Numpy Introduction

Module 3: NumPy

NumPy: Introduction

What is NumPy?

NumPy (Numerical Python) is one of the most fundamental packages for numerical computations in Python. It provides support for large multi-dimensional arrays and matrices, along with a collection of mathematical functions to operate on these arrays.

Why use NumPy?

Efficient storage and manipulation of numerical arrays is fundamental to many forms of scientific and data analysis.
It’s much faster than traditional Python lists for numerical operations.
Behind the scenes optimizations written in C.

# First, let's import NumPy
import numpy as np

Creating Arrays

To create an array we use the function np.array(). We must pass a list or tuple to the function.

# Create a simple array from a list
ls = [1, 2, 3, 4, 5]
arr = np.array(ls)
print(arr)

# Single line
# Create a simple array from a list
arr = np.array([1, 2, 3, 4, 5])
print(arr)

# Tuples work too!
tup = (1, 2, 3, 4, 5)
arr = np.array(tup)
print(arr)

# Single line
arr = np.array((1, 2, 3, 4, 5))
print(arr)

What happens when we initialize a numpy array with a set?

The np.array documentation says that the object argument must be “an array, any object exposing the array interface, an object whose __array__ method returns an array, or any (nested) sequence”.

A set is not a sequence. Specifically, sets are unordered and do not support the __getitem__ method. Hence we cannot create an array from a set like we trying to with the list.

# Just do not do this...
arr = np.array({1, 2, 3, 4, 5})

print(arr)

Exercise

What will be the value of arr? Which value type will their elements have?

arr = np.array((1, 2.0, 3))

print(arr)

The elements of the array will all be of type float64. This is because NumPy automatically upcasts the integer values to floats to ensure that all elements in the array have the same data type, which is necessary for efficient computation.

Exercise

What will be the value of arr? Which value type will their elements have?

arr = np.array(1, 2.0, 3)

print(arr)

The np.array function expects a single argument, (typically a list or a tuple). In this case, we are passing three separate arguments instead of a single iterable.

Array Types

NumPy arrays are restricted to a single data type. The hierarchy of data types is as follows: string > float > integer. Other data types are not permitted.

# Array of strings
a = np.array(["a", "b", "c"])

print(a)

# Array of floats
a = np.array([1, 2.2, 3.3])

# Notice how '1' is turned into a float '1.'
print(a)

# Mixed array
a = np.array([1, "two", 3])

# Notice that '1' and '3' are turned into strings!
print(a)

We can specify the NumPy array type in the second argument of np.array().

a = np.array([1, 2, 3], float)
print(a)

a = np.array([1, 2, 3], str)
print(a)

Multi-Dimensional Arrays

Nested list are used to create multi-dimensional arrays.

# Create a 2D array (Matrix)
# We will use nested lists
row_0 = [1, 2, 3]
row_1 = [4, 5, 6]
row_2 = [7, 8, 9]
ls_matrix = [row_0, row_1, row_2]
matrix = np.array(ls_matrix)
print(matrix)

# All nested list must have the same length!
row_0 = [1, 2, 3]
row_1 = [4, 5, 6, 0]
row_2 = [7, 8, 9]
ls_matrix = [row_0, row_1, row_2]
matrix = np.array(ls_matrix)
print(matrix)

# Single line
# Create a 2D array (Matrix)
matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
print(matrix)

# Best format - helps clarity
# Create a 2D array (Matrix)
matrix = np.array([
  [1, 2, 3],
  [4, 5, 6],
  [7, 8, 9]
])
print(matrix)

In NumPy, the order of dimensions is indicated by the nesting of brackets. The outermost bracket (purple) represents the first dimension, indexed as dimension 0. The next level of brackets inside (blue) represents the second dimension, or dimension 1. If you continue adding more nested brackets, each new level represents a higher dimension. For example, the third level of brackets would be dimension 2, and so on. This hierarchical structure helps in understanding the shape and structure of multi-dimensional arrays.

Exercise

How would we create a 3D matrix?

# Try it!

Some Functions to Quickly Create Arrays

# Create an array of zeros
zeros = np.zeros(5)
print(zeros)

# Create a 2x3 matrix of ones
shape = (2, 3)
ones = np.ones(shape)
print(ones)

# Single line
# Create a 2x3 matrix of ones
ones = np.ones((2, 3))
print(ones)

# Create an array of a range of numbers
range_array = np.arange(0, 10, 2)  # start, stop, step
print(range_array)

# Create an array of a range of numbers
range_array = np.linspace(0, 10, 6)  # start, stop, length
print(range_array)

Exercise

Exercise: Create an array containing all odd numbers between 25 and 50.

