Multilayer Perceptron (MLP) in Time Series Forecasting: Unveiling Trends, Predicting the Future, and New Insights from Data Mining

# 1. Fundamentals of Time Series Forecasting

Time series forecasting is the process of predicting future values of a time series data, which appears as a sequence of observations ordered over time. It is widely used in many fields such as financial forecasting, weather prediction, and medical diagnosis.

Time series data has the following characteristics:

- **Trend:** Data values show an overall trend over time, which can be increasing, decreasing, or stable.
- **Seasonality:** Data values exhibit predictable patterns at specific times within a year.
- **Cyclicity:** Data values repeat predictable patterns at specific time intervals, which may be longer than seasonal patterns.

# 2. Theoretical Foundations of Multilayer Perceptrons (MLP)

### 2.1 Structure and Working Principle of MLP

#### 2.1.1 Neuron Model

A Multilayer Perceptron (MLP) is a feedforward neural network consisting of multiple layers of neurons. Each neuron receives input signals, performs a weighted sum, and applies a nonlinear activation function to produce an output signal.

The mathematical expression of the neuron model is as follows:

output = activation_function(weight * input + bias)

Where:

* `input`: The input signal to the neuron
* `weight`: The weight of the neuron
* `bias`: The bias of the neuron
* `activation_function`: The activation function of the neuron

Commonly used activation functions include:

* Sigmoid: `activation_function(x) = 1 / (1 + exp(-x))`
* Tanh: `activation_function(x) = (exp(x) - exp(-x)) / (exp(x) + exp(-x))`
* ReLU: `activation_function(x) = max(0, x)`

#### 2.1.2 Network Architecture

An MLP consists of an input layer, an output layer, and multiple hidden layers. The input layer receives input data, and the output layer produces the forecasted results. The hidden layers are responsible for extracting features from the input data and performing nonlinear transformations.

The network architecture of an MLP can be represented as:

Input Layer -> Hidden Layer 1 -> Hidden Layer 2 -> ... -> Hidden Layer N -> Output Layer

Where:

* Input Layer: Receives raw input data
* Hidden Layer: Extracts features and performs nonlinear transformations
* Output Layer: Produces forecasted results

### 2.2 MLP Learning Algorithm

#### 2.2.1 Backpropagation Algorithm

The backpropagation algorithm is a commonly used method for training MLPs. This algorithm calculates the gradient of the loss function and uses the gradient descent method to update the weights and biases of the neurons to minimize the loss function.

The steps of the backpropagation algorithm are as follows:

1. Forward Propagation: Compute the output of the neural network
2. Loss Function Calculation: Compare the output with the target values for difference
3. Backpropagation: Calculate the gradient of the loss function with respect to each weight and bias
4. Update Weights and Biases: Use the gradient descent method to update weights and biases to minimize the loss function

#### 2.2.2 Optimization Alg***

***monly used optimization algorithms include:

* Gradient Descent: `weight -= learning_rate * gradient`
* Momentum Method: `velocity = momentum * velocity + learning_rate * gradient`
* RMSprop: `velocity = decay_rate * velocity + (1 - decay_rate) * gradient**2`
* Adam: `velocity = momentum * velocity + learning_rate * gradient / sqrt(decay_rate * velocity**2 + epsilon)`

### 2.3 Hyperparameter Tuning of MLP

#### 2.3.1 Number of Layers and Neurons

The number of layers and neurons is an important hyperparameter for MLPs. The more layers, the more complex the model and the stronger the ability to extract features. The more neurons, the better the model can fit complex data.

#### 2.3.2 Activat***

***monly used activation functions include ReLU, Tanh, and Sigmoid. Regularization methods include L1 regularization and L2 regularization.

# 3.1 Time Series Data Preprocessing

Time series data preprocessing is a crucial step in time series forecasting, which can effectively improve the prediction accuracy of the model. It mainly includes data cleaning and normalization, feature engineering.

#### 3.1.1 Data Cleaning and Normalization

**Data Cleaning**

The main purpose of data cleaning is to remove outliers, missing values, and noise from the data. Outliers are values that significantly deviate from the normal data range, possibly caused by sensor failures or data recording err