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Single layer classification example

Problem statement

In this example, we consider a dataset where each input vector \( X = ( x , y) \) is assocated to a class A or B respectively with values of +1 and -1. The following figure illustrates the classification problem:

Single layer ANN classification dataset

Network architecture

The single layer architecture is the following:

Architecture of the single layer neural network

As we need to distinguish class A from class B, we have to use an activation function that can separate classes. In this example, hyperbolic tangent has been selected:

Hyperbolic tangent transfert function of the neural network

The choice of hyperbolic tangent is motivated by the fact that this function output a value between -1 and +1. Output can be interpretated in two ways, in terme of binary classes (A or B) or in term of probabilities.

Binary interpretation

To determine if the sample belongs to class A or B, ones can specify the following rule: positive outputs belongs to class A, while negative to class B. Mathematicaly, we add the following function after the output of the network:

Probabilistic interpretation

The second option to interpret the output of the network is to consider it as a probability to belonging to classes A or B. When the output is equal to +1, the probability for the sample to be classify in class A or B is respectively one and zero. The following equations generalize this concept, and convert the network output into probablities:

Probablity to be in class A:

$$ p_A = \frac{o+1}{2} $$

Probablity to be in class B:

$$ p_B = \frac{1-o}{2} $$

Note that sum of probablities is always equal to one ( \( p_A + p_B = 1\) ).

Results

The following figures shows how the space is splitted to separate classes:

Single layer classification results

The following figures is an overview of training results.

3D chart of the classification result for the single layer ann

Source code and download

See also


Last update : 03/11/2020