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""" Basic functions for manipulating 2d arrays
"""
__all__ = ['diag','diagflat','eye','fliplr','flipud','rot90','tri','triu',
'tril','vander','histogram2d','mask_indices',
'tril_indices','tril_indices_from','triu_indices','triu_indices_from',
]
from numpy.core.numeric import asanyarray, equal, subtract, arange, \
zeros, greater_equal, multiply, ones, asarray, alltrue, where, \
empty
def fliplr(m):
"""
Flip array in the left/right direction.
Flip the entries in each row in the left/right direction.
Columns are preserved, but appear in a different order than before.
Parameters
----------
m : array_like
Input array.
Returns
-------
f : ndarray
A view of `m` with the columns reversed. Since a view
is returned, this operation is :math:`\\mathcal O(1)`.
See Also
--------
flipud : Flip array in the up/down direction.
rot90 : Rotate array counterclockwise.
Notes
-----
Equivalent to A[:,::-1]. Does not require the array to be
two-dimensional.
Examples
--------
>>> A = np.diag([1.,2.,3.])
>>> A
array([[ 1., 0., 0.],
[ 0., 2., 0.],
[ 0., 0., 3.]])
>>> np.fliplr(A)
array([[ 0., 0., 1.],
[ 0., 2., 0.],
[ 3., 0., 0.]])
>>> A = np.random.randn(2,3,5)
>>> np.all(numpy.fliplr(A)==A[:,::-1,...])
True
"""
m = asanyarray(m)
if m.ndim < 2:
raise ValueError, "Input must be >= 2-d."
return m[:, ::-1]
def flipud(m):
"""
Flip array in the up/down direction.
Flip the entries in each column in the up/down direction.
Rows are preserved, but appear in a different order than before.
Parameters
----------
m : array_like
Input array.
Returns
-------
out : array_like
A view of `m` with the rows reversed. Since a view is
returned, this operation is :math:`\\mathcal O(1)`.
See Also
--------
fliplr : Flip array in the left/right direction.
rot90 : Rotate array counterclockwise.
Notes
-----
Equivalent to ``A[::-1,...]``.
Does not require the array to be two-dimensional.
Examples
--------
>>> A = np.diag([1.0, 2, 3])
>>> A
array([[ 1., 0., 0.],
[ 0., 2., 0.],
[ 0., 0., 3.]])
>>> np.flipud(A)
array([[ 0., 0., 3.],
[ 0., 2., 0.],
[ 1., 0., 0.]])
>>> A = np.random.randn(2,3,5)
>>> np.all(np.flipud(A)==A[::-1,...])
True
>>> np.flipud([1,2])
array([2, 1])
"""
m = asanyarray(m)
if m.ndim < 1:
raise ValueError, "Input must be >= 1-d."
return m[::-1,...]
def rot90(m, k=1):
"""
Rotate an array by 90 degrees in the counter-clockwise direction.
The first two dimensions are rotated; therefore, the array must be at
least 2-D.
Parameters
----------
m : array_like
Array of two or more dimensions.
k : integer
Number of times the array is rotated by 90 degrees.
Returns
-------
y : ndarray
Rotated array.
See Also
--------
fliplr : Flip an array horizontally.
flipud : Flip an array vertically.
Examples
--------
>>> m = np.array([[1,2],[3,4]], int)
>>> m
array([[1, 2],
[3, 4]])
>>> np.rot90(m)
array([[2, 4],
[1, 3]])
>>> np.rot90(m, 2)
array([[4, 3],
[2, 1]])
"""
m = asanyarray(m)
if m.ndim < 2:
raise ValueError, "Input must >= 2-d."
k = k % 4
if k == 0: return m
elif k == 1: return fliplr(m).swapaxes(0,1)
elif k == 2: return fliplr(flipud(m))
else: return fliplr(m.swapaxes(0,1)) # k==3
def eye(N, M=None, k=0, dtype=float):
"""
Return a 2-D array with ones on the diagonal and zeros elsewhere.
Parameters
----------
N : int
Number of rows in the output.
M : int, optional
Number of columns in the output. If None, defaults to `N`.
k : int, optional
Index of the diagonal: 0 refers to the main diagonal, a positive value
refers to an upper diagonal, and a negative value to a lower diagonal.
dtype : dtype, optional
Data-type of the returned array.
Returns
-------
I : ndarray (N,M)
An array where all elements are equal to zero, except for the `k`-th
diagonal, whose values are equal to one.
