j] = aij
# new.dtype.char = self.dtype.char
elif isinstance(other, dok_matrix):
if other.shape != self.shape:
raise ValueError("matrix dimensions are not equal")
# We could alternatively set the dimensions to the largest of
# the two matrices to be summed. Would this be a good idea?
res_dtype = upcast(self.dtype, other.dtype)
new = dok_matrix(self.shape, dtype=res_dtype)
new.update(self)
for key in other.keys():
new[key] += other[key]
elif isspmatrix(other):
csc = self.tocsc()
new = csc + other
elif isdense(other):
new = self.todense() + other
else:
return NotImplemented
return new
def __radd__(self, other):
# First check if argument is a scalar
if isscalarlike(other):
new = dok_matrix(self.shape, dtype=self.dtype)
# Add this scalar to every element.
M, N = self.shape
for i in xrange(M):
for j in xrange(N):
aij = self.get((i, j), 0) + other
if aij != 0:
new[i, j] = aij
elif isinstance(other, dok_matrix):
if other.shape != self.shape:
raise ValueError("matrix dimensions are not equal")
new = dok_matrix(self.shape, dtype=self.dtype)
new.update(self)
for key in other:
new[key] += other[key]
elif isspmatrix(other):
csc = self.tocsc()
new = csc + other
elif isdense(other):
new = other + self.todense()
else:
return NotImplemented
return new
def __add__(self, other):
if isscalarlike(other):
res_dtype = upcast_scalar(self.dtype, other)
new = dok_matrix(self.shape, dtype=res_dtype)
# Add this scalar to every element.
M, N = self.shape
for key in itertools.product(xrange(M), xrange(N)):
aij = dict.get(self, (key), 0) + other
if aij:
new[key] = aij
# new.dtype.char = self.dtype.char
elif isspmatrix_dok(other):
if other.shape != self.shape:
raise ValueError("Matrix dimensions are not equal.")
# We could alternatively set the dimensions to the largest of
# the two matrices to be summed. Would this be a good idea?
res_dtype = upcast(self.dtype, other.dtype)
new = dok_matrix(self.shape, dtype=res_dtype)
dict.update(new, self)
with np.errstate(over='ignore'):
dict.update(new,
((k, new[k] + other[k]) for k in iterkeys(other)))
elif isspmatrix(other):
csc = self.tocsc()
new = csc + other
elif isdense(other):
new = self.todense() + other
else:
return NotImplemented
return new
def extend(self, xi, yi, orders=None):
"""
Extend the PiecewisePolynomial by a list of points
Parameters
----------
xi : array_like
A sorted list of x-coordinates.
yi : list of lists of length N1
``yi[i]`` (if ``axis == 0``) is the list of derivatives known
at ``xi[i]``.
orders : int or list of ints, optional
A list of polynomial orders, or a single universal order.
"""
if self._y_axis == 0:
# allow yi to be a ragged list
for i in xrange(len(xi)):
if orders is None or _isscalar(orders):
self.append(xi[i],yi[i],orders)
else:
self.append(xi[i],yi[i],orders[i])
else:
preslice = (slice(None,None,None),) * self._y_axis
for i in xrange(len(xi)):
if orders is None or _isscalar(orders):
self.append(xi[i],yi[preslice + (i,)],orders)
else:
self.append(xi[i],yi[preslice + (i,)],orders[i])
def test_cascade(self):
for J in xrange(1, 7):
for i in xrange(1, 5):
lpcoef = wavelets.daub(i)
k = len(lpcoef)
x, phi, psi = wavelets.cascade(lpcoef, J)
assert_(len(x) == len(phi) == len(psi))
assert_equal(len(x), (k - 1) * 2 ** J)
def _get_arrayXarray(self, row, col):
# inner indexing
i, j = map(np.atleast_2d, np.broadcast_arrays(row, col))
newdok = dok_matrix(i.shape, dtype=self.dtype)
for key in itertools.product(xrange(i.shape[0]), xrange(i.shape[1])):
v = dict.get(self, (i[key], j[key]), 0)
if v:
dict.__setitem__(newdok, key, v)
return newdok
def test_correspond_2_and_up(self):
# Tests correspond(Z, y) on linkage and CDMs over observation sets of
# different sizes.
for i in xrange(2, 4):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_(correspond(Z, y))
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_(correspond(Z, y))
def test_improvement(self):
import time
start = time.time()
for i in xrange(100):
quad(self.lib.sin, 0, 100)
fast = time.time() - start
start = time.time()
for i in xrange(100):
quad(math.sin, 0, 100)
slow = time.time() - start
assert_(fast < 0.5*slow, (fast, slow))
def piecefuncgen(num):
Mk = order // 2 - num
if (Mk < 0):
return 0 # final function is 0
coeffs = [(1 - 2 * (k % 2)) * float(comb(order + 1, k, exact=1)) / fval
for k in xrange(Mk + 1)]
shifts = [-bound - k for k in xrange(Mk + 1)]
def thefunc(x):
res = 0.0
for k in range(Mk + 1):
res += coeffs[k] * (x + shifts[k]) ** order
return res
return thefunc
def test_nd_simplex(self):
# simple smoke test: triangulate a n-dimensional simplex
for nd in xrange(2, 8):
points = np.zeros((nd+1, nd))
for j in xrange(nd):
points[j,j] = 1.0
points[-1,:] = 1.0
tri = qhull.Delaunay(points)
tri.vertices.sort()
assert_equal(tri.vertices, np.arange(nd+1, dtype=int)[None,:])
assert_equal(tri.neighbors, -1 + np.zeros((nd+1), dtype=int)[None,:])
def _printresmat(function, interval, resmat):
# Print the Romberg result matrix.
i = j = 0
print('Romberg integration of', repr(function), end=' ')
print('from', interval)
print('')
print('%6s %9s %9s' % ('Steps', 'StepSize', 'Results'))
for i in xrange(len(resmat)):
print('%6d %9f' % (2**i, (interval[1]-interval[0])/(2.**i)), end=' ')
for j in xrange(i+1):
print('%9f' % (resmat[i][j]), end=' ')
print('')
print('')
print('The final result is', resmat[i][j], end=' ')
print('after', 2**(len(resmat)-1)+1, 'function evaluations.')
def test_improvement(self):
def myfunc(x): # Euler's constant integrand
return -exp(-x)*log(x)
import time
start = time.time()
for i in xrange(20):
quad(self.lib._multivariate_indefinite, 0, 100)
fast = time.time() - start
start = time.time()
for i in xrange(20):
quad(myfunc, 0, 100)
slow = time.time() - start
# 2+ times faster speeds generated by nontrivial ctypes
# function (single variable)
assert_(fast < 0.5*slow, (fast, slow))
def test_num_obs_linkage_4_and_up(self):
# Tests num_obs_linkage(Z) on linkage on observation sets between sizes
# 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_equal(num_obs_linkage(Z), i)
def setUp(self):
self.tck = splrep([0, 1, 2, 3, 4, 5], [0, 10, -1, 3, 7, 2], s=0)
self.test_xs = np.linspace(-1, 6, 100)
self.spline_ys = splev(self.test_xs, self.tck)
self.spline_yps = splev(self.test_xs, self.tck, der=1)
self.xi = np.unique(self.tck[0])
self.yi = [[splev(x, self.tck, der=j) for j in xrange(3)] for x in self.xi]
def test_exponential(self):
degree = 5
p = approximate_taylor_polynomial(np.exp, 0, degree, 1, 15)
for i in xrange(degree + 1):
assert_almost_equal(p(0), 1)
p = p.deriv()
assert_almost_equal(p(0), 0)
def test_num_obs_linkage_multi_matrix(self):
# Tests num_obs_linkage with observation matrices of multiple sizes.
for n in xrange(2, 10):
X = np.random.rand(n, 4)
Y = pdist(X)
Z = linkage(Y)
assert_equal(num_obs_linkage(Z), n)
def test_vector(self):
xs = [0, 1, 2]
ys = np.array([[0, 1], [1, 0], [2, 1]])
P = BarycentricInterpolator(xs, ys)
Pi = [BarycentricInterpolator(xs, ys[:, i]) for i in xrange(ys.shape[1])]
test_xs = np.linspace(-1, 3, 100)
assert_almost_equal(P(test_xs), np.rollaxis(np.asarray([p(test_xs) for p in Pi]), -1))
def test_call(self):
poly = []
for n in xrange(5):
poly.extend([x.strip() for x in
("""
orth.jacobi(%(n)d,0.3,0.9)
orth.sh_jacobi(%(n)d,0.3,0.9)
orth.genlaguerre(%(n)d,0.3)
orth.laguerre(%(n)d)
orth.hermite(%(n)d)
orth.hermitenorm(%(n)d)
orth.gegenbauer(%(n)d,0.3)
orth.chebyt(%(n)d)
orth.chebyu(%(n)d)
orth.chebyc(%(n)d)
orth.chebys(%(n)d)
orth.sh_chebyt(%(n)d)
orth.sh_chebyu(%(n)d)
orth.legendre(%(n)d)
orth.sh_legendre(%(n)d)
""" % dict(n=n)).split()
])
olderr = np.seterr(all='ignore')
try:
for pstr in poly:
p = eval(pstr)
assert_almost_equal(p(0.315), np.poly1d(p)(0.315), err_msg=pstr)
finally:
np.seterr(**olderr)
def test_is_valid_linkage_4_and_up(self):
# Tests is_valid_linkage(Z) on linkage on observation sets between
# sizes 4 and 15 (step size 3).
for i in xrange(4, 15, 3):
y = np.random.rand(i*(i-1)//2)
Z = linkage(y)
assert_(is_valid_linkage(Z) == True)
def test_concurrent_ok(self):
f = lambda t, y: 1.0
for k in xrange(3):
for sol in ('vode', 'zvode', 'lsoda', 'dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.1)
assert_allclose(r2.y, 0.2)
for sol in ('dopri5', 'dop853'):
r = ode(f).set_integrator(sol)
r.set_initial_value(0, 0)
r2 = ode(f).set_integrator(sol)
r2.set_initial_value(0, 0)
r.integrate(r.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
r.integrate(r.t + 0.1)
r2.integrate(r2.t + 0.1)
assert_allclose(r.y, 0.3)
assert_allclose(r2.y, 0.2)
def _get_row_ranges(self, rows, col_slice):
"""
Fast path for indexing in the case where column index is slice.
This gains performance improvement over brute force by more
efficient skipping of zeros, by accessing the elements
column-wise in order.
Parameters
----------
rows : sequence or xrange
Rows indexed. If xrange, must be within valid bounds.
col_slice : slice
Columns indexed
"""
j_start, j_stop, j_stride = col_slice.indices(self.shape[1])
col_range = xrange(j_start, j_stop, j_stride)
nj = len(col_range)
new = lil_matrix((len(rows), nj), dtype=self.dtype)
_csparsetools.lil_get_row_ranges(self.shape[0], self.shape[1],
self.rows, self.data,
new.rows, new.data,
rows,
j_start, j_stop, j_stride, nj)
return new
def _major_slice(self, idx, copy=False):
"""Index along the major axis where idx is a slice object.
