Numerical

Function in the numerical sub-module can be called by importing the sub-module first.

min, argmin, max, argmax

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.min.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.max.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argmax.html

WARNING: Difference to numpy: the out keyword argument is not implemented.

These functions follow the same pattern, and work with generic iterables, and ndarrays. min, and max return the minimum or maximum of a sequence. If the input array is two-dimensional, the axis keyword argument can be supplied, in which case the minimum/maximum along the given axis will be returned. If axis=None (this is also the default value), the minimum/maximum of the flattened array will be determined.

argmin/argmax return the position (index) of the minimum/maximum in the sequence.

# code to be run in micropython

import ulab as np

a = np.array([1, 2, 3])
print(a)
print(a[-1:-1:-3])
try:
    sa = list(a[-1:-1:-3])
    la = len(sa)
except IndexError as e:
    sa = str(e)
    la = -1

print(sa, la)

a[-1:-1:-3] = np.ones(0)
print(a)

b = np.ones(0) + 1
print(b)
# print('b', b.shape())

array([1.0, 2.0, 3.0], dtype=float)
array([], dtype=float)
[] 0
array([1.0, 2.0, 3.0], dtype=float)
array([], dtype=float)

# code to be run in micropython

import ulab as np
a = np.array([1, 2, 3])
print(a[0:1:-3])

0, 1, -3array([], dtype=float)

# code to be run in CPython

l = list(range(13))

l[0:10:113]

[0]

# code to be run in CPython

a = np.array([1, 2, 3])
np.ones(0, dtype=uint8) / np.zeros(0, dtype=uint16)
np.ones(0).shape

(0,)

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([1, 2, 0, 1, 10])
print('a:', a)
print('min of a:', numerical.min(a))
print('argmin of a:', numerical.argmin(a))

b = np.array([[1, 2, 0], [1, 10, -1]])
print('\nb:\n', b)
print('min of b (flattened):', numerical.min(b))
print('min of b (axis=0):', numerical.min(b, axis=0))
print('min of b (axis=1):', numerical.min(b, axis=1))

a: array([1.0, 2.0, 0.0, 1.0, 10.0], dtype=float)
min of a: 0.0
argmin of a: 2

b:
 array([[1.0, 2.0, 0.0],
     [1.0, 10.0, -1.0]], dtype=float)
min of b (flattened): -1.0
min of b (axis=0): array([1.0, 2.0, -1.0], dtype=float)
min of b (axis=1): array([0.0, -1.0], dtype=float)

sum, std, mean

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sum.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.std.html

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.mean.html

These three functions follow the same pattern: if the axis keyword is not specified, it assumes the default value of None, and returns the result of the computation for the flattened array. Otherwise, the calculation is along the given axis.

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print('a: \n', a)

print('sum, flat array: ', numerical.sum(a))

print('mean, horizontal: ', numerical.mean(a, axis=1))

print('std, vertical: ', numerical.std(a, axis=0))

a:
 array([[1.0, 2.0, 3.0],
     [4.0, 5.0, 6.0],
     [7.0, 8.0, 9.0]], dtype=float)
sum, flat array:  45.0
mean, horizontal:  array([2.0, 5.0, 8.0], dtype=float)
std, vertical:  array([2.44949, 2.44949, 2.44949], dtype=float)

roll

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.roll.html

The roll function shifts the content of a vector by the positions given as the second argument. If the axis keyword is supplied, the shift is applied to the given axis.

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([1, 2, 3, 4, 5, 6, 7, 8])
print("a:\t\t\t", a)

numerical.roll(a, 2)
print("a rolled to the left:\t", a)

# this should be the original vector
numerical.roll(a, -2)
print("a rolled to the right:\t", a)

a:                   array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)
a rolled to the left:        array([3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 1.0, 2.0], dtype=float)
a rolled to the right:       array([1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0], dtype=float)

Rolling works with matrices, too. If the axis keyword is 0, the matrix is rolled along its vertical axis, otherwise, horizontally.

