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 = 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]: 0.00068010881743107965
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 = 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 the covariance 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]: 0.013479040400098754
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 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 groupmin: lowest rank in the groupmax: highest rank in the groupfirst: 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 windowmin_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 argumenttime_rulewas used instead offreqthat referred to the legacy time rule constantshow: optionally specify method for down or re-sampling. Default is is min forrolling_min(), max forrolling_max(), median forrolling_median(), and mean for all other rolling functions. SeeDataFrame.resample()‘s how argument for more information.
These functions can be applied to ndarrays or Series objects:
In [38]: ts = pd.Series(np.random.randn(1000), index=pd.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 0x7f81cab9f490>
In [41]: pd.rolling_mean(ts, 60).plot(style='k')
Out[41]: <matplotlib.axes._subplots.AxesSubplot at 0x7f81cab9f490>
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 = pd.DataFrame(np.random.randn(1000, 4), index=ts.index,
....: columns=['A', 'B', 'C', 'D'])
....:
In [43]: df = df.cumsum()
In [44]: pd.rolling_sum(df, 60).plot(subplots=True)
Out[44]:
array([<matplotlib.axes._subplots.AxesSubplot object at 0x7f81ce3e5fd0>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f81cc2edf10>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f81cc8dc2d0>,
<matplotlib.axes._subplots.AxesSubplot object at 0x7f81de276310>], 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]: pd.rolling_apply(ts, 60, mad).plot(style='k')
Out[46]: <matplotlib.axes._subplots.AxesSubplot at 0x7f81cc4dd750>
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:
boxcartriangblackmanhammingbartlettparzenbohmanblackmanharrisnuttallbarthannkaiser(needs beta)gaussian(needs std)general_gaussian(needs power, width)slepian(needs width).
In [47]: ser = pd.Series(np.random.randn(10), index=pd.date_range('1/1/2000', periods=10))
In [48]: pd.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]: pd.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]: pd.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]: pd.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]: pd.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]: pd.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]: pd.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 argumentpairwise=Trueis passed then computes the statistic for each pair of columns, returning aPanelwhoseitemsare the dates in question (see the next section).
For example:
In [55]: df2 = df[:20]
In [56]: pd.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 = pd.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 = pd.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 0x7f81ddf63750>
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]: pd.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]: pd.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. NoNaNswill be output oncemin_periodsnon-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 argumenttime_rulewas used instead offreqthat 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 0x7f81cc425210>
In [65]: pd.expanding_mean(ts).plot(style='k')
Out[65]: <matplotlib.axes._subplots.AxesSubplot at 0x7f81cc425210>
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 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)^i
which gives
y_t = \frac{x_t + (1 - \alpha)x_{t-1} + (1 - \alpha)^2 x_{t-2} + ... + (1 - \alpha)^t x_{0}}{1 + (1 - \alpha) + (1 - \alpha)^2 + ... + (1 - \alpha)^t}
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, 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]: ts.plot(style='k--')
Out[66]: <matplotlib.axes._subplots.AxesSubplot at 0x7f81ddf05fd0>
In [67]: pd.ewma(ts, span=20).plot(style='k')
Out[67]: <matplotlib.axes._subplots.AxesSubplot at 0x7f81ddf05fd0>
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.