Scipy 简明教程

SciPy - ODR

ODR 代表 Orthogonal Distance Regression ，用于回归研究。基本的线性回归通常用于通过在图表上绘制最佳拟合线来估计两个变量 y 和 x 之间的关系。

ODR stands for Orthogonal Distance Regression, which is used in the regression studies. Basic linear regression is often used to estimate the relationship between the two variables y and x by drawing the line of best fit on the graph.

为此使用的数学方法称为 Least Squares ，其目的是最小化每个点的平方误差之和。这里关键的问题是，如何计算每个点的误差（也称为残差）？

The mathematical method that is used for this is known as Least Squares, and aims to minimize the sum of the squared error for each point. The key question here is how do you calculate the error (also known as the residual) for each point?

在标准线性回归中，目标是从 X 值预测 Y 值，因此明智的做法是计算 Y 值的误差（在下图中显示为灰色线）。然而，有时考虑 X 和 Y 中的误差更明智（如下面的图像中的虚线红线所示）。

In a standard linear regression, the aim is to predict the Y value from the X value – so the sensible thing to do is to calculate the error in the Y values (shown as the gray lines in the following image). However, sometimes it is more sensible to take into account the error in both X and Y (as shown by the dotted red lines in the following image).

例如 − 当您知道您的 X 测量值不确定，或者当您不想关注一个变量相对于另一个变量的误差时。

For example − When you know your measurements of X are uncertain, or when you do not want to focus on the errors of one variable over another.

orthogonal distance linear regression

正交距离回归（ODR）是一种可以做到这一点的方法（在这种上下文中，正交表示垂直 – 因此它计算垂直于线的误差，而不是只是“垂直”的误差）。

Orthogonal Distance Regression (ODR) is a method that can do this (orthogonal in this context means perpendicular – so it calculates errors perpendicular to the line, rather than just ‘vertically’).

scipy.odr Implementation for Univariate Regression

以下示例演示了 scipy.odr 在单变量回归中的实现。

The following example demonstrates scipy.odr implementation for univariate regression.

import numpy as np
import matplotlib.pyplot as plt
from scipy.odr import *
import random

# Initiate some data, giving some randomness using random.random().
x = np.array([0, 1, 2, 3, 4, 5])
y = np.array([i**2 + random.random() for i in x])

# Define a function (quadratic in our case) to fit the data with.
def linear_func(p, x):
   m, c = p
   return m*x + c

# Create a model for fitting.
linear_model = Model(linear_func)

# Create a RealData object using our initiated data from above.
data = RealData(x, y)

# Set up ODR with the model and data.
odr = ODR(data, linear_model, beta0=[0., 1.])

# Run the regression.
out = odr.run()

# Use the in-built pprint method to give us results.
out.pprint()

上述程序将生成以下输出。

The above program will generate the following output.

Beta: [ 5.51846098 -4.25744878]
Beta Std Error: [ 0.7786442 2.33126407]

Beta Covariance: [
   [ 1.93150969 -4.82877433]
   [ -4.82877433 17.31417201
]]

Residual Variance: 0.313892697582
Inverse Condition #: 0.146618499389
Reason(s) for Halting:
   Sum of squares convergence