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:
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', 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
(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:
pop[('California', 2010):('Texas', 2000)]
(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:
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: 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:
index = pd.MultiIndex.from_tuples(index)
index
MultiIndex([('California', 2000),
('California', 2010),
( 'New York', 2000),
( 'New York', 2010),
( 'Texas', 2000),
( 'Texas', 2010)],
) 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:
pop = pop.reindex(index)
pop
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:
pop[:, 2010]
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:
pop_df = pop.unstack()
pop_df
| 2000 | 2010 | |
|---|---|---|
| California | 33871648 | 37253956 |
| New York | 18976457 | 19378102 |
| Texas | 20851820 | 25145561 |
Naturally, the stack() method provides the opposite operation:
pop_df.stack()
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:
pop_df = pd.DataFrame({'total': pop,
'under18': [9267089, 9284094,
4687374, 4318033,
5906301, 6879014]})
pop_df
| 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:
f_u18 = pop_df['under18'] / pop_df['total']
f_u18.unstack()
| 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:
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 | 0.708767 | 0.907432 |
| 2 | 0.432888 | 0.807112 | |
| b | 1 | 0.517452 | 0.510478 |
| 2 | 0.313438 | 0.281365 |
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', 2000): 33871648,
('California', 2010): 37253956,
('Texas', 2000): 20851820,
('Texas', 2010): 25145561,
('New York', 2000): 18976457,
('New York', 2010): 19378102}
pd.Series(data)
California 2000 33871648
2010 37253956
Texas 2000 20851820
2010 25145561
New York 2000 18976457
2010 19378102
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:
pd.MultiIndex.from_arrays([['a', 'a', 'b', 'b'], [1, 2, 1, 2]])
MultiIndex([('a', 1),
('a', 2),
('b', 1),
('b', 2)],
) 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 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):
pd.MultiIndex(levels=[['a', 'b'], [1, 2]],
labels=[[0, 0, 1, 1], [0, 1, 0, 1]])
--------------------------------------------------------------------------- TypeError Traceback (most recent call last) Cell In[17], line 1 ----> 1 pd.MultiIndex(levels=[['a', 'b'], [1, 2]], 2 labels=[[0, 0, 1, 1], [0, 1, 0, 1]]) TypeError: MultiIndex.__new__() got an unexpected keyword argument 'labels'
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:
pop.index.names = ['state', 'year']
pop
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:
# 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 | 29.0 | 36.1 | 55.0 | 35.9 | 36.0 | 37.5 |
| 2 | 44.0 | 37.9 | 29.0 | 38.1 | 29.0 | 38.3 | |
| 2014 | 1 | 35.0 | 37.3 | 34.0 | 38.0 | 39.0 | 37.0 |
| 2 | 33.0 | 38.1 | 31.0 | 36.2 | 30.0 | 36.4 | |
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:
health_data['Guido']
| type | HR | Temp | |
|---|---|---|---|
| year | visit | ||
| 2013 | 1 | 55.0 | 35.9 |
| 2 | 29.0 | 38.1 | |
| 2014 | 1 | 34.0 | 38.0 |
| 2 | 31.0 | 36.2 |
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:
pop
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:
pop['California', 2000]
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:
pop['California']
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):
pop.loc['California':'New York']
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:
pop[:, 2000]
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:
pop[pop > 22000000]
state year
California 2000 33871648
2010 37253956
Texas 2010 25145561
dtype: int64 Selection based on fancy indexing also works:
pop[['California', 'Texas']]
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:
health_data
| subject | Bob | Guido | Sue | ||||
|---|---|---|---|---|---|---|---|
| type | HR | Temp | HR | Temp | HR | Temp | |
| year | visit | ||||||
| 2013 | 1 | 29.0 | 36.1 | 55.0 | 35.9 | 36.0 | 37.5 |
| 2 | 44.0 | 37.9 | 29.0 | 38.1 | 29.0 | 38.3 | |
| 2014 | 1 | 35.0 | 37.3 | 34.0 | 38.0 | 39.0 | 37.0 |
| 2 | 33.0 | 38.1 | 31.0 | 36.2 | 30.0 | 36.4 | |
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 55.0
2 29.0
2014 1 34.0
2 31.0
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 | 29.0 | 36.1 |
| 2 | 44.0 | 37.9 | |
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 29.0
2 44.0
2014 1 35.0
2 33.0
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')]
Cell In[32], 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 | 29.0 | 55.0 | 36.0 |
| 2014 | 1 | 35.0 | 34.0 | 39.0 |
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:
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 0.380549
2 0.434967
c 1 0.680160
2 0.116969
b 1 0.788898
2 0.440712
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(type(e))
print(e)
<class 'pandas.errors.UnsortedIndexError'> '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:
data = data.sort_index()
data
char int
a 1 0.380549
2 0.434967
b 1 0.788898
2 0.440712
c 1 0.680160
2 0.116969
dtype: float64 With the index sorted in this way, partial slicing will work as expected:
data['a':'b']
char int
a 1 0.380549
2 0.434967
b 1 0.788898
2 0.440712
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 | |||
| 2000 | 33871648 | 18976457 | 20851820 |
| 2010 | 37253956 | 19378102 | 25145561 |
pop.unstack(level=1)
| 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:
pop.unstack().stack()
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:
pop_flat = pop.reset_index(name='population')
pop_flat
| 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:
pop_flat.set_index(['state', 'year'])
| 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:
health_data
| subject | Bob | Guido | Sue | ||||
|---|---|---|---|---|---|---|---|
| type | HR | Temp | HR | Temp | HR | Temp | |
| year | visit | ||||||
| 2013 | 1 | 29.0 | 36.1 | 55.0 | 35.9 | 36.0 | 37.5 |
| 2 | 44.0 | 37.9 | 29.0 | 38.1 | 29.0 | 38.3 | |
| 2014 | 1 | 35.0 | 37.3 | 34.0 | 38.0 | 39.0 | 37.0 |
| 2 | 33.0 | 38.1 | 31.0 | 36.2 | 30.0 | 36.4 | |
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:
data_mean = health_data.mean(level='year')
data_mean
/tmp/ipykernel_839/2845460446.py:1: FutureWarning: Using the level keyword in DataFrame and Series aggregations is deprecated and will be removed in a future version. Use groupby instead. df.median(level=1) should use df.groupby(level=1).median(). data_mean = health_data.mean(level='year')
| subject | Bob | Guido | Sue | |||
|---|---|---|---|---|---|---|
| type | HR | Temp | HR | Temp | HR | Temp |
| year | ||||||
| 2013 | 36.5 | 37.0 | 42.0 | 37.0 | 32.5 | 37.9 |
| 2014 | 34.0 | 37.7 | 32.5 | 37.1 | 34.5 | 36.7 |
By further making use of the axis keyword, we can take the mean among levels on the columns as well:
data_mean.mean(axis=1, level='type')
/tmp/ipykernel_839/1641729190.py:1: FutureWarning: Using the level keyword in DataFrame and Series aggregations is deprecated and will be removed in a future version. Use groupby instead. df.median(level=1) should use df.groupby(level=1).median(). data_mean.mean(axis=1, level='type')
| type | HR | Temp |
|---|---|---|
| year | ||
| 2013 | 37.000000 | 37.300000 |
| 2014 | 33.666667 | 37.166667 |
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.