Neural Network System for Predicting Anomalous Data in Applied Sensor Systems

Abstract

This article advances the research on the intelligent monitoring and control of helicopter turboshaft engines in onboard conditions. The proposed neural network system for anomaly prediction functions as a module within the helicopter turboshaft engine monitoring and control expert system. A SARIMAX-based preprocessor model was developed to determine autocorrelation and partial autocorrelation in training data, accounting for dynamic changes and external factors, achieving a prediction accuracy of up to 97.9%. A modified LSTM-based predictor model with Dropout and Dense layers predicted sensor data, with a tested error margin of 0.218% for predicting the TV3-117 aircraft engine gas temperature values before the compressor turbine during one minute of helicopter flight. A reconstructor model restored missing time series values and replaced outliers with synthetic values, achieving up to 98.73% accuracy. An anomaly detector model using the concept of dissonance successfully identified two anomalies: a sensor malfunction and a sharp temperature drop within two minutes of sensor activity, with type I and II errors below 1.12 and 1.01% and a detection time under 1.611 s. The system’s AUC-ROC value of 0.818 confirms its strong ability to differentiate between normal and anomalous data, ensuring reliable and accurate anomaly detection. The limitations involve the dependency on the quality of data from onboard sensors, affected by malfunctions or noise, with the LSTM network’s accuracy (up to 97.9%) varying with helicopter conditions, and the model’s high computational demand potentially limiting real-time use in resource-constrained environments.

1. Introduction

1.1. Relevance of the Research

In the context of rapid technological development and the increasing amounts of data collected by sensor systems [1], there is a need to develop methods for detecting anomalous data. Predicting anomalous data in sensor systems is vital in critical areas such as aviation [2,3], medicine [4,5], energy [6], and industrial production [7,8]. Sensor systems collect and analyze large amounts of data in real time, enabling the control and monitoring of various processes and conditions [9]. Due to equipment wear, interference, or calibration errors, data anomalies can occur, indicating potential failures or malfunctions that lead to economic losses or safety threats [10]. Therefore, developing methods for effectively predicting and detecting anomalous data is crucial for ensuring the reliability and safety of sensor systems.

Modern approaches to anomaly prediction use machine learning methods [11,12] and artificial intelligence [13,14] to identify the hidden patterns and trends in large amounts of data. These methods adapt to changing conditions and provide timely warnings of potential issues, allowing for preventive measures and risk minimization. This article’s scientific relevance stems from the need to improve algorithms’ accuracy and speed, necessitating research on model optimization and data-processing enhancement. In the context of the increasing digitization and reliance on automated systems, anomaly prediction is becoming a key element in maintaining their stable and efficient operation.

1.2. State of the Art

In recent years, researchers have actively applied machine learning and artificial intelligence methods, such as neural networks, clustering, and outlier detection techniques. Machine learning [11,12] and artificial intelligence [13,14] play a key role in predicting anomalous data. Deep learning algorithms, including recurrent [15,16] and convolutional neural networks [17,18], have been proven effective in processing time series data from sensors. Studies [15,16,17,18] show that combining different neural network architectures can enhance anomaly prediction accuracy and reliability. For example, using autoencoders to identify hidden patterns in data ensures high accuracy in detecting anomalies in complex systems, such as aviation and industrial installations [19,20].

Alongside machine learning methods, statistical methods [21,22] and clustering techniques [22,23] are widely used for predicting anomalous data. Statistical approaches, such as probability density estimation and principal component analysis, identify deviations from normal data behavior. Clustering methods, like k-means and hierarchical clustering, group data and detect anomalies based on differences between clusters. These methods are often combined with machine learning to enhance prediction quality.

Research also focuses on the practical integration of anomaly prediction methods into industrial applications [24]. Implementing such systems significantly improves equipment monitoring and diagnostics, preventing failures and reducing downtime. For example, in aviation, anomaly prediction systems help detect potential faults in engines and other critical components, ensuring higher levels of safety and reliability [25]. Similarly, in the energy sector, these systems optimize control processes and reduce the risk of emergencies [26].

The application of neural networks for predicting anomalous data in sensor systems is scientifically justified due to their ability to efficiently process large amounts of data and identify complex, nonlinear dependencies [27,28] that are difficult to detect using traditional methods [29,30]. Their capability to train from historical data enables adaptive prediction [31], which is crucial in changing, dynamic systems, such as aviation and industrial applications. Moreover, the ability to automatically extract features from data makes neural networks a powerful tool for anomaly detection, minimizing the need for manual tuning and data preprocessing [32]. This enhances anomaly detection accuracy and reliability, which is critical for ensuring the safety and resilience of sensor systems.

1.3. Main Attributes of the Research

The object of the research is the sensor data.

The subject of the research includes the neural network system for predicting anomalous data in sensor systems.

The research aim is to develop a neural network system for predicting anomalous data in sensor systems, which will provide enhanced accuracy and speed in detecting data deviations, allowing for timely responses to potential failures and malfunctions.

To achieve this aim, the following scientific and practical tasks were solved:

• Development of the proposed neural network system for predicting anomalous data in sensor systems.
• Development of the preprocessor model.
• Development of the predictor model.
• Development of the reconstructor model.
• Development of the anomaly detector model.
• Conducting a computational experiment involving the simulation of predictive anomalies for a sensor in a helicopter turboshaft engine (TE).

The main contribution of this research lies in developing a neural network system for predicting anomalous data in sensor systems, which will, in turn, enhance the reliability and safety of critical systems, minimize economic losses by preventing emergencies, and optimize maintenance and operation processes. The developed system will be capable of adapting to changing conditions and system dynamics, providing more effective real-time control and monitoring.

