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.. currentmodule:: pandas

.. ipython:: python
   :suppress:

   import numpy as np
   np.random.seed(123456)
   from pandas import *
   import pandas.util.testing as tm
   randn = np.random.randn
   np.set_printoptions(precision=4, suppress=True)
   import matplotlib.pyplot as plt
   plt.close('all')
   options.display.mpl_style='default'
   options.display.max_rows=15

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).

.. ipython:: python

   ser = Series(randn(8))

   ser.pct_change()


.. ipython:: python

   df = DataFrame(randn(10, 4))

   df.pct_change(periods=3)

Covariance

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

.. ipython:: python

   s1 = Series(randn(1000))
   s2 = Series(randn(1000))
   s1.cov(s2)

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.

.. ipython:: python

   frame = DataFrame(randn(1000, 5), columns=['a', 'b', 'c', 'd', 'e'])
   frame.cov()

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.

.. ipython:: python

   frame = DataFrame(randn(20, 3), columns=['a', 'b', 'c'])
   frame.ix[:5, 'a'] = np.nan
   frame.ix[5:10, 'b'] = np.nan

   frame.cov()

   frame.cov(min_periods=12)

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 :ref:`caveats <computation.covariance.caveats>` associated with this method of calculating correlation matrices in the :ref:`covariance section <computation.covariance>`.

.. ipython:: python

   frame = DataFrame(randn(1000, 5), columns=['a', 'b', 'c', 'd', 'e'])
   frame.ix[::2] = np.nan

   # Series with Series
   frame['a'].corr(frame['b'])
   frame['a'].corr(frame['b'], method='spearman')

   # Pairwise correlation of DataFrame columns
   frame.corr()

Note that non-numeric columns will be automatically excluded from the correlation calculation.

Like cov, corr also supports the optional min_periods keyword:

.. ipython:: python

   frame = DataFrame(randn(20, 3), columns=['a', 'b', 'c'])
   frame.ix[:5, 'a'] = np.nan
   frame.ix[5:10, 'b'] = np.nan

   frame.corr()

   frame.corr(min_periods=12)

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

.. ipython:: python

   index = ['a', 'b', 'c', 'd', 'e']
   columns = ['one', 'two', 'three', 'four']
   df1 = DataFrame(randn(5, 4), index=index, columns=columns)
   df2 = DataFrame(randn(4, 4), index=index[:4], columns=columns)
   df1.corrwith(df2)
   df2.corrwith(df1, axis=1)

Data ranking

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

.. ipython:: python

   s = Series(np.random.randn(5), index=list('abcde'))
   s['d'] = s['b'] # so there's a tie
   s.rank()

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.

.. ipython:: python

   df = DataFrame(np.random.randn(10, 6))
   df[4] = df[2][:5] # some ties
   df
   df.rank(1)

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
.. currentmodule:: pandas


.. currentmodule:: pandas.stats.api

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 :mod:`pandas` namespace, but otherwise they can be found in :mod:`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 :ref:`frequency string <timeseries.alias>` or :ref:`DateOffset <timeseries.offsets>` 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 :meth:`DataFrame.resample`'s how argument for more information.

These functions can be applied to ndarrays or Series objects:

.. ipython:: python

   ts = Series(randn(1000), index=date_range('1/1/2000', periods=1000))
   ts = ts.cumsum()

   ts.plot(style='k--')

   @savefig rolling_mean_ex.png
   rolling_mean(ts, 60).plot(style='k')

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:

.. ipython:: python
   :suppress:

   plt.close('all')


.. ipython:: python

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

   @savefig rolling_mean_frame.png
   rolling_sum(df, 60).plot(subplots=True)

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:

.. ipython:: python

   mad = lambda x: np.fabs(x - x.mean()).mean()
   @savefig rolling_apply_ex.png
   rolling_apply(ts, 60, mad).plot(style='k')

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).
.. ipython:: python

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

   rolling_window(ser, 5, 'triang')

Note that the boxcar window is equivalent to rolling_mean:

.. ipython:: python

   rolling_window(ser, 5, 'boxcar')

   rolling_mean(ser, 5)

For some windowing functions, additional parameters must be specified:

.. ipython:: python

   rolling_window(ser, 5, 'gaussian', std=0.1)

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.

.. ipython:: python

   rolling_window(ser, 5, 'boxcar')

   rolling_window(ser, 5, 'boxcar', center=True)

   rolling_mean(ser, 5, center=True)

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 :ref:`the next section <stats.moments.corr_pairwise>`).

For example:

.. ipython:: python

   df2 = df[:20]
   rolling_corr(df2, df2['B'], window=5)

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 :ref:`covariance section <computation.covariance>` for :ref:`caveats <computation.covariance.caveats>` associated with this method of calculating covariance and correlation matrices.

.. ipython:: python

   covs = rolling_cov(df[['B','C','D']], df[['A','B','C']], 50, pairwise=True)
   covs[df.index[-50]]


.. ipython:: python

   correls = rolling_corr(df, 50)
   correls[df.index[-50]]

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:

.. ipython:: python
   :suppress:

   plt.close('all')


.. ipython:: python

   @savefig rolling_corr_pairwise_ex.png
   correls.ix[:, 'A', 'C'].plot()

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:

.. ipython:: python

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

   expanding_mean(df)[:5]

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 :ref:`frequency string <timeseries.alias>` or :ref:`DateOffset <timeseries.offsets>` 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:

.. ipython:: python
   :suppress:

   plt.close('all')


.. ipython:: python

   ts.plot(style='k--')

   @savefig expanding_mean_frame.png
   expanding_mean(ts).plot(style='k')

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 at 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. 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.

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:

.. ipython:: python

   plt.close('all')
   ts.plot(style='k--')

   @savefig ewma_ex.png
   ewma(ts, span=20).plot(style='k')

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.