# Fraud Detection Using Neural Networks: A Case Study of Income Tax
## 3 Methodology
### Model Architecture  
In Artificial Neural Networks, just as it is critical to have high-quality data, it is also critical to select appropriate parameters for the problem at hand. The activation function is one of the most important parameters because it plays a significant role in aggregating signals into the output signal that is propagated to the other neurons in the network, as well as processing information and passing it throughout the network. Figure 1: A neuron structure and sample ANN Multi-layer perceptron structure [21; 22]. (**a**) A neuron structure; (**b**) Sample ANN structure. According to [25], without an activation function, any ANN is just a simple linear function. Despite the fact that linear equations are simple and straightforward to solve, they are limited in their complexity, and they lack the ability to learn and discover complicated data mappings. Most of the time, a neural network without an activation function acts as a linear regression model with low performance and capacity. It is preferable for a neural network to perform tasks other than learning and computing a linear function, such as modelling complex tasks. Sigmoid, ReLu, softmax, softsign, linear, hard-sigmoid, softplus, and others were tested using the grid search to determine which activation function is best suited for income tax fraud detection. Sigmoid is the most commonly utilized activation function because it is a non-linear function that may convert and squash numbers in the range of 0 to 1. It is critical to have a sigmoid activation function on the output layer in cases of binary classification [23]. A rectified linear unit (ReLU), also known as a piecewise linear function, works in such a way that it will output the input directly if it is positive, or zero otherwise. This function is widely used in neural networks and is widely assumed to be more efficient than others since neurons are not activated all at once, but instead, a subset of neurons are activated at a time. It is recommended that this function be tried first because numerous studies have shown that it performs well in a variety of tasks [23; 26]. Softmax is a function that converts numbers into probabilities. A Softmax's output is a vector containing the probabilities of each possible outcome. For all possible outcomes or classes, the probabilities in a vector sum to one. It is widely used as a neural network's final activation function to standardize network output to a probability distribution over expected output classes. We can consider sigmoid function values as probabilities of data points from a certain class because we know they range from 0 to 1. The softmax function, unlike sigmoid functions, can be used for multiclass classification tasks [23; 26]. The softsign function has more robust synchronization than softmax due to its smoother asymptotic line and comparatively slow and soft saturation. The softsign function activation value is distributed uniformly in a large number of nonlinear yet good area gradient flows. Because the softsign activation function is nonlinear, it detects errors more effectively [27]. Among other commonly used activation functions is a hard-sigmoid. This activation function has a lower computation cost than a sigmoid activation function. It has shown promising results on binary classification tasks when implemented. A hard sigmoid is a sigmoid activation function generalization. It is almost linear, so it is much faster and less expensive to compute than a standard sigmoid activation function [28]. Finally, the softplus activation function was used; this function has several advantages, including being smooth in the definition domain. This property makes the softplus function more stable regardless of whether it is estimated in the positive or negative direction. Another benefit is that the softplus unit has a non-zero gradient even when the unit's input is negative. Gradients can be propagated by the softplus function across all inputs [29].