Python Numpy Tutorial (with Jupyter and Colab) (2024)

This tutorial was originally contributed by Justin Johnson.

We will use the Python programming language for all assignments in this course.Python is a great general-purpose programming language on its own, but with thehelp of a few popular libraries (numpy, scipy, matplotlib) it becomes a powerfulenvironment for scientific computing.

We expect that many of you will have some experience with Python and numpy;for the rest of you, this section will serve as a quick crash course on boththe Python programming language and its use for scientificcomputing. We’ll also introduce notebooks, which are a very convenient wayof tinkering with Python code. Some of you may have previous knowledge in adifferent language, in which case we also recommend referencing:NumPy for Matlab users,Python for R users, and/orPython for SAS users.

Table of Contents

Jupyter and Colab Notebooks
Python
- Python versions
- Basic data types
- Containers
  - Lists
  - Dictionaries
  - Sets
  - Tuples
- Functions
- Classes
Numpy
- Arrays
- Array indexing
- Datatypes
- Array math
- Broadcasting
- Numpy Documentation
SciPy
- Image operations
- MATLAB files
- Distance between points
Matplotlib
- Plotting
- Subplots
- Images

Jupyter and Colab Notebooks

Before we dive into Python, we’d like to briefly talk about notebooks.A Jupyter notebook lets you write and executePython code locally in your web browser. Jupyter notebooksmake it very easy to tinker with code and execute it in bitsand pieces; for this reason they are widely used in scientificcomputing.Colab on the other hand is Google’s flavor ofJupyter notebooks that is particularly suited for machinelearning and data analysis and that runs entirely in the cloud.Colab is basically Jupyter notebook on steroids: it’s free, requires no setup,comes preinstalled with many packages, is easy to share with the world,and benefits from free access to hardware accelerators like GPUs and TPUs (with some caveats).

Run Tutorial in Colab (recommended). If you wish to run this tutorial entirely in Colab, click the Open in Colab badge at the very top of this page.

Run Tutorial in Jupyter Notebook. If you wish to run the notebook locally with Jupyter, make sure your virtual environment is installed correctly (as per the setup instructions), activate it, then run pip install notebook to install Jupyter notebook. Next, open the notebook and download it to a directory of your choice by right-clicking on the page and selecting Save Page As. Then cd to that directory and run jupyter notebook.

This should automatically launch a notebook server at http://localhost:8888.If everything worked correctly, you should see a screen like this, showing allavailable notebooks in the current directory. Click jupyter-notebook-tutorial.ipynband follow the instructions in the notebook. Otherwise, you can continue reading thetutorial with code snippets below.

Python

Python is a high-level, dynamically typed multiparadigm programming language.Python code is often said to be almost like pseudocode, since it allows youto express very powerful ideas in very few lines of code while being veryreadable. As an example, here is an implementation of the classic quicksortalgorithm in Python:

def quicksort(arr): if len(arr) <= 1: return arr pivot = arr[len(arr) // 2] left = [x for x in arr if x < pivot] middle = [x for x in arr if x == pivot] right = [x for x in arr if x > pivot] return quicksort(left) + middle + quicksort(right)print(quicksort([3,6,8,10,1,2,1]))# Prints "[1, 1, 2, 3, 6, 8, 10]"

Python versions

As of Janurary 1, 2020, Python has officially dropped support for python2.For this class all code will use Python 3.7. Ensure you have gone through the setup instructionsand correctly installed a python3 virtual environment before proceeding with this tutorial.You can double-check your Python version at the command line after activating your environmentby running python --version.

Basic data types

Like most languages, Python has a number of basic types including integers,floats, booleans, and strings. These data types behave in ways that arefamiliar from other programming languages.

Numbers: Integers and floats work as you would expect from other languages:

x = 3print(type(x)) # Prints "<class 'int'>"print(x) # Prints "3"print(x + 1) # Addition; prints "4"print(x - 1) # Subtraction; prints "2"print(x * 2) # Multiplication; prints "6"print(x ** 2) # Exponentiation; prints "9"x += 1print(x) # Prints "4"x *= 2print(x) # Prints "8"y = 2.5print(type(y)) # Prints "<class 'float'>"print(y, y + 1, y * 2, y ** 2) # Prints "2.5 3.5 5.0 6.25"

Note that unlike many languages, Python does not have unary increment (x++)or decrement (x--) operators.

