![[Stable]](figures/lifecycle-stable.svg)
Generates a list of arguments as required by rstan::sampling or
rstan::vb by combining the required options, with data, and type of
initialisation. Initialisation defaults to random but it is expected that
create_initial_conditions will be used.
Usage
create_stan_args(
stan = stan_opts(),
data = NULL,
init = "random",
verbose = FALSE
)Arguments
- stan
A list of stan options as generated by
stan_opts(). Defaults tostan_opts(). Can be used to overridedata,init, andverbosesettings if desired.- data
A list of stan data as created by
create_stan_data- init
Initial conditions passed to
rstan. Defaults to "random" but can also be a function (as supplied bycreate_intitial_conditions).- verbose
Logical, defaults to
FALSE. Should verbose progress messages be returned.
Examples
# default settings
create_stan_args()
#> $data
#> NULL
#>
#> $init
#> [1] "random"
#>
#> $refresh
#> [1] 0
#>
#> $object
#> S4 class stanmodel 'estimate_infections' coded as follows:
#> functions {
#> // convolve two vectors as a backwards dot product
#> // y vector should be reversed
#> // limited to the length of x and backwards looking for x indexes
#> vector convolve_with_rev_pmf(vector x, vector y, int len) {
#> int xlen = num_elements(x);
#> int ylen = num_elements(y);
#> vector[len] z;
#> if (xlen + ylen <= len) {
#> reject("convolve_with_rev_pmf: len is longer then x and y combined");
#> }
#> for (s in 1:len) {
#> z[s] = dot_product(
#> x[max(1, (s - ylen + 1)):min(s, xlen)],
#> y[max(1, ylen - s + 1):min(ylen, ylen + xlen - s)]
#> );
#> }
#> return(z);
#> }
#> // convolve latent infections to reported (but still unobserved) cases
#> vector convolve_to_report(vector infections,
#> vector delay_rev_pmf,
#> int seeding_time) {
#> int t = num_elements(infections);
#> vector[t - seeding_time] reports;
#> vector[t] unobs_reports = infections;
#> int delays = num_elements(delay_rev_pmf);
#> if (delays) {
#> unobs_reports = convolve_with_rev_pmf(unobs_reports, delay_rev_pmf, t);
#> reports = unobs_reports[(seeding_time + 1):t];
#> } else {
#> reports = infections[(seeding_time + 1):t];
#> }
#> return(reports);
#> }
#> // Calculate the daily probability of reporting using parametric
#> // distributions up to the maximum observed delay.
#> // If sigma is 0 all the probability mass is put on n.
#> // Adapted from https://github.com/epiforecasts/epinowcast
#> // @author Sam Abbott
#> // @author Adrian Lison
#> vector discretised_pmf(real mu, real sigma, int n, int dist) {
#> vector[n] lpmf;
#> if (sigma > 0) {
#> vector[n] upper_lcdf;
#> if (dist == 0) {
#> for (i in 1:n) {
#> upper_lcdf[i] = lognormal_lcdf(i | mu, sigma);
#> }
#> } else if (dist == 1) {
#> real alpha = mu^2 / sigma^2;
#> real beta = mu / sigma^2;
#> for (i in 1:n) {
#> upper_lcdf[i] = gamma_lcdf(i | alpha, beta);
#> }
#> } else {
#> reject("Unknown distribution function provided.");
#> }
#> // discretise
#> lpmf[1] = upper_lcdf[1];
#> lpmf[2:n] = log_diff_exp(upper_lcdf[2:n], upper_lcdf[1:(n-1)]);
#> // normalize
#> lpmf = lpmf - upper_lcdf[n];
#> } else {
#> // delta function
#> lpmf = rep_vector(negative_infinity(), n);
#> lpmf[n] = 0;
#> }
#> return(exp(lpmf));
#> }
#> // reverse a mf
#> vector reverse_mf(vector pmf) {
#> int max_pmf = num_elements(pmf);
#> vector[max_pmf] rev_pmf;
#> for (d in 1:max_pmf) {
#> rev_pmf[d] = pmf[max_pmf - d + 1];
#> }
#> return rev_pmf;
#> }
#> vector rev_seq(int base, int len) {
#> vector[len] seq;
#> for (i in 1:len) {
#> seq[i] = len + base - i;
#> }
#> return(seq);
#> }
#> real rev_pmf_mean(vector rev_pmf, int base) {
#> int len = num_elements(rev_pmf);
#> vector[len] rev_pmf_seq = rev_seq(base, len);
#> return(dot_product(rev_pmf_seq, rev_pmf));
#> }
#> real rev_pmf_var(vector rev_pmf, int base, real mean) {
#> int len = num_elements(rev_pmf);
#> vector[len] rev_pmf_seq = rev_seq(base, len);
#> for (i in 1:len) {
#> rev_pmf_seq[i] = pow(rev_pmf_seq[i], 2);
#> }
#> return(dot_product(rev_pmf_seq, rev_pmf) - pow(mean, 2));
#> }
#> array[] int get_delay_type_max(
#> int delay_types, array[] int delay_types_p, array[] int delay_types_id,
#> array[] int delay_types_groups, array[] int delay_max, array[] int delay_np_pmf_groups
#> ) {
#> array[delay_types] int ret;
#> for (i in 1:delay_types) {
#> ret[i] = 1;
#> for (j in delay_types_groups[i]:(delay_types_groups[i + 1] - 1)) {
#> if (delay_types_p[j]) { // parametric
#> ret[i] += delay_max[delay_types_id[j]] - 1;
#> } else { // nonparametric
#> ret[i] += delay_np_pmf_groups[delay_types_id[j] + 1] -
#> delay_np_pmf_groups[delay_types_id[j]] - 1;
#> }
#> }
#> }
#> return ret;
#> }
#> vector get_delay_rev_pmf(
#> int delay_id, int len, array[] int delay_types_p, array[] int delay_types_id,
#> array[] int delay_types_groups, array[] int delay_max,
#> vector delay_np_pmf, array[] int delay_np_pmf_groups,
#> array[] real delay_mean, array[] real delay_sigma, array[] int delay_dist,
#> int left_truncate, int reverse_pmf, int cumulative
#> ) {
#> // loop over delays
#> vector[len] pmf = rep_vector(0, len);
#> int current_len = 1;
#> int new_len;
#> for (i in delay_types_groups[delay_id]:(delay_types_groups[delay_id + 1] - 1)) {
#> if (delay_types_p[i]) { // parametric
#> vector[delay_max[delay_types_id[i]]] new_variable_pmf =
#> discretised_pmf(
#> delay_mean[delay_types_id[i]],
#> delay_sigma[delay_types_id[i]],
#> delay_max[delay_types_id[i]],
#> delay_dist[delay_types_id[i]]
#> );
#> new_len = current_len + delay_max[delay_types_id[i]] - 1;
#> if (current_len == 1) { // first delay
#> pmf[1:new_len] = new_variable_pmf;
#> } else { // subsequent delay to be convolved
#> pmf[1:new_len] = convolve_with_rev_pmf(
#> pmf[1:current_len], reverse_mf(new_variable_pmf), new_len
#> );
#> }
#> } else { // nonparametric
#> int start = delay_np_pmf_groups[delay_types_id[i]];
#> int end = delay_np_pmf_groups[delay_types_id[i] + 1] - 1;
#> new_len = current_len + end - start;
#> if (current_len == 1) { // first delay
#> pmf[1:new_len] = delay_np_pmf[start:end];
#> } else { // subsequent delay to be convolved
#> pmf[1:new_len] = convolve_with_rev_pmf(
#> pmf[1:current_len], reverse_mf(delay_np_pmf[start:end]), new_len
#> );
#> }
#> }
#> current_len = new_len;
#> }
#> if (left_truncate) {
#> pmf = append_row(
#> rep_vector(0, left_truncate),
#> pmf[(left_truncate + 1):len] / sum(pmf[(left_truncate + 1):len])
#> );
#> }
#> if (cumulative) {
#> pmf = cumulative_sum(pmf);
#> }
#> if (reverse_pmf) {
#> pmf = reverse_mf(pmf);
#> }
#> return pmf;
#> }
#> void delays_lp(array[] real delay_mean, array[] real delay_mean_mean, array[] real delay_mean_sd,
#> array[] real delay_sd, array[] real delay_sd_mean, array[] real delay_sd_sd,
#> array[] int delay_dist, array[] int weight) {
#> int mean_delays = num_elements(delay_mean);
#> int sd_delays = num_elements(delay_sd);
#> if (mean_delays) {
#> for (s in 1:mean_delays) {
#> if (delay_mean_sd[s] > 0) {
#> // uncertain mean
#> target += normal_lpdf(delay_mean[s] | delay_mean_mean[s], delay_mean_sd[s]) * weight[s];
#> // if a distribution with postive support only truncate the prior
#> if (delay_dist[s]) {
#> target += -normal_lccdf(0 | delay_mean_mean[s], delay_mean_sd[s]) * weight[s];
#> }
#> }
#> }
#> }
#> if (sd_delays) {
#> for (s in 1:sd_delays) {
#> if (delay_sd_sd[s] > 0) {
#> // uncertain sd
#> target += normal_lpdf(delay_sd[s] | delay_sd_mean[s], delay_sd_sd[s]) * weight[s];
#> target += -normal_lccdf(0 | delay_sd_mean[s], delay_sd_sd[s]) * weight[s];
#> }
#> }
#> }
#> }
#> // eigenvalues for approximate hilbert space gp
#> // see here for details: https://arxiv.org/pdf/2004.11408.pdf
#> real lambda(real L, int m) {
#> real lam;
#> lam = ((m*pi())/(2*L))^2;
#> return lam;
#> }
#> // eigenfunction for approximate hilbert space gp
#> // see here for details: https://arxiv.org/pdf/2004.11408.pdf
#> vector phi(real L, int m, vector x) {
#> vector[rows(x)] fi;
#> fi = 1/sqrt(L) * sin(m*pi()/(2*L) * (x+L));
#> return fi;
#> }
