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notes_estom/Python/pandas/user_guide/computation.md
estomm d31be4f219 matplotlib & pandas
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# Computational tools
## Statistical functions
### Percent change
``Series`` and ``DataFrame`` have a method
[``pct_change()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.pct_change.html#pandas.DataFrame.pct_change) to compute the percent change over a given number
of periods (using ``fill_method`` to fill NA/null values *before* computing
the percent change).
``` python
In [1]: ser = pd.Series(np.random.randn(8))
In [2]: ser.pct_change()
Out[2]:
0 NaN
1 -1.602976
2 4.334938
3 -0.247456
4 -2.067345
5 -1.142903
6 -1.688214
7 -9.759729
dtype: float64
```
``` python
In [3]: df = pd.DataFrame(np.random.randn(10, 4))
In [4]: df.pct_change(periods=3)
Out[4]:
0 1 2 3
0 NaN NaN NaN NaN
1 NaN NaN NaN NaN
2 NaN NaN NaN NaN
3 -0.218320 -1.054001 1.987147 -0.510183
4 -0.439121 -1.816454 0.649715 -4.822809
5 -0.127833 -3.042065 -5.866604 -1.776977
6 -2.596833 -1.959538 -2.111697 -3.798900
7 -0.117826 -2.169058 0.036094 -0.067696
8 2.492606 -1.357320 -1.205802 -1.558697
9 -1.012977 2.324558 -1.003744 -0.371806
```
### Covariance
[``Series.cov()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.Series.cov.html#pandas.Series.cov) can be used to compute covariance between series
(excluding missing values).
``` python
In [5]: s1 = pd.Series(np.random.randn(1000))
In [6]: s2 = pd.Series(np.random.randn(1000))
In [7]: s1.cov(s2)
Out[7]: 0.000680108817431082
```
Analogously, [``DataFrame.cov()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cov.html#pandas.DataFrame.cov) to compute pairwise covariances among the
series in the DataFrame, also excluding NA/null values.
::: tip Note
Assuming the missing data are missing at random this results in an estimate
for the covariance matrix which is unbiased. However, for many applications
this estimate may not be acceptable because the estimated covariance matrix
is not guaranteed to be positive semi-definite. This could lead to
estimated correlations having absolute values which are greater than one,
and/or a non-invertible covariance matrix. See [Estimation of covariance
matrices](http://en.wikipedia.org/w/index.php?title=Estimation_of_covariance_matrices)
for more details.
:::
``` python
In [8]: frame = pd.DataFrame(np.random.randn(1000, 5),
...: columns=['a', 'b', 'c', 'd', 'e'])
...:
In [9]: frame.cov()
Out[9]:
a b c d e
a 1.000882 -0.003177 -0.002698 -0.006889 0.031912
b -0.003177 1.024721 0.000191 0.009212 0.000857
c -0.002698 0.000191 0.950735 -0.031743 -0.005087
d -0.006889 0.009212 -0.031743 1.002983 -0.047952
e 0.031912 0.000857 -0.005087 -0.047952 1.042487
```
``DataFrame.cov`` also supports an optional ``min_periods`` keyword that
specifies the required minimum number of observations for each column pair
in order to have a valid result.
``` python
In [10]: frame = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c'])
In [11]: frame.loc[frame.index[:5], 'a'] = np.nan
In [12]: frame.loc[frame.index[5:10], 'b'] = np.nan
In [13]: frame.cov()
Out[13]:
a b c
a 1.123670 -0.412851 0.018169
b -0.412851 1.154141 0.305260
c 0.018169 0.305260 1.301149
In [14]: frame.cov(min_periods=12)
Out[14]:
a b c
a 1.123670 NaN 0.018169
b NaN 1.154141 0.305260
c 0.018169 0.305260 1.301149
```
### Correlation
Correlation may be computed using the [``corr()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.corr.html#pandas.DataFrame.corr) method.
Using the ``method`` parameter, several methods for computing correlations are
provided:
Method name | Description
---|---
pearson (default) | Standard correlation coefficient
kendall | Kendall Tau correlation coefficient
spearman | Spearman rank correlation coefficient
All of these are currently computed using pairwise complete observations.
Wikipedia has articles covering the above correlation coefficients:
- [Pearson correlation coefficient](https://en.wikipedia.org/wiki/Pearson_correlation_coefficient)
- [Kendall rank correlation coefficient](https://en.wikipedia.org/wiki/Kendall_rank_correlation_coefficient)
- [Spearman’s rank correlation coefficient](https://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient)
::: tip Note
Please see the [caveats](#computation-covariance-caveats) associated
with this method of calculating correlation matrices in the
[covariance section](#computation-covariance).
:::
``` python
In [15]: frame = pd.DataFrame(np.random.randn(1000, 5),
....: columns=['a', 'b', 'c', 'd', 'e'])
....:
In [16]: frame.iloc[::2] = np.nan
# Series with Series
In [17]: frame['a'].corr(frame['b'])
Out[17]: 0.013479040400098794
In [18]: frame['a'].corr(frame['b'], method='spearman')
Out[18]: -0.007289885159540637
# Pairwise correlation of DataFrame columns
In [19]: frame.corr()
Out[19]:
a b c d e
a 1.000000 0.013479 -0.049269 -0.042239 -0.028525
b 0.013479 1.000000 -0.020433 -0.011139 0.005654
c -0.049269 -0.020433 1.000000 0.018587 -0.054269
d -0.042239 -0.011139 0.018587 1.000000 -0.017060
e -0.028525 0.005654 -0.054269 -0.017060 1.000000
```
Note that non-numeric columns will be automatically excluded from the
correlation calculation.
Like ``cov``, ``corr`` also supports the optional ``min_periods`` keyword:
``` python
In [20]: frame = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c'])
In [21]: frame.loc[frame.index[:5], 'a'] = np.nan
In [22]: frame.loc[frame.index[5:10], 'b'] = np.nan
In [23]: frame.corr()
Out[23]:
a b c
a 1.000000 -0.121111 0.069544
b -0.121111 1.000000 0.051742
c 0.069544 0.051742 1.000000
In [24]: frame.corr(min_periods=12)
Out[24]:
a b c
a 1.000000 NaN 0.069544
b NaN 1.000000 0.051742
c 0.069544 0.051742 1.000000
```
*New in version 0.24.0.*
The ``method`` argument can also be a callable for a generic correlation
calculation. In this case, it should be a single function
that produces a single value from two ndarray inputs. Suppose we wanted to
compute the correlation based on histogram intersection:
``` python
# histogram intersection
In [25]: def histogram_intersection(a, b):
....: return np.minimum(np.true_divide(a, a.sum()),
....: np.true_divide(b, b.sum())).sum()
....:
In [26]: frame.corr(method=histogram_intersection)
Out[26]:
a b c
a 1.000000 -6.404882 -2.058431
b -6.404882 1.000000 -19.255743
c -2.058431 -19.255743 1.000000
```
A related method [``corrwith()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.corrwith.html#pandas.DataFrame.corrwith) is implemented on DataFrame to
compute the correlation between like-labeled Series contained in different
DataFrame objects.
