# Computational tools

## Statistical Functions

### Percent Change

`Series`, `DataFrame`, and `Panel` all have a method `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).

    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

    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

The `Series` object has a method `cov` to compute covariance between series (excluding NA/null values).

    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]: 0.00068010881743107965

Analogously, `DataFrame` has a method `cov` to compute pairwise covariances among the series in the DataFrame, also excluding NA/null values.

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 for more details.

    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.

    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

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.

Note

Please see the caveats associated with this method of calculating correlation matrices in the covariance section.

    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]: 0.013479040400098752

    In [18]: frame['a'].corr(frame['b'], method='spearman')
    Out[18]: -0.0072898851595406371

    # 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:

    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

A related method `corrwith` is implemented on DataFrame to compute the correlation between like-labeled Series contained in different DataFrame objects.

    In [25]: index = ['a', 'b', 'c', 'd', 'e']

    In [26]: columns = ['one', 'two', 'three', 'four']

    In [27]: df1 = pd.DataFrame(np.random.randn(5, 4), index=index, columns=columns)

    In [28]: df2 = pd.DataFrame(np.random.randn(4, 4), index=index[:4], columns=columns)

    In [29]: df1.corrwith(df2)
    Out[29]:
    one     -0.125501
    two     -0.493244
    three    0.344056
    four     0.004183
    dtype: float64

    In [30]: df2.corrwith(df1, axis=1)
    Out[30]:
    a   -0.675817
    b    0.458296
    c    0.190809
    d   -0.186275
    e         NaN
    dtype: float64

### Data ranking

The `rank` method produces a data ranking with ties being assigned the mean of the ranks (by default) for the group:

    In [31]: s = pd.Series(np.random.np.random.randn(5), index=list('abcde'))

    In [32]: s['d'] = s['b'] # so there's a tie

    In [33]: s.rank()
    Out[33]:
    a    5.0
    b    2.5
    c    1.0
    d    2.5
    e    4.0
    dtype: float64

`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.

    In [34]: df = pd.DataFrame(np.random.np.random.randn(10, 6))

    In [35]: df[4] = df[2][:5] # some ties

    In [36]: df
    Out[36]:
              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 [37]: df.rank(1)
    Out[37]:
         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

Warning

Prior to version 0.18.0, `pd.rolling_*`, `pd.expanding_*`, and `pd.ewm*` were module level functions and are now deprecated. These are replaced by using the `Rolling`, `Expanding` and `EWM`. objects and a corresponding method call.

The deprecation warning will show the new syntax, see an example here You can view the previous documentation here

For working with data, a number of windows 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.

Starting in version 0.18.1, the `rolling()` and `expanding()` functions can be used directly from DataFrameGroupBy objects, see the groupby docs.

Note

The API for window statistics is quite similar to the way one works with `GroupBy` objects, see the documentation here

We work with `rolling`, `expanding` and `exponentially weighted` data through the corresponding objects, `Rolling`, `Expanding` and `EWM`.

    In [38]: s = pd.Series(np.random.randn(1000), index=pd.date_range('1/1/2000', periods=1000))

    In [39]: s = s.cumsum()

    In [40]: s
    Out[40]:
    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
    2000-01-06    -1.869391
    2000-01-07     0.563674
                    ...
    2002-09-20   -68.233054
    2002-09-21   -66.765687
    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`.

    In [41]: r = s.rolling(window=60)

    In [42]: r
    Out[42]: Rolling [window=60,center=False,axis=0]

These object provide tab-completion of the avaible methods and properties.

    In [14]: r.
    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)

Warning

The `freq` and `how` arguments were in the API prior to 0.18.0 changes. These are deprecated in the new API. You can simply resample the input prior to creating a window function.

For example, instead of `s.rolling(window=5,freq='D').max()` to get the max value on a rolling 5 Day window, one could use `s.resample('D').max().rolling(window=5).max()`, which first resamples the data to daily data, then provides a rolling 5 day window.

