Machine Learning

0. Python Tutorial

About the notebook/slides

  • The slides are programmed as a Jupyter/IPython notebook.
  • Feel free to try them and experiment on your own by launching the notebooks.

  • You can run the notebook online: Binder

Introduction

  • Python is a great general-purpose programming language,...
  • ...and with the help of a libraries such as numpy, scipy and matplotlib (among many others) it becomes a powerful environment for scientific computing.
 
 

Scripta volant, codex manent

Python main characteristics:

  • dynamic type system
  • interpreted (actually: compiled to bytecode, *.pyc files)
  • multi-paradigm: imperative, procedural, object-oriented, (functional), literate; do whatever you want
  • indentation is important!

I assume that you have some programming experience:

  • Java (static type system, compiled, object-oriented, verbose)
  • C/C++ (static type system, compiled, multi-paradigm, low-level)
  • Matlab? R?

In this tutorial

  • Basic Python: Basic data types, containers, loops, functions and classes.
  • numpy: arrays, array indexing, datatypes, array math and broadcasting.
  • matplotlib: plotting, subplots, images.
  • Jupyter and IPython: Creating notebooks and typical workflows.

Basics of Python

  • Python is a high-level, dynamically typed multiparadigm programming language.
  • Python code is often said to be almost like pseudocode, since it allows you to express very powerful ideas in very few lines of code while being very readable.

Required Python packages (libraries) to run this notebook:

  • Python
  • Jupyter/IPython - interactive Python shell
  • numpy - linear algebra library
  • matplotlib - plotting library

Scientific Python Distributions (available for almost every platform):

As an example, here is an implementation of the classic quicksort algorithm in Python:

In [1]:
def quicksort(arr):
    if len(arr) <= 1:
        return arr
    pivot = arr[int(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)
In [2]:
quicksort([3,6,8,10,1,2,1])
Out[2]:
[1, 1, 2, 3, 6, 8, 10]

Python versions

  • There are currently two different supported versions of Python, 2.x and 3.x.
  • Python 3.0 introduced many backwards-incompatible changes to the language, so code written for 2.x may not work under 3.x and vice versa.
  • In our class we will be using Python 3.x.

Note: You can check your Python version at the command line by running python --version.

Basic data types

Numeric types

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

In [3]:
x = 3
print(x, type(x))
3 <class 'int'>
In [4]:
print(x + 1)   # Addition;
print(x - 1)   # Subtraction;
print(x * 2)   # Multiplication;
print(x ** 2)  # Exponentiation.
4
2
6
9

Autoincrements:

In [5]:
x += 1
print(x)  # Prints "4"
x *= 2
print(x)  # Prints "8"
4
8

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

Python also has built-in types for long integers and complex numbers; you can find all of the details in the documentation.

Real numbers (float)

In [6]:
y = 2.5
print(type(y)) # Prints "<class 'float'>"
print(y, y + 1, y * 2, y ** 2) # Prints "2.5 3.5 5.0 6.25"
<class 'float'>
2.5 3.5 5.0 6.25

Booleans

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

In [7]:
t, f, aa, bb = True, False, True, False # Hmm... yes you can do this in Python
print(t, f, type(t))
True False <class 'bool'>

Now we let's look at the operations:

In [8]:
print(t and f) # Logical AND;
print(t or f)  # Logical OR;
print(not t)   # Logical NOT;
print(t != f)  # Logical XOR;
False
True
False
True

Booleans and truth Testing

  • In programming we invariably need some concept of conditions to allow a program to react differently to different situations.
  • Python uses a combination of Boolean variables, which evaluate to either True or False, and if statements.
In [5]:
day = "Sunday"
if day == 'Sunday':
    print('Sleep!!!')
else:
    print('Go to work')
Sleep!!!

There is no switch or case statements

In [10]:
if day == 'Monday':
    print('Week end is over!')
elif day == 'Sunday' or day =='Saturday':
    print('Sleep!')
else:
    print('Meeeh')
Week end is over!
In [11]:
var_1 = 234
if var_1:
    print('Do something with', var_1)
else:
    print('Nothing to do')
Do something with 234
In [12]:
1 == 2
Out[12]:
False
In [13]:
50 == 2*25
Out[13]:
True

There is another boolean operator is, that tests whether two objects are the same type:

In [14]:
1 is 1
Out[14]:
True

But not...

