What are the responses of the output unit Z with respect to various input combinations?
Requirements:
Consider the McCulloch-Pitts neural network shown in Fig. All the units, except those at the
input level, have the activation function
What are the responses of the output unit Z with respect to various input combinations? We assume
the inputs are binary. What logical function the whole network realizes?
Ans.
2. We need to design a simple neural network to realize a two input AND function and one output unit. A
bias of 0.4 is used. The inputs and outputs are in bipolar form. The structure of the required neural net is as follows.
If Hebb’s Learning Rule is used what will be the values of the weights after the completion of a single epoch. Explain the entire process in brief along with a suitable truth table(If the initial weights are taken as 0.2 each).
During the training process all the weights are initialized to :
w0=w1=w2=0.2—————–(i)
At each training instance, the weights are changed according to the following formula:
wiNew= wiOld+wi———————(ii)
The increment is computed as following according to Hebb’s Rule:
wi=xi * t , where t is the target output.
Let the inputs be x0, x1 and x2. The output of the summation unit is calculated as follows:
y_ini= x0i*w0i+ x1i*w0i + x2i*W2i————(iii)
y_outi=ϕ(y_ini)———————(iv)
3. A student wants to train a perceptron to realize the logical AND function using input and outputs in the bipolar form. The structure of the perceptron is as given below. The activation function
for the output unit is :
The student considers the initial weight to be 0. The learning rate is also initialized by the student as 1. The student needs to find the final weights after the application of the perceptron learning rule at the end of an epoch.
4. A student wants to train a perceptron to realize the logical AND function using input and outputs in the bipolar form. The structure of the perceptron is as given below. The activation function
for the output unit is :
The student considers the initial weight to be 0.3. Learning rate is also initialised by the student as 0.2. The student needs to find the final weights after the application of the LMS learning rule at the end of an epoch.
5. A researcher is working on his first neural network to implement multilayer perceptron. There are two units in the Input Layer, two units in the Hidden Layer and two units in the Output Layer. The w1,w2,w2,…,w8 represent the respective weights. b1 and b2 are the biases for Hidden Layer and Output Layer, respectively. The network and the corresponding values are as follows:
Now we pass this weighted sum through the logistic function (sigmoid function) so as to squash the weighted sum into the range (0 and +1). The logistic function is an activation function for our example neural network.
The researcher wants to find out the total error at the end of one epoch in the network and the updated weights of w1 and w5 at the end of the backward pass. He considers sigmoid function as the activation function for the network he has designed.
6. A researcher is working on his first neural network to implement a multilayer perceptron. There are two units in the Input Layer, two units in the Hidden Layer and two units in the Output Layer. The w1,w2,w2,…,w8 represent the respective weights. b1 and b2 are the biases for Hidden Layer and Output Layer, respectively. The network and the corresponding values are as follows:
The researcher wants to find out the total error at the end of one epoch in the network and the updated weights of w1 and w5 at the end of the backward pass. He considers sigmoid function as the activation function for the network he has designed.
Input values
X1=0.05
X2=0.10
Initial weight
W1=0.15 w5=0.40
W2=0.20 w6=0.45
W3=0.25 w7=0.50
W4=0.30 w8=0.55
Bias Values
b1=0.35 b2=0.60
Target Values
T1=0.01
T2=0.99
Now, we first calculate the values of H1 and H2 by a forward pass.
Forward Pass
To find the value of H1 we first multiply the input value from the weights as
H1=x1×w1+x2×w2+b1
H1=0.05×0.15+0.10×0.20+0.35
H1=0.3775
To calculate the final result of H1, we performed the sigmoid function as
We will calculate the value of H2 in the same way as H1
H2=x1×w3+x2×w4+b1
H2=0.05×0.25+0.10×0.30+0.35
H2=0.3925
To calculate the final result of H1, we performed the sigmoid function as
Now, we calculate the values of y1 and y2 in the same way as we calculate the H1 and H2.
To find the value of y1, we first multiply the input value i.e., the outcome of H1 and H2 from the weights as
y1=H1×w5+H2×w6+b2
y1=0.593269992×0.40+0.596884378×0.45+0.60
y1=1.10590597
To calculate the final result of y1 we performed the sigmoid function as
We will calculate the value of y2 in the same way as y1
y2=H1×w7+H2×w8+b2
y2=0.593269992×0.50+0.596884378×0.55+0.60
y2=1.2249214
To calculate the final result of H1, we performed the sigmoid function as
Our target values are 0.01 and 0.99. Our y1 and y2 value is not matched with our target values T1 and T2.
Now, we will find the total error, which is simply the difference between the outputs from the target outputs. The total error is calculated as
So, the total error is
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