""" Basic functions for manipulating 2d arrays """ from __future__ import division, absolute_import, print_function from numpy.core.numeric import ( asanyarray, arange, zeros, greater_equal, multiply, ones, asarray, where, int8, int16, int32, int64, empty, promote_types ) from numpy.core import iinfo __all__ = [ 'diag', 'diagflat', 'eye', 'fliplr', 'flipud', 'rot90', 'tri', 'triu', 'tril', 'vander', 'histogram2d', 'mask_indices', 'tril_indices', 'tril_indices_from', 'triu_indices', 'triu_indices_from', ] i1 = iinfo(int8) i2 = iinfo(int16) i4 = iinfo(int32) def _min_int(low, high): """ get small int that fits the range """ if high <= i1.max and low >= i1.min: return int8 if high <= i2.max and low >= i2.min: return int16 if high <= i4.max and low >= i4.min: return int32 return int64 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, must be at least 2-D. 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]. Requires the array to be at least 2-D. 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(np.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: # k == 3 return fliplr(m.swapaxes(0, 1)) 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 (the default) refers to the main diagonal, a positive value refers to an upper diagonal, and a negative value to a lower diagonal. dtype : data-type, optional Data-type of the returned array. Returns ------- I : ndarray of shape (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 -------- identity : (almost) equivalent function diag : diagonal 2-D array from 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. See the more detailed documentation for numpy.diagonal if you use this function to extract a diagonal and wish to write to the resulting array; whether it returns a copy or a view depends on what version of numpy you are using. 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 triangle 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: return v.diagonal(k) 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; 0, the default, corresponds to the "main" diagonal, a positive (negative) k giving the number of the diagonal above (below) the main. Returns ------- out : ndarray The 2-D output array. See Also -------- diag : MATLAB work-alike 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): """ An array with ones at and below the given diagonal and zeros elsewhere. 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 at and 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 ------- tri : ndarray of shape (N, M) Array with its lower triangle filled with ones and zero elsewhere; in other words T[i,j] == 1 for i <= j + k, 0 otherwise. 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.outer(arange(N, dtype=_min_int(0, N)), arange(-k, M-k, dtype=_min_int(-k, M - k))) # Avoid making a copy if the requested type is already bool m = m.astype(dtype, copy=False) return m 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, optional Diagonal above which to zero elements. k = 0 (the default) is the main diagonal, k < 0 is below it and k > 0 is above. Returns ------- tril : ndarray, shape (M, N) Lower triangle of m, of same shape and data-type as m. See Also -------- triu : same thing, only for the upper triangle 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) mask = tri(*m.shape[-2:], k=k, dtype=bool) return where(mask, m, zeros(1, m.dtype)) def triu(m, k=0): """ Upper triangle of an array. Return a copy of a matrix with the elements below the k-th diagonal zeroed. Please refer to the documentation for tril for further details. See Also -------- tril : lower triangle of an array 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) mask = tri(*m.shape[-2:], k=k-1, dtype=bool) return where(mask, zeros(1, m.dtype), m) # Originally borrowed from John Hunter and matplotlib def vander(x, N=None, increasing=False): """ Generate a Vandermonde matrix. The columns of the output matrix are powers of the input vector. The order of the powers is determined by the increasing boolean argument. Specifically, when increasing is False, the i-th output column is the input vector raised element-wise to the power of N - i - 1. Such a matrix with a geometric progression in each row is named for Alexandre- Theophile Vandermonde. Parameters ---------- x : array_like 1-D input array. N : int, optional Number of columns in the output. If N is not specified, a square array is returned (N = len(x)). increasing : bool, optional Order of the powers of the columns. If True, the powers increase from left to right, if False (the default) they are reversed. .. versionadded:: 1.9.0 Returns ------- out : ndarray Vandermonde matrix. If increasing is False, the first column is x^(N-1), the second x^(N-2) and so forth. If increasing is True, the columns are x^0, x^1, ..., x^(N-1). See Also -------- polynomial.polynomial.polyvander 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]]) >>> np.vander(x, increasing=True) array([[ 1, 1, 1, 1], [ 1, 2, 4, 8], [ 1, 3, 9, 27], [ 1, 5, 25, 125]]) 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 x.ndim != 1: raise ValueError("x must be a one-dimensional array or sequence.") if N is None: N = len(x) v = empty((len(x), N), dtype=promote_types(x.dtype, int)) tmp = v[:, ::-1] if not increasing else v if N > 0: tmp[:, 0] = 1 if N > 1: tmp[:, 1:] = x[:, None] multiply.accumulate(tmp[:, 1:], out=tmp[:, 1:], axis=1) return v 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,) An array containing the x coordinates of the points to be histogrammed. y : array_like, shape (N,) An array containing the y coordinates of the points to be histogrammed. 