#> // spectral density of the exponential quadratic kernal
#> real spd_se(real alpha, real rho, real w) {
#> real S;
#> S = (alpha^2) * sqrt(2*pi()) * rho * exp(-0.5*(rho^2)*(w^2));
#> return S;
#> }
#> // spectral density of the Matern 3/2 kernel
#> real spd_matern(real alpha, real rho, real w) {
#> real S;
#> S = 4*alpha^2 * (sqrt(3)/rho)^3 * 1/((sqrt(3)/rho)^2 + w^2)^2;
#> return S;
#> }
#> // setup gaussian process noise dimensions
#> int setup_noise(int ot_h, int t, int horizon, int estimate_r,
#> int stationary, int future_fixed, int fixed_from) {
#> int noise_time = estimate_r > 0 ? (stationary > 0 ? ot_h : ot_h - 1) : t;
#> int noise_terms = future_fixed > 0 ? (noise_time - horizon + fixed_from) : noise_time;
#> return(noise_terms);
#> }
#> // setup approximate gaussian process
#> matrix setup_gp(int M, real L, int dimension) {
#> vector[dimension] time;
#> matrix[dimension, M] PHI;
#> real half_dim = dimension / 2.0;
#> for (s in 1:dimension) {
#> time[s] = (s - half_dim) / half_dim;
#> }
#> for (m in 1:M){
#> PHI[,m] = phi(L, m, time);
#> }
#> return(PHI);
#> }
#> // update gaussian process using spectral densities
#> vector update_gp(matrix PHI, int M, real L, real alpha,
#> real rho, vector eta, int type) {
#> vector[M] diagSPD; // spectral density
#> vector[M] SPD_eta; // spectral density * noise
#> int noise_terms = rows(PHI);
#> vector[noise_terms] noise = rep_vector(1e-6, noise_terms);
#> real unit_rho = rho / noise_terms;
#> // GP in noise - spectral densities
#> if (type == 0) {
#> for(m in 1:M){
#> diagSPD[m] = sqrt(spd_se(alpha, unit_rho, sqrt(lambda(L, m))));
#> }
#> }else if (type == 1) {
#> for(m in 1:M){
#> diagSPD[m] = sqrt(spd_matern(alpha, unit_rho, sqrt(lambda(L, m))));
#> }
#> }
#> SPD_eta = diagSPD .* eta;
#> noise = noise + PHI[,] * SPD_eta;
#> return(noise);
#> }
#> // priors for gaussian process
#> void gaussian_process_lp(real rho, real alpha, vector eta,
#> real ls_meanlog, real ls_sdlog,
#> real ls_min, real ls_max, real alpha_sd) {
#> if (ls_sdlog > 0) {
#> rho ~ lognormal(ls_meanlog, ls_sdlog) T[ls_min, ls_max];
#> } else {
#> rho ~ inv_gamma(1.499007, 0.057277 * ls_max) T[ls_min, ls_max];
#> }
#> alpha ~ normal(0, alpha_sd);
#> eta ~ std_normal();
#> }
#> // update a vector of Rts
#> vector update_Rt(int t, real log_R, vector noise, array[] int bps,
#> array[] real bp_effects, int stationary) {
#> // define control parameters
#> int bp_n = num_elements(bp_effects);
#> int bp_c = 0;
#> int gp_n = num_elements(noise);
#> // define result vectors
#> vector[t] bp = rep_vector(0, t);
#> vector[t] gp = rep_vector(0, t);
#> vector[t] R;
#> // initialise breakpoints
#> if (bp_n) {
#> for (s in 1:t) {
#> if (bps[s]) {
#> bp_c += bps[s];
#> bp[s] = bp_effects[bp_c];
#> }
#> }
#> bp = cumulative_sum(bp);
#> }
#> //initialise gaussian process
#> if (gp_n) {
#> if (stationary) {
#> gp[1:gp_n] = noise;
#> // fix future gp based on last estimated
#> if (t > gp_n) {
#> gp[(gp_n + 1):t] = rep_vector(noise[gp_n], t - gp_n);
#> }
#> }else{
#> gp[2:(gp_n + 1)] = noise;
#> gp = cumulative_sum(gp);
#> }
#> }
#> // Calculate Rt
#> R = rep_vector(log_R, t) + bp + gp;
#> R = exp(R);
#> return(R);
#> }
#> // Rt priors
#> void rt_lp(vector log_R, array[] real initial_infections, array[] real initial_growth,
#> array[] real bp_effects, array[] real bp_sd, int bp_n, int seeding_time,
#> real r_logmean, real r_logsd, real prior_infections,
#> real prior_growth) {
#> // prior on R
#> log_R ~ normal(r_logmean, r_logsd);
#> //breakpoint effects on Rt
#> if (bp_n > 0) {
#> bp_sd[1] ~ normal(0, 0.1) T[0,];
#> bp_effects ~ normal(0, bp_sd[1]);
#> }
#> // initial infections
#> initial_infections ~ normal(prior_infections, 0.2);
#> if (seeding_time > 1) {
#> initial_growth ~ normal(prior_growth, 0.2);
#> }
#> }
#> // calculate infectiousness (weighted sum of the generation time and infections)
#> // for a single time point
#> real update_infectiousness(vector infections, vector gt_rev_pmf,
#> int seeding_time, int index){
#> int gt_max = num_elements(gt_rev_pmf);
#> // work out where to start the convolution of past infections with the
#> // generation time distribution: (current_time - maximal generation time) if
#> // that is >= 1, otherwise 1
#> int inf_start = max(1, (index + seeding_time - gt_max + 1));
#> // work out where to end the convolution: current_time
#> int inf_end = (index + seeding_time);
#> // number of indices of the generation time to sum over (inf_end - inf_start + 1)
#> int pmf_accessed = min(gt_max, index + seeding_time);
#> // calculate the elements of the convolution
#> real new_inf = dot_product(
#> infections[inf_start:inf_end], tail(gt_rev_pmf, pmf_accessed)
#> );
#> return(new_inf);
#> }
#> // generate infections by using Rt = Rt-1 * sum(reversed generation time pmf * infections)
#> vector generate_infections(vector oR, int uot, vector gt_rev_pmf,
#> array[] real initial_infections, array[] real initial_growth,
#> int pop, int ht) {
#> // time indices and storage
#> int ot = num_elements(oR);
#> int nht = ot - ht;
#> int t = ot + uot;
#> vector[ot] R = oR;
#> real exp_adj_Rt;
#> vector[t] infections = rep_vector(0, t);
#> vector[ot] cum_infections;
#> vector[ot] infectiousness;
#> // Initialise infections using daily growth
#> infections[1] = exp(initial_infections[1]);
#> if (uot > 1) {
#> real growth = exp(initial_growth[1]);
#> for (s in 2:uot) {
#> infections[s] = infections[s - 1] * growth;
#> }
#> }
#> // calculate cumulative infections
#> if (pop) {
#> cum_infections[1] = sum(infections[1:uot]);
#> }
#> // iteratively update infections
#> for (s in 1:ot) {
#> infectiousness[s] = update_infectiousness(infections, gt_rev_pmf, uot, s);
#> if (pop && s > nht) {
#> exp_adj_Rt = exp(-R[s] * infectiousness[s] / (pop - cum_infections[nht]));
#> exp_adj_Rt = exp_adj_Rt > 1 ? 1 : exp_adj_Rt;
#> infections[s + uot] = (pop - cum_infections[s]) * (1 - exp_adj_Rt);
#> }else{
#> infections[s + uot] = R[s] * infectiousness[s];
#> }
#> if (pop && s < ot) {
#> cum_infections[s + 1] = cum_infections[s] + infections[s + uot];
#> }
#> }
#> return(infections);
#> }
#> // backcalculate infections using mean shifted cases and non-parametric noise
#> vector deconvolve_infections(vector shifted_cases, vector noise, int fixed,
#> int prior) {
#> int t = num_elements(shifted_cases);
#> vector[t] infections = rep_vector(1e-5, t);
#> if(!fixed) {
#> vector[t] exp_noise = exp(noise);
#> if (prior == 1) {
#> infections = infections + shifted_cases .* exp_noise;
#> }else if (prior == 0) {
#> infections = infections + exp_noise;
#> }else if (prior == 2) {
#> infections[1] = infections[1] + shifted_cases[1] * exp_noise[1];
#> for (i in 2:t) {
#> infections[i] = infections[i - 1] * exp_noise[i];
#> }
#> }
#> }else{
#> infections = infections + shifted_cases;
#> }
#> return(infections);
#> }
#> // apply day of the week effect
#> vector day_of_week_effect(vector reports, array[] int day_of_week, vector effect) {
#> int t = num_elements(reports);
#> int wl = num_elements(effect);
#> // scale day of week effect
#> vector[wl] scaled_effect = wl * effect;
#> vector[t] scaled_reports;
#> for (s in 1:t) {
#> // add reporting effects (adjust for simplex scale)
#> scaled_reports[s] = reports[s] * scaled_effect[day_of_week[s]];
#> }
#> return(scaled_reports);
#> }
#> // Scale observations by fraction reported and update log density of
#> // fraction reported
#> vector scale_obs(vector reports, real frac_obs) {
#> int t = num_elements(reports);
#> vector[t] scaled_reports;
#> scaled_reports = reports * frac_obs;
#> return(scaled_reports);
#> }
#> // Truncate observed data by some truncation distribution
#> vector truncate(vector reports, vector trunc_rev_cmf, int reconstruct) {
#> int t = num_elements(reports);
#> vector[t] trunc_reports = reports;
#> // Calculate cmf of truncation delay
#> int trunc_max = min(t, num_elements(trunc_rev_cmf));
#> int first_t = t - trunc_max + 1;
#> // Apply cdf of truncation delay to truncation max last entries in reports
