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%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
# Following is optional: set plotting styles
import seaborn; seaborn.set()
While the Python language is an excellent tool for general-purpose programming, with a highly readable syntax, rich and powerful data types (strings, lists, sets, dictionaries, arbitrary length integers, etc) and a very comprehensive standard library, it was not designed specifically for mathematical and scientific computing. Neither the language nor its standard library have facilities for the efficient representation of multidimensional datasets, tools for linear algebra and general matrix manipulations (an essential building block of virtually all technical computing), nor any data visualization facilities.
In particular, Python lists are very flexible containers that can be nested arbitrarily deep and which can hold any Python object in them, but they are poorly suited to represent efficiently common mathematical constructs like vectors and matrices. In contrast, much of our modern heritage of scientific computing has been built on top of libraries written in the Fortran language, which has native support for vectors and matrices as well as a library of mathematical functions that can efficiently operate on entire arrays at once.
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L = [1, 2, 3, 4, 5]
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# Zero-based Indexing
print(L[0], L[1])
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# Indexing from the end
print(L[-1], L[-2])
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# Slicing
L[0:3]
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# The 0 can be left-out
L[:3]
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# Slicing by a step size
L[0:5:2]
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# Reversing with a negative step size
L[::-1]
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# Lists of multiple types
L2 = [1, 'two', 3.14]
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# Adding lists together will append them:
L + L2
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import math
# make a large list of theta values
theta = [0.01 * i for i in range(1000000)]
sin_theta = [math.sin(t) for t in theta]
sin_theta[:10]
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%timeit [math.sin(t) for t in theta]
Let's take a look at doing essentially the same operation using NumPy.
By convention, we'll import numpy under the shorthand np:
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import numpy as np
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theta = 0.01 * np.arange(1E6)
sin_theta = np.sin(theta)
sin_theta[:10]
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%timeit np.sin(theta)
NumPy's version of this is nearly 10x faster than the list-based Python version, and it is arguably simpler as well!
There is a lot of info out there about how to use numpy. Here I want to just briefly go over some of the key concepts as we progress to talking about using Python for real-world data.
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# from a list
np.array([1, 2, 3, 4])
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# range of numbers, like Python's range()
np.arange(0, 10, 0.5)
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# range of numbers between two limits
np.linspace(0, 10, 5)
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# array of zeros
np.zeros(10)
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# array of ones
np.ones(10)
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# array of random values
np.random.rand(10)
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# define some arrays
x = np.arange(5)
y = np.random.random(5)
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# addition – add 1 to each
x + 1
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# multiplication – multiply each by 2
y * 2
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# two arrays: everything is element-wise
x / y
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# exponentiation
np.exp(x)
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# trigonometric functions
np.sin(x)
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# combining operations
np.cos(x) + np.sin(2 * np.pi * (x - y))
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x
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x[0], x[1]
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x[:3]
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x[::2]
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x[::-1]
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Unlike lists, NumPy arrays can have multiple dimensions, and the indexing and slicing works efficiently!
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M = np.arange(20).reshape(4, 5)
M
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M[1, 2]
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M[:2, :2]
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M[:, 1:3]
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M
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M < 8
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Boolean mask arrays can be used to select portions of a larger array, and operate on them
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M[M < 8] = 0
M
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M[M == 12] *= 2
M
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M[M % 2 == 0] = 999
M
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As I mentioned, there is much more to numpy arrays, but this has covered the basic pieces needed here!
For data-intensive work in Python the Pandas library has become essential. Pandas originally meant Panel Data, though many users probably don't know that.
Pandas can be thought of as NumPy with built-in labels for rows and columns, but it's also much, much more than that.
Pandas does this through two fundamental object types, both built upon NumPy arrays: the Series object, and the DataFrame object.
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s = pd.Series([0.1, 0.2, 0.3, 0.4])
The series has a built-in concept of an index, which by default is the numbers 0 through N - 1
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s.index
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We can access series values via the index, just like for NumPy arrays:
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s[0]
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Unlike the NumPy array, though, this index can be something other than integers:
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s2 = pd.Series(np.arange(4), index=['a', 'b', 'c', 'd'])
s2
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s2['c']
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In this way, a Series object can be thought of as similar to an ordered dictionary mapping one typed value to another typed value.
In fact, it's possible to construct a series directly from a Python dictionary:
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pop_dict = {'California': 38332521,
'Texas': 26448193,
'New York': 19651127,
'Florida': 19552860,
'Illinois': 12882135}
populations = pd.Series(pop_dict)
populations
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Note that because Python dictionaries are an unordered object, the order of the resulting series will not match the order of the dictionary definition.
We can index or slice the populations as expected:
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populations['California']
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populations['California':'Illinois']
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A dataframe, essentially, is a multi-dimensional object to hold labeled data. You can think of it as multiple Series object which share the same index.
One of the most common ways of creating a dataframe is from a dictionary of arrays or lists. Note that in the IPython notebook with the correct settings, the dataframe will display in a rich HTML view:
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data = {'state': ['California', 'Texas', 'New York', 'Florida', 'Illinois'],
'population': [38332521, 26448193, 19651127, 19552860, 12882135],
'area':[423967, 695662, 141297, 170312, 149995]}
states = pd.DataFrame(data)
states
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If we don't like what the index looks like, we can reset it:
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states = states.set_index('state')
states
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To access a Series representing a column in the data, use typical slicing syntax:
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states['area']
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To access a row, you need to use a special row-location operator:
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states.loc['California']
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As you play around with DataFrames, you'll notice that many operations which work on NumPy arrays will also work on dataframes.
