Multi-source data fusion-based knowledge transfer for unmanned aerial vehicle flight data anomaly detection and recovery

Abstract

Flight data anomaly detection (AD) is essential for unmanned aerial vehicle (UAV) health management. Despite the current dominance of data-driven approaches, their effectiveness often requires sufficient data for model training. However, in practice, it is inevitable to face the situation of limited data, such as the high cost of data acquisition and the difficulty of collecting data in special scenarios, resulting in the performance degradation of the traditional data-driven methods with limited samples. This paper proposes an innovative data-driven approach leveraging transfer learning to detect and recover abnormal UAV flight data with limited samples through multi-source data fusion. First, a data-driven framework based on one-dimensional convolutional neural network and bi-directional long short-term memory (1D CNN-BiLSTM) with parameter selection and residual smoothing (1DCB-PSRS) is proposed. It employs the designed 1D CNN-BiLSTM prediction model for fully extracting spatiotemporal features of flight data, the maximum information coefficient (MIC) for parameter selection, and the exponentially weighted moving average (EWMA) for residual smoothing, thereby improving the AD and recovery performance. Second, multiple source domains with sufficient data are fused to pre-train the model to gain initialized parameters for the target domain. Then, the model is fine-tuned using limited training samples in the target domain through model-based transfer learning method and is evaluated using test data of the target domain. Finally, the effectiveness of the proposed method is verified on real UAV flight data.

Introduction

In recent years, the widespread application of unmanned aerial vehicles (UAVs) in fields such as the internet of things^1,2, emergency monitoring³, and precision agriculture⁴ has propelled the rapid expansion of the UAV market^5,6,7. However, as the UAV market continues to flourish, there is a growing demand for enhanced safety and reliability of UAVs. Compared with manned aircraft, UAVs have a higher accident rate due to factors such as strong autonomy, complex flight environments, and greater susceptibility to external disturbances, which results in significant economic losses for relevant countries and companies⁸. Thus, ensuring the flight safety and reliability of UAVs have become a crucial topic⁹. Flight data reflects the operational status of the UAV and the health status of each key component, providing an important basis for UAV health management operation and maintenance. By monitoring and analyzing flight data, potential faults or anomalies can be detected, and necessary preventive measures can be taken to reduce flight risks. Therefore, conducting research on flight data anomaly detection (AD) and recovery plays a crucial role in improving the safety and reliability of UAVs.

In recent years, many studies have been conducted on UAV flight data AD, including knowledge- and model-based, and data-driven approaches⁸. Knowledge- and model-based approaches rely on expert knowledge or accurate mathematical models to detect anomalies in flight data^10,11,12,13. However, due to the complexity of UAV structures^14,15, it is very difficult to obtain rich expert knowledge or build accurate mathematical models, making these methods limited in practical applications. In contrast, data-driven methods can effectively learn and detect anomalies from extensive UAV flight data without the need for expert knowledge or precise mathematical models and have received much attention and research. For example, He et al.¹⁶ proposed a structured sparse subspace learning (SSL) AD (SSLAD) algorithm to detect and identify anomalous UAV flight data. Furthermore, they developed an online method based on subspace tracking to realized high-accuracy UAV AD and recovery performance¹⁷. Zhong et al.¹⁸ presented a spatiotemporal correlation based on long short-term memory (STC-LSTM) method that effectively identified anomalies in UAV roll angle. Guo et al.¹⁹ proposed an approach based on uncertainty-aware LSTM (UA-LSTM) for UAV actuator fault detection. Wang et al.²⁰ designed a data-driven UAV fault detection method using stacked LSTM. He et al.²¹ used an improved graph attention network (GAT) method for UAV fault detection. Park et al.²² proposed an unsupervised fault detection model based on a stacked auto-encoder (AE), which was trained using normal UAV flight data and differentiated between safe and abnormal states based on reconstruction errors. Yang et al.²³ designed a model based on AE and LSTM and was verified on real flight data. Bae et al.²⁴ used an model based on AE and LSTM to detect abnormal flight parameters. Bell et al.²⁵ proposed an LSTM-based deep learning autoencoder combined with a dynamic thresholding algorithm for detecting abnormal flight data. Wang et al.²⁶ used an LSTM model with residual filtering (LSTM-RF) to detect bias and drift anomalies of UAV flight data. Yang et al.^27,28 used multivariate regression analysis to detect anomalies in real flight data and achieved good detection performance. Lu et al.²⁹ proposed a novel multirate-aware LSTM data-driven approach for AD of UAV flight data. Wang et al.³⁰ proposed a deeply integrated AE-based AD method to detect UAV flight data injected with anomalies. Jiang et al.³¹ introduced a robust spatialtemporal AE model and validated its effectiveness on real flight data. Despite some progress in existing data-driven approaches, their effectiveness often relies on the availability of sufficient data for model training. However, obtaining sufficient data for model training is not always feasible. For example, during the test flight of UAVs, data samples are limited because only specific flight phases or key performance indicators are tested, rather than conducting long-duration flights. In addition, formulating new flight plans to acquire sufficient flight data may also require expensive flight costs. This may make it difficult for existing methods to capture complex and diverse anomaly patterns from limited samples, resulting in lower AD accuracy. Therefore, enhancing flight data AD performance with limited UAV samples is still a major challenge and problem at present.

