# AIC for a model with multiple dependent variables

I am performing a model selection analysis with some variations of the same 3-equation dynamical system of ODEs:

$$\dot{x} = f(x,y,z)\\ \dot{y} = g(x,y,z)\\ \dot{z} = h(x,y,z)$$

The particularity of the problem is that I don't have the same number of data points for each variable. Let's call $$n_x$$ the number of observations for variable $$x$$, $$n_y$$ for $$y$$ and $$n_Z$$ for $$z$$. I obtain the best fit via minimizing least squares, which for this problem would be

$$SSR = SSR_x + SSR_y+SSR_z=\sum_{i=1}^{n_x}(x_i-\hat{x}_i)^2+\sum_{i=1}^{n_y}(y_i-\hat{y}_i)^2+\sum_{i=1}^{n_z}(z_i-\hat{z}_i)^2$$ where the hat denotes data point. So I fit each model and compare quality of fit with the Akaike Information Criterion, which, following this question, can be computed from the SSR as

$$AIC=2K+n\ln RSS,$$ where $$K$$ is the number of parameters and $$n$$ is the number of observations. where , and the +1 comes from including the variance of the statistical model as an estimated parameter (I guess we can drop this since it's a constant). My issue is of course what to plug in $$n$$. My guess is that, since the likelihood is additive, we could simply split in three terms as

$$AIC=2K+\sum_{i\in{x,y,z}} n_i \ln RSS_i,$$

Does this make sense or am I missing something?

• Hi and welcome to SE! A few comments: the exact definition of the AIC is $2K-2\log L$. In the special case where model errors are i.i.d. and normally distributed, then the AIC becomes $2K+n \log(RSS/n)$ (see here for the BIC: en.wikipedia.org/wiki/…). $K$ should include all parameters of your model, including the error variance, there is no need to add a term for the variance of the statistical model. Sep 14, 2022 at 14:47
• Yes! Thank you very much. K would in this case then be the number of model parameters plus 3? For the three error variances, one for each variable. Sep 15, 2022 at 7:34
• There is no need to "add" anything to $K$. This is the number of parameters of the model, including its error variance, which does not change whether you use the likelihood or the RSS. Sep 15, 2022 at 7:46
• Excellent, thank you. What about the sum? Does it make sense? Sep 16, 2022 at 8:24
• Can you update your question so as to add more details about your model? Especially, what are the three variables you are mentioning, what are the three ODEs, and what are your observations? Thanks! Sep 16, 2022 at 14:58