Horizontal rolls are faster, because they require fewer steps, and larger memory chunks are copied, however, they also require more RAM: basically the whole row must be stored internally. Most expensive are the None keyword values, because with axis = None, the array is flattened first, hence the row’s length is the size of the whole matrix.

Vertical rolls require two internal copies of single columns.

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])
print("a:\n", a)

numerical.roll(a, 2)
print("\na rolled to the left:\n", a)

numerical.roll(a, -1, axis=1)
print("\na rolled up:\n", a)

numerical.roll(a, 1, axis=None)
print("\na rolled with None:\n", a)

a:
 array([[1.0, 2.0, 3.0, 4.0],
     [5.0, 6.0, 7.0, 8.0]], dtype=float)

a rolled to the left:
 array([[3.0, 4.0, 5.0, 6.0],
     [7.0, 8.0, 1.0, 2.0]], dtype=float)

a rolled up:
 array([[6.0, 3.0, 4.0, 5.0],
     [2.0, 7.0, 8.0, 1.0]], dtype=float)

a rolled with None:
 array([[3.0, 4.0, 5.0, 2.0],
     [7.0, 8.0, 1.0, 6.0]], dtype=float)

Simple running weighted average

As a demonstration of the conciseness of ulab/numpy operations, we will calculate an exponentially weighted running average of a measurement vector in just a couple of lines. I chose this particular example, because I think that this can indeed be used in real-life applications.

# code to be run in micropython

import ulab as np
from ulab import numerical
from ulab import vector

def dummy_adc():
    # dummy adc function, so that the results are reproducible
    return 2

n = 10
# These are the normalised weights; the last entry is the most dominant
weight = vector.exp([1, 2, 3, 4, 5])
weight = weight/numerical.sum(weight)

print(weight)
# initial array of samples
samples = np.array([0]*n)

for i in range(n):
    # a new datum is inserted on the right hand side. This simply overwrites whatever was in the last slot
    samples[-1] = dummy_adc()
    print(numerical.mean(samples[-5:]*weight))
    print(samples[-5:])
    # the data are shifted by one position to the left
    numerical.roll(samples, 1)

array([0.01165623031556606, 0.03168492019176483, 0.08612854033708572, 0.234121635556221, 0.6364086270332336], dtype=float)
0.2545634508132935
array([0.0, 0.0, 0.0, 0.0, 2.0], dtype=float)
0.3482121050357819
array([0.0, 0.0, 0.0, 2.0, 2.0], dtype=float)
0.3826635211706161
array([0.0, 0.0, 2.0, 2.0, 2.0], dtype=float)
0.3953374892473221
array([0.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)
0.3999999813735485
array([2.0, 2.0, 2.0, 2.0, 2.0], dtype=float)

flip

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.flip.html

The flip function takes one positional, an ndarray, and one keyword argument, axis = None, and reverses the order of elements along the given axis. If the keyword argument is None, the matrix’ entries are flipped along all axes. flip returns a new copy of the array.

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([1, 2, 3, 4, 5])
print("a: \t", a)
print("a flipped:\t", np.flip(a))

a = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]], dtype=np.uint8)
print("\na flipped horizontally\n", numerical.flip(a, axis=1))
print("\na flipped vertically\n", numerical.flip(a, axis=0))
print("\na flipped horizontally+vertically\n", numerical.flip(a))

a:   array([1.0, 2.0, 3.0, 4.0, 5.0], dtype=float)
a flipped:   array([5.0, 4.0, 3.0, 2.0, 1.0], dtype=float)

a flipped horizontally
 array([[3, 2, 1],
     [6, 5, 4],
     [9, 8, 7]], dtype=uint8)

a flipped vertically
 array([[7, 8, 9],
     [4, 5, 6],
     [1, 2, 3]], dtype=uint8)

a flipped horizontally+vertically
 array([[9, 8, 7],
     [6, 5, 4],
     [3, 2, 1]], dtype=uint8)

diff

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.diff.html

The diff function returns the numerical derivative of the forward scheme, or more accurately, the differences of an ndarray along a given axis. The order of derivative can be stipulated with the n keyword argument, which should be between 0, and 9. Default is 1. If higher order derivatives are required, they can be gotten by repeated calls to the function. The axis keyword argument should be -1 (last axis, in ulab equivalent to the second axis, and this also happens to be the default value), 0, or 1.

