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Exponential Regression Exponential Regression

How to Calculate Exponential Regression?

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Exponential regression refers to the process of arriving at an equation for the exponential curve that best fits a set of data. It is very similar to linear regression, where we try to arrive at an equation for the (straight) line that best fits a set of data.

Often, we come across situations where a set of data doesn’t follow a straight line or a parabola. In such situations, the exponential curve is generally the best fit. This includes situations where there is slow growth initially and then a quick acceleration of growth, or in situations where there is rapid decay initially and then a sudden deceleration of decay.

In this article, we will learn about exponential regression and the formula used to explain the decaying or growth relationship. Let’s begin.

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What is Exponential Regression?

It is a model that explains processes that experiences growth at a double rate. It is used for situations where the growth begins slowly but rapidly speeds up without bounds or where the decay starts rapidly but slows down to get to zero. 

The equation used to express Exponential Regression is 

y = ab^x  

It refers to the process of arriving at an equation for the exponential curve that best fits a set of data. This regression is very similar to linear regression, where we try to arrive at an equation for the (straight) line that best fits a set of data.

Often, we come across situations where a set of data doesn’t follow a straight line or a parabola. In such situations, the exponential curve is generally the best fit. This includes situations where there is slow growth initially and then a quick acceleration of growth, or in situations where there is rapid decay initially and then a sudden deceleration of decay.

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 is the slope and c is the intercept of the linear regression model best fitted to the data (x, ln(y)).

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How to calculate exponential regression with a GDC?

The following example shows how the regression of a data set can be found using a graphing calculator (GDC). 

Let’s consider the following set of data:

x

y

0

4

1

9

2

12

3

25

4

53

5

98

In order to determine the exponential regression for the set, you must first enter the x-coordinates and y-coordinates of the set into your graphic calculator and calculate the regression. When you plot the graph, it will look something like this:

Exponential Regression Exponential Regression

The equation of the function that best fits the dataset should appear on your screen next to the line, which in this case is y=4.0511.877x.

As a part of the results, your calculator will display a number labeled by the variable R or R2. 

This variable denotes the correlation coefficient; the relative predictive power of your exponential model. Its value can lie between 0 and 1; the closer the value is to 1, the more accurate the model is, and the closer the value is to 0, the more inaccurate the model is.

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Wrapping up;

In the real world, financial investments, natural resources, or population are such factors where the phenomena of doubling time can be seen. In those cases, exponential regression is the appropriate model to explain the gradual growth or decay of the research subject.

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