Python also has built-in types for complex numbers;you can find all of the detailsin the documentation.

Booleans: Python implements all of the usual operators for Boolean logic,but uses English words rather than symbols (&&, ||, etc.):

t = Truef = Falseprint(type(t)) # Prints "<class 'bool'>"print(t and f) # Logical AND; prints "False"print(t or f) # Logical OR; prints "True"print(not t) # Logical NOT; prints "False"print(t != f) # Logical XOR; prints "True"

Strings: Python has great support for strings:

hello = 'hello' # String literals can use single quotesworld = "world" # or double quotes; it does not matter.print(hello) # Prints "hello"print(len(hello)) # String length; prints "5"hw = hello + ' ' + world # String concatenationprint(hw) # prints "hello world"hw12 = '%s %s %d' % (hello, world, 12) # sprintf style string formattingprint(hw12) # prints "hello world 12"

String objects have a bunch of useful methods; for example:

s = "hello"print(s.capitalize()) # Capitalize a string; prints "Hello"print(s.upper()) # Convert a string to uppercase; prints "HELLO"print(s.rjust(7)) # Right-justify a string, padding with spaces; prints " hello"print(s.center(7)) # Center a string, padding with spaces; prints " hello "print(s.replace('l', '(ell)')) # Replace all instances of one substring with another; # prints "he(ell)(ell)o"print(' world '.strip()) # Strip leading and trailing whitespace; prints "world"

You can find a list of all string methods in the documentation.

Containers

Python includes several built-in container types: lists, dictionaries, sets, and tuples.

Lists

A list is the Python equivalent of an array, but is resizeableand can contain elements of different types:

xs = [3, 1, 2] # Create a listprint(xs, xs[2]) # Prints "[3, 1, 2] 2"print(xs[-1]) # Negative indices count from the end of the list; prints "2"xs[2] = 'foo' # Lists can contain elements of different typesprint(xs) # Prints "[3, 1, 'foo']"xs.append('bar') # Add a new element to the end of the listprint(xs) # Prints "[3, 1, 'foo', 'bar']"x = xs.pop() # Remove and return the last element of the listprint(x, xs) # Prints "bar [3, 1, 'foo']"

As usual, you can find all the gory details about listsin the documentation.

Slicing:In addition to accessing list elements one at a time, Python providesconcise syntax to access sublists; this is known as slicing:

nums = list(range(5)) # range is a built-in function that creates a list of integersprint(nums) # Prints "[0, 1, 2, 3, 4]"print(nums[2:4]) # Get a slice from index 2 to 4 (exclusive); prints "[2, 3]"print(nums[2:]) # Get a slice from index 2 to the end; prints "[2, 3, 4]"print(nums[:2]) # Get a slice from the start to index 2 (exclusive); prints "[0, 1]"print(nums[:]) # Get a slice of the whole list; prints "[0, 1, 2, 3, 4]"print(nums[:-1]) # Slice indices can be negative; prints "[0, 1, 2, 3]"nums[2:4] = [8, 9] # Assign a new sublist to a sliceprint(nums) # Prints "[0, 1, 8, 9, 4]"

We will see slicing again in the context of numpy arrays.

Loops: You can loop over the elements of a list like this:

animals = ['cat', 'dog', 'monkey']for animal in animals: print(animal)# Prints "cat", "dog", "monkey", each on its own line.

