The Ransac algorithm, also known as the Random Sampling Consensus Algorithm, is an iterative method for estimating mathematical models from a set of data containing noise or outliers. The Ransac algorithm is particularly suitable for linear regression problems because of its ability to handle data sets containing outliers and estimate the optimal linear model.
1 Introduction
In the world of data analysis and machine learning, linear regression is a widely used predictive model. However, traditional linear regression methods can suffer severely when the data set contains outliers or noise. To solve this problem, the Ransac linear regression algorithm provides a robust method to estimate linear model parameters.
The core idea of the Ransac algorithm is to randomly select a subset from the data set as the basic sample, and use this subset to estimate the parameters of the linear model. It then calculates the error for all data points to this model and determines whether the model is acceptable based on a preset threshold. If the model is accepted, the Ransac algorithm continues to optimize the model parameters; otherwise, it selects another subset and repeats the process.
In Ransac linear regression, the basic steps of the algorithm are as follows:
- Randomly select a subset of the data set as the base sample for the model. The size of the base sample is usually set by the user, usually as a proportion of the data set size.
- Estimate parameters of a linear model, such as slope and intercept, using the base sample.
- Calculate the model error, which is the distance between each point in the data set and the value predicted by the model.
- Determine whether the stopping criterion is met, that is, whether a good enough model has been found. If satisfied, exit the algorithm; otherwise, continue iteration.
- Select the points in the data set that are most inconsistent with the current model as outliers and remove them from the data set.
- Repeat steps 1-5 until a good enough model is found or the maximum number of iterations is reached.
The advantage of the Ransac algorithm is that it can handle data sets containing outliers and estimate the best linear model. Its disadvantage is that the number of iterations may be larger and the computational complexity is higher. In addition, the Ransac algorithm is sensitive to the distribution assumptions of the data. If the data distribution does not meet the assumptions, the performance of the algorithm may decrease.
2 Python code
# -*- coding: utf-8 -*-
"""
@Time ： 2023/10/17 14:13
@Auth ： RS迷途小书童
@File ：Ransac线性回归.py
@IDE ：PyCharm
"""
import numpy as np # 导入numpy库，用于进行数值计算和处理数组
import matplotlib.pyplot as plt # 导入matplotlib库的pyplot模块，用于绘制图形
import random # 导入random库，用于生成随机数
# 定义生成数据集的参数
SIZE = 500 # 数据点的总数
OUT = 230 # 数据的上限
X = np.linspace(0, 100, SIZE) # 生成从0到100，共SIZE个数据点的等差数列
Y = [] # 创建一个空列表，用于存储所有的数据值
# 对于X中的每一个元素，执行以下操作
for i in X:
# 生成一个0到10之间的随机整数，如果这个数大于5，执行下面的if语句，否则执行else语句
if random.randint(0, 10) > 5:
# 从0到OUT之间随机生成一个整数，并添加到Y列表中
Y.append(random.randint(0, OUT))
else:
# 再次生成一个0到10之间的随机整数，如果这个数大于5，执行下面的if语句，否则执行else语句
if random.randint(0, 10) > 5:
# 根据当前元素i和随机生成的数值计算出一个新的y值，并添加到Y列表中
Y.append(3 * i + 10 + 3 * random.random())
else:
Y.append(3 * i + 10 - 3 * random.random()) # 同上，只是计算公式略有不同
list_x = np.array(X) # 将X转换为numpy数组，方便后续的数据处理和计算
list_y = np.array(Y) # 将Y转换为numpy数组，方便后续的数据处理和计算
# 使用matplotlib库绘制原始数据点的散点图
plt.scatter(list_x, list_y) # 在二维平面上绘制原始数据点，使用散点图展示
plt.show() # 显示绘制的图形
def linear_regression(list_x, list_y):
# 进行迭代操作，寻找最佳的线性回归模型参数a和b
iters = 10000 # 迭代次数
epsilon = 3 # 内点的误差阈值
threshold = (SIZE - OUT) / SIZE + 0.01 # 阈值，用于控制早停（early stopping）策略
best_a, best_b = 0, 0 # 最佳线性回归模型的参数，初始值为0
pre_total = 0 # 内点数量的初始值，初始为0
# 进行迭代操作，寻找最佳的线性回归模型参数a和b
for i in range(iters):
# 从SIZE个数据点中随机选择两个点，索引存储在sample_index中
sample_index = random.sample(range(SIZE), 2)
x_1 = list_x[sample_index[0]] # 获取第一个点的x值
x_2 = list_x[sample_index[1]] # 获取第二个点的x值
y_1 = list_y[sample_index[0]] # 获取第一个点的y值
y_2 = list_y[sample_index[1]] # 获取第二个点的y值
# 根据两个点的坐标计算出线性回归模型的斜率a和截距b
a = (y_2 - y_1) / (x_2 - x_1) # 计算斜率a
b = y_1 - a * x_1 # 计算截距b
total_in = 0 # 内点计数器，初始值为0
# 对于每一个数据点，计算其对应的预测值，并与真实值进行比较，如果误差小于epsilon，则认为此点为内点，计数器加1
for index in range(SIZE):
y_estimate = a * list_x[index] + b # 根据线性回归模型计算出预测值
if abs(y_estimate - list_y[index]) < epsilon: # 判断预测值与真实值的误差是否小于epsilon
total_in += 1 # 如果小于epsilon，则此点为内点，计数器加1
# 如果当前的内点数量大于之前所有的内点数量，则更新最佳参数a和b，以及内点数量pre_total
if total_in > pre_total: # 记录最大内点数与对应的参数
pre_total = total_in
best_a = a
best_b = b
# 如果当前的内点数量大于设定的阈值所对应的人数，则跳出循环，不再进行迭代
if total_in > SIZE * threshold: # 如果当前内点数量大于阈值所设定的人数，则跳出循环
break # 跳出循环
print("迭代{}次,a = {}, b = {}".format(i, best_a, best_b)) # 输出当前迭代的次数，以及对应的线性回归模型参数a和b
x_line = list_x # 获取x轴的数据
y_line = best_a * x_line + best_b # 根据最佳线性回归模型计算出y轴的数据
plt.plot(x_line, y_line, c='r') # 使用matplotlib库绘制出线性回归模型的直线图，并用红色表示
plt.scatter(list_x, list_y) # 使用matplotlib库绘制出原始数据的散点图，并用其他颜色表示
plt.show() # 显示绘制的图形
linear_regression(list_x, list_y)
3 Summary
Ransac linear regression is a powerful linear regression method, especially suitable for processing data sets containing outliers and noise. Through the principle of random sampling consistency, the Ransac algorithm can screen out reliable basic samples from the data and estimate the parameters of the linear model based on this. Compared with traditional linear regression, the Ransac algorithm has better robustness, flexibility, computational efficiency and interpretability. In practical applications, Ransac linear regression has been widely used in various fields, such as regression prediction, feature selection, and anomaly detection. By combining with other technologies and methods, Ransac linear regression is also expected to further expand its application scope and performance in the future. In short, Ransac linear regression is a linear regression method worthy of in-depth study and application.