"""
if idx == slice(None):
return self.copy() if copy else self
M, N = self._swap(self.shape)
start, stop, step = idx.indices(M)
M = len(xrange(start, stop, step))
new_shape = self._swap((M, N))
if M == 0:
return self.__class__(new_shape)
row_nnz = np.diff(self.indptr)
idx_dtype = self.indices.dtype
res_indptr = np.zeros(M+1, dtype=idx_dtype)
np.cumsum(row_nnz[idx], out=res_indptr[1:])
if step == 1:
all_idx = slice(self.indptr[start], self.indptr[stop])
res_indices = np.array(self.indices[all_idx], copy=copy)
res_data = np.array(self.data[all_idx], copy=copy)
else:
nnz = res_indptr[-1]
res_indices = np.empty(nnz, dtype=idx_dtype)
res_data = np.empty(nnz, dtype=self.dtype)
csr_row_slice(start, stop, step, self.indptr, self.indices,
self.data, res_indices, res_data)
return self.__class__((res_data, res_indices, res_indptr),
shape=new_shape, copy=False)
def _check_dot(self, jac_cls, complex=False, tol=1e-6, **kw):
np.random.seed(123)
N = 7
def rand(*a):
q = np.random.rand(*a)
if complex:
q = q + 1j*np.random.rand(*a)
return q
def assert_close(a, b, msg):
d = abs(a - b).max()
f = tol + abs(b).max()*tol
if d > f:
raise AssertionError('%s: err %g' % (msg, d))
self.A = rand(N, N)
# initialize
x0 = np.random.rand(N)
jac = jac_cls(**kw)
jac.setup(x0, self._func(x0), self._func)
# check consistency
for k in xrange(2*N):
v = rand(N)
if hasattr(jac, '__array__'):
Jd = np.array(jac)
if hasattr(jac, 'solve'):
Gv = jac.solve(v)
Gv2 = np.linalg.solve(Jd, v)
assert_close(Gv, Gv2, 'solve vs array')
if hasattr(jac, 'rsolve'):
Gv = jac.rsolve(v)
Gv2 = np.linalg.solve(Jd.T.conj(), v)
assert_close(Gv, Gv2, 'rsolve vs array')
if hasattr(jac, 'matvec'):
Jv = jac.matvec(v)
Jv2 = np.dot(Jd, v)
assert_close(Jv, Jv2, 'dot vs array')
if hasattr(jac, 'rmatvec'):
Jv = jac.rmatvec(v)
Jv2 = np.dot(Jd.T.conj(), v)
assert_close(Jv, Jv2, 'rmatvec vs array')
if hasattr(jac, 'matvec') and hasattr(jac, 'solve'):
Jv = jac.matvec(v)
Jv2 = jac.solve(jac.matvec(Jv))
assert_close(Jv, Jv2, 'dot vs solve')
if hasattr(jac, 'rmatvec') and hasattr(jac, 'rsolve'):
Jv = jac.rmatvec(v)
Jv2 = jac.rmatvec(jac.rsolve(Jv))
assert_close(Jv, Jv2, 'rmatvec vs rsolve')
x = rand(N)
jac.update(x, self._func(x))
def test_more_barycentric_transforms(self):
# Triangulate some "nasty" grids
eps = np.finfo(float).eps
npoints = {2: 70, 3: 11, 4: 5, 5: 3}
_is_32bit_platform = np.intp(0).itemsize < 8
for ndim in xrange(2, 6):
# Generate an uniform grid in n-d unit cube
x = np.linspace(0, 1, npoints[ndim])
grid = np.c_[list(map(np.ravel, np.broadcast_arrays(*np.ix_(*([x]*ndim)))))].T
err_msg = "ndim=%d" % ndim
# Check using regular grid
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri, err_msg=err_msg,
unit_cube=True)
# Check with eps-perturbations
np.random.seed(1234)
m = (np.random.rand(grid.shape[0]) < 0.2)
grid[m,:] += 2*eps*(np.random.rand(*grid[m,:].shape) - 0.5)
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri, err_msg=err_msg,
unit_cube=True,
unit_cube_tol=2*eps)
# Check with duplicated data
tri = qhull.Delaunay(np.r_[grid, grid])
self._check_barycentric_transforms(tri, err_msg=err_msg,
unit_cube=True,
unit_cube_tol=2*eps)
if not _is_32bit_platform:
# test numerically unstable, and reported to fail on 32-bit
# installs
# Check with larger perturbations
np.random.seed(4321)
m = (np.random.rand(grid.shape[0]) < 0.2)
grid[m,:] += 1000*eps*(np.random.rand(*grid[m,:].shape) - 0.5)
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri, err_msg=err_msg,
unit_cube=True,
unit_cube_tol=1500*eps)
# Check with yet larger perturbations
np.random.seed(4321)
m = (np.random.rand(grid.shape[0]) < 0.2)
grid[m,:] += 1e6*eps*(np.random.rand(*grid[m,:].shape) - 0.5)
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri, err_msg=err_msg,
unit_cube=True,
unit_cube_tol=1e7*eps)
def test_incremental(self):
with warnings.catch_warnings():
warnings.filterwarnings("ignore", category=DeprecationWarning)
P = PiecewisePolynomial([self.xi[0]], [self.yi[0]], 3)
for i in xrange(1, len(self.xi)):
P.append(self.xi[i], self.yi[i], 3)
assert_almost_equal(P(self.test_xs), self.spline_ys)
def test_inverse(self):
for n in xrange(1, 10):
a = hilbert(n)
b = invhilbert(n)
# The Hilbert matrix is increasingly badly conditioned,
# so take that into account in the test
c = cond(a)
assert_allclose(a.dot(b), eye(n), atol=1e-15*c, rtol=1e-15*c)
def hermite_recursion(n, nodes):
H = np.zeros((n, nodes.size))
H[0,:] = np.pi**(-0.25) * np.exp(-0.5*nodes**2)
if n > 1:
H[1,:] = sqrt(2.0) * nodes * H[0,:]
for k in xrange(2, n):
H[k,:] = sqrt(2.0/k) * nodes * H[k-1,:] - sqrt((k-1.0)/k) * H[k-2,:]
return H
def solve(self):
"""
Runs the DifferentialEvolutionSolver.
Returns
-------
res : OptimizeResult
The optimization result represented as a ``OptimizeResult`` object.
Important attributes are: ``x`` the solution array, ``success`` a
Boolean flag indicating if the optimizer exited successfully and
``message`` which describes the cause of the termination. See
`OptimizeResult` for a description of other attributes. If `polish`
was employed, and a lower minimum was obtained by the polishing,
then OptimizeResult also contains the ``jac`` attribute.
"""
nit, warning_flag = 0, False
# dictionary that holds standard status messages of optimizers
status_message = _status_message['success']
# The population may have just been initialized (all entries are
# np.inf). If it has you have to calculate the initial energies.
# Although this is also done in the evolve generator it's possible
# that someone can set maxiter=0, at which point we still want the
# initial energies to be calculated (the following loop isn't run).
#np.all checks that there are no 0's in the array
if self.maxiter == 0:
if np.all(np.isinf(self.population_energies)):
if self.disp:
print("Calculating initial energies when maxiter = 0")
self._calculate_population_energies()
# for i in range(self.num_population_members):
# print(self.population[i,:])
# do the optimisation.
for nit in xrange(1, self.maxiter + 1):
if self.disp:
print("iter: ", nit)
# evolve the population by a generation
try:
next(self)
except StopIteration:
warning_flag = True
status_message = _status_message['maxfev']
break
print("differential_evolution step %d: f(x)= %g" %
(nit, self.population_energies[0]))
#save populations at each iter and rank to analyze after
# np.save("before_rank"+str(self.rank)+"iter"+str(nit), self.population)
#migrate
self.migration()
# np.save("after_rank"+str(self.rank)+"iter"+str(nit), self.population)
# should the solver terminate?
# print("Checking if should converge")
# convergence = self.convergence
#
# if (self.callback and
# self.callback(self._scale_parameters(self.population[0]),
# convergence=self.tol / convergence) is True):
#
# warning_flag = True
# status_message = ('callback function requested stop early '
# 'by returning True')
# break
# print("checking if tolerance level reached")
## intol = (np.std(self.population_energies) <=
## self.atol +
## self.tol * np.abs(np.mean(self.population_energies)))
#
# intol = self.population_energies[0] <= self.mse_thresh
# if warning_flag or intol:
# print("stopping iterations")
# break
print("Starting next iter")
else:
status_message = _status_message['maxiter']
warning_flag = True
DE_result = OptimizeResult(x=self.x,
fun=self.population_energies[0],
nfev=self._nfev,
nit=nit,
message=status_message,
success=(warning_flag is not True))
print("done iters")
if self.polish:
print("performing final polishing")
result = minimize(self.func,
np.copy(DE_result.x),
method='L-BFGS-B',
bounds=self.limits.T,
args=self.args)
self._nfev += result.nfev
DE_result.nfev = self._nfev
if result.fun < DE_result.fun:
DE_result.fun = result.fun
DE_result.x = result.x
DE_result.jac = result.jac
# to keep internal state consistent
self.population_energies[0] = result.fun
self.population[0] = self._unscale_parameters(result.x)
return DE_result
def create_individual(data):
return [random.randint(0, 1) for _ in xrange(len(data))]
def _get_sliceXslice(self, row, col):
row = xrange(*row.indices(self.shape[0]))
return self._get_row_ranges(row, col)
def romb(y, dx=1.0, axis=-1, show=False):
"""
Romberg integration using samples of a function.
Parameters
----------
y : array_like
A vector of ``2**k + 1`` equally-spaced samples of a function.
dx : float, optional
The sample spacing. Default is 1.
axis : int, optional
The axis along which to integrate. Default is -1 (last axis).
show : bool, optional
When `y` is a single 1-D array, then if this argument is True
print the table showing Richardson extrapolation from the
samples. Default is False.
Returns
-------
romb : ndarray
The integrated result for `axis`.
See also
--------
quad : adaptive quadrature using QUADPACK
romberg : adaptive Romberg quadrature
quadrature : adaptive Gaussian quadrature
fixed_quad : fixed-order Gaussian quadrature
dblquad : double integrals
tplquad : triple integrals
simps : integrators for sampled data
cumtrapz : cumulative integration for sampled data
ode : ODE integrators
odeint : ODE integrators
Examples
--------
>>> from scipy import integrate
>>> x = np.arange(10, 14.25, 0.25)
>>> y = np.arange(3, 12)
>>> integrate.romb(y)
56.0
>>> y = np.sin(np.power(x, 2.5))
>>> integrate.romb(y)
-0.742561336672229
>>> integrate.romb(y, show=True)
Richardson Extrapolation Table for Romberg Integration
====================================================================
-0.81576
4.63862 6.45674
-1.10581 -3.02062 -3.65245
-2.57379 -3.06311 -3.06595 -3.05664
-1.34093 -0.92997 -0.78776 -0.75160 -0.74256
====================================================================
-0.742561336672229
"""
y = np.asarray(y)
nd = len(y.shape)
Nsamps = y.shape[axis]
Ninterv = Nsamps - 1
n = 1
k = 0
while n < Ninterv:
n <<= 1
k += 1
if n != Ninterv:
raise ValueError("Number of samples must be one plus a "
"non-negative power of 2.")
R = {}
slice_all = (slice(None), ) * nd
slice0 = tupleset(slice_all, axis, 0)
slicem1 = tupleset(slice_all, axis, -1)
h = Ninterv * np.asarray(dx, dtype=float)
R[(0, 0)] = (y[slice0] + y[slicem1]) / 2.0 * h
slice_R = slice_all
start = stop = step = Ninterv
for i in xrange(1, k + 1):
start >>= 1
slice_R = tupleset(slice_R, axis, slice(start, stop, step))
step >>= 1
R[(i, 0)] = 0.5 * (R[(i - 1, 0)] + h * y[slice_R].sum(axis=axis))
for j in xrange(1, i + 1):
prev = R[(i, j - 1)]
R[(i, j)] = prev + (prev - R[(i - 1, j - 1)]) / ((1 <<
(2 * j)) - 1)
h /= 2.0
if show:
if not np.isscalar(R[(0, 0)]):
print("*** Printing table only supported for integrals" +
" of a single data set.")
else:
try:
precis = show[0]
except (TypeError, IndexError):
precis = 5
try:
width = show[1]
except (TypeError, IndexError):
width = 8
formstr = "%%%d.%df" % (width, precis)
title = "Richardson Extrapolation Table for Romberg Integration"
print("", title.center(68), "=" * 68, sep="\n", end="\n")
for i in xrange(k + 1):
for j in xrange(i + 1):
print(formstr % R[(i, j)], end=" ")
print()
print("=" * 68)
print()
return R[(k, k)]
def quadrature(func,
a,
b,
args=(),
tol=1.49e-8,
rtol=1.49e-8,
maxiter=50,
vec_func=True,
miniter=1):
"""
Compute a definite integral using fixed-tolerance Gaussian quadrature.
Integrate `func` from `a` to `b` using Gaussian quadrature
with absolute tolerance `tol`.
Parameters
----------
func : function
A Python function or method to integrate.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
args : tuple, optional
Extra arguments to pass to function.
tol, rtol : float, optional
Iteration stops when error between last two iterates is less than
`tol` OR the relative change is less than `rtol`.
maxiter : int, optional
Maximum order of Gaussian quadrature.
vec_func : bool, optional
True or False if func handles arrays as arguments (is
a "vector" function). Default is True.
miniter : int, optional
Minimum order of Gaussian quadrature.
Returns
-------
val : float
Gaussian quadrature approximation (within tolerance) to integral.
err : float
Difference between last two estimates of the integral.
See also
--------
romberg: adaptive Romberg quadrature
fixed_quad: fixed-order Gaussian quadrature
quad: adaptive quadrature using QUADPACK
dblquad: double integrals
tplquad: triple integrals
romb: integrator for sampled data
simps: integrator for sampled data
cumtrapz: cumulative integration for sampled data
ode: ODE integrator
odeint: ODE integrator
Examples
--------
>>> from scipy import integrate
>>> f = lambda x: x**8
>>> integrate.quadrature(f, 0.0, 1.0)
(0.11111111111111106, 4.163336342344337e-17)
>>> print(1/9.0) # analytical result
0.1111111111111111
>>> integrate.quadrature(np.cos, 0.0, np.pi/2)
(0.9999999999999536, 3.9611425250996035e-11)
>>> np.sin(np.pi/2)-np.sin(0) # analytical result
1.0
"""
if not isinstance(args, tuple):
args = (args, )
vfunc = vectorize1(func, args, vec_func=vec_func)
val = np.inf
err = np.inf
maxiter = max(miniter + 1, maxiter)
for n in xrange(miniter, maxiter + 1):
newval = fixed_quad(vfunc, a, b, (), n)[0]
err = abs(newval - val)
val = newval
if err < tol or err < rtol * abs(val):
break
else:
warnings.warn(
"maxiter (%d) exceeded. Latest difference = %e" % (maxiter, err),
AccuracyWarning)
return val, err
def scalar_search_wolfe2(phi, derphi=None, phi0=None,
old_phi0=None, derphi0=None,
c1=1e-4, c2=0.9, amax=None,
extra_condition=None, maxiter=10):
"""Find alpha that satisfies strong Wolfe conditions.
alpha > 0 is assumed to be a descent direction.
