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# 03.03 Operations in Pandas

One of the essential pieces of NumPy is the ability to perform quick element-wise operations, both with basic arithmetic (addition, subtraction, multiplication, etc.) and with more sophisticated operations (trigonometric functions, exponential and logarithmic functions, etc.). Pandas inherits much of this functionality from NumPy, and the ufuncs that we introduced in Computation on NumPy Arrays: Universal Functions are key to this.
Pandas includes a couple useful twists, however: for unary operations like negation and trigonometric functions, these ufuncs will preserve index and column labels in the output, and for binary operations such as addition and multiplication, Pandas will automatically align indices when passing the objects to the ufunc. This means that keeping the context of data and combining data from different sources–both potentially error-prone tasks with raw NumPy arrays–become essentially foolproof ones with Pandas. We will additionally see that there are well-defined operations between one-dimensional Series structures and two-dimensional DataFrame structures.

### Ufuncs: Index Preservation

Because Pandas is designed to work with NumPy, any NumPy ufunc will work on Pandas Series and DataFrame objects. Let's start by defining a simple Series and DataFrame on which to demonstrate this:
import pandas as pd
import numpy as np
rng = np.random.RandomState(42)
ser = pd.Series(rng.randint(0, 10, 4))
ser
0 6 1 3 2 7 3 4 dtype: int64
df = pd.DataFrame(rng.randint(0, 10, (3, 4)),
columns=['A', 'B', 'C', 'D'])
df
If we apply a NumPy ufunc on either of these objects, the result will be another Pandas object with the indices preserved:
np.exp(ser)
0 403.428793 1 20.085537 2 1096.633158 3 54.598150 dtype: float64
Or, for a slightly more complex calculation:
np.sin(df * np.pi / 4)
Any of the ufuncs discussed in Computation on NumPy Arrays: Universal Functions can be used in a similar manner.

### UFuncs: Index Alignment

For binary operations on two Series or DataFrame objects, Pandas will align indices in the process of performing the operation. This is very convenient when working with incomplete data, as we'll see in some of the examples that follow.

#### Index alignment in Series

As an example, suppose we are combining two different data sources, and find only the top three US states by area and the top three US states by population:
area = pd.Series({'Alaska': 1723337, 'Texas': 695662,
'California': 423967}, name='area')
population = pd.Series({'California': 38332521, 'Texas': 26448193,
'New York': 19651127}, name='population')
Let's see what happens when we divide these to compute the population density:
population / area
Alaska NaN California 90.413926 New York NaN Texas 38.018740 dtype: float64
The resulting array contains the union of indices of the two input arrays, which could be determined using standard Python set arithmetic on these indices:
area.index | population.index
Index(['Alaska', 'California', 'New York', 'Texas'], dtype='object')
Any item for which one or the other does not have an entry is marked with NaN, or "Not a Number," which is how Pandas marks missing data (see further discussion of missing data in Handling Missing Data). This index matching is implemented this way for any of Python's built-in arithmetic expressions; any missing values are filled in with NaN by default:
A = pd.Series([2, 4, 6], index=[0, 1, 2])
B = pd.Series([1, 3, 5], index=[1, 2, 3])
A + B
0 NaN 1 5.0 2 9.0 3 NaN dtype: float64
If using NaN values is not the desired behavior, the fill value can be modified using appropriate object methods in place of the operators. For example, calling A.add(B) is equivalent to calling A + B, but allows optional explicit specification of the fill value for any elements in A or B that might be missing:
A.add(B, fill_value=0)
0 2.0 1 5.0 2 9.0 3 5.0 dtype: float64

#### Index alignment in DataFrame

A similar type of alignment takes place for both columns and indices when performing operations on DataFrames:
A = pd.DataFrame(rng.randint(0, 20, (2, 2)),
columns=list('AB'))
A
B = pd.DataFrame(rng.randint(0, 10, (3, 3)),
columns=list('BAC'))
B
A + B
Notice that indices are aligned correctly irrespective of their order in the two objects, and indices in the result are sorted. As was the case with Series, we can use the associated object's arithmetic method and pass any desired fill_value to be used in place of missing entries. Here we'll fill with the mean of all values in A (computed by first stacking the rows of A):
fill = A.stack().mean()
A.add(B, fill_value=fill)
The following table lists Python operators and their equivalent Pandas object methods:
Python OperatorPandas Method(s)
+add()
-sub(), subtract()
*mul(), multiply()
/truediv(), div(), divide()
//floordiv()
%mod()
**pow()

### Ufuncs: Operations Between DataFrame and Series

When performing operations between a DataFrame and a Series, the index and column alignment is similarly maintained. Operations between a DataFrame and a Series are similar to operations between a two-dimensional and one-dimensional NumPy array. Consider one common operation, where we find the difference of a two-dimensional array and one of its rows:
A = rng.randint(10, size=(3, 4))
A
array([[3, 8, 2, 4], [2, 6, 4, 8], [6, 1, 3, 8]])
A - A[0]
array([[ 0, 0, 0, 0], [-1, -2, 2, 4], [ 3, -7, 1, 4]])
According to NumPy's broadcasting rules (see Computation on Arrays: Broadcasting), subtraction between a two-dimensional array and one of its rows is applied row-wise.
In Pandas, the convention similarly operates row-wise by default:
df = pd.DataFrame(A, columns=list('QRST'))
df - df.iloc[0]
If you would instead like to operate column-wise, you can use the object methods mentioned earlier, while specifying the axis keyword:
df.subtract(df['R'], axis=0)
Note that these DataFrame/Series operations, like the operations discussed above, will automatically align indices between the two elements:
halfrow = df.iloc[0, ::2]
halfrow
Q 3 S 2 Name: 0, dtype: int64
df - halfrow