1. Curve Fitting and Interpolation: Lecture (IV)
Chapter 5
Curve Fitting and Interpolation: Lecture (IV)
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2. Outline Least-square regression Linear (least-square) regression
Introduction-What is regression? When do we use it?
Linear (least-square) regression
What is linear regression?
What is “least-square”?
Accuracy of linear regression
Dr. Jie Zou PHY 3320

3. Introduction Regression
What is it? - To derive an approximating function or curve that represents the general trend of the data. The curve does not necessarily pass through all the data points.
When do we use it? - Usually used when the data appear to have significant error.
Linear regression
Non-linear regression
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4. Linear least-square regression
Linear regression: Fit a “best” line to the data.
Fitting function: y = a0 + a1x
Parameters a0 and a1 are to be determined.
Residual error:
ei = yi – a0 – a1xi, i= 1, 2, …
ei: residual error at each data point
Least-square criterion for a “best” fit:
Minimize with respect to the parameters a0 and a1.
Residual error
Inadequate criterion: Minimize
Dr. Jie Zou PHY 3320

5. Derivation of the linear least-square regression
Best straight line: y = a0 + a1x
Determine a0 and a1: Apply the least-square criterion
Minimize with respect to a0 and a1
Set
Solve the simultaneous linear equations for a0 and a1:
Dr. Jie Zou PHY 3320

6. Example: Linear regression
Use least-square regression, fit a straight line to the values in the table below.
v (m/s) 10 20 30 40 50 60 70 80
F (N) 25
380
550
610
1220
830
1450
Best fit line:
F = v
Dr. Jie Zou PHY 3320

7. Accuracy of linear regression
To quantify the “goodness” of our fit:
The residual error before regression:
The residual error after regression:
The difference between S0 and S provides a measure of the accuracy of regression or the extent of improvement achieved by the least-square fit.
Correlation coefficient:
A good least-square fit is indicated by a large value or r; rmax = 1.0.
Before
After
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8. Example: Correlation coefficient
In the previous example on Slide #6, calculate the correlation coefficient r for the best fit line.
Answer: r = (0.8805)1/2 =
“These results indicate that 88.05% of the original uncertainty has been explained by the linear model” (textbook by Chapra, p. 299).
Dr. Jie Zou PHY 3320

9. Implementation on a computer
For the example on Slide #6, write an M-file, Mylinearregression.m, to find the best fit line using the method of least-squares.
Plot the original discrete data points (in open circles) and the best fit line on the same figure.
Dr. Jie Zou PHY 3320