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numpy
, scipy
and matplotlib
(among many others) it becomes a powerful environment for scientific computing.Python main characteristics:
*.pyc
files)I assume that you have some programming experience:
I also 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 both on the Python programming language and on the use of Python for scientific computing.
Some of you may have previous knowledge in Matlab, in which case we also recommend the numpy for Matlab users page (https://docs.scipy.org/doc/numpy-dev/user/numpy-for-matlab-users.html).
numpy
: arrays, array indexing, datatypes, array math and broadcasting.matplotlib
: plotting, subplots, images.Jupyter
and IPython
: Creating notebooks and typical workflows.Required Python packages (libraries) to run this notebook:
Python
Jupyter
/IPython
- interactive Python shellnumpy
- linear algebra librarymatplotlib
- plotting libraryScientific Python Distributions (available for almost every platform):
As an example, here is an implementation of the classic quicksort algorithm in Python:
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)
quicksort([3,6,8,10,1,2,1])
[1, 1, 2, 3, 6, 8, 10]
Note: You can check your Python version at the command line by running python --version
.
Integers and floats work as you would expect from other languages:
x = 3
print(x, type(x))
3 <class 'int'>
print(x + 1) # Addition;
print(x - 1) # Subtraction;
print(x * 2) # Multiplication;
print(x ** 2) # Exponentiation.
4 2 6 9
Autoincrements:
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
)
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
Python implements all of the usual operators for Boolean algebra, but uses English words rather than symbols (&&
, ||
, !
, etc.):
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:
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
True
or False
, and if
statements.day = "Sunday"
if day == 'Sunday':
print('Sleep!!!')
else:
print('Go to work')
Sleep!!!
There is no switch
or case
statements
if day == 'Monday':
print('Week end is over!')
elif day == 'Sunday' or day =='Saturday':
print('Sleep!')
else:
print('Meeeh')
Week end is over!
var_1 = 234
if var_1:
print('Do something with', var_1)
else:
print('Nothing to do')
Do something with 234
1 == 2
False
50 == 2*25
True
There is another boolean operator is
, that tests whether two objects are the same type:
1 is 1
True
But not...
1 is int
False
print(type(1), type(int))
<class 'int'> <class 'type'>
1 is 1.0
False
String literals can use single quotes (''
) or or double quotes(""
); it does not matter.
hello = 'hello'
world = "world"
print(hello, len(hello))
hello 5
hw = hello + ' ' + world # String concatenation
print(hw)
hello world
String formatting
hw12 = '%s %s! your number is: %d' % (hello, world, 12) # sprintf style string formatting
print(hw12)
hello world! your number is: 12
Note: Checkout https://docs.python.org/3/library/string.html#formatspec for string formatting specs.
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"
Hello HELLO hello hello he(ell)(ell)o world
Again, you can find a list of all string methods in the documentation.
Python includes several built-in container types: lists, dictionaries, sets, and tuples.
A list is the Python equivalent of an array, but is resizeable and can contain elements of different types:
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
xs[2] = 'foo' # Lists can contain elements of different types
print(xs)
[3, 1, 'foo']
xs.append('bar') # Add a new element to the end of the list
print(xs)
[3, 1, 'foo', 'bar']
xs = xs + ['thing1', 'thing2'] # Adding lists (the += op works too)
print(xs)
[3, 1, 'foo', 'bar', 'thing1', 'thing2']
x = xs.pop() # Remove and return the last element of the list
print(x, xs)
thing2 [3, 1, 'foo', 'bar', 'thing1']
As usual, you can find all the gory details about lists in the documentation.
In addition to accessing list elements one at a time, Python provides concise syntax to access sublists; this is known as slicing:
nums = list(range(5)) # range is a built-in function (more on this later)
print(nums)
[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 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]
You can loop over the elements of a list like this:
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:
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
In Python we have only a while
loop.
i = 0
while i < 3:
print(i)
i += 1
0 1 2
When programming, frequently we want to transform one type of data into another.
