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# 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 = 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

### 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]: 0.00068010881743109993

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 = 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

`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

    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 = 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]: 0.013479040400098801

    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

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

    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 = 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

    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 = 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 0xad5fc40c>

    In [41]: rolling_mean(ts, 60).plot(style='k')
    Out[41]: <matplotlib.axes.AxesSubplot at 0xad5fc40c>

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 0xad3188ac>,
           <matplotlib.axes.AxesSubplot object at 0xadaf56ac>,
           <matplotlib.axes.AxesSubplot object at 0xadb597ec>,
           <matplotlib.axes.AxesSubplot object at 0xad8a19ac>], 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 0xad5d470c>

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`: 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]: 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 = 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 = 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.AxesSubplot at 0xad7a9bec>

## 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]: 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]: 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.AxesSubplot at 0xad7f446c>

    In [65]: expanding_mean(ts).plot(style='k')
    Out[65]: <matplotlib.axes.AxesSubplot at 0xad7f446c>

## 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 [66]: plt.close('all')

    In [67]: ts.plot(style='k--')
    Out[67]: <matplotlib.axes.AxesSubplot at 0xad5d494c>

    In [68]: ewma(ts, span=20).plot(style='k')
    Out[68]: <matplotlib.axes.AxesSubplot at 0xad5d494c>

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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