#> if (reconstruct) {
#> trunc_reports[first_t:t] ./= trunc_rev_cmf[1:trunc_max];
#> } else {
#> trunc_reports[first_t:t] .*= trunc_rev_cmf[1:trunc_max];
#> }
#> return(trunc_reports);
#> }
#> // Truncation distribution priors
#> void truncation_lp(array[] real truncation_mean, array[] real truncation_sd,
#> array[] real trunc_mean_mean, array[] real trunc_mean_sd,
#> array[] real trunc_sd_mean, array[] real trunc_sd_sd) {
#> int truncation = num_elements(truncation_mean);
#> if (truncation) {
#> if (trunc_mean_sd[1] > 0) {
#> // uncertain mean
#> truncation_mean ~ normal(trunc_mean_mean, trunc_mean_sd);
#> }
#> if (trunc_sd_sd[1] > 0) {
#> // uncertain sd
#> truncation_sd ~ normal(trunc_sd_mean, trunc_sd_sd);
#> }
#> }
#> }
#> // update log density for reported cases
#> void report_lp(array[] int cases, vector reports,
#> array[] real rep_phi, real phi_mean, real phi_sd,
#> int model_type, real weight) {
#> if (model_type) {
#> real sqrt_phi; // the reciprocal overdispersion parameter (phi)
#> rep_phi[model_type] ~ normal(phi_mean, phi_sd) T[0,];
#> sqrt_phi = 1 / sqrt(rep_phi[model_type]);
#> if (weight == 1) {
#> cases ~ neg_binomial_2(reports, sqrt_phi);
#> } else {
#> target += neg_binomial_2_lpmf(cases | reports, sqrt_phi) * weight;
#> }
#> } else {
#> if (weight == 1) {
#> cases ~ poisson(reports);
#> } else {
#> target += poisson_lpmf(cases | reports) * weight;
#> }
#> }
#> }
#> // update log likelihood (as above but not vectorised and returning log likelihood)
#> vector report_log_lik(array[] int cases, vector reports,
#> array[] real rep_phi, int model_type, real weight) {
#> int t = num_elements(reports);
#> vector[t] log_lik;
#> // defer to poisson if phi is large, to avoid overflow
#> if (model_type == 0) {
#> for (i in 1:t) {
#> log_lik[i] = poisson_lpmf(cases[i] | reports[i]) * weight;
#> }
#> } else {
#> real sqrt_phi = 1 / sqrt(rep_phi[model_type]);
#> for (i in 1:t) {
#> log_lik[i] = neg_binomial_2_lpmf(cases[i] | reports[i], sqrt_phi) * weight;
#> }
#> }
#> return(log_lik);
#> }
#> // sample reported cases from the observation model
#> array[] int report_rng(vector reports, array[] real rep_phi, int model_type) {
#> int t = num_elements(reports);
#> array[t] int sampled_reports;
#> real sqrt_phi = 1e5;
#> if (model_type) {
#> sqrt_phi = 1 / sqrt(rep_phi[model_type]);
#> }
#>
#> for (s in 1:t) {
#> if (reports[s] < 1e-8) {
#> sampled_reports[s] = 0;
#> } else {
#> // defer to poisson if phi is large, to avoid overflow
#> if (sqrt_phi > 1e4) {
#> sampled_reports[s] = poisson_rng(reports[s] > 1e8 ? 1e8 : reports[s]);
#> } else {
#> sampled_reports[s] = neg_binomial_2_rng(reports[s] > 1e8 ? 1e8 : reports[s], sqrt_phi);
#> }
#> }
#> }
#> return(sampled_reports);
#> }
#> // calculate Rt directly from inferred infections
#> vector calculate_Rt(vector infections, int seeding_time,
#> vector gt_rev_pmf, int smooth) {
#> int t = num_elements(infections);
#> int ot = t - seeding_time;
#> vector[ot] R;
#> vector[ot] sR;
#> vector[ot] infectiousness = rep_vector(1e-5, ot);
#> // calculate Rt using Cori et al. approach
#> for (s in 1:ot) {
#> infectiousness[s] += update_infectiousness(
#> infections, gt_rev_pmf, seeding_time, s
#> );
#> R[s] = infections[s + seeding_time] / infectiousness[s];
#> }
#> if (smooth) {
#> for (s in 1:ot) {
#> real window = 0;
#> sR[s] = 0;
#> for (i in max(1, s - smooth):min(ot, s + smooth)) {
#> sR[s] += R[i];
#> window += 1;
#> }
#> sR[s] = sR[s] / window;
#> }
#> }else{
#> sR = R;
#> }
#> return(sR);
#> }
#> // Convert an estimate of Rt to growth
#> array[] real R_to_growth(vector R, real gt_mean, real gt_var) {
#> int t = num_elements(R);
#> array[t] real r;
#> if (gt_var > 0) {
#> real k = gt_var / pow(gt_mean, 2);
#> for (s in 1:t) {
#> r[s] = (pow(R[s], k) - 1) / (k * gt_mean);
#> }
#> } else {
#> // limit as gt_sd -> 0
#> for (s in 1:t) {
#> r[s] = log(R[s]) / gt_mean;
#> }
#> }
#> return(r);
#> }
#> }
#> data {
#> int t; // unobserved time
#> int seeding_time; // time period used for seeding and not observed
#> int horizon; // forecast horizon
#> int future_time; // time in future for Rt
#> array[t - horizon - seeding_time] int<lower = 0> cases; // observed cases
#> vector<lower = 0>[t] shifted_cases; // prior infections (for backcalculation)
#> int<lower = 0> delay_n; // number of delay distributions
#> int<lower = 0> delay_n_p; // number of parametric delay distributions
#> int<lower = 0> delay_n_np; // number of nonparametric delay distributions
#> array[delay_n_p] real delay_mean_mean; // prior mean of mean delay distribution
#> array[delay_n_p] real<lower = 0> delay_mean_sd; // prior sd of mean delay distribution
#> array[delay_n_p] real<lower = 0> delay_sd_mean; // prior sd of sd of delay distribution
#> array[delay_n_p] real<lower = 0> delay_sd_sd; // prior sd of sd of delay distribution
#> array[delay_n_p] int<lower = 1> delay_max; // maximum delay distribution
#> array[delay_n_p] int<lower = 0> delay_dist; // 0 = lognormal; 1 = gamma
#> int<lower = 0> delay_np_pmf_max; // number of nonparametric pmf elements
#> vector<lower = 0, upper = 1>[delay_np_pmf_max] delay_np_pmf; // ragged array of fixed PMFs
#> array[delay_n_np + 1] int<lower = 1> delay_np_pmf_groups; // links to ragged array
#> array[delay_n_p] int<lower = 0> delay_weight;
#> int<lower = 0> delay_types; // number of delay types
#> array[delay_n] int<lower = 0> delay_types_p; // whether delay types are parametric
#> array[delay_n] int<lower = 0> delay_types_id; // whether delay types are parametric
#> array[delay_types + 1] int<lower = 0> delay_types_groups; // index of each delay (parametric or non)
#> real L; // boundary value for infections gp
#> int<lower=1> M; // basis functions for infections gp
#> real ls_meanlog; // meanlog for gp lengthscale prior
#> real ls_sdlog; // sdlog for gp lengthscale prior
#> real<lower=0> ls_min; // Lower bound for the lengthscale
#> real<lower=0> ls_max; // Upper bound for the lengthscale
#> real alpha_sd; // standard deviation of the alpha gp kernal parameter
#> int gp_type; // type of gp, 0 = squared exponential, 1 = 3/2 matern
#> int stationary; // is underlying gaussian process first or second order
#> int fixed; // should a gaussian process be used
#> int estimate_r; // should the reproduction no be estimated (1 = yes)
#> real prior_infections; // prior for initial infections
#> real prior_growth; // prior on initial growth rate
#> real <lower = 0> r_mean; // prior mean of reproduction number
#> real <lower = 0> r_sd; // prior standard deviation of reproduction number
#> int bp_n; // no of breakpoints (0 = no breakpoints)
#> array[t - seeding_time] int breakpoints; // when do breakpoints occur
#> int future_fixed; // is underlying future Rt assumed to be fixed
#> int fixed_from; // Reference date for when Rt estimation should be fixed
#> int pop; // Initial susceptible population
#> int<lower = 0> gt_id; // id of generation time
#> int backcalc_prior; // Prior type to use for backcalculation
#> int rt_half_window; // Half the moving average window used when calculating Rt
#> array[t - seeding_time] int day_of_week; // day of the week indicator (1 - 7)
#> int model_type; // type of model: 0 = poisson otherwise negative binomial
#> real phi_mean; // Mean and sd of the normal prior for the
#> real phi_sd; // reporting process
#> int week_effect; // length of week effect
#> int obs_scale; // logical controlling scaling of observations
#> real obs_scale_mean; // mean scaling factor for observations
#> real obs_scale_sd; // standard deviation of observation scaling
#> real obs_weight; // weight given to observation in log density
#> int likelihood; // Should the likelihood be included in the model
#> int return_likelihood; // Should the likehood be returned by the model
#> int<lower = 0> trunc_id; // id of truncation
#> int<lower = 0> delay_id; // id of delay
#> }
#> transformed data{
#> // observations