Transfer learning is an effective solution that significantly improves the performance of target domain tasks by transferring the knowledge learned from a source domain with abundant data to a target domain with limited data³². The benefits of transfer learning have been widely demonstrated in time series prediction³³ and fault diagnosis^34,35. For example, Zhang et al.³³ used a bi-directional long short-term memory (BiLSTM) recurrent neural network-based transfer learning for remaining useful life (RUL) estimation. Lu et al.³⁶ proposed a model-based transfer learning approach to model the target prediction task using knowledge acquired from the source task, enabling heat load prediction. Tariq et al.³⁷ developed a residual neural network-based transfer learning method for predicting subway platform PM2.5 health risk level sequences. Li et al.³⁵ utilized a priori knowledge from multiple fault classification tasks with known operating conditions to achieve bearing fault diagnosis with unknown operating conditions. Zhang et al.³⁸ used a decentralized joint transfer learning approach aided by blockchain technology for collaborative mechanical fault diagnosis. These studies provide valuable experience and insights for UAV flight data AD with limited samples. However, the performance of these methods often relies on single source domain information, which limits their applicability in a wider range of scenarios. This is because data from a single source domain may lack sufficient representative features, especially when there is a significant distribution difference between the source and target domains, making it difficult to obtain desirable results. Multi-source data fusion-based transfer learning methods serve as an effective extension and supplement. By integrating information from multiple source domains, they reduce over-reliance on data from any single source domain, thereby enhancing model performance³⁹. However, in the field of UAV flight data AD, research on transfer learning methods based on multi-source data fusion is still limited.

To address the above challenges and inspired by transfer learning, a data-driven knowledge transfer approach based on multi-source data fusion is proposed in this paper for detecting and recovering abnormal UAV flight data. First, to reduce the negative impact of irrelevant flight parameters on the model, a correlation analysis method based on maximum information coefficient (MIC) is adopted to select flight parameters with correlation as model inputs. Second, combining the advantages of one-dimensional convolutional (1D CNN) in local feature extraction and BiLSTM in capturing long-term dependencies, a prediction model based on 1D CNN-BiLSTM is designed to fully extract the spatiotemporal features of flight data and obtain the predicted values of the monitored parameter. Third, multiple source domains with sufficient data are fused for pre-training the model, and the limited training samples and test data of the target domain are used to fine-tune and evaluate the model. Finally, the exponentially weighted moving average (EWMA) is further introduced to smooth the residuals, thus reducing the interference of random noise and improving the AD performance. Specifically, the main contributions of this paper are described as follows:

• 1. A data-driven framework based on 1D CNN-BiLSTM with parameter selection and residual smoothing (1DCB-PSRS) is proposed. It inherits the strengths of 1D CNN and BiLSTM in extracting local features and capturing long-term dependencies, utilizes MIC to select relevant flight parameters as model inputs, and employs EWMA to smooth residuals, thereby fully extracting the spatiotemporal features of flight data and reducing the interference of irrelevant flight parameters and random noise on the model performance.

