.. currentmodule:: pandas
.. ipython:: python
:suppress:
import numpy as np
np.random.seed(123456)
np.set_printoptions(precision=4, suppress=True)
import pandas as pd
import matplotlib
matplotlib.style.use('ggplot')
import matplotlib.pyplot as plt
plt.close('all')
pd.options.display.max_rows=15
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 = pd.Series(np.random.randn(8)) ser.pct_change()
.. ipython:: python df = pd.DataFrame(np.random.randn(10, 4)) df.pct_change(periods=3)
The Series object has a method cov to compute covariance between series
(excluding NA/null values).
.. ipython:: python s1 = pd.Series(np.random.randn(1000)) s2 = pd.Series(np.random.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 = pd.DataFrame(np.random.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 = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c']) frame.loc[frame.index[:5], 'a'] = np.nan frame.loc[frame.index[5:10], 'b'] = np.nan frame.cov() frame.cov(min_periods=12)
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 = pd.DataFrame(np.random.randn(1000, 5), columns=['a', 'b', 'c', 'd', 'e']) frame.iloc[::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 = pd.DataFrame(np.random.randn(20, 3), columns=['a', 'b', 'c']) frame.loc[frame.index[:5], 'a'] = np.nan frame.loc[frame.index[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 = pd.DataFrame(np.random.randn(5, 4), index=index, columns=columns) df2 = pd.DataFrame(np.random.randn(4, 4), index=index[:4], columns=columns) df1.corrwith(df2) df2.corrwith(df1, axis=1)
The rank method produces a data ranking with ties being assigned the mean
of the ranks (by default) for the group:
.. ipython:: python
s = pd.Series(np.random.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 = pd.DataFrame(np.random.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 groupmin: lowest rank in the groupmax: highest rank in the groupfirst: ranks assigned in the order they appear in the array
.. currentmodule:: pandas.core.window
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 :class:`~pandas.core.window.Rolling`, :class:`~pandas.core.window.Expanding` and :class:`~pandas.core.window.EWM`. objects and a corresponding method call.
The deprecation warning will show the new syntax, see an example :ref:`here <whatsnew_0180.window_deprecations>` You can view the previous documentation here
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.
Starting in version 0.18.1, the rolling() and expanding()
functions can be used directly from DataFrameGroupBy objects,
see the :ref:`groupby docs <groupby.transform.window_resample>`.
Note
The API for window statistics is quite similar to the way one works with GroupBy objects, see the documentation :ref:`here <groupby>`
We work with rolling, expanding and exponentially weighted data through the corresponding
objects, :class:`~pandas.core.window.Rolling`, :class:`~pandas.core.window.Expanding` and :class:`~pandas.core.window.EWM`.
.. ipython:: python
s = pd.Series(np.random.randn(1000), index=pd.date_range('1/1/2000', periods=1000))
s = s.cumsum()
s
These are created from methods on Series and DataFrame.
.. ipython:: python r = s.rolling(window=60) r
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:
window: size of moving windowmin_periods: threshold of non-null data points to require (otherwise result is NA)center: boolean, whether to set the labels at the center (default is False)
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:
.. ipython:: python r.mean()
.. ipython:: python s.plot(style='k--') @savefig rolling_mean_ex.png r.mean().plot(style='k')
.. ipython:: python
:suppress:
plt.close('all')
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
df = pd.DataFrame(np.random.randn(1000, 4),
index=pd.date_range('1/1/2000', periods=1000),
columns=['A', 'B', 'C', 'D'])
df = df.cumsum()
@savefig rolling_mean_frame.png
df.rolling(window=60).sum().plot(subplots=True)
We provide a number of the common statistical functions:
.. currentmodule:: pandas.core.window
| Method | Description |
|---|---|
| :meth:`~Rolling.count` | Number of non-null observations |
| :meth:`~Rolling.sum` | Sum of values |
| :meth:`~Rolling.mean` | Mean of values |
| :meth:`~Rolling.median` | Arithmetic median of values |
| :meth:`~Rolling.min` | Minimum |
| :meth:`~Rolling.max` | Maximum |
| :meth:`~Rolling.std` | Bessel-corrected sample standard deviation |
| :meth:`~Rolling.var` | Unbiased variance |
| :meth:`~Rolling.skew` | Sample skewness (3rd moment) |
| :meth:`~Rolling.kurt` | Sample kurtosis (4th moment) |
| :meth:`~Rolling.quantile` | Sample quantile (value at %) |
| :meth:`~Rolling.apply` | Generic apply |
| :meth:`~Rolling.cov` | Unbiased covariance (binary) |
| :meth:`~Rolling.corr` | Correlation (binary) |
The :meth:`~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 s.rolling(window=60).apply(mad).plot(style='k')
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 |
|---|---|
| :meth:`~Window.sum` | Sum of values |
| :meth:`~Window.mean` | Mean of values |
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).