If you want access to the index of each element within the body of a loop,use the built-in enumerate function:

animals = ['cat', 'dog', 'monkey']for idx, animal in enumerate(animals): print('#%d: %s' % (idx + 1, animal))# Prints "#1: cat", "#2: dog", "#3: monkey", each on its own line

List comprehensions:When programming, frequently we want to transform one type of data into another.As a simple example, consider the following code that computes square numbers:

nums = [0, 1, 2, 3, 4]squares = []for x in nums: squares.append(x ** 2)print(squares) # Prints [0, 1, 4, 9, 16]

You can make this code simpler using a list comprehension:

nums = [0, 1, 2, 3, 4]squares = [x ** 2 for x in nums]print(squares) # Prints [0, 1, 4, 9, 16]

List comprehensions can also contain conditions:

nums = [0, 1, 2, 3, 4]even_squares = [x ** 2 for x in nums if x % 2 == 0]print(even_squares) # Prints "[0, 4, 16]"

Dictionaries

A dictionary stores (key, value) pairs, similar to a Map in Java oran object in Javascript. You can use it like this:

d = {'cat': 'cute', 'dog': 'furry'} # Create a new dictionary with some dataprint(d['cat']) # Get an entry from a dictionary; prints "cute"print('cat' in d) # Check if a dictionary has a given key; prints "True"d['fish'] = 'wet' # Set an entry in a dictionaryprint(d['fish']) # Prints "wet"# print(d['monkey']) # KeyError: 'monkey' not a key of dprint(d.get('monkey', 'N/A')) # Get an element with a default; prints "N/A"print(d.get('fish', 'N/A')) # Get an element with a default; prints "wet"del d['fish'] # Remove an element from a dictionaryprint(d.get('fish', 'N/A')) # "fish" is no longer a key; prints "N/A"

You can find all you need to know about dictionariesin the documentation.

Loops: It is easy to iterate over the keys in a dictionary:

d = {'person': 2, 'cat': 4, 'spider': 8}for animal in d: legs = d[animal] print('A %s has %d legs' % (animal, legs))# Prints "A person has 2 legs", "A cat has 4 legs", "A spider has 8 legs"

If you want access to keys and their corresponding values, use the items method:

d = {'person': 2, 'cat': 4, 'spider': 8}for animal, legs in d.items(): print('A %s has %d legs' % (animal, legs))# Prints "A person has 2 legs", "A cat has 4 legs", "A spider has 8 legs"

Dictionary comprehensions:These are similar to list comprehensions, but allow you to easily constructdictionaries. For example:

nums = [0, 1, 2, 3, 4]even_num_to_square = {x: x ** 2 for x in nums if x % 2 == 0}print(even_num_to_square) # Prints "{0: 0, 2: 4, 4: 16}"

Sets

A set is an unordered collection of distinct elements. As a simple example, considerthe following:

animals = {'cat', 'dog'}print('cat' in animals) # Check if an element is in a set; prints "True"print('fish' in animals) # prints "False"animals.add('fish') # Add an element to a setprint('fish' in animals) # Prints "True"print(len(animals)) # Number of elements in a set; prints "3"animals.add('cat') # Adding an element that is already in the set does nothingprint(len(animals)) # Prints "3"animals.remove('cat') # Remove an element from a setprint(len(animals)) # Prints "2"

As usual, everything you want to know about sets can be foundin the documentation.

Loops:Iterating over a set has the same syntax as iterating over a list;however since sets are unordered, you cannot make assumptions about the orderin which you visit the elements of the set:

animals = {'cat', 'dog', 'fish'}for idx, animal in enumerate(animals): print('#%d: %s' % (idx + 1, animal))# Prints "#1: fish", "#2: dog", "#3: cat"

Set comprehensions:Like lists and dictionaries, we can easily construct sets using set comprehensions:

from math import sqrtnums = {int(sqrt(x)) for x in range(30)}print(nums) # Prints "{0, 1, 2, 3, 4, 5}"

Tuples

A tuple is an (immutable) ordered list of values.A tuple is in many ways similar to a list; one of the most important differences is thattuples can be used as keys in dictionaries and as elements of sets, while lists cannot.Here is a trivial example:

d = {(x, x + 1): x for x in range(10)} # Create a dictionary with tuple keyst = (5, 6) # Create a tupleprint(type(t)) # Prints "<class 'tuple'>"print(d[t]) # Prints "5"print(d[(1, 2)]) # Prints "1"

The documentation has more information about tuples.

