update ANN

ann/index.md

@@ -7,20 +7,32 @@ output: html_document
 
 # (partially) Recurrent Neural Network (pRNN)
 
-A single (i.e., shallow) hidden-layer with 3 (hidden) neurons ANN is built to replicate a runoff hydrograph.  The inputs are lagged by three days and a neural network is *"partially recurrent"*, in that the prediction made in a previous $n$ timesteps is added as input to the following prediction:
+A deep feed-forward (DFF) Artificial Neural Network (ANN) is built to replicate the runoff hydrograph.  The inputs are $n$-day sequences of precipitation $P$ and discharge $Q$. The neural network is *"partially recurrent"* in that the discharge prediction made in a previous time step becomes the input to the following time step. The result is a trained network with the ability to make flow predictions with precipitation only. Training proceeds in 2 stages.
 
-<!-- $$
-    Q(t)=Q(t-1) + Q(t-2) +...+ Q(t-n) + R(t) + R(t-1) + ... + R(t-n) + f(\text{day of year})
-$$ -->
+
+### Seed stage
+
+To initialize the network, discharge is predicted using known discharge of the preceeding $n$-days:
 
 $$
-    \hat{Q}_t=E \left\{ Q_t | Q_{t-1}, Q_{t-2}, \dots , \\ Q_{t-n}, R_t, R_{t-1}, \dots , R_{t-n}, f(\cdot) \right\}
+    \hat{Q}_t = E \left\{ Q_t | Q_{t-1}, Q_{t-2}, \dots , Q_{t-n-1}, P_t, P_{t-1}, \dots , P_{t-n}, f(\cdot) \right\}
 $$
 
-In addition to the recurrency, $n$-day antecedent precipitation is added to the input along a sinusoidal function $(f(\cdot))$ ranging from 0 to 1 at the winter and summer solstices, respectively. In all, there are $2(n+1)$ inputs, $n$ hidden layer nodes and 1 output: Runoff.
+where $\hat{Q}_t$ is the predicted discharge, $P$ is precipitation at time $t$, $Q_{t-1}$ is the dsicharge observe on the day previous, and $f(\cdot)$ is a sinusoidal function ranging from 0 to 1 at the winter and summer solstices, respectively. In all, there are $2n+1$ inputs and 1 output: Runoff. User selects the number of hidden dense layers and the number of nodes per layer.
+
+### Training stage
+
+Once seeded, the model continuous to train for the complete data record set, for a user-set number of epochs.
+
+$$
+    \hat{Q}_t = E \left\{ Q_t | \hat{Q}_{t-1}, \hat{Q}_{t-2}, \dots , \hat{Q}_{t-n-1}, P_t, P_{t-1}, \dots , P_{t-n}, f(\cdot) \right\}
+$$
+
+Notice now the outputs $\hat{Q}$ are now fed into the input of the proceeding training round. This makes half of the ANN's inputs recurrent, making the network itself, partially recurrent. (Note: this formulation differs from typical Recurrent Neural Networks--RNN--where recurrency is applied at the node-level.) 
+
 
-Here $n=3$.
+<br>
 
-### source code
+## Source code
 
 [GitHub](https://github.com/maseology/goANN)