``` python
In [27]: index = ['a', 'b', 'c', 'd', 'e']
In [28]: columns = ['one', 'two', 'three', 'four']
In [29]: df1 = pd.DataFrame(np.random.randn(5, 4), index=index, columns=columns)
In [30]: df2 = pd.DataFrame(np.random.randn(4, 4), index=index[:4], columns=columns)
In [31]: df1.corrwith(df2)
Out[31]:
one -0.125501
two -0.493244
three 0.344056
four 0.004183
dtype: float64
In [32]: df2.corrwith(df1, axis=1)
Out[32]:
a -0.675817
b 0.458296
c 0.190809
d -0.186275
e NaN
dtype: float64
```
### Data ranking
The [``rank()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.Series.rank.html#pandas.Series.rank) method produces a data ranking with ties being
assigned the mean of the ranks (by default) for the group:
``` python
In [33]: s = pd.Series(np.random.np.random.randn(5), index=list('abcde'))
In [34]: s['d'] = s['b'] # so there's a tie
In [35]: s.rank()
Out[35]:
a 5.0
b 2.5
c 1.0
d 2.5
e 4.0
dtype: float64
```
[``rank()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.rank.html#pandas.DataFrame.rank) is also a DataFrame method and can rank either the rows
(``axis=0``) or the columns (``axis=1``). ``NaN`` values are excluded from the
ranking.
``` python
In [36]: df = pd.DataFrame(np.random.np.random.randn(10, 6))
In [37]: df[4] = df[2][:5] # some ties
In [38]: df
Out[38]:
0 1 2 3 4 5
0 -0.904948 -1.163537 -1.457187 0.135463 -1.457187 0.294650
1 -0.976288 -0.244652 -0.748406 -0.999601 -0.748406 -0.800809
2 0.401965 1.460840 1.256057 1.308127 1.256057 0.876004
3 0.205954 0.369552 -0.669304 0.038378 -0.669304 1.140296
4 -0.477586 -0.730705 -1.129149 -0.601463 -1.129149 -0.211196
5 -1.092970 -0.689246 0.908114 0.204848 NaN 0.463347
6 0.376892 0.959292 0.095572 -0.593740 NaN -0.069180
7 -1.002601 1.957794 -0.120708 0.094214 NaN -1.467422
8 -0.547231 0.664402 -0.519424 -0.073254 NaN -1.263544
9 -0.250277 -0.237428 -1.056443 0.419477 NaN 1.375064
In [39]: df.rank(1)
Out[39]:
0 1 2 3 4 5
0 4.0 3.0 1.5 5.0 1.5 6.0
1 2.0 6.0 4.5 1.0 4.5 3.0
2 1.0 6.0 3.5 5.0 3.5 2.0
3 4.0 5.0 1.5 3.0 1.5 6.0
4 5.0 3.0 1.5 4.0 1.5 6.0
5 1.0 2.0 5.0 3.0 NaN 4.0
6 4.0 5.0 3.0 1.0 NaN 2.0
7 2.0 5.0 3.0 4.0 NaN 1.0
8 2.0 5.0 3.0 4.0 NaN 1.0
9 2.0 3.0 1.0 4.0 NaN 5.0
```
``rank`` optionally takes a parameter ``ascending`` which by default is true;
when false, data is reverse-ranked, with larger values assigned a smaller rank.
``rank`` supports different tie-breaking methods, specified with the ``method``
parameter:
- ``average`` : average rank of tied group
- ``min`` : lowest rank in the group
- ``max`` : highest rank in the group
- ``first`` : ranks assigned in the order they appear in the array
## Window Functions
For working with data, a number of window functions are provided for
computing common *window* or *rolling* statistics. Among these are count, sum,
mean, median, correlation, variance, covariance, standard deviation, skewness,
and kurtosis.
The ``rolling()`` and ``expanding()``
functions can be used directly from DataFrameGroupBy objects,
see the [groupby docs](groupby.html#groupby-transform-window-resample).
::: tip Note
The API for window statistics is quite similar to the way one works with ``GroupBy`` objects, see the documentation [here](groupby.html#groupby).
:::
We work with ``rolling``, ``expanding`` and ``exponentially weighted`` data through the corresponding
objects, ``Rolling``, ``Expanding`` and ``EWM``.
``` python
In [40]: s = pd.Series(np.random.randn(1000),
....: index=pd.date_range('1/1/2000', periods=1000))
....:
In [41]: s = s.cumsum()
In [42]: s
Out[42]:
2000-01-01 -0.268824
2000-01-02 -1.771855
2000-01-03 -0.818003
2000-01-04 -0.659244
2000-01-05 -1.942133
...
2002-09-22 -67.457323
2002-09-23 -69.253182
2002-09-24 -70.296818
2002-09-25 -70.844674
2002-09-26 -72.475016
Freq: D, Length: 1000, dtype: float64
```
These are created from methods on ``Series`` and ``DataFrame``.
``` python
In [43]: r = s.rolling(window=60)
In [44]: r
Out[44]: Rolling [window=60,center=False,axis=0]
```
These object provide tab-completion of the available methods and properties.
``` python
In [14]: r.<TAB> # noqa: E225, E999
r.agg r.apply r.count r.exclusions r.max r.median r.name r.skew r.sum
r.aggregate r.corr r.cov r.kurt r.mean r.min r.quantile r.std r.var
```
Generally these methods all have the same interface. They all
accept the following arguments:
- ``window``: size of moving window
- ``min_periods``: threshold of non-null data points to require (otherwise
result is NA)
- ``center``: boolean, whether to set the labels at the center (default is False)
We can then call methods on these ``rolling`` objects. These return like-indexed objects:
``` python
In [45]: r.mean()
Out[45]:
2000-01-01 NaN
2000-01-02 NaN
2000-01-03 NaN
2000-01-04 NaN
2000-01-05 NaN
...