See Also
--------
diag : Return a diagonal 2-D array using a 1-D array specified by the user.
Examples
--------
>>> np.eye(2, dtype=int)
array([[1, 0],
[0, 1]])
>>> np.eye(3, k=1)
array([[ 0., 1., 0.],
[ 0., 0., 1.],
[ 0., 0., 0.]])
"""
if M is None:
M = N
m = zeros((N, M), dtype=dtype)
if k >= M:
return m
if k >= 0:
i = k
else:
i = (-k) * M
m[:M-k].flat[i::M+1] = 1
return m
def diag(v, k=0):
"""
Extract a diagonal or construct a diagonal array.
Parameters
----------
v : array_like
If `v` is a 2-D array, return a copy of its `k`-th diagonal.
If `v` is a 1-D array, return a 2-D array with `v` on the `k`-th
diagonal.
k : int, optional
Diagonal in question. The default is 0. Use `k>0` for diagonals
above the main diagonal, and `k<0` for diagonals below the main
diagonal.
Returns
-------
out : ndarray
The extracted diagonal or constructed diagonal array.
See Also
--------
diagonal : Return specified diagonals.
diagflat : Create a 2-D array with the flattened input as a diagonal.
trace : Sum along diagonals.
triu : Upper triangle of an array.
tril : Lower triange of an array.
Examples
--------
>>> x = np.arange(9).reshape((3,3))
>>> x
array([[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])
>>> np.diag(x)
array([0, 4, 8])
>>> np.diag(x, k=1)
array([1, 5])
>>> np.diag(x, k=-1)
array([3, 7])
>>> np.diag(np.diag(x))
array([[0, 0, 0],
[0, 4, 0],
[0, 0, 8]])
"""
v = asarray(v)
s = v.shape
if len(s) == 1:
n = s[0]+abs(k)
res = zeros((n,n), v.dtype)
if k >= 0:
i = k
else:
i = (-k) * n
res[:n-k].flat[i::n+1] = v
return res
elif len(s) == 2:
if k >= s[1]:
return empty(0, dtype=v.dtype)
if v.flags.f_contiguous:
# faster slicing
v, k, s = v.T, -k, s[::-1]
if k >= 0:
i = k
else:
i = (-k) * s[1]
return v[:s[1]-k].flat[i::s[1]+1]
else:
raise ValueError, "Input must be 1- or 2-d."
def diagflat(v,k=0):
"""
Create a two-dimensional array with the flattened input as a diagonal.
Parameters
----------
v : array_like
Input data, which is flattened and set as the `k`-th
diagonal of the output.
k : int, optional
Diagonal to set. The default is 0.
Returns
-------
out : ndarray
The 2-D output array.
See Also
--------
diag : Matlab workalike for 1-D and 2-D arrays.
diagonal : Return specified diagonals.
trace : Sum along diagonals.
Examples
--------
>>> np.diagflat([[1,2], [3,4]])
array([[1, 0, 0, 0],
[0, 2, 0, 0],
[0, 0, 3, 0],
[0, 0, 0, 4]])
>>> np.diagflat([1,2], 1)
array([[0, 1, 0],
[0, 0, 2],
[0, 0, 0]])
"""
try:
wrap = v.__array_wrap__
except AttributeError:
wrap = None
v = asarray(v).ravel()
s = len(v)
n = s + abs(k)
res = zeros((n,n), v.dtype)
if (k>=0):
i = arange(0,n-k)
fi = i+k+i*n
else:
i = arange(0,n+k)
fi = i+(i-k)*n
res.flat[fi] = v
if not wrap:
return res
return wrap(res)
def tri(N, M=None, k=0, dtype=float):
"""
Construct an array filled with ones at and below the given diagonal.
Parameters
----------
N : int
Number of rows in the array.
M : int, optional
Number of columns in the array.
By default, `M` is taken equal to `N`.
k : int, optional
The sub-diagonal below which the array is filled.
`k` = 0 is the main diagonal, while `k` < 0 is below it,
and `k` > 0 is above. The default is 0.
dtype : dtype, optional
Data type of the returned array. The default is float.
Returns
-------
T : (N,M) ndarray
Array with a lower triangle filled with ones, in other words
``T[i,j] == 1`` for ``i <= j + k``.