Parameters
----------
phi : callable f(x)
Objective scalar function.
derphi : callable f'(x), optional
Objective function derivative (can be None)
phi0 : float, optional
Value of phi at s=0
old_phi0 : float, optional
Value of phi at previous point
derphi0 : float, optional
Value of derphi at s=0
c1 : float, optional
Parameter for Armijo condition rule.
c2 : float, optional
Parameter for curvature condition rule.
amax : float, optional
Maximum step size
extra_condition : callable, optional
A callable of the form ``extra_condition(alpha, phi_value)``
returning a boolean. The line search accepts the value
of ``alpha`` only if this callable returns ``True``.
If the callable returns ``False`` for the step length,
the algorithm will continue with new iterates.
The callable is only called for iterates satisfying
the strong Wolfe conditions.
maxiter : int, optional
Maximum number of iterations to perform
Returns
-------
alpha_star : float or None
Best alpha, or None if the line search algorithm did not converge.
phi_star : float
phi at alpha_star
phi0 : float
phi at 0
derphi_star : float or None
derphi at alpha_star, or None if the line search algorithm
did not converge.
Notes
-----
Uses the line search algorithm to enforce strong Wolfe
conditions. See Wright and Nocedal, 'Numerical Optimization',
1999, pg. 59-60.
For the zoom phase it uses an algorithm by [...].
"""
if phi0 is None:
phi0 = phi(0.)
if derphi0 is None and derphi is not None:
derphi0 = derphi(0.)
alpha0 = 0
if old_phi0 is not None and derphi0 != 0:
alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0)
else:
alpha1 = 1.0
if alpha1 < 0:
alpha1 = 1.0
phi_a1 = phi(alpha1)
#derphi_a1 = derphi(alpha1) evaluated below
phi_a0 = phi0
derphi_a0 = derphi0
if extra_condition is None:
extra_condition = lambda alpha, phi: True
for i in xrange(maxiter):
if alpha1 == 0 or (amax is not None and alpha0 == amax):
# alpha1 == 0: This shouldn't happen. Perhaps the increment has
# slipped below machine precision?
alpha_star = None
phi_star = phi0
phi0 = old_phi0
derphi_star = None
if alpha1 == 0:
msg = 'Rounding errors prevent the line search from converging'
else:
msg = "The line search algorithm could not find a solution " + \
"less than or equal to amax: %s" % amax
warn(msg, LineSearchWarning)
break
if (phi_a1 > phi0 + c1 * alpha1 * derphi0) or \
((phi_a1 >= phi_a0) and (i > 1)):
alpha_star, phi_star, derphi_star = \
_zoom(alpha0, alpha1, phi_a0,
phi_a1, derphi_a0, phi, derphi,
phi0, derphi0, c1, c2, extra_condition)
break
derphi_a1 = derphi(alpha1)
if (abs(derphi_a1) <= -c2*derphi0):
if extra_condition(alpha1, phi_a1):
alpha_star = alpha1
phi_star = phi_a1
derphi_star = derphi_a1
break
if (derphi_a1 >= 0):
alpha_star, phi_star, derphi_star = \
_zoom(alpha1, alpha0, phi_a1,
phi_a0, derphi_a1, phi, derphi,
phi0, derphi0, c1, c2, extra_condition)
break
alpha2 = 2 * alpha1 # increase by factor of two on each iteration
if amax is not None:
alpha2 = min(alpha2, amax)
alpha0 = alpha1
alpha1 = alpha2
phi_a0 = phi_a1
phi_a1 = phi(alpha1)
derphi_a0 = derphi_a1
else:
# stopping test maxiter reached
alpha_star = alpha1
phi_star = phi_a1
derphi_star = None
warn('The line search algorithm did not converge', LineSearchWarning)
return alpha_star, phi_star, phi0, derphi_star
def nonlin_solve(F,
x0,
jacobian='krylov',
iter=None,
verbose=False,
maxiter=None,
f_tol=None,
f_rtol=None,
x_tol=None,
x_rtol=None,
tol_norm=None,
line_search='armijo',
callback=None,
full_output=False,
raise_exception=True):
"""
Find a root of a function, in a way suitable for large-scale problems.
Parameters
----------
%(params_basic)s
jacobian : Jacobian
A Jacobian approximation: `Jacobian` object or something that
`asjacobian` can transform to one. Alternatively, a string specifying
which of the builtin Jacobian approximations to use:
krylov, broyden1, broyden2, anderson
diagbroyden, linearmixing, excitingmixing
%(params_extra)s
full_output : bool
If true, returns a dictionary `info` containing convergence
information.
raise_exception : bool
If True, a `NoConvergence` exception is raise if no solution is found.
See Also
--------
asjacobian, Jacobian
Notes
-----
This algorithm implements the inexact Newton method, with
backtracking or full line searches. Several Jacobian
approximations are available, including Krylov and Quasi-Newton
methods.
References
----------
.. [KIM] C. T. Kelley, \"Iterative Methods for Linear and Nonlinear
Equations\". Society for Industrial and Applied Mathematics. (1995)
https://archive.siam.org/books/kelley/fr16/
"""
# Can't use default parameters because it's being explicitly passed as None
# from the calling function, so we need to set it here.
tol_norm = maxnorm if tol_norm is None else tol_norm
condition = TerminationCondition(f_tol=f_tol,
f_rtol=f_rtol,
x_tol=x_tol,
x_rtol=x_rtol,
iter=iter,
norm=tol_norm)
x0 = _as_inexact(x0)
func = lambda z: _as_inexact(F(_array_like(z, x0))).flatten()
x = x0.flatten()
dx = np.inf
Fx = func(x)
Fx_norm = norm(Fx)
jacobian = asjacobian(jacobian)
jacobian.setup(x.copy(), Fx, func)
if maxiter is None:
if iter is not None:
maxiter = iter + 1
else:
maxiter = 100 * (x.size + 1)
if line_search is True:
line_search = 'armijo'
elif line_search is False:
line_search = None
if line_search not in (None, 'armijo', 'wolfe'):
raise ValueError("Invalid line search")
# Solver tolerance selection
gamma = 0.9
eta_max = 0.9999
eta_treshold = 0.1
eta = 1e-3
for n in xrange(maxiter):
status = condition.check(Fx, x, dx)
if status:
break
# The tolerance, as computed for scipy.sparse.linalg.* routines
tol = min(eta, eta * Fx_norm)
dx = -jacobian.solve(Fx, tol=tol)
if norm(dx) == 0:
raise ValueError("Jacobian inversion yielded zero vector. "
"This indicates a bug in the Jacobian "
"approximation.")
# Line search, or Newton step
if line_search:
s, x, Fx, Fx_norm_new = _nonlin_line_search(
func, x, Fx, dx, line_search)
else:
s = 1.0
x = x + dx
Fx = func(x)
Fx_norm_new = norm(Fx)
jacobian.update(x.copy(), Fx)
if callback:
callback(x, Fx)
# Adjust forcing parameters for inexact methods
eta_A = gamma * Fx_norm_new**2 / Fx_norm**2
if gamma * eta**2 < eta_treshold:
eta = min(eta_max, eta_A)
else:
eta = min(eta_max, max(eta_A, gamma * eta**2))
Fx_norm = Fx_norm_new
# Print status
if verbose:
sys.stdout.write("%d: |F(x)| = %g; step %g\n" %
(n, tol_norm(Fx), s))
sys.stdout.flush()
else:
if raise_exception:
raise NoConvergence(_array_like(x, x0))
else:
status = 2
if full_output:
info = {
'nit': condition.iteration,
'fun': Fx,
'status': status,
'success': status == 1,
'message': {
1: 'A solution was found at the specified '
'tolerance.',
2: 'The maximum number of iterations allowed '
'has been reached.'
}[status]
}
return _array_like(x, x0), info
else:
return _array_like(x, x0)
def test_more_barycentric_transforms(self):
# Triangulate some "nasty" grids
eps = np.finfo(float).eps
npoints = {2: 70, 3: 11, 4: 5, 5: 3}
_is_32bit_platform = np.intp(0).itemsize < 8
for ndim in xrange(2, 6):
# Generate an uniform grid in n-d unit cube
x = np.linspace(0, 1, npoints[ndim])
grid = np.c_[list(
map(np.ravel, np.broadcast_arrays(*np.ix_(*([x] * ndim)))))].T
err_msg = "ndim=%d" % ndim
# Check using regular grid
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True)
# Check with eps-perturbations
np.random.seed(1234)
m = (np.random.rand(grid.shape[0]) < 0.2)
grid[m, :] += 2 * eps * (np.random.rand(*grid[m, :].shape) - 0.5)
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True,
unit_cube_tol=2 * eps)
# Check with duplicated data
tri = qhull.Delaunay(np.r_[grid, grid])
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True,
unit_cube_tol=2 * eps)
if not _is_32bit_platform:
# test numerically unstable, and reported to fail on 32-bit
# installs
# Check with larger perturbations
np.random.seed(4321)
m = (np.random.rand(grid.shape[0]) < 0.2)
grid[m, :] += 1000 * eps * (np.random.rand(*grid[m, :].shape) -
0.5)
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True,
unit_cube_tol=1500 * eps)
# Check with yet larger perturbations
np.random.seed(4321)
m = (np.random.rand(grid.shape[0]) < 0.2)
grid[m, :] += 1e6 * eps * (np.random.rand(*grid[m, :].shape) -
0.5)
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True,
unit_cube_tol=1e7 * eps)
def random(m, n, density=0.01, format='coo', dtype=None,
random_state=None, data_rvs=None):
"""Generate a sparse matrix of the given shape and density with randomly
distributed values.
Parameters
----------
m, n : int
shape of the matrix
density : real, optional
density of the generated matrix: density equal to one means a full
matrix, density of 0 means a matrix with no non-zero items.
format : str, optional
sparse matrix format.
dtype : dtype, optional
type of the returned matrix values.
random_state : {numpy.random.RandomState, int}, optional
Random number generator or random seed. If not given, the singleton
numpy.random will be used. This random state will be used
for sampling the sparsity structure, but not necessarily for sampling
the values of the structurally nonzero entries of the matrix.
data_rvs : callable, optional
Samples a requested number of random values.
This function should take a single argument specifying the length
of the ndarray that it will return. The structurally nonzero entries
of the sparse random matrix will be taken from the array sampled
by this function. By default, uniform [0, 1) random values will be
sampled using the same random state as is used for sampling
the sparsity structure.
Examples
--------
>>> from scipy.sparse import construct
>>> from scipy import stats
>>> class CustomRandomState(object):
... def randint(self, k):
... i = np.random.randint(k)
... return i - i % 2
>>> rs = CustomRandomState()
>>> rvs = stats.poisson(25, loc=10).rvs
>>> S = construct.random(3, 4, density=0.25, random_state=rs, data_rvs=rvs)
>>> S.A
array([[ 36., 0., 33., 0.],
[ 0., 0., 0., 0.],
[ 0., 0., 36., 0.]])
Notes
-----
Only float types are supported for now.
"""
if density < 0 or density > 1:
raise ValueError("density expected to be 0 <= density <= 1")
if dtype and (dtype not in [np.float32, np.float64, np.longdouble]):
raise NotImplementedError("type %s not supported" % dtype)
mn = m * n
tp = np.intc
if mn > np.iinfo(tp).max:
tp = np.int64
if mn > np.iinfo(tp).max:
msg = """\
Trying to generate a random sparse matrix such as the product of dimensions is
greater than %d - this is not supported on this machine
"""
raise ValueError(msg % np.iinfo(tp).max)
# Number of non zero values
k = int(density * m * n)
if random_state is None:
random_state = np.random
elif isinstance(random_state, (int, np.integer)):
random_state = np.random.RandomState(random_state)
if data_rvs is None:
data_rvs = random_state.rand
# Use the algorithm from python's random.sample for k < mn/3.
if mn < 3*k:
# We should use this line, but choice is only available in numpy >= 1.7
# ind = random_state.choice(mn, size=k, replace=False)
ind = random_state.permutation(mn)[:k]
else:
ind = np.empty(k, dtype=tp)
selected = set()
for i in xrange(k):
j = random_state.randint(mn)
while j in selected:
j = random_state.randint(mn)
selected.add(j)
ind[i] = j
j = np.floor(ind * 1. / m).astype(tp)
i = (ind - j * m).astype(tp)
vals = data_rvs(k).astype(dtype)
return coo_matrix((vals, (i, j)), shape=(m, n)).asformat(format)
def gcrotmk(A,
b,
x0=None,
tol=1e-5,
maxiter=1000,
M=None,
callback=None,
m=20,
k=None,
CU=None,
discard_C=False,
truncate='oldest',
atol=None):
"""
Solve a matrix equation using flexible GCROT(m,k) algorithm.
