course Mth 272 August 13, 2009 at 9:26 AM Question: `qQuery problem 7.7.4 points (1,0), (2,0), (3,0), (3,1), (4,1), (4,2), (5,2), (6,2)
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Given Solution: `a The text gives you equations related to the sum of the x terms, sum of y values, sum of x^2, sum of y^2 etc, into which you can plug the given information. To use partial derivatives and get the same results. The strategy is to assume that the equation is y = a x + b and write an expression for the sum of the squared errors, then minimize this expression with respect to a and b, which are treated as variables. If y = a x + b then the errors at the four points are respectively | (a * 1 + b) - 0 |, | (a * 2 + b) - 0 |, | (a * 3 + b) - 0 |, | (a * 3 + b) - 1 |, | (a * 4 + b) - 1 |, | (a * 4 + b) - 2 |, | (a * 5 + b) - 2 |, and | (a * 6 + b) - 2 |. The sum of the squared errors is therefore sum of squared errors: ( (a * 1 + b) - 0 )^2+( (a * 2 + b) - 0 )^2+( (a * 3 + b) - 0 )^2+( (a * 3 + b) - 1 )^2+( (a * 4 + b) - 1 )^2+( (a * 4 + b) - 2 )^2+( (a * 5 + b) - 2 )^2+( (a * 6 + b) - 2 )^2. It is straightforward if a little tedious to simplify this expression, but after simplifying all terms, squaring and then collecting like terms we get 116Aa^2 + 2AaA(28Ab - 37) + 8Ab^2 - 16Ab + 14. We minimize this expression by finding the derivatives with respect to a and b: The derivatives of this expression with respect to a and b are respectively 56Aa + 16Ab - 16 and 232Aa + 56Ab - 74. Setting both derivatives equal to zero we get the system 56Aa + 16Ab - 16 = 0 232Aa + 56Ab - 74 = 0. Solving this system for a and b we get a = 1/2, b = - 3/4. So see that this is a minimum we have to evaluate the expression f_aa * f_bb - 4 f_ab^2. f_aa = 56 and f_bb = 56, while f_ab = 0 so f_aa * f_bb - 4 f_ab^2 is positive, telling us we have a minimum. Thus our equation is y = a x + b or y = 1/2 x - 3/4. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `qQuery problem 7.7.6 (was 7.7.16) use partial derivatives,etc., to find least-squares line for (-3,0), (-1,1), (1,1), (3,2) YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I did the same thing for this problem (use calculator) y=ax+b a=.3 B=1 Y=.3x+1 confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a If y = a x + b then the errors at the four points are respectively | (a * -3 + b) - 0 |, | (a * -1 + b) - 1 |, | (a * 1 + b) - 1 | and | (a * 3 + b) - 2 |. The sum of the squared errors is therefore ( (a * -3 + b) - 0 )^2 + ( (a * -1 + b) - 1 )^2 + ( (a * 1 + b) - 1 )^2 + ( (a * 3 + b) - 2 )^2 = [ 9 a^2 - 6 ab + b^2 ] + [ (a^2 - 2 a b + b^2) - 2 ( -a + b) + 1 ] + [ a^2 + 2 ab + b^2 - 2 ( a + b) + 1 ] + [ 9 a^2 + 6 ab + b^2 - 4 ( 3a + b) + 4 ] = 20Aa^2 - 12Aa + 4Ab^2 - 8Ab + 6. This expression is to be minimized with respect to variables a and b. The derivative with respect to a is 40 a - 12 and the derivative with respect to b is 8 b - 8. 40 a - 12 = 0 if a = 12/40 = .3. 8b - 8 = 0 if b = 1. The second derivatives with respect to and and b are both positive; the derivative with respect to a then b is zero. So the test for max, min or saddle point yields a max or min, and since both derivatives are positive the critical point gives a min. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: What was your expression for the sum of the squared errors? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I do not know how to do this problem. I have been using the calculator because the book doesn’t explain this section very well. Any information would be greatly appreciated. Thanks confidence rating: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a Right, for the values of a and b you correctly obtained above. The expression for squared errors is 20Aa^2 - 12Aa + 4Ab^2 - 8Ab + 6. For a = .3 and b = 1 this expression gives 1.8 - 3.6 + 4 - 8 + 6 = .2. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: "