Introduction to Artificial Intelligence $($CIS$316)$

Tutorial

Machine Learning $-$ Neural Networks

Exercise $1.$

The input to a single input perceptron is $2\mathrm{.}0,$ its weight is $2\mathrm{.}3$ and its bias $($threshold$)$ is $-3.($a$)$ What is the net input to the transfer function? $($b$)$ What is the output if the transfer function is a $($i$)$ a step function $($ii$)$ a sigmoid function? $($Answers: $1,0\mathrm{.}8320)$

Exercise $2.$

Given a two$-$input neuron with the following parameters: bias, weight and input vectors are: and, calculate the neuron output for the transfer functions $($a$)$ step function $($b$)$ sigmoid function. $($Answer: input to the transfer function is $-1\mathrm{.}8)$

Exercise $3.$

A single layer neural network is to have six inputs and two outputs. The outputs are to be limited to and continuous over the range $0$ to $1.($a$)$ How many neurons are required? $($b$)$ Draw the neural network, showing the interconnections $($c$)$ What are the dimensions of the weight matrix? $($d$)$ What kind of transfer function can be used?

Exercise $4.$

Design single$-$perceptron network $($manually$)$ for the following three classification problems by finding weight and bias values for the perceptron. Assume step transfer function.

Exercise $5.$

Use perceptron training rule to train a perceptron network to classify the following input vectors correctly. Apply each input vector in order, and repeat as many times as is required to train the network.

$,,,$

Use the initial weights and bias: and $b=0.$

$($Answer: $)$

Exercise $6.($Not covered$)$

A classification problem has four classes of input vector shown below. Design a perceptron network for it$.$ Class $1$: $,$ class $2$:$,$ class $3$: $,$ class $4$:

$($Answer: $)$

Exercise $4.($Solution Hint$)$

First we draw a line between each set of dark and light data points.

The next step is to find the weights and biases. The weight vectors must be orthogonal to the decision boundaries, and pointing in the direction of

points to be classified as $1($the dark points$).$ The weight vectors can have any length we like.

Here is one set of choices for the weight vectors:

$()1()1()1$

Now we find the bias values for each perceptron by picking a point on the decision boundary and satisfying Eq$.(4\mathrm{.}15)$

$w^{T}p+b=0$

$b={?}_{-1}{w}^{T}p$

This gives us the following three biases:

$(a)b=-\left[\begin{array}{cc}-2& 1\end{array}\right]\left[\begin{array}{c}0\\ 0\end{array}\right]=0,(b)b=-\left[\begin{array}{cc}0& -2\end{array}\right]\left[\begin{array}{c}0\\ -1\end{array}\right]=-2,(c)b=-\left[\begin{array}{cc}2& -2\end{array}\right]\left[\begin{array}{c}-2\\ 1\end{array}\right]=6$