Gauss and the Method of Least Squares

1. Gauss and the Method of Least Squares Teddy Petrou Hongxiao Zhu

2. Outline • Who was Gauss? • Why was there controversy in finding the method of least squares? • Gauss’ treatment of error • Gauss’ derivation of the method of least squares • Gauss’ derivation by modern matrix notation • Gauss-Markov theorem • Limitations of the method of least squares • References

3. Johann Carl Friedrich Gauss Born:1777 Brunswick, Germany Died: February 23, 1855, Göttingen, Germany By the age of eight during arithmetic class he astonished his teachers by being able to instantly find the sum of the first hundred integers.

4. Facts about Gauss • Attended Brunswick College in 1792, where he discovered many important theorems before even reaching them in his studies • Found a square root in two different ways to fifty decimal places by ingenious expansions and interpolations • Constructed a regular 17 sided polygon, the first advance in this matter in two millennia. He was only 18 when he made the discovery

5. Ideas of Gauss • Gauss was a mathematical scientist with interests in so many areas as a young man including theory of numbers, to algebra, analysis, geometry, probability, and the theory of errors. • His interests grew, including observational astronomy, celestial mechanics, surveying, geodesy, capillarity, geomagnetism, electromagnetism, mechanism optics, and actuarial science.

6. Intellectual Personality and Controversy • Those who knew Gauss best found him to be cold and uncommunicative. • He only published half of his ideas and found no one to share his most valued thoughts. • In 1805 Adrien-Marie Legendre published a paper on the method of least squares. His treatment, however, lacked a ‘formal consideration of probability and it’s relationship to least squares’, making it impossible to determine the accuracy of the method when applied to real observations. • Gauss claimed that he had written colleagues concerning the use of least squares dating back to 1795

7. Formal Arrival of Least Squares • Gauss • Published ‘The theory of the Motion of Heavenly Bodies’in 1809. He gave a probabilistic justification of the method,which was based on the assumption of a normal distribution of errors. Gauss himself later abandoned the use of normal error function. • Published‘Theory of the Combination of Observations Least Subject to Errors’in 1820s. He substituted the root mean square error for Laplace’s mean absolute error. • Laplace Derived the method of least squares (between1802 and 1820) from the principle that the best estimate should have the smallest ‘mean error’ -the mean of the absolute value of the error.

8. Treatment of Errors • Using probability theory to describe error • Error will be treated as a random variable • Two types of errors • Constant-associated with calibration • Random error

9. Error Assumptions • Gauss began his study by making two assumptions • Random errors of measurements of the same type lie within fixed limits • All errors within these limits are possible, but not necessarily with equal likelihood

10. Density Function

11. Mean and Variance • Define . In many cases assume k=0 • Define mean square error as • If k=0 then the variance will equal

12. Reasons for • is always positive and is simple. • The function is differentiable and integrable unlike the absolute value function. • The function approximates the average value in cases where large numbers of observations are being considered,and is simple to use when considering small numbers of observations.

13. More on Variance If then variance equals . Suppose we have independent random variables with standard deviation 1 and expected value 0. The linear function of total errors is given by Now the variance of E is given as This is assuming every error falls within standard deviations from the mean

14. Gauss’ Derivation of the Method of Least Squares • Suppose a quantity, V=f(x), where V, x are unknown. We estimate V by an observation L. • If x is calculated by L, L~f(x), error will occur. • But if several quantities V,V’,V’’…depend on the same unknown x and they are determined by inexact observations, then we can recover x by some combinations of the observations. • Similar situations occur when we observe several quantities that depend on several unknowns.

15. Gauss’ Derivation of the Method of Least Squares

16. Gauss’ Derivation of the Method of Least Squares

18. Gauss’ Derivation of the Method of Least Squares • Solutions: • It’s still not obvious: How do these results relate with the least squares estimation?

19. Gauss’ Derivation of the Method of Least Squares • It can be proved that

20. Gauss’ derivation by modern matrix notation:

21. Gauss’ derivation by modern matrix notation: Gauss’ results are equivalent to the following lemma:

23. Gauss-Markov theorem

24. Limitation of the Method of Least Squares • Nothing is perfect: • This method is very sensitive to the presence of unusual data points. One or two outliers can sometimes seriously skew the results of a least squares analysis.

25. References • Gauss, Carl Friedrich, Translated by G. W. Stewart. 1995. Theory of the Combination of Observations Least Subject to Errors: Part One, Part Two, Supplement. Philadelphia: Society for Industrial and Applied Mathematics. • Plackett, R. L. 1949. A Historical Note on the Method of Least Squares. Biometrika. 36:458–460. • Stephen M. Stiger, Gauss and the Invention of Least Squares. The Annals of Statistics, Vol.9, No.3(May,1981),465-474. • Plackett, Robin L. 1972. The Discovery of the Method of Least Squares. Plackett, Robin L. 1972. The Discovery of the Method of Least Squares. • Belinda B.Brand, Guass’ Method of Least Squares: A historically-based introduction. August 2003 • http://www.infoplease.com/ce6/people/A0820346.html • http://www.stetson.edu/~efriedma/periodictable/html/Ga.html