#> int ot = t - seeding_time - horizon; // observed time
#> int ot_h = ot + horizon; // observed time + forecast horizon
#> // gaussian process
#> int noise_terms = setup_noise(ot_h, t, horizon, estimate_r, stationary, future_fixed, fixed_from);
#> matrix[noise_terms, M] PHI = setup_gp(M, L, noise_terms); // basis function
#> // Rt
#> real r_logmean = log(r_mean^2 / sqrt(r_sd^2 + r_mean^2));
#> real r_logsd = sqrt(log(1 + (r_sd^2 / r_mean^2)));
#> array[delay_types] int delay_type_max = get_delay_type_max(
#> delay_types, delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf_groups
#> );
#> }
#> parameters{
#> // gaussian process
#> array[fixed ? 0 : 1] real<lower = ls_min,upper=ls_max> rho; // length scale of noise GP
#> array[fixed ? 0 : 1] real<lower = 0> alpha; // scale of of noise GP
#> vector[fixed ? 0 : M] eta; // unconstrained noise
#> // Rt
#> vector[estimate_r] log_R; // baseline reproduction number estimate (log)
#> array[estimate_r] real initial_infections ; // seed infections
#> array[estimate_r && seeding_time > 1 ? 1 : 0] real initial_growth; // seed growth rate
#> array[bp_n > 0 ? 1 : 0] real<lower = 0> bp_sd; // standard deviation of breakpoint effect
#> array[bp_n] real bp_effects; // Rt breakpoint effects
#> // observation model
#> array[delay_n_p] real delay_mean; // mean of delays
#> array[delay_n_p] real<lower = 0> delay_sd; // sd of delays
#> simplex[week_effect] day_of_week_simplex;// day of week reporting effect
#> array[obs_scale] real<lower = 0, upper = 1> frac_obs; // fraction of cases that are ultimately observed
#> array[model_type] real<lower = 0> rep_phi; // overdispersion of the reporting process
#> }
#> transformed parameters {
#> vector[fixed ? 0 : noise_terms] noise; // noise generated by the gaussian process
#> vector<lower = 0, upper = 10 * r_mean>[estimate_r > 0 ? ot_h : 0] R; // reproduction number
#> vector[t] infections; // latent infections
#> vector[ot_h] reports; // estimated reported cases
#> vector[ot] obs_reports; // observed estimated reported cases
#> vector[delay_type_max[gt_id]] gt_rev_pmf;
#> // GP in noise - spectral densities
#> if (!fixed) {
#> noise = update_gp(PHI, M, L, alpha[1], rho[1], eta, gp_type);
#> }
#> // Estimate latent infections
#> if (estimate_r) {
#> gt_rev_pmf = get_delay_rev_pmf(
#> gt_id, delay_type_max[gt_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean, delay_sd, delay_dist,
#> 1, 1, 0
#> );
#> R = update_Rt(
#> ot_h, log_R[estimate_r], noise, breakpoints, bp_effects, stationary
#> );
#> infections = generate_infections(
#> R, seeding_time, gt_rev_pmf, initial_infections, initial_growth, pop,
#> future_time
#> );
#> } else {
#> // via deconvolution
#> infections = deconvolve_infections(
#> shifted_cases, noise, fixed, backcalc_prior
#> );
#> }
#> // convolve from latent infections to mean of observations
#> if (delay_id) {
#> vector[delay_type_max[delay_id]] delay_rev_pmf = get_delay_rev_pmf(
#> delay_id, delay_type_max[delay_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean, delay_sd, delay_dist,
#> 0, 1, 0
#> );
#> reports = convolve_to_report(infections, delay_rev_pmf, seeding_time);
#> } else {
#> reports = infections[(seeding_time + 1):t];
#> }
#> // weekly reporting effect
#> if (week_effect > 1) {
#> reports = day_of_week_effect(reports, day_of_week, day_of_week_simplex);
#> }
#> // scaling of reported cases by fraction observed
#> if (obs_scale) {
#> reports = scale_obs(reports, frac_obs[1]);
#> }
#> // truncate near time cases to observed reports
#> if (trunc_id) {
#> vector[delay_type_max[trunc_id]] trunc_rev_cmf = get_delay_rev_pmf(
#> trunc_id, delay_type_max[trunc_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean, delay_sd, delay_dist,
#> 0, 1, 1
#> );
#> obs_reports = truncate(reports[1:ot], trunc_rev_cmf, 0);
#> } else {
#> obs_reports = reports[1:ot];
#> }
#> }
#> model {
#> // priors for noise GP
#> if (!fixed) {
#> gaussian_process_lp(
#> rho[1], alpha[1], eta, ls_meanlog, ls_sdlog, ls_min, ls_max, alpha_sd
#> );
#> }
#> // penalised priors for delay distributions
#> delays_lp(
#> delay_mean, delay_mean_mean,
#> delay_mean_sd, delay_sd, delay_sd_mean, delay_sd_sd,
#> delay_dist, delay_weight
#> );
#> if (estimate_r) {
#> // priors on Rt
#> rt_lp(
#> log_R, initial_infections, initial_growth, bp_effects, bp_sd, bp_n,
#> seeding_time, r_logmean, r_logsd, prior_infections, prior_growth
#> );
#> }
#> // prior observation scaling
#> if (obs_scale) {
#> frac_obs[1] ~ normal(obs_scale_mean, obs_scale_sd) T[0, 1];
#> }
#> // observed reports from mean of reports (update likelihood)
#> if (likelihood) {
#> report_lp(
#> cases, obs_reports, rep_phi, phi_mean, phi_sd, model_type, obs_weight
#> );
#> }
#> }
#> generated quantities {
#> array[ot_h] int imputed_reports;
#> vector[estimate_r > 0 ? 0: ot_h] gen_R;
#> array[ot_h] real r;
#> real gt_mean;
#> real gt_var;
#> vector[return_likelihood ? ot : 0] log_lik;
#> if (estimate_r){
#> // estimate growth from estimated Rt
#> gt_mean = rev_pmf_mean(gt_rev_pmf, 1);
#> gt_var = rev_pmf_var(gt_rev_pmf, 1, gt_mean);
#> r = R_to_growth(R, gt_mean, gt_var);
#> } else {
#> // sample generation time
#> array[delay_n_p] real delay_mean_sample =
#> normal_rng(delay_mean_mean, delay_mean_sd);
#> array[delay_n_p] real delay_sd_sample =
#> normal_rng(delay_sd_mean, delay_sd_sd);
#> vector[delay_type_max[gt_id]] sampled_gt_rev_pmf = get_delay_rev_pmf(
#> gt_id, delay_type_max[gt_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean_sample, delay_sd_sample,
#> delay_dist, 1, 1, 0
#> );
#> gt_mean = rev_pmf_mean(sampled_gt_rev_pmf, 1);
#> gt_var = rev_pmf_var(sampled_gt_rev_pmf, 1, gt_mean);
#> // calculate Rt using infections and generation time
#> gen_R = calculate_Rt(
#> infections, seeding_time, sampled_gt_rev_pmf, rt_half_window
#> );
#> // estimate growth from calculated Rt
#> r = R_to_growth(gen_R, gt_mean, gt_var);
#> }
#> // simulate reported cases
#> imputed_reports = report_rng(reports, rep_phi, model_type);
#> // log likelihood of model
#> if (return_likelihood) {
#> log_lik = report_log_lik(
#> cases, obs_reports, rep_phi, model_type, obs_weight
#> );
#> }
#> }
#>
#> $method
#> [1] "sampling"
#>
#> $cores
#> [1] 1
#>
#> $warmup
#> [1] 250
#>
#> $chains
#> [1] 4
#>
#> $save_warmup
#> [1] FALSE
#>
#> $seed
#> [1] 35453657
#>
#> $future
#> [1] FALSE
#>
#> $max_execution_time
#> [1] Inf
#>
#> $control
#> $control$adapt_delta
#> [1] 0.95
#>
#> $control$max_treedepth
#> [1] 15
#>
#>
#> $iter
#> [1] 750
#>
# increasing warmup
create_stan_args(stan = stan_opts(warmup = 1000))
#> $data
#> NULL
#>
#> $init
#> [1] "random"
#>
#> $refresh
#> [1] 0
#>
#> $object
#> S4 class stanmodel 'estimate_infections' coded as follows:
#> functions {
#> // convolve two vectors as a backwards dot product
#> // y vector should be reversed
#> // limited to the length of x and backwards looking for x indexes
#> vector convolve_with_rev_pmf(vector x, vector y, int len) {
#> int xlen = num_elements(x);
#> int ylen = num_elements(y);
#> vector[len] z;
#> if (xlen + ylen <= len) {
#> reject("convolve_with_rev_pmf: len is longer then x and y combined");
#> }
#> for (s in 1:len) {
#> z[s] = dot_product(
#> x[max(1, (s - ylen + 1)):min(s, xlen)],
#> y[max(1, ylen - s + 1):min(ylen, ylen + xlen - s)]
#> );
#> }
#> return(z);
#> }
#> // convolve latent infections to reported (but still unobserved) cases
#> vector convolve_to_report(vector infections,
#> vector delay_rev_pmf,
#> int seeding_time) {
#> int t = num_elements(infections);
#> vector[t - seeding_time] reports;
#> vector[t] unobs_reports = infections;
#> int delays = num_elements(delay_rev_pmf);
#> if (delays) {
#> unobs_reports = convolve_with_rev_pmf(unobs_reports, delay_rev_pmf, t);
#> reports = unobs_reports[(seeding_time + 1):t];
#> } else {
#> reports = infections[(seeding_time + 1):t];
#> }
#> return(reports);
#> }
#> // Calculate the daily probability of reporting using parametric
#> // distributions up to the maximum observed delay.
