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# Computational tools

## Statistical functions

### Percent Change

Both `Series` and `DataFrame` has a method `pct_change` to compute the percent change over a given number of periods (using `fill_method` to fill NA/null values).

    In [1]: ser = Series(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 = DataFrame(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

    [10 rows x 4 columns]

### Covariance

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

    In [5]: s1 = Series(randn(1000))

    In [6]: s2 = Series(randn(1000))

    In [7]: s1.cov(s2)
    Out[7]: 0.0006801088174310957

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

    In [8]: frame = DataFrame(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

    [5 rows x 5 columns]

`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 = DataFrame(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

    [3 rows x 3 columns]

    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

    [3 rows x 3 columns]

### Correlation

Several methods for computing correlations are provided. Several kinds of correlation methods 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.

    In [15]: frame = DataFrame(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]: 0.013479040400098763

    In [18]: frame['a'].corr(frame['b'], method='spearman')
    Out[18]: -0.0072898851595406388

    # 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

    [5 rows x 5 columns]

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 = DataFrame(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

    [3 rows x 3 columns]

    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

    [3 rows x 3 columns]

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 = DataFrame(randn(5, 4), index=index, columns=columns)

    In [28]: df2 = DataFrame(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 = Series(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 = DataFrame(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

    [10 rows x 6 columns]

    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

    [10 rows x 6 columns]

`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_corr_pairwise` | Pairwise correlation of DataFrame columns
`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

These functions can be applied to ndarrays or Series objects:

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

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

    In [40]: ts.plot(style='k--')
    Out[40]: <matplotlib.axes.AxesSubplot at 0x7abc7d0>

    In [41]: rolling_mean(ts, 60).plot(style='k')
    Out[41]: <matplotlib.axes.AxesSubplot at 0x7abc7d0>

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

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

    In [44]: rolling_sum(df, 60).plot(subplots=True)
    Out[44]:
    array([<matplotlib.axes.AxesSubplot object at 0x7c24dd0>,
           <matplotlib.axes.AxesSubplot object at 0x5c4fbd0>,
           <matplotlib.axes.AxesSubplot object at 0x699b750>,
           <matplotlib.axes.AxesSubplot object at 0x5dfb3d0>], 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]: rolling_apply(ts, 60, mad).plot(style='k')
    Out[46]: <matplotlib.axes.AxesSubplot at 0x60054d0>

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 = Series(randn(10), index=date_range('1/1/2000', periods=10))

    In [48]: 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   -0.622722
    2000-01-06   -0.460623
    2000-01-07   -0.229918
    2000-01-08   -0.237308
    2000-01-09   -0.335064
    2000-01-10   -0.403449
    Freq: D, dtype: float64

Note that the `boxcar` window is equivalent to `rolling_mean`:

    In [49]: 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]: 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]: 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   -0.261998
    2000-01-06   -0.230600
    2000-01-07    0.121276
    2000-01-08   -0.136220
    2000-01-09   -0.057945
    2000-01-10   -0.199326
    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]: 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]: 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]: 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

### 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`: compute statistic for matching column names, returning a DataFrame

For example:

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

    In [56]: 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
    2000-01-08  0.840416   1  0.796505 -0.498672
    2000-01-09 -0.135275   1  0.753895 -0.634445
    2000-01-10 -0.346229   1 -0.682232 -0.645681
    2000-01-11 -0.365524   1 -0.775831 -0.561991
    2000-01-12 -0.204761   1 -0.855874 -0.382232
    2000-01-13  0.575218   1 -0.747531  0.167892
    2000-01-14  0.519499   1 -0.687277  0.192822
    2000-01-15  0.048982   1  0.167669 -0.061463
                     ... ...       ...       ...

    [20 rows x 4 columns]

### Computing rolling pairwise correlations

In financial data analysis and other fields it’s common to compute correlation matrices for a collection of time series. More difficult is to compute a moving-window correlation matrix. This can be done using the `rolling_corr_pairwise` function, which yields a `Panel` whose `items` are the dates in question:

    In [57]: correls = rolling_corr_pairwise(df, 50)

    In [58]: correls[df.index[-50]]
    Out[58]:
              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

    [4 rows x 4 columns]

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

    In [59]: correls.ix[:, 'A', 'C'].plot()
    Out[59]: <matplotlib.axes.AxesSubplot at 0x5b5cb90>

## 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 [60]: rolling_mean(df, window=len(df), min_periods=1)[:5]
    Out[60]:
                       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

    [5 rows x 4 columns]

    In [61]: expanding_mean(df)[:5]
    Out[61]:
                       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

    [5 rows x 4 columns]

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)
`expanding_corr_pairwise` | Pairwise correlation of DataFrame columns

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 [62]: ts.plot(style='k--')
    Out[62]: <matplotlib.axes.AxesSubplot at 0x731fa50>

    In [63]: expanding_mean(ts).plot(style='k')
    Out[63]: <matplotlib.axes.AxesSubplot at 0x731fa50>

## Exponentially weighted moment functions

A related set of functions are exponentially weighted versions of many of the above statistics. A number of EW (exponentially weighted) functions are provided using the blending method. For example, where is the result and the input, we compute an exponentially weighted moving average as

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

Note

the equation above is sometimes written in the form

where .

You can pass one of the three to these functions but not more. 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 the list of functions available:

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

Here are an example for a univariate time series:

    In [64]: plt.close('all')

    In [65]: ts.plot(style='k--')
    Out[65]: <matplotlib.axes.AxesSubplot at 0x7470f50>

    In [66]: ewma(ts, span=20).plot(style='k')
    Out[66]: <matplotlib.axes.AxesSubplot at 0x7470f50>

Note

The EW functions perform a standard adjustment to the initial observations whereby if there are fewer observations than called for in the span, those observations are reweighted accordingly.

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