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Published byNoel Preston Modified over 9 years ago
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Linear Least-Squares Approximation Ellen and Jason
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Problem the Algorithm Solves Finds a function that most clearly passes through a set of points Algorithm is used for – Summarizing data – Predicting data
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Standard variable names & expressions p = X k q = Y k r = X k Y k s = X k ) 2 d = (m + 1) s - (p) 2 a = [(m + 1) r - pq] / d b =[sq – pr] / d y = ax + b
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Standard Terms & Definitions l 1 Approximation: ax k + b – y k l 2 Approximation: a, b ax k + b – y k 2
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Principle behind algorithm/How it was derived (y i – ax i - b) 2 This equation gives us the error.
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Principle behind algorithm/How it was derived Now we take the derivative with respect to m, and with respect to b. df/dm = 2(y i - mx i - b)(-x i ) = -2x i y i + 2mx i 2 + 2bx i df/db = 2(Y i - mx i - b)(-1) = -2y i + 2mx i + 2b
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Principle behind algorithm/How it was derived Now we want to set the equations to 0 because we are trying to find the minimum error. On the graph you can see that the minimum is at a point where the derivative would equal 0. -2y i + 2mx i + 2b = -2y i ) + m (2x i ) + b (2) -2x i y i + 2mx i 2 + 2bx i (-2x i y i ) + m (2x i 2 ) + b (2x i )
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Principle behind algorithm/How it was derived (-2x i y i ) + m (2x i 2 ) + b (2x i ) = 0 m (2x i 2 ) + b (2x i ) = (2x i y i ) m (x i 2 ) + b (x i ) = (x i y i ) -2y i ) + m (2x i ) + b (2) = 0 m (2x i ) + b (2) = (2y i ) m (x i ) + b = (y i )
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Principle behind algorithm/How it was derived Am + Bb = E Cm + Db = F To get rid of variables A, C and m we multiply the first equation by C and the second equation by -A C(Am +Bb = E) -A(Cm + Db = F) => ACm + BCb = EC -ACm - ADb = -AF ----------------------- (BC – AD)b = EC – AF b = (EC-AF)/(BC-AD)
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Principle behind algorithm/How it was derived Am + Bb = E Cm + Db = F Now, in order to get rid of b, we multiply the first equation by D and the second equation by (-B) D(Am +Bb = E) -B(Cm + Db = F) => ADm + BDb = ED -BCm - BDb = -BF ----------------------- (AD - BC)m = ED - BF m = (AD - BC)/(ED - BF)
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Example Find the linear least-squares solution for the table of values p = 1 + 3 + 4 + 9 = 17 q = 2 + 6 + 9 + 8 = 25 r = (1*2) + (3*6) + (4*9) + (9*8) = 128 s = (1 2 ) + (3 2 ) + (4 2 ) + (9 2 ) = 107 d =(3+1)*107 - (17 2 ) = 139 a = [(3+1)*128- (17*25)] / 139 = 87/139 b = [(107*25) - (17*128)] / 139 = 499/139 Y = ax+b Y = (87/139)x + (499/139)
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Example: Graph
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Example: Error Point y - (ax +b) Point 1: [2 - (0.6259*1 +3.5900)] 2 = 4.9102 Point 2: [6 - (0.6259*3 + 3.5900)] 2 = 0.2833 Point 3: [9 - (0.6259*4 + 3.5900)] 2 = 8.4472 Point 4: [8 - (0.6259*9 + 3.5900)] 2 = 1.4960 Error: 15.1367
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Advantages Least Squares RegressionLeast Absolute Deviation Regressions Not Very RobustRobust Stable SolutionUnstable Solution Always One SolutionPossibly Multiple Solutions
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Disadvantages Limitations in shapes – May not be effective for data that is nonlinear. For example, a linear function would not represent the sets of points in this graph very well.
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