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}
The real or complex N-by-N matrix of the linear system.
b : {array, matrix}
Right hand side of the linear system. Has shape (N,) or (N,1).
x0 : {array, matrix}
Starting guess for the solution.
tol, atol : float, optional
Tolerances for convergence, ``norm(residual) <= max(tol*norm(b), atol)``.
The default for ``atol`` is `tol`.
.. warning::
The default value for `atol` will be changed in a future release.
For future compatibility, specify `atol` explicitly.
maxiter : int, optional
Maximum number of iterations. Iteration will stop after maxiter
steps even if the specified tolerance has not been achieved.
M : {sparse matrix, dense matrix, LinearOperator}, optional
Preconditioner for A. The preconditioner should approximate the
inverse of A. gcrotmk is a 'flexible' algorithm and the preconditioner
can vary from iteration to iteration. Effective preconditioning
dramatically improves the rate of convergence, which implies that
fewer iterations are needed to reach a given error tolerance.
callback : function, optional
User-supplied function to call after each iteration. It is called
as callback(xk), where xk is the current solution vector.
m : int, optional
Number of inner FGMRES iterations per each outer iteration.
Default: 20
k : int, optional
Number of vectors to carry between inner FGMRES iterations.
According to [2]_, good values are around m.
Default: m
CU : list of tuples, optional
List of tuples ``(c, u)`` which contain the columns of the matrices
C and U in the GCROT(m,k) algorithm. For details, see [2]_.
The list given and vectors contained in it are modified in-place.
If not given, start from empty matrices. The ``c`` elements in the
tuples can be ``None``, in which case the vectors are recomputed
via ``c = A u`` on start and orthogonalized as described in [3]_.
discard_C : bool, optional
Discard the C-vectors at the end. Useful if recycling Krylov subspaces
for different linear systems.
truncate : {'oldest', 'smallest'}, optional
Truncation scheme to use. Drop: oldest vectors, or vectors with
smallest singular values using the scheme discussed in [1,2].
See [2]_ for detailed comparison.
Default: 'oldest'
Returns
-------
x : array or matrix
The solution found.
info : int
Provides convergence information:
* 0 : successful exit
* >0 : convergence to tolerance not achieved, number of iterations
References
----------
.. [1] E. de Sturler, ''Truncation strategies for optimal Krylov subspace
methods'', SIAM J. Numer. Anal. 36, 864 (1999).
.. [2] J.E. Hicken and D.W. Zingg, ''A simplified and flexible variant
of GCROT for solving nonsymmetric linear systems'',
SIAM J. Sci. Comput. 32, 172 (2010).
.. [3] M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti,
''Recycling Krylov subspaces for sequences of linear systems'',
SIAM J. Sci. Comput. 28, 1651 (2006).
"""
A, M, x, b, postprocess = make_system(A, M, x0, b)
if not np.isfinite(b).all():
raise ValueError("RHS must contain only finite numbers")
if truncate not in ('oldest', 'smallest'):
raise ValueError("Invalid value for 'truncate': %r" % (truncate, ))
if atol is None:
warnings.warn(
"scipy.sparse.linalg.gcrotmk called without specifying `atol`. "
"The default value will change in the future. To preserve "
"current behavior, set ``atol=tol``.",
category=DeprecationWarning,
stacklevel=2)
atol = tol
matvec = A.matvec
psolve = M.matvec
if CU is None:
CU = []
if k is None:
k = m
axpy, dot, scal = None, None, None
r = b - matvec(x)
axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'],
(x, r))
b_norm = nrm2(b)
if discard_C:
CU[:] = [(None, u) for c, u in CU]
# Reorthogonalize old vectors
if CU:
# Sort already existing vectors to the front
CU.sort(key=lambda cu: cu[0] is not None)
# Fill-in missing ones
C = np.empty((A.shape[0], len(CU)), dtype=r.dtype, order='F')
us = []
j = 0
while CU:
# More memory-efficient: throw away old vectors as we go
c, u = CU.pop(0)
if c is None:
c = matvec(u)
C[:, j] = c
j += 1
us.append(u)
# Orthogonalize
Q, R, P = qr(C, overwrite_a=True, mode='economic', pivoting=True)
del C
# C := Q
cs = list(Q.T)
# U := U P R^-1, back-substitution
new_us = []
for j in xrange(len(cs)):
u = us[P[j]]
for i in xrange(j):
u = axpy(us[P[i]], u, u.shape[0], -R[i, j])
if abs(R[j, j]) < 1e-12 * abs(R[0, 0]):
# discard rest of the vectors
break
u = scal(1.0 / R[j, j], u)
new_us.append(u)
# Form the new CU lists
CU[:] = list(zip(cs, new_us))[::-1]
if CU:
axpy, dot = get_blas_funcs(['axpy', 'dot'], (r, ))
# Solve first the projection operation with respect to the CU
# vectors. This corresponds to modifying the initial guess to
# be
#
# x' = x + U y
# y = argmin_y || b - A (x + U y) ||^2
#
# The solution is y = C^H (b - A x)
for c, u in CU:
yc = dot(c, r)
x = axpy(u, x, x.shape[0], yc)
r = axpy(c, r, r.shape[0], -yc)
# GCROT main iteration
for j_outer in xrange(maxiter):
# -- callback
if callback is not None:
callback(x)
beta = nrm2(r)
# -- check stopping condition
beta_tol = max(atol, tol * b_norm)
if beta <= beta_tol and (j_outer > 0 or CU):
# recompute residual to avoid rounding error
r = b - matvec(x)
beta = nrm2(r)
if beta <= beta_tol:
j_outer = -1
break
ml = m + max(k - len(CU), 0)
cs = [c for c, u in CU]
try:
Q, R, B, vs, zs, y, pres = _fgmres(matvec,
r / beta,
ml,
rpsolve=psolve,
atol=max(atol, tol * b_norm) /
beta,
cs=cs)
y *= beta
except LinAlgError:
# Floating point over/underflow, non-finite result from
# matmul etc. -- report failure.
break
#
# At this point,
#
# [A U, A Z] = [C, V] G; G = [ I B ]
# [ 0 H ]
#
# where [C, V] has orthonormal columns, and r = beta v_0. Moreover,
#
# || b - A (x + Z y + U q) ||_2 = || r - C B y - V H y - C q ||_2 = min!
#
# from which y = argmin_y || beta e_1 - H y ||_2, and q = -B y
#
#
# GCROT(m,k) update
#
# Define new outer vectors
# ux := (Z - U B) y
ux = zs[0] * y[0]
for z, yc in zip(zs[1:], y[1:]):
ux = axpy(z, ux, ux.shape[0], yc) # ux += z*yc
by = B.dot(y)
for cu, byc in zip(CU, by):
c, u = cu
ux = axpy(u, ux, ux.shape[0], -byc) # ux -= u*byc
# cx := V H y
hy = Q.dot(R.dot(y))
cx = vs[0] * hy[0]
for v, hyc in zip(vs[1:], hy[1:]):
cx = axpy(v, cx, cx.shape[0], hyc) # cx += v*hyc
# Normalize cx, maintaining cx = A ux
# This new cx is orthogonal to the previous C, by construction
try:
alpha = 1 / nrm2(cx)
if not np.isfinite(alpha):
raise FloatingPointError()
except (FloatingPointError, ZeroDivisionError):
# Cannot update, so skip it
continue
cx = scal(alpha, cx)
ux = scal(alpha, ux)
# Update residual and solution
gamma = dot(cx, r)
r = axpy(cx, r, r.shape[0], -gamma) # r -= gamma*cx
x = axpy(ux, x, x.shape[0], gamma) # x += gamma*ux
# Truncate CU
if truncate == 'oldest':
while len(CU) >= k and CU:
del CU[0]
elif truncate == 'smallest':
if len(CU) >= k and CU:
# cf. [1,2]
D = solve(R[:-1, :].T, B.T).T
W, sigma, V = svd(D)
# C := C W[:,:k-1], U := U W[:,:k-1]
new_CU = []
for j, w in enumerate(W[:, :k - 1].T):
c, u = CU[0]
c = c * w[0]
u = u * w[0]
for cup, wp in zip(CU[1:], w[1:]):
cp, up = cup
c = axpy(cp, c, c.shape[0], wp)
u = axpy(up, u, u.shape[0], wp)
# Reorthogonalize at the same time; not necessary
# in exact arithmetic, but floating point error
# tends to accumulate here
for cp, up in new_CU:
alpha = dot(cp, c)
c = axpy(cp, c, c.shape[0], -alpha)
u = axpy(up, u, u.shape[0], -alpha)
alpha = nrm2(c)
c = scal(1.0 / alpha, c)
u = scal(1.0 / alpha, u)
new_CU.append((c, u))
CU[:] = new_CU
# Add new vector to CU
CU.append((cx, ux))
# Include the solution vector to the span
CU.append((None, x.copy()))
if discard_C:
CU[:] = [(None, uz) for cz, uz in CU]
return postprocess(x), j_outer + 1
def invhilbert(n, exact=False):
"""
Compute the inverse of the Hilbert matrix of order `n`.
The entries in the inverse of a Hilbert matrix are integers. When `n`
is greater than 14, some entries in the inverse exceed the upper limit
of 64 bit integers. The `exact` argument provides two options for
dealing with these large integers.
Parameters
----------
n : int
The order of the Hilbert matrix.
exact : bool, optional
If False, the data type of the array that is returned is np.float64,
and the array is an approximation of the inverse.
If True, the array is the exact integer inverse array. To represent
the exact inverse when n > 14, the returned array is an object array
of long integers. For n <= 14, the exact inverse is returned as an
array with data type np.int64.
Returns
-------
invh : (n, n) ndarray
The data type of the array is np.float64 if `exact` is False.
If `exact` is True, the data type is either np.int64 (for n <= 14)
or object (for n > 14). In the latter case, the objects in the
array will be long integers.
See Also
--------
hilbert : Create a Hilbert matrix.
Notes
-----
.. versionadded:: 0.10.0
Examples
--------
>>> from scipy.linalg import invhilbert
>>> invhilbert(4)
array([[ 16., -120., 240., -140.],
[ -120., 1200., -2700., 1680.],
[ 240., -2700., 6480., -4200.],
[ -140., 1680., -4200., 2800.]])
>>> invhilbert(4, exact=True)
array([[ 16, -120, 240, -140],
[ -120, 1200, -2700, 1680],
[ 240, -2700, 6480, -4200],
[ -140, 1680, -4200, 2800]], dtype=int64)
>>> invhilbert(16)[7,7]
4.2475099528537506e+19
>>> invhilbert(16, exact=True)[7,7]
42475099528537378560L
"""
from scipy.special import comb
if exact:
if n > 14:
dtype = object
else:
dtype = np.int64
else:
dtype = np.float64
invh = np.empty((n, n), dtype=dtype)
for i in xrange(n):
for j in xrange(0, i + 1):
s = i + j
invh[i, j] = ((-1)**s * (s + 1) * comb(n + i, n - j - 1, exact) *
comb(n + j, n - i - 1, exact) * comb(s, i, exact)**2)
if i != j:
invh[j, i] = invh[i, j]
return invh
def lobpcg(A,
X,
B=None,
M=None,
Y=None,
tol=None,
maxiter=20,
largest=True,
verbosityLevel=0,
retLambdaHistory=False,
retResidualNormsHistory=False):
"""Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)
LOBPCG is a preconditioned eigensolver for large symmetric positive
definite (SPD) generalized eigenproblems.
Parameters
----------
A : {sparse matrix, dense matrix, LinearOperator}
The symmetric linear operator of the problem, usually a
sparse matrix. Often called the "stiffness matrix".
X : array_like
Initial approximation to the k eigenvectors. If A has
shape=(n,n) then X should have shape shape=(n,k).
B : {dense matrix, sparse matrix, LinearOperator}, optional
the right hand side operator in a generalized eigenproblem.
by default, B = Identity
often called the "mass matrix"
M : {dense matrix, sparse matrix, LinearOperator}, optional
preconditioner to A; by default M = Identity
M should approximate the inverse of A
Y : array_like, optional
n-by-sizeY matrix of constraints, sizeY < n
The iterations will be performed in the B-orthogonal complement
of the column-space of Y. Y must be full rank.
