# -*- coding: utf-8 -*- """ A Theil-Sen Estimator for Multiple Linear Regression Model """ # Author: Florian Wilhelm # # License: BSD 3 clause from __future__ import division, print_function, absolute_import import warnings from itertools import combinations import numpy as np from scipy import linalg from scipy.special import binom from scipy.linalg.lapack import get_lapack_funcs from .base import LinearModel from ..base import RegressorMixin from ..utils import check_random_state from ..utils import check_X_y, _get_n_jobs from ..externals.joblib import Parallel, delayed from ..externals.six.moves import xrange as range from ..exceptions import ConvergenceWarning _EPSILON = np.finfo(np.double).eps def _modified_weiszfeld_step(X, x_old): """Modified Weiszfeld step. This function defines one iteration step in order to approximate the spatial median (L1 median). It is a form of an iteratively re-weighted least squares method. Parameters ---------- X : array, shape = [n_samples, n_features] Training vector, where n_samples is the number of samples and n_features is the number of features. x_old : array, shape = [n_features] Current start vector. Returns ------- x_new : array, shape = [n_features] New iteration step. References ---------- - On Computation of Spatial Median for Robust Data Mining, 2005 T. Kärkkäinen and S. Äyrämö http://users.jyu.fi/~samiayr/pdf/ayramo_eurogen05.pdf """ diff = X - x_old diff_norm = np.sqrt(np.sum(diff ** 2, axis=1)) mask = diff_norm >= _EPSILON # x_old equals one of our samples is_x_old_in_X = int(mask.sum() < X.shape[0]) diff = diff[mask] diff_norm = diff_norm[mask][:, np.newaxis] quotient_norm = linalg.norm(np.sum(diff / diff_norm, axis=0)) if quotient_norm > _EPSILON: # to avoid division by zero new_direction = (np.sum(X[mask, :] / diff_norm, axis=0) / np.sum(1 / diff_norm, axis=0)) else: new_direction = 1. quotient_norm = 1. return (max(0., 1. - is_x_old_in_X / quotient_norm) * new_direction + min(1., is_x_old_in_X / quotient_norm) * x_old) def _spatial_median(X, max_iter=300, tol=1.e-3): """Spatial median (L1 median). The spatial median is member of a class of so-called M-estimators which are defined by an optimization problem. Given a number of p points in an n-dimensional space, the point x minimizing the sum of all distances to the p other points is called spatial median. Parameters ---------- X : array, shape = [n_samples, n_features] Training vector, where n_samples is the number of samples and n_features is the number of features. max_iter : int, optional Maximum number of iterations. Default is 300. tol : float, optional Stop the algorithm if spatial_median has converged. Default is 1.e-3. Returns ------- spatial_median : array, shape = [n_features] Spatial median. n_iter : int Number of iterations needed. References ---------- - On Computation of Spatial Median for Robust Data Mining, 2005 T. Kärkkäinen and S. Äyrämö http://users.jyu.fi/~samiayr/pdf/ayramo_eurogen05.pdf """ if X.shape[1] == 1: return 1, np.median(X.ravel()) tol **= 2 # We are computing the tol on the squared norm spatial_median_old = np.mean(X, axis=0) for n_iter in range(max_iter): spatial_median = _modified_weiszfeld_step(X, spatial_median_old) if np.sum((spatial_median_old - spatial_median) ** 2) < tol: break else: spatial_median_old = spatial_median else: warnings.warn("Maximum number of iterations {max_iter} reached in " "spatial median for TheilSen regressor." "".format(max_iter=max_iter), ConvergenceWarning) return n_iter, spatial_median def _breakdown_point(n_samples, n_subsamples): """Approximation of the breakdown point. Parameters ---------- n_samples : int Number of samples. n_subsamples : int Number of subsamples to consider. Returns ------- breakdown_point : float Approximation of breakdown point. """ return 1 - (0.5 ** (1 / n_subsamples) * (n_samples - n_subsamples + 1) + n_subsamples - 1) / n_samples def _lstsq(X, y, indices, fit_intercept): """Least