2. Materials and Methods

2.1. Development of the Proposed Neural Network System for Predicting Anomalous Data in Sensor Systems

The block diagram of the proposed neural network system for predicting anomalous data in sensor systems (hereinafter referred to as the “system”) is shown in Figure 1. The system is created for each sensor of the sensor system under research and includes a preprocessor, predictor, reconstructor, and anomaly detector.

The preprocessor prepares sensor data for subsequent processing. The predictor uses a neural network to predict the next sensor value based on historical data. The reconstructor checks if the current value is an outlier and replaces it with the predicted value from the predictor if necessary. The anomaly detector identifies anomalous sequences in the data to alert the operator. The working cycle of the sensor data cleaning module is as follows. The preprocessor operates at a frequency set by the operator, extracting raw data from storage and forming a training dataset for the predictor’s neural network. The predictor runs according to the data acquisition frequency, predicting the current value. If the sensor returns an empty value, it is replaced with the predicted one. Otherwise, the reconstructor classifies the value as either an outlier or normal. If an anomalous value is recognized, it is replaced with the predicted one. The system then transmits this value for storage. At the end of the cycle, the anomaly detector determines if an anomalous sequence is ending and notifies the operator if necessary.

When the predicting system detects anomalous data, it performs a series of actions to ensure accurate reporting and system integrity. Upon identifying an anomaly, the system initially replaces the anomalous data with the predicted value from the neural network. Following this replacement, the anomaly detector assesses whether the anomalous sequence is concluding. If the sequence is deemed significant, the system generates a notification for the operator, providing a detailed analysis of the detected anomalies. This analysis includes information on the anomalies and the nature of the potential impacts on the sensor system, enabling the operator to make informed decisions regarding further actions or adjustments. Thus, the system not only corrects the data but also actively informs the operator about potential issues, enhancing overall system reliability and response capability.

2.2. Development of the Preprocessor Model

The preprocessor (Figure 2) prepares the training dataset for the neural network in the predictor subsystem and consists of the following components: an analyzer, restorer, outlier detector, and normalizer. The analyzer extracts sensor data from storage and converts it into a format suitable for further processing. The restorer then replaces missing values with synthetic ones. The outlier detector identifies anomalous points and replaces them with NULL. The processed data passes through the restorer again for outlier replacement. The normalizer forms normalized subsequences, which become the training dataset for the neural network.

The restorer uses the SARIMAX model to predict missing values and replace them with synthetic ones [33,34]. The SARIMAX model is defined by the equation:

$$y_t=c+{\phi }_1\cdot y_{t-1}+{\phi }_2\cdot y_{t-2}+\dots +{\phi }_p\cdot y_{t-p}+{\epsilon }_t+{\theta }_1\cdot {\epsilon }_{t-1}+\dots +{\theta }_q\cdot {\epsilon }_{t-q}+\beta \cdot X_t,$$

(1)

where c is a constant, φ₁, …, φ_p are the autoregressive coefficients, θ₁, …, θ_q are the moving average coefficients, ε_t is the white noise, and X_t are the exogenous regressors. If t_i = NULL, then

$$t_i={\widehat{y}}_i,$$

(2)

where ${\widehat{y}}_i$ is the predicted value obtained using the SARIMAX model. To check stationarity, the Dickey–Fuller and KPSS tests are used [35,36]. If the series is not stationary, transformations such as logarithms are applied to prepare the data for analysis.

The analyzer retrieves the time series T from the data store, where T = (t₁, t₂, …, t_m), and each value t_i belongs to the real numbers ℝ set or is empty (equal to NULL), that is

$$T=\left\{\left.t_i\right|t_i\in ℝ \mathrm{or} t_i=NULL, i=1,2,\dots ,m\right\}.$$

(3)

The normalizer then creates a training dataset consisting of “readings sequence” and “predicted value” pairs. The first element of a pair is the fixed-length sensor readings normalized subsequence, and the second element is a single normalized sensor reading following this subsequence. The normalizer transforms the time series into a set of normalized subsequences using min–max normalization. The sequence T_i,n of the time series T with length n is defined as

T_i,n = (t_i, …, t_i+n−1), 1 ≤ i ≤ m − n + 1.

(4)

Using min–max normalization, for a subsequence T_i,n, its normalized version ${\tilde{T}}_{i,n}$ is calculated as

$${\tilde{T}}_{i,n}=\left({\tilde{t}}_i,\dots ,{\tilde{t}}_{i+n-1}\right),$$

(5)

where

$${\tilde{t}}_i={\displaystyle \frac{t_i-t_{\min }}{t_{\max }-t_{\min }}},j\in \{i,i+1,\dots ,i+n-1\},$$

(6)

where t_min and t_max are the minimum and maximum values in T_i,n.

For a normalized subsequence ${\tilde{T}}_{i,n}$, the element ${\tilde{t}}_{n+1}$ is considered as its predicted value. The training sample is formed from pairs (subsequence, predicted value):

$$S_T^n=\left\{\left({\tilde{T}}_{i,n},{\tilde{t}}_{n+1}\right)\left|1\le i\le m-n\right.\right\}.$$

(7)

The subsequence length n (n << m) is the Cleanup Module parameter and is calculated as

n = f_sensor · horizon,

(8)

where f_sensor is the sensor frequency and horizon is the historical horizon (the time interval length in the past) used by the predictor and selected by the human operator.

2.3. Development of the Predictor Model

The predictor is a recurrent neural network (RNN) with a long short-term memory (LSTM) layer [37,38] (Figure 3). The RNN training is carried out on the dataset prepared by the preprocessor. During use, the neural network takes a subsequence of actual sensor values preceding the current value as input and outputs a predicted value.