Returns
-------
w : array
Array of k eigenvalues
v : array
An array of k eigenvectors. V has the same shape as X.
Other Parameters
----------------
tol : scalar, optional
Solver tolerance (stopping criterion)
by default: tol=n*sqrt(eps)
maxiter : integer, optional
maximum number of iterations
by default: maxiter=min(n,20)
largest : bool, optional
when True, solve for the largest eigenvalues, otherwise the smallest
verbosityLevel : integer, optional
controls solver output. default: verbosityLevel = 0.
retLambdaHistory : boolean, optional
whether to return eigenvalue history
retResidualNormsHistory : boolean, optional
whether to return history of residual norms
Examples
--------
Solve A x = lambda B x with constraints and preconditioning.
>>> from scipy.sparse import spdiags, issparse
>>> from scipy.sparse.linalg import lobpcg, LinearOperator
>>> n = 100
>>> vals = [np.arange(n, dtype=np.float64) + 1]
>>> A = spdiags(vals, 0, n, n)
>>> A.toarray()
array([[ 1., 0., 0., ..., 0., 0., 0.],
[ 0., 2., 0., ..., 0., 0., 0.],
[ 0., 0., 3., ..., 0., 0., 0.],
...,
[ 0., 0., 0., ..., 98., 0., 0.],
[ 0., 0., 0., ..., 0., 99., 0.],
[ 0., 0., 0., ..., 0., 0., 100.]])
Constraints.
>>> Y = np.eye(n, 3)
Initial guess for eigenvectors, should have linearly independent
columns. Column dimension = number of requested eigenvalues.
>>> X = np.random.rand(n, 3)
Preconditioner -- inverse of A (as an abstract linear operator).
>>> invA = spdiags([1./vals[0]], 0, n, n)
>>> def precond( x ):
... return invA * x
>>> M = LinearOperator(matvec=precond, shape=(n, n), dtype=float)
Here, ``invA`` could of course have been used directly as a preconditioner.
Let us then solve the problem:
>>> eigs, vecs = lobpcg(A, X, Y=Y, M=M, tol=1e-4, maxiter=40, largest=False)
>>> eigs
array([ 4., 5., 6.])
Note that the vectors passed in Y are the eigenvectors of the 3 smallest
eigenvalues. The results returned are orthogonal to those.
Notes
-----
If both retLambdaHistory and retResidualNormsHistory are True,
the return tuple has the following format
(lambda, V, lambda history, residual norms history).
In the following ``n`` denotes the matrix size and ``m`` the number
of required eigenvalues (smallest or largest).
The LOBPCG code internally solves eigenproblems of the size 3``m`` on every
iteration by calling the "standard" dense eigensolver, so if ``m`` is not
small enough compared to ``n``, it does not make sense to call the LOBPCG
code, but rather one should use the "standard" eigensolver,
e.g. numpy or scipy function in this case.
If one calls the LOBPCG algorithm for 5``m``>``n``,
it will most likely break internally, so the code tries to call the standard
function instead.
It is not that n should be large for the LOBPCG to work, but rather the
ratio ``n``/``m`` should be large. It you call the LOBPCG code with ``m``=1
and ``n``=10, it should work, though ``n`` is small. The method is intended
for extremely large ``n``/``m``, see e.g., reference [28] in
https://arxiv.org/abs/0705.2626
The convergence speed depends basically on two factors:
1. How well relatively separated the seeking eigenvalues are
from the rest of the eigenvalues.
One can try to vary ``m`` to make this better.
2. How well conditioned the problem is. This can be changed by using proper
preconditioning. For example, a rod vibration test problem (under tests
directory) is ill-conditioned for large ``n``, so convergence will be
slow, unless efficient preconditioning is used.
For this specific problem, a good simple preconditioner function would
be a linear solve for A, which is easy to code since A is tridiagonal.
*Acknowledgements*
lobpcg.py code was written by Robert Cimrman.
Many thanks belong to Andrew Knyazev, the author of the algorithm,
for lots of advice and support.
References
----------
.. [1] A. V. Knyazev (2001),
Toward the Optimal Preconditioned Eigensolver: Locally Optimal
Block Preconditioned Conjugate Gradient Method.
SIAM Journal on Scientific Computing 23, no. 2,
pp. 517-541. :doi:`10.1137/S1064827500366124`
.. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov (2007),
Block Locally Optimal Preconditioned Eigenvalue Xolvers (BLOPEX)
in hypre and PETSc. https://arxiv.org/abs/0705.2626
.. [3] A. V. Knyazev's C and MATLAB implementations:
https://bitbucket.org/joseroman/blopex
"""
blockVectorX = X
blockVectorY = Y
residualTolerance = tol
maxIterations = maxiter
if blockVectorY is not None:
sizeY = blockVectorY.shape[1]
else:
sizeY = 0
# Block size.
if len(blockVectorX.shape) != 2:
raise ValueError('expected rank-2 array for argument X')
n, sizeX = blockVectorX.shape
if sizeX > n:
raise ValueError('X column dimension exceeds the row dimension')
A = _makeOperator(A, (n, n))
B = _makeOperator(B, (n, n))
M = _makeOperator(M, (n, n))
if (n - sizeY) < (5 * sizeX):
# warn('The problem size is small compared to the block size.' \
# ' Using dense eigensolver instead of LOBPCG.')
if blockVectorY is not None:
raise NotImplementedError('The dense eigensolver '
'does not support constraints.')
# Define the closed range of indices of eigenvalues to return.
if largest:
eigvals = (n - sizeX, n - 1)
else:
eigvals = (0, sizeX - 1)
A_dense = A(np.eye(n))
B_dense = None if B is None else B(np.eye(n))
return eigh(A_dense, B_dense, eigvals=eigvals, check_finite=False)
if residualTolerance is None:
residualTolerance = np.sqrt(1e-15) * n
maxIterations = min(n, maxIterations)
if verbosityLevel:
aux = "Solving "
if B is None:
aux += "standard"
else:
aux += "generalized"
aux += " eigenvalue problem with"
if M is None:
aux += "out"
aux += " preconditioning\n\n"
aux += "matrix size %d\n" % n
aux += "block size %d\n\n" % sizeX
if blockVectorY is None:
aux += "No constraints\n\n"
else:
if sizeY > 1:
aux += "%d constraints\n\n" % sizeY
else:
aux += "%d constraint\n\n" % sizeY
print(aux)
##
# Apply constraints to X.
if blockVectorY is not None:
if B is not None:
blockVectorBY = B(blockVectorY)
else:
blockVectorBY = blockVectorY
# gramYBY is a dense array.
gramYBY = np.dot(blockVectorY.T.conj(), blockVectorBY)
try:
# gramYBY is a Cholesky factor from now on...
gramYBY = cho_factor(gramYBY)
except:
raise ValueError('cannot handle linearly dependent constraints')
_applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)
##
# B-orthonormalize X.
blockVectorX, blockVectorBX = _b_orthonormalize(B, blockVectorX)
##
# Compute the initial Ritz vectors: solve the eigenproblem.
blockVectorAX = A(blockVectorX)
gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
_lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
ii = np.argsort(_lambda)[:sizeX]
if largest:
ii = ii[::-1]
_lambda = _lambda[ii]
eigBlockVector = np.asarray(eigBlockVector[:, ii])
blockVectorX = np.dot(blockVectorX, eigBlockVector)
blockVectorAX = np.dot(blockVectorAX, eigBlockVector)
if B is not None:
blockVectorBX = np.dot(blockVectorBX, eigBlockVector)
##
# Active index set.
activeMask = np.ones((sizeX, ), dtype=bool)
lambdaHistory = [_lambda]
residualNormsHistory = []
previousBlockSize = sizeX
ident = np.eye(sizeX, dtype=A.dtype)
ident0 = np.eye(sizeX, dtype=A.dtype)
##
# Main iteration loop.
blockVectorP = None # set during iteration
blockVectorAP = None
blockVectorBP = None
for iterationNumber in xrange(maxIterations):
if verbosityLevel > 0:
print('iteration %d' % iterationNumber)
aux = blockVectorBX * _lambda[np.newaxis, :]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
residualNorms = np.sqrt(aux)
residualNormsHistory.append(residualNorms)
ii = np.where(residualNorms > residualTolerance, True, False)
activeMask = activeMask & ii
if verbosityLevel > 2:
print(activeMask)
currentBlockSize = activeMask.sum()
if currentBlockSize != previousBlockSize:
previousBlockSize = currentBlockSize
ident = np.eye(currentBlockSize, dtype=A.dtype)
if currentBlockSize == 0:
break
if verbosityLevel > 0:
print('current block size:', currentBlockSize)
print('eigenvalue:', _lambda)
print('residual norms:', residualNorms)
if verbosityLevel > 10:
print(eigBlockVector)
activeBlockVectorR = as2d(blockVectorR[:, activeMask])
if iterationNumber > 0:
activeBlockVectorP = as2d(blockVectorP[:, activeMask])
activeBlockVectorAP = as2d(blockVectorAP[:, activeMask])
activeBlockVectorBP = as2d(blockVectorBP[:, activeMask])
if M is not None:
# Apply preconditioner T to the active residuals.
activeBlockVectorR = M(activeBlockVectorR)
##
# Apply constraints to the preconditioned residuals.
if blockVectorY is not None:
_applyConstraints(activeBlockVectorR, gramYBY, blockVectorBY,
blockVectorY)
##
# B-orthonormalize the preconditioned residuals.
aux = _b_orthonormalize(B, activeBlockVectorR)
activeBlockVectorR, activeBlockVectorBR = aux
activeBlockVectorAR = A(activeBlockVectorR)
if iterationNumber > 0:
aux = _b_orthonormalize(B,
activeBlockVectorP,
activeBlockVectorBP,
retInvR=True)
activeBlockVectorP, activeBlockVectorBP, invR = aux
activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)
##
# Perform the Rayleigh Ritz Procedure:
# Compute symmetric Gram matrices:
xaw = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
waw = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)
xbw = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)
if iterationNumber > 0:
xap = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
wap = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
pap = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
xbp = np.dot(blockVectorX.T.conj(), activeBlockVectorBP)
wbp = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)
gramA = np.bmat([[np.diag(_lambda), xaw, xap],
[xaw.T.conj(), waw, wap],
[xap.T.conj(), wap.T.conj(), pap]])
gramB = np.bmat([[ident0, xbw, xbp], [xbw.T.conj(), ident, wbp],
[xbp.T.conj(), wbp.T.conj(), ident]])
else:
gramA = np.bmat([[np.diag(_lambda), xaw], [xaw.T.conj(), waw]])
gramB = np.bmat([[ident0, xbw], [xbw.T.conj(), ident]])
_assert_symmetric(gramA)
_assert_symmetric(gramB)
if verbosityLevel > 10:
save(gramA, 'gramA')
save(gramB, 'gramB')
# Solve the generalized eigenvalue problem.
_lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False)
ii = np.argsort(_lambda)[:sizeX]
if largest:
ii = ii[::-1]
if verbosityLevel > 10:
print(ii)
_lambda = _lambda[ii]
eigBlockVector = eigBlockVector[:, ii]
lambdaHistory.append(_lambda)
if verbosityLevel > 10:
print('lambda:', _lambda)
## # Normalize eigenvectors!
## aux = np.sum( eigBlockVector.conjugate() * eigBlockVector, 0 )
## eigVecNorms = np.sqrt( aux )
## eigBlockVector = eigBlockVector / eigVecNorms[np.newaxis,:]
# eigBlockVector, aux = _b_orthonormalize( B, eigBlockVector )
if verbosityLevel > 10:
print(eigBlockVector)
pause()
##
# Compute Ritz vectors.
if iterationNumber > 0:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:sizeX + currentBlockSize]
eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
pp += np.dot(activeBlockVectorP, eigBlockVectorP)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
app += np.dot(activeBlockVectorAP, eigBlockVectorP)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
bpp += np.dot(activeBlockVectorBP, eigBlockVectorP)
else:
eigBlockVectorX = eigBlockVector[:sizeX]
eigBlockVectorR = eigBlockVector[sizeX:]
pp = np.dot(activeBlockVectorR, eigBlockVectorR)
app = np.dot(activeBlockVectorAR, eigBlockVectorR)
bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
if verbosityLevel > 10:
print(pp)
print(app)
print(bpp)
pause()
blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp
blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp
aux = blockVectorBX * _lambda[np.newaxis, :]
blockVectorR = blockVectorAX - aux
aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
residualNorms = np.sqrt(aux)
if verbosityLevel > 0:
print('final eigenvalue:', _lambda)
print('final residual norms:', residualNorms)
if retLambdaHistory:
if retResidualNormsHistory:
return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
else:
return _lambda, blockVectorX, lambdaHistory
else:
if retResidualNormsHistory:
return _lambda, blockVectorX, residualNormsHistory
else:
return _lambda, blockVectorX
def scalar_search_wolfe1(phi, derphi, phi0=None, old_phi0=None, derphi0=None,
c1=1e-4, c2=0.9,
amax=50, amin=1e-8, xtol=1e-14):
"""
Scalar function search for alpha that satisfies strong Wolfe conditions
alpha > 0 is assumed to be a descent direction.
