Python’s Numpy module

From Python for Data Analysis, 2nd Ed, chapter 4:

• enables working with $n$-dimensional arrays
• math functions without needing to loop over arrays
• reading/writing to files
• advanced math: linear algebra, random numbers, etc

Numpy overview

• Creating and manipulating ndarray objects and doing math on them
• Data types for efficient storage and use
• Indexing and slicing; with boolean expressions and fancy indexing
• Unary and binary math functions

Linear algebra basics:

Vectors and matrices

Basics: Vectors

Vectors: $\vec{x} = [ 1, 2, 3 ]$

• Why? Most data comes in vectors

Can do bulk operations using math magic:

• Adding or subtracting a scalar: $$ \vec{x} + 1 = [ 2, 3, 4 ] $$
• Multiplying or dividing by a scalar: $$ \vec{x} \times 2 = [ 2, 4, 6 ] $$
• Adding two vectors (of same size): $$ \vec{x} + \vec{x} = [ 2, 4, 6 ] $$

Vector math

inner/dot product : $$ \vec{x} \cdot \vec{y} = \sum x_i y_i $$

• Calculates “length of projection”
• Multiply corresponding elements and sum to result in scalar
• Useful in calculating weighted sums, scaling data elements, etc.

outer product : $$ \vec{x} \times \vec{y} = [x_i y_j]_{ij} $$

• element-by-element multiplication, results in $ n \times m $ size matrix
• useful when duplicating rows or columns, or scaling them

Basics: Matrices

$$ A=\left[ \begin{array}{ccc} a_{11} & \cdots & a_{1n} \newline \vdots & \ddots & \vdots \newline a_{m1} & \cdots & a_{mn} \newline \end{array} \right] $$

Uses:

• Aggregation of many vectors
• Bases for transformation spaces
• Image data and manipulation

Matrix multiplication

Must have matching inner dimensions, results in a matrix: $$ A_{m\times n} \times B_{n\times o} = C_{m\times o} $$

Each element of output matrix is the result of one inner product: $$ c_{ij} = \sum_k a_{ik} b_{kj} $$

Rows of $A$ matched to columns of $B$ to create single elements of $C$: $$ \left[ \begin{array}{c} \bbox[5px,yellow,border:2px solid red]{\begin{array}{ccc} a_{11} & \cdots & a_{1n} \end{array}} \newline \begin{array}{ccc} \vdots & \ddots & \vdots \newline a_{m1} & \cdots & a_{mn} \newline \end{array} \end{array} \right] \times \left[ \begin{array}{cc} \bbox[5px,yellow,border:2px solid red]{\begin{array}{c} b_{11} \newline \vdots \newline b_{n1} \end{array}} & \begin{array}{cc} \cdots & b_{1o} \newline \ddots & \vdots \newline \cdots & b_{no} \newline \end{array} \end{array} \right] = \left[ \begin{array}{ccc} \bbox[5px,yellow,border:2px solid red]{c_{11}} & \cdots & c_{1o} \newline \vdots & \ddots & \vdots \newline c_{m1} & \cdots & c_{mo} \newline \end{array} \right] $$

Useful transforming rows of data, image operations, 3D rotations, machine learning, etc.

NumPy: conventional looping versus vector operations

Loop through your data and calculate mean and standard deviation (or regression, min, max, etc.).

vector = [1,2,3]
sum = 0
for element in vector:
    sum += element
mean = sum / len(vector)

Use vector operations to do it shorter and more efficiently. $$ \mu = \sum_{i=1..N} x_i / N $$

import numpy as np
vector = np.array([1,2,3])
mean = np.sum(vector) / len(vector)

Numpy exercise

Calculate standard deviation $$ \sigma = \sqrt{ \sum_{i=1..N} ( x_i - \mu )^2 / ( N - 1 ) } $$ where $N$ is the number of elements in $ \vec{x} $ and $ \mu $ is its mean.

Submit here

Make a Colab notebook with your solution

• Work in groups as before
• Log into Google Colab with any Google account
• Create a new Python notebook
• Write some text and code blocks to explain your standard deviation code
• Compare your result to the output of np.std(vector, ddof=1) in your notebook
• Share your link