Hierarchical Indexing

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. Early Pandas versions provided Panel and Panel4D objects that could be thought of as 3D or 4D analogs to the 2D DataFrame, but they were somewhat clunky to use in practice. A far more common pattern for handling higher-dimensional data 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. (If you’re interested in true N-dimensional arrays with Pandas-style flexible indices, you can look into the excellent Xarray package.)

In this chapter, 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 data.

We begin with the standard imports:

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:

index = [('California', 2010), ('California', 2020),
         ('New York', 2010), ('New York', 2020),
         ('Texas', 2010), ('Texas', 2020)]
populations = [37253956, 39538223,
               19378102, 20201249,
               25145561, 29145505]
pop = pd.Series(populations, index=index)
pop

(California, 2010)    37253956
(California, 2020)    39538223
(New York, 2010)      19378102
(New York, 2020)      20201249
(Texas, 2010)         25145561
(Texas, 2020)         29145505
dtype: int64

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

pop[('California', 2020):('Texas', 2010)]

(California, 2020)    39538223
(New York, 2010)      19378102
(New York, 2020)      20201249
(Texas, 2010)         25145561
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:

pop[[i for i in pop.index if i[1] == 2010]]

(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: The 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 types of operations we wish to have. We can create a multi-index from the tuples as follows:

index = pd.MultiIndex.from_tuples(index)

The MultiIndex represents 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 reindex our series with this MultiIndex, we see the hierarchical representation of the data:

pop = pop.reindex(index)
pop

California  2010    37253956
            2020    39538223
New York    2010    19378102
            2020    20201249
Texas       2010    25145561
            2020    29145505
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 2020, we can use the Pandas slicing notation:

pop[:, 2020]

California    39538223
New York      20201249
Texas         29145505
dtype: int64

The result is a singly indexed Series 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 hierarchically 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:

pop_df = pop.unstack()
pop_df

	2010	2020
California	37253956	39538223
New York	19378102	20201249
Texas	25145561	29145505

Naturally, the stack method provides the opposite operation:

pop_df.stack()

California  2010    37253956
            2020    39538223
New York    2010    19378102
            2020    20201249
Texas       2010    25145561
            2020    29145505
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 manipulate two-dimensional data within a one-dimensional Series, we can also use it to manipulate 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:

pop_df = pd.DataFrame({'total': pop,
                       'under18': [9284094, 8898092,
                                   4318033, 4181528,
                                   6879014, 7432474]})
pop_df

		total	under18
California	2010	37253956	9284094
California	2020	39538223	8898092
New York	2010	19378102	4318033
New York	2020	20201249	4181528
Texas	2010	25145561	6879014
Texas	2020	29145505	7432474

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:

f_u18 = pop_df['under18'] / pop_df['total']
f_u18.unstack()

	2010	2020
California	0.249211	0.225050
New York	0.222831	0.206994
Texas	0.273568	0.255013

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:

df = pd.DataFrame(np.random.rand(4, 2),
                  index=[['a', 'a', 'b', 'b'], [1, 2, 1, 2]],
                  columns=['data1', 'data2'])
df

		data1	data2
a	1	0.748464	0.561409
a	2	0.379199	0.622461
b	1	0.701679	0.687932
b	2	0.436200	0.950664

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:

data = {('California', 2010): 37253956,
        ('California', 2020): 39538223,
        ('New York', 2010): 19378102,
        ('New York', 2020): 20201249,
        ('Texas', 2010): 25145561,
        ('Texas', 2020): 29145505}
pd.Series(data)

California  2010    37253956
            2020    39538223
New York    2010    19378102
            2020    20201249
Texas       2010    25145561
            2020    29145505
dtype: int64

Nevertheless, it is sometimes useful to explicitly create a MultiIndex; we’ll look at a couple of methods for doing this next.

Explicit MultiIndex Constructors#

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

pd.MultiIndex.from_arrays([['a', 'a', 'b', 'b'], [1, 2, 1, 2]])

MultiIndex([('a', 1),
            ('a', 2),
            ('b', 1),
            ('b', 2)],
           )

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

pd.MultiIndex.from_tuples([('a', 1), ('a', 2), ('b', 1), ('b', 2)])

MultiIndex([('a', 1),
            ('a', 2),
            ('b', 1),
            ('b', 2)],
           )

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

pd.MultiIndex.from_product([['a', 'b'], [1, 2]])

MultiIndex([('a', 1),
            ('a', 2),
            ('b', 1),
            ('b', 2)],
           )

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

pd.MultiIndex(levels=[['a', 'b'], [1, 2]],
              codes=[[0, 0, 1, 1], [0, 1, 0, 1]])

MultiIndex([('a', 1),
            ('a', 2),
            ('b', 1),
            ('b', 2)],
           )

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 previously discussed MultiIndex constructors, or by setting the names attribute of the index after the fact:

pop.index.names = ['state', 'year']
pop

state       year
California  2010    37253956
            2020    39538223
New York    2010    19378102
            2020    20201249
Texas       2010    25145561
            2020    29145505
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:

# 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

	subject	Bob		Guido		Sue
	type	HR	Temp	HR	Temp	HR	Temp
year	visit
2013	1	30.0	38.0	56.0	38.3	45.0	35.8
2013	2	47.0	37.1	27.0	36.0	37.0	36.4
2014	1	51.0	35.9	24.0	36.7	32.0	36.2
2014	2	49.0	36.3	48.0	39.2	31.0	35.7

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:

health_data['Guido']

	type	HR	Temp
year	visit
2013	1	56.0	38.3
2013	2	27.0	36.0
2014	1	24.0	36.7
2014	2	48.0	39.2

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 DataFrame objects.

