pandas
python
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).
python
ser = pd.Series(np.random.randn(8))
ser.pct_change()
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).
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.
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.
python
frame = pd.DataFrame(np.random.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)
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 <computation.covariance.caveats> associated with this method of calculating correlation matrices in the covariance section <computation.covariance>.
python
frame = pd.DataFrame(np.random.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:
python
frame = pd.DataFrame(np.random.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.
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:
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.
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
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 ~pandas.core.window.Rolling, ~pandas.core.window.Expanding and ~pandas.core.window.EWM. objects and a corresponding method call.
The deprecation warning will show the new syntax, see an example 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 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 here <groupby>
We work with rolling, expanding and exponentially weighted data through the corresponding objects, ~pandas.core.window.Rolling, ~pandas.core.window.Expanding and ~pandas.core.window.EWM.
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.
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.varGenerally 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:
python
r.mean()
python
s.plot(style='k--')
@savefig rolling_mean_ex.png r.mean().plot(style='k')
python
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:
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:
pandas.core.window
| Method | 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 |
Bessel-corrected sample standard deviation |
~Rolling.var |
Unbiased variance |
~Rolling.skew |
Sample skewness (3rd moment) |
~Rolling.kurt |
Sample kurtosis (4th moment) |
~Rolling.quantile |
Sample quantile (value at %) |
~Rolling.apply |
Generic apply |
~Rolling.cov |
Unbiased covariance (binary) |
~Rolling.corr |
Correlation (binary) |
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:
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 |
|---|---|
~Window.sum |
Sum of values |
~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).
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 ~Rolling.mean.
python
ser.rolling(window=5, win_type='boxcar').mean() ser.rolling(window=5).mean()
For some windowing functions, additional parameters must be specified:
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.
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.
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.
python
dft.rolling(2).sum() dft.rolling(2, min_periods=1).sum()
Specifying an offset allows a more intuitive specification of the rolling frequency.
python
dft.rolling('2s').sum()
Using a non-regular, but still monotonic index, rolling with an integer window does not impart any special calculation.
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.
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.
python
dft = dft.reset_index() dft dft.rolling('2s', on='foo').sum()
Using .rolling() with a time-based index is quite similar to 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.
python
ser.rolling(window=5).mean() ser.rolling(window=5, center=True).mean()
~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 (seethe next section <stats.moments.corr_pairwise>).
For example:
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 covariance section
<computation.covariance> for caveats
<computation.covariance.caveats> associated with this method of calculating covariance and correlation matrices.
python
covs = df[['B','C','D']].rolling(window=50).cov(df[['A','B','C']], pairwise=True) covs[df.index[-50]]
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:
python
plt.close('all')
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. These operations are similar to the aggregating API <basics.aggregate>, groupby aggregates <groupby.aggregate>, and resample API <timeseries.aggregate>.
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.
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:
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.
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:
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:
python
- 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
python
r.agg({'A' : 'sum', 'B' : 'std'})
Furthermore you can pass a nested dict to indicate different aggregations on different columns.
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 ~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:
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.
pandas.core.window
| 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 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. 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:
python
plt.close('all')
python
s.plot(style='k--')
@savefig expanding_mean_frame.png s.expanding().mean().plot(style='k')
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 ~pandas.core.window.EWM object. A number of expanding EW (exponentially weighted) methods are provided:
pandas.core.window
| Function | Description |
|---|---|
~EWM.mean |
EW moving average |
~EWM.var |
EW moving variance |
~EWM.std |
EW moving standard deviation |
~EWM.corr |
EW moving correlation |
~EWM.cov |
EW moving covariance |
In general, a weighted moving average is calculated as
where xt is the input and yt is the result.
The EW functions support two variants of exponential weights. The default, adjust=True, uses the weights wi = (1 − α)i which gives
When adjust=False is specified, moving averages are calculated as
which is equivalent to using weights
Note
These equations are sometimes written in terms of α′ = 1 − α, e.g.
yt = α′yt − 1 + (1 − α′)xt.
The difference between the above two variants arises because we are dealing with series which have finite history. Consider a series of infinite history:
Noting that the denominator is a geometric series with initial term equal to 1 and a ratio of 1 − α we have
which shows the equivalence of the above two variants for infinite series. When adjust=True we have y0 = x0 and from the last representation above we have yt = αxt + (1 − α)yt − 1, therefore there is an assumption that x0 is not an ordinary value but rather an exponentially weighted moment of the infinite series up to that point.
One must have 0 < α ≤ 1, and while since version 0.18.0 it has been possible to pass α directly, it's often easier to think about either the span, center of mass (com) or half-life of an EW moment:
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:
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
Whereas if ignore_na=True, the weighted average would be calculated as
The ~Ewm.var, ~Ewm.std, and ~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
(For wi = 1, this reduces to the usual N/(N − 1) factor, with N = t + 1.) See Weighted Sample Variance for further details.