Sliding window of M-by-N shape numpy.ndarray
Question:
I have a Numpy array of shape (6,2):
[[ 0, 1],
[10,11],
[20,21],
[30,31],
[40,41],
[50,51]]
I need a sliding window with step size 1 and window size 3 like this:
[[ 0, 1,10,11,20,21],
[10,11,20,21,30,31],
[20,21,30,31,40,41],
[30,31,40,41,50,51]]
I’m looking for a Numpy solution. If your solution could parametrise the shape of the original array as well as the window size and step size, that’d be great.
I found this related answer Using strides for an efficient moving average filter but I don’t see how to specify the stepsize there and how to collapse the window from the 3d to a continuous 2d array. Also this Rolling or sliding window iterator? but that’s in Python and I’m not sure how efficient that is. Also, it supports elements but does not join them together in the end if each element has multiple features.
Answers:
In [1]: import numpy as np
In [2]: a = np.array([[00,01], [10,11], [20,21], [30,31], [40,41], [50,51]])
In [3]: w = np.hstack((a[:-2],a[1:-1],a[2:]))
In [4]: w
Out[4]:
array([[ 0, 1, 10, 11, 20, 21],
[10, 11, 20, 21, 30, 31],
[20, 21, 30, 31, 40, 41],
[30, 31, 40, 41, 50, 51]])
You could write this in as a function as so:
def window_stack(a, stepsize=1, width=3):
n = a.shape[0]
return np.hstack( a[i:1+n+i-width:stepsize] for i in range(0,width) )
This doesn’t really depend on the shape of the original array, as long as a.ndim = 2. Note that I never use either lengths in the interactive version. The second dimension of the shape is irrelevant; each row can be as long as you want. Thanks to @Jaime’s suggestion, you can do it without checking the shape at all:
def window_stack(a, stepsize=1, width=3):
return np.hstack( a[i:1+i-width or None:stepsize] for i in range(0,width) )
You can do a vectorized sliding window in numpy using fancy indexing.
>>> import numpy as np
>>> a = np.array([[00,01], [10,11], [20,21], [30,31], [40,41], [50,51]])
>>> a
array([[ 0, 1],
[10, 11],
[20, 21], #define our 2d numpy array
[30, 31],
[40, 41],
[50, 51]])
>>> a = a.flatten()
>>> a
array([ 0, 1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51]) #flattened numpy array
>>> indexer = np.arange(6)[None, :] + 2*np.arange(4)[:, None]
>>> indexer
array([[ 0, 1, 2, 3, 4, 5],
[ 2, 3, 4, 5, 6, 7], #sliding window indices
[ 4, 5, 6, 7, 8, 9],
[ 6, 7, 8, 9, 10, 11]])
>>> a[indexer]
array([[ 0, 1, 10, 11, 20, 21],
[10, 11, 20, 21, 30, 31], #values of a over sliding window
[20, 21, 30, 31, 40, 41],
[30, 31, 40, 41, 50, 51]])
>>> np.sum(a[indexer], axis=1)
array([ 63, 123, 183, 243]) #sum of values in 'a' under the sliding window.
Explanation for what this code is doing.
The np.arange(6)[None, :] creates a row vector 0 through 6, and np.arange(4)[:, None] creates a column vector 0 through 4. This results in a 4×6 matrix where each row (six of them) represents a window, and the number of rows (four of them) represents the number of windows. The multiple of 2 makes the sliding window slide 2 units at a time which is necessary for sliding over each tuple. Using numpy array slicing you can pass the sliding window into the flattened numpy array and do aggregates on them like sum.
One solution is
np.lib.stride_tricks.as_strided(a, shape=(4,6), strides=(8,4)).
Using strides is intuitive when you start thinking in terms of pointers/addresses.
The as_strided() method has 3 arguments.
- data
- shape
- strides
data is the array on which we would operate.
To use as_strided() for implementing sliding window functions, we must compute the shape of the output beforehand. In the question, (4,6) is the shape of output. If the dimensions are not correct, we end up reading garbage values. This is because we are accessing data by moving the pointer by a couple of bytes (depending on data type).
Determining the correct value of strides is essential to get expected results.
Before calculating strides, find out the memory occupied by each element using arr.strides[-1]. In this example, the memory occupied by one element is 4 bytes.
Numpy arrays are created in row major fashion. The first element of the next row is right next to the last element of the current row.
Ex:
0 , 1 | 10, 11 | ...
10 is right next to 1.
