Consider again the same $10$ instances.

P$1=(4,1)$

P$2=(2,1)$

P$3=(2,2)$

P$4=(5,5)$

P$5=(3,2)$

N$1=(3,4)$

N$2=(1,1)$

N$3=(3,6)$

N$4=(3,1)$

N$5=(2,3)$

A neural network comprising of $2$ input neurons $($for x and y values$),2$ hidden layer neurons $($h$1$ and

h$2),$ and a single output neuron, was trained to classify the data. The hidden and output neurons use

the threshold activation function, which outputs either $0($for a negative instance$)$ or $1$ for a positive

instance.

y

x

h$2$

h$1$

o$1$

wh$2$

$2=1$

wh$1$

$2=-1$

wh$2$

$1=3$

wh$1$

$1=1$

wo$1$

$2=2$

wo$1$

$1=2$

wh$2$

$0=-3$

wh$1$

$0=1$

wo$1$

$0=-3$

Figure $3$: Three$-$layer neural network with weights and bias values.

$6$

COS$4852/$A$2$

i$)$ Calculate the output for each of the three neurons h$1,$ h$2,$ and o$1$ when N$3$ is given as input to

the network. Use the weights and bias values as shown in Figure $3.$

ii$)$ Does the network classify N$3$ correctly?

iii$)$ What do you observe about the outputs of h$1$ and h$2$ when N$3$ is given as input? Could the

network still correctly classify N$3?$ Explain in detail.

iv$)$ Draw the decision boundaries for each of the three neurons, h$1,$ h$2,$ and o$1.$ Show all your

steps and calculations. Explain what you did to arrive at the diagram for o$1.$