I worked out an example usage case for the unit_vector variable type, but while it seems to sample and recover simulated parameters well, I tend to get divergence warnings. The scenario being modeled is one where you have noisy samples from a sine wave with a known rate but unknown phase, amplitude & noise:

```
data{
int n;
vector[n] y;
vector[n] x;
real<lower=0> rate ;
}
parameters{
unit_vector[2] phase_helper;
real<lower=0> amplitude;
real<lower=0> noise;
}
transformed parameters{
real phase = atan2(phase_helper[1],phase_helper[2]) ;
}
model{
amplitude ~ weibull(2,2) ;
noise ~ normal(0,1) ;
y ~ normal( amplitude * sin( x*rate*2*pi()+phase ) , noise ) ;
}
```

And here’s R code to generate and sample the model:

```
library(rstan)
rstan_options(auto_write = TRUE)
n = 1e3
x = seq(0,10,length.out = n)
rate = .5
noise = .1
phase = pi/2
amplitude = 1
f = amplitude*sin(x*rate*2*pi+phase)
y = f + rnorm(n,0,noise)
plot(x,y,type='l')
lines(x,f,col='green',lwd=2)
post = rstan::stan(
file = 'periodic.stan'
, data = list(
n = length(y)
, x = x
, y = y
, rate = rate
)
, chains = 4
, cores = 4 #set to the number of physical cores on your system
, iter = 1e3
)
rstan::traceplot(post)
```

Examination of the pairs plot (below) reveals that the dependencies are between the phase-related variables, which I think is expected, so possibly the calculation of the divergence warning needs to be tweaked to avoid this false alarm?