247 lines
7.0 KiB
Python
247 lines
7.0 KiB
Python
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'''Tools for multivariate analysis
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Author : Josef Perktold
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License : BSD-3
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TODO:
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- names of functions, currently just "working titles"
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'''
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import numpy as np
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from statsmodels.tools.tools import Bunch
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def partial_project(endog, exog):
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'''helper function to get linear projection or partialling out of variables
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endog variables are projected on exog variables
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Parameters
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----------
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endog : ndarray
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array of variables where the effect of exog is partialled out.
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exog : ndarray
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array of variables on which the endog variables are projected.
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Returns
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-------
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res : instance of Bunch with
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- params : OLS parameter estimates from projection of endog on exog
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- fittedvalues : predicted values of endog given exog
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- resid : residual of the regression, values of endog with effect of
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exog partialled out
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Notes
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-----
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This is no-frills mainly for internal calculations, no error checking or
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array conversion is performed, at least for now.
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'''
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x1, x2 = endog, exog
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params = np.linalg.pinv(x2).dot(x1)
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predicted = x2.dot(params)
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residual = x1 - predicted
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res = Bunch(params=params,
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fittedvalues=predicted,
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resid=residual)
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return res
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def cancorr(x1, x2, demean=True, standardize=False):
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'''canonical correlation coefficient beween 2 arrays
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Parameters
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----------
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x1, x2 : ndarrays, 2_D
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two 2-dimensional data arrays, observations in rows, variables in columns
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demean : bool
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If demean is true, then the mean is subtracted from each variable
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standardize : bool
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If standardize is true, then each variable is demeaned and divided by
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its standard deviation. Rescaling does not change the canonical
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correlation coefficients.
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Returns
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-------
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ccorr : ndarray, 1d
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canonical correlation coefficients, sorted from largest to smallest.
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Note, that these are the square root of the eigenvalues.
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Notes
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-----
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This is a helper function for other statistical functions. It only
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calculates the canonical correlation coefficients and does not do a full
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canoncial correlation analysis
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The canonical correlation coefficient is calculated with the generalized
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matrix inverse and does not raise an exception if one of the data arrays
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have less than full column rank.
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See Also
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--------
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cc_ranktest
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cc_stats
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CCA not yet
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'''
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#x, y = x1, x2
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if demean or standardize:
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x1 = (x1 - x1.mean(0))
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x2 = (x2 - x2.mean(0))
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if standardize:
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#std does not make a difference to canonical correlation coefficients
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x1 /= x1.std(0)
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x2 /= x2.std(0)
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t1 = np.linalg.pinv(x1).dot(x2)
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t2 = np.linalg.pinv(x2).dot(x1)
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m = t1.dot(t2)
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cc = np.sqrt(np.linalg.eigvals(m))
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cc.sort()
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return cc[::-1]
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def cc_ranktest(x1, x2, demean=True, fullrank=False):
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'''rank tests based on smallest canonical correlation coefficients
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Anderson canonical correlations test (LM test) and
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Cragg-Donald test (Wald test)
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Assumes homoskedasticity and independent observations, overrejects if
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there is heteroscedasticity or autocorrelation.
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The Null Hypothesis is that the rank is k - 1, the alternative hypothesis
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is that the rank is at least k.
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Parameters
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----------
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x1, x2 : ndarrays, 2_D
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two 2-dimensional data arrays, observations in rows, variables in columns
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demean : bool
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If demean is true, then the mean is subtracted from each variable.
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fullrank : bool
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If true, then only the test that the matrix has full rank is returned.
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If false, the test for all possible ranks are returned. However, no
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the p-values are not corrected for the multiplicity of tests.
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Returns
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-------
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value : float
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value of the test statistic
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p-value : float
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p-value for the test Null Hypothesis tha the smallest canonical
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correlation coefficient is zero. based on chi-square distribution
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df : int
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degrees of freedom for thechi-square distribution in the hypothesis test
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ccorr : ndarray, 1d
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All canonical correlation coefficients sorted from largest to smallest.
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Notes
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-----
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Degrees of freedom for the distribution of the test statistic are based on
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number of columns of x1 and x2 and not on their matrix rank.
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(I'm not sure yet what the interpretation of the test is if x1 or x2 are of
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reduced rank.)
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See Also
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--------
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cancorr
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cc_stats
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'''
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from scipy import stats
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nobs1, k1 = x1.shape
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nobs2, k2 = x2.shape
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cc = cancorr(x1, x2, demean=demean)
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cc2 = cc * cc
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if fullrank:
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df = np.abs(k1 - k2) + 1
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value = nobs1 * cc2[-1]
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w_value = nobs1 * (cc2[-1] / (1. - cc2[-1]))
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return value, stats.chi2.sf(value, df), df, cc, w_value, stats.chi2.sf(w_value, df)
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else:
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r = np.arange(min(k1, k2))[::-1]
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df = (k1 - r) * (k2 - r)
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values = nobs1 * cc2[::-1].cumsum()
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w_values = nobs1 * (cc2 / (1. - cc2))[::-1].cumsum()
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return values, stats.chi2.sf(values, df), df, cc, w_values, stats.chi2.sf(w_values, df)
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def cc_stats(x1, x2, demean=True):
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'''MANOVA statistics based on canonical correlation coefficient
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Calculates Pillai's Trace, Wilk's Lambda, Hotelling's Trace and
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Roy's Largest Root.
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Parameters
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----------
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x1, x2 : ndarrays, 2_D
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two 2-dimensional data arrays, observations in rows, variables in columns
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demean : bool
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If demean is true, then the mean is subtracted from each variable.
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Returns
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-------
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res : dict
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Dictionary containing the test statistics.
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Notes
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-----
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same as `canon` in Stata
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missing: F-statistics and p-values
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TODO: should return a results class instead
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produces nans sometimes, singular, perfect correlation of x1, x2 ?
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'''
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nobs1, k1 = x1.shape # endogenous ?
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nobs2, k2 = x2.shape
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cc = cancorr(x1, x2, demean=demean)
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cc2 = cc**2
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lam = (cc2 / (1 - cc2)) # what if max cc2 is 1 ?
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# Problem: ccr might not care if x1 or x2 are reduced rank,
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# but df will depend on rank
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df_model = k1 * k2 # df_hypothesis (we do not include mean in x1, x2)
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df_resid = k1 * (nobs1 - k2 - demean)
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s = min(df_model, k1)
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m = 0.5 * (df_model - k1)
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n = 0.5 * (df_resid - k1 - 1)
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df1 = k1 * df_model
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df2 = k2
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pt_value = cc2.sum() # Pillai's trace
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wl_value = np.product(1 / (1 + lam)) # Wilk's Lambda
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ht_value = lam.sum() # Hotelling's Trace
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rm_value = lam.max() # Roy's largest root
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#from scipy import stats
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# what's the distribution, the test statistic ?
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res = {}
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res['canonical correlation coefficient'] = cc
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res['eigenvalues'] = lam
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res["Pillai's Trace"] = pt_value
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res["Wilk's Lambda"] = wl_value
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res["Hotelling's Trace"] = ht_value
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res["Roy's Largest Root"] = rm_value
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res['df_resid'] = df_resid
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res['df_m'] = m
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return res
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