Functions

Python functions are defined using the def keyword. For example:

def sign(x): if x > 0: return 'positive' elif x < 0: return 'negative' else: return 'zero'for x in [-1, 0, 1]: print(sign(x))# Prints "negative", "zero", "positive"

We will often define functions to take optional keyword arguments, like this:

def hello(name, loud=False): if loud: print('HELLO, %s!' % name.upper()) else: print('Hello, %s' % name)hello('Bob') # Prints "Hello, Bob"hello('Fred', loud=True) # Prints "HELLO, FRED!"

There is a lot more information about Python functionsin the documentation.

Classes

The syntax for defining classes in Python is straightforward:

class Greeter(object): # Constructor def __init__(self, name): self.name = name # Create an instance variable # Instance method def greet(self, loud=False): if loud: print('HELLO, %s!' % self.name.upper()) else: print('Hello, %s' % self.name)g = Greeter('Fred') # Construct an instance of the Greeter classg.greet() # Call an instance method; prints "Hello, Fred"g.greet(loud=True) # Call an instance method; prints "HELLO, FRED!"

You can read a lot more about Python classesin the documentation.

Numpy

Numpy is the core library for scientific computing in Python.It provides a high-performance multidimensional array object, and tools for working with thesearrays. If you are already familiar with MATLAB, you might findthis tutorial useful to get started with Numpy.

Arrays

A numpy array is a grid of values, all of the same type, and is indexed by a tuple ofnonnegative integers. The number of dimensions is the rank of the array; the shapeof an array is a tuple of integers giving the size of the array along each dimension.

We can initialize numpy arrays from nested Python lists,and access elements using square brackets:

import numpy as npa = np.array([1, 2, 3]) # Create a rank 1 arrayprint(type(a)) # Prints "<class 'numpy.ndarray'>"print(a.shape) # Prints "(3,)"print(a[0], a[1], a[2]) # Prints "1 2 3"a[0] = 5 # Change an element of the arrayprint(a) # Prints "[5, 2, 3]"b = np.array([[1,2,3],[4,5,6]]) # Create a rank 2 arrayprint(b.shape) # Prints "(2, 3)"print(b[0, 0], b[0, 1], b[1, 0]) # Prints "1 2 4"

Numpy also provides many functions to create arrays:

import numpy as npa = np.zeros((2,2)) # Create an array of all zerosprint(a) # Prints "[[ 0. 0.] # [ 0. 0.]]"b = np.ones((1,2)) # Create an array of all onesprint(b) # Prints "[[ 1. 1.]]"c = np.full((2,2), 7) # Create a constant arrayprint(c) # Prints "[[ 7. 7.] # [ 7. 7.]]"d = np.eye(2) # Create a 2x2 identity matrixprint(d) # Prints "[[ 1. 0.] # [ 0. 1.]]"e = np.random.random((2,2)) # Create an array filled with random valuesprint(e) # Might print "[[ 0.91940167 0.08143941] # [ 0.68744134 0.87236687]]"

You can read about other methods of array creationin the documentation.

Array indexing

Numpy offers several ways to index into arrays.

Slicing:Similar to Python lists, numpy arrays can be sliced.Since arrays may be multidimensional, you must specify a slice for each dimensionof the array:

import numpy as np# Create the following rank 2 array with shape (3, 4)# [[ 1 2 3 4]# [ 5 6 7 8]# [ 9 10 11 12]]a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])# Use slicing to pull out the subarray consisting of the first 2 rows# and columns 1 and 2; b is the following array of shape (2, 2):# [[2 3]# [6 7]]b = a[:2, 1:3]# A slice of an array is a view into the same data, so modifying it# will modify the original array.print(a[0, 1]) # Prints "2"b[0, 0] = 77 # b[0, 0] is the same piece of data as a[0, 1]print(a[0, 1]) # Prints "77"

