Deep-Learning-Powered GRU Model for Flight Ticket Fare Forecasting

Abstract

Forecasting flight fares is a critical task in the rapidly expanding civil aviation industry and involves numerous factors. However, traditional airfare prediction systems are ineffective due to the complex and nonlinear relationships of multiple factors, which are not able to accurately account for the impact of different attributes such as time period. To tackle these issues, in this study, we proposed a novel approach that utilizes a deep-learning model, specifically, the Gated Recurrent Unit (GRU), by incorporating 44 decision features. The proposed model is able to capture the intricate relationships between various factors effectively and predict air ticket prices with high accuracy. In the experiments, it was found that the GRU model significantly outperforms not only classic machine learning models but also the MLP and LSTM in terms of assessment indicators of mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R²). The GRU model is thus promising concerning the fare prediction of flight tickets.

1. Introduction

Finance is critical in our daily economic activities, and accurate financial forecasts may help economic growth [1,2,3,4]. However, the modern global market is complicated, chaotic, and dynamic and may be influenced by diverse factors; thus, price prediction becomes a challenging task [4,5]. Recently, some studies have been conducted to address this issue, with a focus on machine learning techniques [6,7]. Approaches to price prediction are inclined toward machine learning (ML) as machine learning algorithms can not only learn to identify patterns or correlations within training data but also achieve excellent predictive capability for unseen data. Machine learning techniques have been used to predict customer behavior [8], market trends, and subsequent prices based on historical data [9,10,11]. On the other hand, decision-making and decision support in business are other essential issues in finance but are not easy to address since non-optimal short-term decisions are sometimes preferred or required in order to achieve long-term strategic goals [3,12]. ML models have also been proposed to optimize business policies to make sequential decisions toward a long-term objective [13].

Due to the excellent performance and nobility of deep learning [9,14,15], it has recently attracted much attention in computer vision, natural language processing, and finical prediction [15,16]. Deep learning research has undergone rapid development in recent years and has been successfully applied to a wide variety of areas [17]. It has achieved good performance in quantitative investment, risk control, and other areas [18]. For price prediction, various deep learning algorithms have been used, and RNN is one of them [19]. RNN is well-suited to time series forecasting since it can process variable-length sequences of incoming data in its internal memory. RNN-based models have been successfully applied to the predictions of future prices [20], and long short-term memory (LSTM) and gated recurrent unit (GRU) are the top options [18,21]. They can also be used together with conventional ML-based prediction models, such as decision trees, genetic algorithms [22], and support vector machines [23], for the prediction of prices. Notice that the trend exhibited by the flight ticket fare is highly complex and nonlinear [24]. Conventional forecasting methods based on a linear model, such as linear regression, therefore faces a number of challenges. A deep neural network that is able to better model highly nonlinear changes is hence promising for such purposes.

Additionally, financial prediction is complicated due to the fact that there are many features of the series that have varying degrees of influence or importance [13]. Among these features, however, only a few have higher impacts on the trend of the series and convey the most relevant information overall. It is, therefore, necessary for prediction or analysis methods to identify salient and influential features from a large amount of data. For RNN, the gated recurrent unit was shown to be able to prioritize relevant features, allowing for more efficient extraction in short-term mode and preventing data loss due to lengthy time series [25]. Therefore, in this research, we propose the use of the GRU model for RNN, considering its ability to learn to capture correlations between flight ticket fares and relevant features.

On the other hand, a lack of appropriate data is a major impediment to the advancement of studies on projecting future fares of flight tickets [22]. Since ticket prices are extremely sensitive to business interests, they are seldom disclosed, and the vast majority of airlines do not make their pricing plans known to the public. Despite the limited resources available, a few publications have developed a range of methodologies for predicting ticket prices [26,27]. In this research, we used historical flight data from Ethiopian Airlines, which is the oldest and leading airline in Africa, for the purpose of this study. The data ranged from January 2018 to July 2022, and there were more than 1,000,000 records from 22 domestic and international airline destinations. The sufficiency and quality of the data make the analysis results credible.

The key contributions of this study are summarized as follows:

• The results obtained from the GRU model demonstrate a significant impact and provide valuable insights for ongoing research in this area.
• The research introduces a novel approach employing a deep-learning GRU model for fare prediction. This method addresses the limitations of traditional machine learning techniques that rely heavily on statistical variables in their models.
• The GRU model leverages its unique architecture to capture temporal dependencies in flight data, resulting in improved predictive performance.

