3D Multiple Regression Graph with rgl package in R

In this article, we are going to walk through the process of creating a 3D multiple regression graph using the rgl package in R programming language in detail.

What is Multiple Regression?

Regression analysis is a powerful statistical tool for understanding the relationship between variables. One of the well-known regression techniques is Linear Regression, which deals with predicting a dependent variable using only one independent variable.

Multiple Linear Regression also known as Multiple Regression, is the prediction of the dependent variable using more than one independent variable. In simple words, it is a statistical technique used to analyze the relationship between a single dependent variable and multiple independent variables.

The equation of multiple regression is:

Y = b0 + b1 * x1 + b2 * x2 + b3 * x3 + …… bn * xn

Where,

Y -> dependent variable

x1, x2, x3, …… xn -> multiple independent variables

n -> number of features

b0 -> y- intercept

bn -> regression coefficient of nth feature

RGL Package

The rgl package in R is a great tool for creating interactive 3D visualizations, which makes it ideal for representing complex relationships in multiple regression models. It enables users to explore data interactively, visualize regression planes, and customize plots, thereby enhancing the understanding of the relationships between multiple dependent and independent variables.

Steps in Creating 3D Plot

Let us see step by step, how we can create a 3D multiple regression graph in R using the rgl package.

Install and load the rgl package

First, we install and then load the RGL package in our R script.

R

install.packages("rgl")
library(rgl)

The rgl Package in R provides us with various functionalities to customize our 3D plots.

Preparing Data

We’ll start by generating some sample data for demonstration purposes. Here we’ll create two independent variables, x1 and x2, and one dependent variable, y.

Replace this sample data with your own dataset when working on your specific analysis.

R

# Set a random seed for reproducibility
set.seed(123)
# Generate sample data using rnorm() function
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n)
y <- 2 * x1 + 3 * x2 + rnorm(n)

rnorm is the R function that simulates random variates having a specified normal distribution.

Fitting the Multiple Linear Regression Model

Next, we fit a multiple linear regression model to our data using the ‘lm’ function. This model will help us understand how x1 and x2 collectively influence the dependent variable y.

lm() function which stands for linear model. This function can be used to create a simple regression model. The same function can be used to generate linear regression model, with change of parameters.

R

# Fit a multiple linear regression model
lm_model <- lm(y ~ x1 + x2)

The model is specified using the formula y ~ x1 + x2, where the dependent variable y is being estimated using two independent variables, x1 and x2, in a multiple linear regression analysis.

Creating the 3D Plot

Now, let’s create an interactive 3D plot using the rgl package. We’ll start by initializing an empty 3D plot, adding data points, and then visualizing the regression plane.

#Create an empty 3D plot

open3d()

#3d plot using the lm_model

plot3d(lm_model)

# Add a title

title3d(“Multiple Linear Regression 3D Plot”)

Examples of Creating 3D Multiple Regression Graph

Now let us see an example to understand the concept. The code follows the same procedure as given in the above steps.

A 3D Plot for Multiple Linear Regression without Squared Variable

R

#loading the rgl package into R-studio
library(rgl)

#creates a 3d plot in new window
open3d()

#multiple linear regression model- without squared variable
lm_model= lm(mpg ~ wt + qsec, data = mtcars)

#lm model plotted on the 3d plot we created
plot3d(lm_model, plane.col='red')

#add a title
title3d("Multiple Linear Regression 3D Plot")

# add animation to the plot for visual clarity
play3d(spin3d(axis = c(0, 0, 1)), duration = 30)

Output:

3D Multiple Regression Graph with rgl package in R

The outputs for this code will appear in a new small window because of the function ‘open_3d’ and not the console of R studio.

It looks like this:

Output in R- studio

• Firstly, it installs and loads the ‘rgl‘ package, which enables 3D visualization capabilities. Next, it opens a new 3D plotting window because of ‘open3d‘ for visualization. The code then proceeds to create a multiple linear regression model (`lm_model`) using the ‘mtcars‘ dataset, with ‘mpg‘ (miles per gallon) as the response variable and ‘wt‘ (weight) and ‘qsec‘ (quarter mile time) as predictor variables.
• This model does not include a squared term, meaning it assumes a linear relationship between the predictors and the response. This implies, it may not effectively capture nonlinear patterns in the data. The code then generates a 3D plot of the regression model, representing the regression plane in red because of ‘plot3d‘ function, and labels it as “Multiple Linear Regression 3D Plot” using the ‘title3d’ function. Lastly, it adds an animation using the ‘play3d‘ function that rotates the 3D plot around the z-axis for 30 seconds, allowing for a dynamic exploration of the plot from various angles. You may even change the duration of the animation to occur.
• We can observe from the output that the plot shows the regression plot, which means the plane there is going to predict the mpg of the car using the input wt and qsec when provided.

A 3D Plot for Multiple Linear Regression with Squared Variable

R

#loading the package rgl
library(rgl)

# initialize new 3d plot window
open3d()

#multiple linear regression model- with squared variable
lm_model= lm(mpg ~ wt + I(wt^2) + qsec, data = mtcars)

#3d plot of model in the new window
plot3d(lm_model, plane.col='green')

#add title for the 3d plot
title3d("Multiple Linear Regression 3D Plot")

# add animation to spin for 30 seconds
play3d(spin3d(axis = c(0, 0, 1)), duration = 30)

Output:

3D Multiple Regression Graph with rgl package in R

In the above code, we used the same dataset of mtcars. The output looks like this in the R studio:

Output in R studio

• We used ‘wt^2‘ as one of the independent variable parameters, which is the square of ‘wt‘ variable. This may be necessary many times. In some real-world data, the relationship between the dependent variable and an independent variable may not be linear. Squaring the variable allows us to capture nonlinear relationships. Hence, we squared it. We can see, in the graph that a non-linearity is produced in the graph, which touches most of the points in the plot, thereby a better prediction/regression model.

Visualizing Multiple Linear Regression in 3D with R

We are going to create our own dataset to use for the multiple regression by using vectors concept in R and create a graph. The data we are going to use has two independent variables – age and experience and the dependent variable is income. Here the procedure same as the above codes.

Loading Packages

R

#loading the package rgl
library(rgl)

Building Linear Regression Model

R

#data to be used in the plot
age <- c(25,30,47,32,43,51,28,33,37,39,29,47,54,51,44,41,58,23,44,37)
exp <- c(1,3,2,5,10,7,5,4,5,8,1,9,5,4,12,6,17,1,9,10)
income <- c(30450,35670,31580,40130,47830,41630,41340,37650,40250,45150,27840,46110,
            36720,34800,5130,38900,63600,30870,44190,48700)

#multiple linear regression model
lm_model= lm(income ~ age + exp)
print(lm_model)

Output:

Call:
lm(formula = income ~ age + exp)
Coefficients:
(Intercept)          age          exp  
   28982.46        68.95      1082.37  

Visualizing Regression Graph

R

#create new window for 3d plotting
open3d()

#plotting the model in the created window
plot3d(lm_model, plane.col='blue')

#title of the graph
title3d("Multiple Linear Regression 3D Plot")

# add animation for 30 sec duration
play3d(spin3d(axis = c(0, 0, 1)), duration = 30)

Output:

3D Multiple Regression Graph with rgl package in R

The output shown here helps us to predict the income when the experience and income is given. We can also observe that the dataset had an outlier, which may cause the model performance to decrease.

The output in R-studio looks like this:

Output in R studio

Conclusion

By following the steps outlined in this article, you can effectively generate interactive 3D plots that help you gain insights into your regression models.