For example there's arithmetic. Let's compute the area in square miles and add a column to the data
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states['density'] = states['population'] / states['area']
states
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We can even use masking the way we did in NumPy:
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states[states['density'] > 100]
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And we can do things like sorting the items in the array, and indexing to take the first two rows:
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states.sort_index(by='density', ascending=False)[:3]
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One useful method to use is the describe method, which computes summary statistics for each column:
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states.describe()
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There are many, many more interesting operations that can be done on Series and DataFrame objects, but rather than continue using this toy data, we'll instead move to a real-world example, and illustrate some of the advanced concepts along the way.
This example is drawn from Wes McKinney's excellent book on the Pandas library, O'Reilly's Python for Data Analysis.
We'll be taking a look at a freely available dataset: the database of names given to babies in the United States over the last century.
First things first, we need to download the data, which can be found at http://www.ssa.gov/oact/babynames/limits.html. If you uncomment the following commands, it will do this automatically (note that these are linux shell commands; they will not work on Windows):
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# !curl -O http://www.ssa.gov/oact/babynames/names.zip
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# !mkdir -p data/names
# !mv names.zip data/names/
# !cd data/names/ && unzip names.zip
Now we should have a data/names directory which contains a number of text files, one for each year of data:
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!ls data/names
Let's take a quick look at one of these files:
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!head data/names/yob1880.txt
Each file is just a comma-separated list of names, genders, and counts of babies with that name in each year.
We can load these files using pd.read_csv, which is specifically designed for this:
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names1880 = pd.read_csv('data/names/yob1880.txt')
names1880.head()
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Oops! Something went wrong. Our algorithm tried to be smart, and use the first line as index labels. Let's fix this by specifying the index names manually:
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names1880 = pd.read_csv('data/names/yob1880.txt',
names=['name', 'gender', 'births'])
names1880.head()
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That looks better. Now we can start playing with the data a bit.
First let's think about how we might count the total number of females and males born in the US in 1880.
If you're used to NumPy, you might be tempted to use masking like this:
First, we can get a mask over all females & males, and then use it to select a subset of the data:
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males = names1880[names1880.gender == 'M']
females = names1880[names1880.gender == 'F']
Now we can take the sum of the births for each of these:
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males.births.sum(), females.births.sum()
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But there's an easier way to do this, using one of Pandas' very powerful features: groupby:
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grouped = names1880.groupby('gender')
grouped
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This grouped object is now an abstract representation of the data, where the data is split on the given column. In order to actually do something with this data, we need to specify an aggregation operation to do across the data. In this case, what we want is the sum:
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grouped.sum()
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We can do other aggregations as well:
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grouped.size()
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grouped.mean()
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Or, if we wish, we can get a description of the grouping:
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grouped.describe()
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But here we've just been looking at a single year. Let's try to put together all the data in all the years.
To do this, we'll have to use pandas concat function to concatenate all the data together.
First we'll create a function which loads the data as we did the above data:
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def load_year(year):
data = pd.read_csv('data/names/yob{0}.txt'.format(year),
names=['name', 'gender', 'births'])
data['year'] = year
return data
Now let's load all the data into a list, and call pd.concat on that list:
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names = pd.concat([load_year(year) for year in range(1880, 2014)])
names.head()
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It looks like we've done it!
Let's start with something easy: we'll use groupby again to see the total number of births per year:
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births = names.groupby('year').births.sum()
births.head()
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We can use the plot() method to see a quick plot of these (note that because we used the %matplotlib inline magic at the start of the notebook, the resulting plot will be shown inline within the notebook).
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births.plot();
The so-called "baby boom" generation after the second world war is abundantly clear!
We can also use other aggregates: let's see how many names are used each year:
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names.groupby('year').births.count().plot();
Apparently there's been a huge increase of the diversity of names with time!
groupby can also be used to add columns to the data: think of it as a view of the data that you're modifying. Let's add a column giving the frequency of each name within each year & gender:
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def add_frequency(group):
group['birth_freq'] = group.births / group.births.sum()
return group
names = names.groupby(['year', 'gender']).apply(add_frequency)
names.head()
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Notice that the apply() function iterates over each group, and calls a function which modifies the group.
This result is then re-constructed into a container which looks ike the original dataframe.
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men = names[names.gender == 'M']
women = names[names.gender == 'W']
And then we could proceed as above, using groupby to group on the year.
But we would end up with two different views of the data. A better way to do this is to use a pivot_table, which is essentially a groupby in multiple dimensions at once:
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births = names.pivot_table('births',
index='year', columns='gender',
aggfunc=sum)
births.head()
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Note that this has grouped the index by the value of year, and grouped the columns by the value of gender.
Let's plot the results now:
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births.plot(title='Total Births');
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names_to_check = ['Allison', 'Alison']
# filter on just the names we're interested in
births = names[names.name.isin(names_to_check)]
# pivot table to get year vs. gender
births = births.pivot_table('births', index='year', columns='gender')
# fill all NaNs with zeros
births = births.fillna(0)
# normalize along columns
births = births.div(births.sum(1), axis=0)
births.plot(title='Fraction of babies named Allison');
We can see that prior to about 1905, all babies named Allison were male. Over the 20th century, this reversed, until the end of the century nearly all Allisons were female!
There's some noise in this data: we can smooth it out a bit by using a 5-year rolling mean:
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pd.rolling_mean(births, 5).plot(title="Allisons: 5-year moving average");
This gives a smoother picture of the transition, and is an example of the bias/variance tradeoff that we'll often see in modeling: a smoother model has less variance (variation due to sampling or other noise) but at the expense of more bias (the model systematically mis-represents the data slightly).
We'll discuss this type of tradeoff more in coming sessions.
We've just scratched the surface of what can be done with Pandas, but we'll get a chance to play with this more in the breakout session coming up.
For more information on using Pandas, check out the pandas documentation or the book Python for Data Analysis by Pandas creator Wes McKinney.