2. Related Work

With advancements in machine learning, price forecasting techniques have made significant strides in diverse industries [28], such as agriculture, stock, bitcoin, and oil [29,30,31]. Recent studies on flight fare prediction have utilized mainly machine learning approaches. A study discussing the challenge of air ticket fare prediction shows that its variability and dependence on various factors lead to revenue losses and customer dissatisfaction [32]. This research proposed an ensemble model using neural networks, XGBoost regressor, and light gradient boosting machine regression to estimate flight ticket fares with the least mean absolute error. The dataset contained flight ticket data from March to June 2019 for various airlines. Airlines can plan their strategies of operations and allocate necessary resources according to predicted ticket prices so as to impact specific market segments. Another study compares the prediction performances of flight ticket prices based on eight regression models, including multilayer perceptrons, generalized regression neural networks, extreme learning machines, random forest regression trees, and so on [33]. This study aimed to identify the factors that impact airfare, such as departure and arrival times, the amount of free luggage allowed, the days before departure, and so on. The models were trained on a dataset containing 1814 flights for a single international route, and the results indicated that the model “bagging regression tree” achieved the highest accuracy, followed by the random forest model. Additionally, in another study, random forest and multilayer perceptron models were trained individually and combined with assigned weights to form a stacked prediction model [34], which was shown to outperform the individual models. The data used to train the models consisted of 51,000 records for a 7-day round trip nonstop flight with three domestic airlines. Twelve features were extracted from the data, including the airline, flight number, date of purchase, departure date and time, and so on. The stacked prediction model outperformed the random forest and multilayer perceptron models with 4.4% and 7.7% improvement, respectively, based on the R² metric.

Furthermore, a new model was proposed to predict the minimum available price for a particular itinerary, i.e., a specific flight on a given route for a specific departure date [26]. To achieve this, a modified ensemble-based learning algorithm was proposed to learn from past patterns of price changes and make forecasts for future prices. This study utilized a recursive approach to predict future prices sequentially. The model used various features, including prices of the same itinerary, recent itineraries, and itineraries for the same day of the week or month. The experimental results demonstrated that the model performed best on diverse routes, where different flights have independent pricing behaviors. A comparison of the algorithms showed that the proposed model based on ensemble learning was superior to KNN.

Table 1. Summary of related works on the prediction of flight ticket fares.

Ref.	Methods	Features	Dataset	Metrics
Tuli et al. [32]	linear regression, support vector regressor, k-neighbor regressor, decision tree regressor, bagging regressor, XGBoost regressor, light gradient boosting machine, extra tree regressor, and artificial neural network	airline, date of journey, source, destination, route, departure time, arrival time, duration, and total stops	10,683 records	mean absolute error and R2
Prasath et al. [35]	k-nearest neighbors	origin, destination, date of departure, time of departure, time of arrival, total fare, airways, and date of departure	not given	root mean square error and R2
Tziridis et al. [33]	multilayer perceptrons, generalized regression neural network, extreme learning machine, random forest regression tree, regression tree, bagging regression tree, regression SVM and linear regression	days before departure, arrival time, amount of free luggage, departure time, number of intermediate stops, holiday, time of day and day of the week, and number of intermediates	1814 flights for an international route	mean square error mean absolute error and R2
Vu et al. [34]	random forest and multilayer perceptrons	departure date, arrival date, departure time, arrival time, fare class, date of purchase, number of stops, price, departure airport, arrival airport, airline, and flight number	51,000 records	mean absolute percentage error and R2
Chen et al. [26]	ensemble-based learning algorithm	Prices of the same itinerary, prices of recent itineraries before the target day, prices of itineraries with the same day of the week, and prices of itineraries with the same date of the month	five international routes for 110 days	mean absolute percentage error

shows the summary of related works.

3. Methodology

3.1. Data Source

A combination of high-quality data sources and effective feature engineering can significantly improve the results of machine learning research and help to achieve accurate and meaningful predictions [23]. In this research, the data were sourced from Ethiopian Airlines. Despite its humble beginnings, Ethiopian Airlines has had a successful 75-year journey that has positioned it as Africa’s leading aviation group [8,36]. The data used in this study covered a period from January 2018 to July 2022 and comprised a total of 1,032,215 records from 22 domestic and international destinations served by the airline. This extensive dataset provided a robust foundation for machine learning research and could yield valuable business insights.