Imagine the 2D array reshaped to 1D (This is acceptable as the data is stored in a row-major format). The first element of each row in the output is the odd indexed element in the 1D array.
0, 10, 20, 30, ..
The number of steps in memory we need to take to move from 0 to 10, 10 to 20, and so on is 2 * mem size of element. Each row has a stride of 2 * 4bytes = 8.
For a given row in the output, all the elements are adjacent to each other in our imaginary 1D array. To get the next element in a row, just take one stride equal to the size of an element. The value of column stride is 4 bytes.
Therefore, strides=(8,4)
An alternate explanation:
The output has a shape of (4,6). Column stride 4. So, the first row elements start at index 0 and have 6 elements each spaced 4 bytes apart.
After the first row is collected, the second row starts 8 bytes away from the starting of the current row. The third row starts 8 bytes away from the starting point of the second row and so on.
The shape determines the number of rows and columns we need. strides define the memory steps to start a row and collect a column element
A short list comprehension is possible with more_itertools.windowed1:
Given
import numpy as np
import more_itertools as mit
a = [["00","01"],
["10","11"],
["20","21"],
["30","31"],
["40","41"],
["50","51"]]
b = np.array(a)
Code
np.array([list(mit.flatten(w)) for w in mit.windowed(a, n=3)])
or
np.array([[i for item in w for i in item] for w in mit.windowed(a, n=3)])
or
np.array(list(mit.windowed(b.ravel(), n=6)))
Output
array([['00', '01', '10', '11', '20', '21'],
['10', '11', '20', '21', '30', '31'],
['20', '21', '30', '31', '40', '41'],
['30', '31', '40', '41', '50', '51']],
dtype='<U2')
Sliding windows of size n=3 are created and flattened. Note the default step size is more_itertools.windowed(..., step=1).
Performance
As an array, the accepted answer is fastest.
%timeit np.hstack((a[:-2], a[1:-1], a[2:]))
# 37.5 µs ± 1.88 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit np.hstack((b[:-2], b[1:-1], b[2:]))
# 12.9 µs ± 166 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
%timeit np.array([list(mit.flatten(w)) for w in mit.windowed(a, n=3)])
# 23.2 µs ± 1.73 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit np.array([[i for item in w for i in item] for w in mit.windowed(a, n=3)])
# 21.2 µs ± 999 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit np.array(list(mit.windowed(b.ravel(), n=6)))
# 43.4 µs ± 374 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
A third-party library that implements itertool recipes and many helpful tools.
This is a pure Python implementation:
def sliding_window(arr, window=3):
i = iter(arr)
a = []
for e in range(0, window): a.append(next(i))
yield a
for e in i:
a = a[1:] + [e]
yield a
An example:
# flatten array
flatten = lambda l: [item for sublist in l for item in sublist]
a = [[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]]
w = sliding_window(a, width=3)
print( list(map(flatten,w)) )
[[0, 1, 10, 11, 20, 21], [10, 11, 20, 21, 30, 31], [20, 21, 30, 31, 40, 41], [30, 31, 40, 41, 50, 51]]
Benchmark
import timeit
def benchmark():
a = [[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]]
sliding_window(a, width=3)
times = timeit.Timer(benchmark).repeat(3, number=1000)
time_taken = min(times) / 1000
print(time_taken)
1.0944640007437556e-06
Here is One-liner using Numpy >= v1.17
rowsJoined = 3
splits = np.vstack(np.split(x,np.array([[i, i + rowsJoined] for i in range(x.shape[0] - (rowsJoined - 1))]).reshape(-1))).reshape(-1, rowsJoined * x.shape[1])
Test
x = np.array([[00,1],
[10,11],
[20,21],
[30,31],
[40,41],
[50,51]])
Result
[[ 0 1 10 11 20 21]
[10 11 20 21 30 31]
[20 21 30 31 40 41]
[30 31 40 41 50 51]]
Test Performance On Large Array
import numpy as np
import time
x = np.array(range(1000)).reshape(-1, 2)
rowsJoined = 3
all_t = 0.
for i in range(1000):
start_ = time.time()
np.vstack(
numpy.split(x,np.array([[i, i + rowsJoined] for i in range(x.shape[0] - (rowsJoined - 1))])
.reshape(-1))).reshape(-1, rowsJoined * x.shape[1])
all_t += time.time() - start_
print('Average Time of 1000 Iterations on Array of Shape '
'1000 x 2 is: {} Seconds.'.format(all_t/1000.))
Performance Result
Average Time of 1000 Iterations on Array of Shape 1000 x 2 is: 0.0016909 Seconds.
