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Chapter 5

Curve Fitting and Interpolation: Lecture (IV)

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Outline

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

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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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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.

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Residual error

Inadequate criterion: Minimize

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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:

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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 70 380 550 610 1220

830 1450

Best fit line:

F = -234.2857 + 19.47024 v

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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.

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Before

After

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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 = 0.9383 “These results indicate that 88.05% of

the original uncertainty has been explained by the linear model” (textbook by Chapra, p. 299).

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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.