# index_utils¶

## Module: index_utils¶

Utilities for indexing into 2-d arrays, brought in from numpy 1.4, to support use of older versions of numpy

## Functions¶

nitime.index_utils.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. 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.

histogram
1D histogram
histogramdd
Multidimensional histogram

Notes

When normed is True, then the returned histogram is the sample density, defined such that:

where H is the histogram array and 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,))

nitime.index_utils.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. indices : tuple of arrays. The n arrays of indices corresponding to the locations where mask_func(np.ones((n, n)), k) is True.

Notes

New in version 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])

nitime.index_utils.tri(N, M=None, k=0, dtype=<type '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. 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.]])

nitime.index_utils.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. L : ndarray, shape (M, N) Lower triangle of m, of same shape and data-type as m.

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]])

nitime.index_utils.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). 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.

triu_indices
similar function, for upper-triangular.
mask_indices
generic function accepting an arbitrary mask function.

Notes

New in version 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]])

nitime.index_utils.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).

Notes

New in version 1.4.0.

nitime.index_utils.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.

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]])

nitime.index_utils.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). 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.

tril_indices
similar function, for lower-triangular.
mask_indices
generic function accepting an arbitrary mask function.

Notes

New in version 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]])

nitime.index_utils.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).

Notes

New in version 1.4.0.

nitime.index_utils.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)). 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

 [R1] 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]])