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 bin_count / sample_count / 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 the sum over bins of the product bin_value * bin_area is 1. Please note that the histogram does not follow the Cartesian convention where x values are on the abscissa 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 -------- >>> import matplotlib as mpl >>> import matplotlib.pyplot as plt Construct a 2D-histogram with variable bin width. First define the bin edges: >>> xedges = [0, 1, 1.5, 3, 5] >>> yedges = [0, 2, 3, 4, 6] Next we create a histogram H with random bin content: >>> x = np.random.normal(3, 1, 100) >>> y = np.random.normal(1, 1, 100) >>> H, xedges, yedges = np.histogram2d(y, x, bins=(xedges, yedges)) Or we fill the histogram H with a determined bin content: >>> H = np.ones((4, 4)).cumsum().reshape(4, 4) >>> print H[::-1] # This shows the bin content in the order as plotted [[ 13. 14. 15. 16.] [ 9. 10. 11. 12.] [ 5. 6. 7. 8.] [ 1. 2. 3. 4.]] Imshow can only do an equidistant representation of bins: >>> fig = plt.figure(figsize=(7, 3)) >>> ax = fig.add_subplot(131) >>> ax.set_title('imshow: equidistant') >>> im = plt.imshow(H, interpolation='nearest', origin='low', extent=[xedges[0], xedges[-1], yedges[0], yedges[-1]]) pcolormesh can display exact bin edges: >>> ax = fig.add_subplot(132) >>> ax.set_title('pcolormesh: exact bin edges') >>> X, Y = np.meshgrid(xedges, yedges) >>> ax.pcolormesh(X, Y, H) >>> ax.set_aspect('equal') NonUniformImage displays exact bin edges with interpolation: >>> ax = fig.add_subplot(133) >>> ax.set_title('NonUniformImage: interpolated') >>> im = mpl.image.NonUniformImage(ax, interpolation='bilinear') >>> xcenters = xedges[:-1] + 0.5 * (xedges[1:] - xedges[:-1]) >>> ycenters = yedges[:-1] + 0.5 * (yedges[1:] - yedges[:-1]) >>> im.set_data(xcenters, ycenters, H) >>> ax.images.append(im) >>> ax.set_xlim(xedges[0], xedges[-1]) >>> ax.set_ylim(yedges[0], yedges[-1]) >>> ax.set_aspect('equal') >>> 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, m=None): """ Return the indices for the lower-triangle of an (n, m) array. Parameters ---------- n : int The row dimension of the arrays for which the returned indices will be valid. k : int, optional Diagonal offset (see tril for details). m : int, optional .. versionadded:: 1.9.0 The column dimension of the arrays for which the returned arrays will be valid. By default m is taken equal to n. 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 where(tri(n, m, k=k, dtype=bool)) def tril_indices_from(arr, k=0): """ Return the indices for the lower-triangle of arr. See tril_indices for full details. Parameters ---------- arr : array_like The indices will be valid for square arrays whose dimensions are the same as arr. k : int, optional Diagonal offset (see tril for details). See Also -------- tril_indices, tril Notes ----- .. versionadded:: 1.4.0 """ if arr.ndim != 2: raise ValueError("input array must be 2-d") return tril_indices(arr.shape[-2], k=k, m=arr.shape[-1]) def triu_indices(n, k=0, m=None): """ Return the indices for the upper-triangle of an (n, m) array. Parameters ---------- n : int The size of the arrays for which the returned indices will be valid. k : int, optional Diagonal offset (see triu for details). m : int, optional .. versionadded:: 1.9.0 The column dimension of the arrays for which the returned arrays will be valid. By default m is taken equal to n. Returns ------- inds : tuple, shape(2) of ndarrays, shape(n) The indices for the triangle. The returned tuple contains two arrays, each with the indices along one dimension of the array. Can be used to slice a ndarray of shape(n, n). 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 where(~tri(n, m, k=k-1, dtype=bool)) def triu_indices_from(arr, k=0): """ Return the indices for the upper-triangle of arr. See triu_indices for full details. Parameters ---------- arr : ndarray, shape(N, N) The indices will be valid for square arrays. k : int, optional Diagonal offset (see triu for details). Returns ------- triu_indices_from : tuple, shape(2) of ndarray, shape(N) Indices for the upper-triangle of arr. See Also -------- triu_indices, triu Notes ----- .. versionadded:: 1.4.0 """ if arr.ndim != 2: raise ValueError("input array must be 2-d") return triu_indices(arr.shape[-2], k=k, m=arr.shape[-1])