The LSTM layer consists of identical LSTM blocks (Figure 4), whose number matches the subsequence length n defined during preprocessing. These blocks collectively form a hidden state vector h, with its length set by the system operator. A Dropout layer randomly deactivates a specified neuron fraction in vector h to prevent overfitting, typically deactivating 20%, though this parameter can be adjusted by the operator. The Dense layer applies the modified (special) rectified linear unit (ReLU) activation function (Appendix A) to transform the data and obtain the predicted value i_i+1.

Each LSTM block comprises a block state and several filter layers. The block state is a vector carrying information from previous time steps and propagates through the entire chain of LSTM blocks. An LSTM block contains three types of filters (“gates”), an input gate, a forget gate, and an output gate, which regulate the data amount to be saved, forgotten, and output, respectively.

Figure 4 illustrates the following notations: the “Control gate” is a control unit with a sigmoidal activation function; the “Recording gate” is a recording unit with a tangential activation function; the “Output gate” is an output unit with a sigmoidal activation function; c_t−1 is the memory tensor at the previous time step; c_t is the memory tensor at the current time step; h_t−1 is the network output tensor at the previous time step; h_t is the network output tensor at the current time step; x_t is the input tensor at the current time step; n_t is the tensor at the control unit output; r_t is the tensor at the recording unit output; and W_cn, W_hn, W_xn, W_hr, W_xr, W_ch, W_hh, and W_xh are the weight coefficients connecting inputs to units.

The LSTM mechanism addresses the vanishing gradient problem by regulating the flow of new information into the state vector c_t, as well as the network’s output state h_t, and updating its state c_t. The vector filters are determined by the following expressions:

$$i_t=\sigma \cdot \left(U_i\cdot x_t+W_i\cdot h_{t-1}\right),$$

(9)

$$f_t=\sigma \cdot \left(U_f\cdot x_t+W_f\cdot h_{f-1}\right),$$

(10)

$$o_t=\sigma \cdot \left(U_o\cdot x_t+W_o\cdot h_{o-1}\right),$$

(11)

$$g_t=\sigma \cdot \left(U_g\cdot x_t+W_g\cdot h_{g-1}\right),$$

(12)

where σ is the sigmoid function; x is the input sequence; h is the network cell hidden state vector; and U_i, U_f, U_o, U_g, W_i, W_f, W_o, W_g are the network filter weight matrices for the training sequence element indices i, f, o, g, t.

Based on the network filter values, its internal state (internal memory) and hidden state vectors are determined as follows:

$$c_t=\tanh \left(i_t\circ g_t+f_t\circ c_{t-1}\right),$$

(13)

$$h_t=o_t\circ c_t,$$

(14)

where c_t represents the network cell state vector, h_t denotes the hidden state vector, and “∘” indicates element-wise multiplication.

In the Dropout layer, certain elements of the activation vector are randomly set to zero to prevent overfitting. For an input activation vector h of size t and a mask vector m, where mask elements are either 0 or 1, the Dropout is expressed as

$$h_{dropout}=h\odot m,$$

(15)

where “⊙” denotes element-wise multiplication, and m is the mask vector created randomly with probability p (the deactivated neuron proportion):

$$m_i=\left\{\begin{array}{c}{\displaystyle \frac{1}{1-p}} \mathrm{with} \mathrm{probability} 1-p\\ 0 \mathrm{with} \mathrm{probability} p\end{array}\right.,$$

(16)

where p is the probability that the element will be deactivated, and ${\displaystyle \frac{1}{1-p}}$ is the scaling factor for maintaining the activation level during training.

After applying Dropout during training, the output vector h_dropout is scaled by ${\displaystyle \frac{1}{1-p}}$ to preserve the expected output activation:

$$h_{dropout}={\displaystyle \frac{h\odot m}{1-p}}.$$

(17)

The Dense layer (or fully connected layer) transforms the input data through a linear combination followed by an activation function. An input activation vector h_dropout of size n is transformed into an output activation vector h_dense of size m using a weight matrix W and a bias b. The linear transformation is expressed as follows:

z = W · h_dropout + b,

(18)

where W is an m × n weight matrix, b is a displacement vector of size m, and z is an intermediate output vector of size m.

The modified (special) ReLU activation function is applied to the intermediate output z to obtain the final output vector h_dense:

h_dense = ReLU(z),

(19)

that is

h_dense = ReLU(z₁), ReLU(z₂), …, ReLU(z_m),

(20)

and

ReLU(z_i) = max(0, z_i).

(21)

Thus, for each element z_i in vector z, if z_i > 0, it remains unchanged, and if z_i ≤ 0, then it is replaced by 0.

2.4. Development of the Reconstructor Model

The reconstructor receives the current sensor reading and checks whether it is significantly different from the denormalized synthetic value received from the predictor. If the difference is significant, the real value is replaced with a synthetic one, which is then stored in the database. The reconstructor implementation is based on the prediction error probability distribution use [39]. In accordance with this approach, the difference threshold ε > 0 between the real sensor reading t_i+1 and the predicted synthetic sensor value ${\widehat{t}}_{i+1}$ is determined. The prediction error at step i + 1 is defined as

$${\epsilon }_{i+1}=\left|t_{i+1}-{\widehat{t}}_{i+1}\right|.$$

(22)

The difference threshold ϵ is calculated as

ϵ = μ + k · δ,

(23)

where μ is the predicting error average value calculated from historical data; δ is the predicting error standard deviation calculated from historical data; and k is a parameter determined by the system operator (for example, k = 3).