In [15]:
1 is int
Out[15]:
False
In [16]:
print(type(1), type(int))
<class 'int'> <class 'type'>
In [17]:
1 is 1.0
Out[17]:
False

Strings

String literals can use single quotes ('') or or double quotes(""); it does not matter.

In [18]:
hello = 'hello'
world = "world"
print(hello, len(hello))
hello 5
In [19]:
hw = hello + ' ' + world  # String concatenation
print(hw)
hello world

String formatting

In [20]:
hw12 = '%s %s! your number is: %d' % (hello, world, 12)  # sprintf style string formatting
print(hw12)
hello world! your number is: 12

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

In [21]:
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"
Hello
HELLO
  hello
 hello 
he(ell)(ell)o
world

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 resizeable and can contain elements of different types:

In [22]:
xs = [3, 1, 2]    # Create a list
print(xs, xs[2])
print(xs[-1])     # Negative indices count from the end of the list; prints "2"
[3, 1, 2] 2
2
In [23]:
xs[2] = 'foo'    # Lists can contain elements of different types
print(xs)
[3, 1, 'foo']
In [24]:
xs.append('bar') # Add a new element to the end of the list
print(xs)
[3, 1, 'foo', 'bar']
In [25]:
xs =  xs + ['thing1', 'thing2'] # Adding lists (the += op works too)
print(xs)
[3, 1, 'foo', 'bar', 'thing1', 'thing2']

Slicing

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

In [27]:
nums = list(range(5)) # range is a built-in function (more on this later)
print(nums)
[0, 1, 2, 3, 4]
In [28]:
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 slice
print(nums)         # Prints "[0, 1, 8, 9, 4]"
[2, 3]
[2, 3, 4]
[0, 1]
[0, 1, 2, 3, 4]
[0, 1, 2, 3]
[0, 1, 8, 9, 4]

Loops

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

In [29]:
animals = ['cat', 'dog', 'monkey']
for animal in animals:
    aa = animal + ' :)'
    print(aa)
cat :)
dog :)
monkey :)

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

In [30]:
animals = ['cat', 'dog', 'monkey']
for idx, animal in enumerate(animals):
    print('Item number %d is a %s' % (idx + 1, animal))
Item number 1 is a cat
Item number 2 is a dog
Item number 3 is a monkey

Conditional loops

In Python we have only a while loop.

In [31]:
i = 0
while i < 3:
    print(i)
    i += 1
0
1
2

List comprehensions

When programming, frequently we want to transform one type of data into another.

For example, consider the following code that computes square numbers:

In [32]:
nums = [0, 1, 2, 3, 4]
squares = []
for x in nums:
    squares.append(x ** 2)
print(squares)
[0, 1, 4, 9, 16]

You can make this code simpler using a list comprehension:

In [33]:
nums = [0, 1, 2, 3, 4]
squares = [x ** 2 for x in nums]
print(squares)
[0, 1, 4, 9, 16]

List comprehensions can also contain conditions:

In [34]:
nums = [0, 1, 2, 3, 4]
even_squares = [x ** 2 for x in nums if x % 2 == 0]
print(even_squares)
[0, 4, 16]

Dictionaries

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

In [35]:
d = {'cat': 'cute', 'dog': 'furry'}  # Create a new dictionary with some data
In [36]:
print(d['cat'])       # Get an entry from a dictionary; prints "cute"
print('cat' in d)     # Check if a dictionary has a given key; prints "True"
cute
True
In [37]:
d['fish'] = 'wet'     # Set an entry in a dictionary
print(d['fish'])      # Prints "wet"
wet

Handling not found keys:

In [38]:
print(d['monkey'])  # KeyError: 'monkey' not a key of d
---------------------------------------------------------------------------
KeyError                                  Traceback (most recent call last)
<ipython-input-38-e3ac4f3aa8c2> in <module>()
----> 1 print(d['monkey'])  # KeyError: 'monkey' not a key of d

KeyError: 'monkey'
In [39]:
print(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"
N/A
wet
In [40]:
del d['fish']               # Remove an element from a dictionary
print(d.get('fish', 'N/A')) # "fish" is no longer a key; prints "N/A"
N/A

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

In [7]:
d = {'person': 2, 'cat': 4, 'spider': 8}
for animal in d:
    legs = d[animal]
    print('A %s has %d legs' % (animal, legs))
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 construct dictionaries. For example:

In [42]:
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)
{0: 0, 2: 4, 4: 16}