You can also mix integer indexing with slice indexing.However, doing so will yield an array of lower rank than the original array.Note that this is quite different from the way that MATLAB handles arrayslicing:

import numpy as np# Create the following rank 2 array with shape (3, 4)# [[ 1 2 3 4]# [ 5 6 7 8]# [ 9 10 11 12]]a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])# Two ways of accessing the data in the middle row of the array.# Mixing integer indexing with slices yields an array of lower rank,# while using only slices yields an array of the same rank as the# original array:row_r1 = a[1, :] # Rank 1 view of the second row of arow_r2 = a[1:2, :] # Rank 2 view of the second row of aprint(row_r1, row_r1.shape) # Prints "[5 6 7 8] (4,)"print(row_r2, row_r2.shape) # Prints "[[5 6 7 8]] (1, 4)"# We can make the same distinction when accessing columns of an array:col_r1 = a[:, 1]col_r2 = a[:, 1:2]print(col_r1, col_r1.shape) # Prints "[ 2 6 10] (3,)"print(col_r2, col_r2.shape) # Prints "[[ 2] # [ 6] # [10]] (3, 1)"

Integer array indexing:When you index into numpy arrays using slicing, the resulting array viewwill always be a subarray of the original array. In contrast, integer arrayindexing allows you to construct arbitrary arrays using the data from anotherarray. Here is an example:

import numpy as npa = np.array([[1,2], [3, 4], [5, 6]])# An example of integer array indexing.# The returned array will have shape (3,) andprint(a[[0, 1, 2], [0, 1, 0]]) # Prints "[1 4 5]"# The above example of integer array indexing is equivalent to this:print(np.array([a[0, 0], a[1, 1], a[2, 0]])) # Prints "[1 4 5]"# When using integer array indexing, you can reuse the same# element from the source array:print(a[[0, 0], [1, 1]]) # Prints "[2 2]"# Equivalent to the previous integer array indexing exampleprint(np.array([a[0, 1], a[0, 1]])) # Prints "[2 2]"

One useful trick with integer array indexing is selecting or mutating oneelement from each row of a matrix:

import numpy as np# Create a new array from which we will select elementsa = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])print(a) # prints "array([[ 1, 2, 3], # [ 4, 5, 6], # [ 7, 8, 9], # [10, 11, 12]])"# Create an array of indicesb = np.array([0, 2, 0, 1])# Select one element from each row of a using the indices in bprint(a[np.arange(4), b]) # Prints "[ 1 6 7 11]"# Mutate one element from each row of a using the indices in ba[np.arange(4), b] += 10print(a) # prints "array([[11, 2, 3], # [ 4, 5, 16], # [17, 8, 9], # [10, 21, 12]])

Boolean array indexing:Boolean array indexing lets you pick out arbitrary elements of an array.Frequently this type of indexing is used to select the elements of an arraythat satisfy some condition. Here is an example:

import numpy as npa = np.array([[1,2], [3, 4], [5, 6]])bool_idx = (a > 2) # Find the elements of a that are bigger than 2; # this returns a numpy array of Booleans of the same # shape as a, where each slot of bool_idx tells # whether that element of a is > 2.print(bool_idx) # Prints "[[False False] # [ True True] # [ True True]]"# We use boolean array indexing to construct a rank 1 array# consisting of the elements of a corresponding to the True values# of bool_idxprint(a[bool_idx]) # Prints "[3 4 5 6]"# We can do all of the above in a single concise statement:print(a[a > 2]) # Prints "[3 4 5 6]"

For brevity we have left out a lot of details about numpy array indexing;if you want to know more you shouldread the documentation.

Datatypes

Every numpy array is a grid of elements of the same type.Numpy provides a large set of numeric datatypes that you can use to construct arrays.Numpy tries to guess a datatype when you create an array, but functions that constructarrays usually also include an optional argument to explicitly specify the datatype.Here is an example:

import numpy as npx = np.array([1, 2]) # Let numpy choose the datatypeprint(x.dtype) # Prints "int64"x = np.array([1.0, 2.0]) # Let numpy choose the datatypeprint(x.dtype) # Prints "float64"x = np.array([1, 2], dtype=np.int64) # Force a particular datatypeprint(x.dtype) # Prints "int64"

You can read all about numpy datatypesin the documentation.

