Rstan - Gaussian Random Walk

Question

I'm trying to make a parameter be sampled according to a Gaussian Random Walk. In R the code would look like this:

#simulate a Gaussian random walk
#N  :  number of steps
#x0 :  initial offset
#mu :  drift velocity
#variance : step size
Gauss_RandomWalk <- function(N, x0, mu, variance) {
  z <- cumsum(rnorm(n=N, mean=mu, sd=sqrt(variance)))
  t <- 1:N
  x <- (x0 + t*mu + z)
  return(x)
}

Actually, results when setting x0=0. , mu=0. , variance=0.035**2 look good and reasonable:

[1]  0.040703269  0.009159686  0.052360030  0.059352074  0.092739218  0.098240752  0.113957813  0.064187776
[9]  0.062757728  0.063948224  0.034591074  0.004828493  0.019809969  0.032135111  0.025692763 -0.031678858
[17] -0.048033007 -0.020708105 -0.032231674  0.004917305  0.030961430  0.099054042  0.043441737 -0.010513085

Whenever I'm trying to do it in Stan, like for example according to what is represented here :

// model to be fitted
model {
  sqrtQ ~ student_t(2, 0, 0.035);
  // here we define the random walk for the log_Rt parameter
  log_Rt[1] ~ normal(0. , 0.035);
    for (t in 2:number_days) {
        log_Rt[t] ~ normal(log_Rt[t-1], sqrtQ);
  }
  print(log_Rt);
(...)
}

Results aren't that good. Look for example the jump -0.839043 -> 1.91956 which is about 85 times the standard deviation and statistically would be impossible... But why do such jumps occur?

Chain 1: [0.382303,-0.489057,0.33374,-0.839043,1.91956,0.249953,-1.88793,1.61106,1.11189,-0.725063,-0.513174,1.79012,1.57758,0.75819,0.525524,1.44762,1.19118,0.485563,-1.48318,-1.36389,1.96355,0.321416,-0.365132,-0.644287,0.0981577,1.02943,-1.27993,-1.98085,-1.75191,-1.76489,1.60888,1.48925,-0.0452427,-0.92583,-1.21594,-0.906329,-0.700237,-0.208039,0.493656,0.490295,-1.61091,1.94587,0.758567,-1.02318,-1.92659,-0.999492,1.78042,0.214125,-0.0158054,-0.753422, .......

EDIT: I tried as well:

// model to be fitted
model {
  // here we define the random walk for the log_Rt parameter
  log_Rt[1] ~ normal(0. , 0.035);
    for (t in 2:number_days) {
        log_Rt[t] ~ normal(log_Rt[t-1], 0.035);
  }
  print(log_Rt);
(...)
}

But things do not change at all.

Answer 1

The normal function you use actually tries to estimate the probability of log_Rt and therefor produces strange results. You could use generate quantities to do something similar to what you try in r:

require(rstan)
#> Loading required package: rstan
#> Loading required package: StanHeaders
#> Loading required package: ggplot2
#> rstan (Version 2.21.2, GitRev: 2e1f913d3ca3)
#> For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores()).
#> To avoid recompilation of unchanged Stan programs, we recommend calling
#> rstan_options(auto_write = TRUE)
scode<-"
generated quantities {
 real y[10];
  y[1] = normal_rng(0, .035);
    for (t in 2:10) {
        y[t] = normal_rng(y[t-1], 0.035);
  }}"
m <- stan_model(model_code = scode) 
f<-sampling(m, iter=1, chain=1, algorithm='Fixed_param')
#> 
#> SAMPLING FOR MODEL '82a947a4909cd091541c3476a3eab1f0' NOW (CHAIN 1).
#> Chain 1: Iteration: 1 / 1 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 0 seconds (Warm-up)
#> Chain 1:                1.4e-05 seconds (Sampling)
#> Chain 1:                1.4e-05 seconds (Total)
#> Chain 1:
extract(f,'y')
#> $y
#>           
#> iterations       [,1]       [,2]       [,3]       [,4]         [,5]        [,6]
#>       [1,] 0.06017424 0.07683703 0.08551969 0.04428875 -0.001519853 0.000953893
#>           
#> iterations        [,7]        [,8]        [,9]       [,10]
#>       [1,] 0.008921459 -0.06060675 -0.04087786 -0.02217786

^{Created on 2021-09-28 by the reprex package (v2.0.1)}