A Markov Christmas chain

Let’s generate colorful Christmas chains that fulfill the Markov property!

I was discussing potential math-related Christmas decorations with Christoffer Andersen, when he suggested a Markov Christmas chain. I thought that could be a fun little visualization project, so here it is!

For this project, I will use R, and the packages ggplot2 for plotting and ggforce to get geom_ellipse for the chains themselves.

library(ggplot2)
library(ggforce)

First we need to generate the actual Markov chain. We decide on some colors for our chain:

colors <- c("darkred", "#1A3E10", "olivedrab", "#5F3B71")

and then we also need to specify the length of the chain, let’s do 20 chains for now.

chain_length <- 20

Then we set up the transition matrix for the Markov chain. In this case, I don’t want any colors repeating directly after each other, so I set all the diagonal entries to zero.

n_colors <- length(colors)

transition_probs <- matrix(c(0.0, 1/3, 1/3, 1/3,
                             1/3, 0.0, 1/3, 1/3,
                             1/3, 1/3, 0.0, 1/3, 
                             1/3, 1/3, 1/3, 0.0), 
                           nrow = n_colors, byrow = TRUE, 
                           dimnames = list(colors, colors))

To visualize the transition probabilities, we can use the package markovchain:

library(markovchain)
plot_transitions <- new("markovchain", transitionMatrix = transition_probs)
plot(plot_transitions)

For the starting color, I just choose a random color with equal probability.

starting_index <- sample(size = 1, 1:n_colors, prob = rep(1/n_colors, n_colors))

Now we set up a loop to generate a realization from this Markov chain. For each iterations, the color from the previous iteration decides the probabilities for the next color, according to the transition matrix above.

index_list <- rep(NA, chain_length)
index_list[1] <- starting_index

for(chain in 2:chain_length){
  current_chain <- sample(size = 1, 1:n_colors, 
                          prob = transition_probs[index_list[chain-1], ])
  index_list[chain] <- current_chain
}

chain_data <- data.frame(chain = 1:chain_length, 
                         color_id = as.factor(index_list))

Once we have our chain realization, we can plot it!

ggplot(chain_data) +
  geom_ellipse(aes(x0 = chain, y0 = 0, a = 0.7, b = 0.5, angle = 0, 
                   color = color_id), 
               linewidth = 1.5, fill = NA) +
  scale_color_manual(values = colors) +
  coord_fixed(ylim = c(1, -1)) +
  theme_void() +
  theme(legend.position = "none") 

And there you have it!