Examples
--------
>>> np.tri(3, 5, 2, dtype=int)
array([[1, 1, 1, 0, 0],
[1, 1, 1, 1, 0],
[1, 1, 1, 1, 1]])
>>> np.tri(3, 5, -1)
array([[ 0., 0., 0., 0., 0.],
[ 1., 0., 0., 0., 0.],
[ 1., 1., 0., 0., 0.]])
"""
if M is None: M = N
m = greater_equal(subtract.outer(arange(N), arange(M)),-k)
return m.astype(dtype)
def tril(m, k=0):
"""
Lower triangle of an array.
Return a copy of an array with elements above the `k`-th diagonal zeroed.
Parameters
----------
m : array_like, shape (M, N)
Input array.
k : int
Diagonal above which to zero elements.
`k = 0` is the main diagonal, `k < 0` is below it and `k > 0` is above.
Returns
-------
L : ndarray, shape (M, N)
Lower triangle of `m`, of same shape and data-type as `m`.
See Also
--------
triu
Examples
--------
>>> np.tril([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 0, 0, 0],
[ 4, 0, 0],
[ 7, 8, 0],
[10, 11, 12]])
"""
m = asanyarray(m)
out = multiply(tri(m.shape[0], m.shape[1], k=k, dtype=int),m)
return out
def triu(m, k=0):
"""
Upper triangle of an array.
Construct a copy of a matrix with elements below the k-th diagonal zeroed.
Please refer to the documentation for `tril`.
See Also
--------
tril
Examples
--------
>>> np.triu([[1,2,3],[4,5,6],[7,8,9],[10,11,12]], -1)
array([[ 1, 2, 3],
[ 4, 5, 6],
[ 0, 8, 9],
[ 0, 0, 12]])
"""
m = asanyarray(m)
out = multiply((1-tri(m.shape[0], m.shape[1], k-1, int)),m)
return out
# borrowed from John Hunter and matplotlib
def vander(x, N=None):
"""
Generate a Van der Monde matrix.
The columns of the output matrix are decreasing powers of the input
vector. Specifically, the i-th output column is the input vector to
the power of ``N - i - 1``. Such a matrix with a geometric progression
in each row is named Van Der Monde, or Vandermonde matrix, from
Alexandre-Theophile Vandermonde.
Parameters
----------
x : array_like
1-D input array.
N : int, optional
Order of (number of columns in) the output. If `N` is not specified,
a square array is returned (``N = len(x)``).
Returns
-------
out : ndarray
Van der Monde matrix of order `N`. The first column is ``x^(N-1)``,
the second ``x^(N-2)`` and so forth.
References
----------
.. [1] Wikipedia, "Vandermonde matrix",
http://en.wikipedia.org/wiki/Vandermonde_matrix
Examples
--------
>>> x = np.array([1, 2, 3, 5])
>>> N = 3
>>> np.vander(x, N)
array([[ 1, 1, 1],
[ 4, 2, 1],
[ 9, 3, 1],
[25, 5, 1]])
>>> np.column_stack([x**(N-1-i) for i in range(N)])
array([[ 1, 1, 1],
[ 4, 2, 1],
[ 9, 3, 1],
[25, 5, 1]])
>>> x = np.array([1, 2, 3, 5])
>>> np.vander(x)
array([[ 1, 1, 1, 1],
[ 8, 4, 2, 1],
[ 27, 9, 3, 1],
[125, 25, 5, 1]])
The determinant of a square Vandermonde matrix is the product
of the differences between the values of the input vector:
>>> np.linalg.det(np.vander(x))
48.000000000000043
>>> (5-3)*(5-2)*(5-1)*(3-2)*(3-1)*(2-1)
48
"""
x = asarray(x)
if N is None: N=len(x)
X = ones( (len(x),N), x.dtype)
for i in range(N-1):
X[:,i] = x**(N-i-1)
return X
def histogram2d(x,y, bins=10, range=None, normed=False, weights=None):
"""
Compute the bi-dimensional histogram of two data samples.
Parameters
----------
x : array_like, shape(N,)
A sequence of values to be histogrammed along the first dimension.
y : array_like, shape(M,)
A sequence of values to be histogrammed along the second dimension.
bins : int or [int, int] or array_like or [array, array], optional
The bin specification:
* If int, the number of bins for the two dimensions (nx=ny=bins).
* If [int, int], the number of bins in each dimension (nx, ny = bins).
* If array_like, the bin edges for the two dimensions (x_edges=y_edges=bins).
