--- title: "Fitting Markov Modulated Point Process Models" output: rmarkdown::html_vignette author: Owen G. Ward. vignette: > %\VignetteIndexEntry{Fitting Markov Modulated Point Process Models} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, include = FALSE} knitr::opts_chunk$set( collapse = TRUE, eval = FALSE, comment = "#>" ) set.seed(100) # to make it reproducible options(rmarkdown.html_vignette.check_title = FALSE) ``` ```{r setup} library(ppdiag) library(rstan) library(cmdstanr) library(tidyverse) library(bayesplot) ``` Functions for fitting standard Hawkes and Poisson point processes to data are included in `ppdiag`. However, currently, we do not include fitting algorithms for Markov modulated point processes, as these rely on the use of `rstan` or (more recently), `cmdstanr` to perform Bayesian inference for the model parameters. Here we provide instructions on how to fit these models so that they can easily be used in conjunction with the diagnostic tools of `ppdiag`. # Installing RStan or Cmdstanr - Instructions for installing RStan can be found at https://github.com/stan-dev/rstan/wiki/RStan-Getting-Started - Instructions for installing CmdStanR can be found at https://mc-stan.org/cmdstanr/articles/cmdstanr.html. As mentioned in this link there are some differences between RStan and CmdStanR. In our experience, CmdStanR does seem to be maybe more reliable and less prone to crashing. - If you are having trouble installing or using Stan there is lots of great help available at https://discourse.mc-stan.org/, where someone else has probably already answered your question! # The Stan code for MMPP/MMHP We first include the stan code to fit each of these Markov modulated models to a temporal point process. ```{r mmpp-stan-code} mmpp_stan_code <- " data{ int num_events; //maximum of number of events for each pair each window => max(unlist(lapply(return_df$event.times,length)))) vector[num_events+1] time_matrix; // include termination time in the last entry } parameters{ real lambda0; //baseline rate for each pair real c; //baseline rate for each pair real w1; //CTMC transition rate real w2; //CTMC transition rate } transformed parameters{ real q1; real q2; q1 = (lambda0).*w1; q2 = (lambda0).*w2; } model{ real integ; // Placeholder variable for calculating integrals row_vector[2] forward[num_events]; // Forward variables from forward-backward algorithm row_vector[2] forward_termination; row_vector[2] probs_1[num_events+1]; // Probability vector for transition to state 1 (active state) row_vector[2] probs_2[num_events+1]; // Probability vector for transition to state 2 (inactive state) vector[num_events+1] interevent; //priors c ~ lognormal(0,1); w1 ~ beta(0.5,0.5); w2 ~ beta(0.5,0.5); lambda0 ~ gamma(1, 1); interevent = time_matrix; // ---- prepare for forward algorithm // --- log probability of Markov transition logP_ij(t) for(n in 1:(num_events + 1)){ probs_1[n][1] = log(q2/(q1+q2)+q1/(q1+q2)*exp(-(q1+q2)*interevent[n])); //1->1 probs_2[n][2] = log(q1/(q1+q2)+q2/(q1+q2)*exp(-(q1+q2)*interevent[n])); //2->2 probs_1[n][2] = log1m_exp(probs_2[n][2]); //2->1 probs_2[n][1] = log1m_exp(probs_1[n][1]); //1->2 } //consider n = 1 integ = interevent[1]*lambda0; forward[1][1] = log_sum_exp(probs_1[1]) + log(lambda0*(1+c)) - integ*(1+c); forward[1][2] = log_sum_exp(probs_2[1]) + log(lambda0) - integ; if(num_events>1){ for(n in 2:num_events){ integ = interevent[n]*lambda0; forward[n][1] = log_sum_exp(forward[n-1] + probs_1[n]) + log(lambda0*(1+c))- integ*(1+c); forward[n][2] = log_sum_exp(forward[n-1] + probs_2[n]) + log(lambda0) - integ; } } integ = interevent[num_events]*lambda0; forward_termination[1] = log_sum_exp(forward[num_events] + probs_1[num_events]) - integ*(1+c); forward_termination[2] = log_sum_exp(forward[num_events] + probs_2[num_events]) - integ; target += log_sum_exp(forward_termination); } " ``` ```{r mmhp-stan-code} mmhp_stan_code <- " data{ int num_events; //number of events vector[num_events+1] time_matrix; // include termination time as last entry } parameters{ real lambda0; //baseline rate for each pair real w_lambda; real w_q1; //CTMC transition rate real w_q2; // real alpha; real beta_delta; real delta_1; // P(initial state = 1) } transformed parameters{ real lambda1; real q1; real q2; row_vector[2] log_delta; real beta; lambda1 = (lambda0).*(1+w_lambda); q2 = (lambda0).*w_q2; q1 = (lambda0).*w_q1; log_delta[1] = log(delta_1); log_delta[2] = log(1-delta_1); beta = alpha*(1+beta_delta); } model{ real integ; // Placeholder variable for calculating integrals row_vector[2] forward[num_events]; // Forward variables from forward-backward algorithm row_vector[2] forward_termination; // Forward variables at termination time row_vector[2] probs_1[num_events+1]; // Probability vector for transition to state 1 (active state) row_vector[2] probs_2[num_events+1]; // Probability vector for transition to state 2 (inactive state) row_vector[2] int_1[num_events+1]; // Integration of lambda when state transit to 1 (active state) row_vector[2] int_2[num_events+1]; // Integration of lambda when state transit to 2 (inactive state) real R[num_events+1]; // record variable for Hawkes process vector[num_events+1] interevent; real K0; real K1; real K2; real K3; real K4; real K5; //priors w_lambda ~ gamma(1,1); alpha ~ gamma(1,1);//lognormal(0,1); beta_delta ~ lognormal(0,2);//normal(0,10); //delta_1 ~ beta(2,2); w_q1 ~ beta(2,2); w_q2 ~ beta(2,2); lambda0 ~ gamma(1,1); interevent = time_matrix; if(num_events==0){ // there is no event occured in this period //--- prepare for forward calculation probs_1[1][1] = log(q2/(q1+q2)+q1/(q1+q2)*exp(-(q1+q2)*interevent[1])); //1->1 probs_2[1][2] = log(q1/(q1+q2)+q2/(q1+q2)*exp(-(q1+q2)*interevent[1])); //2->2 probs_1[1][2] = log1m_exp(probs_2[1][2]); //2->1 probs_2[1][1] = log1m_exp(probs_1[1][1]); //1->2 R[1] = 0; K0 = exp(-(q1+q2)*interevent[1]); K1 = (1-exp(-(q1+q2)*interevent[1]))/(q1+q2); K2 = (1-exp(-(q1+q2)*interevent[1]))/(q1+q2); int_1[1][1] = ((q2^2*lambda1+q2*q1*lambda0)*interevent[1] + (q1^2*lambda1+q2*q1*lambda0)*K0*interevent[1] + (lambda1-lambda0)*q2*q1*K1 + (lambda1-lambda0)*q2*q1*K2)/(q1+q2)^2/exp(probs_1[1][1]); //1->1 int_1[1][2] = ((q2^2*lambda1+lambda0*q1*q2)*interevent[1] - (lambda1*q1*q2+lambda0*q2^2)*K0*interevent[1] + (lambda0-lambda1)*q2^2*K1 + (lambda1-lambda0)*q1*q2*K2)/(q1+q2)^2/exp(probs_1[1][2]); //2->1 int_2[1][1] = ((q1*q2*lambda1+q1^2*lambda0)*interevent[1] - (q1^2*lambda1+q1*q2*lambda0)*K0*interevent[1] + (lambda1-lambda0)*q1^2*K1 + q1*q2*(lambda0-lambda1)*K2)/(q1+q2)^2/exp(probs_2[1][1]); //1->2 