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Math 4030 – 11b Method of Least Squares
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Model: Dependent (response) Variable Independent (control) Variable Random Error Objectives: Find (estimated) values of the coefficients and (based on the sample data) Evaluate the model efficiency; Predict or estimate the Y values for “un-tested” x values. XY x1x1 y1y1 x2x2 y2y2 …… xnxn ynyn Raw data:
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Regression Line (Best Fitting Line) Estimated Y value for given x Estimated coefficients for and Error Finding the equation of the best fitting line is to find the coefficients a and b, the estimated values of and . How?
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The Method of Least Squares Method of Least Squares finds the line (or a and b) such that sum of squared errors is minimized (among all choices of a and b.
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Calculation: Solve a system of 2 linear equations.
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Solution by Cramer’s Rule:
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Square-Sum Notations:
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Solutions: Estimate : Estimate : Equation for the Regression Line: Residual sum of squares (or error sum of squares): Note: exchange x and y will end in different regression line.
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Curvilinear Regression (Sec. 11.3): Exponential: Reciprocal: Power:
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Objectives: Find (estimated) values of the coefficients i ‘s (based on the sample data) Predict or estimate the Y values for “un-tested” x values. XY x1x1 y1y1 x2x2 y2y2 …… xnxn ynyn Raw data: Polynomial Regression:
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Calculation: Solve a system of p + 1 linear equations to solve p + 1 unknowns. If all x i values are distinct, the system has the unique solution.
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