Broadcasting in python

• Broadcasting in python
  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • General Principle

Example 1

Carolies from Carb, Proteins, Fats in 100g of different foods:

                  Apples    Beef      Eggs    Potatoes

$\begin{matrix}\text{Carb}\\\text{Protein}\\\text{Fat}\end{matrix} \begin{bmatrix} 56.0 & 0.0 & 4.4 & 68.0 \\ 1.2 & 104.0 & 52.0 & 8.0 \\ 1.8 & 135.0 & 99.0 &0.9 \end{bmatrix}$

• Calculate % of calories for carb, protein, and fat for each product without explicit for-loop?

$\begin{matrix}\text{Carb}\\\text{Protein}\\\text{Fat}\end{matrix} \begin{bmatrix} 56.0 & 0.0 & 4.4 & 68.0 \\ 1.2 & 104.0 & 52.0 & 8.0 \\ 1.8 & 135.0 & 99.0 &0.9 \end{bmatrix}=A$ where A is $\mathbb{R}^{3\times 4}$

> import numpy as np
> A=np.array([[56.0,0.0,4.4,68.0],
>           [1.2,104.0,52.0,8.0],
>           [1.8,135.0,99.0,0.9]])
> print(A)

Prints out

[[ 56.    0.    4.4  68. ]
 [  1.2 104.   52.    8. ]
 [  1.8 135.   99.    0.9]]

Calculate sum for each column

> cal=A.sum(axis=0) #axis=0 vertically
> print(cal)

Prints out

[ 59.  239.  155.4  76.9]

Then calculate percentages

> percentage=100*A/cal.reshape(1,4)
> print(percentage)

Prints out

[[94.91525424  0.          2.83140283 88.42652796]
 [ 2.03389831 43.51464435 33.46203346 10.40312094]
 [ 3.05084746 56.48535565 63.70656371  1.17035111]]

cal.reshape(1,4) is an example of python broadcasting, where where you take a matrix A. So this is a (3,4) matrix and you divide it by a (1,4) matrix. And technically, after this first line of codes cal, the variable cal, is already a (1,4) matrix. So technically you don’t need to call reshape here again

Example 2

$\begin{bmatrix}1\\2\\3\\4\end{bmatrix} + 100$ Python will auto expand 100 to a (1,4) matrix of 100.

$\begin{bmatrix}1\\2\\3\\4\end{bmatrix} + \begin{bmatrix}100\\100\\100\\100\end{bmatrix} = \begin{bmatrix}101\\102\\103\\104\end{bmatrix}$

Example 3

$\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix} + \begin{bmatrix}100&200&300\end{bmatrix}$

$(m,n)$                  Python will make $(1,n) \to (m,n)$

$\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix} + \begin{bmatrix}100&200&300\\ 100&200&300 \end{bmatrix}=\begin{bmatrix}101&202&303\\ 104&205&306 \end{bmatrix}$

Example 4

$\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix} + \begin{bmatrix}100\\200\end{bmatrix}$

     $(m,n)$           Python will make $(m,1) \to (m,n)$

$\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix} + \begin{bmatrix}100&100&100\\200&200&200\end{bmatrix}=\begin{bmatrix}101&202&303\\ 104&205&306 \end{bmatrix}$

General Principle

$(m,n)$     $+-*/$      $(1,n)$ $\to (m,n)$

$(m,n)$     $+-*/$      $(m,1)$ $\to (m,n)$

$(m,1)$     $+-*/$      $\mathbb{R}$ $\to (m,1)$

$\begin{bmatrix}1\\2\\3\end{bmatrix}+100=\begin{bmatrix}101\\102\\103\end{bmatrix}$