#> // If sigma is 0 all the probability mass is put on n.
#> // Adapted from https://github.com/epiforecasts/epinowcast
#> // @author Sam Abbott
#> // @author Adrian Lison
#> vector discretised_pmf(real mu, real sigma, int n, int dist) {
#> vector[n] lpmf;
#> if (sigma > 0) {
#> vector[n] upper_lcdf;
#> if (dist == 0) {
#> for (i in 1:n) {
#> upper_lcdf[i] = lognormal_lcdf(i | mu, sigma);
#> }
#> } else if (dist == 1) {
#> real alpha = mu^2 / sigma^2;
#> real beta = mu / sigma^2;
#> for (i in 1:n) {
#> upper_lcdf[i] = gamma_lcdf(i | alpha, beta);
#> }
#> } else {
#> reject("Unknown distribution function provided.");
#> }
#> // discretise
#> lpmf[1] = upper_lcdf[1];
#> lpmf[2:n] = log_diff_exp(upper_lcdf[2:n], upper_lcdf[1:(n-1)]);
#> // normalize
#> lpmf = lpmf - upper_lcdf[n];
#> } else {
#> // delta function
#> lpmf = rep_vector(negative_infinity(), n);
#> lpmf[n] = 0;
#> }
#> return(exp(lpmf));
#> }
#> // reverse a mf
#> vector reverse_mf(vector pmf) {
#> int max_pmf = num_elements(pmf);
#> vector[max_pmf] rev_pmf;
#> for (d in 1:max_pmf) {
#> rev_pmf[d] = pmf[max_pmf - d + 1];
#> }
#> return rev_pmf;
#> }
#> vector rev_seq(int base, int len) {
#> vector[len] seq;
#> for (i in 1:len) {
#> seq[i] = len + base - i;
#> }
#> return(seq);
#> }
#> real rev_pmf_mean(vector rev_pmf, int base) {
#> int len = num_elements(rev_pmf);
#> vector[len] rev_pmf_seq = rev_seq(base, len);
#> return(dot_product(rev_pmf_seq, rev_pmf));
#> }
#> real rev_pmf_var(vector rev_pmf, int base, real mean) {
#> int len = num_elements(rev_pmf);
#> vector[len] rev_pmf_seq = rev_seq(base, len);
#> for (i in 1:len) {
#> rev_pmf_seq[i] = pow(rev_pmf_seq[i], 2);
#> }
#> return(dot_product(rev_pmf_seq, rev_pmf) - pow(mean, 2));
#> }
#> array[] int get_delay_type_max(
#> int delay_types, array[] int delay_types_p, array[] int delay_types_id,
#> array[] int delay_types_groups, array[] int delay_max, array[] int delay_np_pmf_groups
#> ) {
#> array[delay_types] int ret;
#> for (i in 1:delay_types) {
#> ret[i] = 1;
#> for (j in delay_types_groups[i]:(delay_types_groups[i + 1] - 1)) {
#> if (delay_types_p[j]) { // parametric
#> ret[i] += delay_max[delay_types_id[j]] - 1;
#> } else { // nonparametric
#> ret[i] += delay_np_pmf_groups[delay_types_id[j] + 1] -
#> delay_np_pmf_groups[delay_types_id[j]] - 1;
#> }
#> }
#> }
#> return ret;
#> }
#> vector get_delay_rev_pmf(
#> int delay_id, int len, array[] int delay_types_p, array[] int delay_types_id,
#> array[] int delay_types_groups, array[] int delay_max,
#> vector delay_np_pmf, array[] int delay_np_pmf_groups,
#> array[] real delay_mean, array[] real delay_sigma, array[] int delay_dist,
#> int left_truncate, int reverse_pmf, int cumulative
#> ) {
#> // loop over delays
#> vector[len] pmf = rep_vector(0, len);
#> int current_len = 1;
#> int new_len;
#> for (i in delay_types_groups[delay_id]:(delay_types_groups[delay_id + 1] - 1)) {
#> if (delay_types_p[i]) { // parametric
#> vector[delay_max[delay_types_id[i]]] new_variable_pmf =
#> discretised_pmf(
#> delay_mean[delay_types_id[i]],
#> delay_sigma[delay_types_id[i]],
#> delay_max[delay_types_id[i]],
#> delay_dist[delay_types_id[i]]
#> );
#> new_len = current_len + delay_max[delay_types_id[i]] - 1;
#> if (current_len == 1) { // first delay
#> pmf[1:new_len] = new_variable_pmf;
#> } else { // subsequent delay to be convolved
#> pmf[1:new_len] = convolve_with_rev_pmf(
#> pmf[1:current_len], reverse_mf(new_variable_pmf), new_len
#> );
#> }
#> } else { // nonparametric
#> int start = delay_np_pmf_groups[delay_types_id[i]];
#> int end = delay_np_pmf_groups[delay_types_id[i] + 1] - 1;
#> new_len = current_len + end - start;
#> if (current_len == 1) { // first delay
#> pmf[1:new_len] = delay_np_pmf[start:end];
#> } else { // subsequent delay to be convolved
#> pmf[1:new_len] = convolve_with_rev_pmf(
#> pmf[1:current_len], reverse_mf(delay_np_pmf[start:end]), new_len
#> );
#> }
#> }
#> current_len = new_len;
#> }
#> if (left_truncate) {
#> pmf = append_row(
#> rep_vector(0, left_truncate),
#> pmf[(left_truncate + 1):len] / sum(pmf[(left_truncate + 1):len])
#> );
#> }
#> if (cumulative) {
#> pmf = cumulative_sum(pmf);
#> }
#> if (reverse_pmf) {
#> pmf = reverse_mf(pmf);
#> }
#> return pmf;
#> }
#> void delays_lp(array[] real delay_mean, array[] real delay_mean_mean, array[] real delay_mean_sd,
#> array[] real delay_sd, array[] real delay_sd_mean, array[] real delay_sd_sd,
#> array[] int delay_dist, array[] int weight) {
#> int mean_delays = num_elements(delay_mean);
#> int sd_delays = num_elements(delay_sd);
#> if (mean_delays) {
#> for (s in 1:mean_delays) {
#> if (delay_mean_sd[s] > 0) {
#> // uncertain mean
#> target += normal_lpdf(delay_mean[s] | delay_mean_mean[s], delay_mean_sd[s]) * weight[s];
#> // if a distribution with postive support only truncate the prior
#> if (delay_dist[s]) {
#> target += -normal_lccdf(0 | delay_mean_mean[s], delay_mean_sd[s]) * weight[s];
#> }
#> }
#> }
#> }
#> if (sd_delays) {
#> for (s in 1:sd_delays) {
#> if (delay_sd_sd[s] > 0) {
#> // uncertain sd
#> target += normal_lpdf(delay_sd[s] | delay_sd_mean[s], delay_sd_sd[s]) * weight[s];
#> target += -normal_lccdf(0 | delay_sd_mean[s], delay_sd_sd[s]) * weight[s];
#> }
#> }
#> }
#> }
#> // eigenvalues for approximate hilbert space gp
#> // see here for details: https://arxiv.org/pdf/2004.11408.pdf
#> real lambda(real L, int m) {
#> real lam;
#> lam = ((m*pi())/(2*L))^2;
#> return lam;
#> }
#> // eigenfunction for approximate hilbert space gp
#> // see here for details: https://arxiv.org/pdf/2004.11408.pdf
#> vector phi(real L, int m, vector x) {
#> vector[rows(x)] fi;
#> fi = 1/sqrt(L) * sin(m*pi()/(2*L) * (x+L));
#> return fi;
#> }
#> // spectral density of the exponential quadratic kernal
#> real spd_se(real alpha, real rho, real w) {
#> real S;
#> S = (alpha^2) * sqrt(2*pi()) * rho * exp(-0.5*(rho^2)*(w^2));
#> return S;
#> }
#> // spectral density of the Matern 3/2 kernel
#> real spd_matern(real alpha, real rho, real w) {
#> real S;
#> S = 4*alpha^2 * (sqrt(3)/rho)^3 * 1/((sqrt(3)/rho)^2 + w^2)^2;
#> return S;
#> }
#> // setup gaussian process noise dimensions
#> int setup_noise(int ot_h, int t, int horizon, int estimate_r,
#> int stationary, int future_fixed, int fixed_from) {
#> int noise_time = estimate_r > 0 ? (stationary > 0 ? ot_h : ot_h - 1) : t;
#> int noise_terms = future_fixed > 0 ? (noise_time - horizon + fixed_from) : noise_time;
#> return(noise_terms);
#> }
#> // setup approximate gaussian process
#> matrix setup_gp(int M, real L, int dimension) {
#> vector[dimension] time;
#> matrix[dimension, M] PHI;
#> real half_dim = dimension / 2.0;
#> for (s in 1:dimension) {
#> time[s] = (s - half_dim) / half_dim;
#> }
#> for (m in 1:M){
#> PHI[,m] = phi(L, m, time);
#> }
#> return(PHI);
#> }
#> // update gaussian process using spectral densities
#> vector update_gp(matrix PHI, int M, real L, real alpha,
#> real rho, vector eta, int type) {
#> vector[M] diagSPD; // spectral density
#> vector[M] SPD_eta; // spectral density * noise
#> int noise_terms = rows(PHI);
