Computational tools — pandas 0.18.1 documentation (original) (raw)

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]: 0.00068010881743110698

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 matricesfor 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 thecovariance 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]: 0.013479040400098789

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 = 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 4.0 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 methodparameter:

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 hereYou can view the previous documentationhere

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.

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, 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:

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, dtype: float64

In [44]: s.plot(style='k--') Out[44]: <matplotlib.axes._subplots.AxesSubplot at 0x11ce37110>

In [45]: r.mean().plot(style='k') Out[45]: <matplotlib.axes._subplots.AxesSubplot at 0x11ce37110>

_images/rolling_mean_ex.png

They can also be applied to DataFrame objects. This is really just syntactic sugar for applying the moving window operator to all of the DataFrame’s columns:

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 0x11cdc2890>, <matplotlib.axes._subplots.AxesSubplot object at 0x120dfef50>, <matplotlib.axes._subplots.AxesSubplot object at 0x120cd4110>, <matplotlib.axes._subplots.AxesSubplot object at 0x1127f6e90>], dtype=object)

_images/rolling_mean_frame.png

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 0x112f25950>

_images/rolling_apply_ex.png

Rolling Windows

Passing win_type to .rolling generates a generic rolling window computation, that is weighted according the win_type. The following methods are available:

Method Description
sum() 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:

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 awin_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.

Centering Windows

By default the labels are set to the right edge of the window, but acenter keyword is available so the labels can be set at the center.

In [56]: ser.rolling(window=5).mean() Out[56]: 2000-01-01 NaN 2000-01-02 NaN 2000-01-03 NaN 2000-01-04 NaN 2000-01-05 -0.841164 2000-01-06 -0.779948 2000-01-07 -0.565487 2000-01-08 -0.502815 2000-01-09 -0.553755 2000-01-10 -0.472211 Freq: D, dtype: float64

In [57]: ser.rolling(window=5, center=True).mean() Out[57]: 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 orDataFrame/DataFrame. Here is the behavior in each case:

For example:

In [58]: df2 = df[:20]

In [59]: df2.rolling(window=5).corr(df2['B']) Out[59]: 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 0.334449 0.193380 2000-01-06 -0.083745 1.0 -0.521587 -0.556126 2000-01-07 -0.292940 1.0 -0.658532 -0.458128 ... ... ... ... ... 2000-01-14 0.519499 1.0 -0.687277 0.192822 2000-01-15 0.048982 1.0 0.167669 -0.061463 2000-01-16 0.217190 1.0 0.167564 -0.326034 2000-01-17 0.641180 1.0 -0.164780 -0.111487 2000-01-18 0.130422 1.0 0.322833 0.632383 2000-01-19 0.317278 1.0 0.384528 0.813656 2000-01-20 0.293598 1.0 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 ofDataFrame 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 [60]: covs = df[['B','C','D']].rolling(window=50).cov(df[['A','B','C']], pairwise=True)

In [61]: covs[df.index[-50]] Out[61]: 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 [62]: correls = df.rolling(window=50).corr()

In [63]: correls[df.index[-50]] Out[63]: 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

You can efficiently retrieve the time series of correlations between two columns using .loc indexing:

In [64]: correls.loc[:, 'A', 'C'].plot() Out[64]: <matplotlib.axes._subplots.AxesSubplot at 0x11ea521d0>

_images/rolling_corr_pairwise_ex.png

Aggregation

Once the Rolling, Expanding or EWM objects have been created, several methods are available to perform multiple computations on the data. This is very similar to a .groupby(...).agg seen here.

In [65]: dfa = pd.DataFrame(np.random.randn(1000, 3), ....: index=pd.date_range('1/1/2000', periods=1000), ....: columns=['A', 'B', 'C']) ....:

In [66]: r = dfa.rolling(window=60,min_periods=1)

In [67]: r Out[67]: 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 [68]: r.aggregate(np.sum) Out[68]: A B C 2000-01-01 0.314226 -0.001675 0.071823 2000-01-02 1.206791 0.678918 -0.267817 2000-01-03 1.421701 0.600508 -0.445482 2000-01-04 1.912539 -0.759594 1.146974 2000-01-05 2.919639 -0.061759 -0.743617 2000-01-06 2.665637 1.298392 -0.803529 2000-01-07 2.513985 1.923089 -1.928308 ... ... ... ... 2002-09-20 1.447669 -12.360302 2.734381 2002-09-21 1.871783 -13.896542 3.086102 2002-09-22 2.540658 -12.594402 3.162542 2002-09-23 2.974674 -12.727703 3.861005 2002-09-24 1.391366 -13.584590 3.790683 2002-09-25 2.027313 -15.083214 3.377896 2002-09-26 1.290363 -13.569459 3.809884