Parameters
----------
phi : callable phi(alpha)
Function at point `alpha`
derphi : callable dphi(alpha)
Derivative `d phi(alpha)/ds`. Returns a scalar.
phi0 : float, optional
Value of `f` at 0
old_phi0 : float, optional
Value of `f` at the previous point
derphi0 : float, optional
Value `derphi` at 0
c1, c2 : float, optional
Wolfe parameters
amax, amin : float, optional
Maximum and minimum step size
xtol : float, optional
Relative tolerance for an acceptable step.
Returns
-------
alpha : float
Step size, or None if no suitable step was found
phi : float
Value of `phi` at the new point `alpha`
phi0 : float
Value of `phi` at `alpha=0`
Notes
-----
Uses routine DCSRCH from MINPACK.
"""
if phi0 is None:
phi0 = phi(0.)
if derphi0 is None:
derphi0 = derphi(0.)
if old_phi0 is not None and derphi0 != 0:
alpha1 = min(1.0, 1.01*2*(phi0 - old_phi0)/derphi0)
if alpha1 < 0:
alpha1 = 1.0
else:
alpha1 = 1.0
phi1 = phi0
derphi1 = derphi0
isave = np.zeros((2,), np.intc)
dsave = np.zeros((13,), float)
task = b'START'
maxiter = 100
for i in xrange(maxiter):
stp, phi1, derphi1, task = minpack2.dcsrch(alpha1, phi1, derphi1,
c1, c2, xtol, task,
amin, amax, isave, dsave)
if task[:2] == b'FG':
alpha1 = stp
phi1 = phi(stp)
derphi1 = derphi(stp)
else:
break
else:
# maxiter reached, the line search did not converge
stp = None
if task[:5] == b'ERROR' or task[:4] == b'WARN':
stp = None # failed
return stp, phi1, phi0
def solve(self):
"""
Runs the DifferentialEvolutionSolver.
Returns
-------
res : OptimizeResult
The optimization result represented as a ``OptimizeResult`` object.
Important attributes are: ``x`` the solution array, ``success`` a
Boolean flag indicating if the optimizer exited successfully and
``message`` which describes the cause of the termination. See
`OptimizeResult` for a description of other attributes. If `polish`
was employed, and a lower minimum was obtained by the polishing,
then OptimizeResult also contains the ``jac`` attribute.
"""
nit, warning_flag = 0, False
status_message = _status_message['success']
# The population may have just been initialized (all entries are
# np.inf). If it has you have to calculate the initial energies.
# Although this is also done in the evolve generator it's possible
# that someone can set maxiter=0, at which point we still want the
# initial energies to be calculated (the following loop isn't run).
if np.all(np.isinf(self.population_energies)):
self.population_energies[:] = self._calculate_population_energies(
self.population)
self._promote_lowest_energy()
# do the optimisation.
for nit in xrange(1, self.maxiter + 1):
# evolve the population by a generation
try:
next(self)
except StopIteration:
warning_flag = True
if self._nfev > self.maxfun:
status_message = _status_message['maxfev']
elif self._nfev == self.maxfun:
status_message = ('Maximum number of function evaluations'
' has been reached.')
break
if self.disp:
print("differential_evolution step %d: f(x)= %g"
% (nit,
self.population_energies[0]))
# should the solver terminate?
convergence = self.convergence
if (self.callback and
self.callback(self._scale_parameters(self.population[0]),
convergence=self.tol / convergence) is True):
warning_flag = True
status_message = ('callback function requested stop early '
'by returning True')
break
if np.any(np.isinf(self.population_energies)):
intol = False
else:
intol = (np.std(self.population_energies) <=
self.atol +
self.tol * np.abs(np.mean(self.population_energies)))
if warning_flag or intol:
break
else:
status_message = _status_message['maxiter']
warning_flag = True
DE_result = OptimizeResult(
x=self.x,
fun=self.population_energies[0],
nfev=self._nfev,
nit=nit,
message=status_message,
success=(warning_flag is not True))
if self.polish:
result = minimize(self.func,
np.copy(DE_result.x),
method='L-BFGS-B',
bounds=self.limits.T)
self._nfev += result.nfev
DE_result.nfev = self._nfev
if result.fun < DE_result.fun:
DE_result.fun = result.fun
DE_result.x = result.x
DE_result.jac = result.jac
# to keep internal state consistent
self.population_energies[0] = result.fun
self.population[0] = self._unscale_parameters(result.x)
return DE_result
def _identify_ridge_lines(matr, max_distances, gap_thresh):
"""
Identify ridges in the 2-D matrix.
Expect that the width of the wavelet feature increases with increasing row
number.
Parameters
----------
matr : 2-D ndarray
Matrix in which to identify ridge lines.
max_distances : 1-D sequence
At each row, a ridge line is only connected
if the relative max at row[n] is within
`max_distances`[n] from the relative max at row[n+1].
gap_thresh : int
If a relative maximum is not found within `max_distances`,
there will be a gap. A ridge line is discontinued if
there are more than `gap_thresh` points without connecting
a new relative maximum.
Returns
-------
ridge_lines : tuple
Tuple of 2 1-D sequences. `ridge_lines`[ii][0] are the rows of the
ii-th ridge-line, `ridge_lines`[ii][1] are the columns. Empty if none
found. Each ridge-line will be sorted by row (increasing), but the
order of the ridge lines is not specified.
References
----------
Bioinformatics (2006) 22 (17): 2059-2065.
:doi:`10.1093/bioinformatics/btl355`
http://bioinformatics.oxfordjournals.org/content/22/17/2059.long
Examples
--------
>>> data = np.random.rand(5,5)
>>> ridge_lines = _identify_ridge_lines(data, 1, 1)
Notes
-----
This function is intended to be used in conjunction with `cwt`
as part of `find_peaks_cwt`.
"""
if(len(max_distances) < matr.shape[0]):
raise ValueError('Max_distances must have at least as many rows '
'as matr')
all_max_cols = _boolrelextrema(matr, np.greater, axis=1, order=1)
# Highest row for which there are any relative maxima
has_relmax = np.where(all_max_cols.any(axis=1))[0]
if(len(has_relmax) == 0):
return []
start_row = has_relmax[-1]
# Each ridge line is a 3-tuple:
# rows, cols,Gap number
ridge_lines = [[[start_row],
[col],
0] for col in np.where(all_max_cols[start_row])[0]]
final_lines = []
rows = np.arange(start_row - 1, -1, -1)
cols = np.arange(0, matr.shape[1])
for row in rows:
this_max_cols = cols[all_max_cols[row]]
# Increment gap number of each line,
# set it to zero later if appropriate
for line in ridge_lines:
line[2] += 1
# XXX These should always be all_max_cols[row]
# But the order might be different. Might be an efficiency gain
# to make sure the order is the same and avoid this iteration
prev_ridge_cols = np.array([line[1][-1] for line in ridge_lines])
# Look through every relative maximum found at current row
# Attempt to connect them with existing ridge lines.
for ind, col in enumerate(this_max_cols):
# If there is a previous ridge line within
# the max_distance to connect to, do so.
# Otherwise start a new one.
line = None
if(len(prev_ridge_cols) > 0):
diffs = np.abs(col - prev_ridge_cols)
closest = np.argmin(diffs)
if diffs[closest] <= max_distances[row]:
line = ridge_lines[closest]
if(line is not None):
# Found a point close enough, extend current ridge line
line[1].append(col)
line[0].append(row)
line[2] = 0
else:
new_line = [[row],
[col],
0]
ridge_lines.append(new_line)
# Remove the ridge lines with gap_number too high
# XXX Modifying a list while iterating over it.
# Should be safe, since we iterate backwards, but
# still tacky.
for ind in xrange(len(ridge_lines) - 1, -1, -1):
line = ridge_lines[ind]
if line[2] > gap_thresh:
final_lines.append(line)
del ridge_lines[ind]
out_lines = []
for line in (final_lines + ridge_lines):
sortargs = np.array(np.argsort(line[0]))
rows, cols = np.zeros_like(sortargs), np.zeros_like(sortargs)
rows[sortargs] = line[0]
cols[sortargs] = line[1]
out_lines.append([rows, cols])
return out_lines
def test_derivative(self):
P = KroghInterpolator(self.xs,self.ys)
m = 10
r = P.derivatives(self.test_xs,m)
for i in xrange(m):
assert_almost_equal(P.derivative(self.test_xs,i),r[i])
def __getitem__(self, index):
"""Return the element(s) index=(i, j), where j may be a slice.
This always returns a copy for consistency, since slices into
Python lists return copies.
"""
# Scalar fast path first
if isinstance(index, tuple) and len(index) == 2:
i, j = index
# Use isinstance checks for common index types; this is
# ~25-50% faster than isscalarlike. Other types are
# handled below.
if ((isinstance(i, int) or isinstance(i, np.integer)) and
(isinstance(j, int) or isinstance(j, np.integer))):
v = _csparsetools.lil_get1(self.shape[0], self.shape[1],
self.rows, self.data,
i, j)
return self.dtype.type(v)
# Utilities found in IndexMixin
i, j = self._unpack_index(index)
# Proper check for other scalar index types
i_intlike = isintlike(i)
j_intlike = isintlike(j)
if i_intlike and j_intlike:
v = _csparsetools.lil_get1(self.shape[0], self.shape[1],
self.rows, self.data,
i, j)
return self.dtype.type(v)
elif j_intlike or isinstance(j, slice):
# column slicing fast path
if j_intlike:
j = self._check_col_bounds(j)
j = slice(j, j+1)
if i_intlike:
i = self._check_row_bounds(i)
i = xrange(i, i+1)
i_shape = None
elif isinstance(i, slice):
i = xrange(*i.indices(self.shape[0]))
i_shape = None
else:
i = np.atleast_1d(i)
i_shape = i.shape
if i_shape is None or len(i_shape) == 1:
return self._get_row_ranges(i, j)
i, j = self._index_to_arrays(i, j)
if i.size == 0:
return lil_matrix(i.shape, dtype=self.dtype)
new = lil_matrix(i.shape, dtype=self.dtype)
i, j = _prepare_index_for_memoryview(i, j)
_csparsetools.lil_fancy_get(self.shape[0], self.shape[1],
self.rows, self.data,
new.rows, new.data,
i, j)
return new
def test_high_derivative(self):
P = KroghInterpolator(self.xs, self.ys)
for i in xrange(len(self.xs), 2 * len(self.xs)):
assert_almost_equal(P.derivative(self.test_xs, i),
np.zeros(len(self.test_xs)))
def solve(self):
"""
Runs the DifferentialEvolutionSolver.
Returns
-------
res : OptimizeResult
The optimization result represented as a ``OptimizeResult`` object.
Important attributes are: ``x`` the solution array, ``success`` a
Boolean flag indicating if the optimizer exited successfully and
``message`` which describes the cause of the termination. See
`OptimizeResult` for a description of other attributes. If `polish`
was employed, and a lower minimum was obtained by the polishing,
then OptimizeResult also contains the ``jac`` attribute.
