Gaussian distribution

Mixture Models in R

Victor Medina

Researcher at The University of Edinburgh

Mixture model to Gender dataset

Packages for fitting mixture models

mixtools
- The Poisson distribution is not implemented.
bayesmix
- Bayesian inference is outside the scope of the course.
EMCluster
- Only Gaussian distributions.
flexmix
- Has all the distributions we need and gives you the flexibility to perform more complex models.

Properties of Gaussian distribution

Mean

Standard deviation

Sample from a Gaussian distribution

To generate samples from a Gaussian distribution:

rnorm(n, mean, sd)

Example: Generate 100 values from a Gaussian distribution with a mean of 10 and a standard deviation of 5

> population_sample <- rnorm(n = 100, mean = 10, sd = 5)
> head(population_sample)

[1]  6.248874  9.564190 16.006521  9.139647 10.114969 16.423538

Estimation of the mean

Don't know the mean and the standard deviation, only know the observations
- Need to be estimated from the observations
To estimate the mean, we can calculate the sample mean

> mean_estimate <- mean(population_sample)

10.35759

To estimate the sd, we perform the following procedure

$$value_i\rightarrow (. -mean\_estimate)\rightarrow (.)^2\rightarrow mean (.)\rightarrow \sqrt{(.)}$$

> population_sample %>%  
+   subtract(mean_estimate) %>%
+   raise_to_power(2) %>% mean() %>% sqrt()

5.318641

Using the sd function

> standard_deviation_estimate <- sd(population_sample)
> standard_deviation_estimate

5.345435

Visualizing the estimated Gaussian distribution

# Transform the sample into a data frame
population_sample <- data.frame(x = population_sample)

# Plot the histogram
ggplot(data = population_sample) + 
   geom_histogram(aes(x = x, y = ..density..)) +
   stat_function(geom = "line", 
                 fun = dnorm, 
                 args = list(mean = mean_estimate, 
                             sd = standard_deviation_estimate))

Let's practice!

Mixture Models in R