Sets

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

In [43]:
animals = {'cat', 'dog'}
print(animals)
{'dog', 'cat'}
In [44]:
print('cat' in animals)   # Check if an element is in a set; prints "True"
print('fish' in animals)  # prints "False"
True
False
In [45]:
animals.add('fish')      # Add an element to a set
print('fish' in animals)
print(len(animals))       # Number of elements in a set;
True
3
In [46]:
animals.add('cat')           # Adding an element that is already in the set does nothing
print(animals, len(animals))       
animals.remove('cat')        # Remove an element from a set
print(animals,len(animals))       
{'dog', 'cat', 'fish'} 3
{'dog', 'fish'} 2

Loops in sets

  • Iterating over a set has the same syntax as iterating over a list;
  • however since sets are unordered, you cannot make assumptions about the order in which you visit the elements of the set.
In [47]:
animals = {'cat', 'dog', 'fish'}
for idx, animal in enumerate(animals):
    print('#%d: %s' % (idx + 1, animal))
# Prints "#1: fish", "#2: dog", "#3: cat"
#1: dog
#2: cat
#3: fish

Set comprehensions

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

In [48]:
from math import sqrt
print('Set:', {int(sqrt(x)) for x in range(30)})
print('List:', [int(sqrt(x)) for x in range(30)])
Set: {0, 1, 2, 3, 4, 5}
List: [0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5]

Tuples

  • An (immutable) ordered list of values;
  • is in many ways similar to a list;
  • one of the most important differences is that:
    • tuples can be used as keys in dictionaries and
    • as elements of sets,

Here is a trivial example:

In [9]:
t = (5, 6)
print(t, type(t))
(5, 6) <class 'tuple'>
5 6
In [50]:
d = {(x, x + 1): x for x in range(10)}  # Create a dictionary with tuple keys
print(d)
{(0, 1): 0, (1, 2): 1, (2, 3): 2, (3, 4): 3, (4, 5): 4, (5, 6): 5, (6, 7): 6, (7, 8): 7, (8, 9): 8, (9, 10): 9}
In [51]:
print(d[t])       
print(d[(1, 2)])
5
1
In [52]:
t[0] = 1 # Produces TypeError 'tuple' object does not support item assignment
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-52-5757bc7a932f> in <module>()
----> 1 t[0] = 1 # Produces TypeError 'tuple' object does not support item assignment

TypeError: 'tuple' object does not support item assignment

Functions

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

In [53]:
def sign(x):
    if x > 0:
        return 'positive'
    elif x < 0:
        return 'negative'
    else:
        return 'zero'
In [54]:
for x in [-1, 0, 1]:
    print(sign(x))
negative
zero
positive

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

In [55]:
def hello(name, loud=False):
    if loud:
        print('HELLO, %s' % name.upper())
    else:
        print('Hello, %s!' % name)
In [56]:
hello('Bob')
hello('Fred', loud=True)
Hello, Bob!
HELLO, FRED

$\lambda$-constructors (inline functions)

In [57]:
func = lambda x: x**x + 2
print(type(func))
<class 'function'>
In [58]:
func(3)
Out[58]:
29
In [59]:
x = [1, 5, 3, 2, 7, 8, 3, 6]

print([func(elem) for elem in x])
list(map(lambda x: x**x + 2, x)) # note the map() function
[3, 3127, 29, 6, 823545, 16777218, 29, 46658]
Out[59]:
[3, 3127, 29, 6, 823545, 16777218, 29, 46658]

Classes and object oriented programming

  • The syntax for defining classes in Python is straightforward.
  • Remember to include self as the first parameter of the class methods.
In [20]:
class Greeter():
    # 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)
In [23]:
g = Greeter('Fred')  # Construct an instance of the Greeter class
g.greet()            # Call an instance method; prints "Hello, Fred"
g.greet(loud=True)   # Call an instance method; prints "HELLO, FRED!"
Hello, Fred
HELLO, FRED!

Exceptions

As in many other modern languages Python can handle exceptions.

In [25]:
def my_func(a):
    return a * a
In [26]:
print(my_func(2))
4
In [27]:
my_func('AAAAA')
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-27-c43b8d3fab9a> in <module>()
----> 1 my_func('AAAAA')

<ipython-input-25-77fcce2d5fdd> in my_func(a)
      1 def my_func(a):
----> 2     return a * a

TypeError: can't multiply sequence by non-int of type 'str'
In [64]:
try:
    print(my_func("A"))
except TypeError as e:
    print('Caught exception:', e)
Caught exception: can't multiply sequence by non-int of type 'str'

The import statement

  • We have seen already the import statement in action.
  • Python has a huge number of libraries included with the distribution.
  • Most of these variables and functions are not accessible from a normal Python interactive session.
  • Instead, you have to import them.

