# 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.ix[:5, 'a'] = np.nan

    In [12]: frame.ix[5:10, 'b'] = np.nan

    In [13]: frame.cov()
    Out[13]:
              a         b         c
    a  1.210090 -0.430629  0.018002
    b -0.430629  1.240960  0.347188
    c  0.018002  0.347188  1.301149

    In [14]: frame.cov(min_periods=12)
    Out[14]:
              a         b         c
    a  1.210090       NaN  0.018002
    b       NaN  1.240960  0.347188
    c  0.018002  0.347188  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.ix[::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.ix[:5, 'a'] = np.nan

    In [22]: frame.ix[5:10, 'b'] = np.nan

    In [23]: frame.corr()
    Out[23]:
              a         b         c
    a  1.000000 -0.076520  0.160092
    b -0.076520  1.000000  0.135967
    c  0.160092  0.135967  1.000000

    In [24]: frame.corr(min_periods=12)
    Out[24]:
              a         b         c
    a  1.000000       NaN  0.160092
    b       NaN  1.000000  0.135967
    c  0.160092  0.135967  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  3  1.5  5  1.5  6
    1  2  6  4.5  1  4.5  3
    2  1  6  3.5  5  3.5  2
    3  4  5  1.5  3  1.5  6
    4  5  3  1.5  4  1.5  6
    5  1  2  5.0  3  NaN  4
    6  4  5  3.0  1  NaN  2
    7  2  5  3.0  4  NaN  1
    8  2  5  3.0  4  NaN  1
    9  2  3  1.0  4  NaN  5

`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

## Moving (rolling) statistics / moments

For working with time series data, a number of functions are provided for computing common moving or rolling statistics. Among these are count, sum, mean, median, correlation, variance, covariance, standard deviation, skewness, and kurtosis. All of these methods are in the `pandas` namespace, but otherwise they can be found in `pandas.stats.moments`.

Function | Description
--- | ---
`rolling_count()` | Number of non-null observations
`rolling_sum()` | Sum of values
`rolling_mean()` | Mean of values
`rolling_median()` | Arithmetic median of values
`rolling_min()` | Minimum
`rolling_max()` | Maximum
`rolling_std()` | Unbiased standard deviation
`rolling_var()` | Unbiased variance
`rolling_skew()` | Unbiased skewness (3rd moment)
`rolling_kurt()` | Unbiased kurtosis (4th moment)
`rolling_quantile()` | Sample quantile (value at %)
`rolling_apply()` | Generic apply
`rolling_cov()` | Unbiased covariance (binary)
`rolling_corr()` | Correlation (binary)
`rolling_window()` | Moving window function

Generally these methods all have the same interface. The binary operators (e.g. `rolling_corr()`) take two Series or DataFrames. Otherwise, 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)
> - `freq`: optionally specify a frequency string or DateOffset to pre-conform the data to. Note that prior to pandas v0.8.0, a keyword argument `time_rule` was used instead of `freq` that referred to the legacy time rule constants
> - `how`: optionally specify method for down or re-sampling. Default is is min for `rolling_min()`, max for `rolling_max()`, median for `rolling_median()`, and mean for all other rolling functions. See `DataFrame.resample()`‘s how argument for more information.

These functions can be applied to ndarrays or Series objects:

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

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

    In [40]: ts.plot(style='k--')
    Out[40]: <matplotlib.axes._subplots.AxesSubplot at 0x7f095de1be50>

    In [41]: pd.rolling_mean(ts, 60).plot(style='k')
    Out[41]: <matplotlib.axes._subplots.AxesSubplot at 0x7f095de1be50>

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 [42]: df = pd.DataFrame(np.random.randn(1000, 4), index=ts.index,
       ....:                columns=['A', 'B', 'C', 'D'])
       ....:

    In [43]: df = df.cumsum()

    In [44]: pd.rolling_sum(df, 60).plot(subplots=True)
    Out[44]:
    array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f095f038a50>,
           <matplotlib.axes._subplots.AxesSubplot object at 0x7f095f05c190>,
           <matplotlib.axes._subplots.AxesSubplot object at 0x7f095dba32d0>,
           <matplotlib.axes._subplots.AxesSubplot object at 0x7f0960d0e5d0>], dtype=object)

The `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:

    In [45]: mad = lambda x: np.fabs(x - x.mean()).mean()

    In [46]: pd.rolling_apply(ts, 60, mad).plot(style='k')
    Out[46]: <matplotlib.axes._subplots.AxesSubplot at 0x7f095d4b01d0>

The `rolling_window()` function performs a generic rolling window computation on the input data. The weights used in the window are specified by the `win_type` keyword. The list of recognized types are:

> - `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 [47]: ser = pd.Series(np.random.randn(10), index=pd.date_range('1/1/2000', periods=10))

    In [48]: pd.rolling_window(ser, 5, 'triang')
    Out[48]:
    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 `rolling_mean()`.

    In [49]: pd.rolling_window(ser, 5, 'boxcar')
    Out[49]:
    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 [50]: pd.rolling_mean(ser, 5)
    Out[50]:
    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 [51]: pd.rolling_window(ser, 5, 'gaussian', std=0.1)
    Out[51]:
    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

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. This keyword is available in other rolling functions as well.

    In [52]: pd.rolling_window(ser, 5, 'boxcar')
    Out[52]:
    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 [53]: pd.rolling_window(ser, 5, 'boxcar', center=True)
    Out[53]:
    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

    In [54]: pd.rolling_mean(ser, 5, center=True)
    Out[54]:
    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

Note

In rolling sum mode (`mean=False`) there is no normalization done to the weights. 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 rolling mean calculation (`mean=True`)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.

### Binary rolling moments

`rolling_cov()` and `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 `Panel` whose `items` are the dates in question (see the next section).

For example:

    In [55]: df2 = df[:20]

    In [56]: pd.rolling_corr(df2, df2['B'], window=5)
    Out[56]:
                       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 -0.262853   1  0.334449  0.193380
    2000-01-06 -0.083745   1 -0.521587 -0.556126
    2000-01-07 -0.292940   1 -0.658532 -0.458128
    ...              ...  ..       ...       ...
    2000-01-14  0.519499   1 -0.687277  0.192822
    2000-01-15  0.048982   1  0.167669 -0.061463
    2000-01-16  0.217190   1  0.167564 -0.326034
    2000-01-17  0.641180   1 -0.164780 -0.111487
    2000-01-18  0.130422   1  0.322833  0.632383
    2000-01-19  0.317278   1  0.384528  0.813656
    2000-01-20  0.293598   1  0.159538  0.742381

    [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 `Panel` whose `items` 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 [57]: covs = pd.rolling_cov(df[['B','C','D']], df[['A','B','C']], 50, pairwise=True)

    In [58]: covs[df.index[-50]]
    Out[58]:
              A         B          C
    B  2.667506  1.671711   1.938634
    C  8.513843  1.938634  10.556436
    D -7.714737 -1.434529  -7.082653

    In [59]: correls = pd.rolling_corr(df, 50)

    In [60]: correls[df.index[-50]]
    Out[60]:
              A         B         C         D
    A  1.000000  0.604221  0.767429 -0.776170
    B  0.604221  1.000000  0.461484 -0.381148
    C  0.767429  0.461484  1.000000 -0.748863
    D -0.776170 -0.381148 -0.748863  1.000000

Note

Prior to version 0.14 this was available through `rolling_corr_pairwise` which is now simply syntactic sugar for calling `rolling_corr(..., pairwise=True)` and deprecated. This is likely to be removed in a future release.

You can efficiently retrieve the time series of correlations between two columns using `ix` indexing:

    In [61]: correls.ix[:, 'A', 'C'].plot()
    Out[61]: <matplotlib.axes._subplots.AxesSubplot at 0x7f0970b629d0>

## Expanding window moment functions

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. As these calculations are a special case of rolling statistics, they are implemented in pandas such that the following two calls are equivalent:

    In [62]: pd.rolling_mean(df, window=len(df), min_periods=1)[:5]
    Out[62]:
                       A         B         C         D
    2000-01-01 -1.388345  3.317290  0.344542 -0.036968
    2000-01-02 -1.123132  3.622300  1.675867  0.595300
    2000-01-03 -0.628502  3.626503  2.455240  1.060158
    2000-01-04 -0.768740  3.888917  2.451354  1.281874
    2000-01-05 -0.824034  4.108035  2.556112  1.140723

    In [63]: pd.expanding_mean(df)[:5]
    Out[63]:
                       A         B         C         D
    2000-01-01 -1.388345  3.317290  0.344542 -0.036968
    2000-01-02 -1.123132  3.622300  1.675867  0.595300
    2000-01-03 -0.628502  3.626503  2.455240  1.060158
    2000-01-04 -0.768740  3.888917  2.451354  1.281874
    2000-01-05 -0.824034  4.108035  2.556112  1.140723

Like the `rolling_` functions, the following methods are included in the `pandas` namespace or can be located in `pandas.stats.moments`.