If the prediction error e_i+1 exceeds the threshold ϵ, then the current reading t_i+1 is considered an outlier, that is

If e_i+1 ≥ ϵ, then t_i+1 is an outlier.

(24)

If t_i+1 is an outlier, then it is replaced with the predicted value ${\widehat{t}}_{i+1}$, which is then stored in the data store, that is

$$t_{i+1}\leftarrow {\widehat{t}}_{i+1}.$$

(25)

2.5. Development of the Anomaly Detector Model

The anomaly detector assesses whether a sensor reading subsequence, ending with the current value, constitutes an anomaly. If it is identified as such, the detector generates a notification for the monitoring sensor system operator. The specified subsequence length is defined by values corresponding to significant time intervals, which are calculated based on the sensor’s sampling frequency. The implementation of the anomaly detector uses the concept of dissonance [40]. Dissonance, unlike most anomaly detection algorithms, requires only one intuitive parameter, the anomalous subsequence length, and is formally defined as follows: Let T be the sensor reading time series; $S_T^n$ is the set of all possible subsequences of length n in T; T_i,n is the subsequence of length n starting at position i; D is the subsequence under investigation for anomalies; M_D is the set of all subsequences that are not trivial matches with D; and k is the parameter defined by the system operator indicating the significance level. The Euclidean distance between the two subsequences T_i,n and T_j,n is defined as

$$ED\left(T_{i,n},T_{j,n}\right)=\sqrt{{\displaystyle \sum _{i=1}^n{\left(T_{i+t-1}-T_{j+t-1}\right)}^2}},$$

(26)

where T_j,n is the subsequence of length n starting at position j.

Subsequences T_i,n and T_j,n are not trivial matches if there exists a subsequence $T_{p,n}\in S_T^n$ (where i < p < j) such that

ED(T_i,n, T_j,n) < ED(T_i,n, T_p,n).

(27)

Subsequence $T_{p,n}\in S_T^n$ is called dissonance if

$$\forall C\in S_T^n, \min \left(ED\right(D,M_D\left)\right)>\min \left(ED\right(C,M_C\left)\right),$$

(28)

where M_C is the set of all subsequences that are not trivial matches to C.

Subsequence $D\in S_T^n$ is called the most significant k-th dissonance in T if

$$\min \left(ED\left(D,M_D^k\right)\right)\mathrm{is}\mathrm{the}\mathrm{largest}\mathrm{among}\mathrm{all}\mathrm{subsequences},$$

(29)

where $M_D^k$ is the set of the k-th closest subsequence that is not a trivial match to D.

Let t_i+1 be the current sensor reading considering a subsequence of length n ending with t_i+1. It is determined whether this subsequence D is a dissonance with a significance degree of at least k if

$$\min \left(ED\left(D,M_D^k\right)\right)\ge \min \left(ED\left(D,M_C^k\right)\right),$$

(30)

then a notification is generated for the human operator that the subsequence is an anomaly. If the condition for dissonance is satisfied, then

Notice: The subsequence is an anomaly.

(31)

3. Results

3.1. Experiment Description

The research object in this study is the TV3-117 aircraft engine, which belongs to the helicopter TE class. The sensor in the sensory system consists of 14 T-101 thermocouples, which record the gas temperature before the compressor turbine $T_G^{\ast }$ value on board the helicopter (1 value per second) (

Table 1. Fragment of the training dataset (readings of the TV3-117 aircraft engine gas temperature in front of the compressor turbine sensor) (author’s research).

Number	1	…	35	…	84	…	116	…	152	…	198	…	255	256
Value	0.932	…	0.948	…	0.953	…	0.929	…	0.968	…	0.970	…	0.952	0.953

) [41]. The preprocessor is implemented using Python libraries as follows. The analyzer uses the standard libraries “openpyxl” and “pandas”. The outlier detector is based on algorithms from the “adtk” (Anomaly Detection Toolkit) library [42]. The SARIMAX model, used by the restorer, is implemented in the standard “statsmodels” library. The normalizer is implemented using the standard “sklearn” library. The predictor is implemented based on the Keras library [43] and the TensorFlow framework [44]. For training the recurrent neural network, subsequences of length n = 256 were used, corresponding to 256 s of sensor operation. The RNN hidden state was taken as a column vector of length ∣h∣ = 32. The following commonly accepted parameters were used for network training: the loss function was MSE (Mean Square Error), the optimizer was Adam, and the epochs and batch size number were 20 and 32, respectively. The anomaly detector was implemented using the “MatrixProfile library” for the Python 3.12.4 programming language [45].

The training dataset (Table 1) homogeneity evaluation using Fisher–Pearson [46] and Fisher–Snedecor [47] criteria is detailed in [41]. According to these criteria, the training dataset of gas temperature readings before the compressor turbine for the TV3-117 aircraft engine is homogeneous, as the calculated values for the Fisher–Pearson and Fisher–Snedecor criteria are below their critical values, i.e., ${\chi }_{T_G^{*}}^2=4.645<{\chi }_{critical}^2=6.6$ и $F_{T_G^{*}}=4.645<F_{critical}=6.6$. To assess the representativeness of the training dataset, ref. [41] employed cluster analysis (k-means [48,49]). Using random sampling, the training and test datasets were divided in a 2:1 ratio (67% and 33%, corresponding to 172 and 84 elements, respectively). The cluster analysis results for the training dataset (Table 1) identified eight classes (classes I …VIII), indicating the presence of eight groups and demonstrating the similarity between the training and test datasets (Figure 5).

