def odd_ext(x, n, axis=-1):
"""Extends `x` along with `axis` by odd-extension.
This function was previously a part of "scipy.signal.signaltools" but is no
longer exposed.
Args:
x : input array
n : the number of points to be added to the both end
axis: the axis to be extended
"""
if n < 1:
return x
if n > x.shape[axis] - 1:
raise ValueError(
f"The extension length n ({n}) is too big. "
f"It must not exceed x.shape[axis]-1, which is {x.shape[axis] - 1}."
)
left_end = lax.slice_in_dim(x, 0, 1, axis=axis)
left_ext = jnp.flip(lax.slice_in_dim(x, 1, n + 1, axis=axis), axis=axis)
right_end = lax.slice_in_dim(x, -1, None, axis=axis)
right_ext = jnp.flip(lax.slice_in_dim(x, -(n + 1), -1, axis=axis),
axis=axis)
ext = jnp.concatenate(
(2 * left_end - left_ext, x, 2 * right_end - right_ext), axis=axis)
return ext
def __getitem__(self, key):
if not isinstance(key, tuple):
key = (key,)
params = [self.axis, self.ndmin, self.trans1d, -1]
if isinstance(key[0], str):
# split off the directive
directive, *key = key # pytype: disable=bad-unpacking
# check two special cases: matrix directives
if directive == "r":
params[-1] = 0
elif directive == "c":
params[-1] = 1
else:
vec = directive.split(",")
k = len(vec)
if k < 4:
vec += params[k:]
else:
# ignore everything after the first three comma-separated ints
vec = vec[:3] + params[-1]
try:
params = list(map(int, vec))
except ValueError as err:
raise ValueError(
f"could not understand directive {directive!r}"
) from err
axis, ndmin, trans1d, matrix = params
output = []
for item in key:
if isinstance(item, slice):
newobj = _make_1d_grid_from_slice(item, op_name=self.op_name)
elif isinstance(item, str):
raise ValueError("string directive must be placed at the beginning")
else:
newobj = item
newobj = array(newobj, copy=False, ndmin=ndmin)
if trans1d != -1 and ndmin - np.ndim(item) > 0:
shape_obj = list(range(ndmin))
# Calculate number of left shifts, with overflow protection by mod
num_lshifts = ndmin - abs(ndmin + trans1d + 1) % ndmin
shape_obj = tuple(shape_obj[num_lshifts:] + shape_obj[:num_lshifts])
newobj = transpose(newobj, shape_obj)
output.append(newobj)
res = concatenate(tuple(output), axis=axis)
if matrix != -1 and res.ndim == 1:
# insert 2nd dim at axis 0 or 1
res = expand_dims(res, matrix)
return res
def setxor1d(ar1, ar2, assume_unique=False):
_check_arraylike("setxor1d", ar1, ar2)
ar1 = core.concrete_or_error(None, ar1, "The error arose in setxor1d()")
ar2 = core.concrete_or_error(None, ar2, "The error arose in setxor1d()")
ar1 = ravel(ar1)
ar2 = ravel(ar2)
if not assume_unique:
ar1 = unique(ar1)
ar2 = unique(ar2)
aux = concatenate((ar1, ar2))
if aux.size == 0:
return aux
aux = sort(aux)
flag = concatenate((array([True]), aux[1:] != aux[:-1], array([True])))
return aux[flag[1:] & flag[:-1]]
def union1d(ar1, ar2, *, size=None, fill_value=None):
_check_arraylike("union1d", ar1, ar2)
if size is None:
ar1 = core.concrete_or_error(None, ar1, "The error arose in union1d()")
ar2 = core.concrete_or_error(None, ar2, "The error arose in union1d()")
else:
size = core.concrete_or_error(operator.index, size,
"The error arose in union1d()")
return unique(concatenate((ar1, ar2), axis=None),
size=size,
fill_value=fill_value)
def _intersect1d_sorted_mask(ar1, ar2, return_indices=False):
"""
Helper function for intersect1d which is jit-able
"""
ar = concatenate((ar1, ar2))
if return_indices:
iota = lax.broadcasted_iota(np.int64, np.shape(ar), dimension=0)
aux, indices = lax.sort_key_val(ar, iota)
else:
aux = sort(ar)
mask = aux[1:] == aux[:-1]
if return_indices:
return aux, mask, indices
else:
return aux, mask
def polyint(p, m=1, k=None):
m = core.concrete_or_error(operator.index, m, "'m' argument of jnp.polyint")
k = 0 if k is None else k
_check_arraylike("polyint", p, k)
p, k = _promote_dtypes_inexact(p, k)
if m < 0:
raise ValueError("Order of integral must be positive (see polyder)")
k = atleast_1d(k)
if len(k) == 1:
k = full((m,), k[0])
if k.shape != (m,):
raise ValueError("k must be a scalar or a rank-1 array of length 1 or m.")