Beyond the output array, the function requires only a couple of bytes of extra RAM for the differentiation stencil. (The stencil is an int8 array, one byte longer than n. This also explains, why the highest order is 9: the coefficients of a ninth-order stencil all fit in signed bytes, while 10 would require int16.) Note that as usual in numerical differentiation (and also in numpy), the length of the respective axis will be reduced by n after the operation. If n is larger than, or equal to the length of the axis, an empty array will be returned.

WARNING: the diff function does not implement the prepend and append keywords that can be found in numpy.

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array(range(9), dtype=np.uint8)
print('a:\n', a)

print('\nfirst derivative:\n', numerical.diff(a, n=1))
print('\nsecond derivative:\n', numerical.diff(a, n=2))

c = np.array([[1, 2, 3, 4], [4, 3, 2, 1], [1, 4, 9, 16], [0, 0, 0, 0]])
print('\nc:\n', c)
print('\nfirst derivative, first axis:\n', numerical.diff(c, axis=0))
print('\nfirst derivative, second axis:\n', numerical.diff(c, axis=1))

a:
 array([0, 1, 2, 3, 4, 5, 6, 7, 8], dtype=uint8)

first derivative:
 array([1, 1, 1, 1, 1, 1, 1, 1], dtype=uint8)

second derivative:
 array([0, 0, 0, 0, 0, 0, 0], dtype=uint8)

c:
 array([[1.0, 2.0, 3.0, 4.0],
     [4.0, 3.0, 2.0, 1.0],
     [1.0, 4.0, 9.0, 16.0],
     [0.0, 0.0, 0.0, 0.0]], dtype=float)

first derivative, first axis:
 array([[3.0, 1.0, -1.0, -3.0],
     [-3.0, 1.0, 7.0, 15.0],
     [-1.0, -4.0, -9.0, -16.0]], dtype=float)

first derivative, second axis:
 array([[1.0, 1.0, 1.0],
     [-1.0, -1.0, -1.0],
     [3.0, 5.0, 7.0],
     [0.0, 0.0, 0.0]], dtype=float)

median

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.median.html

The function computes the median along the specified axis, and returns the median of the array elements. If the axis keyword argument is None, the arrays is flattened first. The dtype of the results is always float.

# code to be run in micropython

import ulab as np

a = np.array(range(12), dtype=np.int8).reshape((3, 4))
print('a:\n', a)
print('\nmedian of the flattened array: ', np.median(a))
print('\nmedian along the vertical axis: ', np.median(a, axis=0))
print('\nmedian along the horizontal axis: ', np.median(a, axis=1))

a:
 array([[0, 1, 2, 3],
       [4, 5, 6, 7],
       [8, 9, 10, 11]], dtype=int8)

median of the flattened array:  5.5

median along the vertical axis:  array([4.0, 5.0, 6.0, 7.0], dtype=float)

median along the horizontal axis:  array([1.5, 5.5, 9.5], dtype=float)

sort

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.sort.html

The sort function takes an ndarray, and sorts its elements in ascending order along the specified axis using a heap sort algorithm. As opposed to the .sort() method discussed earlier, this function creates a copy of its input before sorting, and at the end, returns this copy. Sorting takes place in place, without auxiliary storage. The axis keyword argument takes on the possible values of -1 (the last axis, in ulab equivalent to the second axis, and this also happens to be the default value), 0, 1, or None. The first three cases are identical to those in diff, while the last one flattens the array before sorting.

If descending order is required, the result can simply be flipped, see flip.

WARNING: numpy defines the kind, and order keyword arguments that are not implemented here. The function in ulab always uses heap sort, and since ulab does not have the concept of data fields, the order keyword argument would have no meaning.

