Exponential Regression in Python (Step-by-Step)

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 between x and y

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

Step 1: Create the Data

First, let’s create some fake data for two variables: x and y:

import numpy as np

x = np.arange(1, 21, 1)
y = np.array([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, let’s create a quick scatterplot to visualize the relationship between x and y:

import matplotlib.pyplot as plt

plt.scatter(x, y)
plt.show()

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 as opposed to a linear regression model.

Step 3: Fit the Exponential Regression Model

Next, we’ll use the polyfit() 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
fit = np.polyfit(x, np.log(y), 1)

#view the output of the model
print(fit)

[0.2041002  0.98165772]

Based on the output, the fitted exponential regression equation can be written as:

ln(y) = 0.9817 + 0.2041(x)

Applying 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, y, based on the value of the predictor variable, x. For example, if x = 12, then we would predict that y would be 30.897:

y = 2.6689 * 1.2264¹² = 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 Python
How to Perform Polynomial Regression in Python
How to Perform Quantile Regression in Python