.. ipython:: python
ser = pd.Series(np.random.randn(10), index=pd.date_range('1/1/2000', periods=10))
ser.rolling(window=5, win_type='triang').mean()
Note that the boxcar window is equivalent to :meth:`~Rolling.mean`.
.. ipython:: python ser.rolling(window=5, win_type='boxcar').mean() ser.rolling(window=5).mean()
For some windowing functions, additional parameters must be specified:
.. ipython:: python ser.rolling(window=5, win_type='gaussian').mean(std=0.1)
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 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 .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.
.. versionadded:: 0.19.0
New in version 0.19.0 are the ability to pass an offset (or convertible) to a .rolling() method and have it produce
variable sized windows based on the passed time window. For each time point, this includes all preceding values occurring
within the indicated time delta.
This can be particularly useful for a non-regular time frequency index.
.. ipython:: python
dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
index=pd.date_range('20130101 09:00:00', periods=5, freq='s'))
dft
This is a regular frequency index. Using an integer window parameter works to roll along the window frequency.
.. ipython:: python dft.rolling(2).sum() dft.rolling(2, min_periods=1).sum()
Specifying an offset allows a more intuitive specification of the rolling frequency.
.. ipython:: python
dft.rolling('2s').sum()
Using a non-regular, but still monotonic index, rolling with an integer window does not impart any special calculation.
.. ipython:: python
dft = pd.DataFrame({'B': [0, 1, 2, np.nan, 4]},
index = pd.Index([pd.Timestamp('20130101 09:00:00'),
pd.Timestamp('20130101 09:00:02'),
pd.Timestamp('20130101 09:00:03'),
pd.Timestamp('20130101 09:00:05'),
pd.Timestamp('20130101 09:00:06')],
name='foo'))
dft
dft.rolling(2).sum()
Using the time-specification generates variable windows for this sparse data.
.. ipython:: python
dft.rolling('2s').sum()
Furthermore, we now allow an optional on parameter to specify a column (rather than the
default of the index) in a DataFrame.
.. ipython:: python
dft = dft.reset_index()
dft
dft.rolling('2s', on='foo').sum()
Using .rolling() with a time-based index is quite similar to :ref:`resampling <timeseries.resampling>`. They
both operate and perform reductive operations on time-indexed pandas objects.
When using .rolling() with an offset. The offset is a time-delta. Take a backwards-in-time looking window, and
aggregate all of the values in that window (including the end-point, but not the start-point). This is the new value
at that point in the result. These are variable sized windows in time-space for each point of the input. You will get
a same sized result as the input.
When using .resample() with an offset. Construct a new index that is the frequency of the offset. For each frequency
bin, aggregate points from the input within a backwards-in-time looking window that fall in that bin. The result of this
aggregation is the output for that frequency point. The windows are fixed size size in the frequency space. Your result
will have the shape of a regular frequency between the min and the max of the original input object.
To summarize, .rolling() is a time-based window operation, while .resample() is a frequency-based window operation.
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.
.. ipython:: python ser.rolling(window=5).mean() ser.rolling(window=5, center=True).mean()
:meth:`~Rolling.cov` and :meth:`~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 :ref:`the next section <stats.moments.corr_pairwise>`).
For example:
.. ipython:: python df2 = df[:20] df2.rolling(window=5).corr(df2['B'])
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 = df[['B','C','D']].rolling(window=50).cov(df[['A','B','C']], pairwise=True) covs[df.index[-50]]
.. ipython:: python correls = df.rolling(window=50).corr() correls[df.index[-50]]
You can efficiently retrieve the time series of correlations between two
columns using .loc indexing:
.. ipython:: python
:suppress:
plt.close('all')
.. ipython:: python @savefig rolling_corr_pairwise_ex.png correls.loc[:, 'A', 'C'].plot()
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 :ref:`here <groupby.aggregate>`.
.. ipython:: python
dfa = pd.DataFrame(np.random.randn(1000, 3),
index=pd.date_range('1/1/2000', periods=1000),
columns=['A', 'B', 'C'])
r = dfa.rolling(window=60,min_periods=1)
r
We can aggregate by passing a function to the entire DataFrame, or select a Series (or multiple Series) via standard getitem.
.. ipython:: python r.aggregate(np.sum) r['A'].aggregate(np.sum) r[['A','B']].aggregate(np.sum)
As you can see, the result of the aggregation will have the selected columns, or all columns if none are selected.
