.. currentmodule:: pandas .. _computation: .. ipython:: python :suppress: import numpy as np np.random.seed(123456) from pandas import * import pandas.util.testing as tm randn = np.random.randn np.set_printoptions(precision=4, suppress=True) import matplotlib.pyplot as plt plt.close('all') Computational tools =================== Statistical functions --------------------- .. _computation.pct_change: Percent Change ~~~~~~~~~~~~~~ Both Series and DataFrame has a method pct_change to compute the percent change over a given number of periods (using fill_method to fill NA/null values). .. ipython:: python ser = Series(randn(8)) ser.pct_change() .. ipython:: python df = DataFrame(randn(10, 4)) df.pct_change(periods=3) .. _computation.covariance: Covariance ~~~~~~~~~~ The Series object has a method cov to compute covariance between series (excluding NA/null values). .. ipython:: python s1 = Series(randn(1000)) s2 = Series(randn(1000)) s1.cov(s2) Analogously, DataFrame has a method cov to compute pairwise covariances among the series in the DataFrame, also excluding NA/null values. .. ipython:: python frame = DataFrame(randn(1000, 5), columns=['a', 'b', 'c', 'd', 'e']) frame.cov() .. _computation.correlation: Correlation ~~~~~~~~~~~ Several methods for computing correlations are provided. Several kinds of correlation methods are provided: .. csv-table:: :header: "Method name", "Description" :widths: 20, 80 pearson (default), Standard correlation coefficient kendall, Kendall Tau correlation coefficient spearman, Spearman rank correlation coefficient .. \rho = \cov(x, y) / \sigma_x \sigma_y All of these are currently computed using pairwise complete observations. .. ipython:: python frame = DataFrame(randn(1000, 5), columns=['a', 'b', 'c', 'd', 'e']) frame.ix[::2] = np.nan # Series with Series frame['a'].corr(frame['b']) frame['a'].corr(frame['b'], method='spearman') # Pairwise correlation of DataFrame columns frame.corr() Note that non-numeric columns will be automatically excluded from the correlation calculation. A related method corrwith is implemented on DataFrame to compute the correlation between like-labeled Series contained in different DataFrame objects. .. ipython:: python index = ['a', 'b', 'c', 'd', 'e'] columns = ['one', 'two', 'three', 'four'] df1 = DataFrame(randn(5, 4), index=index, columns=columns) df2 = DataFrame(randn(4, 4), index=index[:4], columns=columns) df1.corrwith(df2) df2.corrwith(df1, axis=1) .. _computation.ranking: Data ranking ~~~~~~~~~~~~ The rank method produces a data ranking with ties being assigned the mean of the ranks (by default) for the group: .. ipython:: python s = Series(np.random.randn(5), index=list('abcde')) s['d'] = s['b'] # so there's a tie s.rank() rank is also a DataFrame method and can rank either the rows (axis=0) or the columns (axis=1). NaN values are excluded from the ranking. .. ipython:: python df = DataFrame(np.random.randn(10, 6)) df[4] = df[2][:5] # some ties df df.rank(1) rank optionally takes a parameter ascending which by default is true; when false, data is reverse-ranked, with larger values assigned a smaller rank. rank supports different tie-breaking methods, specified with the method parameter: - average : average rank of tied group - min : lowest rank in the group - max : highest rank in the group - first : ranks assigned in the order they appear in the array .. note:: These methods are significantly faster (around 10-20x) than scipy.stats.rankdata. .. currentmodule:: pandas .. currentmodule:: pandas.stats.api .. _stats.moments: Moving (rolling) statistics / moments ------------------------------------- For working with time series data, a number of functions are provided for computing common *moving* or *rolling* statistics. Among these are count, sum, mean, median, correlation, variance, covariance, standard deviation, skewness, and kurtosis. All of these methods are in the :mod:pandas namespace, but otherwise they can be found in :mod:pandas.stats.moments. .. csv-table:: :header: "Function", "Description" :widths: 20, 80 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_corr_pairwise, Pairwise correlation of DataFrame columns Generally these methods all have the same interface. The binary operators (e.g. rolling_corr) take two Series or DataFrames. Otherwise, they all accept the following arguments: - window: size of moving window - min_periods: threshold of non-null data points to require (otherwise result is NA) - freq: optionally specify a :ref:frequency string  or :ref: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 These functions can be applied to ndarrays or Series objects: .. ipython:: python ts = Series(randn(1000), index=date_range('1/1/2000', periods=1000)) ts = ts.cumsum() ts.plot(style='k--') @savefig rolling_mean_ex.png width=4.5in rolling_mean(ts, 60).plot(style='k') They can also be applied to DataFrame objects. This is really just syntactic sugar for applying the moving window operator to all of the DataFrame's columns: .. ipython:: python :suppress: plt.close('all') .. ipython:: python df = DataFrame(randn(1000, 