Squares Estimator for TheilSenRegressor class. This function calculates the least squares method on a subset of rows of X and y defined by the indices array. Optionally, an intercept column is added if intercept is set to true. Parameters ---------- X : array, shape = [n_samples, n_features] Design matrix, where n_samples is the number of samples and n_features is the number of features. y : array, shape = [n_samples] Target vector, where n_samples is the number of samples. indices : array, shape = [n_subpopulation, n_subsamples] Indices of all subsamples with respect to the chosen subpopulation. fit_intercept : bool Fit intercept or not. Returns ------- weights : array, shape = [n_subpopulation, n_features + intercept] Solution matrix of n_subpopulation solved least square problems. """ fit_intercept = int(fit_intercept) n_features = X.shape[1] + fit_intercept n_subsamples = indices.shape[1] weights = np.empty((indices.shape[0], n_features)) X_subpopulation = np.ones((n_subsamples, n_features)) # gelss need to pad y_subpopulation to be of the max dim of X_subpopulation y_subpopulation = np.zeros((max(n_subsamples, n_features))) lstsq, = get_lapack_funcs(('gelss',), (X_subpopulation, y_subpopulation)) for index, subset in enumerate(indices): X_subpopulation[:, fit_intercept:] = X[subset, :] y_subpopulation[:n_subsamples] = y[subset] weights[index] = lstsq(X_subpopulation, y_subpopulation)[1][:n_features] return weights class TheilSenRegressor(LinearModel, RegressorMixin): """Theil-Sen Estimator: robust multivariate regression model. The algorithm calculates least square solutions on subsets with size n_subsamples of the samples in X. Any value of n_subsamples between the number of features and samples leads to an estimator with a compromise between robustness and efficiency. Since the number of least square solutions is "n_samples choose n_subsamples", it can be extremely large and can therefore be limited with max_subpopulation. If this limit is reached, the subsets are chosen randomly. In a final step, the spatial median (or L1 median) is calculated of all least square solutions. Read more in the :ref:`User Guide `. Parameters ---------- fit_intercept : boolean, optional, default True Whether to calculate the intercept for this model. If set to false, no intercept will be used in calculations. copy_X : boolean, optional, default True If True, X will be copied; else, it may be overwritten. max_subpopulation : int, optional, default 1e4 Instead of computing with a set of cardinality 'n choose k', where n is the number of samples and k is the number of subsamples (at least number of features), consider only a stochastic subpopulation of a given maximal size if 'n choose k' is larger than max_subpopulation. For other than small problem sizes this parameter will determine memory usage and runtime if n_subsamples is not changed. n_subsamples : int, optional, default None Number of samples to calculate the parameters. This is at least the number of features (plus 1 if fit_intercept=True) and the number of samples as a maximum. A lower number leads to a higher breakdown point and a low efficiency while a high number leads to a low breakdown point and a high efficiency. If None, take the minimum number of subsamples leading to maximal robustness. If n_subsamples is set to n_samples, Theil-Sen is identical to least squares. max_iter : int, optional, default 300 Maximum number of iterations for the calculation of spatial median. tol : float, optional, default 1.e-3 Tolerance when calculating spatial median. random_state : int, RandomState instance or None, optional, default None A random number generator instance to define the state of the random permutations generator. If int, random_state is the seed used by the random number generator; If RandomState instance, random_state is the random number generator; If None, the random number generator is the RandomState instance used