Table 2. A few samples of the flight data.

No.	Origin	Destination	Flight Type
			Domestic	International
1.	Addis Ababa (ADD)	ASO (Asosa)	Yes	No
2.	Djibouti (JIB) Djibouti	Addis Ababa (ADD)	No	Yes
3.	Addis Ababa (ADD)	Entebbe (EBB) Uganda	No	Yes
4.	Hargeisa (HGA) Somalia	Addis Ababa (ADD)	No	Yes
5.	Addis Ababa (ADD)	Hawassa (AWA)	Yes	No
6.	Bahir Dar (BJR)	Addis Ababa (ADD)	Yes	No
7.	Dire Dawa (DIR)	Addis Ababa (ADD)	Yes	No

displays a few samples that contain a part of the data, including the origin, destination, and type of flight.

3.2. Preliminary Data Analysis

Data visualization is essential for interpreting complex datasets, allowing users to make sense of a large amount of information quickly and efficiently [37,38]. The data ranged from January 2018 to July 2022, and there were more than 1,000,000 records from 22 domestic and international airline destinations. As we dove into analyzing this vast dataset, it became crucial to present the findings in an easily interpretable way so as to bring about the insights intuitively. We employed data visualization techniques to reveal patterns, trends, and correlations that would otherwise have been challenging to discern. Figure 1 shows the relationship between travel dates and flight ticket fares (in USD), with six subfigures depicting different aspects of the data. Figure 1a–e display the monthly averages of flight ticket fares for the year 2018 through the year 2022 (up to July), which reveal information regarding price changes over time. Figure 1f further presents a more granular view of the data that depicts the daily fares of flight tickets for the entire dataset. Collectively, these subfigures give an overview of trends and fluctuations concerning prices for the whole dataset.

Additionally, Figure 2 presents a boxenplot of flight fares versus booking classes. The boxenplot is a commonly used statistical tool that displays the distribution of a dataset and highlights such key features as the median, quartiles, and outliers. Unlike a traditional boxplot, the boxenplot divides the data into smaller subgroups, allowing for a more detailed view of the distribution. In this case, we used the boxenplot to examine the relationship between the flight fare and the booking class, which is the letter code used by the airline to identify the fare type and restrictions of a ticket. By analyzing the boxenplot, we were able to identify the ranges of ticket fares as well as the potential outliers or anomalies that might have an impact on our analysis. This initial exploration served as a foundation for our subsequent pilot analysis, allowing us to better understand the data beforehand. Furthermore, the correlations between the features were computed, and a few of them are selected and displayed in

Table 3. A sample of the correlations between features.

	Holiday	Booking Class B	Week Day	Point of Ticket Issuance	Booking Class H	Seg_Dest	Season	Class of Service	Op Flt Num	Actual Fare
Holiday	1	−0.71	0.18	0.75	0.19	0.31	0.51	0.73	0.23	0.97
Booking Class B	−0.71	1	0.21	0.36	−0.05	0.63	0.32	0.67	0.01	0.44
Week Day	0.18	0.21	1	0.30	0.02	0.16	0.01	0.95	−0.01	0.75
Point of Ticket Issuance	0.75	0.36	0.30	1	0.72	−0.06	0.53	0.57	0.02	0. 78
Booking Class H	0.19	−0.05	0.02	0.72	1	−0.16	0.05	0.16	0.08	−0.11
Seg_Dest	0.31	0.63	0.16	0.05	−0.16	1	0.79	0.08	0.51	0.92
Season	0.51	0.32	0.01	0.53	0.05	0.79	1	0.58	0.07	0.67
Class of Service	0.73	0.67	0.95	0.57	0.16	0.08	0.58	1	0.19	0.84
Op Flt Num	0.23	0.01	−0.01	0.02	0.08	0.51	0.07	0.19	1	0.61
Actual Fare	0.97	0.44	0.75	0. 78	−0.11	0.92	0.67	0.84	0.61	1

. As can be observed from Table 3, some features, such as Holiday, Week Day, Seg_Dest, and Class of Service, are highly correlated to the Actual Fare. The preliminary data analysis, consisting of descriptive statistics and correlation techniques, has provided valuable insights into the key factors that impact flight ticket prices. These insights were instrumental in the development of an accurate prediction model.

