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._subplots.AxesSubplot at 0xad2fd04c>

In [41]: rolling_mean(ts, 60).plot(style='k')
Out[41]: <matplotlib.axes._subplots.AxesSubplot at 0xad2fd04c>
_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 [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._subplots.AxesSubplot object at 0xad3a944c>,
       <matplotlib.axes._subplots.AxesSubplot object at 0xacfa1acc>,
       <matplotlib.axes._subplots.AxesSubplot object at 0xacb74b4c>,
       <matplotlib.axes._subplots.AxesSubplot object at 0xacb6da0c>], dtype=object)
_images/rolling_mean_frame.png

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._subplots.AxesSubplot at 0xacd3276c>
_images/rolling_apply_ex.png

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   -1.037870
2000-01-06   -0.767705
2000-01-07   -0.383197
2000-01-08   -0.395513
2000-01-09   -0.558440
2000-01-10   -0.672416
Freq: D, dtype: float64

Note that the boxcar window is equivalent to rolling_mean.

In [49]: 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   -1.309989
2000-01-06   -1.153000
2000-01-07    0.606382
2000-01-08   -0.681101
2000-01-09   -0.289724
2000-01-10   -0.996632
Freq: D, dtype: float64

By default the labels are set to the right edge of the window, but a center keyword is available so the labels can be set at the center. This keyword is available in other rolling functions as well.

In [52]: 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

Note

In rolling sum mode (mean=False) there is no normalization done to the weights. Passing custom weights of [1, 1, 1] will yield a different result than passing weights of [2, 2, 2], for example. When passing a win_type instead of explicitly specifying the weights, the weights are already normalized so that the largest weight is 1.

In contrast, the nature of the rolling mean calculation (mean=True)is such that the weights are normalized with respect to each other. Weights of [1, 1, 1] and [2, 2, 2] yield the same result.

Binary rolling moments

rolling_cov and rolling_corr can compute moving window statistics about two Series or any combination of DataFrame/Series or DataFrame/DataFrame. Here is the behavior in each case:

  • two Series: compute the statistic for the pairing.
  • DataFrame/Series: compute the statistics for each column of the DataFrame with the passed Series, thus returning a DataFrame.
  • DataFrame/DataFrame: by default compute the statistic for matching column names, returning a DataFrame. If the keyword argument pairwise=True is passed then computes the statistic for each pair of columns, returning a Panel whose items are the dates in question (see the next section).

For example:

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

In [56]: 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._subplots.AxesSubplot at 0xacd6ffcc>
_images/rolling_corr_pairwise_ex.png

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._subplots.AxesSubplot at 0xad09564c>

In [65]: expanding_mean(ts).plot(style='k')
Out[65]: <matplotlib.axes._subplots.AxesSubplot at 0xad09564c>
_images/expanding_mean_frame.png

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:

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

In [67]: ts.plot(style='k--')
Out[67]: <matplotlib.axes._subplots.AxesSubplot at 0xacb59d4c>

In [68]: ewma(ts, span=20).plot(style='k')
Out[68]: <matplotlib.axes._subplots.AxesSubplot at 0xacb59d4c>
_images/ewma_ex.png

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.