if m == 0:
return p
else:
coeff = maximum(1, arange(len(p) + m, 0, -1)[np.newaxis, :] - 1 - arange(m)[:, np.newaxis]).prod(0)
return true_divide(concatenate((p, k)), coeff)
def logpdf(x, alpha):
x, alpha = _promote_dtypes_inexact(x, alpha)
if alpha.ndim != 1:
raise ValueError(
f"`alpha` must be one-dimensional; got alpha.shape={alpha.shape}"
)
if x.shape[0] not in (alpha.shape[0], alpha.shape[0] - 1):
raise ValueError(
"`x` must have either the same number of entries as `alpha` "
f"or one entry fewer; got x.shape={x.shape}, alpha.shape={alpha.shape}"
)
one = lax._const(x, 1)
if x.shape[0] != alpha.shape[0]:
x = jnp.concatenate([x, lax.sub(one, x.sum(0, keepdims=True))], axis=0)
normalize_term = jnp.sum(gammaln(alpha)) - gammaln(jnp.sum(alpha))
if x.ndim > 1:
alpha = lax.broadcast_in_dim(alpha, alpha.shape + (1,) * (x.ndim - 1), (0,))
log_probs = lax.sub(jnp.sum(xlogy(lax.sub(alpha, one), x), axis=0), normalize_term)
return jnp.where(_is_simplex(x), log_probs, -jnp.inf)
def _cofactor_solve(a, b):
"""Equivalent to det(a)*solve(a, b) for nonsingular mat.
Intermediate function used for jvp and vjp of det.
This function borrows heavily from jax.numpy.linalg.solve and
jax.numpy.linalg.slogdet to compute the gradient of the determinant
in a way that is well defined even for low rank matrices.
This function handles two different cases:
* rank(a) == n or n-1
* rank(a) < n-1
For rank n-1 matrices, the gradient of the determinant is a rank 1 matrix.
Rather than computing det(a)*solve(a, b), which would return NaN, we work
directly with the LU decomposition. If a = p @ l @ u, then
det(a)*solve(a, b) =
prod(diag(u)) * u^-1 @ l^-1 @ p^-1 b =
prod(diag(u)) * triangular_solve(u, solve(p @ l, b))
If a is rank n-1, then the lower right corner of u will be zero and the
triangular_solve will fail.
Let x = solve(p @ l, b) and y = det(a)*solve(a, b).
Then y_{n}
x_{n} / u_{nn} * prod_{i=1...n}(u_{ii}) =
x_{n} * prod_{i=1...n-1}(u_{ii})
So by replacing the lower-right corner of u with prod_{i=1...n-1}(u_{ii})^-1
we can avoid the triangular_solve failing.
To correctly compute the rest of y_{i} for i != n, we simply multiply
x_{i} by det(a) for all i != n, which will be zero if rank(a) = n-1.
For the second case, a check is done on the matrix to see if `solve`
returns NaN or Inf, and gives a matrix of zeros as a result, as the
gradient of the determinant of a matrix with rank less than n-1 is 0.
This will still return the correct value for rank n-1 matrices, as the check
is applied *after* the lower right corner of u has been updated.
Args:
a: A square matrix or batch of matrices, possibly singular.
b: A matrix, or batch of matrices of the same dimension as a.