These results enabled the optimal sample sizes for the determination of gas temperature before the compressor turbine for the TV3-117 aircraft engine: the training dataset comprises256 elements (100%), the validation dataset comprises 172 elements (67% of the training dataset), and the test dataset comprises84 elements (33% of the training dataset).

After the processing of data by the preprocessor, the first 172 values (67%) from the sensor (control dataset) were used in two roles: as a dataset for tuning the SARIMAX model parameters used by the reconstructor and as the training dataset for the recurrent neural network in the predictor subsystem. The remaining 84 values (33%) from the sensor (test dataset) were used for the data cleaning module operation simulated under normal conditions.

Testing the dataset with ADF and KPSS tests confirmed its non-stationarity. To achieve stationarity, techniques were sequentially applied, including first differencing to remove the trend and seasonal differencing to eliminate seasonality [50]. The corresponding autocorrelation function (ACF) and partial autocorrelation function (PACF) diagrams are shown in Figure 6. The final diagrams in Figure 6c allow for the SARIMAX model parameters’ following range selections: autoregressive order p ∈ [0, 7], differencing order d = 1, moving average order q ∈ [0, 7], seasonal autoregressive order P = 1, seasonal differencing order D = 1, seasonal moving average order Q = 0, and seasonal period s = 96.

Clarification: In the context of sensor data registration on board helicopters during flight, the term “seasonality” refers to regular and predictable changes in the data caused by time periods within the year (autumn–winter and spring–summer flights), which must be factored for accurate real-time interpretation and analysis. The impact on the sensor readings for gas temperatures before the compressor turbine in helicopter flight during the autumn–winter and spring–summer periods is linked to variations in the external temperature and pressure conditions, leading to changes in the thermodynamic properties of the air entering the engine [51,52,53]. In this research, the parameters have been adjusted for helicopter flight conditions during the spring–summer period.

In addition to accounting for “seasonality”, the calibration of the predicting system incorporates actual environmental conditions such as temperature, humidity, and atmospheric pressure. These factors significantly influence the sensor data interpretation accuracy, as they affect the engine’s thermodynamic properties and the sensor readings. By integrating these real-time environmental parameters into the calibration process, the system can more precisely adjust for variations and improve the reliability of its predictions under varying flight conditions.

Among all the obtained models, the model with SARIMAX parameters s = 96, (p = 6, d = 1, q = 5) (P = 0, D = 1, Q = 1) was selected for the data cleaning module’s routine operation. This model was chosen due to its lowest values for the Akaike information criterion (AIC) [54] and Bayesian information criterion (BIC) [55] compared to other models, and the residual series (the difference between actual and predicted values) exhibits unbiasedness properties, stationarity, and a lack of autocorrelation. Figure 7 presents a 4 min prediction for helicopter flight, with the prediction error having an MSE = 0.218%. In Figure 7, the “blue curve” represents the actual data recorded by the sensor, and the “red curve” represents the predicted data.

Since the proposed SARIMAX model predicts values for the gas temperature from the TV3-117 aircraft engine’s in front of the compressor turbine sensor with high accuracy, it can be used to replace missing values with plausible synthetic values. Examples of the reconstructor’s performance are shown in Figure 8. In Figure 8, the “blue curve” represents the actual data recorded by the sensor, the “red curve” represents restored data, the “red dots” indicate outliers, and the “green dots” represent imputed values.

3.2. Evaluation of Data Restoring Accuracy

In the experiments, the TV3-117 aircraft engine gas temperature in front of the compressor turbine sensor readings, synthesized by the data cleaning module, were compared with the actual values obtained from the data repository to assess accuracy. The accuracy was measured using the MSE metric, defined as

$$MSE={\displaystyle \frac{1}{\left|B\right|}}\cdot {\displaystyle \sum _{i=1}^{\left|B\right|}{\left(t_i-{\widehat{t}}_i\right)}^2},$$

(32)

where t_i and ${\widehat{t}}_i$ are the actual and synthetic sensor values, respectively, and ∣B∣ denotes the sensor reading block length used for comparison. The experiments results, where the block length varied with helicopter flights from one to two minutes, are shown in Figure 9. As seen in Figure 9, the developed module ensures high and stable accuracy, with the MSE value not exceeding 0.025 (2.5%).

Figure 10 shows examples of the module’s operation simulation for different lengths of the sensor reading block. It can be seen from Figure 10 that the module adequately predicts normal values and detects point emissions in the data. In Figure 10, the “blue curve” represents the actual data recorded by the sensor, and the “red curve” represents the predicted data.

3.3. Results of Data Anomaly Prediction

Figure 11 illustrates an example of two anomalies detected during the data cleaning module’s simulation, corresponding to two minutes of the TV3-117 aircraft engine gas temperature before the compressor turbine sensor activity under helicopter flight conditions. The first anomaly is a “Top 1 dissonance” found in the test dataset, indicating a temporary malfunction of the sensor. The second anomaly is a “Top 10 dissonance” and points to a rapid decrease in gas temperature before the compressor turbine due to increased airflow speed and changes in external air pressure. In both cases, the detected anomalies require a response from the helicopter’s flight commander (human operator). These anomalies prompt an immediate assessment by the flight commander to determine the appropriate corrective actions or adjustments needed. Such proactive responses help maintain optimal engine performance and ensure operational safety by addressing potential issues before they escalate. The system’s ability to accurately detect and classify anomalies in real time significantly enhances the reliability and responsiveness of the monitoring process.