"""
nit, warning_flag = 0, False
status_message = _status_message['success']
# The population may have just been initialized (all entries are
# np.inf). If it has you have to calculate the initial energies.
# Although this is also done in the evolve generator it's possible
# that someone can set maxiter=0, at which point we still want the
# initial energies to be calculated (the following loop isn't run).
if np.all(np.isinf(self.population_energies)):
self._calculate_population_energies()
for nmig in xrange(1, self.number_of_migrations + 1):
if nmig != 1:
# Get the host node
host = int(self.island_marker[-1])
# Get all the neighbors list
neighbors = self.topology.neighbors(host)
neighbor_results = {}
neighbor_energy_results = {}
for each_neighbor in neighbors:
replacement = client.get(self.key + str(each_neighbor))
if replacement is None:
for _ in range(int(self.wait_time / self.poll_time)):
replacement = client.get(self.key +
str(each_neighbor))
if replacement is None:
print("POLLING!!!")
time.sleep(self.poll_time)
else:
break
if replacement is not None:
neighbor_results[each_neighbor] = np.array([
float(items)
for items in replacement.split(",")
])
neighbor_energy_results[each_neighbor] = self.func(
neighbor_results[each_neighbor], *self.args)
total_computed_neighbors = len(neighbor_results)
energies = []
for each_neighbor in neighbor_results.keys():
energies.append((neighbor_results[each_neighbor],
neighbor_energy_results[each_neighbor]))
for pop_index in range(1, total_computed_neighbors + 1):
energies.append((self.population[pop_index],
self.population_energies[pop_index]))
energies.sort(key=lambda x: x[-1])
energies = energies[:total_computed_neighbors]
for pop_index in range(1, total_computed_neighbors + 1):
self.population[pop_index] = energies[pop_index - 1][0]
self.population_energies[pop_index] = energies[pop_index -
1][1]
# do the optimisation.
is_optimisation_complete = False
for nit in xrange(1, self.maxiter + 1):
# evolve the population by a generation
try:
next(self)
except StopIteration:
warning_flag = True
status_message = _status_message['maxfev']
#is_optimisation_complete = False
break
if self.disp:
print("differential_evolution step %d: f(x)= %g" %
(nit, self.population_energies[0]))
# should the solver terminate?
convergence = self.convergence
if (self.callback and self.callback(
self._scale_parameters(self.population[0]),
convergence=self.tol / convergence) is True):
warning_flag = True
status_message = ('callback function requested stop early '
'by returning True')
is_optimisation_complete = False
break
intol = (np.std(self.population_energies) <= self.atol +
self.tol * np.abs(np.mean(self.population_energies)))
if intol:
is_optimisation_complete = False
if warning_flag or intol:
break
else:
status_message = _status_message['maxiter']
warning_flag = True
client.set(self.island_marker,
",".join([str(items) for items in self.x]))
print("MARKED IN MEMCACHE")
print(self.island_marker,
",".join([str(items) for items in self.x]))
if not is_optimisation_complete:
#break
print("Exited due to some break condition above!!",
status_message)
DE_result = OptimizeResult(x=self.x,
fun=self.population_energies[0],
nfev=self._nfev,
nit=nit,
message=status_message,
success=(warning_flag is not True))
if self.polish:
result = minimize(self.func,
np.copy(DE_result.x),
method='L-BFGS-B',
bounds=self.limits.T,
args=self.args)
self._nfev += result.nfev
DE_result.nfev = self._nfev
if result.fun < DE_result.fun:
DE_result.fun = result.fun
DE_result.x = result.x
DE_result.jac = result.jac
# to keep internal state consistent
self.population_energies[0] = result.fun
self.population[0] = self._unscale_parameters(result.x)
return DE_result
def test_low_derivatives(self):
P = KroghInterpolator(self.xs,self.ys)
D = P.derivatives(self.test_xs,len(self.xs)+2)
for i in xrange(D.shape[0]):
assert_almost_equal(self.true_poly.deriv(i)(self.test_xs),
D[i])
def _fgmres(matvec,
v0,
m,
atol,
lpsolve=None,
rpsolve=None,
cs=(),
outer_v=(),
prepend_outer_v=False):
"""
FGMRES Arnoldi process, with optional projection or augmentation
Parameters
----------
matvec : callable
Operation A*x
v0 : ndarray
Initial vector, normalized to nrm2(v0) == 1
m : int
Number of GMRES rounds
atol : float
Absolute tolerance for early exit
lpsolve : callable
Left preconditioner L
rpsolve : callable
Right preconditioner R
CU : list of (ndarray, ndarray)
Columns of matrices C and U in GCROT
outer_v : list of ndarrays
Augmentation vectors in LGMRES
prepend_outer_v : bool, optional
Whether augmentation vectors come before or after
Krylov iterates
Raises
------
LinAlgError
If nans encountered
Returns
-------
Q, R : ndarray
QR decomposition of the upper Hessenberg H=QR
B : ndarray
Projections corresponding to matrix C
vs : list of ndarray
Columns of matrix V
zs : list of ndarray
Columns of matrix Z
y : ndarray
Solution to ||H y - e_1||_2 = min!
res : float
The final (preconditioned) residual norm
"""
if lpsolve is None:
lpsolve = lambda x: x
if rpsolve is None:
rpsolve = lambda x: x
axpy, dot, scal, nrm2 = get_blas_funcs(['axpy', 'dot', 'scal', 'nrm2'],
(v0, ))
vs = [v0]
zs = []
y = None
res = np.nan
m = m + len(outer_v)
# Orthogonal projection coefficients
B = np.zeros((len(cs), m), dtype=v0.dtype)
# H is stored in QR factorized form
Q = np.ones((1, 1), dtype=v0.dtype)
R = np.zeros((1, 0), dtype=v0.dtype)
eps = np.finfo(v0.dtype).eps
breakdown = False
# FGMRES Arnoldi process
for j in xrange(m):
# L A Z = C B + V H
if prepend_outer_v and j < len(outer_v):
z, w = outer_v[j]
elif prepend_outer_v and j == len(outer_v):
z = rpsolve(v0)
w = None
elif not prepend_outer_v and j >= m - len(outer_v):
z, w = outer_v[j - (m - len(outer_v))]
else:
z = rpsolve(vs[-1])
w = None
if w is None:
w = lpsolve(matvec(z))
else:
# w is clobbered below
w = w.copy()
w_norm = nrm2(w)
# GCROT projection: L A -> (1 - C C^H) L A
# i.e. orthogonalize against C
for i, c in enumerate(cs):
alpha = dot(c, w)
B[i, j] = alpha
w = axpy(c, w, c.shape[0], -alpha) # w -= alpha*c
# Orthogonalize against V
hcur = np.zeros(j + 2, dtype=Q.dtype)
for i, v in enumerate(vs):
alpha = dot(v, w)
hcur[i] = alpha
w = axpy(v, w, v.shape[0], -alpha) # w -= alpha*v
hcur[i + 1] = nrm2(w)
with np.errstate(over='ignore', divide='ignore'):
# Careful with denormals
alpha = 1 / hcur[-1]
if np.isfinite(alpha):
w = scal(alpha, w)
if not (hcur[-1] > eps * w_norm):
# w essentially in the span of previous vectors,
# or we have nans. Bail out after updating the QR
# solution.
breakdown = True
vs.append(w)
zs.append(z)
# Arnoldi LSQ problem
# Add new column to H=Q*R, padding other columns with zeros
Q2 = np.zeros((j + 2, j + 2), dtype=Q.dtype, order='F')
Q2[:j + 1, :j + 1] = Q
Q2[j + 1, j + 1] = 1
R2 = np.zeros((j + 2, j), dtype=R.dtype, order='F')
R2[:j + 1, :] = R
Q, R = qr_insert(Q2,
R2,
hcur,
j,
which='col',
overwrite_qru=True,
check_finite=False)
# Transformed least squares problem
# || Q R y - inner_res_0 * e_1 ||_2 = min!
# Since R = [R'; 0], solution is y = inner_res_0 (R')^{-1} (Q^H)[:j,0]
# Residual is immediately known
res = abs(Q[0, -1])
# Check for termination
if res < atol or breakdown:
break
if not np.isfinite(R[j, j]):
# nans encountered, bail out
raise LinAlgError()
# -- Get the LSQ problem solution
# The problem is triangular, but the condition number may be
# bad (or in case of breakdown the last diagonal entry may be
# zero), so use lstsq instead of trtrs.
y, _, _, _, = lstsq(R[:j + 1, :j + 1], Q[0, :j + 1].conj())
B = B[:, :j + 1]
return Q, R, B, vs, zs, y, res
def getrow(self, i):
"""Returns the i-th row as a (1 x n) DOK matrix."""
new = dok_matrix((1, self.shape[1]), dtype=self.dtype)
dict.update(new, (((0, j), self[i, j]) for j in xrange(self.shape[1])))
return new
def __iter__(self):
for r in xrange(self.shape[0]):
yield self[r, :]
def getcol(self, j):
"""Returns the j-th column as a (m x 1) DOK matrix."""
new = dok_matrix((self.shape[0], 1), dtype=self.dtype)
dict.update(new, (((i, 0), self[i, j]) for i in xrange(self.shape[0])))
return new
def test_daub(self):
for i in xrange(1, 15):
assert_equal(len(wavelets.daub(i)), i * 2)
def __getitem__(self, index):
"""If key=(i, j) is a pair of integers, return the corresponding
element. If either i or j is a slice or sequence, return a new sparse
matrix with just these elements.
"""
zero = self.dtype.type(0)
i, j = self._unpack_index(index)
i_intlike = isintlike(i)
j_intlike = isintlike(j)
if i_intlike and j_intlike:
i = int(i)
j = int(j)
if i < 0:
i += self.shape[0]
if i < 0 or i >= self.shape[0]:
raise IndexError('Index out of bounds.')
if j < 0:
j += self.shape[1]
if j < 0 or j >= self.shape[1]:
raise IndexError('Index out of bounds.')
return dict.get(self, (i,j), zero)
elif ((i_intlike or isinstance(i, slice)) and
(j_intlike or isinstance(j, slice))):
# Fast path for slicing very sparse matrices
i_slice = slice(i, i+1) if i_intlike else i
j_slice = slice(j, j+1) if j_intlike else j
i_indices = i_slice.indices(self.shape[0])
j_indices = j_slice.indices(self.shape[1])
i_seq = xrange(*i_indices)
j_seq = xrange(*j_indices)
newshape = (len(i_seq), len(j_seq))
newsize = _prod(newshape)
if len(self) < 2*newsize and newsize != 0:
# Switch to the fast path only when advantageous
# (count the iterations in the loops, adjust for complexity)
#
# We also don't handle newsize == 0 here (if
# i/j_intlike, it can mean index i or j was out of
# bounds)
return self._getitem_ranges(i_indices, j_indices, newshape)
i, j = self._index_to_arrays(i, j)
if i.size == 0:
return dok_matrix(i.shape, dtype=self.dtype)
min_i = i.min()
if min_i < -self.shape[0] or i.max() >= self.shape[0]:
raise IndexError('Index (%d) out of range -%d to %d.' %
(i.min(), self.shape[0], self.shape[0]-1))
if min_i < 0:
i = i.copy()
i[i < 0] += self.shape[0]
min_j = j.min()
if min_j < -self.shape[1] or j.max() >= self.shape[1]:
raise IndexError('Index (%d) out of range -%d to %d.' %
(j.min(), self.shape[1], self.shape[1]-1))
if min_j < 0:
j = j.copy()
j[j < 0] += self.shape[1]
newdok = dok_matrix(i.shape, dtype=self.dtype)
for key in itertools.product(xrange(i.shape[0]), xrange(i.shape[1])):
v = dict.get(self, (i[key], j[key]), zero)
if v:
dict.__setitem__(newdok, key, v)
return newdok
def romberg(function,
a,
b,
args=(),
tol=1.48e-8,
rtol=1.48e-8,
show=False,
divmax=10,
vec_func=False):
"""
Romberg integration of a callable function or method.
Returns the integral of `function` (a function of one variable)
over the interval (`a`, `b`).
If `show` is 1, the triangular array of the intermediate results
will be printed. If `vec_func` is True (default is False), then
`function` is assumed to support vector arguments.
Parameters
----------
function : callable
Function to be integrated.
a : float
Lower limit of integration.
b : float
Upper limit of integration.
Returns
-------
results : float
Result of the integration.
Other Parameters
----------------
args : tuple, optional
Extra arguments to pass to function. Each element of `args` will
be passed as a single argument to `func`. Default is to pass no
extra arguments.
tol, rtol : float, optional
The desired absolute and relative tolerances. Defaults are 1.48e-8.
show : bool, optional
Whether to print the results. Default is False.
divmax : int, optional
Maximum order of extrapolation. Default is 10.
vec_func : bool, optional
Whether `func` handles arrays as arguments (i.e whether it is a
"vector" function). Default is False.
See Also
--------
fixed_quad : Fixed-order Gaussian quadrature.
quad : Adaptive quadrature using QUADPACK.
dblquad : Double integrals.
tplquad : Triple integrals.
romb : Integrators for sampled data.
simps : Integrators for sampled data.
cumtrapz : Cumulative integration for sampled data.
ode : ODE integrator.
odeint : ODE integrator.
References
----------
.. [1] 'Romberg's method' https://en.wikipedia.org/wiki/Romberg%27s_method
Examples
--------
Integrate a gaussian from 0 to 1 and compare to the error function.
>>> from scipy import integrate
>>> from scipy.special import erf
>>> gaussian = lambda x: 1/np.sqrt(np.pi) * np.exp(-x**2)
>>> result = integrate.romberg(gaussian, 0, 1, show=True)
Romberg integration of <function vfunc at ...> from [0, 1]
::
Steps StepSize Results
1 1.000000 0.385872
2 0.500000 0.412631 0.421551
4 0.250000 0.419184 0.421368 0.421356
8 0.125000 0.420810 0.421352 0.421350 0.421350
16 0.062500 0.421215 0.421350 0.421350 0.421350 0.421350
32 0.031250 0.421317 0.421350 0.421350 0.421350 0.421350 0.421350
The final result is 0.421350396475 after 33 function evaluations.
