What is Exponential Regression?
Exponential Regression using a Linear Model
Basic Concepts
Sometimes linear regression can be used with relationships that are not inherently linear but can be made to be linear after a transformation. In particular, we consider the following exponential model:

Taking the natural log (see Exponentials and Logs) of both sides of the equation, we have the following equivalent equation:

This equation has the form of a linear regression model (where I have added an error term ε):

Observations
Since αeβ(x+1) = αeβx · eβ, we note that an increase in x of 1 unit results in y being multiplied by eβ.
A model of the form ln y = βx + δ is referred to as a log-level regression model. Clearly, any such model can be expressed as an exponential regression model of form y = αeβx by setting α = eδ.
X | Y | X-Xֿ | LnY | (X-Xֿ)*(X-Xֿ) | (LnY-LnYֿ) | (LnY-LnYֿ)*(LnY-LnYֿ) | (X-Xֿ)*(LnY-LnYֿ) |
0 | 2 | -2.25 | 0.693147 | 5.0625 | -2.26013 | 5.1082 | 5.0853 |
1 | 5 | -1.25 | 1.609438 | 1.5625 | -1.34384 | 1.805912 | 1.6798 |
3 | 45 | 0.75 | 3.806662 | 0.5625 | 0.853382 | 0.728262 | 0.64004 |
5 | 300 | 2.75 | 5.703782 | 7.5625 | 2.750502 | 7.565264 | 7.56388 |
2.25 | 88 | 0 | 2.953258 | 14.75 | -9E-05 | 15.20764 | 14.969 |
Square root of (14.75 ) 3.840572874 | Square root of (15.20764) 3.899697 |
b 1.014848822
r | 0.999462 |
r*r or Square of r | 0.998925 |
Exponential Regression Y=aebx
y = 1.9539e1.0148x
R² = 0.9989
X | Y | X-Xֿ | LnY | (X-Xֿ)*(X-Xֿ) | (LnY-LnYֿ) | (LnY-LnYֿ)*(LnY-LnYֿ) | (X-Xֿ)*(LnY-LnYֿ) |
2 | 4 | ||||||
3 | 9 | ||||||
4 | 17 | ||||||
5 | 30 | ||||||
y = 1.0407e0.6856x
R² = 0.9928
It is a model that explains processes that experiences growth/decay at a double rate. It is used for situations where the change begins slowly but rapidly speeds up without bounds.
The equation to express Exponential Regression is
y = ab^x
How to calculate exponential regression?
The exponential equation is y=ab^x. Let’s take a look at some of the characteristics of this equation:
‘b’ must be greater than 0 and not equal to one; b>0, b1
The model takes the initial value of ‘a’. The ‘a’ does not equal one; a0.
If b>1, the function models exponential growth.
If 0<b<1, the function models exponential decay.
Coefficients a and b have to be chosen so that the equation corresponds to the exponential curve that best fits the data set.
Steps to calculate the equation
You can follow these three steps to calculate your data to find the value of ‘y’.
Step 1: Transform the data so that it allows for the linear model.
Step 2: Use the method of least squares to determine the linear model.
Step 3: Transform the data and the model back to their original form.
For the data (x,y), the formula is:
y=exp(c)exp(mx)
In this equation,
m → slope
c → intercept of the linear regression model best fitted to the data (x, ln(y)).
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