*76*

**Exponential regression** is a type of regression that can be used to model the following situations:

**1. Exponential growth:** Growth begins slowly and then accelerates rapidly without bound.

**2. Exponential decay:** Decay begins rapidly and then slows down to get closer and closer to zero.

The equation of an exponential regression model takes the following form:

y = ab^{x}

where:

**y:**The response variable**x:**The predictor variable**a, b:**The regression coefficients that describe the relationship betweenÂ*x*and*y*

The following step-by-step example shows how to perform exponential regression in R.

**Step 1: Create the Data**

First, letâ€™s create some fake data for two variables:Â *x* andÂ *y*:

x=1:20 y=c(1, 3, 5, 7, 9, 12, 15, 19, 23, 28, 33, 38, 44, 50, 56, 64, 73, 84, 97, 113)

**Step 2: Visualize the Data**

Next, letâ€™s create a quick scatterplot to visualize the relationship betweenÂ *x* andÂ *y*:

plot(x, y)

From the plot we can see that there exists a clear exponential growth pattern between the two variables.

Thus, it seems like a good idea to fit an exponential regression equation to describe the relationship between the variables.

**Step 3: Fit the Exponential Regression Model**

Next, weâ€™ll use the **lm()** function to fit an exponential regression model, using the natural log of *y* as the response variable and *x* as the predictor variable:

#fit the model model log(y)~ x)#view the output of the model summary(model) Call: lm(formula = log(y) ~ x) Residuals: Min 1Q Median 3Q Max -1.1858 -0.1768 0.1104 0.2720 0.3300 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 0.98166 0.17118 5.735 1.95e-05 *** x 0.20410 0.01429 14.283 2.92e-11 *** --- Signif. codes: 0 â€˜***â€™ 0.001 â€˜**â€™ 0.01 â€˜*â€™ 0.05 â€˜.â€™ 0.1 â€˜ â€™ 1 Residual standard error: 0.3685 on 18 degrees of freedom Multiple R-squared: 0.9189, Adjusted R-squared: 0.9144 F-statistic: 204 on 1 and 18 DF, p-value: 2.917e-11

The overall F-value of the model is 204 and the corresponding p-value is extremely small (2.917e-11), which indicates that the model as a whole is useful.

Using the coefficients from the output table, we can see that the fitted exponential regression equation is:

**ln(y) = 0.9817 + 0.2041(x)**

ApplyingÂ *e* to both sides, we can rewrite the equation as:

**y = 2.6689 * 1.2264 ^{x}**

We can use this equation to predict the response variable,Â *y*, based on the value of the predictor variable,Â *x*. For example, if *x* = 12, then we would predict that *y* would beÂ **30.897**:

y = 2.6689 * 1.2264^{12} = 30.897

**Bonus:** Feel free to use this online Exponential Regression Calculator to automatically compute the exponential regression equation for a given predictor and response variable.

**Additional Resources**

How to Perform Simple Linear Regression in R

How to Perform Multiple Linear Regression in R

How to Perform Quadratic Regression in R

How to Perform Polynomial Regression in R