校园网站方案,网站友情链接出售,网站开发用户登陆的安全,烟台网站建设 熊掌号鲁棒线性模型估计 1.RANSAC算法1.1 算法的基本原理1.2 迭代次数N的计算1.3 参考代码 参考文献 当数据中出现较多异常点时#xff0c;常用的线性回归OLS会因为这些异常点的存在无法正确估计线性模型的参数#xff1a; W ( X T X ) − 1 X T Y \qquad \qquad W(X^TX)^{-1}X^T… 鲁棒线性模型估计 1.RANSAC算法1.1 算法的基本原理1.2 迭代次数N的计算1.3 参考代码 参考文献 当数据中出现较多异常点时常用的线性回归OLS会因为这些异常点的存在无法正确估计线性模型的参数 W ( X T X ) − 1 X T Y \qquad \qquad W(X^TX)^{-1}X^TY W(XTX)−1XTY
此时就需要寻找更鲁棒的方法过滤掉异常点以获得更准确的模型预测参数。
1.RANSAC算法
1.1 算法的基本原理
RANSACrandom sample consensus随机抽样一致算法)基本原理如下 ——————————————————————————————————————— 变量 N \quad N N迭代次数 s \quad s s每次迭代过程用于估计模型参数最小的样本点数量 τ \quad \tau τ每次迭代过程用于辨别正常点和异常点的阈值 d \quad d d每次迭代过程对辨别到的正常点进行线性拟合所需的最小个数; b e s t E r r \quad bestErr bestErr最佳误差用于判断是否更新模型参数初始值可设为无穷大值。 b e s t M o d e l \quad bestModel bestModel最佳模型。
算法
进行N次迭代
随机抽取 s s s个样本数据对这 s s s个数据进行线性拟合得到初始模型 m o d e l 1 model_1 model1使用 m o d e l 1 model_1 model1对 s s s个数据中各数据的自变量进行预测获得对应的 y ^ \hat{y} y^计算误差值 ( y − y ^ ) 2 (y-\hat{y})^2 (y−y^)2。如果误差值小于 τ \tau τ则为正常点否则为异常点若正常点个数小于 d d d直接进入下一次循环使用正常点进行线性拟合获取模型 m o d e l 2 model_2 model2对判断出的正常点求均方差 E r r 1 n ∑ ( y − y ^ ) 2 Err\cfrac{1}{n}\sum(y-\hat{y})^2 Errn1∑(y−y^)2 n n n为正常点的个数如果 E r r b e s t E r r ErrbestErr ErrbestErr那么设置 b e s t E r r E r r bestErrErr bestErrErr同时设置 b e s t M o d e l m o d e l 2 bestModelmodel_2 bestModelmodel2。 ————————————————————————————————————————
1.2 迭代次数N的计算
迭代次数 N N N的计算满足下式 1 − ( 1 − ( 1 − e ) s ) N p \qquad 1-(1-(1-e)^s)^Np 1−(1−(1−e)s)Np
其中 e \qquad e e为数据中异常点的概率 p \qquad p p为 N N N次迭代过程中至少存在一次采样数据全为正常点的期望概率如设置为0.99。
因此有 N log ( 1 − p ) log ( 1 − ( 1 − e ) s ) \qquad N\cfrac{\log(1-p)}{\log(1-(1-e)^s)} Nlog(1−(1−e)s)log(1−p)
1.3 参考代码
代码来源于https://en.wikipedia.org/wiki/Random_sample_consensus
from copy import copy
import numpy as np
from numpy.random import default_rng
rng default_rng()class RANSAC:def __init__(self, n10, k100, t0.05, d10, modelNone, lossNone, metricNone):self.n n # n: Minimum number of data points to estimate parametersself.k k # k: Maximum iterations allowedself.t t # t: Threshold value to determine if points are fit wellself.d d # d: Number of close data points required to assert model fits wellself.model model # model: class implementing fit and predictself.loss loss # loss: function of y_true and y_pred that returns a vectorself.metric metric # metric: function of y_true and y_pred and returns a floatself.best_fit Noneself.best_error np.infdef fit(self, X, y):for _ in range(self.k):ids rng.permutation(X.shape[0])maybe_inliers ids[: self.n]maybe_model copy(self.model).fit(X[maybe_inliers], y[maybe_inliers])thresholded (self.loss(y[ids][self.n :], maybe_model.predict(X[ids][self.n :])) self.t)inlier_ids ids[self.n :][np.flatnonzero(thresholded).flatten()]if inlier_ids.size self.d:inlier_points np.hstack([maybe_inliers, inlier_ids])better_model copy(self.model).fit(X[inlier_points], y[inlier_points])this_error self.metric(y[inlier_points], better_model.predict(X[inlier_points]))if this_error self.best_error:self.best_error this_errorself.best_fit better_modelreturn selfdef predict(self, X):return self.best_fit.predict(X)def square_error_loss(y_true, y_pred):return (y_true - y_pred) ** 2def mean_square_error(y_true, y_pred):return np.sum(square_error_loss(y_true, y_pred)) / y_true.shape[0]class LinearRegressor:def __init__(self):self.params Nonedef fit(self, X: np.ndarray, y: np.ndarray):r, _ X.shapeX np.hstack([np.ones((r, 1)), X])self.params np.linalg.inv(X.T X) X.T yreturn selfdef predict(self, X: np.ndarray):r, _ X.shapeX np.hstack([np.ones((r, 1)), X])return X self.paramsif __name__ __main__:regressor RANSAC(modelLinearRegressor(), losssquare_error_loss, metricmean_square_error)X np.array([-0.848,-0.800,-0.704,-0.632,-0.488,-0.472,-0.368,-0.336,-0.280,-0.200,-0.00800,-0.0840,0.0240,0.100,0.124,0.148,0.232,0.236,0.324,0.356,0.368,0.440,0.512,0.548,0.660,0.640,0.712,0.752,0.776,0.880,0.920,0.944,-0.108,-0.168,-0.720,-0.784,-0.224,-0.604,-0.740,-0.0440,0.388,-0.0200,0.752,0.416,-0.0800,-0.348,0.988,0.776,0.680,0.880,-0.816,-0.424,-0.932,0.272,-0.556,-0.568,-0.600,-0.716,-0.796,-0.880,-0.972,-0.916,0.816,0.892,0.956,0.980,0.988,0.992,0.00400]).reshape(-1,1)y np.array([-0.917,-0.833,-0.801,-0.665,-0.605,-0.545,-0.509,-0.433,-0.397,-0.281,-0.205,-0.169,-0.0531,-0.0651,0.0349,0.0829,0.0589,0.175,0.179,0.191,0.259,0.287,0.359,0.395,0.483,0.539,0.543,0.603,0.667,0.679,0.751,0.803,-0.265,-0.341,0.111,-0.113,0.547,0.791,0.551,0.347,0.975,0.943,-0.249,-0.769,-0.625,-0.861,-0.749,-0.945,-0.493,0.163,-0.469,0.0669,0.891,0.623,-0.609,-0.677,-0.721,-0.745,-0.885,-0.897,-0.969,-0.949,0.707,0.783,0.859,0.979,0.811,0.891,-0.137]).reshape(-1,1)regressor.fit(X, y)import matplotlib.pyplot as pltfig, ax plt.subplots(1, 1)ax.set_box_aspect(1)plt.scatter(X, y)line np.linspace(-1, 1, num100).reshape(-1, 1)plt.plot(line, regressor.predict(line), cperu)plt.show(blockTrue)结果如下
参考文献
[1] https://www.cse.psu.edu/~rtc12/CSE486/lecture15.pdf [2] https://en.wikipedia.org/wiki/Random_sample_consensus [3] Overview of the RANSAC Algorithm [4] https://scikit-learn.org/stable/auto_examples/linear_model/plot_ransac.html