2002-09-22 -62.914971
2002-09-23 -63.061867
2002-09-24 -63.213876
2002-09-25 -63.375074
2002-09-26 -63.539734
Freq: D, Length: 1000, dtype: float64
```
``` python
In [46]: s.plot(style='k--')
Out[46]: <matplotlib.axes._subplots.AxesSubplot at 0x7f66048bdef0>
In [47]: r.mean().plot(style='k')
Out[47]: <matplotlib.axes._subplots.AxesSubplot at 0x7f66048bdef0>
```
![rolling_mean_ex](https://static.pypandas.cn/public/static/images/rolling_mean_ex.png)
They can also be applied to DataFrame objects. This is really just syntactic
sugar for applying the moving window operator to all of the DataFrame’s columns:
``` python
In [48]: df = pd.DataFrame(np.random.randn(1000, 4),
....: index=pd.date_range('1/1/2000', periods=1000),
....: columns=['A', 'B', 'C', 'D'])
....:
In [49]: df = df.cumsum()
In [50]: df.rolling(window=60).sum().plot(subplots=True)
Out[50]:
array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f66075a7f60>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f65f79e55c0>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f65f7998588>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f65f794b550>],
dtype=object)
```
![rolling_mean_frame](https://static.pypandas.cn/public/static/images/rolling_mean_frame.png)
### Method summary
We provide a number of common statistical functions:
Method | Description
---|---
[count()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.count.html#pandas.core.window.Rolling.count) | Number of non-null observations
[sum()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.sum.html#pandas.core.window.Rolling.sum) | Sum of values
[mean()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.mean.html#pandas.core.window.Rolling.mean) | Mean of values
[median()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.median.html#pandas.core.window.Rolling.median) | Arithmetic median of values
[min()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.min.html#pandas.core.window.Rolling.min) | Minimum
[max()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.max.html#pandas.core.window.Rolling.max) | Maximum
[std()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.std.html#pandas.core.window.Rolling.std) | Bessel-corrected sample standard deviation
[var()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.var.html#pandas.core.window.Rolling.var) | Unbiased variance
[skew()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.skew.html#pandas.core.window.Rolling.skew) | Sample skewness (3rd moment)
[kurt()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.kurt.html#pandas.core.window.Rolling.kurt) | Sample kurtosis (4th moment)
[quantile()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.quantile.html#pandas.core.window.Rolling.quantile) | Sample quantile (value at %)
[apply()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.apply.html#pandas.core.window.Rolling.apply) | Generic apply
[cov()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.cov.html#pandas.core.window.Rolling.cov) | Unbiased covariance (binary)
[corr()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.corr.html#pandas.core.window.Rolling.corr) | Correlation (binary)
The [``apply()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.apply.html#pandas.core.window.Rolling.apply) function takes an extra ``func`` argument and performs
generic rolling computations. The ``func`` argument should be a single function
that produces a single value from an ndarray input. Suppose we wanted to
compute the mean absolute deviation on a rolling basis:
``` python
In [51]: def mad(x):
....: return np.fabs(x - x.mean()).mean()
....:
In [52]: s.rolling(window=60).apply(mad, raw=True).plot(style='k')
Out[52]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f795fac8>
```
![rolling_apply_ex](https://static.pypandas.cn/public/static/images/rolling_apply_ex.png)
### Rolling windows
Passing ``win_type`` to ``.rolling`` generates a generic rolling window computation, that is weighted according the ``win_type``.
The following methods are available:
Method | Description
---|---
[sum()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Window.sum.html#pandas.core.window.Window.sum) | Sum of values
[mean()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Window.mean.html#pandas.core.window.Window.mean) | Mean of values
The weights used in the window are specified by the ``win_type`` keyword.
The list of recognized types are the [scipy.signal window functions](https://docs.scipy.org/doc/scipy/reference/signal.html#window-functions):
- ``boxcar``
- ``triang``
- ``blackman``
- ``hamming``
- ``bartlett``
- ``parzen``
- ``bohman``
- ``blackmanharris``
- ``nuttall``
- ``barthann``
- ``kaiser`` (needs beta)
- ``gaussian`` (needs std)
- ``general_gaussian`` (needs power, width)
- ``slepian`` (needs width)
- ``exponential`` (needs tau).
``` python
In [53]: ser = pd.Series(np.random.randn(10),
....: index=pd.date_range('1/1/2000', periods=10))
....:
In [54]: ser.rolling(window=5, win_type='triang').mean()
Out[54]:
2000-01-01 NaN
2000-01-02 NaN
2000-01-03 NaN
2000-01-04 NaN
2000-01-05 -1.037870
2000-01-06 -0.767705
2000-01-07 -0.383197
2000-01-08 -0.395513
2000-01-09 -0.558440
2000-01-10 -0.672416
Freq: D, dtype: float64
```
Note that the ``boxcar`` window is equivalent to [``mean()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.mean.html#pandas.core.window.Rolling.mean).
``` python
In [55]: ser.rolling(window=5, win_type='boxcar').mean()
Out[55]:
2000-01-01 NaN
2000-01-02 NaN
2000-01-03 NaN
2000-01-04 NaN
2000-01-05 -0.841164
2000-01-06 -0.779948
2000-01-07 -0.565487
2000-01-08 -0.502815
2000-01-09 -0.553755
2000-01-10 -0.472211
Freq: D, dtype: float64
In [56]: ser.rolling(window=5).mean()
Out[56]:
2000-01-01 NaN
2000-01-02 NaN
2000-01-03 NaN
2000-01-04 NaN
2000-01-05 -0.841164
2000-01-06 -0.779948
2000-01-07 -0.565487
2000-01-08 -0.502815
2000-01-09 -0.553755
2000-01-10 -0.472211
Freq: D, dtype: float64
```
For some windowing functions, additional parameters must be specified:
``` python
In [57]: ser.rolling(window=5, win_type='gaussian').mean(std=0.1)
Out[57]:
2000-01-01 NaN
2000-01-02 NaN
2000-01-03 NaN
2000-01-04 NaN
2000-01-05 -1.309989
2000-01-06 -1.153000
2000-01-07 0.606382
2000-01-08 -0.681101
2000-01-09 -0.289724
2000-01-10 -0.996632
Freq: D, dtype: float64
```
::: tip Note
For ``.sum()`` with a ``win_type``, there is no normalization done to the
weights for the window. Passing custom weights of ``[1, 1, 1]`` will yield a different
result than passing weights of ``[2, 2, 2]``, for example. When passing a
``win_type`` instead of explicitly specifying the weights, the weights are
already normalized so that the largest weight is 1.
