Calculate the Volume Under a Plot of Kernel Bivariate Density Estimation
In this article, we will explore how to calculate the volume under a plot of kernel bivariate density estimation using numerical integration. We’ll start by understanding the basics of kernel density estimation and then dive into the details of calculating the volume under a 2D surface.
Introduction
Kernel density estimation (KDE) is a non-parametric method for estimating the probability density function (PDF) of a random variable. In the context of bivariate data, KDE can be used to estimate the joint PDF of two variables, x and y.
Given a set of points in 2D space, we can use KDE to estimate the underlying distribution by creating a grid of points within a certain range and then fitting a density function to the data. The resulting density function is then used to make predictions about new, unseen data points.
In this article, we will focus on calculating the volume under a plot of kernel bivariate density estimation. Specifically, we’ll explore how to calculate the joint entropy of two variables using numerical integration.
Background
To understand how to calculate the volume under a plot of kernel bivariate density estimation, we need to review some background concepts.
Kernel Density Estimation
Kernel density estimation is a non-parametric method for estimating the PDF of a random variable. The basic idea behind KDE is to create a weighted average of the observed data points, where the weights are determined by a kernel function.
Given a set of n data points $(x_1, y_1), (x_2, y_2), …, (x_n, y_n)$, we can use KDE to estimate the joint PDF $p(x, y)$ as follows:
$$p(x, y) = \frac{1}{n} \sum_{i=1}^n K\left(\frac{x-x_i}{h}\right)K\left(\frac{y-y_i}{h}\right),$$
where $K$ is a kernel function, $h$ is the bandwidth parameter, and $\frac{x-x_i}{h}$ and $\frac{y-y_i}{h}$ are the relative distances between the data points and the point $(x, y)$.
Bivariate Normal Distribution
The bivariate normal distribution is a common distribution used in statistics and machine learning. It has two parameters: the mean vector $\mu = (\mu_x, \mu_y)$ and the covariance matrix $\Sigma$.
Given a dataset of bivariate data points $(x_i, y_i)$, we can estimate the mean vector and covariance matrix using maximum likelihood estimation.
Joint Entropy
The joint entropy of two variables x and y is defined as:
$$H(x, y) = -\sum_{i=1}^n p(x_i, y_i) \log p(x_i, y_i).$$
For a bivariate normal distribution, the joint PDF can be written as:
$$p(x, y) = \frac{1}{2\pi \sigma_x \sigma_y \sqrt{1 - \rho^2}} \exp\left(-\frac{1}{2(1-\rho^2)}\left[\left(\frac{x-\mu_x}{\sigma_x}\right)^2 - 2\rho\left(\frac{x-\mu_x}{\sigma_x}\right)\left(\frac{y-\mu_y}{\sigma_y}\right) + \left(\frac{y-\mu_y}{\sigma_y}\right)^2\right]\right),$$
where $\rho$ is the correlation coefficient between x and y.
Numerical Integration
To calculate the volume under a plot of kernel bivariate density estimation, we need to use numerical integration. In this section, we’ll show how to perform numerical integration using the kde2d function in R.
Loading Libraries and Data
First, we load the necessary libraries and data:
library(MASS)
set.seed(123)
x <- rnorm(1000, mean = 5, sd = 1.5)
y <- rnorm(1000, mean = 3, sd = 1.2)
den <- kde2d(x, y, n = 100)
In this example, we create a dataset of bivariate normal data points $(x_i, y_i)$ and then use the kde2d function to estimate the joint PDF $p(x, y)$. The resulting density is stored in the den variable.
Normalizing Constant
The first step in calculating the volume under the plot of kernel bivariate density estimation is to compute the normalizing constant. This constant represents the average value of the density over the entire domain.
norm <- sum(den$z) * (diff(xlim)/100) * (diff(ylim)/100)
In this example, we calculate the normalizing constant by summing up the values of the density over the entire domain and then dividing by the area of the grid.
Numerical Integration
Now that we have the normalizing constant, we can proceed with numerical integration. The integrand function is used to compute the integrand:
integrand <- den$z * log(den$z) * (-1)
In this example, we define the integrand as the product of the density and its logarithm.
Calculating Volume
Finally, we can calculate the volume under the plot of kernel bivariate density estimation by summing up the values of the integrand over the entire domain:
volume <- sum(integrand) * (diff(xlim)/100) * (diff(ylim)/100)
In this example, we compute the volume by summing up the values of the integrand and then multiplying by the area of the grid.
Self-Normalization
To obtain a meaningful result, it’s often necessary to self-normalize the calculated volume. This involves dividing the volume by the normalizing constant:
volume <- volume / norm
In this example, we divide the calculated volume by the normalizing constant to ensure that the resulting value has a meaningful units.
Verification
To verify our calculations, we can compare the result with the theoretical value obtained using the entropy formula for a bivariate normal distribution.
k <- 2
mu <- c(5, 3)
Sigma <- matrix(c(4, 0, 0, 4), nrow = 2)
det_Sigma <- det(Sigma)
theta <- arctan((sum(diag(Sigma))/2) / (sqrt(prod(diag(Sigma)))))
ent <- (k/2) * (1 + log(2*pi)) + (1/2) * log(det_Sigma)
print(paste("Theoretical entropy:", ent))
In this example, we compute the theoretical value of the joint entropy using the formula for a bivariate normal distribution.
Conclusion
In this article, we’ve explored how to calculate the volume under a plot of kernel bivariate density estimation using numerical integration. We started by reviewing the basics of kernel density estimation and then dove into the details of calculating the volume under a 2D surface.
By following these steps, you should be able to implement numerical integration for kernel bivariate density estimation in your own projects. Remember to verify your results against theoretical values whenever possible to ensure accuracy.
Last modified on 2023-09-13