Returns:
det(a) and cofactor(a)^T*b, aka adjugate(a)*b
"""
a = _promote_arg_dtypes(jnp.asarray(a))
b = _promote_arg_dtypes(jnp.asarray(b))
a_shape = jnp.shape(a)
b_shape = jnp.shape(b)
a_ndims = len(a_shape)
if not (a_ndims >= 2 and a_shape[-1] == a_shape[-2]
and b_shape[-2:] == a_shape[-2:]):
msg = ("The arguments to _cofactor_solve must have shapes "
"a=[..., m, m] and b=[..., m, m]; got a={} and b={}")
raise ValueError(msg.format(a_shape, b_shape))
if a_shape[-1] == 1:
return a[..., 0, 0], b
# lu contains u in the upper triangular matrix and l in the strict lower
# triangular matrix.
# The diagonal of l is set to ones without loss of generality.
lu, pivots, permutation = lax_linalg.lu(a)
dtype = lax.dtype(a)
batch_dims = lax.broadcast_shapes(lu.shape[:-2], b.shape[:-2])
x = jnp.broadcast_to(b, batch_dims + b.shape[-2:])
lu = jnp.broadcast_to(lu, batch_dims + lu.shape[-2:])
# Compute (partial) determinant, ignoring last diagonal of LU
diag = jnp.diagonal(lu, axis1=-2, axis2=-1)
parity = jnp.count_nonzero(pivots != jnp.arange(a_shape[-1]), axis=-1)
sign = jnp.asarray(-2 * (parity % 2) + 1, dtype=dtype)
# partial_det[:, -1] contains the full determinant and
# partial_det[:, -2] contains det(u) / u_{nn}.
partial_det = jnp.cumprod(diag, axis=-1) * sign[..., None]
lu = lu.at[..., -1, -1].set(1.0 / partial_det[..., -2])
permutation = jnp.broadcast_to(permutation, batch_dims + (a_shape[-1], ))
iotas = jnp.ix_(*(lax.iota(jnp.int32, b) for b in batch_dims + (1, )))
# filter out any matrices that are not full rank
d = jnp.ones(x.shape[:-1], x.dtype)
d = lax_linalg.triangular_solve(lu, d, left_side=True, lower=False)
d = jnp.any(jnp.logical_or(jnp.isnan(d), jnp.isinf(d)), axis=-1)
d = jnp.tile(d[..., None, None], d.ndim * (1, ) + x.shape[-2:])
x = jnp.where(d, jnp.zeros_like(x), x) # first filter
x = x[iotas[:-1] + (permutation, slice(None))]
x = lax_linalg.triangular_solve(lu,
x,
left_side=True,
lower=True,
unit_diagonal=True)
x = jnp.concatenate(
(x[..., :-1, :] * partial_det[..., -1, None, None], x[..., -1:, :]),
axis=-2)
x = lax_linalg.triangular_solve(lu, x, left_side=True, lower=False)
x = jnp.where(d, jnp.zeros_like(x), x) # second filter
return partial_det[..., -1], x
def eigh_tridiagonal(d,
e,
*,
eigvals_only=False,
select='a',
select_range=None,
tol=None):
if not eigvals_only:
raise NotImplementedError(
"Calculation of eigenvectors is not implemented")
def _sturm(alpha, beta_sq, pivmin, alpha0_perturbation, x):
"""Implements the Sturm sequence recurrence."""
n = alpha.shape[0]
zeros = jnp.zeros(x.shape, dtype=jnp.int32)
ones = jnp.ones(x.shape, dtype=jnp.int32)
# The first step in the Sturm sequence recurrence
# requires special care if x is equal to alpha[0].
def sturm_step0():
q = alpha[0] - x
count = jnp.where(q < 0, ones, zeros)
q = jnp.where(alpha[0] == x, alpha0_perturbation, q)
return q, count
# Subsequent steps all take this form:
def sturm_step(i, q, count):
q = alpha[i] - beta_sq[i - 1] / q - x
count = jnp.where(q <= pivmin, count + 1, count)
q = jnp.where(q <= pivmin, jnp.minimum(q, -pivmin), q)
return q, count
# The first step initializes q and count.
q, count = sturm_step0()
# Peel off ((n-1) % blocksize) steps from the main loop, so we can run
# the bulk of the iterations unrolled by a factor of blocksize.
blocksize = 16
i = 1
peel = (n - 1) % blocksize
unroll_cnt = peel
def unrolled_steps(args):
start, q, count = args
for j in range(unroll_cnt):
q, count = sturm_step(start + j, q, count)
return start + unroll_cnt, q, count
i, q, count = unrolled_steps((i, q, count))
