Broadcasting · NumPy

Broadcasting is how NumPy combines arrays of different shapes. You've used it already: a * 2 multiplies every element by 2, because the single number is stretched to match the array.

A row across every row

Python

import numpy as np

prices = np.array([[10.0, 20.0, 30.0],
                   [15.0, 25.0, 35.0]])
tax = np.array([0.05, 0.12, 0.18])     # one rate per column

print(prices * (1 + tax))

When shapes line up from the right, NumPy reuses the smaller array along the missing axis instead of copying it.

tax has shape (3,) and prices (2, 3). NumPy repeats tax for each row, without actually copying it, so every column gets its own rate.

A column across every column

Python

import numpy as np

prices = np.array([[10.0, 20.0, 30.0],
                   [15.0, 25.0, 35.0]])
discount = np.array([[0.10],            # shape (2, 1): one value per row
                     [0.50]])

print(prices * (1 - discount))

The rules

NumPy compares the shapes from the right. Two dimensions are compatible when they're equal, or when one of them is 1, which gets stretched:

Shape A	Shape B	Result
`(2, 3)`	`(3,)`	`(2, 3)`
`(2, 3)`	`(2, 1)`	`(2, 3)`
`(3, 1)`	`(1, 4)`	`(3, 4)`
`(2, 3)`	`(2,)`	error: 3 and 2 don't match

Python

import numpy as np

a = np.ones((2, 3))
b = np.array([1, 2])          # shape (2,)
print(a + b)                  # ValueError: shapes (2,3) and (2,) can't broadcast

To add b to each row, it needs to be a column, shape (2, 1): a + b[:, np.newaxis] works.

Building a table: an outer product

Python

import numpy as np

n = np.arange(1, 6)
table = n[:, np.newaxis] * n          # (5, 1) times (5,) gives (5, 5)
print(table)

A column times a row gives every combination: a multiplication table with no loops.

Normalising columns

A real use: rescale each column to have mean 0 and standard deviation 1, so that columns measured in different units become comparable:

Python

import numpy as np

# height in cm, weight in kg, age in years
data = np.array([[170.0, 65.0, 21.0],
                 [182.0, 80.0, 35.0],
                 [158.0, 52.0, 28.0],
                 [175.0, 73.0, 42.0]])

means = data.mean(axis=0)        # shape (3,): one mean per column
stds = data.std(axis=0)
scaled = (data - means) / stds   # (4, 3) with (3,): broadcast down the rows

print(np.round(scaled, 2))
print(np.round(scaled.mean(axis=0), 10))    # every column now has mean 0

Distances from a point

Python

import numpy as np

shops = np.array([[2.0, 3.0],
                  [5.0, 1.0],
                  [1.0, 7.0],
                  [6.0, 6.0]])
home = np.array([3.0, 4.0])

distances = np.sqrt(((shops - home) ** 2).sum(axis=1))
print(np.round(distances, 2))
print("nearest shop:", distances.argmin())

shops - home subtracts the same point from every row in one step.