Gauss and the Method of Least Squares

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Gauss and the Method of Least  Squares

 

A presentation by …..

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   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

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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.

 

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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

 

 

 

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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.

 

 

 

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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

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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.

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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

 

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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

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Density Function

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Mean and Variance

 

• Define                   . In many cases assume k=0
• Define mean square error as

 

 

 

• If k=0 then the variance will equal 

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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.

 

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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  

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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.

 

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Gauss’ Derivation of the Method of Least Squares

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Gauss’ Derivation of the Method of Least Squares

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Gauss’ Derivation of the Method of Least Squares

 

• Solutions:

 

 

 

 

• It’s still not obvious:

     How do these results relate with the least squares estimation?

 

 

 

 

 

 

 

 

 

 

 

 

 

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Gauss’ Derivation of the Method of Least Squares

 

• It can be proved that

 

 

 

 

 

 

 

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 Gauss’ derivation by modern matrix notation:

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Gauss’ derivation by modern matrix notation:

 

Gauss’ results are equivalent to the  following lemma: 

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Gauss-Markov theorem

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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.

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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