# Run the remaining steps of the Sturm sequence using a partially
# unrolled while loop.
unroll_cnt = blocksize
def cond(iqc):
i, q, count = iqc
return jnp.less(i, n)
_, _, count = lax.while_loop(cond, unrolled_steps, (i, q, count))
return count
alpha = jnp.asarray(d)
beta = jnp.asarray(e)
supported_dtypes = (jnp.float32, jnp.float64, jnp.complex64,
jnp.complex128)
if alpha.dtype != beta.dtype:
raise TypeError(
"diagonal and off-diagonal values must have same dtype, "
f"got {alpha.dtype} and {beta.dtype}")
if alpha.dtype not in supported_dtypes or beta.dtype not in supported_dtypes:
raise TypeError(
"Only float32 and float64 inputs are supported as inputs "
"to jax.scipy.linalg.eigh_tridiagonal, got "
f"{alpha.dtype} and {beta.dtype}")
n = alpha.shape[0]
if n <= 1:
return jnp.real(alpha)
if jnp.issubdtype(alpha.dtype, jnp.complexfloating):
alpha = jnp.real(alpha)
beta_sq = jnp.real(beta * jnp.conj(beta))
beta_abs = jnp.sqrt(beta_sq)
else:
beta_abs = jnp.abs(beta)
beta_sq = jnp.square(beta)
# Estimate the largest and smallest eigenvalues of T using the Gershgorin
# circle theorem.
off_diag_abs_row_sum = jnp.concatenate(
[beta_abs[:1], beta_abs[:-1] + beta_abs[1:], beta_abs[-1:]], axis=0)
lambda_est_max = jnp.amax(alpha + off_diag_abs_row_sum)
lambda_est_min = jnp.amin(alpha - off_diag_abs_row_sum)
# Upper bound on 2-norm of T.
t_norm = jnp.maximum(jnp.abs(lambda_est_min), jnp.abs(lambda_est_max))
# Compute the smallest allowed pivot in the Sturm sequence to avoid
# overflow.
finfo = np.finfo(alpha.dtype)
one = np.ones([], dtype=alpha.dtype)
safemin = np.maximum(one / finfo.max, (one + finfo.eps) * finfo.tiny)
pivmin = safemin * jnp.maximum(1, jnp.amax(beta_sq))
alpha0_perturbation = jnp.square(finfo.eps * beta_abs[0])
abs_tol = finfo.eps * t_norm
if tol is not None:
abs_tol = jnp.maximum(tol, abs_tol)
# In the worst case, when the absolute tolerance is eps*lambda_est_max and
# lambda_est_max = -lambda_est_min, we have to take as many bisection steps
# as there are bits in the mantissa plus 1.
# The proof is left as an exercise to the reader.
max_it = finfo.nmant + 1
# Determine the indices of the desired eigenvalues, based on select and
# select_range.
if select == 'a':
target_counts = jnp.arange(n, dtype=jnp.int32)
elif select == 'i':
if select_range[0] > select_range[1]:
raise ValueError('Got empty index range in select_range.')
target_counts = jnp.arange(select_range[0],
select_range[1] + 1,
dtype=jnp.int32)
elif select == 'v':
# TODO(phawkins): requires dynamic shape support.
raise NotImplementedError("eigh_tridiagonal(..., select='v') is not "
"implemented")
else:
raise ValueError("'select must have a value in {'a', 'i', 'v'}.")
# Run binary search for all desired eigenvalues in parallel, starting from
# the interval lightly wider than the estimated
# [lambda_est_min, lambda_est_max].
fudge = 2.1 # We widen starting interval the Gershgorin interval a bit.
