This notebook contains an excerpt from the Python Data Science Handbook by Jake VanderPlas; the content is available on GitHub.

The text is released under the CC-BY-NC-ND license, and code is released under the MIT license. If you find this content useful, please consider supporting the work by buying the book!

Hierarchical Indexing

Up to this point we've been focused primarily on one-dimensional and two-dimensional data, stored in Pandas Series and DataFrame objects, respectively. Often it is useful to go beyond this and store higher-dimensional data–that is, data indexed by more than one or two keys. While Pandas does provide Panel and Panel4D objects that natively handle three-dimensional and four-dimensional data (see Aside: Panel Data), a far more common pattern in practice is to make use of hierarchical indexing (also known as multi-indexing) to incorporate multiple index levels within a single index. In this way, higher-dimensional data can be compactly represented within the familiar one-dimensional Series and two-dimensional DataFrame objects.

In this section, we'll explore the direct creation of MultiIndex objects, considerations when indexing, slicing, and computing statistics across multiply indexed data, and useful routines for converting between simple and hierarchically indexed representations of your data.

We begin with the standard imports:

In [1]:
import pandas as pd
import numpy as np

A Multiply Indexed Series

Let's start by considering how we might represent two-dimensional data within a one-dimensional Series. For concreteness, we will consider a series of data where each point has a character and numerical key.

The bad way

Suppose you would like to track data about states from two different years. Using the Pandas tools we've already covered, you might be tempted to simply use Python tuples as keys:

In [2]:
index = [('California', 2000), ('California', 2010),
         ('New York', 2000), ('New York', 2010),
         ('Texas', 2000), ('Texas', 2010)]
populations = [33871648, 37253956,
               18976457, 19378102,
               20851820, 25145561]
pop = pd.Series(populations, index=index)
pop
Out[2]:
(California, 2000)    33871648
(California, 2010)    37253956
(New York, 2000)      18976457
(New York, 2010)      19378102
(Texas, 2000)         20851820
(Texas, 2010)         25145561
dtype: int64

With this indexing scheme, you can straightforwardly index or slice the series based on this multiple index:

In [3]:
pop[('California', 2010):('Texas', 2000)]
Out[3]:
(California, 2010)    37253956
(New York, 2000)      18976457
(New York, 2010)      19378102
(Texas, 2000)         20851820
dtype: int64

But the convenience ends there. For example, if you need to select all values from 2010, you'll need to do some messy (and potentially slow) munging to make it happen:

In [4]:
pop[[i for i in pop.index if i[1] == 2010]]
Out[4]:
(California, 2010)    37253956
(New York, 2010)      19378102
(Texas, 2010)         25145561
dtype: int64

This produces the desired result, but is not as clean (or as efficient for large datasets) as the slicing syntax we've grown to love in Pandas.

The Better Way: Pandas MultiIndex

Fortunately, Pandas provides a better way. Our tuple-based indexing is essentially a rudimentary multi-index, and the Pandas MultiIndex type gives us the type of operations we wish to have. We can create a multi-index from the tuples as follows:

In [5]:
index = pd.MultiIndex.from_tuples(index)
index
Out[5]:
MultiIndex(levels=[['California', 'New York', 'Texas'], [2000, 2010]],
           labels=[[0, 0, 1, 1, 2, 2], [0, 1, 0, 1, 0, 1]])

Notice that the MultiIndex contains multiple levels of indexing–in this case, the state names and the years, as well as multiple labels for each data point which encode these levels.

If we re-index our series with this MultiIndex, we see the hierarchical representation of the data:

In [6]:
pop = pop.reindex(index)
pop
Out[6]:
California  2000    33871648
            2010    37253956
New York    2000    18976457
            2010    19378102
Texas       2000    20851820
            2010    25145561
dtype: int64

Here the first two columns of the Series representation show the multiple index values, while the third column shows the data. Notice that some entries are missing in the first column: in this multi-index representation, any blank entry indicates the same value as the line above it.

Now to access all data for which the second index is 2010, we can simply use the Pandas slicing notation:

In [7]:
pop[:, 2010]
Out[7]:
California    37253956
New York      19378102
Texas         25145561
dtype: int64

The result is a singly indexed array with just the keys we're interested in. This syntax is much more convenient (and the operation is much more efficient!) than the home-spun tuple-based multi-indexing solution that we started with. We'll now further discuss this sort of indexing operation on hieararchically indexed data.

