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Least Squares Approximation: A Linear Algebra Technique

Least Squares Approximation: A Linear Algebra Technique. By Larry Wong and James Sfregola. Soo……You have a bunch of Data. What can we do with this? Use Least Squares Approximation (LSE) to estimate the relationship of the dependent variable and the independent variable(s).

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Least Squares Approximation: A Linear Algebra Technique

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  1. Least Squares Approximation: A Linear Algebra Technique By Larry Wong and James Sfregola

  2. Soo……You have a bunch of Data. • What can we do with this? • Use Least Squares Approximation (LSE) to estimate the relationship of the dependent variable and the independent variable(s).

  3. What is Least Squares? • Approximates solutions for inconsistent over determined systems • What does inconsistent mean? • Over determined, what’s that? • In other words LSE allows you to describe a model as being represented by the line of best fit. • Where the “best fit” line, curve or polynomial is created from the approximations.

  4. Least Squares it’s Actually Pretty Useful • Least Squares Approximations are highly relevant in various fields • Really, how so? • Still don’t believe us? • Fine we’ll give examples…but not now.

  5. So How Does it Estimate a Solution? • Least Squares Approximation estimates a best fit solution for a system by • Minimizing the magnitude error vector, e. • Estimating the coefficients of the best fit equation

  6. How Do We Ensure Positive Error? • Least Squares • Eliminates negative terms • More susceptible to outlying data • Least Absolute Difference • Difficult to work with in linear algebra

  7. The Least Squares Theorem • For an over determined inconsistent system Ax=b: • This is easily solved using inverse and transpose multiplication • the resulting vector x can be determined in the form x=(AT A) -1 AT b • where x will correspond to the coefficients in your linear or polynomial expression.

  8. How to Approximate the Error Term? • e = b-Ax , where e is the error term • Where e1= the distance from our data point to the best fit approximation given by the orthogonal projection

  9. Application to Economics • Econometrics utilizes Ordinary Least Squares Approximations (OLS) • Similar to LSE except it follows 7 classical assumptions Estimated Regression Equation

  10. Bibliography • Poole, David Linear Algebra: A Modern Introduction. Canada: Thomson Brooks/Cole, 2006. • Studenmund, A.H. Using Econometrics: A Practical Guide. New York: Pearson Education, 2006.

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