For 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)
[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)
[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)
[0, 4, 16]
A dictionary stores (key, value) pairs, similar to a Map
in Java or an object in Javascript. You can use it like this:
d = {'cat': 'cute', 'dog': 'furry'} # Create a new dictionary with some data
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
d['fish'] = 'wet' # Set an entry in a dictionary
print(d['fish']) # Prints "wet"
wet
Handling not found keys:
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'
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
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:
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
These are similar to list comprehensions, but allow you to easily construct dictionaries. 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)
{0: 0, 2: 4, 4: 16}
A set is an unordered collection of distinct elements. As a simple example, consider the following:
animals = {'cat', 'dog'}
print(animals)
{'dog', 'cat'}
print('cat' in animals) # Check if an element is in a set; prints "True"
print('fish' in animals) # prints "False"
True False
animals.add('fish') # Add an element to a set
print('fish' in animals)
print(len(animals)) # Number of elements in a set;
True 3
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
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
Like lists and dictionaries, we can easily construct sets using set comprehensions:
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]
Here is a trivial example:
t = (5, 6)
print(t, type(t))
(5, 6) <class 'tuple'> 5 6
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}
print(d[t])
print(d[(1, 2)])
5 1
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
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))
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')
hello('Fred', loud=True)
Hello, Bob! HELLO, FRED
func = lambda x: x**x + 2
print(type(func))
<class 'function'>
func(3)
29
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]
[3, 3127, 29, 6, 823545, 16777218, 29, 46658]
self
as the first parameter of the class methods.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)
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!
For more information check the Python docs
As in many other modern languages Python can handle exceptions.
def my_func(a):
return a * a
print(my_func(2))
4
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'
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'
import
statement¶import
statement in action.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 whole modules with an additional short name:
import scipy.spatial.distance
scipy.spatial.distance.euclidean((1,1), (2,2))
1.4142135623730951
import scipy.spatial.distance as dists
dists.euclidean((1,1), (2,2))
1.4142135623730951
For example, there is a math
module containing many useful functions. To access, say, the square root function, you can
from math import sqrt
and then
sqrt(81)
9.0
numpy
: scientific computing with Python¶array
object, and tools for working with these arrays.To use numpy
, we first need to import numpy
package:
import numpy as np
A numpy array
is a grid of values, all of the same type, and is indexed by a tuple of nonnegative integers.
scipy
.We can initialize numpy arrays from nested Python lists, and access elements using square brackets:
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]
b = np.array([[1, 2, 3], [4, 5, 6]]) # Create a rank 2 array
print(b)
[[1 2 3] [4 5 6]]
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:
a = np.zeros((2,2)) # Create an array of all zeros
print(a)
[[ 0. 0.] [ 0. 0.]]
b = np.ones((1,2)) # Create an array of all ones
print(b)
[[ 1. 1.]]
c = np.full((2,2), 7)
print(c)
[[7 7] [7 7]]
d = np.eye(2) # Create a 2x2 identity matrix
print(d)
[[ 1. 0.] [ 0. 1.]]
e = np.random.random((2,2)) # Create an array filled with random values
print(e)
[[ 0.09845823 0.09568912] [ 0.18341987 0.72051223]]
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:
# 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
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)
:
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.
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.
# 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:
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:
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
Here is an example:
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,)
.
a[[0, 1, 2], [0, 1, 0]]
array([1, 4, 5])
The above example of integer array indexing is equivalent to this:
np.array([a[0, 0], a[1, 1], a[2, 0]])
array([1, 4, 5])
When using integer array indexing, you can reuse the same element from the source array:
a[[0, 0], [1, 1]]
array([2, 2])
Equivalent to the previous integer array indexing example
np.array([a[0, 1], a[0, 1]])
array([2, 2])
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
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
b = np.array([0, 2, 0, 1])
Select one element from each row of a using the indices in b
.
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
.
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:
a = np.array([[1,2], [3, 4], [5, 6]])
Find the elements of a that are bigger than 2:
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
.
print(a[bool_idx])
[3 4 5 6]
We can do all of the above in a single concise statement:
print(a[a > 2])
[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 should read the documentation.
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:
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.
Basic mathematical functions operate elementwise on arrays, and are available both as operator overloads and as functions in the numpy module:
x = np.array([[1,2],[3,4]], dtype=np.float64)
y = np.array([[5,6],[7,8]], dtype=np.float64)
print(x + y)
[[ 6. 8.] [ 10. 12.]]
print(np.add(x, y))
[[ 6. 8.] [ 10. 12.]]
Elementwise difference
print(x - y)
print(np.subtract(x, y))
[[-4. -4.] [-4. -4.]] [[-4. -4.] [-4. -4.]]
Elementwise product
print(x * y)
print(np.multiply(x, y))
[[ 5. 12.] [ 21. 32.]] [[ 5. 12.] [ 21. 32.]]
Elementwise division - both produce the same result.