In contrast, the nature of the ``.mean()`` calculation is
such that the weights are normalized with respect to each other. Weights
of ``[1, 1, 1]`` and ``[2, 2, 2]`` yield the same result.
:::
### Time-aware rolling
*New in version 0.19.0.*
New in version 0.19.0 are the ability to pass an offset (or convertible) to a ``.rolling()`` method and have it produce
variable sized windows based on the passed time window. For each time point, this includes all preceding values occurring
within the indicated time delta.
This can be particularly useful for a non-regular time frequency index.
``` python
In [58]: dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
....: index=pd.date_range('20130101 09:00:00',
....: periods=5,
....: freq='s'))
....:
In [59]: dft
Out[59]:
B
2013-01-01 09:00:00 0.0
2013-01-01 09:00:01 1.0
2013-01-01 09:00:02 2.0
2013-01-01 09:00:03 NaN
2013-01-01 09:00:04 4.0
```
This is a regular frequency index. Using an integer window parameter works to roll along the window frequency.
``` python
In [60]: dft.rolling(2).sum()
Out[60]:
B
2013-01-01 09:00:00 NaN
2013-01-01 09:00:01 1.0
2013-01-01 09:00:02 3.0
2013-01-01 09:00:03 NaN
2013-01-01 09:00:04 NaN
In [61]: dft.rolling(2, min_periods=1).sum()
Out[61]:
B
2013-01-01 09:00:00 0.0
2013-01-01 09:00:01 1.0
2013-01-01 09:00:02 3.0
2013-01-01 09:00:03 2.0
2013-01-01 09:00:04 4.0
```
Specifying an offset allows a more intuitive specification of the rolling frequency.
``` python
In [62]: dft.rolling('2s').sum()
Out[62]:
B
2013-01-01 09:00:00 0.0
2013-01-01 09:00:01 1.0
2013-01-01 09:00:02 3.0
2013-01-01 09:00:03 2.0
2013-01-01 09:00:04 4.0
```
Using a non-regular, but still monotonic index, rolling with an integer window does not impart any special calculation.
``` python
In [63]: dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
....: index=pd.Index([pd.Timestamp('20130101 09:00:00'),
....: pd.Timestamp('20130101 09:00:02'),
....: pd.Timestamp('20130101 09:00:03'),
....: pd.Timestamp('20130101 09:00:05'),
....: pd.Timestamp('20130101 09:00:06')],
....: name='foo'))
....:
In [64]: dft
Out[64]:
B
foo
2013-01-01 09:00:00 0.0
2013-01-01 09:00:02 1.0
2013-01-01 09:00:03 2.0
2013-01-01 09:00:05 NaN
2013-01-01 09:00:06 4.0
In [65]: dft.rolling(2).sum()
Out[65]:
B
foo
2013-01-01 09:00:00 NaN
2013-01-01 09:00:02 1.0
2013-01-01 09:00:03 3.0
2013-01-01 09:00:05 NaN
2013-01-01 09:00:06 NaN
```
Using the time-specification generates variable windows for this sparse data.
``` python
In [66]: dft.rolling('2s').sum()
Out[66]:
B
foo
2013-01-01 09:00:00 0.0
2013-01-01 09:00:02 1.0
2013-01-01 09:00:03 3.0
2013-01-01 09:00:05 NaN
2013-01-01 09:00:06 4.0
```
Furthermore, we now allow an optional ``on`` parameter to specify a column (rather than the
default of the index) in a DataFrame.
``` python
In [67]: dft = dft.reset_index()
In [68]: dft
Out[68]:
foo B
0 2013-01-01 09:00:00 0.0
1 2013-01-01 09:00:02 1.0
2 2013-01-01 09:00:03 2.0
3 2013-01-01 09:00:05 NaN
4 2013-01-01 09:00:06 4.0
In [69]: dft.rolling('2s', on='foo').sum()
Out[69]:
foo B
0 2013-01-01 09:00:00 0.0
1 2013-01-01 09:00:02 1.0
2 2013-01-01 09:00:03 3.0
3 2013-01-01 09:00:05 NaN
4 2013-01-01 09:00:06 4.0
```
### Rolling window endpoints
*New in version 0.20.0.*
The inclusion of the interval endpoints in rolling window calculations can be specified with the ``closed``
parameter:
closed | Description | Default for
---|---|---
right | close right endpoint | time-based windows
left | close left endpoint |  
both | close both endpoints | fixed windows
neither | open endpoints |  
For example, having the right endpoint open is useful in many problems that require that there is no contamination
from present information back to past information. This allows the rolling window to compute statistics
“up to that point in time”, but not including that point in time.
``` python
In [70]: df = pd.DataFrame({'x': 1},
....: index=[pd.Timestamp('20130101 09:00:01'),
....: pd.Timestamp('20130101 09:00:02'),
....: pd.Timestamp('20130101 09:00:03'),
....: pd.Timestamp('20130101 09:00:04'),
....: pd.Timestamp('20130101 09:00:06')])
....:
In [71]: df["right"] = df.rolling('2s', closed='right').x.sum() # default
In [72]: df["both"] = df.rolling('2s', closed='both').x.sum()
In [73]: df["left"] = df.rolling('2s', closed='left').x.sum()
In [74]: df["neither"] = df.rolling('2s', closed='neither').x.sum()
In [75]: df
Out[75]:
x right both left neither
2013-01-01 09:00:01 1 1.0 1.0 NaN NaN
2013-01-01 09:00:02 1 2.0 2.0 1.0 1.0
2013-01-01 09:00:03 1 2.0 3.0 2.0 1.0
2013-01-01 09:00:04 1 2.0 3.0 2.0 1.0
2013-01-01 09:00:06 1 1.0 2.0 1.0 NaN
```
Currently, this feature is only implemented for time-based windows.
For fixed windows, the closed parameter cannot be set and the rolling window will always have both endpoints closed.
### Time-aware rolling vs. resampling
Using ``.rolling()`` with a time-based index is quite similar to [resampling](timeseries.html#timeseries-resampling). They
both operate and perform reductive operations on time-indexed pandas objects.
When using ``.rolling()`` with an offset. The offset is a time-delta. Take a backwards-in-time looking window, and
aggregate all of the values in that window (including the end-point, but not the start-point). This is the new value
at that point in the result. These are variable sized windows in time-space for each point of the input. You will get
a same sized result as the input.
