Part $1$ AN Representation

Consider a single AN used for classification in a $2$D space with an augmented vector as discussed in the Engelbrecht text. This AN is a summation unit $($SU$)$ and its activation function fAN is a step function with outputs $\backslash$gamma $1=1$ for fAN$($net$)>=0$ and $\backslash$gamma $2=0$ for fAN$($net$)<0.$ Given the weights v$1=1\mathrm{.}0,$ v$2=0\mathrm{.}3,$ and v$3=0\mathrm{.}5,$ draw this AN$.$

Draw $($on graph or engineering paper or by using software$)$ the decision boundary encoded by this AN$.$ Be sure to indicate the $\backslash$gamma $1$ side of the boundary.

Add the following points on the graph you just drew and label the class of each according to the AN$.$

$(1\mathrm{.}0,0\mathrm{.}2)$

$(0\mathrm{.}0,0\mathrm{.}0)$

$(0\mathrm{.}1,0\mathrm{.}5)$

$(1\mathrm{.}7,0\mathrm{.}1)$

$(1\mathrm{.}8,1\mathrm{.}4)$

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. $($That is$,$ given this new data item, there would be no set of possible weights that would allow this AN to correctly classify all data items.$)$ Explain why the AN would necessarily misclassify at least one data item.

Explain how the decision boundary for this AN would change if $\backslash$gamma $2$ 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 $\backslash$gamma affects the decision boundary.$]$

Explain how the decision boundary for this AN would change if fAN were changed to be a sigmoidal logistic function.

Explain how the classification of each of the points listed above would change if fAN were a sigmoidal logistic function.

Part $2$ AN Learning

Consider the same AN given above in $1\mathrm{.}1$ together with the following learning rule $($and ignore the points in $1\mathrm{.}3$ and $1\mathrm{.}4$ and possible modifications to the AN listed in $1\mathrm{.}51\mathrm{.}7)$:

vi$($t$)=$ vi$($t$1)+($tp $$ op$)$ zi$,$p

Explain how the AN weights would be updated given each of the following labeled data points, presented in the following order:

$(0\mathrm{.}1,1\mathrm{.}0)\backslash$gamma $2$

$(0\mathrm{.}2,0\mathrm{.}9)\backslash$gamma $2$

$(0\mathrm{.}6,0\mathrm{.}8)\backslash$gamma $1$

$(0\mathrm{.}0,0\mathrm{.}0)\backslash$gamma $1$

Draw the graph of the data and the decision boundary after each weight update for the data given in $2\mathrm{.}1.$

Explain which of the points in $2\mathrm{.}1$ 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 $2\mathrm{.}1$ are correctly classified by this AN$.($Note that you may answer this question on a scale of "one, two, many." That is$,$ if all of the points are not correctly classified after two more passes through the data, you may stop at a total of three passes and report that it still had not correctly classified all of the points after three passes.$)$ Show your work.

Explain the likely effect of introducing a learning rate parameter $(\backslash$eta $)$ into the equation given.