Curve Fitting of Type y=ab^x Algorithm

In this article we are going to develop an algorithm for fitting curve of type y = ab^x using least square regression method.

Procedure for fitting y = ab^x

We have,

	
y = abx ----- (1)

Taking log on both side of equation (1), we get

	
log(y) = log(abx)

log(y) = log(a) + log(bx)

log(y) = log(a) + x*log(b) ----- (2)

Now let Y = log(y), A = log(a) and B = log(b) 

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 afrom A and bfrom B as:
   a = exp(A)
   b = exp(B)

4. Substitute the value of a and b in 
   y= abx to find line of best fit.

Algorithm for Fitting Curve y = ab^x

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 + Xi
     sumX2 = sumX2 + Xi * Xi
     sumY = sumY + log(Yi)
     sumXY = sumXY + 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 and B to b:
   a = exp(A)
   b = exp(B)
   
8. Display value of a and b

8. Stop

Recommended Readings

1. Algorithm for fitting Curve y = ax^b
2. Pseudocode for fitting y = ax^b
3. C Program for fitting curve y = ax^b
4. C++ Program for fitting curve y = ax^b
5. Algorithm for fitting Curve y = ab^x
6. Pseudocode for fitting y = ab^x
7. C Program for fitting curve y = ab^x
8. C++ Program for fitting curve y = ab^x