# Fitting Markov Modulated Point Process Models

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.

# 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.

mmpp_stan_code <- "
data{
int<lower=1> 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<lower=0> lambda0; //baseline rate for each pair real<lower=0> c; //baseline rate for each pair real<lower=0,upper=1> w1; //CTMC transition rate real<lower=0,upper=1> w2; //CTMC transition rate } transformed parameters{ real<lower=0> q1; real<lower=0> 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); } " mmhp_stan_code <- " data{ int<lower=1> num_events; //number of events vector[num_events+1] time_matrix; // include termination time as last entry } parameters{ real<lower=0> lambda0; //baseline rate for each pair real<lower=0> w_lambda; real<lower=0, upper=1> w_q1; //CTMC transition rate real<lower=0, upper=1> w_q2; // real<lower=0> alpha; real<lower=0> beta_delta; real<lower=0,upper=1> delta_1; // P(initial state = 1) } transformed parameters{ real<lower=0> lambda1; real<lower=0> q1; real<lower=0> q2; row_vector[2] log_delta; real<lower=0> 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. 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. 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.

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.

mmpp_post <- lapply(mmpp_sim, mean)

Similarly, we can fit the MMHP model to simulated data also.

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$end)) #interevent arrival time ) mmhp_rstan <- stan(model_code = mmhp_stan_code, data = mmhp_data, chains = 2) mmhp_sim <- rstan::extract(mmhp_rstan) mmhp_post <- lapply(mmhp_sim, mean) bayesplot::mcmc_hist(as.matrix(mmhp_rstan), pars = c("lambda0", "lambda1", "alpha", "beta")) # Fitting these models in Cmdstanr We can also fit these models using cmdstanr. We include the code to fit these examples below (which is not run here). mmpp_file <- write_stan_file(mmpp_stan_code) mmpp_stan <- cmdstan_model(stan_file = mmpp_file) fit_mmpp <- mmpp_stan$sample(data = mmpp_data,
seed = 123,
chains = 4,
parallel_chains = 4,
refresh = 500)

Similarly, we can fit the MMHP model in cmdstanr.

mmhp_file <- write_stan_file(mmhp_stan_code)
mmhp_stan <- cmdstan_model(stan_file = mmhp_file)

fit_mmhp <- mmhp_stan$sample(data = mmhp_data, seed = 123, chains = 4, parallel_chains = 4, refresh = 500) # Using the Stan output with ppdiag Having obtained estimates from fitting these stan models, we can now pass these parameter estimates into ppdiag to evaluate model fit. mmpp_fit_obj <- pp_mmpp(lambda0 = mmpp_post$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)
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)