Hi guys,

I have successfully been able to run an ANN in Stan. Some background, I want to be able to include **additive** time series parameters to an ANN. In the end I want do some structural time series modelling combined with an ANN. Both have their own strengths (SS = time series, ANN = cross-sectional fit) and I want to combine these because of that. The model posted below still has time dummies, but I will replace that in the end with (probably) a random walk component.

```
data {
int<lower=0> N; // number of observations
int<lower=0> Nt; // number of time periods
int<lower=0> Np; // number of explanatory variables
int<lower=0> Nh; // number of knots (only 1 layer)
vector[N] yVar; // explained variable
int<lower=0,upper=Nt> sell[N]; // period sold, sell = 1, ..., Nt
matrix[N,Np] x; // explanatory variables
}
parameters {
vector[Nt] mu; // time series parameter
ordered[Nh] beta; // bias, CONSTRAINT #1
matrix[Nh,Np] omega; // weights in layer
vector[Nh] lambda; // estimates measurement Eq.
real<lower=0> sigEps; // RMSE of measurement Eq.
}
transformed parameters {
vector[N] yHat;
matrix[N,Nh] h;
for(i in 1:N){
for(k in 1:Nh){
h[i,k] = inv_logit( dot_product(x[i],omega[k]) + beta[k] );
}
yHat[i] = mu[sell[i]] + dot_product(h[i],lambda);
}
}
model {
// to_vector(omega) ~ normal(0,5); // can I do this simultanously with constraint #2?
lambda ~ normal(0,5);
beta ~ normal(0,5);
mu ~ normal(0,5);
sigEps ~ normal(0,1);
for(k in 1:(Nh-1)){
// sum(pow(omega[k],2)) ~ normal(1, 0.001 * Np); // Euclidean norm, CONSTRAINT #2 (doesn't work...)
// sum(omega[k] .* omega[k]) ~ normal(1, 0.001 * Np); // Euclidean norm, CONSTRAINT #2 (doesn't work...)
sum(omega[k]) ~ normal(0, 0.001 * Np); // Zero sum, CONSTRAINT #2 (works!)
}
yVar ~ normal(yHat, sigEps);
}
generated quantities {
// Training data stuff in here [LATER]
}
```

I know there has already been some work done on these ANN in Stan, but I think my code is a bit more straightforward if you only are interested in 1 layer. Anyways, I had two short questions.

- First of all, I would like to know if I can combine a zero sum constraint, with a “normal” prior in the model block. For example, in the model block, I would like both omega ~ normal(0,5);
*and*sum(omega) ~ normal(0, 0.001 * Np), see code as well. However, I don’t think I can do both? Can I? - Secondly, I actually do not want a zero sum constraint, but a Euclidian norm constraint (I believe this is also called a L2 norm?). Naively, this constraint would translate as sum(pow(omega, 2)) ~ normal(1, 0.001*Np). In words, the sum of the parameters squared should equal 1. Is there a way to get this constraint in Stan, without too much hassle? I gave two examples (commented out in the code) that didn’t work.

Note that all but 1 of the columns in omega actually has the constraint. The final column in omega can / will be left unconstraint! This makes it slightly more complex I guess… Any suggestions on how to speed up the code are obviously welcome as well!

Many thanks as always, you are the best!

Alex