4), index=ts.index, columns=['A', 'B', 'C', 'D']) df = df.cumsum() @savefig rolling_mean_frame.png width=4.5in rolling_sum(df, 60).plot(subplots=True) The rolling_apply function takes an extra func argument and performs generic rolling computations. The func argument should be a single function that produces a single value from an ndarray input. Suppose we wanted to compute the mean absolute deviation on a rolling basis: .. ipython:: python mad = lambda x: np.fabs(x - x.mean()).mean() @savefig rolling_apply_ex.png width=4.5in rolling_apply(ts, 60, mad).plot(style='k') .. _stats.moments.binary: 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: compute statistic for matching column names, returning a DataFrame For example: .. ipython:: python df2 = df[:20] rolling_corr(df2, df2['B'], window=5) .. _stats.moments.corr_pairwise: Computing rolling pairwise correlations ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In financial data analysis and other fields it's common to compute correlation matrices for a collection of time series. More difficult is to compute a moving-window correlation matrix. This can be done using the rolling_corr_pairwise function, which yields a Panel whose items are the dates in question: .. ipython:: python correls = rolling_corr_pairwise(df, 50) correls[df.index[-50]] You can efficiently retrieve the time series of correlations between two columns using ix indexing: .. ipython:: python :suppress: plt.close('all') .. ipython:: python @savefig rolling_corr_pairwise_ex.png width=4.5in correls.ix[:, 'A', 'C'].plot() Expanding window moment functions --------------------------------- A common alternative to rolling statistics is to use an *expanding* window, which yields the value of the statistic with all the data available up to that point in time. As these calculations are a special case of rolling statistics, they are implemented in pandas such that the following two calls are equivalent: .. ipython:: python rolling_mean(df, window=len(df), min_periods=1)[:5] expanding_mean(df)[:5] Like the rolling_ functions, the following methods are included in the pandas namespace or can be located in pandas.stats.moments. .. csv-table:: :header: "Function", "Description" :widths: 20, 80 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) expanding_corr_pairwise, Pairwise correlation of DataFrame columns Aside from not having a window parameter, these functions have the same interfaces as their rolling_ counterpart. Like above, the parameters they all accept are: - min_periods: threshold of non-null data points to require. Defaults to minimum needed to compute statistic. No NaNs will be output once min_periods non-null data points have been seen. - freq: optionally specify a :ref:frequency string  or :ref: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: .. ipython:: python :suppress: plt.close('all') .. ipython:: python ts.plot(style='k--') @savefig expanding_mean_frame.png width=4.5in expanding_mean(ts).plot(style='k') Exponentially weighted moment functions --------------------------------------- A related set of functions are exponentially weighted versions of many of the above statistics. A number of EW (exponentially weighted) functions are provided using the blending method. For example, where :math:y_t is the result and :math:x_t the input, we compute an exponentially weighted moving average as .. math:: y_t = \alpha y_{t-1} + (1 - \alpha) x_t One must have :math:0 < \alpha \leq 1, but rather than pass :math:\alpha directly, it's easier to think about either the **span** or **center of mass (com)** of an EW moment: .. math:: \alpha = \begin{cases} \frac{2}{s + 1}, s = \text{span}\\ \frac{1}{c + 1}, c = \text{center of mass} \end{cases} You can pass one or the other to these functions but not both. **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. Here is the list of functions available: .. csv-table:: :header: "Function", "Description" :widths: 20, 80 ewma, EW moving average ewmvar, EW moving variance ewmstd, EW moving standard deviation ewmcorr, EW moving correlation ewmcov, EW moving covariance Here are an example for a univariate time series: .. ipython:: python plt.close('all') ts.plot(style='k--') @savefig ewma_ex.png width=4.5in ewma(ts, span=20).plot(style='k') .. note:: The EW functions perform a standard adjustment to the initial observations whereby if there are fewer observations than called for in the span, those observations are reweighted accordingly. .. _stats.ols: Linear and panel regression --------------------------- .. note:: We plan to move this functionality to statsmodels __ for the next release. Some of the result attributes may change names in order to foster naming consistency with the rest of statsmodels. We will provide every effort to provide compatibility with older versions of pandas, however. We have implemented a very fast set of *moving-window linear regression* classes in pandas. Two different types of regressions are supported: - Standard ordinary least squares (OLS) multiple regression - Multiple regression (OLS-based) on panel