# Is an Arma model equivalent to a 1-layer Recurrent Neural Network without activation function?

Given a time series $$f(t)$$ to forecast, let us consider an Arma model of the form: $$f(t) = c + \sum_{i=1}^p a_i f(t-i) + e(t) + \sum_{j=1}^q b_j e(t-j)$$

where $$e(t)$$ are the forecast errors.

On the train set, if $$f(t)$$ is the ground truth, then we define its estimate obtained with this model as $$\widetilde{f}(t) = f(t) + e(t)$$.

Let $$m = min(p,q)$$, we can rewrite the first equation as: $$\widetilde{f}(t) = c + \sum_{i=1}^m (a_i + b_i) f(t-i) + \sum_{i=m+1}^p a_i f(t-i) - \sum_{j=1}^q b_j \widetilde{f}(t-j)$$ Then after reparametrization can be rewritten as: $$\widetilde{f}(t) = c + \sum_{i=1}^k c_i f(t-i) - \sum_{j=1}^q b_j \widetilde{f}(t-j)$$ Which is the equation of a 1-layer recurrent neural network (RNN) without activation function.

So, are Arma models a subset of RNNs or is there a flaw in this reasoning ?

• very interesting question, wish I could upvote more Feb 23, 2020 at 2:47