[1000 rows x 3 columns]

In [69]: r['A'].aggregate(np.sum) Out[69]: 2000-01-01 0.314226 2000-01-02 1.206791 2000-01-03 1.421701 2000-01-04 1.912539 2000-01-05 2.919639 2000-01-06 2.665637 2000-01-07 2.513985 ...
2002-09-20 1.447669 2002-09-21 1.871783 2002-09-22 2.540658 2002-09-23 2.974674 2002-09-24 1.391366 2002-09-25 2.027313 2002-09-26 1.290363 Freq: D, Name: A, dtype: float64

In [70]: r[['A','B']].aggregate(np.sum) Out[70]: A B 2000-01-01 0.314226 -0.001675 2000-01-02 1.206791 0.678918 2000-01-03 1.421701 0.600508 2000-01-04 1.912539 -0.759594 2000-01-05 2.919639 -0.061759 2000-01-06 2.665637 1.298392 2000-01-07 2.513985 1.923089 ... ... ... 2002-09-20 1.447669 -12.360302 2002-09-21 1.871783 -13.896542 2002-09-22 2.540658 -12.594402 2002-09-23 2.974674 -12.727703 2002-09-24 1.391366 -13.584590 2002-09-25 2.027313 -15.083214 2002-09-26 1.290363 -13.569459

[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 at once

With windowed Series you can also pass a list or dict of functions to do aggregation with, outputting a DataFrame:

In [71]: r['A'].agg([np.sum, np.mean, np.std]) Out[71]: sum mean std 2000-01-01 0.314226 0.314226 NaN 2000-01-02 1.206791 0.603396 0.408948 2000-01-03 1.421701 0.473900 0.365959 2000-01-04 1.912539 0.478135 0.298925 2000-01-05 2.919639 0.583928 0.350682 2000-01-06 2.665637 0.444273 0.464115 2000-01-07 2.513985 0.359141 0.479828 ... ... ... ... 2002-09-20 1.447669 0.024128 1.034827 2002-09-21 1.871783 0.031196 1.031417 2002-09-22 2.540658 0.042344 1.026341 2002-09-23 2.974674 0.049578 1.030021 2002-09-24 1.391366 0.023189 1.024793 2002-09-25 2.027313 0.033789 1.022099 2002-09-26 1.290363 0.021506 1.024751

[1000 rows x 3 columns]

If a dict is passed, the keys will be used to name the columns. Otherwise the function’s name (stored in the function object) will be used.

In [72]: r['A'].agg({'result1' : np.sum, ....: 'result2' : np.mean}) ....: Out[72]: result2 result1 2000-01-01 0.314226 0.314226 2000-01-02 0.603396 1.206791 2000-01-03 0.473900 1.421701 2000-01-04 0.478135 1.912539 2000-01-05 0.583928 2.919639 2000-01-06 0.444273 2.665637 2000-01-07 0.359141 2.513985 ... ... ... 2002-09-20 0.024128 1.447669 2002-09-21 0.031196 1.871783 2002-09-22 0.042344 2.540658 2002-09-23 0.049578 2.974674 2002-09-24 0.023189 1.391366 2002-09-25 0.033789 2.027313 2002-09-26 0.021506 1.290363

[1000 rows x 2 columns]

On a widowed DataFrame, you can pass a list of functions to apply to each column, which produces an aggregated result with a hierarchical index:

In [73]: r.agg([np.sum, np.mean]) Out[73]: A B C
sum mean sum mean sum mean 2000-01-01 0.314226 0.314226 -0.001675 -0.001675 0.071823 0.071823 2000-01-02 1.206791 0.603396 0.678918 0.339459 -0.267817 -0.133908 2000-01-03 1.421701 0.473900 0.600508 0.200169 -0.445482 -0.148494 2000-01-04 1.912539 0.478135 -0.759594 -0.189899 1.146974 0.286744 2000-01-05 2.919639 0.583928 -0.061759 -0.012352 -0.743617 -0.148723 2000-01-06 2.665637 0.444273 1.298392 0.216399 -0.803529 -0.133921 2000-01-07 2.513985 0.359141 1.923089 0.274727 -1.928308 -0.275473 ... ... ... ... ... ... ... 2002-09-20 1.447669 0.024128 -12.360302 -0.206005 2.734381 0.045573 2002-09-21 1.871783 0.031196 -13.896542 -0.231609 3.086102 0.051435 2002-09-22 2.540658 0.042344 -12.594402 -0.209907 3.162542 0.052709 2002-09-23 2.974674 0.049578 -12.727703 -0.212128 3.861005 0.064350 2002-09-24 1.391366 0.023189 -13.584590 -0.226410 3.790683 0.063178 2002-09-25 2.027313 0.033789 -15.083214 -0.251387 3.377896 0.056298 2002-09-26 1.290363 0.021506 -13.569459 -0.226158 3.809884 0.063498