To better interpret the anomaly detection results, it is important to discuss both healthy and unhealthy conditions in detail. By examining the baseline or normal operational parameters alongside the detected anomalies, one can more accurately assess the system’s performance and sensitivity. Healthy conditions may include stable engine performance with consistent sensor readings and normal gas temperatures before the compressor turbine, indicative of optimal operation. Unhealthy conditions, on the other hand, might involve irregular sensor data, such as sudden temperature drops or fluctuating readings due to sensor malfunctions, increased airflow speeds, or changes in external air pressure.

To perform cross-validation (k-fold cross-validation), the training dataset is partitioned into k subsets (folds): D₁, D₂, …, D_k. For each i-th fold (i = 1, 2, …, k), one subset serves as the validation set while the remaining k − 1 folds are utilized for training. The model is trained on the k − 1 subsets and assessed on the remaining subset. In each i-th iteration, the model is trained on data $D_{train}^{\left(i\right)}$(and evaluated on data $D_{val}^{\left(i\right)}$), where

$$D_{train}^{\left(i\right)}={\displaystyle \underset{j\ne i}{\cup }{D}_j}, D_{val}^{\left(i\right)}=D_i$$

(33)

Subsequently, the metric (in this research, MSE) is computed on the validation set for each i-th fold:

$${\mathrm{Metric}}_i=\mathrm{evalute}\left(M^{\left(i\right)},D_{val}^{\left(i\right)}\right).$$

(34)

where M⁽ⁱ⁾ represents a model trained on $D_{val}^{\left(i\right)}$. Upon completion of all k iterations, the mean metric value is computed across all folds:

$$\mathrm{Average} \mathrm{Metric}={\displaystyle \frac{1}{k}}\cdot {\displaystyle \sum _{i=1}^k{\mathrm{Metric}}_i}.$$

(35)

In assessing the effectiveness of the computational experiment, a cross-validation procedure (k-fold cross-validation) was performed. For this, k = 5 equal subsets (folds) of 51 elements each randomly chosen from the training sample. The following metric values were obtained: Metric(k₁) = 0.224%, Metric(k₂) = 0.228%, Metric(k₃) = 0.227%, Metric(k₄) = 0.226%, and Metric(k₅) = 0.229%. The average metric was calculated as 0.227%, which differs from MSE_min = 0.218% by 3.96%. This indicates that the average value of the metric (mean square error) after cross-validation is close to the MSE_min obtained from one fold (in this case, k₃). The 3.96% difference reflects how much the average metric deviates from the best individual result achieved during the cross-validation iterations. The results also suggest that the model demonstrates stability in its predictions across different data subsets (folds), as evidenced by the small difference between the average metric and MSE_min. However, it is crucial to note that the average metric offers a more generalized view of performance across the entire training set, while MSE_min highlights the optimal result obtained from a single data subset. Further analysis could investigate whether additional folds or variations in data distribution might further enhance the model’s accuracy.

3.4. Evaluation of the Effectiveness of the Results Obtained

To evaluate the effectiveness [56,57,58,59,60] of the proposed neural network system for predicting anomalous data in sensor systems (Figure 1), various metrics need to be considered, such as mean squared error (MSE) and mean absolute error (MAE), which reflect the accuracy in predicting sensor values. The determination coefficient (R²) shows how well the model explains data variations, while the correctly replaced outlier percentage and the false positive rate demonstrate the effectiveness of the reconstructor and anomaly detector. Additionally, the missed outlier rate and anomaly detection accuracy provide insight into the system’s ability to correctly identify and classify anomalies, while mean response time to anomalies assesses the system’s operational speed. In the context of a system for predicting anomalous data, Precision reflects the system’s ability to avoid false positives. This is crucial to prevent the operator from being overwhelmed by unnecessary notifications about anomalies that are actually normal data. Recall indicates how well the system can identify all existing anomalies. The F1-score is a combined metric that considers both Precision and Recall. The ROC curve displays the true positive rate (TPR) against the false positive rate (FPR) across various threshold values. These metrics enable a comprehensive evaluation, ensuring the accurate prediction and reliable detection of anomalies in sensor data. The results evaluating the effectiveness of the proposed neural network system for predicting anomalous data in sensor systems are presented in

Table 2. Effectiveness evaluation results (author’s research).