3.3. Data Preprocessing

To improve the quality of the data, data preprocessing techniques were applied to the raw data collected from the sources. Figure 3 shows the overall data processing stages. Some errors in the raw data were first corrected in the data cleaning stage, and feature engineering was then conducted. Feature engineering involves creating new features from existing data fields to enable the model to better capture underlying patterns within data. Based on the data analysis in the previous section, a few new features, including Season, Travel Day, Travel Month, IsWeekend (Yes/No), IsHoliday (Yes/No), and NumOfStops, etc., were created, as listed in

Table 4. A few selected features used in the fare prediction model.

Features	Description
Travel Date	The date on which the passenger is scheduled to travel
Booking class	The letter code used by the airline to identify the fare type and restrictions of the ticket purchased by a passenger
Class of Service	The class of service for the flight (economy, business)
Seg Orig.	The origin or starting point of the travel segment
Seg Dest.	The destination or endpoint of the travel segment
Distance in Miles	The distance in miles between the origin and destination of the travel segment
Duration in Hours	The duration of the travel segment in hours
Duration in Minutes	The duration of the travel segment in minutes
Airline Code	The code of the airline operating the flight
Total stop	The total number of stops or layovers involved in the travel segment
Op Flt Num	The operating flight number of the airline that is providing the travel segment
Flight Type	The type of flight, such as domestic, international, or connecting
Pax	The number of passengers traveling on the booking
Weekend	Whether the travel segment falls on a weekend or not
Holiday	Whether the travel segment falls on a holiday or not
Season	The season during which the travel segment takes place, such as summer, winter, autumn, or spring
Marketing Airline Code	The code for the airline that is marketing the flight
Point of Ticket Issuance	The location where the ticket was issued
Actual Fare	The actual fare or cost of the travel segment

. These features exhibit specific characteristics of the flight data that influence ticket fares and are meaningful for business decisions. For instance, Season and Travel Month might indicate long-term seasonal demand for flights, while IsWeekend could help to identify the price difference between weekends and weekdays. Similarly, IsHoliday can be used to analyze whether or not prices change during holidays. The feature NumOfStops may be helpful for differentiating prices between direct and connecting flights. Overall, those features provide a comprehensive view of flight data and help build a more reliable prediction model since they are correlated to flight ticket fares.

To prepare the data for the machine learning algorithms, we further applied two data transformation techniques: data encoding and data normalization. In data encoding, the categorical features were converted into numerical representations by one-hot encoding in the form of binary vectors such that the features were acceptable for most ML algorithms. Next, numerical data such as ticket fare were normalized into the region between 0 and 1 according to their dynamic range based on the following formula:

$${\mathrm{x}}^{\prime }=\frac{x-X_{min}}{X_{max}-X_{min}}$$

(1)

where $X_{max}$ and $X_{min}$ denote the maximum and minimum values of the feature, respectively. Data normalization ensures that the numerical features of various flights are on compatible scales so as to have similar impacts on the prediction model. Finally, the preprocessed data were split into training, validation, and testing datasets with 44 decision features. The training dataset was used to train the model, the validation dataset was used to evaluate the performance during training, and the unseen testing dataset was used to assess the generalization capability of the model.

3.4. Long Short-Term Memory

Recurrent Neural Networks (RNNs) have gained popularity in the field of machine learning for their ability to model sequential data. In RNNs, information is propagated through a chain of connected nodes, with the input sequence data processed in the direction of sequence evolution [39,40]. RNNs have the inherent capacity to memorize and incorporate past information, which has led to their successful application in time series prediction. However, despite the potential of RNNs, their performances can be affected by certain factors. When the length of the input sequence increases, for example, the network layers greatly expand, which usually leads to the problem of gradient vanishing [41].

Long short-term memory (LSTM) is a type of recurrent neural network with a deep architecture [42,43]. Its unique gate mechanism and structure significantly enhance the model’s memory capacity and alleviate the problem of gradient disappearance that is often encountered when training classical RNNs on very long input sequences [44]. Figure 4 depicts the network layout of one neural unit of LSTM. The LSTM neural network is composed of several gates and variables. The input gate, $i_t$, regulates how much new information is added to the cell state at each time step. The forgetting gate,$f_t$, determines how much information from the previous cell state should be retained. The output gate, $o_t$, regulates how much information should be output from the current cell state. The input channel state, ${\tilde{x}}_t$, is obtained by compressing the messages from the input state, $x_t$, and the previous hidden state, $h_{t-1}$, with the tanh function, and might be added to the cell state, under the control of the input gate $i_t$. The cell state $c_{t-1}$ denotes the long-term memory in general that could be forgotten through the forgetting gate, $f_t$.

