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Exponential Regression
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What is Exponential Regression?
Exponential regression refers to the process of arriving at an equation for the exponential curve that best fits a set of data. Exponential 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.
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The Exponential Regression Equation
The exponential equation is y=abx. 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. adoes not equal to 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.
Calculating the Exponential Regression Equation
The following steps can be used to calculate the exponential regression once we have a set of data;
- 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 exponential regression 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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Determining the Exponential Regression of a Dataset using a Calculator
The following example shows how the exponential 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 do an exponential regression. When you plot the graph, it will look something like this:
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 labelled 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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FAQs on Exponential Regression
Exponential regression can be described as the process of arriving at an equation for the exponential curve that best fits a set of data. It is similar to linear regression, where we find the (straight) line of best fit.
Exponential regression is generally used in situations where we see slow growth in the beginning and then a quick acceleration of growth, or in situations where we see a rapid decay initially and then deceleration of decay.
The exponential equation is y=abx, where a0and b>0, b1.
In order to determine the exponential regression using a graphing calculator, you must first input the x-coordinates and y-coordinates into your calculator and do an exponential regression. When you plot your graph, it will show you your exponential curve alongside the equation of the function that best approximates your dataset.