norm_slack = jnp.array(n, alpha.dtype) * fudge * finfo.eps * t_norm
lower = lambda_est_min - norm_slack - 2 * fudge * pivmin
upper = lambda_est_max + norm_slack + fudge * pivmin
# Pre-broadcast the scalars used in the Sturm sequence for improved
# performance.
target_shape = jnp.shape(target_counts)
lower = jnp.broadcast_to(lower, shape=target_shape)
upper = jnp.broadcast_to(upper, shape=target_shape)
mid = 0.5 * (upper + lower)
pivmin = jnp.broadcast_to(pivmin, target_shape)
alpha0_perturbation = jnp.broadcast_to(alpha0_perturbation, target_shape)
# Start parallel binary searches.
def cond(args):
i, lower, _, upper = args
return jnp.logical_and(jnp.less(i, max_it),
jnp.less(abs_tol, jnp.amax(upper - lower)))
def body(args):
i, lower, mid, upper = args
counts = _sturm(alpha, beta_sq, pivmin, alpha0_perturbation, mid)
lower = jnp.where(counts <= target_counts, mid, lower)
upper = jnp.where(counts > target_counts, mid, upper)
mid = 0.5 * (lower + upper)
return i + 1, lower, mid, upper
_, _, mid, _ = lax.while_loop(cond, body, (0, lower, mid, upper))
return mid
def _spectral_helper(x,
y,
fs=1.0,
window='hann',
nperseg=None,
noverlap=None,
nfft=None,
detrend_type='constant',
return_onesided=True,
scaling='density',
axis=-1,
mode='psd',
boundary=None,
padded=False):
"""LAX-backend implementation of `scipy.signal._spectral_helper`.
Unlike the original helper function, `y` can be None for explicitly
indicating auto-spectral (non cross-spectral) computation. In addition to
this, `detrend` argument is renamed to `detrend_type` for avoiding internal
name overlap.
"""
if mode not in ('psd', 'stft'):
raise ValueError(f"Unknown value for mode {mode}, "
"must be one of: ('psd', 'stft')")
def make_pad(mode, **kwargs):
def pad(x, n, axis=-1):
pad_width = [(0, 0) for unused_n in range(x.ndim)]
pad_width[axis] = (n, n)
return jnp.pad(x, pad_width, mode, **kwargs)
return pad
boundary_funcs = {
'even': make_pad('reflect'),
'odd': odd_ext,
'constant': make_pad('edge'),
'zeros': make_pad('constant', constant_values=0.0),
None: lambda x, *args, **kwargs: x
}
# Check/ normalize inputs
if boundary not in boundary_funcs:
raise ValueError(f"Unknown boundary option '{boundary}', "
f"must be one of: {list(boundary_funcs.keys())}")
axis = jax.core.concrete_or_error(operator.index, axis,
"axis of windowed-FFT")
axis = canonicalize_axis(axis, x.ndim)
if nperseg is not None: # if specified by user
nperseg = jax.core.concrete_or_error(int, nperseg,
"nperseg of windowed-FFT")
if nperseg < 1:
raise ValueError('nperseg must be a positive integer')
# parse window; if array like, then set nperseg = win.shape
win, nperseg = signal_helper._triage_segments(window,
nperseg,
input_length=x.shape[axis])
if noverlap is None:
noverlap = nperseg // 2
else:
noverlap = jax.core.concrete_or_error(int, noverlap,
"noverlap of windowed-FFT")
if nfft is None:
nfft = nperseg
else:
nfft = jax.core.concrete_or_error(int, nfft, "nfft of windowed-FFT")
_check_arraylike("_spectral_helper", x)
x = jnp.asarray(x)
if y is None:
outdtype = jax.dtypes.canonicalize_dtype(
np.result_type(x, np.complex64))
else:
_check_arraylike("_spectral_helper", y)
y = jnp.asarray(y)
outdtype = jax.dtypes.canonicalize_dtype(
np.result_type(x, y, np.complex64))
if mode != 'psd':
raise ValueError(
"two-argument mode is available only when mode=='psd'")
if x.ndim != y.ndim:
raise ValueError(
"two-arguments must have the same rank ({x.ndim} vs {y.ndim})."
