227 lines
8.0 KiB
Python
227 lines
8.0 KiB
Python
import numpy as np
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from statsmodels.robust import mad
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from scipy.optimize import minimize_scalar
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class BoxCox:
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"""
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Mixin class to allow for a Box-Cox transformation.
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"""
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def transform_boxcox(self, x, lmbda=None, method='guerrero', **kwargs):
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"""
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Performs a Box-Cox transformation on the data array x. If lmbda is None,
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the indicated method is used to estimate a suitable lambda parameter.
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Parameters
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----------
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x : array_like
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lmbda : float
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The lambda parameter for the Box-Cox transform. If None, a value
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will be estimated by means of the specified method.
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method : {'guerrero', 'loglik'}
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The method to estimate the lambda parameter. Will only be used if
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lmbda is None, and defaults to 'guerrero', detailed in Guerrero
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(1993). 'loglik' maximizes the profile likelihood.
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**kwargs
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Options for the specified method.
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* For 'guerrero', this entails window_length, the grouping
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parameter, scale, the dispersion measure, and options, to be
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passed to the optimizer.
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* For 'loglik': options, to be passed to the optimizer.
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Returns
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-------
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y : array_like
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The transformed series.
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lmbda : float
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The lmbda parameter used to transform the series.
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References
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----------
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Guerrero, Victor M. 1993. "Time-series analysis supported by power
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transformations". `Journal of Forecasting`. 12 (1): 37-48.
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Guerrero, Victor M. and Perera, Rafael. 2004. "Variance Stabilizing
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Power Transformation for Time Series," `Journal of Modern Applied
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Statistical Methods`. 3 (2): 357-369.
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Box, G. E. P., and D. R. Cox. 1964. "An Analysis of Transformations".
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`Journal of the Royal Statistical Society`. 26 (2): 211-252.
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"""
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x = np.asarray(x)
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if np.any(x <= 0):
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raise ValueError("Non-positive x.")
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if lmbda is None:
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lmbda = self._est_lambda(x,
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method=method,
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**kwargs)
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# if less than 0.01, treat lambda as zero.
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if np.isclose(lmbda, 0.):
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y = np.log(x)
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else:
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y = (np.power(x, lmbda) - 1.) / lmbda
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return y, lmbda
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def untransform_boxcox(self, x, lmbda, method='naive'):
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"""
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Back-transforms the Box-Cox transformed data array, by means of the
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indicated method. The provided argument lmbda should be the lambda
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parameter that was used to initially transform the data.
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Parameters
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----------
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x : array_like
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The transformed series.
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lmbda : float
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The lambda parameter that was used to transform the series.
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method : {'naive'}
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Indicates the method to be used in the untransformation. Defaults
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to 'naive', which reverses the transformation.
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NOTE: 'naive' is implemented natively, while other methods may be
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available in subclasses!
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Returns
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-------
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y : array_like
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The untransformed series.
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"""
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method = method.lower()
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x = np.asarray(x)
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if method == 'naive':
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if np.isclose(lmbda, 0.):
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y = np.exp(x)
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else:
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y = np.power(lmbda * x + 1, 1. / lmbda)
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else:
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raise ValueError(f"Method '{method}' not understood.")
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return y
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def _est_lambda(self, x, bounds=(-1, 2), method='guerrero', **kwargs):
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"""
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Computes an estimate for the lambda parameter in the Box-Cox
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transformation using method.
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Parameters
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----------
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x : array_like
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The untransformed data.
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bounds : tuple
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Numeric 2-tuple, that indicate the solution space for the lambda
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parameter. Default (-1, 2).
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method : {'guerrero', 'loglik'}
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The method by which to estimate lambda. Defaults to 'guerrero', but
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the profile likelihood ('loglik') is also available.
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**kwargs
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Options for the specified method.
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* For 'guerrero': window_length (int), the seasonality/grouping
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parameter. Scale ({'mad', 'sd'}), the dispersion measure. Options
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(dict), to be passed to the optimizer.
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* For 'loglik': Options (dict), to be passed to the optimizer.
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Returns
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-------
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lmbda : float
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The lambda parameter.
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"""
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method = method.lower()
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if len(bounds) != 2:
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raise ValueError("Bounds of length {} not understood."
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.format(len(bounds)))
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elif bounds[0] >= bounds[1]:
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raise ValueError("Lower bound exceeds upper bound.")
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if method == 'guerrero':
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lmbda = self._guerrero_cv(x, bounds=bounds, **kwargs)
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elif method == 'loglik':
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lmbda = self._loglik_boxcox(x, bounds=bounds, **kwargs)
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else:
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raise ValueError(f"Method '{method}' not understood.")
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return lmbda
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def _guerrero_cv(self, x, bounds, window_length=4, scale='sd',
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options={'maxiter': 25}):
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"""
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Computes lambda using guerrero's coefficient of variation. If no
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seasonality is present in the data, window_length is set to 4 (as
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per Guerrero and Perera, (2004)).
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NOTE: Seasonality-specific auxiliaries *should* provide their own
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seasonality parameter.
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Parameters
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----------
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x : array_like
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bounds : tuple
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Numeric 2-tuple, that indicate the solution space for the lambda
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parameter.
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window_length : int
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Seasonality/grouping parameter. Default 4, as per Guerrero and
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Perera (2004). NOTE: this indicates the length of the individual
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groups, not the total number of groups!
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scale : {'sd', 'mad'}
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The dispersion measure to be used. 'sd' indicates the sample
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standard deviation, but the more robust 'mad' is also available.
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options : dict
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The options (as a dict) to be passed to the optimizer.
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"""
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nobs = len(x)
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groups = int(nobs / window_length)
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# remove the first n < window_length observations from consideration.
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grouped_data = np.reshape(x[nobs - (groups * window_length): nobs],
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(groups, window_length))
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mean = np.mean(grouped_data, 1)
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scale = scale.lower()
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if scale == 'sd':
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dispersion = np.std(grouped_data, 1, ddof=1)
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elif scale == 'mad':
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dispersion = mad(grouped_data, axis=1)
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else:
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raise ValueError(f"Scale '{scale}' not understood.")
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def optim(lmbda):
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rat = np.divide(dispersion, np.power(mean, 1 - lmbda)) # eq 6, p 40
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return np.std(rat, ddof=1) / np.mean(rat)
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res = minimize_scalar(optim,
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bounds=bounds,
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method='bounded',
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options=options)
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return res.x
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def _loglik_boxcox(self, x, bounds, options={'maxiter': 25}):
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"""
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Taken from the Stata manual on Box-Cox regressions, where this is the
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special case of 'lhs only'. As an estimator for the variance, the
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sample variance is used, by means of the well-known formula.
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Parameters
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----------
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x : array_like
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options : dict
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The options (as a dict) to be passed to the optimizer.
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"""
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sum_x = np.sum(np.log(x))
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nobs = len(x)
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def optim(lmbda):
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y, lmbda = self.transform_boxcox(x, lmbda)
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return (1 - lmbda) * sum_x + (nobs / 2.) * np.log(np.var(y))
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res = minimize_scalar(optim,
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bounds=bounds,
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method='bounded',
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options=options)
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return res.x
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