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Least Squares Approximation: A Linear Algebra Technique By Larry Wong and James Sfregola Soo……You have a bunch of Data. Attendence Big 4 Derby League Position per capita income adjusted population 39,968 1 0 11 35000 1582564 21,852 1 0 10 36725 137471 24,409 1 0 7 36725 371847 26,770 1 0 19 43600 852013 41,917 1 1 2 43600 1351187 37,162 1 0 6 36725 623160 24,510 1 1 16 43600 724121 43,958 1 1 3 36725 738873 40,699 1 0 14 36725 630043 75,595 1 1 1 36725 1194415 What can we do with this? Use Least Squares Approximation (LSE) to estimate the relationship of the dependent variable and the independent variable(s). 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. 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. 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 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 1 2 3 4 `Petal L.` 5 6 7 0.5 1.0 1.5 `Petal W.` 2.0 2.5 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. 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 Application to Economics Econometrics utilizes Ordinary Least Squares Approximations (OLS) Similar to LSE except it follows 7 classical assumptions Estimated Regression Equation 80000 60000 40000 30000 20000 20000 0 10000 0 -10000 -20000 -30000 50 100 150 Residual 200 250 Actual 300 350 Fitted 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.