• 2. A knowledge transfer method based on multi-source data fusion is proposed for UAV flight data AD and recovery with limited samples. It leverages data distribution information from multiple source domains and performs knowledge transfer through sharing the model parameters to avoid reliance a single source domain data and improve the AD and recovery performance in the target domain with limited samples.

• 3. Taking complex high-dimensional real UAV flight data as the experimental object, the effectiveness of 1DCB-PSRS as the pre-training framework and the knowledge transfer method based on multi-source data fusion is verified.

The rest of this paper are structured as follows. Section "Preliminaries" introduces the fundamental theory. Section "Methodology framework" outlines the proposed methodology framework. Section "Experiments" presents the experimental results and analysis. Section "Conclusion and prospects" summarizes this work and gives the future research directions.

Preliminaries

One-dimensional convolutional neural network

One-dimensional convolutional neural network (1D CNN) is easy to train and implement and has been widely used for time-series data processing and analysis^40,41,42. The structure of 1D CNN is depicted in Fig. 1. The convolution layer is used to capture the local features of the input signal⁴³, which is defined as follows:

$$Z_{j}^{l} = \sigma \left( {\sum\limits_{{i = 1}}^{{N_{{l - 1}} }} {Conv\;1D\;\left( {W_{{i,j}}^{l} ,x_{i}^{{l - 1}} } \right) + b_{j}^{l} } } \right)$$

(1)

where ${Z}_{j}^{l}$ is the j^th attribute mapping of the j^th layer. ${x}_{i}^{l-1}$ and ${N}_{l-1}$ are the i^th attribute mapping and the number of attribute mappings of l-1^th layer, respectively. Conv1D $(\bullet )$ denotes the convolution operation. ${W}_{i,j}^{l-1}$ is the convolution kernel. ${b}_{j}^{l}$ is the bias. $\sigma (\bullet )$ is the activation function. Then, ${Z}_{j}^{l}$ is input into the pooling layer. Commonly used maximum and average pooling operations⁴⁴ are defined as follows:

$$Z_{j}^{l + 1} = Max_{pooling} (Z_{j}^{l} )$$

(2)

$$Z_{j}^{l + 1} = A{\text{ver}}_{pooling} (Z_{j}^{l} )$$

(3)

where ${Z}_{j}^{l+1}$ denotes the j^th attribute mapping of the l + 1^th layer. Max_pooling $(\bullet )$ and Aver_pooling $(\bullet )$ denote maximum and average pooling operations, respectively. Finally, feature integration is performed through the fully connected layer, which is defined as follows:

$$Z^{l} = f(W_{full} x^{l - 1} + b_{full} )$$

(4)

where ${W}_{full}$ is the weights, ${b}_{full}$ is the bias, ${Z}^{l}$ is the output, and f $(\bullet )$ is the nonlinear activation function.

Fig. 1

Structure of 1D CNN.

Bi-directional long short-term memory

Bi-directional long short-term memory (BiLSTM) overcomes the unidirectional information flow limitation of traditional LSTM⁴⁵, thereby reducing the loss of crucial contextual information, as shown in Fig. 2. By processing the input sequences in forward and backward directions, BiLSTM obtains hidden state representations in these two directions. This bi-directional information flow allows BiLSTM to synthesize the information from the input sequences to better learn the feature representation of these sequences. Therefore, BiLSTM shows significant advantages in complex sequence modeling tasks, such as question-answering systems⁴⁶, named entity recognition⁴⁷, and text classification⁴⁸.