data __ including with fixed-effects (also known as entity or individual effects) or time-effects. Both kinds of linear models are accessed through the ols function in the pandas namespace. They all take the following arguments to specify either a static (full sample) or dynamic (moving window) regression: - window_type: 'full sample' (default), 'expanding', or rolling - window: size of the moving window in the window_type='rolling' case. If window is specified, window_type will be automatically set to 'rolling' - min_periods: minimum number of time periods to require to compute the regression coefficients Generally speaking, the ols works by being given a y (response) object and an x (predictors) object. These can take many forms: - y: a Series, ndarray, or DataFrame (panel model) - x: Series, DataFrame, dict of Series, dict of DataFrame or Panel Based on the types of y and x, the model will be inferred to either a panel model or a regular linear model. If the y variable is a DataFrame, the result will be a panel model. In this case, the x variable must either be a Panel, or a dict of DataFrame (which will be coerced into a Panel). Standard OLS regression ~~~~~~~~~~~~~~~~~~~~~~~ Let's pull in some sample data: .. ipython:: python from pandas.io.data import DataReader symbols = ['MSFT', 'GOOG', 'AAPL'] data = dict((sym, DataReader(sym, "yahoo")) for sym in symbols) panel = Panel(data).swapaxes('items', 'minor') close_px = panel['Close'] # convert closing prices to returns rets = close_px / close_px.shift(1) - 1 rets.info() Let's do a static regression of AAPL returns on GOOG returns: .. ipython:: python model = ols(y=rets['AAPL'], x=rets.ix[:, ['GOOG']]) model model.beta If we had passed a Series instead of a DataFrame with the single GOOG column, the model would have assigned the generic name x to the sole right-hand side variable. We can do a moving window regression to see how the relationship changes over time: .. ipython:: python :suppress: plt.close('all') .. ipython:: python model = ols(y=rets['AAPL'], x=rets.ix[:, ['GOOG']], window=250) # just plot the coefficient for GOOG @savefig moving_lm_ex.png width=5in model.beta['GOOG'].plot() It looks like there are some outliers rolling in and out of the window in the above regression, influencing the results. We could perform a simple winsorization __ at the 3 STD level to trim the impact of outliers: .. ipython:: python :suppress: plt.close('all') .. ipython:: python winz = rets.copy() std_1year = rolling_std(rets, 250, min_periods=20) # cap at 3 * 1 year standard deviation cap_level = 3 * np.sign(winz) * std_1year winz[np.abs(winz) > 3 * std_1year] = cap_level winz_model = ols(y=winz['AAPL'], x=winz.ix[:, ['GOOG']], window=250) model.beta['GOOG'].plot(label="With outliers") @savefig moving_lm_winz.png width=5in winz_model.beta['GOOG'].plot(label="Winsorized"); plt.legend(loc='best') So in this simple example we see the impact of winsorization is actually quite significant. Note the correlation after winsorization remains high: .. ipython:: python winz.corrwith(rets) Multiple regressions can be run by passing a DataFrame with multiple columns for the predictors x: .. ipython:: python ols(y=winz['AAPL'], x=winz.drop(['AAPL'], axis=1)) Panel regression ~~~~~~~~~~~~~~~~ We've implemented moving window panel regression on potentially unbalanced panel data (see this article __ if this means nothing to you). Suppose we wanted to model the relationship between the magnitude of the daily return and trading volume among a group of stocks, and we want to pool all the data together to run one big regression. This is actually quite easy: .. ipython:: python # make the units somewhat comparable volume = panel['Volume'] / 1e8 model = ols(y=volume, x={'return' : np.abs(rets)}) model In a panel model, we can insert dummy (0-1) variables for the "entities" involved (here, each of the stocks) to account the a entity-specific effect (intercept): .. ipython:: python fe_model = ols(y=volume, x={'return' : np.abs(rets)}, entity_effects=True) fe_model Because we ran the regression with an intercept, one of the dummy variables must be dropped or the design matrix will not be full rank. If we do not use an intercept, all of the dummy variables will be included: .. ipython:: python fe_model = ols(y=volume, x={'return' : np.abs(rets)}, entity_effects=True, intercept=False) fe_model We can also include *time effects*, which demeans the data cross-sectionally at each point in time (equivalent to including dummy variables for each date). More mathematical care must be taken to properly compute the standard errors in this case: .. ipython:: python te_model = ols(y=volume, x={'return' : np.abs(rets)}, time_effects=True, entity_effects=True) te_model Here the intercept (the mean term) is dropped by default because it will be 0 according to the model assumptions, having subtracted off the group means. Result fields and tests ~~~~~~~~~~~~~~~~~~~~~~~ We'll leave it to the user to explore the docstrings and source, especially as we'll be moving this code into statsmodels in the near future.