With windowed Series you can also pass a list or dict of functions to do aggregation with, outputting a DataFrame:
.. ipython:: python r['A'].agg([np.sum, np.mean, np.std])
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.
.. ipython:: python
r['A'].agg({'result1' : np.sum,
'result2' : np.mean})
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:
.. ipython:: python r.agg([np.sum, np.mean])
Passing a dict of functions has different behavior by default, see the next section.
By passing a dict to aggregate you can apply a different aggregation to the
columns of a DataFrame:
.. ipython:: python
:okexcept:
r.agg({'A' : np.sum,
'B' : lambda x: np.std(x, ddof=1)})
The function names can also be strings. In order for a string to be valid it must be implemented on the windowed object
.. ipython:: python
r.agg({'A' : 'sum', 'B' : 'std'})
Furthermore you can pass a nested dict to indicate different aggregations on different columns.
.. ipython:: python
r.agg({'A' : ['sum','std'], 'B' : ['mean','std'] })
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 :class:`~pandas.core.window.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:
.. ipython:: python df.rolling(window=len(df), min_periods=1).mean()[:5] df.expanding(min_periods=1).mean()[:5]
These have a similar set of methods to .rolling methods.
.. currentmodule:: pandas.core.window
| Function | Description |
|---|---|
| :meth:`~Expanding.count` | Number of non-null observations |
| :meth:`~Expanding.sum` | Sum of values |
| :meth:`~Expanding.mean` | Mean of values |
| :meth:`~Expanding.median` | Arithmetic median of values |
| :meth:`~Expanding.min` | Minimum |
| :meth:`~Expanding.max` | Maximum |
| :meth:`~Expanding.std` | Unbiased standard deviation |
| :meth:`~Expanding.var` | Unbiased variance |
| :meth:`~Expanding.skew` | Unbiased skewness (3rd moment) |
| :meth:`~Expanding.kurt` | Unbiased kurtosis (4th moment) |
| :meth:`~Expanding.quantile` | Sample quantile (value at %) |
| :meth:`~Expanding.apply` | Generic apply |
| :meth:`~Expanding.cov` | Unbiased covariance (binary) |
| :meth:`~Expanding.corr` | Correlation (binary) |
.. currentmodule:: pandas
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:
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.center: boolean, whether to set the labels at the center (default is False)
Note
The output of the .rolling and .expanding methods do not return a
NaN if there are at least min_periods non-null values in the current
window. For example,
.. ipython:: python
sn = pd.Series([1, 2, np.nan, 3, np.nan, 4])
sn
sn.rolling(2).max()
sn.rolling(2, min_periods=1).max()
In case of expanding functions, this differs from :meth:`~DataFrame.cumsum`,
:meth:`~DataFrame.cumprod`, :meth:`~DataFrame.cummax`,
and :meth:`~DataFrame.cummin`, which return NaN in the output wherever
a NaN is encountered in the input. In order to match the output of cumsum
with expanding, use :meth:`~DataFrame.fillna`:
.. ipython:: python
sn.expanding().sum()
sn.cumsum()
sn.cumsum().fillna(method='ffill')
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 :meth:`~core.window.Expanding.mean` output for the previous time series dataset:
.. ipython:: python
:suppress:
plt.close('all')
.. ipython:: python s.plot(style='k--') @savefig expanding_mean_frame.png s.expanding().mean().plot(style='k')
.. currentmodule:: pandas.core.window
A related set of functions are exponentially weighted versions of several of
the above statistics. A similar interface to .rolling and .expanding is accessed
through the .ewm method to receive an :class:`~EWM` object.
A number of expanding EW (exponentially weighted)
methods are provided:
| Function | Description |
|---|---|
| :meth:`~EWM.mean` | EW moving average |
| :meth:`~EWM.var` | EW moving variance |
| :meth:`~EWM.std` | EW moving standard deviation |
| :meth:`~EWM.corr` | EW moving correlation |
| :meth:`~EWM.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)^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, 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-life of 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-life and alpha to the EW functions:
- Span corresponds to what is commonly called an "N-day EW moving average".
- Center of mass has a more physical interpretation and can be thought of in terms of span: c = (s - 1) / 2.
- Half-life is the period of time for the exponential weight to reduce to one half.
- Alpha specifies the smoothing factor directly.
Here is an example for a univariate time series:
.. ipython:: python s.plot(style='k--') @savefig ewma_ex.png s.ewm(span=20).mean().plot(style='k')
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_periods
argument 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 :meth:`~Ewm.var`, :meth:`~Ewm.std`, and :meth:`~Ewm.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 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 Weighted Sample Variance for further details.