Metric	Analytical Expression	Resulting Value	Comment
Mean squared error	$MSE={\displaystyle \frac{1}{\left|B\right|}}\cdot {\displaystyle \sum _{i=1}^{\left|B\right|}{\left(t_i-{\widehat{t}}_i\right)}^2}$	0.218%	It indicates that the neural-network-based anomaly predicting system achieves a high level of precision in predicting sensor data, with minimal average squared deviation between the predicted and actual values.
Mean absolute error	$MAE={\displaystyle \frac{1}{\left|B\right|}}\cdot {\displaystyle \sum _{i=1}^{\left|B\right|}\left|t_i-{\widehat{t}}_i\right|}$	0.232%	It reflects the neural-network-based anomaly predicting system’s effectiveness in maintaining a low average absolute error between the predicted and actual sensor readings.
Determination coefficient	$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^{\left|B\right|}{\left(t_i-{\widehat{t}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^{\left|B\right|}{\left(t_i-\overline{t}\right)}^2}}}$	0.983	It signifies that the neural-network-based anomaly predicting system explains 98.3% of the variance in sensor data, indicating a highly accurate model.
Percentage of outliers correctly replaced by the reconstructor	$POCR={\displaystyle \frac{{\left(TP\right)}_{replaced}}{{\left(FP\right)}_{replaced}+{\left(FP\right)}_{replaced}}}\cdot 100\%$	98.73%	It indicates that the neural-network-based anomaly predicting system accurately replaces nearly 99% of detected outliers, demonstrating high efficacy in handling anomalies.
True positive rate	$TPR={\displaystyle \frac{TP}{TP+FN}}$	0.818	It indicates that the neural-network-based anomaly predicting system correctly identifies 81.8% of actual anomalies, demonstrating a strong sensitivity in detecting anomalies.
False positive rate	$FPR={\displaystyle \frac{FP}{FP+TN}}$	0.0112	It indicates that the neural-network-based anomaly predicting system incorrectly classifies 5% of normal data as anomalies, reflecting a low false positive rate.
False negative rate	$FNR={\displaystyle \frac{FN}{FN+TP}}$	0.0101	It indicates that the neural-network-based anomaly predicting system fails to detect only 1.4% of actual anomalies, highlighting its high detection accuracy.
AUC-ROC	AUC-ROC$={\displaystyle \underset{0}{\overset{0}{\int }}TPR\cdot \left(FP{R}^{-1}\left(t\right)\right)dt}$	0.818	It demonstrates that the neural-network-based anomaly predicting system has a robust capability to distinguish between anomalous and normal data, with an 81.8% area under the ROC curve.
Mean response time	$MRT={\displaystyle \frac{1}{N}}\cdot {\displaystyle \sum _{i=1}^N\left(T_i^{detected}-T_i^{occured}\right)}$	1.611 s	It indicates that the neural-network-based anomaly predicting system performs efficiently, processing data and generating predictions within a short timeframe.
Anomaly detection accuracy	$TPR={\displaystyle \frac{TP+TN}{TP+TN+FP+FN}}$	0.982	It indicates that the neural-network-based predicting system achieves a high level of precision in correctly identifying anomalies, with 98.2% accuracy.
Precision	$Precision={\displaystyle \frac{TP}{TP+FP}}$	0.979	It signifies that the neural-network-based anomaly predicting system correctly identifies 97.9% of detected anomalies, minimizing false positives.
Recall	$Recall={\displaystyle \frac{TP}{TP+FN}}$	1.0	It indicates that the neural-network-based anomaly predicting system achieves complete sensitivity, identifying all actual anomalies without any misses.
F1	$F1={\displaystyle \frac{2\cdot Precision\cdot Recall}{Precision+Recall}}$	0.989	It reflects that the neural-network-based anomaly predicting system balances high precision and recall, achieving an excellent overall performance in anomaly detection.

. Accordingly, the confusion matrix is given in

Table 3. The confusion matrix (author’s research).

Actual/Predicted	Predicted Anomaly	Predicted Normal
Actual Anomaly	TP	FN
Actual Normal	FP	TN

.

True positive (TP) represents the number of cases where the system correctly identified an anomaly, that is, the system classified an event as an anomaly, and it was indeed an anomaly. True negative (TN) represents the number of cases where the system correctly identified data as normal (not an anomaly), that is, the system classified an event as normal, and it was indeed normal. False positive (FP) represents the number of false alarms where the system mistakenly classified normal data as anomalies, that is, the system identified an event as an anomaly, but it was actually not an anomaly. False negative (FN) represents the number of missed anomalies where the system did not recognize an anomaly, that is, the system classified an event as normal, but it was actually an anomaly.

A comparative analysis (

Table 4. Comparative analysis results (author’s research).

Metric	Developed Neural Network System	Neural Network System Based on an Autoassociative Neural Network [2]	Hybrid Mathematical Model Utilizing Runge–Kutta Optimization and Bayesian Hyperparameter Optimization
Mean squared error	0.218%	0.218%	0.172%
Mean absolute error	0.232%	0.233%	0.179%
Determination coefficient	0.983	0.980	0.784
Percentage of outliers correctly replaced by the reconstructor	98.73%	98.71%	78.97%
True positive rate	0.818	0.819	0.655
False positive rate	0.0112	0.0113	0.0182
False negative rate	0.0101	0.0101	0.0177
AUC-ROC	0.818	0.818	0.654
Mean response time	1.611 s	1.613 s	2.012 s
Anomaly detection accuracy	0.982	0.981	0.785
Precision	0.979	0.980	0.784
Recall	1.0	1.0	0.993
F1	0.989	0.988	0.859

) was conducted between the developed neural network system (Figure 1), a neural network system based on an autoassociative neural network (autoencoder) created by the same research team [2], and a hybrid mathematical model utilizing Runge–Kutta optimization and Bayesian hyperparameter optimization [31], according to the quality metrics presented in Table 2.

As shown in Table 4, the quality metric values for the developed neural network system (Figure 1) and the neural network system based on the autoassociative neural network (autoencoder) [2] are approximately equal. This indicates that both systems are interchangeable and can serve as alternatives to one another. The quality metrics for the developed neural network system (Figure 1) are, on average, up to 20% higher than those of the hybrid mathematical model utilizing Runge–Kutta and Bayesian hyperparameter optimization.

For ROC analysis [57,58] across the three approaches (the developed neural network system, the neural network system based on an autoassociative neural network (autoencoder) [2], and the hybrid mathematical model utilizing Runge–Kutta optimization and Bayesian hyperparameter optimization [31]), true positive and false positive rates were calculated for each category and technique, followed by the creation of the respective ROC curves. This procedure involved defining a binary classification for each category, isolating it from the others. The ROC analysis results are presented in

Table 5. Comparative analysis results (author’s research).

Metric	Developed Neural Network System	Neural Network System Based on an Autoassociative Neural Network [2]	Hybrid Mathematical Model Utilizing Runge–Kutta Optimization and Bayesian Hyperparameter Optimization
True positive	98	99	5
True negative	3	2	11
False positive	285	288	293
False negative	18	15	104
True positive rate	0.818	0.819	0.655
False positive rate	0.0112	0.0113	0.0182
False negative rate	0.0101	0.0101	0.0177
AUC-ROC	0.818	0.818	0.654

.

