The following external R packages are used:

library(dplyr)
library(gganimate) # 'gifski' and 'png' package have to be installed
library(tidyr)
include_graphics <- knitr::include_graphics
 

1. Introduction

This is to illustrate the “jumping frog” Markov chain 1 using gganimate package.

2. A jumping frog

Say there is a frog jumping across 20 lily pads numbered from 1 to 20 which are arranged clockwise in a circle. Suppose \(X_n\) is the pad number at the \(n\)th jump. Let’s define the “jumping frog” Markov chain \(\{ X_n \}_{n = 0}^{\infty}\) as a Markov chain whose state space is \(S = \{ 1, 2, \dots, 20\}\), with the initial state \(X_0 = 20\) and initial probabilities \(v_{20} = P(X_0 = 20) = 1\) and \(v_s = 0 \text{ }\) for all \(s = 1, \dots, 19\), and trainsition probabilities \(p_{ij} = P(X_n = j \text{ | } X_{n - 1} = i)= \frac{1}{3}\) whenever \(j = i - 1, i\), or \(i + 1\).

Long story short, a Java Applet regarding this frogwalk written by Prof. Rosenthal at the University of Toronto doesn’t work anymore. This is the effort to simulate this Markov chain using R and gganimate package.

With initial parameters below, we can simulate this Markov chain and populate X with some generated \(X_n\)’s:

state_df will be used to create a tibble that indicates the pad number the frog is located at each iteration. state_space will be used as x-axis, pads_point as y-axis (which is just a dummy), and iterations as a facetting variable:

comparison_df is a tibble that acutally indicates where the frog is at at each iteration. state_space and iterations will be variables that state_df and comparison_df have in common, and value will be an indicator:

We shall now left_join state_df and comparison_df using state_space and iterations columns, and create the whole states_df that is ready for generating a .gif file:

In our gganimate usage, we are basically creating a fast slide show of all the facets. iterations is our facetting variable, which is used in transition_time. Whenever the frog is located at the pad, the image will denote such a state with a red dot, and the rest of the pads will be coloured as black.

Here’s the .gif file:

Here’s a slower version (it takes a bit of time to load… 6 seconds?):

If your sole purpose is to display, then saving the image file and using include_graphics() is unnecessary. For that purpose, you can just use animate().

Session info

R session info:

## R version 3.6.1 (2019-07-05)
## Platform: x86_64-w64-mingw32/x64 (64-bit)
## Running under: Windows 10 x64 (build 18362)
## 
## Matrix products: default
## 
## locale:
## [1] LC_COLLATE=English_Canada.1252 
## [2] LC_CTYPE=English_Canada.1252   
## [3] LC_MONETARY=English_Canada.1252
## [4] LC_NUMERIC=C                   
## [5] LC_TIME=English_Canada.1252    
## 
## attached base packages:
## [1] stats     graphics 
## [3] grDevices utils    
## [5] datasets  methods  
## [7] base     
## 
## other attached packages:
##  [1] tidyr_1.0.0       
##  [2] gganimate_1.0.4   
##  [3] ggConvexHull_0.1.0
##  [4] dplyr_0.8.3       
##  [5] reticulate_1.13   
##  [6] pROC_1.15.3       
##  [7] ggrepel_0.8.1     
##  [8] ggplot2_3.2.1     
##  [9] funpark_0.2.6     
## [10] data.table_1.12.6 
## [11] boot_1.3-22       
## [12] rmarkdown_1.17    
## [13] magrittr_1.5      
## [14] itertools2_0.1.1  
## 
## loaded via a namespace (and not attached):
##  [1] progress_1.2.2   
##  [2] prettydoc_0.3.1  
##  [3] tidyselect_0.2.5 
##  [4] xfun_0.11        
##  [5] purrr_0.3.3      
##  [6] lattice_0.20-38  
##  [7] colorspace_1.4-1 
##  [8] vctrs_0.2.0      
##  [9] htmltools_0.4.0  
## [10] yaml_2.2.0       
## [11] utf8_1.1.4       
## [12] rlang_0.4.2      
## [13] pillar_1.4.2     
## [14] glue_1.3.1       
## [15] withr_2.1.2      
## [16] tweenr_1.0.1     
## [17] lifecycle_0.1.0  
## [18] plyr_1.8.4       
## [19] stringr_1.4.0    
## [20] munsell_0.5.0    
## [21] gtable_0.3.0     
## [22] evaluate_0.14    
## [23] labeling_0.3     
## [24] knitr_1.26       
## [25] gifski_0.8.6     
## [26] fansi_0.4.0      
## [27] Rcpp_1.0.3       
## [28] readr_1.3.1      
## [29] scales_1.1.0     
## [30] backports_1.1.5  
## [31] jsonlite_1.6     
## [32] farver_2.0.1     
## [33] gridExtra_2.3    
## [34] png_0.1-7        
## [35] hms_0.5.2        
## [36] digest_0.6.23    
## [37] stringi_1.4.3    
## [38] grid_3.6.1       
## [39] cli_1.1.0        
## [40] tools_3.6.1      
## [41] lazyeval_0.2.2   
## [42] tibble_2.1.3     
## [43] crayon_1.3.4     
## [44] pkgconfig_2.0.3  
## [45] zeallot_0.1.0    
## [46] MASS_7.3-51.4    
## [47] ellipsis_0.3.0   
## [48] Matrix_1.2-17    
## [49] prettyunits_1.0.2
## [50] xml2_1.2.2       
## [51] assertthat_0.2.1 
## [52] rstudioapi_0.10  
## [53] iterators_1.0.12 
## [54] R6_2.4.1         
## [55] compiler_3.6.1

  1. Rosenthal, J. (2019, April 05). STA447/2006 (Stochastic Processes) Lecture Notes, Winter 2019. Retrieved May 21, 2019, from http://probability.ca/jeff/teaching/1819/sta447/notes.pdf↩