[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 [74]: r.agg({'A' : np.sum, ....: 'B' : lambda x: np.std(x, ddof=1)}) ....: Out[74]: A B 2000-01-01 0.314226 NaN 2000-01-02 1.206791 0.482437 2000-01-03 1.421701 0.417825 2000-01-04 1.912539 0.851468 2000-01-05 2.919639 0.837474 2000-01-06 2.665637 0.935441 2000-01-07 2.513985 0.867770 ... ... ... 2002-09-20 1.447669 1.084259 2002-09-21 1.871783 1.088368 2002-09-22 2.540658 1.084707 2002-09-23 2.974674 1.084936 2002-09-24 1.391366 1.079268 2002-09-25 2.027313 1.091334 2002-09-26 1.290363 1.060255

[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 [75]: r.agg({'A' : 'sum', 'B' : 'std'}) Out[75]: A B 2000-01-01 0.314226 NaN 2000-01-02 1.206791 0.482437 2000-01-03 1.421701 0.417825 2000-01-04 1.912539 0.851468 2000-01-05 2.919639 0.837474 2000-01-06 2.665637 0.935441 2000-01-07 2.513985 0.867770 ... ... ... 2002-09-20 1.447669 1.084259 2002-09-21 1.871783 1.088368 2002-09-22 2.540658 1.084707 2002-09-23 2.974674 1.084936 2002-09-24 1.391366 1.079268 2002-09-25 2.027313 1.091334 2002-09-26 1.290363 1.060255

[1000 rows x 2 columns]

Furthermore you can pass a nested dict to indicate different aggregations on different columns.

In [76]: r.agg({'A' : ['sum','std'], 'B' : ['mean','std'] }) Out[76]: A B
sum std mean std 2000-01-01 0.314226 NaN -0.001675 NaN 2000-01-02 1.206791 0.408948 0.339459 0.482437 2000-01-03 1.421701 0.365959 0.200169 0.417825 2000-01-04 1.912539 0.298925 -0.189899 0.851468 2000-01-05 2.919639 0.350682 -0.012352 0.837474 2000-01-06 2.665637 0.464115 0.216399 0.935441 2000-01-07 2.513985 0.479828 0.274727 0.867770 ... ... ... ... ... 2002-09-20 1.447669 1.034827 -0.206005 1.084259 2002-09-21 1.871783 1.031417 -0.231609 1.088368 2002-09-22 2.540658 1.026341 -0.209907 1.084707 2002-09-23 2.974674 1.030021 -0.212128 1.084936 2002-09-24 1.391366 1.024793 -0.226410 1.079268 2002-09-25 2.027313 1.022099 -0.251387 1.091334 2002-09-26 1.290363 1.024751 -0.226158 1.060255

[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 [77]: df.rolling(window=len(df), min_periods=1).mean()[:5] Out[77]: 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 [78]: df.expanding(min_periods=1).mean()[:5] Out[78]: 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

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:

Note

The output of the .rolling and .expanding methods do not return aNaN if there are at least min_periods non-null values in the current window. This differs from cumsum, cumprod, cummax, andcummin, 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 themean() output for the previous time series dataset:

In [79]: s.plot(style='k--') Out[79]: <matplotlib.axes._subplots.AxesSubplot at 0x118390f10>

In [80]: s.expanding().mean().plot(style='k') Out[80]: <matplotlib.axes._subplots.AxesSubplot at 0x118390f10>

_images/expanding_mean_frame.png

Exponentially Weighted Windows

A related set of functions are exponentially weighted versions of several of the above statistics. A similar interface to .rolling and .expanding is accessed thru 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)^iwhich gives

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

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, 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-lifeof an EW moment:

\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}

One must specify precisely one of span, center of mass, half-lifeand alpha to the EW functions:

Here is an example for a univariate time series:

In [81]: s.plot(style='k--') Out[81]: <matplotlib.axes._subplots.AxesSubplot at 0x1132e2dd0>

In [82]: s.ewm(span=20).mean().plot(style='k') Out[82]: <matplotlib.axes._subplots.AxesSubplot at 0x1132e2dd0>

_images/ewma_ex.png

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

EWM also has 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 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 asewmvar(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 Variancefor further details.