Array math

Basic mathematical functions operate elementwise on arrays, and are availableboth as operator overloads and as functions in the numpy module:

import numpy as npx = np.array([[1,2],[3,4]], dtype=np.float64)y = np.array([[5,6],[7,8]], dtype=np.float64)# Elementwise sum; both produce the array# [[ 6.0 8.0]# [10.0 12.0]]print(x + y)print(np.add(x, y))# Elementwise difference; both produce the array# [[-4.0 -4.0]# [-4.0 -4.0]]print(x - y)print(np.subtract(x, y))# Elementwise product; both produce the array# [[ 5.0 12.0]# [21.0 32.0]]print(x * y)print(np.multiply(x, y))# Elementwise division; both produce the array# [[ 0.2 0.33333333]# [ 0.42857143 0.5 ]]print(x / y)print(np.divide(x, y))# Elementwise square root; produces the array# [[ 1. 1.41421356]# [ 1.73205081 2. ]]print(np.sqrt(x))

Note that unlike MATLAB, * is elementwise multiplication, not matrixmultiplication. We instead use the dot function to compute innerproducts of vectors, to multiply a vector by a matrix, and tomultiply matrices. dot is available both as a function in the numpymodule and as an instance method of array objects:

import numpy as npx = np.array([[1,2],[3,4]])y = np.array([[5,6],[7,8]])v = np.array([9,10])w = np.array([11, 12])# Inner product of vectors; both produce 219print(v.dot(w))print(np.dot(v, w))# Matrix / vector product; both produce the rank 1 array [29 67]print(x.dot(v))print(np.dot(x, v))# Matrix / matrix product; both produce the rank 2 array# [[19 22]# [43 50]]print(x.dot(y))print(np.dot(x, y))

Numpy provides many useful functions for performing computations onarrays; one of the most useful is sum:

import numpy as npx = np.array([[1,2],[3,4]])print(np.sum(x)) # Compute sum of all elements; prints "10"print(np.sum(x, axis=0)) # Compute sum of each column; prints "[4 6]"print(np.sum(x, axis=1)) # Compute sum of each row; prints "[3 7]"

You can find the full list of mathematical functions provided by numpyin the documentation.

Apart from computing mathematical functions using arrays, we frequentlyneed to reshape or otherwise manipulate data in arrays. The simplest exampleof this type of operation is transposing a matrix; to transpose a matrix,simply use the T attribute of an array object:

import numpy as npx = np.array([[1,2], [3,4]])print(x) # Prints "[[1 2] # [3 4]]"print(x.T) # Prints "[[1 3] # [2 4]]"# Note that taking the transpose of a rank 1 array does nothing:v = np.array([1,2,3])print(v) # Prints "[1 2 3]"print(v.T) # Prints "[1 2 3]"

Numpy provides many more functions for manipulating arrays; you can see the full listin the documentation.

Broadcasting

Broadcasting is a powerful mechanism that allows numpy to work with arrays of differentshapes when performing arithmetic operations. Frequently we have a smaller array and alarger array, and we want to use the smaller array multiple times to perform some operationon the larger array.

For example, suppose that we want to add a constant vector to eachrow of a matrix. We could do it like this:

import numpy as np# We will add the vector v to each row of the matrix x,# storing the result in the matrix yx = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])v = np.array([1, 0, 1])y = np.empty_like(x) # Create an empty matrix with the same shape as x# Add the vector v to each row of the matrix x with an explicit loopfor i in range(4): y[i, :] = x[i, :] + v# Now y is the following# [[ 2 2 4]# [ 5 5 7]# [ 8 8 10]# [11 11 13]]print(y)

This works; however when the matrix x is very large, computing an explicit loopin Python could be slow. Note that adding the vector v to each row of the matrixx is equivalent to forming a matrix vv by stacking multiple copies of v vertically,then performing elementwise summation of x and vv. We could implement thisapproach like this:

import numpy as np# We will add the vector v to each row of the matrix x,# storing the result in the matrix yx = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])v = np.array([1, 0, 1])vv = np.tile(v, (4, 1)) # Stack 4 copies of v on top of each otherprint(vv) # Prints "[[1 0 1] # [1 0 1] # [1 0 1] # [1 0 1]]"y = x + vv # Add x and vv elementwiseprint(y) # Prints "[[ 2 2 4 # [ 5 5 7] # [ 8 8 10] # [11 11 13]]"