Fig. 2

Structure of BiLSTM.

Each LSTM unit of the BiLSTM network contains input (${i}_{t}$), forget (${f}_{t}$), and output (${o}_{t}$) gates and a memory cell. At time step t, the two update processes of the LSTM unit, i.e., forward and backward propagation, are defined as follows:

$$\overrightarrow {f}_{t} = \sigma (\overrightarrow {W}_{hf} \overrightarrow {h}_{t - 1} + \overrightarrow {W}_{xf} \vec{x}_{t} + \overrightarrow {b}_{f} )$$

(5)

$$\overrightarrow {i}_{t} = \sigma (\overrightarrow {W}_{hi} \overrightarrow {h}_{t - 1} + \overrightarrow {W}_{xi} \vec{x}_{t} + \overrightarrow {b}_{i} )$$

(6)

$$\vec{c}_{t} = \overrightarrow {f}_{t} \odot \vec{c}_{t - 1} + \overrightarrow {i}_{t} \odot \tanh (\overrightarrow {W}_{hc} \overrightarrow {h}_{t - 1} + \overrightarrow {W}_{xc} \vec{x}_{t} + \overrightarrow {b}_{c} )$$

(7)

$$\vec{o}_{t} = \overrightarrow {W}_{ho} \overrightarrow {h}_{t - 1} + \overrightarrow {W}_{xo} \vec{x}_{t} + \overrightarrow {b}_{o}$$

(8)

$$\overrightarrow {h}_{t} = \vec{o}_{t} \odot \tanh (\vec{c}_{t} )$$

(9)

$$\overleftarrow {f}_{t} = \sigma (\overleftarrow {W}_{hf} \overleftarrow {h}_{t - 1} + \overleftarrow {W}_{xf} \vec{x}_{t} + \overleftarrow {b}_{f} )$$

(10)

$$\overleftarrow {i}_{t} = \sigma (\overleftarrow {W}_{hi} \overleftarrow {h}_{t - 1} + \overleftarrow {W}_{xi} \vec{x}_{t} + \overleftarrow {b}_{i} )$$

(11)

$$\mathop{c}\limits^{\leftarrow} _{t} = \overleftarrow {f}_{t} \odot \mathop{c}\limits^{\leftarrow} _{t - 1} + \overleftarrow {I}_{t} \odot \tanh (\overleftarrow {W}_{hc} \overleftarrow {h}_{t - 1} + \overleftarrow {W}_{xc} \mathop{x}\limits^{\leftarrow} _{t} + \overleftarrow {b}_{c} )$$

(12)

$$\mathop{o}\limits^{\leftarrow} _{t} = \overleftarrow {W}_{ho} \overleftarrow {h}_{t - 1} + \overleftarrow {W}_{xo} \mathop{x}\limits^{\leftarrow} _{t} + \overleftarrow {b}_{o}$$

(13)

$$\overleftarrow {h}_{t} = \mathop{o}\limits^{\leftarrow} _{t} \odot \tanh (\mathop{c}\limits^{\leftarrow} _{t} )$$

(14)

where → and ← denote forward and backward propagation, respectively. ${W}_{hf}$, ${W}_{hi}$, ${W}_{hc}$, ${W}_{ho}$, ${W}_{xf}$, ${W}_{xi}$, ${W}_{xc}$, and ${W}_{xo}$ are the weights in the model update process. ${b}_{f}$, ${b}_{i}$, ${b}_{c}$, and ${b}_{o}$ are the biases in the model update process. ⨀ denotes the product. σ is the activation function. ${c}_{t}$ is the memory cell. After forward and backward propagation, the final output ${h}_{t}$ is defined as follows:

$$h_{t} = \overrightarrow {h}_{t} \odot \overleftarrow {h}_{t}$$

(15)

