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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/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  

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

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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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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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When to use exponential regression?

Exponential regression is a powerful tool to identify and understand growth and decay patterns. It is useful for situations where the dependent variable changes (increases or decreases) rapidly over time. Here are some instances when you can use it for your research. 

  1. When you want to analyze natural processes such as population growth or the spread of infectious diseases. 
  2. In the context of market research, it can help analyze the adoption of new products over time. You can use it to predict the rate by identifying the pattern in which the product gains popularity in the market. 
  3. In financial analysis, it can be a suitable tool to forecast the compounding growth of stock price or economic indicators for a time period. 
  4. It can also be useful to gauge customer response to marketing and advertising efforts and predict customer retention and churn rates. 

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What are the advantages and disadvantages of exponential regression?

A modern data analysis tool must be equipped with exponential regression due to its wide use and advantages. But before you decide to use them in your research process, its best you learn about their pros and cons. 

Advantages of exponential regression: 
1. This regression analysis model can help predict data that follows rapid growth or decay patterns. Thus it provides insights into trends that you may otherwise miss. 

2. For manual calculation, the formula is relatively straightforward, making it accessible for researchers with basic statistical knowledge. 

3. It finds application in a wider range of fields, making it relevant for diverse research. 

Disadvantages of exponential regression:
1. The regression analysis is sensitive to outliers. It can significantly impact the model’s accuracy. 

2. It only applies to a dataset that exhibits growth/decay patterns. 

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Example of real-world applications of exponential regression

Let’s look into some examples of how exponential regression can be useful in different research goals. 

1. Market research: To predict the market growth of smartphones. 

Companies can use this regression analysis type to forecast the growth of smartphone sales. They can analyze historical sales data and identify the growth pattern to estimate the expected increase in demand for smartphones over the next few years. 

2. Customer experience: Forecasting customer churn in subscription-based services. 

A company offering subscription-based service can utilize exponential regression to predict customer attrition rate. It can analyze historical customer behavior, such as usage patterns, subscription duration, customer service interaction, etc, to identify the possibility of churn.  

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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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