Dealing with unknown data with rstan/ODE

Modeling
,
1

I have a general question with regard to using stan to solve ODE problems. In the simple harmonic oscillator example, both y[1] and y[2] are supplied during the model fitting. In case of when only one variable can be observed (i.e. only y[1] is known), how should the code by modified in order to model the system? Can you put the whole variable in the parameters chunk and how would achieve that? A lot thanks!

functions {
  real[] sho(real t, real[] y, real[] theta, real[] x_r, int[] x_i) {
    real dy1_dt = y[2];
    real dy2_dt = -y[1] - theta[1] * y[2];
    return { dy1_dt, dy2_dt };
  }
}
data {
  int<lower = 0> T;
  real t0;
  real<lower = t0> ts[T];
  real y_hat[T, 2];
  real y0[2];
}
transformed data {
  real x_r[0];
  int x_i[0];
}
parameters {
  real theta[1];
  real<lower = 0> sigma;
}
transformed parameters {
  real y[T, 2] = integrate_ode_bdf(sho, y0, 0.0, ts, theta, x_r, x_i);
}
model {
  sigma ~ normal(0.1, 0.1);
  theta ~ normal(0.15, 0.1);
  y0 ~ normal({1.0, 0.0}, 0.1);
 
  y_hat[, 1] ~ normal(y[, 1],  sigma);
  y_hat[, 2] ~ normal(y[, 2],  sigma);
}
generated quantities {
}
2

If you only observe the first component, that’s easy.

Change:

y_hat[, 1] ~ normal(y[, 1],  sigma);
y_hat[, 2] ~ normal(y[, 2],  sigma);

to

y_hat[, 1] ~ normal(y[, 1],  sigma);

The question becomes is this enough information to infer all the parameters? That’s something problem dependent. In this case it probably is.

3

Thanks for your reply. This clear my misunderstanding that all data (y) in user defined function need to be supplied when calling stan/sampling function, but I should only need the initial value for the integrator to work and anything I want to fit to the model. I get similar solution by talking to jandraor in his post on a different topic. Thanks to both of you.