* If [array, array], the bin edges in each dimension (x_edges, y_edges = bins).
range : array_like, shape(2,2), optional
The leftmost and rightmost edges of the bins along each dimension
(if not specified explicitly in the `bins` parameters):
``[[xmin, xmax], [ymin, ymax]]``. All values outside of this range
will be considered outliers and not tallied in the histogram.
normed : bool, optional
If False, returns the number of samples in each bin. If True, returns
the bin density, i.e. the bin count divided by the bin area.
weights : array_like, shape(N,), optional
An array of values ``w_i`` weighing each sample ``(x_i, y_i)``. Weights
are normalized to 1 if `normed` is True. If `normed` is False, the
values of the returned histogram are equal to the sum of the weights
belonging to the samples falling into each bin.
Returns
-------
H : ndarray, shape(nx, ny)
The bi-dimensional histogram of samples `x` and `y`. Values in `x`
are histogrammed along the first dimension and values in `y` are
histogrammed along the second dimension.
xedges : ndarray, shape(nx,)
The bin edges along the first dimension.
yedges : ndarray, shape(ny,)
The bin edges along the second dimension.
See Also
--------
histogram: 1D histogram
histogramdd: Multidimensional histogram
Notes
-----
When `normed` is True, then the returned histogram is the sample density,
defined such that:
.. math::
\\sum_{i=0}^{nx-1} \\sum_{j=0}^{ny-1} H_{i,j} \\Delta x_i \\Delta y_j = 1
where `H` is the histogram array and :math:`\\Delta x_i \\Delta y_i`
the area of bin `{i,j}`.
Please note that the histogram does not follow the Cartesian convention
where `x` values are on the abcissa and `y` values on the ordinate axis.
Rather, `x` is histogrammed along the first dimension of the array
(vertical), and `y` along the second dimension of the array (horizontal).
This ensures compatibility with `histogramdd`.
Examples
--------
>>> x, y = np.random.randn(2, 100)
>>> H, xedges, yedges = np.histogram2d(x, y, bins=(5, 8))
>>> H.shape, xedges.shape, yedges.shape
((5,8), (6,), (9,))
We can now use the Matplotlib to visualize this 2-dimensional histogram:
>>> extent = [xedges[0], xedges[-1], yedges[0], yedges[-1]]
>>> import matplotlib.pyplot as plt
>>> plt.imshow(H, extent=extent)
>>> plt.show()
"""
from numpy import histogramdd
try:
N = len(bins)
except TypeError:
N = 1
if N != 1 and N != 2:
xedges = yedges = asarray(bins, float)
bins = [xedges, yedges]
hist, edges = histogramdd([x,y], bins, range, normed, weights)
return hist, edges[0], edges[1]
def mask_indices(n,mask_func,k=0):
"""
Return the indices to access (n, n) arrays, given a masking function.
Assume `mask_func` is a function that, for a square array a of size
``(n, n)`` with a possible offset argument `k`, when called as
``mask_func(a, k)`` returns a new array with zeros in certain locations
(functions like `triu` or `tril` do precisely this). Then this function
returns the indices where the non-zero values would be located.
Parameters
----------
n : int
The returned indices will be valid to access arrays of shape (n, n).
mask_func : callable
A function whose call signature is similar to that of `triu`, `tril`.
That is, ``mask_func(x, k)`` returns a boolean array, shaped like `x`.
`k` is an optional argument to the function.
k : scalar
An optional argument which is passed through to `mask_func`. Functions
like `triu`, `tril` take a second argument that is interpreted as an
offset.
Returns
-------
indices : tuple of arrays.
The `n` arrays of indices corresponding to the locations where
``mask_func(np.ones((n, n)), k)`` is True.
See Also
--------
triu, tril, triu_indices, tril_indices
Notes
-----
.. versionadded:: 1.4.0
Examples
--------
These are the indices that would allow you to access the upper triangular
part of any 3x3 array:
>>> iu = np.mask_indices(3, np.triu)
For example, if `a` is a 3x3 array:
>>> a = np.arange(9).reshape(3, 3)
>>> a
array([[0, 1, 2],
[3, 4, 5],
[6, 7, 8]])
>>> a[iu]
array([0, 1, 2, 4, 5, 8])
An offset can be passed also to the masking function. This gets us the
indices starting on the first diagonal right of the main one:
>>> iu1 = np.mask_indices(3, np.triu, 1)
with which we now extract only three elements:
>>> a[iu1]
array([1, 2, 5])
"""
m = ones((n,n),int)
a = mask_func(m,k)
return where(a != 0)
def tril_indices(n,k=0):
"""
Return the indices for the lower-triangle of an (n, n) array.