Starting in Numpy 1.20, using the new sliding_window_view to slide/roll over windows of elements, and based on the same idea as user42541’s answer, we can do:
import numpy as np
from numpy.lib.stride_tricks import sliding_window_view
# values = np.array([[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]])
sliding_window_view(values.flatten(), window_shape = 2*3)[::2]
# array([[ 0, 1, 10, 11, 20, 21],
# [10, 11, 20, 21, 30, 31],
# [20, 21, 30, 31, 40, 41],
# [30, 31, 40, 41, 50, 51]])
where 2 is the size of sub-arrays and 3 the window.
Details of the intermediate steps:
# values = np.array([[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]])
# Flatten the array (concatenate sub-arrays):
values.flatten()
# array([ 0, 1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51])
# Slide through windows of size 2*3=6:
sliding_window_view(values.flatten(), 2*3)
# array([[ 0, 1, 10, 11, 20, 21],
# [ 1, 10, 11, 20, 21, 30],
# [10, 11, 20, 21, 30, 31],
# [11, 20, 21, 30, 31, 40],
# [20, 21, 30, 31, 40, 41],
# [21, 30, 31, 40, 41, 50],
# [30, 31, 40, 41, 50, 51]])
# Only keep even rows (1 row in 2 - if sub-arrays have a size of x, then replace 2 with x):
sliding_window_view(values.flatten(), 2*3)[::2]
# array([[ 0, 1, 10, 11, 20, 21],
# [10, 11, 20, 21, 30, 31],
# [20, 21, 30, 31, 40, 41],
# [30, 31, 40, 41, 50, 51]])
As of NumPy version 1.20.0 this can be done using
np.lib.stride_tricks.sliding_window_view(arr, winsize)
Example:
>>> arr = np.arange(0, 9).reshape((3, 3))
>>> np.lib.stride_tricks.sliding_window_view(arr, (2, 2))
array([[[[0, 1],
[3, 4]],
[[1, 2],
[4, 5]]],
[[[3, 4],
[6, 7]],
[[4, 5],
[7, 8]]]])
You can read more about it here.
I have a Numpy array of shape (6,2):
[[ 0, 1],
[10,11],
[20,21],
[30,31],
[40,41],
[50,51]]
I need a sliding window with step size 1 and window size 3 like this:
[[ 0, 1,10,11,20,21],
[10,11,20,21,30,31],
[20,21,30,31,40,41],
[30,31,40,41,50,51]]
I’m looking for a Numpy solution. If your solution could parametrise the shape of the original array as well as the window size and step size, that’d be great.
I found this related answer Using strides for an efficient moving average filter but I don’t see how to specify the stepsize there and how to collapse the window from the 3d to a continuous 2d array. Also this Rolling or sliding window iterator? but that’s in Python and I’m not sure how efficient that is. Also, it supports elements but does not join them together in the end if each element has multiple features.
In [1]: import numpy as np
In [2]: a = np.array([[00,01], [10,11], [20,21], [30,31], [40,41], [50,51]])
In [3]: w = np.hstack((a[:-2],a[1:-1],a[2:]))
In [4]: w
Out[4]:
array([[ 0, 1, 10, 11, 20, 21],
[10, 11, 20, 21, 30, 31],
[20, 21, 30, 31, 40, 41],
[30, 31, 40, 41, 50, 51]])
You could write this in as a function as so:
def window_stack(a, stepsize=1, width=3):
n = a.shape[0]
return np.hstack( a[i:1+n+i-width:stepsize] for i in range(0,width) )
This doesn’t really depend on the shape of the original array, as long as a.ndim = 2. Note that I never use either lengths in the interactive version. The second dimension of the shape is irrelevant; each row can be as long as you want. Thanks to @Jaime’s suggestion, you can do it without checking the shape at all:
def window_stack(a, stepsize=1, width=3):
return np.hstack( a[i:1+i-width or None:stepsize] for i in range(0,width) )
You can do a vectorized sliding window in numpy using fancy indexing.