MultiIndex as extra dimension

You might notice something else here: we could easily have stored the same data using a simple DataFrame with index and column labels. In fact, Pandas is built with this equivalence in mind. The unstack() method will quickly convert a multiply indexed Series into a conventionally indexed DataFrame:

In [8]:
pop_df = pop.unstack()
pop_df
Out[8]:
2000 2010
California 33871648 37253956
New York 18976457 19378102
Texas 20851820 25145561

Naturally, the stack() method provides the opposite operation:

In [9]:
pop_df.stack()
Out[9]:
California  2000    33871648
            2010    37253956
New York    2000    18976457
            2010    19378102
Texas       2000    20851820
            2010    25145561
dtype: int64

Seeing this, you might wonder why would we would bother with hierarchical indexing at all. The reason is simple: just as we were able to use multi-indexing to represent two-dimensional data within a one-dimensional Series, we can also use it to represent data of three or more dimensions in a Series or DataFrame. Each extra level in a multi-index represents an extra dimension of data; taking advantage of this property gives us much more flexibility in the types of data we can represent. Concretely, we might want to add another column of demographic data for each state at each year (say, population under 18) ; with a MultiIndex this is as easy as adding another column to the DataFrame:

In [10]:
pop_df = pd.DataFrame({'total': pop,
                       'under18': [9267089, 9284094,
                                   4687374, 4318033,
                                   5906301, 6879014]})
pop_df
Out[10]:
total under18
California 2000 33871648 9267089
2010 37253956 9284094
New York 2000 18976457 4687374
2010 19378102 4318033
Texas 2000 20851820 5906301
2010 25145561 6879014

In addition, all the ufuncs and other functionality discussed in Operating on Data in Pandas work with hierarchical indices as well. Here we compute the fraction of people under 18 by year, given the above data:

In [11]:
f_u18 = pop_df['under18'] / pop_df['total']
f_u18.unstack()
Out[11]:
2000 2010
California 0.273594 0.249211
New York 0.247010 0.222831
Texas 0.283251 0.273568

This allows us to easily and quickly manipulate and explore even high-dimensional data.

Methods of MultiIndex Creation

The most straightforward way to construct a multiply indexed Series or DataFrame is to simply pass a list of two or more index arrays to the constructor. For example:

In [12]:
df = pd.DataFrame(np.random.rand(4, 2),
                  index=[['a', 'a', 'b', 'b'], [1, 2, 1, 2]],
                  columns=['data1', 'data2'])
df
Out[12]:
data1 data2
a 1 0.554233 0.356072
2 0.925244 0.219474
b 1 0.441759 0.610054
2 0.171495 0.886688

The work of creating the MultiIndex is done in the background.

Similarly, if you pass a dictionary with appropriate tuples as keys, Pandas will automatically recognize this and use a MultiIndex by default:

In [13]:
data = {('California', 2000): 33871648,
        ('California', 2010): 37253956,
        ('Texas', 2000): 20851820,
        ('Texas', 2010): 25145561,
        ('New York', 2000): 18976457,
        ('New York', 2010): 19378102}
pd.Series(data)
Out[13]:
California  2000    33871648
            2010    37253956
New York    2000    18976457
            2010    19378102
Texas       2000    20851820
            2010    25145561
dtype: int64

Nevertheless, it is sometimes useful to explicitly create a MultiIndex; we'll see a couple of these methods here.

Explicit MultiIndex constructors

For more flexibility in how the index is constructed, you can instead use the class method constructors available in the pd.MultiIndex. For example, as we did before, you can construct the MultiIndex from a simple list of arrays giving the index values within each level:

In [14]:
pd.MultiIndex.from_arrays([['a', 'a', 'b', 'b'], [1, 2, 1, 2]])
Out[14]:
MultiIndex(levels=[['a', 'b'], [1, 2]],
           labels=[[0, 0, 1, 1], [0, 1, 0, 1]])

You can construct it from a list of tuples giving the multiple index values of each point:

In [15]:
pd.MultiIndex.from_tuples([('a', 1), ('a', 2), ('b', 1), ('b', 2)])
Out[15]:
MultiIndex(levels=[['a', 'b'], [1, 2]],
           labels=[[0, 0, 1, 1], [0, 1, 0, 1]])

You can even construct it from a Cartesian product of single indices:

In [16]:
pd.MultiIndex.from_product([['a', 'b'], [1, 2]])
Out[16]:
MultiIndex(levels=[['a', 'b'], [1, 2]],
           labels=[[0, 0, 1, 1], [0, 1, 0, 1]])

Similarly, you can construct the MultiIndex directly using its internal encoding by passing levels (a list of lists containing available index values for each level) and labels (a list of lists that reference these labels):

In [17]:
pd.MultiIndex(levels=[['a', 'b'], [1, 2]],
              labels=[[0, 0, 1, 1], [0, 1, 0, 1]])
Out[17]:
MultiIndex(levels=[['a', 'b'], [1, 2]],
           labels=[[0, 0, 1, 1], [0, 1, 0, 1]])

Any of these objects can be passed as the index argument when creating a Series or Dataframe, or be passed to the reindex method of an existing Series or DataFrame.

