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# Sigmoid Activation Function Tutorial

In this tutorial, we will learn about the sigmoid activation function. The sigmoid function always returns an output between 0 and 1.

## After this tutorial you will know:

- What is an activation function?
- How to implement the sigmoid function in python?
- How to plot the sigmoid function in python?
- Where do we use the sigmoid function?
- What are the problems caused by the sigmoid activation function?
- Better alternatives to the sigmoid activation.

### What is an Activation Function?

An activation function is a mathematical function that controls the output of a neural network. Activation functions help in determining whether a neuron is to be fired or not.

### Popular Activation Functions

- Binary Step
- Linear
- Sigmoid
- Tanh
- ReLU
- Leaky ReLU
- Softmax

Activation is responsible for adding non-linearity to the output of a neural network model. Without an activation function, a neural network is simply a linear regression.

### Sigmoid Activation Function Formula

The mathematical equation for calculating the output of a neural network is:

**Activation Function**

In this tutorial, we will focus on the sigmoid activation function. This function comes from the sigmoid function in maths.

#### The Formula for the Sigmoid Activation Function

Mathematically you can represent the sigmoid activation function as:

**Formula**

You can see that the denominator will always be greater than 1, therefore the output will always be between 0 and 1.

### Implementing the Sigmoid Activation Function in Python

In this section, we will learn how to implement the sigmoid activation function in Python.

import numpy as np def sig(x): return 1/(1 + np.exp(-x))

Let’s try running the function on some inputs.

import numpy as np def sig(x): return 1/(1 + np.exp(-x)) x = 1.0 print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x))) x = -10.0 print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x))) x = 0.0 print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x))) x = 15.0 print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x))) x = -2.0 print('Applying Sigmoid Activation on (%.1f) gives %.1f' % (x, sig(x)))

#### Output:

- Applying Sigmoid Activation on (1.0) gives 0.7
- Applying Sigmoid Activation on (-10.0) gives 0.0
- Applying Sigmoid Activation on (0.0) gives 0.5
- Applying Sigmoid Activation on (15.0) gives 1.0
- Applying Sigmoid Activation on (-2.0) gives 0.1

### Plotting Sigmoid Activation using Python

To plot sigmoid activation we’ll use the Numpy library:

import numpy as np import matplotlib.pyplot as plt x = np.linspace(-10, 10, 50) p = sig(x) plt.xlabel("x") plt.ylabel("Sigmoid(x)") plt.plot(x, p) plt.show()

#### Output:

**Sigmoid**

We can see that the output is between 0 and 1.

The sigmoid function is commonly used for predicting probabilities since the probability is always between 0 and 1.

One of the disadvantages of the sigmoid function is that towards the end regions the Y values respond very less to the change in X values.

This results in a problem known as the vanishing gradient problem.

Vanishing gradient slows down the learning process and hence is undesirable.

### Alternatives to Sigmoid Activation Function

#### ReLu Activation Function

A better alternative that solves this problem of vanishing gradient is the ReLu activation function.

The ReLu activation function returns 0 if the input is negative otherwise return the input as it is.

Mathematically it is represented as:

**Relu**

You can implement it in Python as follows:

def relu(x): return max(0.0, x)

Let’s see how it works on some inputs.

def relu(x): return max(0.0, x) x = 1.0 print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x))) x = -10.0 print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x))) x = 0.0 print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x))) x = 15.0 print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x))) x = -20.0 print('Applying Relu on (%.1f) gives %.1f' % (x, relu(x)))

#### Output:

- Applying Relu on (1.0) gives 1.0
- Applying Relu on (-10.0) gives 0.0
- Applying Relu on (0.0) gives 0.0
- Applying Relu on (15.0) gives 15.0
- Applying Relu on (-20.0) gives 0.0

#### Leaky ReLu Activation Function

The leaky ReLu addresses the problem of zero gradients for negative values, by giving an extremely small linear component of x to negative inputs.

Mathematically we can define it as:

- f(x) = 0.01x, x < 0
- f(x) = x, x >= 0

You can implement it in Python using:

def leaky_relu(x): if x > 0 : return x else : return 0.01*x x = 1.0 print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x))) x = -10.0 print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x))) x = 0.0 print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x))) x = 15.0 print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x))) x = -20.0 print('Applying Leaky Relu on (%.1f) gives %.1f' % (x, leaky_relu(x)))

#### Output:

- Applying Leaky Relu on (1.0) gives 1.0
- Applying Leaky Relu on (-10.0) gives -0.1
- Applying Leaky Relu on (0.0) gives 0.0
- Applying Leaky Relu on (15.0) gives 15.0
- Applying Leaky Relu on (-20.0) gives -0.2

## Conclusion

This tutorial was about the Sigmoid activation function. We learned how to implement and plot the function in python.

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