>>> import numpy as np
>>> a = np.array([[00,01], [10,11], [20,21], [30,31], [40,41], [50,51]])
>>> a
array([[ 0, 1],
[10, 11],
[20, 21], #define our 2d numpy array
[30, 31],
[40, 41],
[50, 51]])
>>> a = a.flatten()
>>> a
array([ 0, 1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51]) #flattened numpy array
>>> indexer = np.arange(6)[None, :] + 2*np.arange(4)[:, None]
>>> indexer
array([[ 0, 1, 2, 3, 4, 5],
[ 2, 3, 4, 5, 6, 7], #sliding window indices
[ 4, 5, 6, 7, 8, 9],
[ 6, 7, 8, 9, 10, 11]])
>>> a[indexer]
array([[ 0, 1, 10, 11, 20, 21],
[10, 11, 20, 21, 30, 31], #values of a over sliding window
[20, 21, 30, 31, 40, 41],
[30, 31, 40, 41, 50, 51]])
>>> np.sum(a[indexer], axis=1)
array([ 63, 123, 183, 243]) #sum of values in 'a' under the sliding window.
Explanation for what this code is doing.
The np.arange(6)[None, :] creates a row vector 0 through 6, and np.arange(4)[:, None] creates a column vector 0 through 4. This results in a 4×6 matrix where each row (six of them) represents a window, and the number of rows (four of them) represents the number of windows. The multiple of 2 makes the sliding window slide 2 units at a time which is necessary for sliding over each tuple. Using numpy array slicing you can pass the sliding window into the flattened numpy array and do aggregates on them like sum.
One solution is
np.lib.stride_tricks.as_strided(a, shape=(4,6), strides=(8,4)).
Using strides is intuitive when you start thinking in terms of pointers/addresses.
The as_strided() method has 3 arguments.
- data
- shape
- strides
data is the array on which we would operate.
To use as_strided() for implementing sliding window functions, we must compute the shape of the output beforehand. In the question, (4,6) is the shape of output. If the dimensions are not correct, we end up reading garbage values. This is because we are accessing data by moving the pointer by a couple of bytes (depending on data type).
Determining the correct value of strides is essential to get expected results.
Before calculating strides, find out the memory occupied by each element using arr.strides[-1]. In this example, the memory occupied by one element is 4 bytes.
Numpy arrays are created in row major fashion. The first element of the next row is right next to the last element of the current row.
Ex:
0 , 1 | 10, 11 | ...
10 is right next to 1.
Imagine the 2D array reshaped to 1D (This is acceptable as the data is stored in a row-major format). The first element of each row in the output is the odd indexed element in the 1D array.
0, 10, 20, 30, ..
The number of steps in memory we need to take to move from 0 to 10, 10 to 20, and so on is 2 * mem size of element. Each row has a stride of 2 * 4bytes = 8.
For a given row in the output, all the elements are adjacent to each other in our imaginary 1D array. To get the next element in a row, just take one stride equal to the size of an element. The value of column stride is 4 bytes.
Therefore, strides=(8,4)
An alternate explanation:
The output has a shape of (4,6). Column stride 4. So, the first row elements start at index 0 and have 6 elements each spaced 4 bytes apart.
After the first row is collected, the second row starts 8 bytes away from the starting of the current row. The third row starts 8 bytes away from the starting point of the second row and so on.
The shape determines the number of rows and columns we need. strides define the memory steps to start a row and collect a column element
A short list comprehension is possible with more_itertools.windowed1:
Given
import numpy as np
import more_itertools as mit
a = [["00","01"],
["10","11"],
["20","21"],
["30","31"],
["40","41"],
["50","51"]]
b = np.array(a)
Code
np.array([list(mit.flatten(w)) for w in mit.windowed(a, n=3)])
or
np.array([[i for item in w for i in item] for w in mit.windowed(a, n=3)])
or
np.array(list(mit.windowed(b.ravel(), n=6)))
Output
array([['00', '01', '10', '11', '20', '21'],
['10', '11', '20', '21', '30', '31'],
['20', '21', '30', '31', '40', '41'],
['30', '31', '40', '41', '50', '51']],
dtype='<U2')
Sliding windows of size n=3 are created and flattened. Note the default step size is more_itertools.windowed(..., step=1).
Performance
As an array, the accepted answer is fastest.
%timeit np.hstack((a[:-2], a[1:-1], a[2:]))
# 37.5 µs ± 1.88 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit np.hstack((b[:-2], b[1:-1], b[2:]))
# 12.9 µs ± 166 ns per loop (mean ± std. dev. of 7 runs, 100000 loops each)
%timeit np.array([list(mit.flatten(w)) for w in mit.windowed(a, n=3)])
# 23.2 µs ± 1.73 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit np.array([[i for item in w for i in item] for w in mit.windowed(a, n=3)])
# 21.2 µs ± 999 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
%timeit np.array(list(mit.windowed(b.ravel(), n=6)))
# 43.4 µs ± 374 ns per loop (mean ± std. dev. of 7 runs, 10000 loops each)
A third-party library that implements itertool recipes and many helpful tools.