When using ``.resample()`` with an offset. Construct a new index that is the frequency of the offset. For each frequency
bin, aggregate points from the input within a backwards-in-time looking window that fall in that bin. The result of this
aggregation is the output for that frequency point. The windows are fixed size in the frequency space. Your result
will have the shape of a regular frequency between the min and the max of the original input object.
To summarize, ``.rolling()`` is a time-based window operation, while ``.resample()`` is a frequency-based window operation.
### Centering windows
By default the labels are set to the right edge of the window, but a
``center`` keyword is available so the labels can be set at the center.
``` python
In [76]: ser.rolling(window=5).mean()
Out[76]:
2000-01-01 NaN
2000-01-02 NaN
2000-01-03 NaN
2000-01-04 NaN
2000-01-05 -0.841164
2000-01-06 -0.779948
2000-01-07 -0.565487
2000-01-08 -0.502815
2000-01-09 -0.553755
2000-01-10 -0.472211
Freq: D, dtype: float64
In [77]: ser.rolling(window=5, center=True).mean()
Out[77]:
2000-01-01 NaN
2000-01-02 NaN
2000-01-03 -0.841164
2000-01-04 -0.779948
2000-01-05 -0.565487
2000-01-06 -0.502815
2000-01-07 -0.553755
2000-01-08 -0.472211
2000-01-09 NaN
2000-01-10 NaN
Freq: D, dtype: float64
```
### Binary window functions
[``cov()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.cov.html#pandas.core.window.Rolling.cov) and [``corr()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Rolling.corr.html#pandas.core.window.Rolling.corr) can compute moving window statistics about
two ``Series`` or any combination of ``DataFrame/Series`` or
``DataFrame/DataFrame``. Here is the behavior in each case:
- two ``Series``: compute the statistic for the pairing.
- ``DataFrame/Series``: compute the statistics for each column of the DataFrame
with the passed Series, thus returning a DataFrame.
- ``DataFrame/DataFrame``: by default compute the statistic for matching column
names, returning a DataFrame. If the keyword argument ``pairwise=True`` is
passed then computes the statistic for each pair of columns, returning a
``MultiIndexed DataFrame`` whose ``index`` are the dates in question (see [the next section](#stats-moments-corr-pairwise)).
For example:
``` python
In [78]: df = pd.DataFrame(np.random.randn(1000, 4),
....: index=pd.date_range('1/1/2000', periods=1000),
....: columns=['A', 'B', 'C', 'D'])
....:
In [79]: df = df.cumsum()
In [80]: df2 = df[:20]
In [81]: df2.rolling(window=5).corr(df2['B'])
Out[81]:
A B C D
2000-01-01 NaN NaN NaN NaN
2000-01-02 NaN NaN NaN NaN
2000-01-03 NaN NaN NaN NaN
2000-01-04 NaN NaN NaN NaN
2000-01-05 0.768775 1.0 -0.977990 0.800252
... ... ... ... ...
2000-01-16 0.691078 1.0 0.807450 -0.939302
2000-01-17 0.274506 1.0 0.582601 -0.902954
2000-01-18 0.330459 1.0 0.515707 -0.545268
2000-01-19 0.046756 1.0 -0.104334 -0.419799
2000-01-20 -0.328241 1.0 -0.650974 -0.777777
[20 rows x 4 columns]
```
### Computing rolling pairwise covariances and correlations
In financial data analysis and other fields it’s common to compute covariance
and correlation matrices for a collection of time series. Often one is also
interested in moving-window covariance and correlation matrices. This can be
done by passing the ``pairwise`` keyword argument, which in the case of
``DataFrame`` inputs will yield a MultiIndexed ``DataFrame`` whose ``index`` are the dates in
question. In the case of a single DataFrame argument the ``pairwise`` argument
can even be omitted:
::: tip Note
Missing values are ignored and each entry is computed using the pairwise
complete observations. Please see the [covariance section](#computation-covariance) for [caveats](#computation-covariance-caveats) associated with this method of
calculating covariance and correlation matrices.
:::
``` python
In [82]: covs = (df[['B', 'C', 'D']].rolling(window=50)
....: .cov(df[['A', 'B', 'C']], pairwise=True))
....:
In [83]: covs.loc['2002-09-22':]
Out[83]:
B C D
2002-09-22 A 1.367467 8.676734 -8.047366
B 3.067315 0.865946 -1.052533
C 0.865946 7.739761 -4.943924
2002-09-23 A 0.910343 8.669065 -8.443062
B 2.625456 0.565152 -0.907654
C 0.565152 7.825521 -5.367526
2002-09-24 A 0.463332 8.514509 -8.776514
B 2.306695 0.267746 -0.732186
C 0.267746 7.771425 -5.696962
2002-09-25 A 0.467976 8.198236 -9.162599
B 2.307129 0.267287 -0.754080
C 0.267287 7.466559 -5.822650
2002-09-26 A 0.545781 7.899084 -9.326238
B 2.311058 0.322295 -0.844451
C 0.322295 7.038237 -5.684445
```
``` python
In [84]: correls = df.rolling(window=50).corr()
In [85]: correls.loc['2002-09-22':]
Out[85]:
A B C D
2002-09-22 A 1.000000 0.186397 0.744551 -0.769767
B 0.186397 1.000000 0.177725 -0.240802
C 0.744551 0.177725 1.000000 -0.712051
D -0.769767 -0.240802 -0.712051 1.000000
2002-09-23 A 1.000000 0.134723 0.743113 -0.758758
... ... ... ... ...
2002-09-25 D -0.739160 -0.164179 -0.704686 1.000000
2002-09-26 A 1.000000 0.087756 0.727792 -0.736562
B 0.087756 1.000000 0.079913 -0.179477
C 0.727792 0.079913 1.000000 -0.692303
D -0.736562 -0.179477 -0.692303 1.000000
[20 rows x 4 columns]
```
You can efficiently retrieve the time series of correlations between two
columns by reshaping and indexing:
``` python
In [86]: correls.unstack(1)[('A', 'C')].plot()
Out[86]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f8dc79e8>
```
![rolling_corr_pairwise_ex](https://static.pypandas.cn/public/static/images/rolling_corr_pairwise_ex.png)
## Aggregation
Once the ``Rolling``, ``Expanding`` or ``EWM`` objects have been created, several methods are available to
perform multiple computations on the data. These operations are similar to the [aggregating API](https://pandas.pydata.org/pandas-docs/stable/getting_started/basics.html#basics-aggregate),
[groupby API](groupby.html#groupby-aggregate), and [resample API](timeseries.html#timeseries-aggregate).