# Try it!

Exercise

Exercise: Imagine you have a rope that is 12 meters long. You want to cut this rope into 7 equal lengths. Using numpy, create an array that shows at which points (in meters) you should make the cuts to obtain these 7 equal lengths.

# Try it!

Basic Properties of Arrays

Unlike methods, properties don’t require parentheses when accessed. They provide insights into the attributes of the array. Here is a list of some key properties of NumPy arrays.

ndim: Indicates the number of dimensions of the array.

arr = np.zeros((5, 4))

print(arr.ndim)

shape: Returns a tuple representing the dimensions of the array. For a matrix with n rows and m columns, the shape will be (n,m).

# shape returns a tuple
print(arr.shape)

Array Indexing

Indexing 1D-arrays is the same as indexing lists, tuples or strings.

arr = np.array([0, 1, 2, 3, 4, 5, 6])

# Accessing single element (0-indexed)
print(arr[2])

Indexing multi-dimensional arrays is “similar” to indexing nested lists.

matrix = np.array([[0, 1, 2], [3, 4, 5]])

# Accessing element in 2D array (Matrix)
print(matrix[1][2])  # second row, third column

For NumPy arrays, we can include all indices within a single set of parentheses! This is called “indexing with a tuple”.

matrix = np.array([[0, 1, 2], [3, 4, 5]])

# Accessing element in 2D array (Matrix)
print(matrix[1, 2])  # second row, third column

Exercise

What will be the value of x?

matrix = np.array([[0, 1, 2], [3, 4, 5]])

print(matrix[-1, 0])

Exercise

What will be the value of x?

matrix = [[0, 1, 2], [3, 4, 5]]

print(matrix[2, 1])

This raises an error because lists cannot be indexed with tuples.

Exercise

What will be the value of x?

arr_A = np.array([[0, 1, 2], [3, 4, 5]])
arr_B = np.array([[6, 7, 8], [9, 10, 11]])

ls_arrays = [arr_A, arr_B]

x = ls_arrays[1][0, 2]

print(x)

Array Slicing

Slicing NumPy arrays is the same as slicing lists.

arr = np.array([0, 1, 2, 3, 4, 5, 6])

# Accessing a slice of an array
print(arr[1:4])  # Start is inclusive, end is exclusive

matrix = np.array([[0, 1, 2], [3, 4, 5]])

# Accessing rows/columns of a matrix
print(matrix[:, 1])  # All rows, second column

matrix = np.array([[0, 1, 2], [3, 4, 5]])

print(matrix[1, :])  # Second row, all columns

arr = np.array([0, 1, 2, 3, 4, 5, 6])

# Using steps in slicing
print(arr[0:5:2])  # Start at 0, go to 5, step by 2

Exercise

What slicing should we take to retrieve the array [2, 4, 6]?

arr = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8])

Exercise

What slicing should we take to retrieve the array [7, 4, 1]?

arr = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8])

Exercise

What slicing should we take to retrieve the array [2, 4, 6]?

arr = np.array([
  [0, 1, 2],
  [1, 2, 3],
  [2, 3, 4],
  [3, 4, 5],
  [4, 5, 6]
  ])

Exercise

What slicing should we take to retrieve the following matrix?

\[\begin{pmatrix} 1 & 2\\ 3 & 4\\ 5 & 6 \end{pmatrix}\]

arr = np.array([
  [0, 1, 2],
  [1, 2, 3],
  [2, 3, 4],
  [3, 4, 5],
  [4, 5, 6]
  ])

Practice

Deactivate AI assistant tools, and try the following exercises.

Exercise

Exercise: Create an array filled with integers ordered from 1 to 12.

Exercise

Exercise: Reshape it to a $3 \times 4$ matrix.

Exercise

Exercise: Print its value at the first row, fourth column.

