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===== A bit of Python =====
Python is available on any Linux/Unix machine including department machines and Macs.
You can download and install python on your own computer by following
instructions at [[http://www.python.org]]
You can use the Python interpreter interactively by typing //python// at a terminal window.
Ipython is a nicer front end to python that is invoked with
ipython
To quit, type control-d
To run python code in a file //code.py//, either type
run code.py
in //ipython//, or type
python code.py
at the unix command line.
When in //ipython//, you may type python statements or expressions
that are evaluated, or //ipython// commands. See the
[[http://showmedo.com/videotutorials/video?name=1000010&fromSeriesID=100|Video
tutorial on using ipython]], in five parts by Jeff Rush, for help
getting started with //ipython//.
Documentation is immediately available for many things. For example:
> ipython
asa:~$ ipython
Python 2.7.3 (v2.7.3:70274d53c1dd, Apr 9 2012, 20:52:43)
Type "copyright", "credits" or "license" for more information.
IPython 0.13.2 -- An enhanced Interactive Python.
? -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help -> Python's own help system.
object? -> Details about 'object', use 'object??' for extra details.
In [1]: list?
Type: type
Base Class:
String Form:
Namespace: Python builtin
Docstring:
list() -> new list
list(sequence) -> new list initialized from sequence's items
In [2]: help(list)
Help on class list in module __builtin__:
class list(object)
| list() -> new list
| list(sequence) -> new list initialized from sequence's items
|
| Methods defined here:
.
.
.
| append(...)
| L.append(object) -- append object to end
|
.
.
.
|
| sort(...)
| L.sort(cmp=None, key=None, reverse=False) -- stable sort *IN PLACE*;
| cmp(x, y) -> -1, 0, 1
What is the value of $(100\cdot 2 - 12^2) / 7 \cdot 5 + 2\;\;\;$?
In [301]: (100*2 - 12**2) / 7*5 + 2
Out[301]: 42
In order to compute something like $\sin(\pi/2)$ we first need to //import// the //math// module:
In [303]: import math
In [304]: math.sin(math.pi/2)
1.0
How do I find out what other mathematical functions are available?
help("math")
===== Plotting =====
Let's get on to that all important step of visualizing data. We will be using the [[http://matplotlib.org |matplotlib]] Python package for that. Let's start by plotting the function $f(x) = x^2$.
First, let's generate the numbers. Well, there are tons of ways to do so. First, using a''for'' loop.
In [3]: f = []
In [4]: for i in range(10) :
...: f.append(i**2)
...:
In [5]: f
Out[5]: [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
To plot the data, first import the ''pyplot'' module.
In [6]: import matplotlib.pyplot as plt
In [7]: plt.plot(range(10), f)
Out[7]: []
In order to actually see the plot you need to do:
In [8]: plt.show()
As an alternative, you can put matplotlib in interactive mode before plotting using the command ''plt.ion()''.
Python has some nifty syntax for generating lists. Watch this! A [[http://www.secnetix.de/olli/Python/list_comprehensions.hawk|list comprehension]]!!
In [9]: f = [i**2 for i in range(10)]
In [10]: f
Out[10]: [0, 1, 4, 9, 16, 25, 36, 49, 64, 81]
There's an alternative way of doing this using ''numpy'':
In [11]: import numpy as np
In [12]: f = np.arange(10)**2
In [13]: f
Out[13]: array([ 0, 1, 4, 9, 16, 25, 36, 49, 64, 81])
Note that plotting functions to accept either lists or ''numpy'' arrays, so a fast way of doing our plot is
In [14]: plt.plot(np.arange(10), np.arange(10)**2)
For a smoother plot:
In [14]: x = np.arange(10, 0.1)
In [15]: plt.plot(x, x**2, 'ob')
Out[15]: []
/* {{ Notes:plot2.png?400 }}*/
We can add a second plot to the same axes by calling //plot// again:
In [16]: plt.plot(x, x, 'dr')
Out[16]: []
/*{{ Notes:plot3.png?400 }}*/
===== Matrices in Python =====
Can I work with vectors and matrices in python?
Of course! No data analysis tool is worth the bytes it burns if it
doesn't. The ''numpy'' package provides the required magic.
Let's create an array that represents the following matrix:
\[\left ( \begin{array}{cc}
1 & 2\\
3 & 4\\
5 & 6
\end{array} \right ) \]
by doing
In [17]: import numpy as np
In [18]: m = np.array([[1,2], [3,4], [5,6]])
In [19]: m
Out[19]:
array([[1, 2],
[3, 4],
[5, 6]])
Let's construct the matrices
\[a = \left ( \begin{array}{cc}
2 & 2 & 2\\
2 & 2 & 2\\
2 & 2 & 2
\end{array} \right ) \]
and
\[b = \left ( \begin{array}{cc}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9
\end{array} \right ) \]
In [16]: a = np.ones((3,3)) * 2
In [17]: a
Out[17]:
array([[ 2., 2., 2.],
[ 2., 2., 2.],
[ 2., 2., 2.]])
In [18]: b = np.resize(np.arange(9)+1,(3,3))
In [19]: b
Out[19]:
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
What is the value of $a * b$?
In [20]: a * b
Out[21]:
array([[ 2, 4, 6],
[ 8, 10, 12],
[14, 16, 18]])
The ''*'' operator does a component-wise multiplication. Use the
''numpy'' function ''dot'' to do matrix multiplication.
In [22]: np.dot(a,b)
Out[22]:
array([[24, 30, 36],
[24, 30, 36],
[24, 30, 36]])
An array is transposed by
In [23]: b.transpose()
Out[23]:
array([[1, 4, 7],
[2, 5, 8],
[3, 6, 9]])
In [24]: b.T
Out[24]:
array([[1, 4, 7],
[2, 5, 8],
[3, 6, 9]])
Elements and sub-matrices are easily extracted:
In [25]: b
Out[25]:
array([[1, 2, 3],
[4, 5, 6],
[7, 8, 9]])
In [26]: b[0,0]
Out[26]: 1
In [27]: b[0,1]
Out[27]: 2
In [28]: b[0:2, 1:3]
Out[28]:
array([[2, 3],
[5, 6]])
Let's multiply the first row of a $a$ by the second column of $b$.
In [29]: np.dot(a[0], b[:,1])
Out[29]: 30.0
In [30]: np.dot(a[0],b.T[1])
Out[30]: 30.0
How do I find the inverse of a matrix?
In [2]: z = np.array([[2,1,1],[1,2,2],[2,3,4]])
In [3]: z
Out[3]:
array([[2, 1, 1],
[1, 2, 2],
[2, 3, 4]])
In [4]: np.linalg.inv(z)
Out[4]:
array([[ 0.66666667, -0.33333333, 0. ],
[ 0. , 2. , -1. ],
[-0.33333333, -1.33333333, 1. ]])
In [5]: np.dot(z, np.linalg.inv(z))
Out[5]:
array([[ 1., 0., 0.],
[ 0., 1., 0.],
[ 0., 0., 1.]])