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;
print(np.sqrt(x))
[[ 1. 1.41421356] [ 1.73205081 2. ]]
*
is elementwise multiplication, not matrix multiplication.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.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
# Matrix / vector product; both produce the rank 1 array [29 67]
print(x.dot(v))
print(np.dot(x, v))
[29 67] [29 67]
# 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
:
x = np.array([[1,2],[3,4]])
print(x)
[[1 2] [3 4]]
np.sum(x) # Compute sum of all elements; prints "10"
10
np.sum(x, axis=0) # Compute sum of each column; prints "[4 6]"
array([4, 6])
np.sum(x, axis=1) # Compute sum of each row; prints "[3 7]"
array([3, 7])
You can find the full list of mathematical functions provided by numpy in the documentation.
Apart from computing mathematical functions using arrays, we frequently need to reshape or otherwise manipulate data in arrays. The simplest example of this type of operation is transposing a matrix; to transpose a matrix, simply use the T attribute of an array object:
print(x)
print(x.T)
[[1 2] [3 4]] [[1 3] [2 4]]
v = np.array([1,2,3])
print(v)
print(v.T)
[1 2 3] [1 2 3]
Is numpy
actually faster than 'plain' Python?
import random
import numpy as np
%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)
%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 is a powerful mechanism that allows numpy to work with arrays of different shapes when performing arithmetic operations.
For example, suppose that we want to add a constant vector to each row of a matrix. We could do it like this:
x = np.array([[1,2,3], [4,5,6], [7,8,9], [10, 11, 12]])
x
array([[ 1, 2, 3], [ 4, 5, 6], [ 7, 8, 9], [10, 11, 12]])
v = np.array([1, 0, 1])
v
array([1, 0, 1])
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
y
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:
vv = np.tile(v, (4, 1)) # Stack 4 copies of v on top of each other
vv
array([[1, 0, 1], [1, 0, 1], [1, 0, 1], [1, 0, 1]])
y = x + vv # Add x and vv elementwise
y
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:
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
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:
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.
Compute outer product of vectors
v = np.array([1,2,3]) # v has shape (3,)
w = np.array([4,5]) # w has shape (2,)
To compute an outer product, we
v
to be a column vector of shape (3, 1)
, andw
to yield an output of shape (3, 2)
, which is the outer product of v
and w
:print(np.reshape(v, (3, 1)) * w)
[[ 4 5] [ 8 10] [12 15]]
Add a vector to each row of a matrix
x = np.array([[1,2,3], [4,5,6]])
x
has shape (2, 3)
and v
has shape (3,)
so they broadcast to (2, 3)
, producing the following matrix:
print(x + v)
[[2 4 6] [5 7 9]]
Add a vector to each column of a matrix x
has shape (2, 3)
and w
has shape (2,)
.
x
then it has shape (3, 2)
and can be broadcast against w
to yield a result of shape (3, 2)
;(2, 3)
which is the matrix x
with the vector w
added to each column.This gives the following matrix:
print((x.T + w).T)
[[ 5 6 7] [ 9 10 11]]
Another solution is to reshape w
to be a row 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)))
[[ 5 6 7] [ 9 10 11]]
Multiply a matrix by a constant
x
has shape (2, 3)
.numpy
treats scalars as arrays of shape ()
;(2, 3)
,print(x * 2)
[[ 2 4 6] [ 8 10 12]]
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.
numpy
.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.
%matplotlib inline
%config InlineBackend.figure_format = 'retina'
The most important function in matplotlib
is plot
, which allows you to plot 2D data.
Here is a simple example:
# 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:
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']);
You can plot different things in the same figure using the subplot function. Here is an example:
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()
Check out Nicolas P. Rougier's Matplotlib tutorial.
CS231n
Python tutorial by Justin Johnson (http://cs231n.github.io/python-numpy-tutorial/).%load_ext version_information
%version_information scipy, numpy, matplotlib
Software | Version |
---|---|
Python | 3.6.1 64bit [GCC 4.2.1 Compatible Apple LLVM 6.0 (clang-600.0.57)] |
IPython | 6.0.0 |
OS | Darwin 16.5.0 x86_64 i386 64bit |
scipy | 0.19.0 |
numpy | 1.12.1 |
matplotlib | 2.0.0 |
Thu Apr 27 23:03:44 2017 -03 |
# this code is here for cosmetic reasons
from IPython.core.display import HTML
from urllib.request import urlopen
HTML(urlopen('https://raw.githubusercontent.com/lmarti/jupyter_custom/master/custom.include').read().decode('utf-8'))