MultiIndex level names

Sometimes it is convenient to name the levels of the MultiIndex. This can be accomplished by passing the names argument to any of the above MultiIndex constructors, or by setting the names attribute of the index after the fact:

In [18]:
pop.index.names = ['state', 'year']
pop
Out[18]:
state       year
California  2000    33871648
            2010    37253956
New York    2000    18976457
            2010    19378102
Texas       2000    20851820
            2010    25145561
dtype: int64

With more involved datasets, this can be a useful way to keep track of the meaning of various index values.

MultiIndex for columns

In a DataFrame, the rows and columns are completely symmetric, and just as the rows can have multiple levels of indices, the columns can have multiple levels as well. Consider the following, which is a mock-up of some (somewhat realistic) medical data:

In [19]:
# hierarchical indices and columns
index = pd.MultiIndex.from_product([[2013, 2014], [1, 2]],
                                   names=['year', 'visit'])
columns = pd.MultiIndex.from_product([['Bob', 'Guido', 'Sue'], ['HR', 'Temp']],
                                     names=['subject', 'type'])

# mock some data
data = np.round(np.random.randn(4, 6), 1)
data[:, ::2] *= 10
data += 37

# create the DataFrame
health_data = pd.DataFrame(data, index=index, columns=columns)
health_data
Out[19]:
subject Bob Guido Sue
type HR Temp HR Temp HR Temp
year visit
2013 1 31.0 38.7 32.0 36.7 35.0 37.2
2 44.0 37.7 50.0 35.0 29.0 36.7
2014 1 30.0 37.4 39.0 37.8 61.0 36.9
2 47.0 37.8 48.0 37.3 51.0 36.5

Here we see where the multi-indexing for both rows and columns can come in very handy. This is fundamentally four-dimensional data, where the dimensions are the subject, the measurement type, the year, and the visit number. With this in place we can, for example, index the top-level column by the person's name and get a full DataFrame containing just that person's information:

In [20]:
health_data['Guido']
Out[20]:
type HR Temp
year visit
2013 1 32.0 36.7
2 50.0 35.0
2014 1 39.0 37.8
2 48.0 37.3

For complicated records containing multiple labeled measurements across multiple times for many subjects (people, countries, cities, etc.) use of hierarchical rows and columns can be extremely convenient!

Indexing and Slicing a MultiIndex

Indexing and slicing on a MultiIndex is designed to be intuitive, and it helps if you think about the indices as added dimensions. We'll first look at indexing multiply indexed Series, and then multiply-indexed DataFrames.

Multiply indexed Series

Consider the multiply indexed Series of state populations we saw earlier:

In [21]:
pop
Out[21]:
state       year
California  2000    33871648
            2010    37253956
New York    2000    18976457
            2010    19378102
Texas       2000    20851820
            2010    25145561
dtype: int64

We can access single elements by indexing with multiple terms:

In [22]:
pop['California', 2000]
Out[22]:
33871648

The MultiIndex also supports partial indexing, or indexing just one of the levels in the index. The result is another Series, with the lower-level indices maintained:

In [23]:
pop['California']
Out[23]:
year
2000    33871648
2010    37253956
dtype: int64

Partial slicing is available as well, as long as the MultiIndex is sorted (see discussion in Sorted and Unsorted Indices):

In [24]:
pop.loc['California':'New York']
Out[24]:
state       year
California  2000    33871648
            2010    37253956
New York    2000    18976457
            2010    19378102
dtype: int64

With sorted indices, partial indexing can be performed on lower levels by passing an empty slice in the first index:

In [25]:
pop[:, 2000]
Out[25]:
state
California    33871648
New York      18976457
Texas         20851820
dtype: int64

Other types of indexing and selection (discussed in Data Indexing and Selection) work as well; for example, selection based on Boolean masks:

In [26]:
pop[pop > 22000000]
Out[26]:
state       year
California  2000    33871648
            2010    37253956
Texas       2010    25145561
dtype: int64

Selection based on fancy indexing also works:

In [27]:
pop[['California', 'Texas']]
Out[27]:
state       year
California  2000    33871648
            2010    37253956
Texas       2000    20851820
            2010    25145561
dtype: int64

