Artificial neurons (ANs) can be combined in many ways to compute more complex functions than could be computed by a single AN. The most fundamental way is by combining ANs into a layered, feedforward neural network (FFNN). Moreover, FFNNs may be used for many tasks but the two most fundamental are classification and function approximation, of which classification is the easier to visualize. These ANN fundamentals — FFNNs and classification — are the topics of this homework.

The goal of this assignment is to give you experience with basic FFNNs and classification.

- Consider a two-layer FFNN — that is, one with two layers of
computational elements (ANs) — used for classification in a 2D
space with augmented vectors. The ANs in this FFNN are all SUs and their
activation functions are identical to
*f*as given in Homework 1, Subpart 1.1. There are three ANs in the hidden layer and one in the output layer. Given the weights_{AN}*v*_{1,1}=−0.1,*v*_{2,1}=0.2,*v*_{3,1}=−0.9,*v*_{1,2}=0.6,*v*_{2,2}=0.8,*v*_{3,2}=−0.1,*v*_{1,3}=−0.2,*v*_{2,3}=−0.2,*v*_{3,3}=−0.4,*w*_{1}=−0.2,*w*_{2}=0.6,*w*_{3}=0.0, and*w*_{4}=0.1,**draw**this FFNN. **Draw**the decision region encoded by this FFNN. Be sure to indicate the γ_{1}side of the region. Be sure to indicate which portion of the decision region is due to each hidden layer AN. (Hint: If you're having difficulty figuring out exactly how the decision region works out given the weights above, try plotting the points from Exercise 3 immediately below and calculate their classes by plugging their point values into the equations defined by the weights above in Exercise 1.)**Add**the following points on the graph you just drew and**label**the class of each according to the AN.- (0.9, 0.9)
- (−0.6, −0.5)
- (−0.6, 0.2)
- (0.8, −0.8)
- (−0.4, −0.3)
- (0.0, 0.0)

**Explain**the significance of the value of*w*_{4}.**Explain**the significance of the value of*w*_{3}.**Explain**the significance of the relative values of*w*_{1}and*w*_{2}.**Explain**how the decision region for this FFNN would change if the value of*w*_{3}were changed to −0.5 rather than 0.0 and**explain**which points, if any, from those above would be classified differently and which would be classified the same. Be sure to discuss the relative values of*w*_{1},*w*_{2}, and*w*_{3}.**Explain**how the decision region for this FFNN would change if γ_{2}were changed to −1 rather than 0 and**explain**which points, if any, from those above would be classified differently and which would be classified the same. (For this hypothetical, use a value of −0.5 for*w*_{3}.)**Explain**how the decision region for this FFNN would change if*f*for the hidden layer ANs were changed to be a linear activation function, in particular, if_{AN}*f*(_{AN}*net*)=*net*for the hidden layer ANs. (For the output AN,*f*remains unchanged in this hypothetical.)_{AN}**Explain**which points, if any, from those above would be classified differently and which would be classified the same. (For this hypothetical, use a value of −0.5 for*w*_{3}.)

Turn in a neatly handwritten copy of your answers to the exercises for this assignment. The diagrams should be drawn on engineering or graph paper. You may also turn in a scanned electronic copy of this assignment as a backup in case your paper copy is misplaced.