Back propagation algorithm takes five inputs. The first input is training example, each training example is a pair of the form \( x, t \), where \( x \) is a vector of network input values and \( t \) is the vector of target network output values. \( n \) is the learning rate, it is usually set to a small number. In this case, I have given an example like 0.05. It should be set as small as possible, and if you want to converge to the output faster, in that case, you need to set the higher number. \( n_i \) is the number of network inputs, \( n_{hidden} \) is the number of units in the hidden layer, and \( n_{out} \) is the number of output units. The input from unit \( i \) into unit \( j \) is denoted by \( x_{ji} \) in the back propagation algorithm. Similarly, the weight from unit \( i \) to unit \( j \) is denoted as \( w_{ji} \). This is the entire algorithm of the back propagation algorithm: 1. First, we need to create a feed-forward network with \( n_i \) inputs, \( n_{hidden} \) hidden units, and \( n_{out} \) output units. 2. After that, we need to initialize all the network units to a small random number. So first, we will see how to create a feed-forward network with these number of units. For understanding purpose, I have taken ni as equivalent to 3, that is, 3 units in the input layer, n hidden as equivalent to 2, that is, two hidden layer units, and n out as equivalent to 2, that is, 2 output layer units. In this case, x1, x2, x3 are the inputs, and 1, 2, 3, 4, 5, 6, 7 are the units in this case. Next, we need to assign the weights as well as the inputs here. The input from unit 1 to unit 4 is denoted as x4 1, that is nothing but x1 in this case. So, x1 is known to us; that is denoted as x4 1 in back propagation algorithm. Similarly, the weight from 1 to 4 is denoted as w4 1. With the same note, the input from unit 2 to unit 4 is denoted as x4 2, and the weight is denoted as w4 2 in this case. The same thing goes with all the edges over here. Now we need to assign some input and weight to these particular edges. Also, the weight from unit 4 to unit 6 is w64. Similarly, input is x64. The same thing goes with all the edges here. Now, in this case, so we know what is the value of x1, the same thing is assigned to x4 1, x5 1, and we know the value of x2, that is assigned to x4 2 and x5 2. x3 is known to us; that is assigned to x4 3 and x5 3. But we don't have any value or direct input to this unit 4 and 5, so we don't have any values to x6 4, x7 4, x6 5, and x7 5. So, what we do here is we will calculate the output of these units, and those outputs will be given as an input to this particular sixth and seventh unit, and finally, we will come up with the output here. That is, o6 and o7 are the outputs over here. Now, this is how we can construct the feed-forward network. Once you construct it, for every weight, we need to assign some random weights. You can assign any value in the range of minus one to plus one; it should be a very small number. Randomly, we need to assign the weights, and in every iteration, we go on modifying these particular weights unless and until we are happy with these particular outputs here. Next is, we need to perform these particular four steps until the termination condition is satisfied or met. That is, the termination condition may be the number of epochs. You need to perform these particular four steps, let us say, some 50 number of times, that is the number of epochs, or you want to perform these particular four steps, let us say, unless and until you are satisfied with the error rate. Maybe, let us say, that you are expecting an error of less than five percent. Unless and until you have an error more than five percent, you need to perform these steps. Once you achieve the error, which is less than five percent, you can stop it. So, that is one example here. Now, what to do in this particular step is, for each x, t, that is, for each training example, we need to perform these steps. So, what is that we do? We propagate the input forward through the network. That is, input the instance x to the network and compute the output ou for every unit u in the network. That is, you need to give each instance to the network and you need to calculate the output of every unit in the network. I'll just show you that particular thing, how it will be calculated. First, what we need to do is, we have these input and weight vectors. Input is directly given from the instances, and the weight is assigned randomly. Here, we need to take these particular things, and then we need to calculate the output of the fourth unit and fifth unit, that is, the first unit of the hidden layer and the second unit of the hidden layer. So, how to calculate it? To calculate this particular thing, you need to take the summation of a linear combination of weight and input here. That is, first to this particular node, that is, the fourth node, we have three edges coming here. That is, one edge is coming from 1, one edge is coming from 2, as well as one edge is coming from 3. So, we will take the linear combination of weight and input and then we will take the sum of those things. Finally, we apply a sigma term here. So, how it is done is, x4_1 multiplied by w4_1 + x4_2 * w4_2 + x4_3 * w4_3, that is the summation term. Once you get the summation term, we need to apply the sigma term here. Sigma of x is always equivalent to 1 divided by 1 + e raised to the power of -x, that is the sigma term. Once you apply that particular thing, you will get the output at this particular node.