``` python
In [87]: dfa = pd.DataFrame(np.random.randn(1000, 3),
....: index=pd.date_range('1/1/2000', periods=1000),
....: columns=['A', 'B', 'C'])
....:
In [88]: r = dfa.rolling(window=60, min_periods=1)
In [89]: r
Out[89]: Rolling [window=60,min_periods=1,center=False,axis=0]
```
We can aggregate by passing a function to the entire DataFrame, or select a
Series (or multiple Series) via standard ``__getitem__``.
``` python
In [90]: r.aggregate(np.sum)
Out[90]:
A B C
2000-01-01 -0.289838 -0.370545 -1.284206
2000-01-02 -0.216612 -1.675528 -1.169415
2000-01-03 1.154661 -1.634017 -1.566620
2000-01-04 2.969393 -4.003274 -1.816179
2000-01-05 4.690630 -4.682017 -2.717209
... ... ... ...
2002-09-22 2.860036 -9.270337 6.415245
2002-09-23 3.510163 -8.151439 5.177219
2002-09-24 6.524983 -10.168078 5.792639
2002-09-25 6.409626 -9.956226 5.704050
2002-09-26 5.093787 -7.074515 6.905823
[1000 rows x 3 columns]
In [91]: r['A'].aggregate(np.sum)
Out[91]:
2000-01-01 -0.289838
2000-01-02 -0.216612
2000-01-03 1.154661
2000-01-04 2.969393
2000-01-05 4.690630
...
2002-09-22 2.860036
2002-09-23 3.510163
2002-09-24 6.524983
2002-09-25 6.409626
2002-09-26 5.093787
Freq: D, Name: A, Length: 1000, dtype: float64
In [92]: r[['A', 'B']].aggregate(np.sum)
Out[92]:
A B
2000-01-01 -0.289838 -0.370545
2000-01-02 -0.216612 -1.675528
2000-01-03 1.154661 -1.634017
2000-01-04 2.969393 -4.003274
2000-01-05 4.690630 -4.682017
... ... ...
2002-09-22 2.860036 -9.270337
2002-09-23 3.510163 -8.151439
2002-09-24 6.524983 -10.168078
2002-09-25 6.409626 -9.956226
2002-09-26 5.093787 -7.074515
[1000 rows x 2 columns]
```
As you can see, the result of the aggregation will have the selected columns, or all
columns if none are selected.
### Applying multiple functions
With windowed ``Series`` you can also pass a list of functions to do
aggregation with, outputting a DataFrame:
``` python
In [93]: r['A'].agg([np.sum, np.mean, np.std])
Out[93]:
sum mean std
2000-01-01 -0.289838 -0.289838 NaN
2000-01-02 -0.216612 -0.108306 0.256725
2000-01-03 1.154661 0.384887 0.873311
2000-01-04 2.969393 0.742348 1.009734
2000-01-05 4.690630 0.938126 0.977914
... ... ... ...
2002-09-22 2.860036 0.047667 1.132051
2002-09-23 3.510163 0.058503 1.134296
2002-09-24 6.524983 0.108750 1.144204
2002-09-25 6.409626 0.106827 1.142913
2002-09-26 5.093787 0.084896 1.151416
[1000 rows x 3 columns]
```
On a windowed DataFrame, you can pass a list of functions to apply to each
column, which produces an aggregated result with a hierarchical index:
``` python
In [94]: r.agg([np.sum, np.mean])
Out[94]:
A B C
sum mean sum mean sum mean
2000-01-01 -0.289838 -0.289838 -0.370545 -0.370545 -1.284206 -1.284206
2000-01-02 -0.216612 -0.108306 -1.675528 -0.837764 -1.169415 -0.584708
2000-01-03 1.154661 0.384887 -1.634017 -0.544672 -1.566620 -0.522207
2000-01-04 2.969393 0.742348 -4.003274 -1.000819 -1.816179 -0.454045
2000-01-05 4.690630 0.938126 -4.682017 -0.936403 -2.717209 -0.543442
... ... ... ... ... ... ...
2002-09-22 2.860036 0.047667 -9.270337 -0.154506 6.415245 0.106921
2002-09-23 3.510163 0.058503 -8.151439 -0.135857 5.177219 0.086287
2002-09-24 6.524983 0.108750 -10.168078 -0.169468 5.792639 0.096544
2002-09-25 6.409626 0.106827 -9.956226 -0.165937 5.704050 0.095068
2002-09-26 5.093787 0.084896 -7.074515 -0.117909 6.905823 0.115097
[1000 rows x 6 columns]
```
Passing a dict of functions has different behavior by default, see the next
section.
### Applying different functions to DataFrame columns
By passing a dict to ``aggregate`` you can apply a different aggregation to the
columns of a ``DataFrame``:
``` python
In [95]: r.agg({'A': np.sum, 'B': lambda x: np.std(x, ddof=1)})
Out[95]:
A B
2000-01-01 -0.289838 NaN
2000-01-02 -0.216612 0.660747
2000-01-03 1.154661 0.689929
2000-01-04 2.969393 1.072199
2000-01-05 4.690630 0.939657
... ... ...
2002-09-22 2.860036 1.113208
2002-09-23 3.510163 1.132381
2002-09-24 6.524983 1.080963
2002-09-25 6.409626 1.082911
2002-09-26 5.093787 1.136199
[1000 rows x 2 columns]
```
The function names can also be strings. In order for a string to be valid it
must be implemented on the windowed object
``` python
In [96]: r.agg({'A': 'sum', 'B': 'std'})
Out[96]:
A B
2000-01-01 -0.289838 NaN
2000-01-02 -0.216612 0.660747
2000-01-03 1.154661 0.689929
2000-01-04 2.969393 1.072199
2000-01-05 4.690630 0.939657
... ... ...
2002-09-22 2.860036 1.113208
2002-09-23 3.510163 1.132381
2002-09-24 6.524983 1.080963
2002-09-25 6.409626 1.082911
2002-09-26 5.093787 1.136199
[1000 rows x 2 columns]
```
Furthermore you can pass a nested dict to indicate different aggregations on different columns.
``` python
In [97]: r.agg({'A': ['sum', 'std'], 'B': ['mean', 'std']})
Out[97]:
A B
sum std mean std
2000-01-01 -0.289838 NaN -0.370545 NaN
2000-01-02 -0.216612 0.256725 -0.837764 0.660747
2000-01-03 1.154661 0.873311 -0.544672 0.689929
2000-01-04 2.969393 1.009734 -1.000819 1.072199
2000-01-05 4.690630 0.977914 -0.936403 0.939657
... ... ... ... ...
