I am trying to gradually build up some single source of error state space models for revenue forecasting. I’m a relative newbie to STAN and want to ensure that my code is correct. I think it is, but I need feedback.

I’m building upon the multiple source of error (MSOE) state space code here:

```
data {
int<lower=1> n;
vector[n] y;
}
parameters {
vector[n] mu;
vector[n] seasonal;
real<lower=0> sigma_level;
real<lower=0> sigma_seas;
real<lower=0> sigma_irreg;
}
transformed parameters {
vector[n] yhat;
yhat = mu + seasonal;
}
model {
for(t in 4:n)
seasonal[t] ~ normal(- sum(seasonal[t-3:t-1]), sigma_seas);
for(t in 2:n)
mu[t] ~ normal(mu[t-1], sigma_level);
y ~ normal(yhat, sigma_irreg);
}
```

However, that does not use a single source of error, as I want. As a first pass, I want to use Hyndman’s exponential smoothing models as templates (e.g. here). The key difference between the SSOE and MSOE models is that in the code above, sigma_level and sigma_seas are multiples of the residual sigma_irreg. To use just a level example (omitting the seasons for space):

y_t = l_{t-1} + e_t

l_t = l_{t-1} + \alpha e_t

So if we know what l_{t-1} is, the expectation and variance of l_t are l_{t-1} and \alpha^2 \sigma^2, if e_t \sim N(0, \sigma^2). I think. I hope.

Because it lets me update the above code to:

```
data {
int<lower=1> n;
vector[n] y;
int<lower=1> S; // Seasons
}
parameters {
vector[n] mu;
vector[n] seasonal;
real<lower=0, upper=1> alpha; // Level smoother
real<lower=0, upper=1> gamma; // Seasonal smoother
real<lower=0> sigma_irreg;
}
transformed parameters {
real<lower=0> sigma_level;
sigma_level = alpha * sigma_irreg;
real<lower=0> sigma_seas;
sigma_seas = gamma * sigma_irreg;
vector[n] yhat;
yhat = mu + seasonal;
}
model {
// Priors
alpha ~ beta(2, 1);
gamma ~ beta(2, 2);
// Model
for(t in S:n)
seasonal[t] ~ normal(- sum(seasonal[t-S+1:t-1]), sigma_seas);
for(t in 2:n)
mu[t] ~ normal(mu[t-1], sigma_level);
y ~ normal(yhat, sigma_irreg);
}
```

Anyway, I would greatly appreciate feedback. I deeply hope this is correct, because there is no way in hell I’m ever going to program a Kalman filter and I would love to actually use Stan and Bayes in my work.