Numpy broadcasting allows us to perform this computation without actuallycreating multiple copies of v. Consider this version, using broadcasting:

import numpy as np# We will add the vector v to each row of the matrix x,# storing the result in the matrix yx = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])v = np.array([1, 0, 1])y = x + v # Add v to each row of x using broadcastingprint(y) # Prints "[[ 2 2 4] # [ 5 5 7] # [ 8 8 10] # [11 11 13]]"

The line y = x + v works even though x has shape (4, 3) and v has shape(3,) due to broadcasting; this line works as if v actually had shape (4, 3),where each row was a copy of v, and the sum was performed elementwise.

Broadcasting two arrays together follows these rules:

If the arrays do not have the same rank, prepend the shape of the lower rank arraywith 1s until both shapes have the same length.
The two arrays are said to be compatible in a dimension if they have the samesize in the dimension, or if one of the arrays has size 1 in that dimension.
The arrays can be broadcast together if they are compatible in all dimensions.
After broadcasting, each array behaves as if it had shape equal to the elementwisemaximum of shapes of the two input arrays.
In any dimension where one array had size 1 and the other array had size greater than 1,the first array behaves as if it were copied along that dimension

If this explanation does not make sense, try reading the explanationfrom the documentationor this explanation.

Functions that support broadcasting are known as universal functions. You can findthe list of all universal functionsin the documentation.

Here are some applications of broadcasting:

import numpy as np# Compute outer product of vectorsv = np.array([1,2,3]) # v has shape (3,)w = np.array([4,5]) # w has shape (2,)# To compute an outer product, we first reshape v to be a column# vector of shape (3, 1); we can then broadcast it against w to yield# an output of shape (3, 2), which is the outer product of v and w:# [[ 4 5]# [ 8 10]# [12 15]]print(np.reshape(v, (3, 1)) * w)# Add a vector to each row of a matrixx = np.array([[1,2,3], [4,5,6]])# x has shape (2, 3) and v has shape (3,) so they broadcast to (2, 3),# giving the following matrix:# [[2 4 6]# [5 7 9]]print(x + v)# Add a vector to each column of a matrix# x has shape (2, 3) and w has shape (2,).# If we transpose x then it has shape (3, 2) and can be broadcast# against w to yield a result of shape (3, 2); transposing this result# yields the final result of shape (2, 3) which is the matrix x with# the vector w added to each column. Gives the following matrix:# [[ 5 6 7]# [ 9 10 11]]print((x.T + w).T)# Another solution is to reshape w to be a column vector of shape (2, 1);# we can then broadcast it directly against x to produce the same# output.print(x + np.reshape(w, (2, 1)))# Multiply a matrix by a constant:# x has shape (2, 3). Numpy treats scalars as arrays of shape ();# these can be broadcast together to shape (2, 3), producing the# following array:# [[ 2 4 6]# [ 8 10 12]]print(x * 2)

Broadcasting typically makes your code more concise and faster, so youshould strive to use it where possible.

Numpy Documentation

This brief overview has touched on many of the important things that you need toknow about numpy, but is far from complete. Check out thenumpy referenceto find out much more about numpy.

SciPy

Numpy provides a high-performance multidimensional array and basic tools tocompute with and manipulate these arrays.SciPybuilds on this, and providesa large number of functions that operate on numpy arrays and are useful fordifferent types of scientific and engineering applications.

The best way to get familiar with SciPy is tobrowse the documentation.We will highlight some parts of SciPy that you might find useful for this class.

Image operations

SciPy provides some basic functions to work with images.For example, it has functions to read images from disk into numpy arrays,to write numpy arrays to disk as images, and to resize images.Here is a simple example that showcases these functions:

from scipy.misc import imread, imsave, imresize# Read an JPEG image into a numpy arrayimg = imread('assets/cat.jpg')print(img.dtype, img.shape) # Prints "uint8 (400, 248, 3)"# We can tint the image by scaling each of the color channels# by a different scalar constant. The image has shape (400, 248, 3);# we multiply it by the array [1, 0.95, 0.9] of shape (3,);# numpy broadcasting means that this leaves the red channel unchanged,# and multiplies the green and blue channels by 0.95 and 0.9# respectively.img_tinted = img * [1, 0.95, 0.9]# Resize the tinted image to be 300 by 300 pixels.img_tinted = imresize(img_tinted, (300, 300))# Write the tinted image back to diskimsave('assets/cat_tinted.jpg', img_tinted)

Left: The original image. Right: The tinted and resized image.

