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Jan. 31, 2023•0 likes•2 views

Jan. 31, 2023•0 likes•2 views

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Prove that if ten points (xi, yi) all lie on a parabola (y=a2(x^2)+a1x+a0), then the least-squares solution to find the best-fit parabola is in fact this parabola. (hint: show what the normal equations might look like in general, and show that (a0, a1, a2) is guaranteed to be a solution.) Solution y =a0+a1x+a2(x^2) The normal equations are sum(y) =n*a0+a1*sum(x)+a2*sum(x^2) (1) sum(y*x) = a0*sum(x)+a1*sum(x^2)+a2*sum(x^3) (2) sum(y*x^2)= a0*sum(x^2)+a1*sum(x^3)+a2*sum(x^4) (3) These three equations are the normal equations. Here the only unknown quantities are a0, a1 and a2. All other quantities, can be find out from the data given ( xi, yi) , i=1,2, ... , n. (Note that the mathe matical simbol used for \'sum\' is \'sigma\'). Since, there are 3 equations in three unknown quantities (a0, a1 and a2), these simultenous equations can be solved using any standard mathe mathematical technique to obtain (a0, a1, a2). So the solution is guaranteed .

- 1. Prove that if ten points (xi, yi) all lie on a parabola (y=a2(x^2)+a1x+a0), then the least-squares solution to find the best-fit parabola is in fact this parabola. (hint: show what the normal equations might look like in general, and show that (a0, a1, a2) is guaranteed to be a solution.) Solution y =a0+a1x+a2(x^2) The normal equations are sum(y) =n*a0+a1*sum(x)+a2*sum(x^2) (1) sum(y*x) = a0*sum(x)+a1*sum(x^2)+a2*sum(x^3) (2) sum(y*x^2)= a0*sum(x^2)+a1*sum(x^3)+a2*sum(x^4) (3) These three equations are the normal equations. Here the only unknown quantities are a0, a1 and a2. All other quantities, can be find out from the data given ( xi, yi) , i=1,2, ... , n. (Note that the mathe matical simbol used for 'sum' is 'sigma'). Since, there are 3 equations in three unknown quantities (a0, a1 and a2), these simultenous equations can be solved using any standard mathe mathematical technique to obtain (a0, a1, a2). So the solution is guaranteed