#> vector[noise_terms] noise = rep_vector(1e-6, noise_terms);
#> real unit_rho = rho / noise_terms;
#> // GP in noise - spectral densities
#> if (type == 0) {
#> for(m in 1:M){
#> diagSPD[m] = sqrt(spd_se(alpha, unit_rho, sqrt(lambda(L, m))));
#> }
#> }else if (type == 1) {
#> for(m in 1:M){
#> diagSPD[m] = sqrt(spd_matern(alpha, unit_rho, sqrt(lambda(L, m))));
#> }
#> }
#> SPD_eta = diagSPD .* eta;
#> noise = noise + PHI[,] * SPD_eta;
#> return(noise);
#> }
#> // priors for gaussian process
#> void gaussian_process_lp(real rho, real alpha, vector eta,
#> real ls_meanlog, real ls_sdlog,
#> real ls_min, real ls_max, real alpha_sd) {
#> if (ls_sdlog > 0) {
#> rho ~ lognormal(ls_meanlog, ls_sdlog) T[ls_min, ls_max];
#> } else {
#> rho ~ inv_gamma(1.499007, 0.057277 * ls_max) T[ls_min, ls_max];
#> }
#> alpha ~ normal(0, alpha_sd);
#> eta ~ std_normal();
#> }
#> // update a vector of Rts
#> vector update_Rt(int t, real log_R, vector noise, array[] int bps,
#> array[] real bp_effects, int stationary) {
#> // define control parameters
#> int bp_n = num_elements(bp_effects);
#> int bp_c = 0;
#> int gp_n = num_elements(noise);
#> // define result vectors
#> vector[t] bp = rep_vector(0, t);
#> vector[t] gp = rep_vector(0, t);
#> vector[t] R;
#> // initialise breakpoints
#> if (bp_n) {
#> for (s in 1:t) {
#> if (bps[s]) {
#> bp_c += bps[s];
#> bp[s] = bp_effects[bp_c];
#> }
#> }
#> bp = cumulative_sum(bp);
#> }
#> //initialise gaussian process
#> if (gp_n) {
#> if (stationary) {
#> gp[1:gp_n] = noise;
#> // fix future gp based on last estimated
#> if (t > gp_n) {
#> gp[(gp_n + 1):t] = rep_vector(noise[gp_n], t - gp_n);
#> }
#> }else{
#> gp[2:(gp_n + 1)] = noise;
#> gp = cumulative_sum(gp);
#> }
#> }
#> // Calculate Rt
#> R = rep_vector(log_R, t) + bp + gp;
#> R = exp(R);
#> return(R);
#> }
#> // Rt priors
#> void rt_lp(vector log_R, array[] real initial_infections, array[] real initial_growth,
#> array[] real bp_effects, array[] real bp_sd, int bp_n, int seeding_time,
#> real r_logmean, real r_logsd, real prior_infections,
#> real prior_growth) {
#> // prior on R
#> log_R ~ normal(r_logmean, r_logsd);
#> //breakpoint effects on Rt
#> if (bp_n > 0) {
#> bp_sd[1] ~ normal(0, 0.1) T[0,];
#> bp_effects ~ normal(0, bp_sd[1]);
#> }
#> // initial infections
#> initial_infections ~ normal(prior_infections, 0.2);
#> if (seeding_time > 1) {
#> initial_growth ~ normal(prior_growth, 0.2);
#> }
#> }
#> // calculate infectiousness (weighted sum of the generation time and infections)
#> // for a single time point
#> real update_infectiousness(vector infections, vector gt_rev_pmf,
#> int seeding_time, int index){
#> int gt_max = num_elements(gt_rev_pmf);
#> // work out where to start the convolution of past infections with the
#> // generation time distribution: (current_time - maximal generation time) if
#> // that is >= 1, otherwise 1
#> int inf_start = max(1, (index + seeding_time - gt_max + 1));
#> // work out where to end the convolution: current_time
#> int inf_end = (index + seeding_time);
#> // number of indices of the generation time to sum over (inf_end - inf_start + 1)
#> int pmf_accessed = min(gt_max, index + seeding_time);
#> // calculate the elements of the convolution
#> real new_inf = dot_product(
#> infections[inf_start:inf_end], tail(gt_rev_pmf, pmf_accessed)
#> );
#> return(new_inf);
#> }
#> // generate infections by using Rt = Rt-1 * sum(reversed generation time pmf * infections)
#> vector generate_infections(vector oR, int uot, vector gt_rev_pmf,
#> array[] real initial_infections, array[] real initial_growth,
#> int pop, int ht) {
#> // time indices and storage
#> int ot = num_elements(oR);
#> int nht = ot - ht;
#> int t = ot + uot;
#> vector[ot] R = oR;
#> real exp_adj_Rt;
#> vector[t] infections = rep_vector(0, t);
#> vector[ot] cum_infections;
#> vector[ot] infectiousness;
#> // Initialise infections using daily growth
#> infections[1] = exp(initial_infections[1]);
#> if (uot > 1) {
#> real growth = exp(initial_growth[1]);
#> for (s in 2:uot) {
#> infections[s] = infections[s - 1] * growth;
#> }
#> }
#> // calculate cumulative infections
#> if (pop) {
#> cum_infections[1] = sum(infections[1:uot]);
#> }
#> // iteratively update infections
#> for (s in 1:ot) {
#> infectiousness[s] = update_infectiousness(infections, gt_rev_pmf, uot, s);
#> if (pop && s > nht) {
#> exp_adj_Rt = exp(-R[s] * infectiousness[s] / (pop - cum_infections[nht]));
#> exp_adj_Rt = exp_adj_Rt > 1 ? 1 : exp_adj_Rt;
#> infections[s + uot] = (pop - cum_infections[s]) * (1 - exp_adj_Rt);
#> }else{
#> infections[s + uot] = R[s] * infectiousness[s];
#> }
#> if (pop && s < ot) {
#> cum_infections[s + 1] = cum_infections[s] + infections[s + uot];
#> }
#> }
#> return(infections);
#> }
#> // backcalculate infections using mean shifted cases and non-parametric noise
#> vector deconvolve_infections(vector shifted_cases, vector noise, int fixed,
#> int prior) {
#> int t = num_elements(shifted_cases);
#> vector[t] infections = rep_vector(1e-5, t);
#> if(!fixed) {
#> vector[t] exp_noise = exp(noise);
#> if (prior == 1) {
#> infections = infections + shifted_cases .* exp_noise;
#> }else if (prior == 0) {
#> infections = infections + exp_noise;
#> }else if (prior == 2) {
#> infections[1] = infections[1] + shifted_cases[1] * exp_noise[1];
#> for (i in 2:t) {
#> infections[i] = infections[i - 1] * exp_noise[i];
#> }
#> }
#> }else{
#> infections = infections + shifted_cases;
#> }
#> return(infections);
#> }
#> // apply day of the week effect
#> vector day_of_week_effect(vector reports, array[] int day_of_week, vector effect) {
#> int t = num_elements(reports);
#> int wl = num_elements(effect);
#> // scale day of week effect
#> vector[wl] scaled_effect = wl * effect;
#> vector[t] scaled_reports;
#> for (s in 1:t) {
#> // add reporting effects (adjust for simplex scale)
#> scaled_reports[s] = reports[s] * scaled_effect[day_of_week[s]];
#> }
#> return(scaled_reports);
#> }
#> // Scale observations by fraction reported and update log density of
#> // fraction reported
#> vector scale_obs(vector reports, real frac_obs) {
#> int t = num_elements(reports);
#> vector[t] scaled_reports;
#> scaled_reports = reports * frac_obs;
#> return(scaled_reports);
#> }
#> // Truncate observed data by some truncation distribution
#> vector truncate(vector reports, vector trunc_rev_cmf, int reconstruct) {
#> int t = num_elements(reports);
#> vector[t] trunc_reports = reports;
#> // Calculate cmf of truncation delay
#> int trunc_max = min(t, num_elements(trunc_rev_cmf));
#> int first_t = t - trunc_max + 1;
#> // Apply cdf of truncation delay to truncation max last entries in reports
#> if (reconstruct) {
#> trunc_reports[first_t:t] ./= trunc_rev_cmf[1:trunc_max];
#> } else {
#> trunc_reports[first_t:t] .*= trunc_rev_cmf[1:trunc_max];
#> }
#> return(trunc_reports);
#> }
#> // Truncation distribution priors
#> void truncation_lp(array[] real truncation_mean, array[] real truncation_sd,
#> array[] real trunc_mean_mean, array[] real trunc_mean_sd,