MATLAB files

The functions scipy.io.loadmat and scipy.io.savemat allow you to read andwrite MATLAB files. You can read about themin the documentation.

Distance between points

SciPy defines some useful functions for computing distances between sets of points.

The function scipy.spatial.distance.pdist computes the distance between all pairsof points in a given set:

import numpy as npfrom scipy.spatial.distance import pdist, squareform# Create the following array where each row is a point in 2D space:# [[0 1]# [1 0]# [2 0]]x = np.array([[0, 1], [1, 0], [2, 0]])print(x)# Compute the Euclidean distance between all rows of x.# d[i, j] is the Euclidean distance between x[i, :] and x[j, :],# and d is the following array:# [[ 0. 1.41421356 2.23606798]# [ 1.41421356 0. 1. ]# [ 2.23606798 1. 0. ]]d = squareform(pdist(x, 'euclidean'))print(d)

You can read all the details about this functionin the documentation.

A similar function (scipy.spatial.distance.cdist) computes the distance between all pairsacross two sets of points; you can read about itin the documentation.

Matplotlib

Matplotlib is a plotting library.In this section give a brief introduction to the matplotlib.pyplot module,which provides a plotting system similar to that of MATLAB.

Plotting

The most important function in matplotlib is plot,which allows you to plot 2D data. Here is a simple example:

import numpy as npimport matplotlib.pyplot as plt# Compute the x and y coordinates for points on a sine curvex = np.arange(0, 3 * np.pi, 0.1)y = np.sin(x)# Plot the points using matplotlibplt.plot(x, y)plt.show() # You must call plt.show() to make graphics appear.

Running this code produces the following plot:

With just a little bit of extra work we can easily plot multiple linesat once, and add a title, legend, and axis labels:

import numpy as npimport matplotlib.pyplot as plt# Compute the x and y coordinates for points on sine and cosine curvesx = np.arange(0, 3 * np.pi, 0.1)y_sin = np.sin(x)y_cos = np.cos(x)# Plot the points using matplotlibplt.plot(x, y_sin)plt.plot(x, y_cos)plt.xlabel('x axis label')plt.ylabel('y axis label')plt.title('Sine and Cosine')plt.legend(['Sine', 'Cosine'])plt.show()

You can read much more about the plot functionin the documentation.

Subplots

You can plot different things in the same figure using the subplot function.Here is an example:

import numpy as npimport matplotlib.pyplot as plt# Compute the x and y coordinates for points on sine and cosine curvesx = np.arange(0, 3 * np.pi, 0.1)y_sin = np.sin(x)y_cos = np.cos(x)# Set up a subplot grid that has height 2 and width 1,# and set the first such subplot as active.plt.subplot(2, 1, 1)# Make the first plotplt.plot(x, y_sin)plt.title('Sine')# Set the second subplot as active, and make the second plot.plt.subplot(2, 1, 2)plt.plot(x, y_cos)plt.title('Cosine')# Show the figure.plt.show()

You can read much more about the subplot functionin the documentation.

Images

You can use the imshow function to show images. Here is an example:

import numpy as npfrom scipy.misc import imread, imresizeimport matplotlib.pyplot as pltimg = imread('assets/cat.jpg')img_tinted = img * [1, 0.95, 0.9]# Show the original imageplt.subplot(1, 2, 1)plt.imshow(img)# Show the tinted imageplt.subplot(1, 2, 2)# A slight gotcha with imshow is that it might give strange results# if presented with data that is not uint8. To work around this, we# explicitly cast the image to uint8 before displaying it.plt.imshow(np.uint8(img_tinted))plt.show()