--- title: "Numpy Introduction" subtitle: "Module 3: NumPy" format: html --- # NumPy: Introduction ## What is NumPy? NumPy (Numerical Python) is one of the most fundamental packages for numerical computations in Python. It provides support for large multi-dimensional arrays and matrices, along with a collection of mathematical functions to operate on these arrays. <p align="center"> <img src="https://upload.wikimedia.org/wikipedia/commons/thumb/3/31/NumPy_logo_2020.svg/1200px-NumPy_logo_2020.svg.png" width="400" height="auto" /> </p> ### Why use NumPy? - Efficient storage and manipulation of numerical arrays is fundamental to many forms of scientific and data analysis. - It's much faster than traditional Python lists for numerical operations. - Behind the scenes optimizations written in C. ```{python} # First, let's import NumPy import numpy as np ``` ## Creating Arrays To create an array we use the function `np.array()`. We must pass a list or tuple to the function. ```{python} # Create a simple array from a list ls = [1, 2, 3, 4, 5] arr = np.array(ls) print(arr) ``` ```{python} # Single line # Create a simple array from a list arr = np.array([1, 2, 3, 4, 5]) print(arr) ``` ```{python} # Tuples work too! tup = (1, 2, 3, 4, 5) arr = np.array(tup) print(arr) ``` ```{python} # Single line arr = np.array((1, 2, 3, 4, 5)) print(arr) ``` What happens when we initialize a numpy array with a set? The [np.array documentation](https://docs.scipy.org/doc/numpy/reference/generated/numpy.array.html) says that the object argument must be "an array, any object exposing the array interface, an object whose `__array__` method returns an array, **or any (nested) sequence**". A set is not a [sequence](https://docs.python.org/3/glossary.html#term-sequence). Specifically, sets are unordered and do not support the `__getitem__` method. Hence we cannot create an array from a set like we trying to with the list. ```{python} # Just do not do this... arr = np.array({1, 2, 3, 4, 5}) print(arr) ``` ::: {.callout-note} ## Exercise What will be the value of `arr`? Which value type will their elements have? ```{python} arr = np.array((1, 2.0, 3)) print(arr) ``` ::: The elements of the array will all be of type `float64`. This is because NumPy automatically upcasts the integer values to floats to ensure that all elements in the array have the same data type, which is necessary for efficient computation. ::: {.callout-note} ## Exercise What will be the value of `arr`? Which value type will their elements have? ```{python} arr = np.array(1, 2.0, 3) print(arr) ``` ::: The `np.array` function expects a single argument, (typically a list or a tuple). In this case, we are passing three separate arguments instead of a single iterable. --- ### Array Types NumPy arrays are restricted to a single data type. The hierarchy of data types is as follows: string > float > integer. Other data types are not permitted. ```{python} # Array of strings a = np.array(["a", "b", "c"]) print(a) ``` ```{python} # Array of floats a = np.array([1, 2.2, 3.3]) # Notice how '1' is turned into a float '1.' print(a) ``` ```{python} # Mixed array a = np.array([1, "two", 3]) # Notice that '1' and '3' are turned into strings! print(a) ``` We can specify the NumPy array type in the second argument of `np.array()`. ```{python} a = np.array([1, 2, 3], float) print(a) ``` ```{python} a = np.array([1, 2, 3], str) print(a) ``` --- ### Multi-Dimensional Arrays Nested list are used to create multi-dimensional arrays. ```{python} # Create a 2D array (Matrix) # We will use nested lists row_0 = [1, 2, 3] row_1 = [4, 5, 6] row_2 = [7, 8, 9] ls_matrix = [row_0, row_1, row_2] matrix = np.array(ls_matrix) print(matrix) ``` ```{python} # All nested list must have the same length! row_0 = [1, 2, 3] row_1 = [4, 5, 6, 0] row_2 = [7, 8, 9] ls_matrix = [row_0, row_1, row_2] matrix = np.array(ls_matrix) print(matrix) ``` ```{python} # Single line # Create a 2D array (Matrix) matrix = np.array([[1, 2, 3], [4, 5, 6], [7, 8, 9]]) print(matrix) ``` ```{python} # Best format - helps clarity # Create a 2D array (Matrix) matrix = np.array([ [1, 2, 3], [4, 5, 6], [7, 8, 9] ]) print(matrix) ``` In NumPy, the order of dimensions is indicated by the nesting of brackets. The outermost bracket (purple) represents the first dimension, indexed as dimension 0. The next level of brackets inside (blue) represents the second dimension, or dimension 1. If you continue adding more nested brackets, each new level represents a higher dimension. For example, the third level of brackets would be dimension 2, and so on. This hierarchical structure helps in understanding the shape and structure of multi-dimensional