**NOTE: This assignment, like others in this class, is due at the
beginning of the class period. This means that if you are
even a minute late, you lose 20%. If you are worried about
potentially being late, turn in your homework ahead of time.
Do this by submitting them to me during office hours or by sliding it
under my office door. Do not send assignments to me through email or
leave them in my departmental mail box.**

As discussed in class, the fundamental computing element for an artificial neural network (ANN) is the artificial neuron (AN). ANs can be combined in many ways to compute more complex functions but the most fundamental way is by combining them into a layered, feedforward neural network (FFNN). ANs and FFNNs can learn in many ways but the most fundamental way is supervised learning. Moreover, ANs and FFNNs may be used for many tasks but the two most fundamental are classification and function approximation. These ANN fundamentals — ANs, FFNNs, supervised learning, classification, and function approximation — are the topics of this homework.

- Consider a single AN used for classification in a 2D space with an
augmented vector. This AN is a summation unit (SU) and its activation
function
*f*is a step function with outputs γ_{AN}_{1}=1 and γ_{2}=0. Given the weights*v*_{1}=0.3,*v*_{2}=0.2, and*v*_{3}=0.1,**draw**this AN. **Draw**(on graph or engineering paper) the decision boundary encoded by this AN. Be sure to indicate the γ_{1}side of the boundary.**Add**the following points on the graph you just drew and**label**the class of each according to the AN.- (0.5, 0.5)
- (0, 0)
- (−0.25, 0.3)
- (0.9, −0.1)
- (0,0.5)

- Assume that the AN’s classification of each of the points above
is correct.
**List**a new labeled data item which, if added to the data set above, would necessarily cause the AN to misclassify at least one data item.**Explain**why the AN would necessarily misclassify at least one data item. **Explain**how the decision boundary for this AN would change if γ_{2}were changed to −1, rather than 0.

[Note for explanatory questions, like this one: To explain the answer, describe what the answer is — in this case, what changes (if any) there are to the decision boundary — and also**why**that is the correct answer — in this case that answer would be based on how the value of γ affects the decision boundary.]**Explain**how the decision boundary for this AN would change if*f*were changed to be a sigmoidal function._{AN}**Explain**how the classification of each of the points listed above would change if*f*were a sigmoidal function._{AN}

- Consider the same AN given above in 1.1.
**Explain**how its weights would be updated, using the updating rule we covered in class on January 16, given each of the following labeled data points, presented in the following order:- (0.8, 0.8) γ
_{1} - (−0.1, −0.2) γ
_{2} - (0.3, −0.2) γ
_{1} - (0, 0) γ
_{2}

- (0.8, 0.8) γ
**Draw**the graph of the data and the decision boundary after each weight update for the data given in 2.1.**Explain**which of the points in 2.1 are correctly classified by this AN after all of the weight updates from one learning pass through this data and**explain**what this tells us about the use of this learning rule.**Explain**how many more passes through the data are needed before all of the points from 2.1 are correctly classified by this AN.**Show your work.**

- 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 1.1. There are four ANs in the hidden layer and one in the output layer. Given the weights_{AN}*v*_{1,1}=0.6,*v*_{2,1}=0.2,*v*_{3,1}=0.1,*v*_{4,1}=0.0,*v*_{1,2}=0.3,*v*_{2,2}=0.6,*v*_{3,2}=0.7,*v*_{4,2}=0.8,*v*_{1,3}=0.1,*v*_{2,3}=0.2,*v*_{3,3}=0.1,*v*_{4,3}=0.1,*w*_{1}=0.5,*w*_{2}=1.0,*w*_{3}=0.0, and*w*_{4}=0.75,**draw**this FFNN. **Draw**the decision region encoded by this FFNN. Be sure to indicate the γ_{1}side of the region.**Add**the following points on the graph you just drew and**label**the class of each according to the AN.- (0.5, 0.5)
- (0, 0)
- (−0.25, 0.3)
- (0.9, −0.1)
- (0.25, 0.2)

**Explain**how the decision region for this AN 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.

- Consider the FFNN given above in 3.1 but with sigmoidal activation
functions for each AN and η = 0.5.
**Explain**how its weights would be updated, using the backpropagation algorithm we covered in class on January 28, given if presented with the data item (0.8, 0.8) γ_{1}.**Show your work.**Keep track of four significant digits. **Calculate**the output value of the FFNN above if, after learning on (0.8, 0.8) γ_{1}, you were to present this data item to the FFNN again.**Show your work.**Keep track of four significant digits.**Explain**whether the error value for the input (0.8, 0.8) γ_{1}increased or decreased due to learning.

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.