SciPy ODR

by Online Tutorials Library July 14, 2022

SciPy ODR

The ODR is an abbreviation form of Orthogonal Distance Regression. It is used in the regression studies. The basic linear regression is used to estimate the relationship between the two variables y and x by drawing the line of the best fit in the graph. Then the question arises why Orthogonal Distance Regression (ODR) needs? Sometimes measurement error occurred in the independent variable (x), not in the dependent variable (y).

The standard linear regression is focused on predict the Y value from the X value, so the useful thing to do is to calculate the error in the Y values (as we shown by dotted black lines in the below image). However, it is better to be account for the error in both X and Y (as shown by the dotted red lines in the following image).

Orthogonal Distance Regression (ODR) is a method that is used to calculate the error perpendicular to the line rather than vertically.

Orthogonal Distance Regression provides ODRPACK to perform ODR with non-linear functions. It is basically a FORTRAN-77 library. It can do explicit or implicit ODR fits. It can also be used to solve the ordinary least square problem (OLS).

Implementation of scipy.odr for Univariate Regression

The Univariate regression can be defined as determining relationship between one independent variable and one dependent variable. Consider the following example:

  import numpy as np  import matplotlib.pyplot as plt  from scipy.odr import *  import random  # Initiate some data, and generate the random number using random.random().  a = np.array([0, 1, 2, 3, 4, 5, 6, 7])  b = np.array([i**2 + random.random() for i in x])  # Define a quadratic function ( in this case) to fit the data with.  def linear_func(z, a):     m, p = z     return m*a + p  # Creating a model for fitting.  linear_model_fit = Model(linear_func)  # Creating a RealData object using our initiated data from above.  data = RealData(x, y)  # Fixed up ODR with the model and data.  odr = ODR(data, linear_model_fit, beta0=[0., 1.])  # Here we run the regression using the run().  out = odr.run()  # Use the in-built pprint method to give us results.  out.pprint()  

Output:

Beta: [ 7.62787497 -8.53630181]  Beta Std Error: [0.89306061 3.69444539]  Beta Covariance: [[ 1.52116591 -5.32408057]   [-5.32408057 26.0323407 ]]  Residual Variance: 0.5243065494144553  Inverse Condition #: 0.18510252155770376  Reason(s) for Halting:    Sum of squares convergence

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