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = numerical.sort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

c = numerical.sort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)

c = numerical.sort(a, axis=None)
print('\nflattened a sorted:\n', c)

a:
 array([[1.0, 12.0, 3.0, 0.0],
     [5.0, 3.0, 4.0, 1.0],
     [9.0, 11.0, 1.0, 8.0],
     [7.0, 10.0, 0.0, 1.0]], dtype=float)

a sorted along vertical axis:
 array([[1.0, 3.0, 0.0, 0.0],
     [5.0, 10.0, 1.0, 1.0],
     [7.0, 11.0, 3.0, 1.0],
     [9.0, 12.0, 4.0, 8.0]], dtype=float)

a sorted along horizontal axis:
 array([[0.0, 1.0, 3.0, 12.0],
     [1.0, 3.0, 4.0, 5.0],
     [1.0, 8.0, 9.0, 11.0],
     [0.0, 1.0, 7.0, 10.0]], dtype=float)

flattened a sorted:
 array([0.0, 0.0, 1.0, ..., 10.0, 11.0, 12.0], dtype=float)

Heap sort requires \(\sim N\log N\) operations, and notably, the worst case costs only 20% more time than the average. In order to get an order-of-magnitude estimate, we will take the sine of 1000 uniformly spaced numbers between 0, and two pi, and sort them:

# code to be run in micropython

import ulab as np
from ulab import vector
from ulab import numerical

@timeit
def sort_time(array):
    return numerical.sort(array)

b = vector.sin(np.linspace(0, 6.28, num=1000))
print('b: ', b)
sort_time(b)
print('\nb sorted:\n', b)

argsort

numpy: https://docs.scipy.org/doc/numpy/reference/generated/numpy.argsort.html

Similarly to sort, argsort takes a positional, and a keyword argument, and returns an unsigned short index array of type ndarray with the same dimensions as the input, or, if axis=None, as a row vector with length equal to the number of elements in the input (i.e., the flattened array). The indices in the output sort the input in ascending order. The routine in argsort is the same as in sort, therefore, the comments on computational expenses (time and RAM) also apply. In particular, since no copy of the original data is required, virtually no RAM beyond the output array is used.

Since the underlying container of the output array is of type uint16_t, neither of the output dimensions should be larger than 65535. If that happens to be the case, the function will bail out with a ValueError.

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([[1, 12, 3, 0], [5, 3, 4, 1], [9, 11, 1, 8], [7, 10, 0, 1]], dtype=np.float)
print('\na:\n', a)
b = numerical.argsort(a, axis=0)
print('\na sorted along vertical axis:\n', b)

c = numerical.argsort(a, axis=1)
print('\na sorted along horizontal axis:\n', c)

c = numerical.argsort(a, axis=None)
print('\nflattened a sorted:\n', c)

a:
 array([[1.0, 12.0, 3.0, 0.0],
     [5.0, 3.0, 4.0, 1.0],
     [9.0, 11.0, 1.0, 8.0],
     [7.0, 10.0, 0.0, 1.0]], dtype=float)

a sorted along vertical axis:
 array([[0, 1, 3, 0],
     [1, 3, 2, 1],
     [3, 2, 0, 3],
     [2, 0, 1, 2]], dtype=uint16)

a sorted along horizontal axis:
 array([[3, 0, 2, 1],
     [3, 1, 2, 0],
     [2, 3, 0, 1],
     [2, 3, 0, 1]], dtype=uint16)

flattened a sorted:
 array([3, 14, 0, ..., 13, 9, 1], dtype=uint16)

Since during the sorting, only the indices are shuffled, argsort does not modify the input array, as one can verify this by the following example:

# code to be run in micropython

import ulab as np
from ulab import numerical

a = np.array([0, 5, 1, 3, 2, 4], dtype=np.uint8)
print('\na:\n', a)
b = numerical.argsort(a, axis=1)
print('\nsorting indices:\n', b)
print('\nthe original array:\n', a)

a:
 array([0, 5, 1, 3, 2, 4], dtype=uint8)

sorting indices:
 array([0, 2, 4, 3, 5, 1], dtype=uint16)

the original array:
 array([0, 5, 1, 3, 2, 4], dtype=uint8)