Parameters
----------
n : int
Sets the size of the arrays for which the returned indices will be valid.
k : int, optional
Diagonal offset (see `tril` for details).
Returns
-------
inds : tuple of arrays
The indices for the triangle. The returned tuple contains two arrays,
each with the indices along one dimension of the array.
See also
--------
triu_indices : similar function, for upper-triangular.
mask_indices : generic function accepting an arbitrary mask function.
tril, triu
Notes
-----
.. versionadded:: 1.4.0
Examples
--------
Compute two different sets of indices to access 4x4 arrays, one for the
lower triangular part starting at the main diagonal, and one starting two
diagonals further right:
>>> il1 = np.tril_indices(4)
>>> il2 = np.tril_indices(4, 2)
Here is how they can be used with a sample array:
>>> a = np.arange(16).reshape(4, 4)
>>> a
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
Both for indexing:
>>> a[il1]
array([ 0, 4, 5, 8, 9, 10, 12, 13, 14, 15])
And for assigning values:
>>> a[il1] = -1
>>> a
array([[-1, 1, 2, 3],
[-1, -1, 6, 7],
[-1, -1, -1, 11],
[-1, -1, -1, -1]])
These cover almost the whole array (two diagonals right of the main one):
>>> a[il2] = -10
>>> a
array([[-10, -10, -10, 3],
[-10, -10, -10, -10],
[-10, -10, -10, -10],
[-10, -10, -10, -10]])
"""
return mask_indices(n,tril,k)
def tril_indices_from(arr,k=0):
"""
Return the indices for the lower-triangle of an (n, n) array.
See `tril_indices` for full details.
Parameters
----------
n : int
Sets the size of the arrays for which the returned indices will be valid.
k : int, optional
Diagonal offset (see `tril` for details).
See Also
--------
tril_indices, tril
Notes
-----
.. versionadded:: 1.4.0
"""
if not arr.ndim==2 and arr.shape[0] == arr.shape[1]:
raise ValueError("input array must be 2-d and square")
return tril_indices(arr.shape[0],k)
def triu_indices(n,k=0):
"""
Return the indices for the upper-triangle of an (n, n) array.
Parameters
----------
n : int
Sets the size of the arrays for which the returned indices will be valid.
k : int, optional
Diagonal offset (see `triu` for details).
Returns
-------
inds : tuple of arrays
The indices for the triangle. The returned tuple contains two arrays,
each with the indices along one dimension of the array.
See also
--------
tril_indices : similar function, for lower-triangular.
mask_indices : generic function accepting an arbitrary mask function.
triu, tril
Notes
-----
.. versionadded:: 1.4.0
Examples
--------
Compute two different sets of indices to access 4x4 arrays, one for the
upper triangular part starting at the main diagonal, and one starting two
diagonals further right:
>>> iu1 = np.triu_indices(4)
>>> iu2 = np.triu_indices(4, 2)
Here is how they can be used with a sample array:
>>> a = np.arange(16).reshape(4, 4)
>>> a
array([[ 0, 1, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[12, 13, 14, 15]])
Both for indexing:
>>> a[iu1]
array([ 0, 1, 2, 3, 5, 6, 7, 10, 11, 15])
And for assigning values:
>>> a[iu1] = -1
>>> a
array([[-1, -1, -1, -1],
[ 4, -1, -1, -1],
[ 8, 9, -1, -1],
[12, 13, 14, -1]])
These cover only a small part of the whole array (two diagonals right
of the main one):
>>> a[iu2] = -10
>>> a
array([[ -1, -1, -10, -10],
[ 4, -1, -1, -10],
[ 8, 9, -1, -1],
[ 12, 13, 14, -1]])
"""
return mask_indices(n,triu,k)
def triu_indices_from(arr,k=0):
"""
Return the indices for the lower-triangle of an (n, n) array.
See `triu_indices` for full details.
Parameters
----------
n : int
Sets the size of the arrays for which the returned indices will be valid.
k : int, optional
Diagonal offset (see `triu` for details).
See Also
--------
triu_indices, triu
Notes
-----
.. versionadded:: 1.4.0
"""
if not arr.ndim==2 and arr.shape[0] == arr.shape[1]:
raise ValueError("input array must be 2-d and square")
return triu_indices(arr.shape[0],k)