Plotting Density Functions with Different Lengths in R
In this article, we will explore how to create a plot that displays different density functions of continuous and discrete variables. We will cover the basics of density functions, how to generate them, and how to visualize them using ggplot2 and other R packages.
Introduction
Density functions are mathematical descriptions of the probability distribution of a variable. They provide valuable information about the shape and characteristics of the data. In this article, we will focus on creating plots that display continuous and discrete density functions. We will use R as our programming language and explore various methods for generating and visualizing these density functions.
Generating Density Functions
To generate density functions, we need to understand the probability distributions that describe them. There are several types of probability distributions, including:
- Continuous Distributions: These distributions describe variables that can take on any value within a certain range. Examples include the normal distribution and the exponential distribution.
- Discrete Distributions: These distributions describe variables that can only take on specific values. Examples include the binomial distribution and the Poisson distribution.
In this article, we will focus on generating continuous density functions using the dgamma() function in R, which is based on the gamma distribution. We will also use the dpois() function to generate discrete density functions, specifically the Poisson distribution.
Generating Gamma Density Function
The gamma distribution is a continuous probability distribution that is commonly used to model waiting times or other random events. The probability density function (pdf) of the gamma distribution is given by:
$$f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}$$
where $\alpha$ is the shape parameter and $\beta$ is the rate parameter.
To generate a gamma density function in R, we can use the dgamma() function:
# Load necessary libraries
library(plotfunctions)
# Set parameters for gamma distribution
alpha <- 2
beta <- 0.5
# Generate gamma density function
gamma_density <- dgamma(x = seq(0, 10, by = 0.1), shape = alpha, rate = beta)
Generating Poisson Density Function
The Poisson distribution is a discrete probability distribution that describes the number of events occurring in a fixed interval of time or space. The probability mass function (pmf) of the Poisson distribution is given by:
$$P(k; \lambda) = \frac{\lambda^k e^{-\lambda}}{k!}$$
where $k$ is the number of occurrences and $\lambda$ is the mean and variance of the distribution.
To generate a Poisson density function in R, we can use the dpois() function:
# Load necessary libraries
library(plotfunctions)
# Set parameters for Poisson distribution
lamda <- 2
# Generate Poisson density function
poisson_density <- dpois(x = seq(0, 10), lambda = lamda)
Plotting Density Functions
Once we have generated the density functions, we can plot them using ggplot2. We will create a single plot that displays all three density functions.
Creating a Single Plot
To create a single plot, we need to define our x-axis values and y-axis limits. In this case, we want to display values between 0 and 10.
# Load necessary libraries
library(ggplot2)
library(plotfunctions)
# Set parameters for gamma distribution
alpha <- 2
beta <- 0.5
# Set parameters for Poisson distribution
lamda <- 2
# Generate x-axis values
theta <- seq(0, 10, by = 0.1)
# Generate gamma density function
gamma_density <- dgamma(x = theta, shape = alpha, rate = beta)
# Generate Poisson density function
poisson_density <- dpois(x = seq(0, 10), lambda = lamda)
Plotting Density Functions
Now we can use ggplot2 to create our plot:
# Create a new data frame with x and density values
df <- data.frame(x = theta, gamma = gamma_density, poisson = poisson_density)
# Define the plot
p <- ggplot(df, aes(x = x, y = gamma)) +
geom_line() +
geom_point(aes(y = poisson), color = "red") +
labs(title = "Density Functions", x = "x", y = "density") +
theme_minimal()
Adding Likelihood Density
To add the likelihood density, we need to adjust our code slightly. The likelihood density is defined as:
$$L(x; \theta) = f(x; \theta)$$
where $f(x; \theta)$ is the probability density function of the random variable.
In this case, we have two likelihood densities: one for the gamma distribution and one for the Poisson distribution.
# Load necessary libraries
library(ggplot2)
library(plotfunctions)
# Set parameters for gamma distribution
alpha <- 2
beta <- 0.5
# Set parameters for Poisson distribution
lamda <- 2
# Generate x-axis values
theta <- seq(0, 10, by = 0.1)
# Generate gamma density function
gamma_density <- dgamma(x = theta, shape = alpha, rate = beta)
# Generate Poisson density function
poisson_density <- dpois(x = seq(0, 10), lambda = lamda)
# Add likelihood densities to plot
p <- ggplot(df, aes(x = x)) +
geom_line(aes(y = gamma)) +
geom_point(aes(y = poisson), color = "red") +
geom_line(aes(y = gamma_density), color = "black", linetype = "dashed") +
labs(title = "Density Functions", x = "x", y = "density") +
theme_minimal()
Final Plot
The final plot displays all three density functions: the priori, likelihood, and posteriori densities. The priori density is the gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$. The likelihood density is the Poisson distribution with mean $\lambda$. The posteriori density is the gamma distribution with shape parameter $\alpha + 1$ and rate parameter $n + \beta$, where $n$ is the number of observations.
# Final plot
p <- ggplot(df, aes(x = x)) +
geom_line(aes(y = gamma)) +
geom_point(aes(y = poisson), color = "red") +
geom_line(aes(y = gamma_density), color = "black", linetype = "dashed") +
labs(title = "Density Functions", x = "x", y = "density") +
theme_minimal()
print(p)
Last modified on 2024-03-26