arrays. ::: {.callout-note} ## Exercise How would we create a 3D matrix? ```{python} # Try it! ``` ::: --- ### Some Functions to Quickly Create Arrays ```{python} # Create an array of zeros zeros = np.zeros(5) print(zeros) ``` ```{python} # Create a 2x3 matrix of ones shape = (2, 3) ones = np.ones(shape) print(ones) ``` ```{python} # Single line # Create a 2x3 matrix of ones ones = np.ones((2, 3)) print(ones) ``` ```{python} # Create an array of a range of numbers range_array = np.arange(0, 10, 2) # start, stop, step print(range_array) ``` ```{python} # Create an array of a range of numbers range_array = np.linspace(0, 10, 6) # start, stop, length print(range_array) ``` ::: {.callout-note} ## Exercise **Exercise:** Create an array containing all odd numbers between 25 and 50. ```{python} # Try it! ``` ::: ::: {.callout-note} ## Exercise **Exercise:** Imagine you have a rope that is 12 meters long. You want to cut this rope into 7 equal lengths. Using numpy, create an array that shows at which points (in meters) you should make the cuts to obtain these 7 equal lengths. ```{python} # Try it! ``` ::: --- ## Basic Properties of Arrays Unlike methods, properties don't require parentheses when accessed. They provide insights into the attributes of the array. Here is a list of some key properties of NumPy arrays. `ndim`: Indicates the number of dimensions of the array. ```{python} arr = np.zeros((5, 4)) print(arr.ndim) ``` `shape`: Returns a tuple representing the dimensions of the array. For a matrix with n rows and m columns, the shape will be (n,m). ```{python} # shape returns a tuple print(arr.shape) ``` --- ## Array Indexing Indexing 1D-arrays is the same as indexing lists, tuples or strings. ```{python} arr = np.array([0, 1, 2, 3, 4, 5, 6]) # Accessing single element (0-indexed) print(arr[2]) ``` Indexing multi-dimensional arrays is "similar" to indexing nested lists. ```{python} matrix = np.array([[0, 1, 2], [3, 4, 5]]) # Accessing element in 2D array (Matrix) print(matrix[1][2]) # second row, third column ``` For NumPy arrays, we can include all indices within a single set of parentheses! This is called "indexing with a tuple". ```{python} matrix = np.array([[0, 1, 2], [3, 4, 5]]) # Accessing element in 2D array (Matrix) print(matrix[1, 2]) # second row, third column ``` ::: {.callout-note} ## Exercise What will be the value of `x`? ```{python} matrix = np.array([[0, 1, 2], [3, 4, 5]]) print(matrix[-1, 0]) ``` ::: ::: {.callout-note} ## Exercise What will be the value of `x`? ```{python} matrix = [[0, 1, 2], [3, 4, 5]] print(matrix[2, 1]) ``` ::: This raises an error because **lists** cannot be indexed with tuples. ::: {.callout-note} ## Exercise What will be the value of `x`? ```{python} arr_A = np.array([[0, 1, 2], [3, 4, 5]]) arr_B = np.array([[6, 7, 8], [9, 10, 11]]) ls_arrays = [arr_A, arr_B] x = ls_arrays[1][0, 2] print(x) ``` ::: --- ## Array Slicing Slicing NumPy arrays is the same as slicing lists. ```{python} arr = np.array([0, 1, 2, 3, 4, 5, 6]) # Accessing a slice of an array print(arr[1:4]) # Start is inclusive, end is exclusive ``` ```{python} matrix = np.array([[0, 1, 2], [3, 4, 5]]) # Accessing rows/columns of a matrix print(matrix[:, 1]) # All rows, second column ``` ```{python} matrix = np.array([[0, 1, 2], [3, 4, 5]]) print(matrix[1, :]) # Second row, all columns ``` ```{python} arr = np.array([0, 1, 2, 3, 4, 5, 6]) # Using steps in slicing print(arr[0:5:2]) # Start at 0, go to 5, step by 2 ``` ::: {.callout-note} ## Exercise What slicing should we take to retrieve the array `[2, 4, 6]`? ```{python} arr = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8]) ``` ::: ::: {.callout-note} ## Exercise What slicing should we take to retrieve the array `[7, 4, 1]`? ```{python} arr = np.array([0, 1, 2, 3, 4, 5, 6, 7, 8]) ``` ::: ::: {.callout-note} ## Exercise What slicing should we take to retrieve the array `[2, 4, 6]`? ```{python} arr = np.array([ [0, 1, 2], [1, 2, 3], [2, 3, 4], [3, 4, 5], [4, 5, 6] ]) ``` ::: ::: {.callout-note} ## Exercise What slicing should we take to retrieve the following matrix? $$\begin{pmatrix} 1 & 2\\ 3 & 4\\ 5 & 6 \end{pmatrix}$$ ```{python} arr = np.array([ [0, 1, 2], [1, 2, 3], [2, 3, 4], [3, 4, 5], [4, 5, 6] ]) ``` ::: --- ## Practice Deactivate AI assistant tools, and try the following exercises. ::: {.callout-note} ## Exercise **Exercise:** Create an array filled with integers ordered from 1 to 12. ```{python} #| eval: false ``` ::: ::: {.callout-note} ## Exercise **Exercise:** Reshape it to a $3 \times 4$ matrix. ```{python} #| eval: false ``` ::: ::: {.callout-note} ## Exercise **Exercise:** Print its value at the first row, fourth column. ```{python} #| eval: false ``` :::