We can then call methods on these `rolling` objects. These return like-indexed objects:

    In [43]: r.mean()
    Out[43]:
    2000-01-01          NaN
    2000-01-02          NaN
    2000-01-03          NaN
    2000-01-04          NaN
    2000-01-05          NaN
    2000-01-06          NaN
    2000-01-07          NaN
                    ...
    2002-09-20   -62.694135
    2002-09-21   -62.812190
    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

    In [44]: s.plot(style='k--')
    Out[44]: <matplotlib.axes._subplots.AxesSubplot at 0x7faad2276790>

    In [45]: r.mean().plot(style='k')
    Out[45]: <matplotlib.axes._subplots.AxesSubplot at 0x7faad2276790>

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:

    In [46]: df = pd.DataFrame(np.random.randn(1000, 4),
       ....:                   index=pd.date_range('1/1/2000', periods=1000),
       ....:                   columns=['A', 'B', 'C', 'D'])
       ....:

    In [47]: df = df.cumsum()

    In [48]: df.rolling(window=60).sum().plot(subplots=True)
    Out[48]:
    array([<matplotlib.axes._subplots.AxesSubplot object at 0x7faad4c8af90>,
           <matplotlib.axes._subplots.AxesSubplot object at 0x7faad3a3d790>,
           <matplotlib.axes._subplots.AxesSubplot object at 0x7faad32bdfd0>,
           <matplotlib.axes._subplots.AxesSubplot object at 0x7faad3b149d0>], dtype=object)

### Method Summary

We provide a number of the common statistical functions:

Method | Description
--- | ---
`count()` | Number of non-null observations
`sum()` | Sum of values
`mean()` | Mean of values
`median()` | Arithmetic median of values
`min()` | Minimum
`max()` | Maximum
`std()` | Bessel-corrected sample standard deviation
`var()` | Unbiased variance
`skew()` | Sample skewness (3rd moment)
`kurt()` | Sample kurtosis (4th moment)
`quantile()` | Sample quantile (value at %)
`apply()` | Generic apply
`cov()` | Unbiased covariance (binary)
`corr()` | Correlation (binary)

The `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:

    In [49]: mad = lambda x: np.fabs(x - x.mean()).mean()

    In [50]: s.rolling(window=60).apply(mad).plot(style='k')
    Out[50]: <matplotlib.axes._subplots.AxesSubplot at 0x7faad7246450>

### 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()` | Sum of values
`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:

- `boxcar`
- `triang`
- `blackman`
- `hamming`
- `bartlett`
- `parzen`
- `bohman`
- `blackmanharris`
- `nuttall`
- `barthann`
- `kaiser` (needs beta)
- `gaussian` (needs std)
- `general_gaussian` (needs power, width)
- `slepian` (needs width).

    In [51]: ser = pd.Series(np.random.randn(10), index=pd.date_range('1/1/2000', periods=10))

    In [52]: ser.rolling(window=5, win_type='triang').mean()
    Out[52]:
    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()`.

    In [53]: ser.rolling(window=5, win_type='boxcar').mean()
    Out[53]:
    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 [54]: ser.rolling(window=5).mean()
    Out[54]:
    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:

    In [55]: ser.rolling(window=5, win_type='gaussian').mean(std=0.1)
    Out[55]:
    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

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.

    In [56]: dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
       ....:                    index=pd.date_range('20130101 09:00:00', periods=5, freq='s'))
       ....:

    In [57]: dft
    Out[57]:
                           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.

    In [58]: dft.rolling(2).sum()
    Out[58]:
                           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 [59]: dft.rolling(2, min_periods=1).sum()
    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  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.

    In [60]: dft.rolling('2s').sum()
    Out[60]:
                           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.

    In [61]: 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 [62]: dft
    Out[62]:
                           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 [63]: dft.rolling(2).sum()
    Out[63]:
                           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.

    In [64]: dft.rolling('2s').sum()
    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  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.