Modules

  • A module allows you to logically organize your Python code;
  • groups related code into a module makes the code easier to understand and use.

  • A module is a file consisting of Python code, can define functions, classes and variables.

  • A module can also include runnable code.

Note: A module is also a Python object with arbitrarily named attributes that you can bind and reference.

Importing complete modules

Importing whole modules with an additional short name:

In [65]:
import scipy.spatial.distance
In [66]:
scipy.spatial.distance.euclidean((1,1), (2,2))
Out[66]:
1.4142135623730951
In [67]:
import scipy.spatial.distance as dists
In [68]:
dists.euclidean((1,1), (2,2))
Out[68]:
1.4142135623730951

Importing a particular function or class.

For example, there is a math module containing many useful functions. To access, say, the square root function, you can

In [69]:
from math import sqrt

and then

In [70]:
sqrt(81)
Out[70]:
9.0

numpy: scientific computing with Python

  • Numpy is the core library for scientific computing in Python.
  • It provides a high-performance multidimensional array object, and tools for working with these arrays.

To use numpy, we first need to import numpy package:

In [71]:
import numpy as np

Arrays

A numpy array is a grid of values, all of the same type, and is indexed by a tuple of nonnegative integers.

  • The number of dimensions is the rank of the array;
  • The shape of an array is a tuple of integers giving the size of the array along each dimension.
  • useful linear algebra, Fourier transform, random number capabilities and more.
  • Complemented by scipy.

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

In [72]:
a = np.array([1, 2, 3])  # Create a rank 1 array
print(type(a), a.shape, a[0], a[1], a[2])
a[0] = 5  # Change an element of the array
print(a)
<class 'numpy.ndarray'> (3,) 1 2 3
[5 2 3]
In [73]:
b = np.array([[1, 2, 3], [4, 5, 6]])  # Create a rank 2 array
print(b)
[[1 2 3]
 [4 5 6]]
In [74]:
print(b.shape)
print(b[0, 0], b[0, 1], b[1, 0])
(2, 3)
1 2 4

Numpy provides many shortcut functions to create arrays:

In [75]:
a = np.zeros((2,2))  # Create an array of all zeros
print(a)
[[ 0.  0.]
 [ 0.  0.]]
In [76]:
b = np.ones((1,2))   # Create an array of all ones
print(b)
[[ 1.  1.]]
In [77]:
c = np.full((2,2), 7)
print(c)
[[7 7]
 [7 7]]
In [78]:
d = np.eye(2)        # Create a 2x2 identity matrix
print(d)
[[ 1.  0.]
 [ 0.  1.]]
In [79]:
e = np.random.random((2,2)) # Create an array filled with random values
print(e)
[[ 0.09845823  0.09568912]
 [ 0.18341987  0.72051223]]

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 dimension of the array:

In [80]:
# Create the following rank 2 array with shape (3, 4)
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])
a
Out[80]:
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):

In [81]:
b = a[:2, 1:3]
print(b)
[[2 3]
 [6 7]]

A slice of an array is a view into the same data, so modifying it will modify the original array.

In [82]:
print(a[0, 1])  
b[0, 0] = 77      # b[0, 0] is the same piece of data as a[0, 1]
print(a[0, 1]) 
2
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: This is quite different from the way that MATLAB handles array slicing.

In [83]:
# Create the following rank 2 array with shape (3, 4)
a = np.array([[1,2,3,4], [5,6,7,8], [9,10,11,12]])
print(a)
[[ 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:

In [84]:
row_r1 = a[1, :]    # Rank 1 view of the second row of a  
row_r2 = a[1:2, :]  # Rank 2 view of the second row of a
row_r3 = a[[1], :]  # Rank 2 view of the second row of a
print(row_r1, row_r1.shape) 
print(row_r2, row_r2.shape)
print(row_r3, row_r3.shape)
[5 6 7 8] (4,)
[[5 6 7 8]] (1, 4)
[[5 6 7 8]] (1, 4)

We can make the same distinction when accessing columns of an array:

In [85]:
col_r1 = a[:, 1]
col_r2 = a[:, 1:2]
print(col_r1, col_r1.shape)
print(col_r2, col_r2.shape)
[ 2  6 10] (3,)
[[ 2]
 [ 6]
 [10]] (3, 1)

Integer array indexing

  • When you index into numpy arrays using slicing, the resulting array view will always be a subarray of the original array.
  • In contrast, integer array indexing allows you to construct arbitrary arrays using the data from another array.