The process by which every LSTM unit updates its parameters is as follows:

$$i_t=\sigma \left(W_i{x}_t+U_i{h}_{t-1}+b_i\right)$$

(2)

$$f_t=\sigma \left(W_f{x}_t+U_f{h}_{t-1}+b_f\right)$$

(3)

$$o_t=\sigma \left(W_o{x}_t+U_o{h}_{t-1}+b_o\right)$$

(4)

$${\tilde{\mathrm{x}}}_t=tanh\left(W_c{x}_t+U_c{h}_{t-1}+b_c\right)$$

(5)

$$c_t=f_t\ast c_{t-1}+i_t\ast {\tilde{\mathrm{x}}}_t$$

(6)

$$h_t=o_t\ast tanh\left(c_t\right)$$

(7)

where $\sigma$ is the sigmoid function that makes a gate act as a switch. $W_f$, $W_i$, and $W_o$ are the weight matrixes of the forget gate, input gate, and output gate, respectively. $b_f$, $b_i$, $b_o$, and $b_c$ are the biases of the forget gate, input gate, output gate, and input channel, respectively.

3.5. GRU Model

The gated recurrent unit (GRU) is another type of recurrent neural network that can handle sequential regression problems [45,46]. GRU is able to memorize and retain important information from past observations in the sequence and is conceptually similar to LSTM [19,47]. GRU is faster to train and less prone to overfitting, especially when working with large datasets or limited computational resources. This advantage is particularly relevant for airfare prediction, which often involves handling massive datasets with complex patterns. Another advantage of GRU is its efficiency in capturing long-term dependencies and relationships within time series data [48]. In airfare prediction, this is crucial for capturing factors such as seasonal trends, holidays, and economic fluctuations, which can significantly impact prices over time. GRUs have been shown to effectively capture such relationships, resulting in accurate airfare predictions. In addition, GRUs are less sensitive to the gradient vanishing problem, which can occur in traditional RNNs when learning long-term dependencies [49]. This makes GRUs more robust and better suited for tasks involving lengthy sequences, such as airfare prediction. Moreover, GRUs can be combined easily with other neural network architectures or feature engineering techniques to improve prediction accuracy. All these make GRUs highly adaptable and suitable for tackling complex prediction tasks [3,31].

Figure 5 displays the internal network structure of a GRU unit. In Figure 5, $r_t$ represents the reset gate, $z_t$ represents the update gate, and $x_t$ denotes the input state, while $h_{t-1}$ and $h_t$ represent the previous and current hidden states, respectively. The symbols W, U, and b are the learnable weights and biases of the network.

The states of a GRU unit are updated according to the formula below:

$$z_t=\sigma \left(W_z{x}_t+U_z{h}_{t-1}+b_z\right)$$

(8)

$$r_t=\sigma \left(W_r{x}_t+U_r{h}_{t-1}+b_r\right)$$

(9)

$$u_t=tanh\left(W_h{x}_t+U_h(r_t\odot h_{t-1}\right)+b_h)$$

(10)

$$h_t=z_t\odot h_{t-1}+\left(1-z_t\right)\odot u_t$$

(11)

Equation (8) computes the update gate $z_t$, which determines how much of the previous hidden state $h_{t-1}$ should be preserved, and how much should be replaced by the new information in $u_t$. The sigmoid function, $\sigma$, squeezes the input into the range [0, 1]; thus, $z_t$ is a value between 0 and 1. Equation (9) computes the reset gate $r_t$, which determines how much of the previous hidden state $h_{t-1}$ should be forgotten, and how much should be used to compute $u_t$. Like $z_t$, $r_t$ is also a value between 0 and 1. Equation (10) computes the candidate hidden state $u_t$, which is a new value that is mixed with the previous hidden state $h_{t-1}$ based on the update gate $z_t$. The hyperbolic tangent function, tanh, compresses the input into the range [−1, 1]. Finally, Equation (11) computes the new hidden state $h_t$ as a linear combination of the previous hidden state, $h_{t-1}$, and the candidate hidden state, $u_t$, weighted by the update gate, $z_t$. The symbol $\odot$ represents element-wise multiplication, and 1 − $z_t$ represents the complement of $z_t$. This ensures that the new hidden state combines only the relevant information from $u_t$ and $h_{t-1}$. The architecture of the GRU unit, with its gating mechanisms, allows for better control over the flow of information through the network and has been shown to be excellent in a diverse range of tasks. The GRU network structure is presented in Figure 6. The implementation of the GRU model is shown in Algorithm 1, which is based on the Pytorch framework.