    In [65]: dft = dft.reset_index()

    In [66]: dft
    Out[66]:
                      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 [67]: dft.rolling('2s', on='foo').sum()
    Out[67]:
                      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.

    In [68]: 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 [69]: df["right"] = df.rolling('2s', closed='right').x.sum()  # default

    In [70]: df["both"] = df.rolling('2s', closed='both').x.sum()

    In [71]: df["left"] = df.rolling('2s', closed='left').x.sum()

    In [72]: df["neither"] = df.rolling('2s', closed='neither').x.sum()

    In [73]: df
    Out[73]:
                         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. 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 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.

    In [74]: ser.rolling(window=5).mean()
    Out[74]:
    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 [75]: ser.rolling(window=5, center=True).mean()
    Out[75]:
    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()` and `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).

For example:

    In [76]: df = pd.DataFrame(np.random.randn(1000, 4),
       ....:                   index=pd.date_range('1/1/2000', periods=1000),
       ....:                   columns=['A', 'B', 'C', 'D'])
       ....:

    In [77]: df = df.cumsum()

    In [78]: df2 = df[:20]

    In [79]: df2.rolling(window=5).corr(df2['B'])
    Out[79]:
                       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
    2000-01-06  0.744106  1.0 -0.967912  0.830021
    2000-01-07  0.683257  1.0 -0.928969  0.384916
    ...              ...  ...       ...       ...
    2000-01-14 -0.392318  1.0  0.570240 -0.591056
    2000-01-15  0.017217  1.0  0.649900 -0.896258
    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

Warning

Prior to version 0.20.0 if `pairwise=True` was passed, a `Panel` would be returned. This will now return a 2-level MultiIndexed DataFrame, see the whatsnew here

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:

Note

Missing values are ignored and each entry is computed using the pairwise complete observations. Please see the covariance section for caveats associated with this method of calculating covariance and correlation matrices.

    In [80]: covs = df[['B','C','D']].rolling(window=50).cov(df[['A','B','C']], pairwise=True)

    In [81]: covs.loc['2002-09-22':]
    Out[81]:
                         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

    In [82]: correls = df.rolling(window=50).corr()

    In [83]: correls.loc['2002-09-22':]
    Out[83]:
                         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
               B  0.134723  1.000000  0.124683 -0.209934
               C  0.743113  0.124683  1.000000 -0.719088
    ...                ...       ...       ...       ...
    2002-09-25 B  0.075157  1.000000  0.064399 -0.164179
               C  0.731888  0.064399  1.000000 -0.704686
               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:

    In [84]: correls.unstack(1)[('A', 'C')].plot()
    Out[84]: <matplotlib.axes._subplots.AxesSubplot at 0x7faad3d16710>

## 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, groupby API, and resample API.

    In [85]: dfa = pd.DataFrame(np.random.randn(1000, 3),
       ....:                    index=pd.date_range('1/1/2000', periods=1000),
       ....:                    columns=['A', 'B', 'C'])
       ....:

    In [86]: r = dfa.rolling(window=60,min_periods=1)

    In [87]: r
    Out[87]: 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.

    In [88]: r.aggregate(np.sum)
    Out[88]:
                       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
    2000-01-06  3.880630  -4.447700 -1.078947
    2000-01-07  4.001957  -2.884072 -3.116903
    ...              ...        ...       ...
    2002-09-20  2.652493 -10.528875  9.867805
    2002-09-21  0.844497  -9.280944  9.522649
    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 [89]: r['A'].aggregate(np.sum)
    Out[89]:
    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
    2000-01-06    3.880630
    2000-01-07    4.001957
                    ...
    2002-09-20    2.652493
    2002-09-21    0.844497
    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 [90]: r[['A','B']].aggregate(np.sum)
    Out[90]:
                       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
    2000-01-06  3.880630  -4.447700
    2000-01-07  4.001957  -2.884072
    ...              ...        ...
    2002-09-20  2.652493 -10.528875
    2002-09-21  0.844497  -9.280944
    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:

    In [91]: r['A'].agg([np.sum, np.mean, np.std])
    Out[91]:
                     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
    2000-01-06  3.880630  0.646772  1.128883
    2000-01-07  4.001957  0.571708  1.049487
    ...              ...       ...       ...
    2002-09-20  2.652493  0.044208  1.164919
    2002-09-21  0.844497  0.014075  1.148231
    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:

    In [92]: r.agg([np.sum, np.mean])
    Out[92]:
                       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
    2000-01-06  3.880630  0.646772  -4.447700 -0.741283 -1.078947 -0.179825
    2000-01-07  4.001957  0.571708  -2.884072 -0.412010 -3.116903 -0.445272
    ...              ...       ...        ...       ...       ...       ...
    2002-09-20  2.652493  0.044208 -10.528875 -0.175481  9.867805  0.164463
    2002-09-21  0.844497  0.014075  -9.280944 -0.154682  9.522649  0.158711
    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`:

    In [93]: r.agg({'A' : np.sum,
       ....:        'B' : lambda x: np.std(x, ddof=1)})
       ....:
    Out[93]:
                       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
    2000-01-06  3.880630  0.966848
    2000-01-07  4.001957  1.240137
    ...              ...       ...
    2002-09-20  2.652493  1.114814
    2002-09-21  0.844497  1.113220
    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

    In [94]: r.agg({'A' : 'sum', 'B' : 'std'})
    Out[94]:
                       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
    2000-01-06  3.880630  0.966848
    2000-01-07  4.001957  1.240137
    ...              ...       ...
    2002-09-20  2.652493  1.114814
    2002-09-21  0.844497  1.113220
    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.

    In [95]: r.agg({'A' : ['sum','std'], 'B' : ['mean','std'] })
    Out[95]:
                       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
    2000-01-06  3.880630  1.128883 -0.741283  0.966848
    2000-01-07  4.001957  1.049487 -0.412010  1.240137
    ...              ...       ...       ...       ...
    2002-09-20  2.652493  1.164919 -0.175481  1.114814
    2002-09-21  0.844497  1.148231 -0.154682  1.113220
    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:

    In [96]: df.rolling(window=len(df), min_periods=1).mean()[:5]
    Out[96]:
                       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 [97]: df.expanding(min_periods=1).mean()[:5]
    Out[97]:
                       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()` | Number of non-null observations
`sum()` | Sum of values
`mean()` | Mean of values
`median()` | Arithmetic median of values
`min()` | Minimum
`max()` | Maximum
`std()` | Unbiased standard deviation
`var()` | Unbiased variance
`skew()` | Unbiased skewness (3rd moment)
`kurt()` | Unbiased kurtosis (4th moment)
`quantile()` | Sample quantile (value at %)
`apply()` | Generic apply
`cov()` | Unbiased covariance (binary)
`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)

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,

    In [98]: sn = pd.Series([1, 2, np.nan, 3, np.nan, 4])

    In [99]: sn
    Out[99]:
    0    1.0
    1    2.0
    2    NaN
    3    3.0
    4    NaN
    5    4.0
    dtype: float64

    In [100]: sn.rolling(2).max()
    Out[100]:
    0    NaN
    1    2.0
    2    NaN
    3    NaN
    4    NaN
    5    NaN
    dtype: float64

    In [101]: sn.rolling(2, min_periods=1).max()
    Out[101]:
    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()`, `cumprod()`, `cummax()`, and `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()`:

    In [102]: sn.expanding().sum()
    Out[102]:
    0     1.0
    1     3.0
    2     3.0
    3     6.0
    4     6.0
    5    10.0
    dtype: float64

    In [103]: sn.cumsum()
    Out[103]:
    0     1.0
    1     3.0
    2     NaN
    3     6.0
    4     NaN
    5    10.0
    dtype: float64

    In [104]: sn.cumsum().fillna(method='ffill')
    Out[104]:
    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()` output for the previous time series dataset:

    In [105]: s.plot(style='k--')
    Out[105]: <matplotlib.axes._subplots.AxesSubplot at 0x7faadb782e50>

    In [106]: s.expanding().mean().plot(style='k')
    Out[106]: <matplotlib.axes._subplots.AxesSubplot at 0x7faadb782e50>

## 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()` | EW moving average
`var()` | EW moving variance
`std()` | EW moving standard deviation
`corr()` | EW moving correlation
`cov()` | EW moving covariance

In general, a weighted moving average is calculated as

\[y_t = \frac{\sum_{i=0}^t w_i x_{t-i}}{\sum_{i=0}^t w_i},\]

where \(x_t\) is the input and \(y_t\) is the result.

The EW functions support two variants of exponential weights. The default, `adjust=True`, uses the weights \(w_i = (1 - \alpha)^i\) which gives

\[y_t = \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ... + (1 - \alpha)^t x_{0}}{1 + (1 - \alpha) + (1 - \alpha)^2 + ... + (1 - \alpha)^t}\]

When `adjust=False` is specified, moving averages are calculated as

\[\begin{split}y_0 &= x_0 \\ y_t &= (1 - \alpha) y_{t-1} + \alpha x_t,\end{split}\]

which is equivalent to using weights

\[\begin{split}w_i = \begin{cases} \alpha (1 - \alpha)^i & \text{if } i < t \\ (1 - \alpha)^i & \text{if } i = t. \end{cases}\end{split}\]

Note

These equations are sometimes written in terms of \(\alpha' = 1 - \alpha\), e.g.

\[y_t = \alpha' y_{t-1} + (1 - \alpha') x_t.\]

The difference between the above two variants arises because we are dealing with series which have finite history. Consider a series of infinite history:

\[y_t = \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...} {1 + (1 - \alpha) + (1 - \alpha)^2 + ...}\]

Noting that the denominator is a geometric series with initial term equal to 1 and a ratio of \(1 - \alpha\) we have

\[\begin{split}y_t &= \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...} {\frac{1}{1 - (1 - \alpha)}}\\ &= [x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...] \alpha \\ &= \alpha x_t + [(1-\alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ...]\alpha \\ &= \alpha x_t + (1 - \alpha)[x_{t-1} + (1 - \alpha) x_{t-2} + ...]\alpha\\ &= \alpha x_t + (1 - \alpha) y_{t-1}\end{split}\]

which shows the equivalence of the above two variants for infinite series. When `adjust=True` we have \(y_0 = x_0\) and from the last representation above we have \(y_t = \alpha x_t + (1 - \alpha) y_{t-1}\), 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 \(0 < \alpha \leq 1\), 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:

\[\begin{split}\alpha = \begin{cases} \frac{2}{s + 1}, & \text{for span}\ s \geq 1\\ \frac{1}{1 + c}, & \text{for center of mass}\ c \geq 0\\ 1 - \exp^{\frac{\log 0.5}{h}}, & \text{for half-life}\ h > 0 \end{cases}\end{split}\]

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:

    In [107]: s.plot(style='k--')
    Out[107]: <matplotlib.axes._subplots.AxesSubplot at 0x7faae1e9dcd0>

    In [108]: s.ewm(span=20).mean().plot(style='k')
    Out[108]: <matplotlib.axes._subplots.AxesSubplot at 0x7faae1e9dcd0>

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

\[\frac{(1-\alpha)^2 \cdot 3 + 1 \cdot 5}{(1-\alpha)^2 + 1}\]

Whereas if `ignore_na=True`, the weighted average would be calculated as

\[\frac{(1-\alpha) \cdot 3 + 1 \cdot 5}{(1-\alpha) + 1}.\]

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

\[\frac{\left(\sum_{i=0}^t w_i\right)^2}{\left(\sum_{i=0}^t w_i\right)^2 - \sum_{i=0}^t w_i^2}.\]

(For \(w_i = 1\), this reduces to the usual \(N / (N - 1)\) factor, with \(N = t + 1\).) See Weighted Sample Variance for further details.

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