int_2[1][2] = ((q1*q2*lambda1+lambda0*q1^2)*interevent[1] + (q1*q2*lambda1+lambda0*q2^2)*K0*interevent[1] + (lambda0-lambda1)*q1*q2*K1 + (lambda0-lambda1)*q1*q2*K2)/(q1+q2)^2/exp(probs_2[1][2]); //2->2 forward_termination[1] = log_sum_exp(log_delta + probs_1[1] - int_1[1]); forward_termination[2] = log_sum_exp(log_delta + probs_2[1] - int_2[1]); target += log_sum_exp(forward_termination); //target += -lambda0*interevent[1]*delta_1-lambda1*interevent[1]*(1-delta_1); }else{ // there is event occured // ---- prepare for forward algorithm // --- log probability of Markov transition logP_ij(t) for(n in 1:(num_events + 1)){ //changed this probs_1[n][1] = log(q2/(q1+q2)+q1/(q1+q2)*exp(-(q1+q2)*interevent[n])); //1->1 probs_2[n][2] = log(q1/(q1+q2)+q2/(q1+q2)*exp(-(q1+q2)*interevent[n])); //2->2 probs_1[n][2] = log1m_exp(probs_2[n][2]); //2->1 probs_2[n][1] = log1m_exp(probs_1[n][1]); //1->2 } // --- R for Hawkes R[1] = 0; for(n in 2:(num_events + 1)){ // and this R[n] = exp(-beta*interevent[n])*(R[n-1]+1); } // Integration of lambda for(n in 1:(num_events)){ //and this K0 = exp(-(q1+q2)*interevent[n]); K1 = (1-exp(-(q1+q2)*interevent[n]))/(q1+q2); K2 = (1-exp(-(q1+q2)*interevent[n]))/(q1+q2); K3 = R[n]*(exp(beta*interevent[n])-1)/beta; K4 = R[n]*(1-exp(-(beta+q1+q2)*interevent[n]))*exp(beta*interevent[n])/(beta+q1+q2); K5 = R[n]*(1-exp(-(q1+q2-beta)*interevent[n]))/(q1+q2-beta); int_1[n][1] = ((q2^2*lambda1+q2*q1*lambda0)*interevent[n] + (q1^2*lambda1+q2*q1*lambda0)*K0*interevent[n] + (lambda1-lambda0)*q2*q1*K1 + (lambda1-lambda0)*q2*q1*K2 + alpha*K3*(q2^2+q1^2*K0) + alpha*q1*q2*K4 + alpha*q1*q2*K5)/(q1+q2)^2/exp(probs_1[n][1]); //1->1 int_1[n][2] = ((q2^2*lambda1+lambda0*q1*q2)*interevent[n] - (lambda1*q1*q2+lambda0*q2^2)*K0*interevent[n] + (lambda0-lambda1)*q2^2*K1 + (lambda1-lambda0)*q1*q2*K2 + alpha*q2*K3*(q2-q1*K0) - alpha*q2^2*K4 + alpha*q1*q2*K5)/(q1+q2)^2/exp(probs_1[n][2]); //2->1 int_2[n][1] = ((q1*q2*lambda1+q1^2*lambda0)*interevent[n] - (q1^2*lambda1+q1*q2*lambda0)*K0*interevent[n] + (lambda1-lambda0)*q1^2*K1 + q1*q2*(lambda0-lambda1)*K2 + alpha*q1*K3*(q2-q1*K0) + alpha*q1^2*K4 - alpha*q2*q1*K5)/(q1+q2)^2/exp(probs_2[n][1]); //1->2 int_2[n][2] = ((q1*q2*lambda1+lambda0*q1^2)*interevent[n] + (q1*q2*lambda1+lambda0*q2^2)*K0*interevent[n] + (lambda0-lambda1)*q1*q2*K1 + (lambda0-lambda1)*q1*q2*K2 + alpha*q1*q2*K3*(1+K0) - alpha*q1*q2*K4 - alpha*q1*q2*K5)/(q1+q2)^2/exp(probs_2[n][2]); //2->2 } //consider n = 1 forward[1][1] = log(lambda1) + log_sum_exp(probs_1[1]-int_1[1]+log_delta); forward[1][2] = log(lambda0) + log_sum_exp(probs_2[1]-int_2[1]+log_delta); if(num_events>1){ for(n in 2:num_events){ forward[n][1] = log_sum_exp(forward[n-1] + probs_1[n] - int_1[n]) + log(lambda1+alpha*R[n]); forward[n][2] = log_sum_exp(forward[n-1] + probs_2[n] - int_2[n]) + log(lambda0); } } forward_termination[1] = log_sum_exp(forward[num_events] + probs_1[num_events] - int_1[num_events]); forward_termination[2] = log_sum_exp(forward[num_events] + probs_2[num_events] - int_2[num_events]); // lots of places with max_Nm and Nm got rid of the +1 target += log_sum_exp(forward_termination); } } " ``` # Fitting these models in RStan To demonstrate how to use these models, we will simulate some data from each of these models and fit the included `stan` models. ```{r sim-mmpp} Q <- matrix(c(-0.4, 0.4, 0.2, -0.2), ncol = 2, byrow = TRUE) mmpp_obj <- pp_mmpp(Q = Q, lambda0 = 1, c = 1.5, delta = c(1/3, 2/3)) sim_mmpp <- pp_simulate(mmpp_obj, n = 50) ``` We also show how to run these models using `rstan` for the MMPP example. ```{r rstan-mmpp} mmpp_data <- list(num_events = length(sim_mmpp$events) - 1, # as first event is start time (0) time_matrix = diff(c(sim_mmpp$events, sim_mmpp$end)) #interevent arrival time ) mmpp_rstan <- stan(model_code = mmpp_stan_code, data = mmpp_data, chains = 2) mmpp_sim <- rstan::extract(mmpp_rstan) ``` The fit of this model can be evaluated in several ways, including some of the tools in the `bayesplot` package, which are important to ensure the fit is reasonable. ```{r mmpp-plot} bayesplot::mcmc_hist(as.matrix(mmpp_rstan), pars = c("lambda0", "c")) ``` To use the results of this fit with the functions of `ppdiag`, we want to extract an estimate of each of the parameters. This can be achieved using the posterior mean. The desired posterior quantities can then be extracted from this object. ```{r mmpp-post} mmpp_post <- lapply(mmpp_sim, mean) ``` Similarly, we can fit the MMHP model to simulated data also. ```{r sim-mmhp} Q <- matrix(c(-0.4, 0.4, 0.2, -0.2), ncol = 2, byrow = TRUE) mmhp_obj <- pp_mmhp(Q = Q, lambda0 = 0.5, lambda1 = 1.5, alpha = 0.5, beta = 0.75, delta = c(1/3, 2/3)) sim_mmhp <- pp_simulate(mmhp_obj, n = 50) mmhp_data <- list(num_events = length(sim_mmhp$events) - 1, # as first event is start time (0) time_matrix = diff(c(sim_mmhp$events, sim_mmhp$end)) #interevent arrival time ) ``` ```{r rstan-mmhp} mmhp_rstan <- stan(model_code = mmhp_stan_code, data = mmhp_data, chains = 2) mmhp_sim <- rstan::extract(mmhp_rstan) mmhp_post <- lapply(mmhp_sim, mean) ``` ```{r plot-draws-mmhp} bayesplot::mcmc_hist(as.matrix(mmhp_rstan), pars = c("lambda0", "lambda1", "alpha", "beta")) ``` # Fitting these models in Cmdstanr We can also fit these models using `cmdstanr`. We include the code to fit these examples below (which is not run here). ```{r mmpp-fit, eval=FALSE} mmpp_file <- write_stan_file(mmpp_stan_code) mmpp_stan <- cmdstan_model(stan_file = mmpp_file) fit_mmpp <- mmpp_stan$sample(data = mmpp_data, seed = 123, chains = 4, parallel_chains = 4, refresh = 500) ``` Similarly, we can fit the MMHP model in `cmdstanr`. ```{r compile-mmhp, eval=FALSE} mmhp_file <- write_stan_file(mmhp_stan_code) mmhp_stan <- cmdstan_model(stan_file = mmhp_file) fit_mmhp <- mmhp_stan$sample(data = mmhp_data, seed = 123, chains = 4, parallel_chains = 4, refresh = 500) ``` # Using the Stan output with `ppdiag` Having obtained estimates from fitting these stan models, we can now pass these parameter estimates into `ppdiag` to evaluate model fit. ```{r mmpp-ppdiag, fig.width=8} mmpp_fit_obj <- pp_mmpp(lambda0 = mmpp_post$lambda0, c = mmpp_post$c, Q = matrix( c(-mmpp_post$q1, mmpp_post$q1, mmpp_post$q2, -mmpp_post$q2), nrow = 2, ncol = 2, byrow = T) ) pp_diag(mmpp_fit_obj, events = sim_mmpp$events) ``` ```{r mmhp-ppdiag, fig.width=8} mmhp_fit_obj <- pp_mmhp(lambda0 = mmhp_post$lambda0, lambda1 = mmhp_post$lambda1, alpha = mmhp_post$alpha, beta = mmhp_post$beta, Q = matrix( c(-mmhp_post$q1, mmhp_post$q1, mmhp_post$q2, -mmhp_post$q2), nrow = 2, ncol = 2, byrow = T) ) pp_diag(mmhp_fit_obj, events = sim_mmhp$events) ```