二维数组中每个对角线的最大值

我有数组，需要动态窗口最大滚动差异。

a = np.array([8, 18, 5,15,12])
print (a)
[ 8 18  5 15 12]

所以首先我自己创造差异：

b = a - a[:, None]
print (b)
[[  0  10  -3   7   4]
 [-10   0 -13  -3  -6]
 [  3  13   0  10   7]
 [ -7   3 -10   0  -3]
 [ -4   6  -7   3   0]]

然后将上三角矩阵替换为0：

c = np.tril(b)
print (c)
[[  0   0   0   0   0]
 [-10   0   0   0   0]
 [  3  13   0   0   0]
 [ -7   3 -10   0   0]
 [ -4   6  -7   3   0]]

最后需要每个对角线的最大值，所以这意味着：

max([0,0,0,0,0]) = 0  
max([-10,13,-10,3]) = 13
max([3,3,-7]) = 3
max([-7,6]) = 6
max([-4]) = -4

因此，预期输出为：

[0, 13, 3, 6, -4]

什么是好的矢量化解决方案？还是可能以其他方式获得预期的输出？

不知道考虑涉及的高级索引的效率到底是多少，但这是一种实现方法：

import numpy as np

a = np.array([8, 18, 5, 15, 12])
b = a[:, None] - a
# Fill lower triangle with largest negative
b[np.tril_indices(len(a))] = np.iinfo(b.dtype).min  # np.finfo for float
# Put diagonals as rows
s = b.strides[1]
diags = np.ndarray((len(a) - 1, len(a) - 1), b.dtype, b, offset=s, strides=(s, (len(a) + 1) * s))
# Get maximum from each row and add initial zero
c = np.r_[0, diags.max(1)]
print(c)
# [ 0 13  3  6 -4]

编辑：

另一个选择（可能不是您想要的）只是使用Numba，例如：

import numpy as np
import numba as nb

def max_window_diffs_jdehesa(a):
    a = np.asarray(a)
    dtinf = np.iinfo(b.dtype) if np.issubdtype(b.dtype, np.integer) else np.finfo(b.dtype)
    out = np.full_like(a, dtinf.min)
    _pwise_diffs(a, out)
    return out

@nb.njit(parallel=True)
def _pwise_diffs(a, out):
    out[0] = 0
    for w in nb.prange(1, len(a)):
        for i in range(len(a) - w):
            out[w] = max(a[i] - a[i + w], out[w])

a = np.array([8, 18, 5, 15, 12])
print(max_window_diffs(a))
# [ 0 13  3  6 -4]

将这些方法与原始方法进行比较：

import numpy as np
import numba as nb

def max_window_diffs_orig(a):
    a = np.asarray(a)
    b = a - a[:, None]
    out = np.zeros(len(a), b.dtype)
    out[-1] = b[-1, 0]
    for i in range(1, len(a) - 1):
        out[i] = np.diag(b, -i).max()
    return out

def max_window_diffs_jdehesa_np(a):
    a = np.asarray(a)
    b = a[:, None] - a
    dtinf = np.iinfo(b.dtype) if np.issubdtype(b.dtype, np.integer) else np.finfo(b.dtype)
    b[np.tril_indices(len(a))] = dtinf.min
    s = b.strides[1]
    diags = np.ndarray((len(a) - 1, len(a) - 1), b.dtype, b, offset=s, strides=(s, (len(a) + 1) * s))
    return np.concatenate([[0], diags.max(1)])

def max_window_diffs_jdehesa_nb(a):
    a = np.asarray(a)
    dtinf = np.iinfo(b.dtype) if np.issubdtype(b.dtype, np.integer) else np.finfo(b.dtype)
    out = np.full_like(a, dtinf.min)
    _pwise_diffs(a, out)
    return out

@nb.njit(parallel=True)
def _pwise_diffs(a, out):
    out[0] = 0
    for w in nb.prange(1, len(a)):
        for i in range(len(a) - w):
            out[w] = max(a[i] - a[i + w], out[w])

np.random.seed(0)
a = np.random.randint(0, 100, size=100)
r = max_window_diffs_orig(a)
print((max_window_diffs_jdehesa_np(a) == r).all())
# True
print((max_window_diffs_jdehesa_nb(a) == r).all())
# True

%timeit max_window_diffs_orig(a)
# 348 µs ± 986 ns per loop (mean ± std. dev. of 7 runs, 1000 loops each)
%timeit max_window_diffs_jdehesa_np(a)
# 91.7 µs ± 1.3 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit max_window_diffs_jdehesa_nb(a)
# 19.7 µs ± 88.1 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)

np.random.seed(0)
a = np.random.randint(0, 100, size=10000)
%timeit max_window_diffs_orig(a)
# 651 ms ± 26 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
%timeit max_window_diffs_jdehesa_np(a)
# 1.61 s ± 6.19 ms per loop (mean ± std. dev. of 7 runs, 1 loop each)
%timeit max_window_diffs_jdehesa_nb(a)
# 22 ms ± 967 µs per loop (mean ± std. dev. of 7 runs, 10 loops each)

对于较小的阵列，第一个可能会更好一些，但对于较大的阵列则效果不佳。另一方面，Numba在所有情况下都相当不错。

采用 ndarray.diagonal

v = [max(c.diagonal(-i)) for i in range(b.shape[0])]
print(v) # [0, 13, 3, 6, -4]

您可以使用numpy.diagonal：

a = np.array([8, 18, 5,15,12])
b = a - a[:, None]
c = np.tril(b)
for i in range(b.shape[0]):
    print(max(c.diagonal(-i)))

输出：

0
13
3
6
-4

这是带有strides- 的矢量化解决方案

from skimage.util import view_as_windows

n = len(a)
z = np.zeros(n-1,dtype=a.dtype)
p = np.concatenate((a,z))

s = view_as_windows(p,n)
mask = np.tri(n,k=-1,dtype=bool)[:,::-1]
v = s[0]-s
out = np.where(mask,v.min()-1,v).max(1)

通过一个循环提高内存效率-

n = len(a)
out = [max(a[:-i+n]-a[i:]) for i in range(n)]

使用np.max代替max更好地利用阵列存储器。

你可以滥用的事实，重塑形状的非方形阵列(N+1, N)，以(N, N+1)将对角线显示为列

from scipy.linalg import toeplitz
a = toeplitz([1,2,3,4], [1,4,3])
# array([[1, 4, 3],
#        [2, 1, 4],
#        [3, 2, 1],
#        [4, 3, 2]])
a.reshape(3, 4)
# array([[1, 4, 3, 2],
#        [1, 4, 3, 2],
#        [1, 4, 3, 2]])

然后可以使用它（请注意，我已经交换了符号并将下三角形设置为零）

smallv = -10000  # replace this with np.nan if you have floats

a = np.array([8, 18, 5,15,12])
b = a[:, None] - a

b[np.tril_indices(len(b), -1)] = smallv
d = np.vstack((b, np.full(len(b), smallv)))

d.reshape(len(d) - 1, -1).max(0)[:-1]
# array([ 0, 13,  3,  6, -4])