Activation Function

Neuron Structure

An artificial neuron is composed of an adder and an activation function. The input signals (represented by x in the figure) are multiplied by each weight W. Then, the summation y₁ is applied to the activation function f(y₁) to produce the output z₁.

Problem 1

Indicate what is the purpose of the weights? For instance, what would happen, if one weight is very big compared to the other weigths?

Activation Function

The activation function must be a sigmoid (has an S shape), real, continuous, limited and have a positive derivative. Two of the most popular activation functions (for multilayer neural networks) are the hyperbolic tangent (z = tanh(y)) and the logistic function (z = logsig(y)). The ANN shown below has three inputs and one output.

Bias

Each neuron has a fixed input called Bias that always has a value of one. The bias allows the neuron to increase its learning flexibility by shifting the value of y as required for the specific conditions of the problem.

Problem 2

What would happen if the ANNs did not have any Bias (fixed input)? For instance, suppose that y must be five, when all the inputs are zero.

Problem 3

Write the Neural Lab code to plot the logistic function: z = logsig(y).

Solution 3

open Neural Lab and click on

. Set the project name to Logsig and select the Main file only option. Then write the code show below. The code begins by creating a vector y of length 100 with values from -10.0 to 10.0. Then, another vector z is computed using the equation of the logsig function. The code ends by creating a XyChart that will be displayed and saved to disk.

press the button to run the code. If you do not have any errors, the variables y and z will be displayed in the variable list and the graph will be displayed.

Logsig\Main.lab

Vector y;
y.CreateSeries(-10.0, 10.0, 100);
Vector z;
z = 1.0 / (1.0 + exp(-1.0*y));
//________________________________________ Create XY Chart
XyChart chart;
chart.AddGraph(y, z, "logsig", 2, 1, 0, 255, 0);
chart.SetLimits(-10.0, 10.0, 0, 1.0);
chart.SetColorMode(2);
chart.SaveAndShow();

Problem 4

Write the Neural Lab code to plot the tanh function: z = tanh(ay) . Use a value of 1.5 for the constant a. Call your project Tanh.

Escriba un códido de Neura Lab para graficar la función tanh: z = tanh(ay) . Use un valor de 1.5 para la constante a. Llame a su proyecto Tanh.

Problem 5

Indicate the main similarities and differences between the functions: logsig and tanh.

Problem 6

Indicate whether the next statement is true or false. When the neurons of an ANN use the logsig function, the output values of the network cannot exceed a value of one no matter what input is applied to the network. A value outside the range [0 1] is impossible to attain at the output of the ANN. If the required range is bigger than from zero to one, then the output signal must be scaled appropriately.

Problem 7

Indicate whether the next statement is true or false. When the neurons of an ANN operate using the tanh function, it is very important to note that the output signal is limited to the range from -1 to 1. The tanh function has a shape very similar to the shape of the logsig function. However, the lower range for the logsig function is zero when compared with the lower range of the tanh which is -1.

Tip

Always scale the output of an ANN using the logsig function to efficiently use the output range (0 to 1). This means that if there are some occasional big values that will force the scaling to attenuate severely the output signal, these values must be clipped before scaling. For instance, to map the weight of a person, zero would map to the lowest people weight and one should map to the highest people weight. However, if very few people have a very high weight or a very low weight, this range will be underused.

Tip

In some applications, it is very important to compute quickly the activation function of an artificial neuron. This can be accomplished by:

Building a lookup table with the values of the activation function for a set of values uniformly distributed
Building a lookup table with the values of the activation function derivative for a set of values uniformly distributed
Making use of the properties of the activation function. For instance, if the input value is bigger than 5, the activation function is one. If the input value is less than -5, the activation function is 0 for the logsig(y) and -1 for the tanh(y) .

Problem 8

Show that the derivative of the logsig function z=logsig(y) can be expressed as shown below.
Open the project in Problem 3 and click on the "Add File" button on the toolbar to add a new file called Derivative. Edit the file to plot the derivative of the logsig function using 1000 points. Observe that the derivative is expressed as a function of z. Thus, you must first compute the value of z=logsig(y) in order to compute the value of the derivative.
Show that the inverse of the logsig function z=logsig(y) can be expressed as shown below.
Click on the "Add" button on the toolbar to add a new file called Inverse. Edit the file to plot the inverse of the logsig function using 1000 points. Observe that you should handle separately the values for z = 0 and z = 1

Logsig\Inverse.lab

Vector z;
z.CreateSeries(0, 1, 1000);
Vector y;
y.Create(1000);
int i;
for (i = 0; i < 1000; i++)
{
     if (z[i] == 0.0)
     {
          . . .
}
//________________________________________ Create XY Chart
XyChart chartInv;
. . .