Transfer learning

Transfer learning aims to utilize knowledge acquired from one specific task (known as the source task) and apply it to another related task (known as the target task)⁴⁹. Transfer learning includes domain D and task T. The domain D is the learning body that contains $X=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}$ and $P(X)$, where $X=\{{x}_{1},{x}_{2},\dots ,{x}_{n}\}$ is the feature space and $P(X)$ is the the marginal probability distribution. Similarly, for the task T that contains the label space ${l}_{i}$ and the conditional probability distribution $P(Y|X)$, where $Y=\{{y}_{1},{y}_{2},\dots ,{y}_{n}\}$. When solving practical problems, there usually includes the source domain ${D}_{s}=\{{X}_{s},P({X}_{s})\}$, the source task ${T}_{s}=\{{Y}_{s},P({Y}_{s}|{X}_{s})\}$, the target domain ${D}_{t}=\{{X}_{t},P({X}_{t})\}$, and the target task ${T}_{t}=\{{Y}_{t},P({Y}_{t}|{X}_{t})\}$ of transfer learning, as shown in Fig. 3. ${D}_{s}$ contains sufficient data and can train the model well while ${D}_{t}$ is the opposite. In the process of transfer, the knowledge is represented by P(${Y}_{s}|{X}_{s}$) of the source task or f_s(.)³⁶, where P(${Y}_{s}|{X}_{s}$) is the conditional probability distribution and f_s(.) is the prediction function.

Fig. 3

The general process of transfer learning.

Methodology framework

Figure 4 illustrates the proposed methodology framework in this paper. First, the MIC is utilized to select relevant flight parameters as model inputs. Second, the raw multiple UAV flight datasets are normalized. Third, the processed UAV flight datasets are divided into multiple source domains and one target domain. Each source domain has sufficient data and can train the model well, while the target domain is the opposite. Fourth, these source domains are fused into a new source domain to pre-train the 1D CNN-BiLSTM model to obtain initialized parameters for the target domain, and the limited training samples and test data of the target domain are used to fine-tune and test the 1D CNN-BiLSTM model. Then, the training and test residuals are calculated based on the model’s predicted values and the original values. Next, the EWMA is applied to smooth the training and test residuals, and the threshold is calculated based on the smoothed training residuals. Finally, AD is achieved by comparing the smoothed test residuals with the threshold, and data recovery is realized by replacing the anomalous data with the model’s predicted values.

Fig. 4

The proposed methodology framework for UAV flight data AD and recovery in this paper.

Data preprocessing

The normalization method⁵⁰ shown in Eq. (16) is used to normalize the raw flight data ${x}^{raw}$ to eliminate magnitude differences.

$$x_{norm} = \frac{{x^{raw} - x_{\min }^{raw} }}{{x_{\max }^{raw} - x_{\min }^{raw} }}$$

(16)

where ${x}_{norm}$ is the normalized data. ${x}_{max}^{raw}$ is the maximum value and ${x}_{min}^{raw}$ is the minimum value of ${x}^{raw}$, respectively.

Structure of 1D CNN-BiLSTM model

Many existing studies mainly use LSTM for modeling and feature extraction of UAV flight data^18,19,20,26. This may result in problems such as inadequate feature extraction and insufficient flight data modeling capability. This is because LSTM focuses mainly on the extraction of time series features and ignores critical local features. To solve these problems, the prediction 1D CNN-BiLSTM model constructed based on 1D CNN and BiLSTM can better consider temporal and local features, which includes Conv1D, Max pooling, BiLSTM, Dense, and Dropout layers, as depicted in Fig. 5, with input and output sizes as shown in Table 1.

Fig. 5

Structure of the designed 1D CNN-BiLSTM.

Table 1 Input and output sizes of each layer.