This is a pure Python implementation:
def sliding_window(arr, window=3):
i = iter(arr)
a = []
for e in range(0, window): a.append(next(i))
yield a
for e in i:
a = a[1:] + [e]
yield a
An example:
# flatten array
flatten = lambda l: [item for sublist in l for item in sublist]
a = [[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]]
w = sliding_window(a, width=3)
print( list(map(flatten,w)) )
[[0, 1, 10, 11, 20, 21], [10, 11, 20, 21, 30, 31], [20, 21, 30, 31, 40, 41], [30, 31, 40, 41, 50, 51]]
Benchmark
import timeit
def benchmark():
a = [[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]]
sliding_window(a, width=3)
times = timeit.Timer(benchmark).repeat(3, number=1000)
time_taken = min(times) / 1000
print(time_taken)
1.0944640007437556e-06
Here is One-liner using Numpy >= v1.17
rowsJoined = 3
splits = np.vstack(np.split(x,np.array([[i, i + rowsJoined] for i in range(x.shape[0] - (rowsJoined - 1))]).reshape(-1))).reshape(-1, rowsJoined * x.shape[1])
Test
x = np.array([[00,1],
[10,11],
[20,21],
[30,31],
[40,41],
[50,51]])
Result
[[ 0 1 10 11 20 21]
[10 11 20 21 30 31]
[20 21 30 31 40 41]
[30 31 40 41 50 51]]
Test Performance On Large Array
import numpy as np
import time
x = np.array(range(1000)).reshape(-1, 2)
rowsJoined = 3
all_t = 0.
for i in range(1000):
start_ = time.time()
np.vstack(
numpy.split(x,np.array([[i, i + rowsJoined] for i in range(x.shape[0] - (rowsJoined - 1))])
.reshape(-1))).reshape(-1, rowsJoined * x.shape[1])
all_t += time.time() - start_
print('Average Time of 1000 Iterations on Array of Shape '
'1000 x 2 is: {} Seconds.'.format(all_t/1000.))
Performance Result
Average Time of 1000 Iterations on Array of Shape 1000 x 2 is: 0.0016909 Seconds.
Starting in Numpy 1.20, using the new sliding_window_view to slide/roll over windows of elements, and based on the same idea as user42541’s answer, we can do:
import numpy as np
from numpy.lib.stride_tricks import sliding_window_view
# values = np.array([[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]])
sliding_window_view(values.flatten(), window_shape = 2*3)[::2]
# array([[ 0, 1, 10, 11, 20, 21],
# [10, 11, 20, 21, 30, 31],
# [20, 21, 30, 31, 40, 41],
# [30, 31, 40, 41, 50, 51]])
where 2 is the size of sub-arrays and 3 the window.
Details of the intermediate steps:
# values = np.array([[0,1], [10,11], [20,21], [30,31], [40,41], [50,51]])
# Flatten the array (concatenate sub-arrays):
values.flatten()
# array([ 0, 1, 10, 11, 20, 21, 30, 31, 40, 41, 50, 51])
# Slide through windows of size 2*3=6:
sliding_window_view(values.flatten(), 2*3)
# array([[ 0, 1, 10, 11, 20, 21],
# [ 1, 10, 11, 20, 21, 30],
# [10, 11, 20, 21, 30, 31],
# [11, 20, 21, 30, 31, 40],
# [20, 21, 30, 31, 40, 41],
# [21, 30, 31, 40, 41, 50],
# [30, 31, 40, 41, 50, 51]])
# Only keep even rows (1 row in 2 - if sub-arrays have a size of x, then replace 2 with x):
sliding_window_view(values.flatten(), 2*3)[::2]
# array([[ 0, 1, 10, 11, 20, 21],
# [10, 11, 20, 21, 30, 31],
# [20, 21, 30, 31, 40, 41],
# [30, 31, 40, 41, 50, 51]])
As of NumPy version 1.20.0 this can be done using
np.lib.stride_tricks.sliding_window_view(arr, winsize)
Example:
>>> arr = np.arange(0, 9).reshape((3, 3))
>>> np.lib.stride_tricks.sliding_window_view(arr, (2, 2))
array([[[[0, 1],
[3, 4]],
[[1, 2],
[4, 5]]],
[[[3, 4],
[6, 7]],
[[4, 5],
[7, 8]]]])
You can read more about it here.