Algorithm 1. Stacked GRU flight fare prediction model.

#Step 1: Define the StackedGRU class class StackedGRU(nn.Module):  #Step 2: Define the constructor to initialize the model parameters  def __init__(self, input_size, hidden_size, output_size, num_layers, dropout, window_size=1):   super(StackedGRU, self).__init__()   #Step 3: Set the model parameters   self.input_size = input_size # Size of input layer   self.hidden_size = hidden_size # Size of hidden layers   self.output_size = output_size # Size of output layer   self.num_layers = num_layers # Number of layers in the stacked GRU model   self.dropout = dropout # Dropout probability   self.window_size = window_size # Number of time steps to include in each window   #Step 4: Create a list of GRU layers   self.gru_layers = nn.ModuleList()   #Step 5: Add the first GRU layer   self.gru_layers.append(nn.GRU(input_size, hidden_size, batch_first=True, dropout=dropout))   #Step 6: Add additional GRU layers if num_layers > 1   for i in range(num_layers-1):    self.gru_layers.append(nn.GRU(hidden_size, hidden_size, batch_first=True, dropout=dropout))   #Step 7: Create the output layer   self.linear = nn.Linear(hidden_size * window_size, output_size)  #Step 8: Define the forward pass through the stacked GRU model  def forward (self, x):   #Step 9: Reshape the input into windows of size self.window_size   x = x.view(x.shape[0], -1, self.window_size, self.input_size)   #Step 10: Transpose the input to (batch_size, window_size, sequence_length, input_size)   x = x.transpose(1, 2)   #Step 11: Flatten the input to (batch_size * window_size, sequence_length, input_size)   x = x.contiguous().view(-1, x.shape[3], x.shape[2])   #Step 12: Pass the input through each GRU layer in the stacked model   for i in range(self.num_layers):    #Step 12a: Get the current GRU layer    output, _ = self.gru_layers[i](x)    #Step 12b: Apply dropout to the output    output = F.dropout(output, p=self.dropout, training=self.training)    #Step 12c: Set the input for the next layer to be the output of the current layer    x = output   #Step 13: Reshape the output to (batch_size, window_size, hidden_size * sequence_length)   output = output.contiguous().view(-1, self.window_size, output.shape[2] * output.shape[1])   #Step 14: Flatten the output to (batch_size * window_size, hidden_size * sequence_length)   output = output.view(-1, output.shape[2])   #Step 15: Pass the final output through the output layer   out = self.linear(output)    #Step 16: Reshape the output to (batch_size, window_size, output_size)   out = out.view(-1, self.window_size, self.output_size)

A GRU model is a powerful algorithm for predicting time series data, particularly when dealing with nonlinear features [25]. However, building an effective GRU model for prediction tasks is not straightforward. Models with too few layers may struggle to capture the nonlinearity of fluctuating features, while models with too many layers may take too long to train and are prone to overfitting. In this study, we investigated its effectiveness in predicting flight ticket fares, which are influenced by a variety of factors. By leveraging the peculiarities of airline ticket fare series, the predictive abilities of the GRU model can be enhanced. After conducting a number of experiments, we developed a stacked GRU model that balances performance and efficiency.

Our experimental architecture is shown in Figure 7, which involved two main stages: offline learning and online prediction. During offline learning, the model is trained on the training data and evaluated with the validation data. Once the model is trained, it is used in the prediction process to efficiently predict flight ticket fares. In addition, the prediction performance of the flight ticket fares for the model is finally evaluated with the unseen testing data based on mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R²). The hyperparameters chosen for the model are listed in

Table 5. Hyperparameter setting for the prediction model.

Parameters	Value
Hidden Size	824, 512, 256, 128, 64
Batch Size	256
Number of Epochs	800
Drop Out	0.5
Learning Rate	0.001
Out Size	1
Optimizer	Adam
Loss Function	Mean Absolute Error
Layers	GRU × 5

, which will be used in later experiments.