Model-based transfer learning method

Transfer learning includes instance-, feature-, relationship-, and model-based (parameter-based) transfer learning³². Instance-based transfer learning involves reusing data samples to facilitate knowledge transfer. Feature-based transfer learning facilitates knowledge transfer by transforming data features from source and target domains into a common feature space. Relation-based transfer learning leverages inter-domain relationships to facilitate knowledge transfer from the source domain to the target domain. Model-based transfer learning entails transferring a pre-trained model from the source domain to the target domain. Considering aspects such as data volume and task complexity, this paper chooses the model-based transfer learning method. The structure of transfer learning is shown in Fig. 6.

Fig. 6

Structure of transfer learning.

As mentioned earlier, there is sufficient data in the source domain to adequately train the model. However, the data in the target domain is limited and training the model directly on the target domain usually does not achieve the desired results. In addition, if the trained model in the source domain is directly applied to the target domain, the performance of the model may be degraded. For this purpose, a full-layer fine-tuning strategy is used. The modeling effect is enhanced by adjusting the model parameter to adapt as much as possible to the target domain requirement. However, when fine-tuning all parameters, the small amount of data in the target domain may lead to model overfitting. Therefore, to mitigate this problem, the Dropout technology is introduced into the designed network to reduce the model overfitting risk. The parameter passing relationship between the source and target tasks can be defined as follows:

$$\theta_{t} = \theta_{s}$$

(17)

where ${\theta }_{t}$ is the weight of the source model and ${\theta }_{s}$ is the weight of the target model. During the fine-tuning process, the updated model parameter is defined as follows:

$$\theta_{s}^{*} = \arg \mathop {\min }\limits_{\theta s} L(\theta_{s} )$$

(18)

where ${\theta }_{s}^{*}$ is the optimized parameter of the model that minimizes the loss function ${L(\theta }_{s})$.

Model training

For the regression model, the mean square error (MSE), which is the average of the squared difference between the predicted and true values, is used as the loss function for the model training in this paper. Assuming that at time t, the predicted output of the model is $y(t)$ and the true value is $\widehat{y}(t)$, the loss function ${L(\theta }_{s})$ can be calculated by Eq. (19).

$$L(\theta_{s} ) = \frac{1}{l}\sum\nolimits_{i = 1}^{l} {(\hat{y}_{i} - y_{i} )^{2} }$$

(19)

where ${\widehat{y}}_{i}$ and ${y}_{i}$ are the i^th elements of $\widehat{y}(t)$ and $y(t)$, respectively, and l is the sample length.

AD and recovery

In the pre-training model stage, assuming that ${e(t)}_{train}$ are the training residuals, which can be calculated by Eq. (20).

$$e(t)_{train} = \hat{y}(t)_{train} - y(t)_{train}$$

(20)

where ${\widehat{y}(t)}_{train}$ and ${y(t)}_{train}$ are the predicted and true data of the training set, respectively. t is an index value. Since random noise makes the residuals fluctuate dramatically, the exponentially weighted moving average (EWMA)⁵¹ is introduced to smooth the residuals ${e(t)}_{train}$, which can be defined as follows:

$$\tilde{e}(t)_{train} = \alpha \tilde{e}(t - 1)_{train} + (1 - \alpha )e(t)_{train}$$

(21)

where ${\widetilde{e}(t)}_{train}$ and ${e(t)}_{train}$ are the smoothed and actual training residuals of the t^th data point, respectively. α is an adjustable hyperparameter. Then, the mean value ${\mu }_{train}$ of ${\widetilde{e}(t)}_{train}$ can be calculated by Eq. (22).

$$\mu_{train} = \frac{1}{l}\sum\limits_{t = 1}^{l} {\tilde{e}(t)_{train} }$$

(22)

where l is the training data length. Based on Eqs. (21)–(22), the standard deviation of ${\widetilde{e}(t)}_{train}$ can be calculated by Eq. (23).

$$\sigma_{train} = \sqrt {\frac{1}{l - 1}\sum\limits_{t = 1}^{l} {(\tilde{e}(t)_{train} - \mu_{trian} )^{2} } }$$