by `np.random`. n_jobs : integer, optional, default 1 Number of CPUs to use during the cross validation. If ``-1``, use all the CPUs. verbose : boolean, optional, default False Verbose mode when fitting the model. Attributes ---------- coef_ : array, shape = (n_features) Coefficients of the regression model (median of distribution). intercept_ : float Estimated intercept of regression model. breakdown_ : float Approximated breakdown point. n_iter_ : int Number of iterations needed for the spatial median. n_subpopulation_ : int Number of combinations taken into account from 'n choose k', where n is the number of samples and k is the number of subsamples. References ---------- - Theil-Sen Estimators in a Multiple Linear Regression Model, 2009 Xin Dang, Hanxiang Peng, Xueqin Wang and Heping Zhang http://home.olemiss.edu/~xdang/papers/MTSE.pdf """ def __init__(self, fit_intercept=True, copy_X=True, max_subpopulation=1e4, n_subsamples=None, max_iter=300, tol=1.e-3, random_state=None, n_jobs=1, verbose=False): self.fit_intercept = fit_intercept self.copy_X = copy_X self.max_subpopulation = int(max_subpopulation) self.n_subsamples = n_subsamples self.max_iter = max_iter self.tol = tol self.random_state = random_state self.n_jobs = n_jobs self.verbose = verbose def _check_subparams(self, n_samples, n_features): n_subsamples = self.n_subsamples if self.fit_intercept: n_dim = n_features + 1 else: n_dim = n_features if n_subsamples is not None: if n_subsamples > n_samples: raise ValueError("Invalid parameter since n_subsamples > " "n_samples ({0} > {1}).".format(n_subsamples, n_samples)) if n_samples >= n_features: if n_dim > n_subsamples: plus_1 = "+1" if self.fit_intercept else "" raise ValueError("Invalid parameter since n_features{0} " "> n_subsamples ({1} > {2})." "".format(plus_1, n_dim, n_samples)) else: # if n_samples < n_features if n_subsamples != n_samples: raise ValueError("Invalid parameter since n_subsamples != " "n_samples ({0} != {1}) while n_samples " "< n_features.".format(n_subsamples, n_samples)) else: n_subsamples = min(n_dim, n_samples) if self.max_subpopulation <= 0: raise ValueError("Subpopulation must be strictly positive " "({0} <= 0).".format(self.max_subpopulation)) all_combinations = max(1, np.rint(binom(n_samples, n_subsamples))) n_subpopulation = int(min(self.max_subpopulation, all_combinations)) return n_subsamples, n_subpopulation def fit(self, X, y): """Fit linear model. Parameters ---------- X : numpy array of shape [n_samples, n_features] Training data y : numpy array of shape [n_samples] Target values Returns ------- self : returns an instance of self. """ random_state = check_random_state(self.random_state) X, y = check_X_y(X, y, y_numeric=True) n_samples, n_features = X.shape n_subsamples, self.n_subpopulation_ = self._check_subparams(n_samples, n_features) self.breakdown_ = _breakdown_point(n_samples, n_subsamples) if self.verbose: print("Breakdown point: {0}".format(self.breakdown_)) print("Number of samples: {0}".format(n_samples)) tol_outliers = int(self.breakdown_ * n_samples) print("Tolerable outliers: {0}".format(tol_outliers)) print("Number of subpopulations: {0}".format( self.n_subpopulation_)) # Determine indices of subpopulation if np.rint(binom(n_samples, n_subsamples)) <= self.max_subpopulation: indices = list(combinations(range(n_samples), n_subsamples)) else: indices = [random_state.choice(n_samples, size=n_subsamples, replace=False) for _ in range(self.n_subpopulation_)] n_jobs = _get_n_jobs(self.n_jobs) index_list = np.array_split(indices, n_jobs) weights = Parallel(n_jobs=n_jobs, verbose=self.verbose)( delayed(_lstsq)(X, y, index_list[job], self.fit_intercept) for job in range(n_jobs)) weights = np.vstack(weights) self.n_iter_, coefs = _spatial_median(weights, max_iter=self.max_iter, tol=self.tol) if self.fit_intercept: self.intercept_ = coefs[0] self.coef_ = coefs[1:] else: self.intercept_ = 0. self.coef_ = coefs return self