4. Experiments and Discussion

The performance of the proposed GRU model was compared with that of the MLP and LSTM. During the training process, a number of parameters, such as batch size, activation function, number of hidden units, type of optimizer, and number of epochs, were adjusted until the best possible model was obtained. Four optimizers, including Adam, Adamax, RmsProp, and Sgd, were tested to minimize the loss. The losses with respect to the number of epochs for the four optimizers are compared in Figure 8. It can be observed from Figure 8 that the Adam optimizer converged much faster than the other three. Therefore, the Adam optimizer was adopted to train the models in the following experiments.

Furthermore, the prediction performance was evaluated with three metrics, described as follows. Mean absolute error is the average of the absolute difference between the target value and the predicted output.

$$MAE=\frac{1}{N} {\displaystyle \sum }_{i=1}^N\left|y_i-{\widehat{y}}_i\right|$$

(12)

Here, $y_i$ denotes the actual value of the flight ticket fare, ${\widehat{y}}_i$ denotes the predicted value, and N denotes the number of samples. MAE is a commonly used metric since it provides an intuitive indicator of the extent to which the predicted values deviate from the actual values on average. MAE is easy to interpret, suitable for continuous variables, and less sensitive to outliers, which means it is robust and can give a good overview of a model’s performance [50].

Root mean square error (RMSE) is another metric widely used to evaluate prediction models. It is the objective function used for optimizing classic regression models. RMSE is computed as follows:

$$RMSE=\sqrt{\frac{1}{N}{\sum }_{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2},$$

(13)

where $y_i$ represents the actual value of the flight ticket fare, ${\widehat{y}}_i$ represents the predicted value of the flight ticket fare, and N denotes the number of samples. RMSE is the standard deviation of predictions from a stochastic point of view. A small value of RMSE is indicative of a model with excellent performance.

Additionally, the coefficient of determination, usually denoted as ${\mathrm{R}}^2$, is a metric for evaluating how well a regression model performs in a number of analyses [51]. The computation of ${\mathrm{R}}^2$ is formulated as follows:

$${\mathrm{R}}^2=1-\frac{RSS}{TSS}=1-\frac{{\sum }_{i=1}^N{\left(y_i-{\widehat{y}}_i\right)}^2}{{\sum }_{i=1}^N{\left(y_i-{\overline{y}}_i\right)}^2}$$

(14)

where ${\overline{y}}_i$ is the mean of the total amount of data, $y_i$ is the actual value of the flight ticket fare, and ${\widehat{y}}_i$ is the predicted flight ticket fare. $RSS$ is the sum of squared prediction errors for the proposed model, TSS is the squared sum of the errors between the actual value and the mean of all outputs, and N is the number of samples. TSS denotes the error of the simple predictor, which simply takes the mean of all observations as the predicted output. A small ratio of RSS to TSS, corresponding to a high R², means that the error of the prediction model is much smaller than that of that simple predictor, and the model is hence considered to be effective. The value of R² usually falls between 0 and 1; when R² is close to 1, this indicates that the model fits the data well in a number of analyses. A high value of R² means that the model could explain a large portion of the variance in the target variable, while a low value of R² suggests that the model cannot capture much of the variation in the data. In summary, R² is a useful metric for comparing regression models so as to determine the overall fit of a model.

Figure 9 depicts a comparison of the convergence speed of the GRU, LSTM, and MLP models. As can be observed from Figure 9, the GRU model outperformed the others in RMSE. The loss of the GRU model was significantly lower than that of the others.

The superior performance of GRU in this study is due to its ability to capture long-term dependencies in input data. GRU has internal memories, allowing it to remember past inputs and incorporate them into current predictions. Such a capability is crucial for predicting ticket fares, as the pricing of tickets can be influenced by temporal factors such as the day of the week. GRU’s ability to capture these factors and incorporate them into predictions could result in more accurate fare predictions, as demonstrated by the lower root mean square error for the GRU model.

Additionally, GRU’s predictive power was compared with those of LSTM and MLP based on mean absolute error (MAE), root mean square error (RMSE), and coefficient of determination (R²), as presented in

Table 6. Performance comparison of the model prediction results.