#> array[] real trunc_sd_mean, array[] real trunc_sd_sd) {
#> int truncation = num_elements(truncation_mean);
#> if (truncation) {
#> if (trunc_mean_sd[1] > 0) {
#> // uncertain mean
#> truncation_mean ~ normal(trunc_mean_mean, trunc_mean_sd);
#> }
#> if (trunc_sd_sd[1] > 0) {
#> // uncertain sd
#> truncation_sd ~ normal(trunc_sd_mean, trunc_sd_sd);
#> }
#> }
#> }
#> // update log density for reported cases
#> void report_lp(array[] int cases, vector reports,
#> array[] real rep_phi, real phi_mean, real phi_sd,
#> int model_type, real weight) {
#> if (model_type) {
#> real sqrt_phi; // the reciprocal overdispersion parameter (phi)
#> rep_phi[model_type] ~ normal(phi_mean, phi_sd) T[0,];
#> sqrt_phi = 1 / sqrt(rep_phi[model_type]);
#> if (weight == 1) {
#> cases ~ neg_binomial_2(reports, sqrt_phi);
#> } else {
#> target += neg_binomial_2_lpmf(cases | reports, sqrt_phi) * weight;
#> }
#> } else {
#> if (weight == 1) {
#> cases ~ poisson(reports);
#> } else {
#> target += poisson_lpmf(cases | reports) * weight;
#> }
#> }
#> }
#> // update log likelihood (as above but not vectorised and returning log likelihood)
#> vector report_log_lik(array[] int cases, vector reports,
#> array[] real rep_phi, int model_type, real weight) {
#> int t = num_elements(reports);
#> vector[t] log_lik;
#> // defer to poisson if phi is large, to avoid overflow
#> if (model_type == 0) {
#> for (i in 1:t) {
#> log_lik[i] = poisson_lpmf(cases[i] | reports[i]) * weight;
#> }
#> } else {
#> real sqrt_phi = 1 / sqrt(rep_phi[model_type]);
#> for (i in 1:t) {
#> log_lik[i] = neg_binomial_2_lpmf(cases[i] | reports[i], sqrt_phi) * weight;
#> }
#> }
#> return(log_lik);
#> }
#> // sample reported cases from the observation model
#> array[] int report_rng(vector reports, array[] real rep_phi, int model_type) {
#> int t = num_elements(reports);
#> array[t] int sampled_reports;
#> real sqrt_phi = 1e5;
#> if (model_type) {
#> sqrt_phi = 1 / sqrt(rep_phi[model_type]);
#> }
#>
#> for (s in 1:t) {
#> if (reports[s] < 1e-8) {
#> sampled_reports[s] = 0;
#> } else {
#> // defer to poisson if phi is large, to avoid overflow
#> if (sqrt_phi > 1e4) {
#> sampled_reports[s] = poisson_rng(reports[s] > 1e8 ? 1e8 : reports[s]);
#> } else {
#> sampled_reports[s] = neg_binomial_2_rng(reports[s] > 1e8 ? 1e8 : reports[s], sqrt_phi);
#> }
#> }
#> }
#> return(sampled_reports);
#> }
#> // calculate Rt directly from inferred infections
#> vector calculate_Rt(vector infections, int seeding_time,
#> vector gt_rev_pmf, int smooth) {
#> int t = num_elements(infections);
#> int ot = t - seeding_time;
#> vector[ot] R;
#> vector[ot] sR;
#> vector[ot] infectiousness = rep_vector(1e-5, ot);
#> // calculate Rt using Cori et al. approach
#> for (s in 1:ot) {
#> infectiousness[s] += update_infectiousness(
#> infections, gt_rev_pmf, seeding_time, s
#> );
#> R[s] = infections[s + seeding_time] / infectiousness[s];
#> }
#> if (smooth) {
#> for (s in 1:ot) {
#> real window = 0;
#> sR[s] = 0;
#> for (i in max(1, s - smooth):min(ot, s + smooth)) {
#> sR[s] += R[i];
#> window += 1;
#> }
#> sR[s] = sR[s] / window;
#> }
#> }else{
#> sR = R;
#> }
#> return(sR);
#> }
#> // Convert an estimate of Rt to growth
#> array[] real R_to_growth(vector R, real gt_mean, real gt_var) {
#> int t = num_elements(R);
#> array[t] real r;
#> if (gt_var > 0) {
#> real k = gt_var / pow(gt_mean, 2);
#> for (s in 1:t) {
#> r[s] = (pow(R[s], k) - 1) / (k * gt_mean);
#> }
#> } else {
#> // limit as gt_sd -> 0
#> for (s in 1:t) {
#> r[s] = log(R[s]) / gt_mean;
#> }
#> }
#> return(r);
#> }
#> }
#> data {
#> int t; // unobserved time
#> int seeding_time; // time period used for seeding and not observed
#> int horizon; // forecast horizon
#> int future_time; // time in future for Rt
#> array[t - horizon - seeding_time] int<lower = 0> cases; // observed cases
#> vector<lower = 0>[t] shifted_cases; // prior infections (for backcalculation)
#> int<lower = 0> delay_n; // number of delay distributions
#> int<lower = 0> delay_n_p; // number of parametric delay distributions
#> int<lower = 0> delay_n_np; // number of nonparametric delay distributions
#> array[delay_n_p] real delay_mean_mean; // prior mean of mean delay distribution
#> array[delay_n_p] real<lower = 0> delay_mean_sd; // prior sd of mean delay distribution
#> array[delay_n_p] real<lower = 0> delay_sd_mean; // prior sd of sd of delay distribution
#> array[delay_n_p] real<lower = 0> delay_sd_sd; // prior sd of sd of delay distribution
#> array[delay_n_p] int<lower = 1> delay_max; // maximum delay distribution
#> array[delay_n_p] int<lower = 0> delay_dist; // 0 = lognormal; 1 = gamma
#> int<lower = 0> delay_np_pmf_max; // number of nonparametric pmf elements
#> vector<lower = 0, upper = 1>[delay_np_pmf_max] delay_np_pmf; // ragged array of fixed PMFs
#> array[delay_n_np + 1] int<lower = 1> delay_np_pmf_groups; // links to ragged array
#> array[delay_n_p] int<lower = 0> delay_weight;
#> int<lower = 0> delay_types; // number of delay types
#> array[delay_n] int<lower = 0> delay_types_p; // whether delay types are parametric
#> array[delay_n] int<lower = 0> delay_types_id; // whether delay types are parametric
#> array[delay_types + 1] int<lower = 0> delay_types_groups; // index of each delay (parametric or non)
#> real L; // boundary value for infections gp
#> int<lower=1> M; // basis functions for infections gp
#> real ls_meanlog; // meanlog for gp lengthscale prior
#> real ls_sdlog; // sdlog for gp lengthscale prior
#> real<lower=0> ls_min; // Lower bound for the lengthscale
#> real<lower=0> ls_max; // Upper bound for the lengthscale
#> real alpha_sd; // standard deviation of the alpha gp kernal parameter
#> int gp_type; // type of gp, 0 = squared exponential, 1 = 3/2 matern
#> int stationary; // is underlying gaussian process first or second order
#> int fixed; // should a gaussian process be used
#> int estimate_r; // should the reproduction no be estimated (1 = yes)
#> real prior_infections; // prior for initial infections
#> real prior_growth; // prior on initial growth rate
#> real <lower = 0> r_mean; // prior mean of reproduction number
#> real <lower = 0> r_sd; // prior standard deviation of reproduction number
#> int bp_n; // no of breakpoints (0 = no breakpoints)
#> array[t - seeding_time] int breakpoints; // when do breakpoints occur
#> int future_fixed; // is underlying future Rt assumed to be fixed
#> int fixed_from; // Reference date for when Rt estimation should be fixed
#> int pop; // Initial susceptible population
#> int<lower = 0> gt_id; // id of generation time
#> int backcalc_prior; // Prior type to use for backcalculation
#> int rt_half_window; // Half the moving average window used when calculating Rt
#> array[t - seeding_time] int day_of_week; // day of the week indicator (1 - 7)
#> int model_type; // type of model: 0 = poisson otherwise negative binomial
#> real phi_mean; // Mean and sd of the normal prior for the
#> real phi_sd; // reporting process
#> int week_effect; // length of week effect
#> int obs_scale; // logical controlling scaling of observations
