Maximum Likelihood Estimation in R
Maximum Likelihood Estimation (MLE) is a key method in statistical modeling, used to estimate parameters by finding the best fit to the observed data. By looking closely at the data we have, MLE calculates the parameter values that make our observed results most likely based on our model. In this article, we explore how to use MLE with the R Programming Language.
What is Likelihood Estimation?
Likelihood Estimation is a statistical method used to estimate the parameters of a probability distribution or a statistical model based on observed data. Unlike traditional estimation methods that focus on finding the “best-fitting” parameters, likelihood estimation frames the problem in terms of the likelihood function.
What is Maximum Likelihood Estimation (MLE)?
The goal of likelihood estimation is to find the parameter values that maximize the likelihood function. These parameter values are called Maximum Likelihood Estimates (MLEs). Once the MLEs are obtained, they provide estimates of the parameters that best explain the observed data. These estimates can be used for inference, prediction, or further analysis depending on the context of the problem.
[Tex]\hat{\theta}_{\text{MLE}} = \arg \max_{\theta} \, L(\theta \,|\, x) [/Tex]
Where:
- ?^MLE: represents the Maximum Likelihood Estimator.
- ?(?∣?): is the likelihood function, which measures the likelihood of the observed data ? for different values of the parameter ?.
- arg max? : denotes the value of θ that maximizes the likelihood function.
Now we will Calculate Maximum Likelihood Estimation using optim function in R Programming Language.
# Load necessary libraries
library(ggplot2)
# Generate synthetic data from an exponential distribution
set.seed(123)
data <- rexp(100, rate = 0.1)
# Likelihood function for the exponential distribution
log_likelihood <- function(lambda) {
sum(dexp(data, rate = lambda, log = TRUE))
}
# Maximum Likelihood Estimation using optim
result <- optim(par = 0.1, log_likelihood, method = "Brent", lower = 0.001, upper = 10)
mle_lambda <- result$par
# Print the MLE estimate
cat("Maximum Likelihood Estimate for Lambda:", mle_lambda, "\n")
Output:
Maximum Likelihood Estimate for Lambda: 10
First, we generate synthetic data representing the time taken for a task to complete from an exponential distribution with a known rate parameter of 0.1.
- We define a likelihood function that computes the log likelihood of the data given a lambda value.
- We use the
optim()function to find the value of lambda that maximizes the log likelihood. - We print the Maximum Likelihood Estimate (MLE) for the parameter lambda.
Visualize the Maximum Likelihood Estimate (MLE) on a plot
Now we can plot the histogram of the data along with the estimated data points for Maximum Likelihood Estimate.
# Plot the histogram of the data with the estimated lambda
ggplot(data = NULL, aes(x = data)) +
geom_histogram(bins = 15, fill = "skyblue", color = "black", alpha = 0.7) +
geom_vline(xintercept = mle_lambda, color = "red", linetype = "dashed") +
annotate("text", x = mle_lambda + 0.1, y = 10,
label = paste("MLE Lambda:", round(mle_lambda, 2)), color = "red") +
labs(title = "Histogram with Estimated Lambda", x = "Time", y = "Frequency")
Output:
Maximum Likelihood Estimation in R
We generate the histogram of the synthetic data.
- We add a vertical dashed line representing the Maximum Likelihood Estimate (MLE) of the parameter lambda.
- We annotate the plot with the MLE estimate value.
- This visualization allows us to see the MLE estimate visually on the histogram plot of the data.
Conclusion
Maximum Likelihood Estimation (MLE) is a vital tool for statistical modeling, especially in parameter estimation from observed data. In our exploration, we focused on Likelihood Estimation’s essence, implementing it practically using R for linear regression with earthquake data. We covered data preparation, likelihood function definition, optimization, and result interpretation. Assessing model fit involved residual analysis, R-squared, hypothesis testing, and visual assessment. We also discussed handling constraints and bounds for parameter validity. Advanced techniques like regularization, feature engineering, and ensemble methods were highlighted, showcasing MLE’s pivotal role in model building and insight derivation.
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