Model	MAE	RMSE	R2
MLP	43.50	64.93	0.60
LSTM	8.67	13.99	0.70
GRU	3.76	5.93	0.98
ARIMA	165.67	205.73	0.51
SVR	246.76	315.69	0.27

. As can be observed from Table 5, MLP was the worst in performance since it had the highest MAE and RMSE values and the lowest value of R². The GRU model was superior in making accurate predictions, making it a reliable reference for the airline transport industry, both for travelers and aviators. The overall performance of GRU stands out as it surpassed both MLP and LSTM. When compared with MLP, the GRU model shows a decrease in MAE from 43.50 to 3.76, a decrease in RMSE from 64.93 to 5.93, and an increase in the coefficient of determination (R²) from 0.60 to 0.98. When compared to LSTM, the GRU model shows a decrease in MAE from 8.67 to 3.76, a decrease in RMSE from 13.99 to 5.93, and an increase in R² from 0.70 to 0.98.

Furthermore, the experimental results of two popular machine learning approaches, autoregressive integrated moving average (ARIMA) and support vector regression (SVR), are listed in Table 6 for comparison. ARIMA is a combination of the autoregressive model, which captures the dependency of the current observation output on the previous observations, and the moving average model, which captures the dependency on the current and previous inputs. The ARIMA model can be characterized as ARIMA (p, d, q), where p is the degree of autoregression, d is the degree of differencing, and q is the degree of moving average. In this study, we utilized the setting of ARIMA (5, 1, 1) to forecast future prices. In addition, for SVR, the kernel was set as a radial basis function (RBF). It can be observed from Table 6 that the GRU model stands out as a clear choice in fare prediction, providing the airline transport industry with a trustworthy reference.

Furthermore, the average of the predicted ticket fares for one flight based on the three different prediction models, GRU, LSTM, and MLP, were compared with the average of the actual fares in

Table 7. Performance comparison of monthly averages for prediction models.

Average Value of Actual Ticket Fares	Average Value of Predicted Ticket Fares
	GRU	LSTM	MLP
379.16	376.43	352.69	285.83

. The results in Table 7 indicate that the average of the predicted ticket fares for GRU is much closer to the actual average fares than those for LSTM and MLP. This suggests that GRU is a more accurate model for predicting ticket fares. Overall, the above results provide evidence of the superiority of the GRU model in predicting flight ticket fares and suggest that it is a promising approach for further study in this field.

Figure 10, Figure 11, Figure 12 and Figure 13 further show the predicted and actual fares of all the flights for GRU, LSTM, MLP, and all three models, respectively, based on monthly averages. As can be observed from these figures, the predicted values by GRU were much closer to the actual values, as indicated by the lower absolute differences between the predicted and actual values. Given the extensive periods covered by historical flight prices, i.e., the four years and seven months in our dataset, the GRU model’s capabilities make it an appropriate choice for the fare prediction of flight tickets.

In addition, we conducted additional experiments with different prediction horizons to provide a comprehensive analysis of the model’s performance over varying timeframes. These experiments aimed to capture short-, medium-, and long-term forecasting scenarios. We evaluated the accuracy of the GRU model using three different prediction horizons; the time steps of 7, 14, and 30, and the experimental results are displayed in Figure 14. It can be observed from Figure 14 that the variations in the predicted outcomes for different prediction horizons were low, which demonstrates the robustness of the GRU model and provides valuable insights into the predictive capabilities of the model across different forecast periods.

5. Conclusions

In this research, we have proposed using the GRU model as a solution for predicting airline ticket fares. Our analysis has demonstrated that this approach surpasses traditional neural networks in effectively preserving and leveraging historical datasets, resulting in more accurate and reliable predictions. This advancement represents a significant contribution to this field, and the straightforward architecture of the GRU model makes it highly practical and versatile for various applications.

However, it is imperative to acknowledge a crucial aspect that should be noticed. Forecasting the fare of an airline ticket is challenging due to the influence of a wide range of internal and external factors. These factors encompass social aspects, political dynamics, and even the market, which could be influenced by events in different countries. As a consequence, unexpected fluctuations in fare prices can occur. Future enhancements to the model should consider incorporating additional features not yet explored to address this complexity. It is essential to consider and account for such scenarios to further improve the predictive capabilities of the GRU model.

Furthermore, the scarcity of flying data in this domain highlights the importance of the work conducted on this deep-learning GRU model. The insights from this study pave the way for future research endeavors as they contribute to the development of more robust and comprehensive fare prediction models.