This document discusses curve fitting of exponential curves. It defines curve fitting as constructing a mathematical function that best fits a series of data points, subject to constraints. There are two general approaches: least squares regression for scattered data to find a general trend, and interpolation for precise data to pass through each point. There are three cases of curve fitting: linear, quadratic, and exponential/logarithmic curves. For exponential curves with the relationship y=abx, the document describes taking the logarithm to transform it into a linear equation that can be solved for the coefficients a and b to determine the best fitting exponential curve.

1. Curve Fitting of
Exponential Curve
Divyang R. Rathod

2. Definition
• Curve fitting is the process of constructing a curve, or mathematical
function, that has the best fit to a series of data points, possibly subject to
constraints.
• It is a statistical technique use to drive coefficient values for equations that
express the value of one(dependent) variable as a function of another
(independent variable)

3. Curve fitting
• There are two general approaches for curve fitting:
1. Least squares regression:
Data exhibit a significant degree of scatter. The strategy is to derive a
single curve that represents the general trend of the data.
2. Interpolation:
Data is very precise. The strategy is to pass a curve or a series of curve
through each of the points.

4. Curve fitting

5. Curve fitting
• There are 3 cases of curve fitting
1. Fitting of Linear Curve
2. Fitting of Quadratic Curve
3. Fitting of Exponential and logarithmic Curves

6. Fitting of Exponential and logarithmic Curves
• Let (xi, yi), I = 1, 2, …, n be the set of n values and let the relation
between x and y be y = ab.
• Taking logarithm on both the sides of the sides of the equation
𝑦 = 𝑎𝑏 𝑥
,
• Putting 𝑙𝑜𝑔 𝑒 = 𝑌, 𝑙𝑜𝑔 𝑒 𝑎 = 𝐴, x = X, and 𝑙𝑜𝑔 𝑒 𝑏 = 𝐵,
Y = A + BX

7. • This is a linear equation in X and Y. The normal equations are,
𝑌 = 𝑛𝐴 + 𝐵 𝑋
𝑋𝑌 = 𝐴 𝑋 + 𝐵 𝑋2
• Solving these equations, A and B, and, hence, a and b can be found. The best
fitting exponential curve is obtained by substituting the values of a and b in
the equation 𝑦 = 𝑎𝑏 𝑥.
• Similarly, the best fitting exponential curves for the relation 𝑦 = 𝑎𝑥 𝑏 and
𝑦 = 𝑎𝑒 𝑏𝑥 can be obtained.

8. Thank You…