Problem 9

Show that the derivative of the tanh function z=tanh(ay) can be expressed as shown below.
Open the project in Problem 4 and click on the "Add File" button on the toolbar to add a new file called Derivative. Edit the file to plot the derivative of the tanh function using 1000 points and a = 1.5. Observe that the derivative is expressed as a function of z. Thus, you must first compute the value of z=tanh(ay) in order to compute the value of the derivative.
Show that the inverse of the tanh function z=tanh(ay) can be expressed as shown below.
Click on the "Add" button on the toolbar to add a new file called Inverse. Edit the file to plot the inverse of the tanh function using 1000 points and a = 1.5. Observe that you should handle separately the values for z = −1 and z = 1

Problem 10

Compute manually the output value of the neural network shown below. Suppose that the neuron is using the logsig function.

Problem 11

Repeat the last problem using Neural Lab code.

Solution 11

click the button to create a new project. Set the project name to ComputOutput and select the Main File only option. Write the code shown below.

click the button to execute the code. If you do not have any errors, the variables x, W, y and z will be displayed in the variable list.

ComputeOutput\Main.lab

//_________________________________________ Main File
Vector x;
x.Create(4);
x[0] = 7.0;
x[1] = 21.0;
x[2] = -10.0;
x[3] = 1.0;

Matrix W;
W.Create(1, 4);
W[0][0] = -2.0;
W[0][1] = 4.5;
W[0][2] = 8.2;
W[0][3] = 0.5;

Vector y = W*x;
double z = 1.0 / (1.0 + exp(-1.0*y[0]));

Problem 12

Find an expression for the output value z₁ of the neural network shown below. The ANN has n inputs, one neuron in the output layer and one output. Suppose that the neuron uses the logsig function.

Problem 13

Find an expression for the output value of the neural network shown below. The ANN has two inputs, two neurons in the output layer and two outputs. Suppose that the ANN is using the logsig function.

Problem 14

Compute the output value for the neural network shown below (perform the computation manually or use Microsoft Excel). The ANN has two inputs, two neurons in the hidden layer 1, one neuron in the output layer and one output. Suppose that the ANN is using the logsig function.

Problem 15

Repeat the last problem using Neural Lab code your project name must be TwoLayer. After running the program, double click on the z to see the value of the output of the network.

Problem 16

Search over the Internet for other activation functions besides the logsig and tanh.

Problem 17

An ANN was implemented by hardware as shown. A power failure damaged the weight w₁₂. Before the power failure problem, the output network was 0.92129 when the input x was applied. Compute the value of the weight w₁₂ for the network to operate normally. Suppose the activation function is the logsig.

Problem 18

An ANN was implemented by hardware as shown. A power failure damaged the weights w₁₁ and w₁₂. Before the problem, the output network was 0.539915 when the first column of x was applied, and 0.327393 for the second column. Compute the values of the weights w₁₁ and w₁₂ for the network to operate appropriately.

Problem 19

The network of the figure has two hidden layers. (a) Compute the size of the matrices: W1, W2 and W3. (b) Compute the number of weights of this ANN.

Rectified Linear Units

The Rectified Linear Unit or ReLU is a recent activation function using in deep learning networks for the hidden layers. When the input is positive, the output of the ReLU function is the same as the input. When the input is negative or zero, the output of the ReLU function is zero. Research has shown that the ReLU activation function may result in a fast training for deep networks. Unfortunately, neurons using the ReLU activation function may become inactive affecting the performance of the network.

Problem 20

Write the Neural Lab code to plot the ReLU function using 200 points. Call your project Rectified.

Problem 21

Write a small program in your favorite programming language to compute the output of the artificial neural network using an if else statement. Call your program ReLuOutput. Suppose that the ANN is using the ReLU function.

Activation Function

Neuron Structure

Activation Function

Bias

Rectified Linear Units

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