Multiply Indexed Series#

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

pop

state       year
California  2010    37253956
            2020    39538223
New York    2010    19378102
            2020    20201249
Texas       2010    25145561
            2020    29145505
dtype: int64

We can access single elements by indexing with multiple terms:

pop['California', 2010]

37253956

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:

pop['California']

year
2010    37253956
2020    39538223
dtype: int64

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

pop.loc['California':'New York']

state       year
California  2010    37253956
            2020    39538223
New York    2010    19378102
            2020    20201249
dtype: int64

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

pop[:, 2010]

state
California    37253956
New York      19378102
Texas         25145561
dtype: int64

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

pop[pop > 22000000]

state       year
California  2010    37253956
            2020    39538223
Texas       2010    25145561
            2020    29145505
dtype: int64

Selection based on fancy indexing also works:

pop[['California', 'Texas']]

state       year
California  2010    37253956
            2020    39538223
Texas       2010    25145561
            2020    29145505
dtype: int64

Multiply Indexed DataFrames#

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

health_data

	subject	Bob		Guido		Sue
	type	HR	Temp	HR	Temp	HR	Temp
year	visit
2013	1	30.0	38.0	56.0	38.3	45.0	35.8
2013	2	47.0	37.1	27.0	36.0	37.0	36.4
2014	1	51.0	35.9	24.0	36.7	32.0	36.2
2014	2	49.0	36.3	48.0	39.2	31.0	35.7

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:

health_data['Guido', 'HR']

year  visit
2013  1        56.0
      2        27.0
2014  1        24.0
      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:

health_data.iloc[:2, :2]

	subject	Bob
	type	HR	Temp
year	visit
2013	1	30.0	38.0
2013	2	47.0	37.1

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:

health_data.loc[:, ('Bob', 'HR')]

year  visit
2013  1        30.0
      2        47.0
2014  1        51.0
      2        49.0
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:

health_data.loc[(:, 1), (:, 'HR')]

  File "/var/folders/xc/sptt9bk14s34rgxt7453p03r0000gp/T/ipykernel_86488/3311942670.py", 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:

idx = pd.IndexSlice
health_data.loc[idx[:, 1], idx[:, 'HR']]

	subject	Bob	Guido	Sue
	type	HR	HR	HR
year	visit
2013	1	30.0	56.0	45.0
2014	1	51.0	24.0	32.0

As you can see, there are 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-Indexes#

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 I briefly mentioned a caveat, but I should emphasize it more here. Many of the MultiIndex slicing operations will fail if the index is not sorted. Let’s take a closer look.

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

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

char  int
a     1      0.280341
      2      0.097290
c     1      0.206217
      2      0.431771
b     1      0.100183
      2      0.015851
dtype: float64

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

try:
    data['a':'b']
except KeyError as e:
    print("KeyError", e)

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, such as the sort_index and sortlevel methods of the DataFrame. We’ll use the simplest, sort_index, here:

data = data.sort_index()
data

char  int
a     1      0.280341
      2      0.097290
b     1      0.100183
      2      0.015851
c     1      0.206217
      2      0.431771
dtype: float64

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

data['a':'b']

char  int
a     1      0.280341
      2      0.097290
b     1      0.100183
      2      0.015851
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:

pop.unstack(level=0)

state	California	New York	Texas
year
2010	37253956	19378102	25145561
2020	39538223	20201249	29145505

pop.unstack(level=1)

year	2010	2020
state
California	37253956	39538223
New York	19378102	20201249
Texas	25145561	29145505

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

pop.unstack().stack()

state       year
California  2010    37253956
            2020    39538223
New York    2010    19378102
            2020    20201249
Texas       2010    25145561
            2020    29145505
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 state and year columns holding the information that was formerly in the index. For clarity, we can optionally specify the name of the data for the column representation:

pop_flat = pop.reset_index(name='population')
pop_flat

	state	year	population
0	California	2010	37253956
1	California	2020	39538223
2	New York	2010	19378102
3	New York	2020	20201249
4	Texas	2010	25145561
5	Texas	2020	29145505

A common pattern is 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:

pop_flat.set_index(['state', 'year'])

		population
state	year
California	2010	37253956
California	2020	39538223
New York	2010	19378102
New York	2020	20201249
Texas	2010	25145561
Texas	2020	29145505

In practice, this type of reindexing is one of the more useful patterns when exploring real-world datasets.