Here is an example:

In [86]:
a = np.array([[1,2], [3, 4], [5, 6]])
print(a)
[[1 2]
 [3 4]
 [5 6]]

An example of integer array indexing. The returned array will have shape (3,).

In [87]:
a[[0, 1, 2], [0, 1, 0]]
Out[87]:
array([1, 4, 5])

The above example of integer array indexing is equivalent to this:

In [88]:
np.array([a[0, 0], a[1, 1], a[2, 0]])
Out[88]:
array([1, 4, 5])

When using integer array indexing, you can reuse the same element from the source array:

In [89]:
a[[0, 0], [1, 1]]
Out[89]:
array([2, 2])

Equivalent to the previous integer array indexing example

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

Create a new array from which we will select elements

In [91]:
a = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
print(a)
[[ 1  2  3]
 [ 4  5  6]
 [ 7  8  9]
 [10 11 12]]

Create an array of indices

In [92]:
b = np.array([0, 2, 0, 1])

Select one element from each row of a using the indices in b.

In [93]:
print(a[np.arange(4), b])  # Prints "[ 1  6  7 11]"
[ 1  6  7 11]

Mutate one element from each row of a using the indices in b.

In [94]:
a[np.arange(4), b] += 10
print(a)
[[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 array that satisfy some condition. Here is an example:

In [95]:
a = np.array([[1,2], [3, 4], [5, 6]])

Find the elements of a that are bigger than 2:

In [96]:
bool_idx = (a > 2)
print(bool_idx)
[[False False]
 [ True  True]
 [ True  True]]

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.

We use boolean array indexing to construct a rank 1 array consisting of the elements of a corresponding to the True values of bool_idx.

In [97]:
print(a[bool_idx])
[3 4 5 6]

We can do all of the above in a single concise statement:

In [98]:
print(a[a > 2])
[3 4 5 6]

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 construct arrays usually also include an optional argument to explicitly specify the datatype. Here is an example:

In [30]:
import numpy as np
x = np.array([1, 2])                  # Let numpy choose the datatype
y = np.array([1.0, 2.0])              # Let numpy choose the datatype
z = np.array([1, 2], dtype=np.float32)  # Force a particular datatype

print(x.dtype, y.dtype, z.dtype)
int64 float64 float32

You can read all about numpy datatypes in the documentation.

Array math

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

In [100]:
x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64)
In [101]:
print(x + y)
[[  6.   8.]
 [ 10.  12.]]
In [102]:
print(np.add(x, y))
[[  6.   8.]
 [ 10.  12.]]

Elementwise difference

In [103]:
print(x - y)
print(np.subtract(x, y))
[[-4. -4.]
 [-4. -4.]]
[[-4. -4.]
 [-4. -4.]]

Elementwise product

In [104]:
print(x * y)
print(np.multiply(x, y))
[[  5.  12.]
 [ 21.  32.]]
[[  5.  12.]
 [ 21.  32.]]

Elementwise division - both produce the same result.

In [105]:
print(x / y)
print(np.divide(x, y))
[[ 0.2         0.33333333]
 [ 0.42857143  0.5       ]]
[[ 0.2         0.33333333]
 [ 0.42857143  0.5       ]]

Elementwise square root;

In [106]:
print(np.sqrt(x))
[[ 1.          1.41421356]
 [ 1.73205081  2.        ]]

Important note:

  • Unlike MATLAB, * is elementwise multiplication, not matrix multiplication.
  • We instead use the numpy.dot() function to compute inner products of vectors, to multiply a vector by a matrix, and to multiply matrices.
  • dot() is available both as a function in the numpy module and as an instance method of array objects.
In [107]:
x = 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 219
print(v.dot(w))
print(np.dot(v, w))
219
219
In [108]:
# Matrix / vector product; both produce the rank 1 array [29 67]
print(x.dot(v))
print(np.dot(x, v))
[29 67]
[29 67]
In [109]:
# Matrix / matrix product; both produce the rank 2 array
# [[19 22]
#  [43 50]]
print(x.dot(y))
print(np.dot(x, y))
[[19 22]
 [43 50]]
[[19 22]
 [43 50]]

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

In [110]:
x = np.array([[1,2],[3,4]])
print(x)
[[1 2]
 [3 4]]
In [111]:
np.sum(x)  # Compute sum of all elements; prints "10"
Out[111]:
10
In [112]:
np.sum(x, axis=0)  # Compute sum of each column; prints "[4 6]"
Out[112]:
array([4, 6])
In [113]:
np.sum(x, axis=1)  # Compute sum of each row; prints "[3 7]"
Out[113]:
array([3, 7])

Is numpy actually faster than 'plain' Python?