>>> print("%g %g" % (2*result, erf(1)))
0.842701 0.842701
"""
if np.isinf(a) or np.isinf(b):
raise ValueError("Romberg integration only available "
"for finite limits.")
vfunc = vectorize1(function, args, vec_func=vec_func)
n = 1
interval = [a, b]
intrange = b - a
ordsum = _difftrap(vfunc, interval, n)
result = intrange * ordsum
resmat = [[result]]
err = np.inf
last_row = resmat[0]
for i in xrange(1, divmax + 1):
n *= 2
ordsum += _difftrap(vfunc, interval, n)
row = [intrange * ordsum / n]
for k in xrange(i):
row.append(_romberg_diff(last_row[k], row[k], k + 1))
result = row[i]
lastresult = last_row[i - 1]
if show:
resmat.append(row)
err = abs(result - lastresult)
if err < tol or err < rtol * abs(result):
break
last_row = row
else:
warnings.warn(
"divmax (%d) exceeded. Latest difference = %e" % (divmax, err),
AccuracyWarning)
if show:
_printresmat(vfunc, interval, resmat)
return result
def svd_reduce(self, max_rank, to_retain=None):
"""
Reduce the rank of the matrix by retaining some SVD components.
This corresponds to the \"Broyden Rank Reduction Inverse\"
algorithm described in [1]_.
Note that the SVD decomposition can be done by solving only a
problem whose size is the effective rank of this matrix, which
is viable even for large problems.
Parameters
----------
max_rank : int
Maximum rank of this matrix after reduction.
to_retain : int, optional
Number of SVD components to retain when reduction is done
(ie. rank > max_rank). Default is ``max_rank - 2``.
References
----------
.. [1] B.A. van der Rotten, PhD thesis,
\"A limited memory Broyden method to solve high-dimensional
systems of nonlinear equations\". Mathematisch Instituut,
Universiteit Leiden, The Netherlands (2003).
https://web.archive.org/web/20161022015821/http://www.math.leidenuniv.nl/scripties/Rotten.pdf
"""
if self.collapsed is not None:
return
p = max_rank
if to_retain is not None:
q = to_retain
else:
q = p - 2
if self.cs:
p = min(p, len(self.cs[0]))
q = max(0, min(q, p - 1))
m = len(self.cs)
if m < p:
# nothing to do
return
C = np.array(self.cs).T
D = np.array(self.ds).T
D, R = qr(D, mode='economic')
C = dot(C, R.T.conj())
U, S, WH = svd(C, full_matrices=False, compute_uv=True)
C = dot(C, inv(WH))
D = dot(D, WH.T.conj())
for k in xrange(q):
self.cs[k] = C[:, k].copy()
self.ds[k] = D[:, k].copy()
del self.cs[q:]
del self.ds[q:]
def binned_statistic_dd(sample, values, statistic='mean',
bins=10, range=None, expand_binnumbers=False):
"""
Compute a multidimensional binned statistic for a set of data.
This is a generalization of a histogramdd function. A histogram divides
the space into bins, and returns the count of the number of points in
each bin. This function allows the computation of the sum, mean, median,
or other statistic of the values within each bin.
Parameters
----------
sample : array_like
Data to histogram passed as a sequence of D arrays of length N, or
as an (N,D) array.
values : (N,) array_like or list of (N,) array_like
The data on which the statistic will be computed. This must be
the same shape as `sample`, or a list of sequences - each with the
same shape as `sample`. If `values` is such a list, the statistic
will be computed on each independently.
statistic : string or callable, optional
The statistic to compute (default is 'mean').
The following statistics are available:
* 'mean' : compute the mean of values for points within each bin.
Empty bins will be represented by NaN.
* 'median' : compute the median of values for points within each
bin. Empty bins will be represented by NaN.
* 'count' : compute the count of points within each bin. This is
identical to an unweighted histogram. `values` array is not
referenced.
* 'sum' : compute the sum of values for points within each bin.
This is identical to a weighted histogram.
* 'std' : compute the standard deviation within each bin. This
is implicitly calculated with ddof=0.
* 'min' : compute the minimum of values for points within each bin.
Empty bins will be represented by NaN.
* 'max' : compute the maximum of values for point within each bin.
Empty bins will be represented by NaN.
* function : a user-defined function which takes a 1D array of
values, and outputs a single numerical statistic. This function
will be called on the values in each bin. Empty bins will be
represented by function([]), or NaN if this returns an error.
bins : sequence or int, optional
The bin specification must be in one of the following forms:
* A sequence of arrays describing the bin edges along each dimension.
* The number of bins for each dimension (nx, ny, ... = bins).
* The number of bins for all dimensions (nx = ny = ... = bins).
range : sequence, optional
A sequence of lower and upper bin edges to be used if the edges are
not given explicitly in `bins`. Defaults to the minimum and maximum
values along each dimension.
expand_binnumbers : bool, optional
'False' (default): the returned `binnumber` is a shape (N,) array of
linearized bin indices.
'True': the returned `binnumber` is 'unraveled' into a shape (D,N)
ndarray, where each row gives the bin numbers in the corresponding
dimension.
See the `binnumber` returned value, and the `Examples` section of
`binned_statistic_2d`.
.. versionadded:: 0.17.0
Returns
-------
statistic : ndarray, shape(nx1, nx2, nx3,...)
The values of the selected statistic in each two-dimensional bin.
bin_edges : list of ndarrays
A list of D arrays describing the (nxi + 1) bin edges for each
dimension.
binnumber : (N,) array of ints or (D,N) ndarray of ints
This assigns to each element of `sample` an integer that represents the
bin in which this observation falls. The representation depends on the
`expand_binnumbers` argument. See `Notes` for details.
See Also
--------
numpy.digitize, numpy.histogramdd, binned_statistic, binned_statistic_2d
Notes
-----
Binedges:
All but the last (righthand-most) bin is half-open in each dimension. In
other words, if `bins` is ``[1, 2, 3, 4]``, then the first bin is
``[1, 2)`` (including 1, but excluding 2) and the second ``[2, 3)``. The
last bin, however, is ``[3, 4]``, which *includes* 4.
`binnumber`:
This returned argument assigns to each element of `sample` an integer that
represents the bin in which it belongs. The representation depends on the
`expand_binnumbers` argument. If 'False' (default): The returned
`binnumber` is a shape (N,) array of linearized indices mapping each
element of `sample` to its corresponding bin (using row-major ordering).
If 'True': The returned `binnumber` is a shape (D,N) ndarray where
each row indicates bin placements for each dimension respectively. In each
dimension, a binnumber of `i` means the corresponding value is between
(bin_edges[D][i-1], bin_edges[D][i]), for each dimension 'D'.
.. versionadded:: 0.11.0
"""
known_stats = ['mean', 'median', 'count', 'sum', 'std','min','max']
if not callable(statistic) and statistic not in known_stats:
raise ValueError('invalid statistic %r' % (statistic,))
# `Ndim` is the number of dimensions (e.g. `2` for `binned_statistic_2d`)
# `Dlen` is the length of elements along each dimension.
# This code is based on np.histogramdd
try:
# `sample` is an ND-array.
Dlen, Ndim = sample.shape
except (AttributeError, ValueError):
# `sample` is a sequence of 1D arrays.
sample = np.atleast_2d(sample).T
Dlen, Ndim = sample.shape
# Store initial shape of `values` to preserve it in the output
values = np.asarray(values)
input_shape = list(values.shape)
# Make sure that `values` is 2D to iterate over rows
values = np.atleast_2d(values)
Vdim, Vlen = values.shape
# Make sure `values` match `sample`
if(statistic != 'count' and Vlen != Dlen):
raise AttributeError('The number of `values` elements must match the '
'length of each `sample` dimension.')
nbin = np.empty(Ndim, int) # Number of bins in each dimension
edges = Ndim * [None] # Bin edges for each dim (will be 2D array)
dedges = Ndim * [None] # Spacing between edges (will be 2D array)
try:
M = len(bins)
if M != Ndim:
raise AttributeError('The dimension of bins must be equal '
'to the dimension of the sample x.')
except TypeError:
bins = Ndim * [bins]
# Select range for each dimension
# Used only if number of bins is given.
if range is None:
smin = np.atleast_1d(np.array(sample.min(axis=0), float))
smax = np.atleast_1d(np.array(sample.max(axis=0), float))
else:
smin = np.zeros(Ndim)
smax = np.zeros(Ndim)
for i in xrange(Ndim):
smin[i], smax[i] = range[i]
# Make sure the bins have a finite width.
for i in xrange(len(smin)):
if smin[i] == smax[i]:
smin[i] = smin[i] - .5
smax[i] = smax[i] + .5
# Create edge arrays
for i in xrange(Ndim):
if np.isscalar(bins[i]):
nbin[i] = bins[i] + 2 # +2 for outlier bins
edges[i] = np.linspace(smin[i], smax[i], nbin[i] - 1)
else:
edges[i] = np.asarray(bins[i], float)
nbin[i] = len(edges[i]) + 1 # +1 for outlier bins
dedges[i] = np.diff(edges[i])
nbin = np.asarray(nbin)
# Compute the bin number each sample falls into, in each dimension
sampBin = [
np.digitize(sample[:, i], edges[i])
for i in xrange(Ndim)
]
# Using `digitize`, values that fall on an edge are put in the right bin.
# For the rightmost bin, we want values equal to the right
# edge to be counted in the last bin, and not as an outlier.
for i in xrange(Ndim):
# Find the rounding precision
decimal = int(-np.log10(dedges[i].min())) + 6
# Find which points are on the rightmost edge.
on_edge = np.where(np.around(sample[:, i], decimal) ==
np.around(edges[i][-1], decimal))[0]
# Shift these points one bin to the left.
sampBin[i][on_edge] -= 1
# Compute the sample indices in the flattened statistic matrix.
binnumbers = np.ravel_multi_index(sampBin, nbin)
result = np.empty([Vdim, nbin.prod()], float)
if statistic == 'mean':
result.fill(np.nan)
flatcount = np.bincount(binnumbers, None)
a = flatcount.nonzero()
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
result[vv, a] = flatsum[a] / flatcount[a]
elif statistic == 'std':
result.fill(0)
flatcount = np.bincount(binnumbers, None)
a = flatcount.nonzero()
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
flatsum2 = np.bincount(binnumbers, values[vv] ** 2)
result[vv, a] = np.sqrt(flatsum2[a] / flatcount[a] -
(flatsum[a] / flatcount[a]) ** 2)
elif statistic == 'count':
result.fill(0)
flatcount = np.bincount(binnumbers, None)
a = np.arange(len(flatcount))
result[:, a] = flatcount[np.newaxis, :]
elif statistic == 'sum':
result.fill(0)
for vv in xrange(Vdim):
flatsum = np.bincount(binnumbers, values[vv])
a = np.arange(len(flatsum))
result[vv, a] = flatsum
elif statistic == 'median':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.median(values[vv, binnumbers == i])
elif statistic == 'min':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.min(values[vv, binnumbers == i])
elif statistic == 'max':
result.fill(np.nan)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = np.max(values[vv, binnumbers == i])
elif callable(statistic):
with np.errstate(invalid='ignore'), suppress_warnings() as sup:
sup.filter(RuntimeWarning)
try:
null = statistic([])
except Exception:
null = np.nan
result.fill(null)
for i in np.unique(binnumbers):
for vv in xrange(Vdim):
result[vv, i] = statistic(values[vv, binnumbers == i])
# Shape into a proper matrix
result = result.reshape(np.append(Vdim, nbin))
# Remove outliers (indices 0 and -1 for each bin-dimension).
core = tuple([slice(None)] + Ndim * [slice(1, -1)])
result = result[core]
# Unravel binnumbers into an ndarray, each row the bins for each dimension
if(expand_binnumbers and Ndim > 1):
binnumbers = np.asarray(np.unravel_index(binnumbers, nbin))
if np.any(result.shape[1:] != nbin - 2):
raise RuntimeError('Internal Shape Error')
# Reshape to have output (`reulst`) match input (`values`) shape
result = result.reshape(input_shape[:-1] + list(nbin-2))
return BinnedStatisticddResult(result, edges, binnumbers)
des_list.append((image_path, des))
print("Feature Extraction Finished")
descriptors = des_list[0][1]
for image_path, descriptor in des_list[1:]:
descriptors = np.vstack((descriptors, descriptor))
# Perform k-means clustering
k = 100
voc, variance = kmeans(descriptors, k, 1)
print("Clustering K Means Finished")
# Calculate the histogram of features
im_features = np.zeros((len(image_paths), k), "float32")
for i in xrange(len(image_paths)):
words, distance = vq(des_list[i][1], voc)
for w in words:
im_features[i][w] += 1
print("Histogram Finished")
# Perform Tf-Idf vectorization
nbr_occurences = np.sum((im_features > 0) * 1, axis=0)
idf = np.array(
np.log((1.0 * len(image_paths) + 1) / (1.0 * nbr_occurences + 1)),
'float32')
print("Vectorization Finished")
# Scaling the words
stdSlr = StandardScaler().fit(im_features)
im_features = stdSlr.transform(im_features)