)
# Check if we can broadcast the outer axes together
try:
outershape = jnp.broadcast_shapes(tuple_delete(x.shape, axis),
tuple_delete(y.shape, axis))
except ValueError as e:
raise ValueError('x and y cannot be broadcast together.') from e
# Special cases for size == 0
if y is None:
if x.size == 0:
return jnp.zeros(x.shape), jnp.zeros(x.shape), jnp.zeros(x.shape)
else:
if x.size == 0 or y.size == 0:
outshape = tuple_insert(outershape,
min([x.shape[axis], y.shape[axis]]), axis)
emptyout = jnp.zeros(outshape)
return emptyout, emptyout, emptyout
# Move time-axis to the end
if x.ndim > 1:
if axis != -1:
x = jnp.moveaxis(x, axis, -1)
if y is not None and y.ndim > 1:
y = jnp.moveaxis(y, axis, -1)
# Check if x and y are the same length, zero-pad if necessary
if y is not None:
if x.shape[-1] != y.shape[-1]:
if x.shape[-1] < y.shape[-1]:
pad_shape = list(x.shape)
pad_shape[-1] = y.shape[-1] - x.shape[-1]
x = jnp.concatenate((x, jnp.zeros(pad_shape)), -1)
else:
pad_shape = list(y.shape)
pad_shape[-1] = x.shape[-1] - y.shape[-1]
y = jnp.concatenate((y, jnp.zeros(pad_shape)), -1)
if nfft < nperseg:
raise ValueError('nfft must be greater than or equal to nperseg.')
if noverlap >= nperseg:
raise ValueError('noverlap must be less than nperseg.')
nstep = nperseg - noverlap
# Apply paddings
if boundary is not None:
ext_func = boundary_funcs[boundary]
x = ext_func(x, nperseg // 2, axis=-1)
if y is not None:
y = ext_func(y, nperseg // 2, axis=-1)
if padded:
# Pad to integer number of windowed segments
# I.e make x.shape[-1] = nperseg + (nseg-1)*nstep, with integer nseg
nadd = (-(x.shape[-1] - nperseg) % nstep) % nperseg
zeros_shape = list(x.shape[:-1]) + [nadd]
x = jnp.concatenate((x, jnp.zeros(zeros_shape)), axis=-1)
if y is not None:
zeros_shape = list(y.shape[:-1]) + [nadd]
y = jnp.concatenate((y, jnp.zeros(zeros_shape)), axis=-1)
# Handle detrending and window functions
if not detrend_type:
def detrend_func(d):
return d
elif not hasattr(detrend_type, '__call__'):
def detrend_func(d):
return detrend(d, type=detrend_type, axis=-1)
elif axis != -1:
# Wrap this function so that it receives a shape that it could
# reasonably expect to receive.
def detrend_func(d):
d = jnp.moveaxis(d, axis, -1)
d = detrend_type(d)
return jnp.moveaxis(d, -1, axis)
else:
detrend_func = detrend_type
if np.result_type(win, np.complex64) != outdtype:
win = win.astype(outdtype)
# Determine scale
if scaling == 'density':
scale = 1.0 / (fs * (win * win).sum())
elif scaling == 'spectrum':
scale = 1.0 / win.sum()**2
else:
raise ValueError(f'Unknown scaling: {scaling}')
if mode == 'stft':
scale = jnp.sqrt(scale)
# Determine onesided/ two-sided
if return_onesided:
sides = 'onesided'
if jnp.iscomplexobj(x) or jnp.iscomplexobj(y):
sides = 'twosided'
warnings.warn('Input data is complex, switching to '
'return_onesided=False')
else:
sides = 'twosided'
if sides == 'twosided':
freqs = jax.numpy.fft.fftfreq(nfft, 1 / fs)
elif sides == 'onesided':
freqs = jax.numpy.fft.rfftfreq(nfft, 1 / fs)
# Perform the windowed FFTs
result = _fft_helper(x, win, detrend_func, nperseg, noverlap, nfft, sides)
if y is not None:
# All the same operations on the y data
result_y = _fft_helper(y, win, detrend_func, nperseg, noverlap, nfft,
sides)
result = jnp.conjugate(result) * result_y
elif mode == 'psd':
result = jnp.conjugate(result) * result
result *= scale
if sides == 'onesided' and mode == 'psd':
end = None if nfft % 2 else -1
result = result.at[..., 1:end].mul(2)
time = jnp.arange(nperseg / 2, x.shape[-1] - nperseg / 2 + 1,
nperseg - noverlap) / fs
if boundary is not None:
time -= (nperseg / 2) / fs
result = result.astype(outdtype)
# All imaginary parts are zero anyways
if y is None and mode != 'stft':
result = result.real
# Move frequency axis back to axis where the data came from
result = jnp.moveaxis(result, -1, axis)
return freqs, time, result