In [116]:
import random
import numpy as np
In [117]:
%timeit sum([random.random() for _ in range(random.randint(10000,10100))])
2.1 ms ± 93 µs per loop (mean ± std. dev. of 7 runs, 100 loops each)
In [118]:
%timeit np.sum(np.random.random(random.randint(10000,10100)))
177 µs ± 3.67 µs per loop (mean ± std. dev. of 7 runs, 10000 loops each)

Broadcasting

Broadcasting is a powerful mechanism that allows numpy to work with arrays of different shapes when performing arithmetic operations.

  • Frequently, we have a smaller array and a larger array, and
  • we want to use the smaller array multiple times to perform some operation on the larger array.

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

In [119]:
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
x
Out[119]:
array([[ 1,  2,  3],
       [ 4,  5,  6],
       [ 7,  8,  9],
       [10, 11, 12]])
In [120]:
v = np.array([1, 0, 1])
v
Out[120]:
array([1, 0, 1])
In [121]:
y = np.empty_like(x)   # Creates an empty matrix with the same shape as x

# Add the vector v to each row of the matrix x with an explicit loop
for i in range(4):
    y[i, :] = x[i, :] + v
In [122]:
y
Out[122]:
array([[ 2,  2,  4],
       [ 5,  5,  7],
       [ 8,  8, 10],
       [11, 11, 13]])
  • This works; however when the matrix x is very large, computing an explicit loop in Python could be slow.

  • Note that adding the vector v to each row of the matrix x 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 this approach like this:

In [123]:
vv = np.tile(v, (4, 1))  # Stack 4 copies of v on top of each other
vv
Out[123]:
array([[1, 0, 1],
       [1, 0, 1],
       [1, 0, 1],
       [1, 0, 1]])
In [124]:
y = x + vv  # Add x and vv elementwise
y
Out[124]:
array([[ 2,  2,  4],
       [ 5,  5,  7],
       [ 8,  8, 10],
       [11, 11, 13]])

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

In [125]:
x = 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 broadcasting
y
Out[125]:
array([[ 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:

  1. If the arrays do not have the same rank, prepend the shape of the lower rank array with 1s until both shapes have the same length.
  2. The two arrays are said to be compatible in a dimension if they have the same size in the dimension, or if one of the arrays has size 1 in that dimension.
  3. The arrays can be broadcast together if they are compatible in all dimensions.
  4. After broadcasting, each array behaves as if it had shape equal to the elementwise maximum of shapes of the two input arrays.
  5. 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 explanation from the documentation or this explanation.

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

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

So far, we touched on many of the important things that you need to know about numpy, but is far from complete.

  • Check out the numpy reference to find out much more about numpy.
  • If you are already familiar with MATLAB, you might find this tutorial useful to get started with numpy.
  • Nicolas P. Rougier's Numpy tutorial highly recommended.
  • Matplotlib is a plotting library.
  • Provides a plotting system similar -and in my opinion better- to that of MATLAB.
In [133]:
import matplotlib.pyplot as plt

By running this special IPython notebook magic command, we will be displaying plots inline and with appropiate resolutions for retina displays. Check the docs for more options.

In [134]:
%matplotlib inline
%config InlineBackend.figure_format = 'retina'

Plotting

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

In [135]:
# Compute the x and y coordinates for points on a sine curve
x = np.arange(0, 3 * np.pi, 0.1)
y_sin = np.sin(x)

# Plot the points using matplotlib
plt.plot(x, y_sin);

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

In [136]:
y_cos = np.cos(x)
plt.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']);

Subplots

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

In [137]:
plt.subplot(2, 1, 1)
plt.plot(x, y_sin)
plt.title('Sine')
plt.subplot(2, 1, 2)
plt.plot(x, y_cos)
plt.title('Cosine')
plt.tight_layout()

What's next

  • Install Python on your own computer (remember to install version 3.x)
  • Retrive this notebook from the class GitHub repository: http://github.com/lmarti/machine-learning.
    • If you are not familiar with the Git version control system you can use GitHub desktop.
  • Try running this notebook.
 
 

Further reading