Function | Description
--- | ---
`expanding_count()` | Number of non-null observations
`expanding_sum()` | Sum of values
`expanding_mean()` | Mean of values
`expanding_median()` | Arithmetic median of values
`expanding_min()` | Minimum
`expanding_max()` | Maximum
`expanding_std()` | Unbiased standard deviation
`expanding_var()` | Unbiased variance
`expanding_skew()` | Unbiased skewness (3rd moment)
`expanding_kurt()` | Unbiased kurtosis (4th moment)
`expanding_quantile()` | Sample quantile (value at %)
`expanding_apply()` | Generic apply
`expanding_cov()` | Unbiased covariance (binary)
`expanding_corr()` | Correlation (binary)

Aside from not having a `window` parameter, these functions have the same interfaces as their `rolling_` counterpart. 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.
> - `freq`: optionally specify a frequency string or DateOffset to pre-conform the data to. Note that prior to pandas v0.8.0, a keyword argument `time_rule` was used instead of `freq` that referred to the legacy time rule constants

Note

The output of the `rolling_` and `expanding_` functions do not return a `NaN` if there are at least `min_periods` non-null values in the current window. This differs from `cumsum`, `cumprod`, `cummax`, and `cummin`, which return `NaN` in the output wherever a `NaN` is encountered in the input.

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 `expanding_mean()` output for the previous time series dataset:

    In [64]: ts.plot(style='k--')
    Out[64]: <matplotlib.axes._subplots.AxesSubplot at 0x7f0960a14210>

    In [65]: pd.expanding_mean(ts).plot(style='k')
    Out[65]: <matplotlib.axes._subplots.AxesSubplot at 0x7f0960a14210>

## Exponentially weighted moment functions

A related set of functions are exponentially weighted versions of several of the above statistics. A number of expanding EW (exponentially weighted) functions are provided:

Function | Description
--- | ---
`ewma()` | EW moving average
`ewmvar()` | EW moving variance
`ewmstd()` | EW moving standard deviation
`ewmcorr()` | EW moving correlation
`ewmcov()` | 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

y_0 &= x_0 \\ y_t &= (1 - \alpha) y_{t-1} + \alpha x_t,

which is equivalent to using weights

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

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

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}

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, but rather than pass \alpha directly, it’s easier to think about either the span, center of mass (com) or halflife of an EW moment:

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

One must specify precisely one of the three to the EW functions. Span corresponds to what is commonly called a “20-day EW moving average” for example. Center of mass has a more physical interpretation. For example, span = 20 corresponds to com = 9.5. Halflife is the period of time for the exponential weight to reduce to one half.

Here is an example for a univariate time series:

    In [66]: ts.plot(style='k--')
    Out[66]: <matplotlib.axes._subplots.AxesSubplot at 0x7f0970c5b2d0>

    In [67]: pd.ewma(ts, span=20).plot(style='k')
    Out[67]: <matplotlib.axes._subplots.AxesSubplot at 0x7f0970c5b2d0>

All the EW functions have a `min_periods` argument, which has the same meaning it does for all the `expanding_` and `rolling_` functions: no output values will be set until at least `min_periods` non-null values are encountered in the (expanding) window. (This is a change from versions prior to 0.15.0, in which the `min_periods` argument affected only the `min_periods` consecutive entries starting at the first non-null value.)

All the EW functions also have an `ignore_na` argument, which deterines 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` (which reproduces the behavior in versions prior to 0.15.0), 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 `ewmvar()`, `ewmstd()`, and `ewmcov()` 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 http://en.wikipedia.org/wiki/Weighted_arithmetic_mean#Weighted_sample_variance for further details.