Curve Fitting of Type y=ax^b Algorithm
In this article we are going to develop an algorithm for fitting curve of type y = axb using least square regression method.
Procedure for fitting y = axb
y = axb ----- (1)
Taking log on both side of equation (1), we get
log(y) = log(axb) log(y) = log(a) + log(xb) log(y) = log(a) + b*log(x) ----- (2) Now let Y = log(y), A = log(a) and X = log(x) then equation (2) becomes, Y = A + bX ----- (3), Now we fit equation (3) using least square regression as: 1. Form normal equations: ∑Y = nA + b ∑X ∑XY = A∑X + b∑X2 2. Solve normal equations as simulataneous equations for A and b 3. We calculate a from A using: a = exp(A) 4. Substitute the value of a and b in y= axb to find line of best fit.
Algorithm for fitting y = axb
1. Start 2. Read Number of Data (n) 3. For i=1 to n: Read Xi and Yi Next i 4. Initialize: sumX = 0 sumX2 = 0 sumY = 0 sumXY = 0 5. Calculate Required Sum For i=1 to n: sumX = sumX + log(Xi) sumX2 = sumX2 + log(Xi) * log(Xi) sumY = sumY + log(Yi) sumXY = sumXY + log(Xi) * log(Yi) Next i 6. Calculate Required Constant A and b of Y = A + bX: b = (n * sumXY - sumX * sumY)/(n*sumX2 - sumX * sumX) A = (sumY - b*sumX)/n 7. Transformation of A to a: a = exp(A) 8. Display value of a and b 8. Stop