Figure 12 distinctly illustrates the variation in the regions beneath the AUC curve for the developed neural network system.

Based on the obtained AUC-ROC metric values, it is noted that the developed neural network system (Figure 1) and the system based on an autoassociative neural network (autoencoder) [2] deliver high accuracy with a minimal false alarm rate. The use of the model employing Runge–Kutta and Bayesian hyperparameter optimization [31] results in lower accuracy and the highest number of false alarms.

4. Discussion

This article represents a continuation in the research field of the intelligent monitoring and operational control of helicopter TEs under onboard conditions. The proposed neural network system for anomaly prediction can function as a neural network module within the helicopter TE monitoring and operation control onboard expert system. It is well-known that predicting the helicopter TE thermogasdynamics parameters based on data obtained from standard sensors during flight is the critical aspect of helicopter flight performance. The developed system allows for real-time data analysis and decision making based on the predicted parameters, thereby enhancing the reliability and efficiency of helicopter operations. Implementing this neural network system into helicopter TE monitoring and control onboard expert systems can significantly improve the fault diagnosis and prediction quality.

By employing a modified LSTM network with a long short-term memory layer as the predictor, which also implements the reconstructor model, the developed system predicts the helicopter TE thermogasdynamics parameters with an accuracy of up to 97.9%. This allows the data cleaning module to accurately (98.2% accuracy achieved) predict normal values and detect outliers in the data. The mean squared error does not exceed 0.025 (2.5%). Although the achieved prediction accuracy of 97.9% is sufficient for asset maintenance planning, it may not be reliable enough for decision making in safety-critical operations, where higher precision and lower error margins are essential. Further refinement is needed to ensure the system’s robustness in scenarios requiring stringent operational safety [59,60] standards.

The application of the anomaly detector model in the developed system enabled the detection of two anomalies in the TV3-117 engine gas temperature before the compressor turbine sensor readings during helicopter flight, based on two minutes of sensor activity. The first anomaly (“Top 1 dissonance”) indicates a brief sensor malfunction, while the second (“Top 10 dissonance”) reflects a sharp drop in gas temperature before the compressor turbine due to increased airflow speed and changing external pressure. The false positive rate does not exceed 1.12%, the false negative rate does not exceed 1.01%, and the anomaly detection time does not exceed 1.611 s, which highlights the high efficiency of the developed system.

The AUC-ROC value of 0.818 suggests that the developed system has a strong ability to differentiate between normal and anomalous data, making it a reliable tool for accurate anomaly detection. This means the system effectively minimizes both false alarms and missed anomalies, confirming its high performance in real-world operational conditions.

The limitations related to the obtained results include the dependency on the accuracy in predicting and detecting anomalies, itself dependent on the quality of data provided by onboard sensors, which may be affected by technical malfunctions or data noise. The proposed system utilizes a modified LSTM network that demonstrates high accuracy (up to 97.9%); however, this accuracy may vary depending on helicopter operating conditions and sensor status. Additionally, the model requires significant computational resources for real-time data processing, which may limit its applicability in environments with restricted computational capacity.

The prospects for further research in the field of helicopter TE intelligent monitoring and operational control include optimizing and adapting the proposed neural network system to work with a wide range of aircraft and operational conditions. Particular interest lies in integrating various data processing methods and improving the reconstruction and prediction algorithms, which will enhance anomaly detection accuracy and speed. Additionally, exploring the potential of hybrid models that combine the strengths of the neural network and traditional approaches could lead to the creation of more robust and reliable diagnostic systems capable of operating effectively under limited computational resources and changing external factors.

Furthermore, future research prospects include conducting similar computational experiments with an expanded dataset size for generalizability and extending the on-board helicopter condition-monitoring experiment time to improve reliability.

5. Conclusions

The scientific novelty of the obtained results lies in the development of a neural network system for predicting anomalous data in sensor systems, which, through a preprocessor, predictor, reconstructor, and anomaly detector, enables anomaly prediction in sensor data with an accuracy of up to 98.2%. The scientific novelty in this article was achieved as follows:

• A preprocessor model based on the SARIMAX model was developed, allowing for the determination of autocorrelation and partial autocorrelation in the training dataset while accounting for dynamic changes (e.g., seasonal variations) and external regressors and trends, enabling time series prediction with an accuracy of up to 97.9%.
• A predictor model based on a modified LSTM network with additional Dropout and Dense layers was developed, enabling sensor-derived value prediction. An example of the prediction of TV3-117 aircraft engine gas temperature before the compressor turbine values for one minute of helicopter flight experimentally confirmed that the prediction error did not exceed 0.218%.
• A reconstructor model was developed, which, by utilizing the distribution of prediction errors, allowed for the restoration of missing time series values and outlier replacement with plausible synthetic values with an accuracy of up to 98.73%.
• An anomaly detector model was developed that, through the application of the concept of dissonance, allowed for sensor data anomaly prediction. This enabled the detection of two anomalies based on two minutes of activity from the TV3-117 aircraft engine gas temperature before the compressor turbine sensor during flight: a brief sensor malfunction and a sharp temperature drop due to changes in airflow, with type I and type II errors below 1.12 and 1.01%, respectively, and a detection time of less than 1.611 s, confirming the system’s high efficiency.
• An AUC-ROC value of 0.818 indicated that the developed system has a strong capability to distinguish between normal and anomalous data, making it a reliable tool for accurately detecting anomalies while effectively minimizing both false positives and missed detections.