#> real obs_scale_mean; // mean scaling factor for observations
#> real obs_scale_sd; // standard deviation of observation scaling
#> real obs_weight; // weight given to observation in log density
#> int likelihood; // Should the likelihood be included in the model
#> int return_likelihood; // Should the likehood be returned by the model
#> int<lower = 0> trunc_id; // id of truncation
#> int<lower = 0> delay_id; // id of delay
#> }
#> transformed data{
#> // observations
#> int ot = t - seeding_time - horizon; // observed time
#> int ot_h = ot + horizon; // observed time + forecast horizon
#> // gaussian process
#> int noise_terms = setup_noise(ot_h, t, horizon, estimate_r, stationary, future_fixed, fixed_from);
#> matrix[noise_terms, M] PHI = setup_gp(M, L, noise_terms); // basis function
#> // Rt
#> real r_logmean = log(r_mean^2 / sqrt(r_sd^2 + r_mean^2));
#> real r_logsd = sqrt(log(1 + (r_sd^2 / r_mean^2)));
#> array[delay_types] int delay_type_max = get_delay_type_max(
#> delay_types, delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf_groups
#> );
#> }
#> parameters{
#> // gaussian process
#> array[fixed ? 0 : 1] real<lower = ls_min,upper=ls_max> rho; // length scale of noise GP
#> array[fixed ? 0 : 1] real<lower = 0> alpha; // scale of of noise GP
#> vector[fixed ? 0 : M] eta; // unconstrained noise
#> // Rt
#> vector[estimate_r] log_R; // baseline reproduction number estimate (log)
#> array[estimate_r] real initial_infections ; // seed infections
#> array[estimate_r && seeding_time > 1 ? 1 : 0] real initial_growth; // seed growth rate
#> array[bp_n > 0 ? 1 : 0] real<lower = 0> bp_sd; // standard deviation of breakpoint effect
#> array[bp_n] real bp_effects; // Rt breakpoint effects
#> // observation model
#> array[delay_n_p] real delay_mean; // mean of delays
#> array[delay_n_p] real<lower = 0> delay_sd; // sd of delays
#> simplex[week_effect] day_of_week_simplex;// day of week reporting effect
#> array[obs_scale] real<lower = 0, upper = 1> frac_obs; // fraction of cases that are ultimately observed
#> array[model_type] real<lower = 0> rep_phi; // overdispersion of the reporting process
#> }
#> transformed parameters {
#> vector[fixed ? 0 : noise_terms] noise; // noise generated by the gaussian process
#> vector<lower = 0, upper = 10 * r_mean>[estimate_r > 0 ? ot_h : 0] R; // reproduction number
#> vector[t] infections; // latent infections
#> vector[ot_h] reports; // estimated reported cases
#> vector[ot] obs_reports; // observed estimated reported cases
#> vector[delay_type_max[gt_id]] gt_rev_pmf;
#> // GP in noise - spectral densities
#> if (!fixed) {
#> noise = update_gp(PHI, M, L, alpha[1], rho[1], eta, gp_type);
#> }
#> // Estimate latent infections
#> if (estimate_r) {
#> gt_rev_pmf = get_delay_rev_pmf(
#> gt_id, delay_type_max[gt_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean, delay_sd, delay_dist,
#> 1, 1, 0
#> );
#> R = update_Rt(
#> ot_h, log_R[estimate_r], noise, breakpoints, bp_effects, stationary
#> );
#> infections = generate_infections(
#> R, seeding_time, gt_rev_pmf, initial_infections, initial_growth, pop,
#> future_time
#> );
#> } else {
#> // via deconvolution
#> infections = deconvolve_infections(
#> shifted_cases, noise, fixed, backcalc_prior
#> );
#> }
#> // convolve from latent infections to mean of observations
#> if (delay_id) {
#> vector[delay_type_max[delay_id]] delay_rev_pmf = get_delay_rev_pmf(
#> delay_id, delay_type_max[delay_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean, delay_sd, delay_dist,
#> 0, 1, 0
#> );
#> reports = convolve_to_report(infections, delay_rev_pmf, seeding_time);
#> } else {
#> reports = infections[(seeding_time + 1):t];
#> }
#> // weekly reporting effect
#> if (week_effect > 1) {
#> reports = day_of_week_effect(reports, day_of_week, day_of_week_simplex);
#> }
#> // scaling of reported cases by fraction observed
#> if (obs_scale) {
#> reports = scale_obs(reports, frac_obs[1]);
#> }
#> // truncate near time cases to observed reports
#> if (trunc_id) {
#> vector[delay_type_max[trunc_id]] trunc_rev_cmf = get_delay_rev_pmf(
#> trunc_id, delay_type_max[trunc_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean, delay_sd, delay_dist,
#> 0, 1, 1
#> );
#> obs_reports = truncate(reports[1:ot], trunc_rev_cmf, 0);
#> } else {
#> obs_reports = reports[1:ot];
#> }
#> }
#> model {
#> // priors for noise GP
#> if (!fixed) {
#> gaussian_process_lp(
#> rho[1], alpha[1], eta, ls_meanlog, ls_sdlog, ls_min, ls_max, alpha_sd
#> );
#> }
#> // penalised priors for delay distributions
#> delays_lp(
#> delay_mean, delay_mean_mean,
#> delay_mean_sd, delay_sd, delay_sd_mean, delay_sd_sd,
#> delay_dist, delay_weight
#> );
#> if (estimate_r) {
#> // priors on Rt
#> rt_lp(
#> log_R, initial_infections, initial_growth, bp_effects, bp_sd, bp_n,
#> seeding_time, r_logmean, r_logsd, prior_infections, prior_growth
#> );
#> }
#> // prior observation scaling
#> if (obs_scale) {
#> frac_obs[1] ~ normal(obs_scale_mean, obs_scale_sd) T[0, 1];
#> }
#> // observed reports from mean of reports (update likelihood)
#> if (likelihood) {
#> report_lp(
#> cases, obs_reports, rep_phi, phi_mean, phi_sd, model_type, obs_weight
#> );
#> }
#> }
#> generated quantities {
#> array[ot_h] int imputed_reports;
#> vector[estimate_r > 0 ? 0: ot_h] gen_R;
#> array[ot_h] real r;
#> real gt_mean;
#> real gt_var;
#> vector[return_likelihood ? ot : 0] log_lik;
#> if (estimate_r){
#> // estimate growth from estimated Rt
#> gt_mean = rev_pmf_mean(gt_rev_pmf, 1);
#> gt_var = rev_pmf_var(gt_rev_pmf, 1, gt_mean);
#> r = R_to_growth(R, gt_mean, gt_var);
#> } else {
#> // sample generation time
#> array[delay_n_p] real delay_mean_sample =
#> normal_rng(delay_mean_mean, delay_mean_sd);
#> array[delay_n_p] real delay_sd_sample =
#> normal_rng(delay_sd_mean, delay_sd_sd);
#> vector[delay_type_max[gt_id]] sampled_gt_rev_pmf = get_delay_rev_pmf(
#> gt_id, delay_type_max[gt_id], delay_types_p, delay_types_id,
#> delay_types_groups, delay_max, delay_np_pmf,
#> delay_np_pmf_groups, delay_mean_sample, delay_sd_sample,
#> delay_dist, 1, 1, 0
#> );
#> gt_mean = rev_pmf_mean(sampled_gt_rev_pmf, 1);
#> gt_var = rev_pmf_var(sampled_gt_rev_pmf, 1, gt_mean);
#> // calculate Rt using infections and generation time
#> gen_R = calculate_Rt(
#> infections, seeding_time, sampled_gt_rev_pmf, rt_half_window
#> );
#> // estimate growth from calculated Rt
#> r = R_to_growth(gen_R, gt_mean, gt_var);
#> }
#> // simulate reported cases
#> imputed_reports = report_rng(reports, rep_phi, model_type);
#> // log likelihood of model
#> if (return_likelihood) {
#> log_lik = report_log_lik(
#> cases, obs_reports, rep_phi, model_type, obs_weight
#> );
#> }
#> }
#>
#> $method
#> [1] "sampling"
#>
#> $cores
#> [1] 1
#>
#> $warmup
#> [1] 1000
#>
#> $chains
#> [1] 4
#>
#> $save_warmup
#> [1] FALSE
#>
#> $seed
#> [1] 36003562
#>
#> $future
#> [1] FALSE
#>
#> $max_execution_time
#> [1] Inf
#>
#> $control
#> $control$adapt_delta
#> [1] 0.95
#>
#> $control$max_treedepth
#> [1] 15
#>
#>
#> $iter
#> [1] 1500
#>