2002-09-22 2.860036 1.132051 -0.154506 1.113208
2002-09-23 3.510163 1.134296 -0.135857 1.132381
2002-09-24 6.524983 1.144204 -0.169468 1.080963
2002-09-25 6.409626 1.142913 -0.165937 1.082911
2002-09-26 5.093787 1.151416 -0.117909 1.136199
[1000 rows x 4 columns]
```
## Expanding windows
A common alternative to rolling statistics is to use an *expanding* window,
which yields the value of the statistic with all the data available up to that
point in time.
These follow a similar interface to ``.rolling``, with the ``.expanding`` method
returning an ``Expanding`` object.
As these calculations are a special case of rolling statistics,
they are implemented in pandas such that the following two calls are equivalent:
``` python
In [98]: df.rolling(window=len(df), min_periods=1).mean()[:5]
Out[98]:
A B C D
2000-01-01 0.314226 -0.001675 0.071823 0.892566
2000-01-02 0.654522 -0.171495 0.179278 0.853361
2000-01-03 0.708733 -0.064489 -0.238271 1.371111
2000-01-04 0.987613 0.163472 -0.919693 1.566485
2000-01-05 1.426971 0.288267 -1.358877 1.808650
In [99]: df.expanding(min_periods=1).mean()[:5]
Out[99]:
A B C D
2000-01-01 0.314226 -0.001675 0.071823 0.892566
2000-01-02 0.654522 -0.171495 0.179278 0.853361
2000-01-03 0.708733 -0.064489 -0.238271 1.371111
2000-01-04 0.987613 0.163472 -0.919693 1.566485
2000-01-05 1.426971 0.288267 -1.358877 1.808650
```
These have a similar set of methods to ``.rolling`` methods.
### Method summary
Function | Description
---|---
[count()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.count.html#pandas.core.window.Expanding.count) | Number of non-null observations
[sum()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.sum.html#pandas.core.window.Expanding.sum) | Sum of values
[mean()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.mean.html#pandas.core.window.Expanding.mean) | Mean of values
[median()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.median.html#pandas.core.window.Expanding.median) | Arithmetic median of values
[min()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.min.html#pandas.core.window.Expanding.min) | Minimum
[max()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.max.html#pandas.core.window.Expanding.max) | Maximum
[std()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.std.html#pandas.core.window.Expanding.std) | Unbiased standard deviation
[var()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.var.html#pandas.core.window.Expanding.var) | Unbiased variance
[skew()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.skew.html#pandas.core.window.Expanding.skew) | Unbiased skewness (3rd moment)
[kurt()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.kurt.html#pandas.core.window.Expanding.kurt) | Unbiased kurtosis (4th moment)
[quantile()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.quantile.html#pandas.core.window.Expanding.quantile) | Sample quantile (value at %)
[apply()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.apply.html#pandas.core.window.Expanding.apply) | Generic apply
[cov()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.cov.html#pandas.core.window.Expanding.cov) | Unbiased covariance (binary)
[corr()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.corr.html#pandas.core.window.Expanding.corr) | Correlation (binary)
Aside from not having a ``window`` parameter, these functions have the same
interfaces as their ``.rolling`` counterparts. Like above, the parameters they
all accept are:
- ``min_periods``: threshold of non-null data points to require. Defaults to
minimum needed to compute statistic. No ``NaNs`` will be output once
``min_periods`` non-null data points have been seen.
- ``center``: boolean, whether to set the labels at the center (default is False).
::: tip Note
The output of the ``.rolling`` and ``.expanding`` methods do not return a
``NaN`` if there are at least ``min_periods`` non-null values in the current
window. For example:
``` python
In [100]: sn = pd.Series([1, 2, np.nan, 3, np.nan, 4])
In [101]: sn
Out[101]:
0 1.0
1 2.0
2 NaN
3 3.0
4 NaN
5 4.0
dtype: float64
In [102]: sn.rolling(2).max()
Out[102]:
0 NaN
1 2.0
2 NaN
3 NaN
4 NaN
5 NaN
dtype: float64
In [103]: sn.rolling(2, min_periods=1).max()
Out[103]:
0 1.0
1 2.0
2 2.0
3 3.0
4 3.0
5 4.0
dtype: float64
```
In case of expanding functions, this differs from [``cumsum()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cumsum.html#pandas.DataFrame.cumsum),
[``cumprod()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cumprod.html#pandas.DataFrame.cumprod), [``cummax()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cummax.html#pandas.DataFrame.cummax),
and [``cummin()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.cummin.html#pandas.DataFrame.cummin), which return ``NaN`` in the output wherever
a ``NaN`` is encountered in the input. In order to match the output of ``cumsum``
with ``expanding``, use [``fillna()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.DataFrame.fillna.html#pandas.DataFrame.fillna):
``` python
In [104]: sn.expanding().sum()
Out[104]:
0 1.0
1 3.0
2 3.0
3 6.0
4 6.0
5 10.0
dtype: float64
In [105]: sn.cumsum()
Out[105]:
0 1.0
1 3.0
2 NaN
3 6.0
4 NaN
5 10.0
dtype: float64
In [106]: sn.cumsum().fillna(method='ffill')
Out[106]:
0 1.0
1 3.0
2 3.0
3 6.0
4 6.0
5 10.0
dtype: float64
```
:::
An expanding window statistic will be more stable (and less responsive) than
its rolling window counterpart as the increasing window size decreases the
relative impact of an individual data point. As an example, here is the
[``mean()``](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.Expanding.mean.html#pandas.core.window.Expanding.mean) output for the previous time series dataset:
``` python
In [107]: s.plot(style='k--')
Out[107]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6607b14400>
In [108]: s.expanding().mean().plot(style='k')
Out[108]: <matplotlib.axes._subplots.AxesSubplot at 0x7f6607b14400>
```
![expanding_mean_frame](https://static.pypandas.cn/public/static/images/expanding_mean_frame.png)
## Exponentially weighted windows
A related set of functions are exponentially weighted versions of several of
the above statistics. A similar interface to ``.rolling`` and ``.expanding`` is accessed
through the ``.ewm`` method to receive an ``EWM`` object.