Multiply indexed DataFrames

A multiply indexed DataFrame behaves in a similar manner. Consider our toy medical DataFrame from before:

In [28]:
health_data
Out[28]:
subject Bob Guido Sue
type HR Temp HR Temp HR Temp
year visit
2013 1 31.0 38.7 32.0 36.7 35.0 37.2
2 44.0 37.7 50.0 35.0 29.0 36.7
2014 1 30.0 37.4 39.0 37.8 61.0 36.9
2 47.0 37.8 48.0 37.3 51.0 36.5

Remember that columns are primary in a DataFrame, and the syntax used for multiply indexed Series applies to the columns. For example, we can recover Guido's heart rate data with a simple operation:

In [29]:
health_data['Guido', 'HR']
Out[29]:
year  visit
2013  1        32.0
      2        50.0
2014  1        39.0
      2        48.0
Name: (Guido, HR), dtype: float64

Also, as with the single-index case, we can use the loc, iloc, and ix indexers introduced in Data Indexing and Selection. For example:

In [30]:
health_data.iloc[:2, :2]
Out[30]:
subject Bob
type HR Temp
year visit
2013 1 31.0 38.7
2 44.0 37.7

These indexers provide an array-like view of the underlying two-dimensional data, but each individual index in loc or iloc can be passed a tuple of multiple indices. For example:

In [31]:
health_data.loc[:, ('Bob', 'HR')]
Out[31]:
year  visit
2013  1        31.0
      2        44.0
2014  1        30.0
      2        47.0
Name: (Bob, HR), dtype: float64

Working with slices within these index tuples is not especially convenient; trying to create a slice within a tuple will lead to a syntax error:

In [32]:
health_data.loc[(:, 1), (:, 'HR')]
  File "<ipython-input-32-8e3cc151e316>", line 1
    health_data.loc[(:, 1), (:, 'HR')]
                     ^
SyntaxError: invalid syntax

You could get around this by building the desired slice explicitly using Python's built-in slice() function, but a better way in this context is to use an IndexSlice object, which Pandas provides for precisely this situation. For example:

In [33]:
idx = pd.IndexSlice
health_data.loc[idx[:, 1], idx[:, 'HR']]
Out[33]:
subject Bob Guido Sue
type HR HR HR
year visit
2013 1 31.0 32.0 35.0
2014 1 30.0 39.0 61.0

There are so many ways to interact with data in multiply indexed Series and DataFrames, and as with many tools in this book the best way to become familiar with them is to try them out!

Rearranging Multi-Indices

One of the keys to working with multiply indexed data is knowing how to effectively transform the data. There are a number of operations that will preserve all the information in the dataset, but rearrange it for the purposes of various computations. We saw a brief example of this in the stack() and unstack() methods, but there are many more ways to finely control the rearrangement of data between hierarchical indices and columns, and we'll explore them here.

Sorted and unsorted indices

Earlier, we briefly mentioned a caveat, but we should emphasize it more here. Many of the MultiIndex slicing operations will fail if the index is not sorted. Let's take a look at this here.

We'll start by creating some simple multiply indexed data where the indices are not lexographically sorted:

In [34]:
index = pd.MultiIndex.from_product([['a', 'c', 'b'], [1, 2]])
data = pd.Series(np.random.rand(6), index=index)
data.index.names = ['char', 'int']
data
Out[34]:
char  int
a     1      0.003001
      2      0.164974
c     1      0.741650
      2      0.569264
b     1      0.001693
      2      0.526226
dtype: float64

If we try to take a partial slice of this index, it will result in an error:

In [35]:
try:
    data['a':'b']
except KeyError as e:
    print(type(e))
    print(e)
<class 'KeyError'>
'Key length (1) was greater than MultiIndex lexsort depth (0)'

Although it is not entirely clear from the error message, this is the result of the MultiIndex not being sorted. For various reasons, partial slices and other similar operations require the levels in the MultiIndex to be in sorted (i.e., lexographical) order. Pandas provides a number of convenience routines to perform this type of sorting; examples are the sort_index() and sortlevel() methods of the DataFrame. We'll use the simplest, sort_index(), here:

In [36]:
data = data.sort_index()
data
Out[36]:
char  int
a     1      0.003001
      2      0.164974
b     1      0.001693
      2      0.526226
c     1      0.741650
      2      0.569264
dtype: float64

With the index sorted in this way, partial slicing will work as expected:

In [37]:
data['a':'b']
Out[37]:
char  int
a     1      0.003001
      2      0.164974
b     1      0.001693
      2      0.526226
dtype: float64

Stacking and unstacking indices

As we saw briefly before, it is possible to convert a dataset from a stacked multi-index to a simple two-dimensional representation, optionally specifying the level to use:

In [38]:
pop.unstack(level=0)
Out[38]:
state California New York Texas
year
2000 33871648 18976457 20851820
2010 37253956 19378102 25145561
In [39]:
pop.unstack(level=1)
Out[39]:
year 2000 2010
state
California 33871648 37253956
New York 18976457 19378102
Texas 20851820 25145561

The opposite of unstack() is stack(), which here can be used to recover the original series:

In [40]:
pop.unstack().stack()
Out[40]:
state       year
California  2000    33871648
            2010    37253956
New York    2000    18976457
            2010    19378102
Texas       2000    20851820
            2010    25145561
dtype: int64

Index setting and resetting

Another way to rearrange hierarchical data is to turn the index labels into columns; this can be accomplished with the reset_index method. Calling this on the population dictionary will result in a DataFrame with a state and year column holding the information that was formerly in the index. For clarity, we can optionally specify the name of the data for the column representation:

In [41]:
pop_flat = pop.reset_index(name='population')
pop_flat
Out[41]:
state year population
0 California 2000 33871648
1 California 2010 37253956
2 New York 2000 18976457
3 New York 2010 19378102
4 Texas 2000 20851820
5 Texas 2010 25145561

Often when working with data in the real world, the raw input data looks like this and it's useful to build a MultiIndex from the column values. This can be done with the set_index method of the DataFrame, which returns a multiply indexed DataFrame:

In [42]:
pop_flat.set_index(['state', 'year'])
Out[42]:
population
state year
California 2000 33871648
2010 37253956
New York 2000 18976457
2010 19378102
Texas 2000 20851820
2010 25145561

In practice, I find this type of reindexing to be one of the more useful patterns when encountering real-world datasets.

Data Aggregations on Multi-Indices

We've previously seen that Pandas has built-in data aggregation methods, such as mean(), sum(), and max(). For hierarchically indexed data, these can be passed a level parameter that controls which subset of the data the aggregate is computed on.

For example, let's return to our health data:

In [43]:
health_data
Out[43]:
subject Bob Guido Sue
type HR Temp HR Temp HR Temp
year visit
2013 1 31.0 38.7 32.0 36.7 35.0 37.2
2 44.0 37.7 50.0 35.0 29.0 36.7
2014 1 30.0 37.4 39.0 37.8 61.0 36.9
2 47.0 37.8 48.0 37.3 51.0 36.5

Perhaps we'd like to average-out the measurements in the two visits each year. We can do this by naming the index level we'd like to explore, in this case the year:

In [44]:
data_mean = health_data.mean(level='year')
data_mean
Out[44]:
subject Bob Guido Sue
type HR Temp HR Temp HR Temp
year
2013 37.5 38.2 41.0 35.85 32.0 36.95
2014 38.5 37.6 43.5 37.55 56.0 36.70

By further making use of the axis keyword, we can take the mean among levels on the columns as well:

In [45]:
data_mean.mean(axis=1, level='type')
Out[45]:
type HR Temp
year
2013 36.833333 37.000000
2014 46.000000 37.283333

Thus in two lines, we've been able to find the average heart rate and temperature measured among all subjects in all visits each year. This syntax is actually a short cut to the GroupBy functionality, which we will discuss in Aggregation and Grouping. While this is a toy example, many real-world datasets have similar hierarchical structure.

Aside: Panel Data

Pandas has a few other fundamental data structures that we have not yet discussed, namely the pd.Panel and pd.Panel4D objects. These can be thought of, respectively, as three-dimensional and four-dimensional generalizations of the (one-dimensional) Series and (two-dimensional) DataFrame structures. Once you are familiar with indexing and manipulation of data in a Series and DataFrame, Panel and Panel4D are relatively straightforward to use. In particular, the ix, loc, and iloc indexers discussed in Data Indexing and Selection extend readily to these higher-dimensional structures.

We won't cover these panel structures further in this text, as I've found in the majority of cases that multi-indexing is a more useful and conceptually simpler representation for higher-dimensional data. Additionally, panel data is fundamentally a dense data representation, while multi-indexing is fundamentally a sparse data representation. As the number of dimensions increases, the dense representation can become very inefficient for the majority of real-world datasets. For the occasional specialized application, however, these structures can be useful. If you'd like to read more about the Panel and Panel4D structures, see the references listed in Further Resources.