Inverse Linear Regression
Introduction
The goal: fit bi-variate data (x,y) to the curve:
y = 1 / (a*x + b)
We will be able to use the linear regression model with the following transformation:
1/y = a* x + b
y' = 1/y
x' = x
a' = a
b' = b
y' = a' * x' + b'
Let's illustrate this with an example.
Example
Fit the curve y = 1 / (a*x + b) to the data:
(-2, -1.43)
(-1, 0.4)
(0, 0.18)
(1, 0.11)
(2, 0.08)
(4, 0.05)
Transform the data to (x', y'): x' = x, y' = 1/y
(-2, -1/1.43 = 0.6693006993)
(-1, 1/0.4 = 2.5)
(0, 1/0.18 = 5.555555556)
(1, 1/0.11 = 9.090909091)
(2, 1/0.08 = 12.5)
(4, 1/0.05 = 20)
Regression analysis with the transformed data:
Slope (a,m) = 3.443917194
Intercept (b) = 5.861915862
r^2 = 0.998386787
1/y = 3.443917194 * x + 5.861915862
y = 1 / (3.443917194 * x + 5.861915862)
Note: My next blog entry will be posted on September 2, 2019, 12:00 AM Pacific Daylight Time on Labor Day.
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Introduction
The goal: fit bi-variate data (x,y) to the curve:
y = 1 / (a*x + b)
We will be able to use the linear regression model with the following transformation:
1/y = a* x + b
y' = 1/y
x' = x
a' = a
b' = b
y' = a' * x' + b'
Let's illustrate this with an example.
Example
Fit the curve y = 1 / (a*x + b) to the data:
(-2, -1.43)
(-1, 0.4)
(0, 0.18)
(1, 0.11)
(2, 0.08)
(4, 0.05)
Transform the data to (x', y'): x' = x, y' = 1/y
(-2, -1/1.43 = 0.6693006993)
(-1, 1/0.4 = 2.5)
(0, 1/0.18 = 5.555555556)
(1, 1/0.11 = 9.090909091)
(2, 1/0.08 = 12.5)
(4, 1/0.05 = 20)
Regression analysis with the transformed data:
Slope (a,m) = 3.443917194
Intercept (b) = 5.861915862
r^2 = 0.998386787
1/y = 3.443917194 * x + 5.861915862
y = 1 / (3.443917194 * x + 5.861915862)
Note: My next blog entry will be posted on September 2, 2019, 12:00 AM Pacific Daylight Time on Labor Day.
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
