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Unformatted text preview: ugh the sample points such that the sum of squared residuals (SSR) is as small as possible, hence the term least squares. 5 ˆˆ ˆ ˆ ˆˆ The ith residual, by definition, is ui = yi − yi = yi − α − β xi , so choose α , β to minimize SSR, i.e., ˆ ˆ ∑ ( u ) = ∑ ( y − α − βˆ x ) n i =1 2 i n i =1 i 2 i ˆˆ = f (α , β ) The last equality means that this SSR is a function of two unknown ˆ parameters α and β ˆ Minimizing SSR How do we do the minimization? First look at a quadratic function with one single parameter, the minimum (the turnaround point) is where the slop is zero. Figure: a quadratic function f (α ) = 3 − 2α + α 2 has a minimum at α = 1 (solution of ∂f (α ) = −2 + 2α = 0 ). ∂α n 1n f (α ) = ∑ ( yi − α )2 has a minimum at α = Y = ∑ yi Similarly, a quadratic function n i =1 i =1 In this case, the objective function to be minimized, SSR, is a quadratic function in two parameters α and β . Graph has the same shape as parabola, but in 3‐D. The minimum is when both the slope in α ‐direction direction ˆˆ ∂f (α , β ) and the slope in β ‐ ˆ ∂α ˆˆ ∂f (α , β ) , are zero. ˆ ∂β That is, −2 n ˆˆ ∂f (α , β ) ˆˆ = −...
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