Thanks a lot for all your help.

I am trying to adapt my model such as I won’t have this problem.

One of the thing I was thinking of is allowing peak at every point but with a prior that is concentred at 0.

Initially I was thinking of doing a prior that is a mixture model but I don’t think it will work.

Really what I want is to have a heaviside step function with a dynamic “x” and when it is not 0 multiply it by a gamma prior.

I understand it is a bit hard to understand.

I am basically trying to fit something like that:

```
generate_data <- function(n_point=100,noise_mu=1000,noise_sigma=5,n_peak=5,coef_shape=5,coef_rate=1,tau_min=3,tau_max=10){
idx <- rbinom(n_peak,n_point, runif(n_peak,0,1))
noise <- rnorm(n_point,noise_mu,noise_sigma)
init_data <- noise
coef_delta <- rgamma(n_peak,coef_shape,coef_rate)
tau <- runif(n_peak,tau_min,tau_max)
data <- init_data
for (i in seq_along(idx)){
v <- idx[i]
delt <- coef_delta[i]
t <- tau[i]
data <- data + median(init_data) * delt * as.numeric(seq(1,n_point) >= v ) * c( rep(0,v-1), exp(-c(seq(0,n_point-v)/t )))
}
list(
data = data,
tau = tau,
delta = coef_delta,
peak_idx = idx,
noise = noise
)
}
```

Where any of the parameters from the function could change.

i.e.

```
generated <- generate_data()
data <- generated$data
plot(data,t="l")
```

The model I came up with is this one:

```
model.code <- "
data{
int<lower=1> T;
real y[T];
}
transformed data{
real<lower=0> median; //median of y
int middle; // idx of middle data
real y_sorted[T];
middle = T / 2;
y_sorted= sort_asc(y);
if (T % 2)
{
median = (y_sorted[middle] + y_sorted[middle + 1]) / 2.0;
}
else
{
median = y_sorted[middle ];
}
}
parameters {
real<lower=0> delta[T];
real<lower=0.0000001> tau[T];
real<lower=0> sigma;
}
transformed parameters {
vector[T] decay; // decay
for (i in 1:T){
decay[i] = 0;
for(j in 0:(i-1)){
decay[i] = decay[i] + median * delta[i-j] * exp(-j / tau[i-j]);
}
}
}
model{
real ps[T];
sigma ~ cauchy(0,5);
delta ~ exponential(1);
y ~ normal( decay , sigma);
}
generated quantities{
real ps_rep[T];
real y_rep[T];
for (i in 1:T){
ps_rep[i] = 0;
for(j in 0:(i-1)){
ps_rep[i] = ps_rep[i] + median * delta[i-j] * exp(-j / tau[i-j]);
}
}
for (nn in 1:T){
y_rep[nn] = ps_rep[nn] ;
}
}
"
```

However, as you can see I am not really handling the delta as I should.

And I end up having my sampler not happy:

```
model.dat <- list(T=100,y=data)
stan.fit <- stan(model_code=model.code,
model_name="lm_decay",
data=model.dat)
```

```
There were 4000 divergent transitions after warmup. Increasing adapt_delta above 0.8 may help. See
http://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmupThere were 3 chains where the estimated Bayesian Fraction of Missing Information was low. See
http://mc-stan.org/misc/warnings.html#bfmi-lowExamine the pairs() plot to diagnose sampling problems
```

More importantly the fit is just bad

```
library(bayesplot)
y_rep <- rstan::extract(stan.fit,"y_rep")
ppc_dens_overlay(y = data,
yrep = y_rep[[1]][1:100,])
```

Could someone advise me on how I could solve this or just give me a hint?