A number of expanding EW (exponentially weighted)
methods are provided:
Function | Description
---|---
[mean()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.mean.html#pandas.core.window.EWM.mean) | EW moving average
[var()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.var.html#pandas.core.window.EWM.var) | EW moving variance
[std()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.std.html#pandas.core.window.EWM.std) | EW moving standard deviation
[corr()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.corr.html#pandas.core.window.EWM.corr) | EW moving correlation
[cov()](https://pandas.pydata.org/pandas-docs/stable/reference/api/pandas.core.window.EWM.cov.html#pandas.core.window.EWM.cov) | EW moving covariance
In general, a weighted moving average is calculated as
<div class="math notranslate nohighlight">
\[y_t = \frac{\sum_{i=0}^t w_i x_{t-i}}{\sum_{i=0}^t w_i},\]
</div>
where \\\(x_t\\\) is the input, \\(y_t\\) is the result and the \\(w_i\\)
are the weights.
The EW functions support two variants of exponential weights.
The default, ``adjust=True``, uses the weights \\(w_i = (1 - \alpha)^i\\)
which gives
<div class="math notranslate nohighlight">
\[y_t = \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...
+ (1 - \alpha)^t x_{0}}{1 + (1 - \alpha) + (1 - \alpha)^2 + ...
+ (1 - \alpha)^t}\]
</div>
When ``adjust=False`` is specified, moving averages are calculated as
<div class="math notranslate nohighlight">
\[\begin{split}y_0 &amp;= x_0 \\
y_t &amp;= (1 - \alpha) y_{t-1} + \alpha x_t,\end{split}\]
</div>
which is equivalent to using weights
<div class="math notranslate nohighlight">
\[\begin{split}w_i = \begin{cases}
\alpha (1 - \alpha)^i &amp; \text{if } i &lt; t \\
(1 - \alpha)^i &amp; \text{if } i = t.
\end{cases}\end{split}\]
</div>
::: tip Note
These equations are sometimes written in terms of \\(\alpha' = 1 - \alpha\\), e.g.
<div class="math notranslate nohighlight">
\[y_t = \alpha' y_{t-1} + (1 - \alpha') x_t.\]
</div>
:::
The difference between the above two variants arises because we are
dealing with series which have finite history. Consider a series of infinite
history, with ``adjust=True``:
<div class="math notranslate nohighlight">
\[y_t = \alpha' y_{t-1} + (1 - \alpha') x_t.\]
</div>
Noting that the denominator is a geometric series with initial term equal to 1
and a ratio of \\(1 - \alpha\\) we have
<div class="math notranslate nohighlight">
\[\begin{split}y_t &amp;= \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...}
{\frac{1}{1 - (1 - \alpha)}}\\
&amp;= [x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...] \alpha \\
&amp;= \alpha x_t + [(1-\alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...]\alpha \\
&amp;= \alpha x_t + (1 - \alpha)[x_{t-1} + (1 - \alpha) x_{t-2} + ...]\alpha\\
&amp;= \alpha x_t + (1 - \alpha) y_{t-1}\end{split}\]
</div>
which is the same expression as ``adjust=False`` above and therefore
shows the equivalence of the two variants for infinite series.
When ``adjust=False``, we have \\(y_0 = x_0\\) and
\\(y_t = \alpha x_t + (1 - \alpha) y_{t-1}\\).
Therefore, there is an assumption that \\(x_0\\) is not an ordinary value
but rather an exponentially weighted moment of the infinite series up to that
point.
One must have \\(0 &lt; \alpha \leq 1\\), and while since version 0.18.0
it has been possible to pass \\(\alpha\\) directly, it’s often easier
to think about either the **span**, **center of mass (com)** or **half-life**
of an EW moment:
<div class="math notranslate nohighlight">
\[\begin{split}\alpha =
\begin{cases}
\frac{2}{s + 1}, &amp; \text{for span}\ s \geq 1\\
\frac{1}{1 + c}, &amp; \text{for center of mass}\ c \geq 0\\
1 - \exp^{\frac{\log 0.5}{h}}, &amp; \text{for half-life}\ h &gt; 0
\end{cases}\end{split}\]
</div>
One must specify precisely one of **span**, **center of mass**, **half-life**
and **alpha** to the EW functions:
- **Span** corresponds to what is commonly called an “N-day EW moving average”.
- **Center of mass** has a more physical interpretation and can be thought of
in terms of span: \\(c = (s - 1) / 2\\).
- **Half-life** is the period of time for the exponential weight to reduce to
one half.
- **Alpha** specifies the smoothing factor directly.
Here is an example for a univariate time series:
``` python
In [109]: s.plot(style='k--')
Out[109]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f7537278>
In [110]: s.ewm(span=20).mean().plot(style='k')
Out[110]: <matplotlib.axes._subplots.AxesSubplot at 0x7f65f7537278>
```
![ewma_ex](https://static.pypandas.cn/public/static/images/ewma_ex.png)
EWM has a ``min_periods`` argument, which has the same
meaning it does for all the ``.expanding`` and ``.rolling`` methods:
no output values will be set until at least ``min_periods`` non-null values
are encountered in the (expanding) window.
EWM also has an ``ignore_na`` argument, which determines how
intermediate null values affect the calculation of the weights.
When ``ignore_na=False`` (the default), weights are calculated based on absolute
positions, so that intermediate null values affect the result.
When ``ignore_na=True``,
weights are calculated by ignoring intermediate null values.
For example, assuming ``adjust=True``, if ``ignore_na=False``, the weighted
average of ``3, NaN, 5`` would be calculated as
<div class="math notranslate nohighlight">
\[\frac{(1-\alpha)^2 \cdot 3 + 1 \cdot 5}{(1-\alpha)^2 + 1}.\]
</div>
Whereas if ``ignore_na=True``, the weighted average would be calculated as
<div class="math notranslate nohighlight">
\[\frac{(1-\alpha) \cdot 3 + 1 \cdot 5}{(1-\alpha) + 1}.\]
</div>
The ``var()``, ``std()``, and ``cov()`` functions have a ``bias`` argument,
specifying whether the result should contain biased or unbiased statistics.
For example, if ``bias=True``, ``ewmvar(x)`` is calculated as
``ewmvar(x) = ewma(x**2) - ewma(x)**2``;
whereas if ``bias=False`` (the default), the biased variance statistics
are scaled by debiasing factors
<div class="math notranslate nohighlight">
\[\frac{\left(\sum_{i=0}^t w_i\right)^2}{\left(\sum_{i=0}^t w_i\right)^2 - \sum_{i=0}^t w_i^2}.\]
</div>
(For \\(w_i = 1\\), this reduces to the usual \\(N / (N - 1)\\) factor,
with \\(N = t + 1\\).)
See [Weighted Sample Variance](http://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Weighted_sample_variance)
on Wikipedia for further details.
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