(23)

Based on Eqs. (22) and (23), the anomaly determination threshold can be calculated by Eq. (24).

$$T = \mu_{train} + \ell \sigma_{train}$$

(24)

where ${\ell}$ is a constant, which is set to 1 in this paper. For the test set, the smoothed test residuals ${\widetilde{e}(t)}_{test}$ are also obtained based on Eq. (21). Then, AD can be calculated by Eq. (25).

$$AD = \left\{ \begin{gathered} \tilde{e}(t)_{test}^{\lambda } > T, \, abnormal \hfill \\ \tilde{e}(t)_{test}^{\lambda } \le T, \, normal \hfill \\ \end{gathered} \right.$$

(25)

where ${\widetilde{e}(t)}_{test}^{\lambda }$ is the λ^th smoothed test residual. If ${\widetilde{e}(t)}_{test}^{\lambda }$ is greater than T, it is recognized as abnormal and vice versa. For data recovery, it can be defined as follows:

$$\hat{y}(t)_{test} = f_{{1D{\text{ CNN - BiLSTM}}}} (x(t)_{test} )$$

(26)

where ${\widehat{y}(t)}_{test}$ is the predicted data for the test set ${x(t)}_{test}$. ${f}_{1D CNN-BiLSTM}(\bullet )$ is the mapping function.

Experiments

Data description

This study uses UAV flight datasets without anomalies that are publicly available from the University of Minnesota^52,53,54. These datasets include the Thor-type UAV flight data with different flight times and scenarios. Table 2 shows the information related to the datasets used in this paper in detail, including dataset name, flight number, raw parameter length, raw parameter number, experiment range, experiment length, sampling frequency, and domain type. It should be noted that the source domains Flight97 and Flight104 are fused to a new source domain, denoted as Flight97-104 dataset, and the specific fusion strategy will be described subsequently.

Table 2 Description of the experimental datasets in this paper.

Given the high cost and extreme difficulty of obtaining real abnormal UAV flight data is extremely difficult, an anomaly injection method is used to generate anomalous data. This is in line with most research practices in the field at this stage^{18,21,26,27,28}. The anomaly injection method can simulate many anomaly types and prevent accidents from occurring in UAVs caused by collecting abnormal data. In this paper, bias and drift anomalies are injected into real UAV flight data as shown in Eqs. (27)–(28), respectively, to validate the model performance. Specifically, bias or drift anomalies in UAV flight data mainly result from fixed or dynamic factors, leading to continuous or gradual deviations from the expected values.

$$y(t)_{bias} = y(t) + \xi$$

(27)

$$y(t)_{drift} = y(t) + \vartheta (t)$$

(28)

where y(t) is the raw data and ξ is the deviation constant. ϑ(t) is a function of t.

NAV north velocity (navvn) is critical to the navigation precision and stability of UAVs, and it is selected as the detection parameter in this paper. In fact, ξ and ϑ(t) are not accurately known when the UAV suffers from bias or drift anomalies. Therefore, in this paper, the values of ξ are injected into the flight97, flight98, and flight104 datasets with values of 5, 3, and 5, respectively, to simulate bias anomaly, and the values of ϑ(t) are uniformly taken in the range of [3,4,5,6], and [5,6], respectively, to simulate drift anomaly. Specifically, anomalies are injected into the last 1000 data points of the Flight97, Flight98, and Flight104 datasets, with anomaly rates of 0.0741, 0.0625, and 0.0714, respectively. Figure 7 illustrates the parameter navvn after injection of bias and drift anomalies, with the blue curves representing normal data and the red curves representing abnormal data.

Fig. 7

Flight data after injecting bias and drift anomalies.

Since many irrelevant flight parameters could influence the model’s performance, selecting mission-relevant flight parameters for experiments before model training is necessary. Therefore, the maximal information coefficient (MIC)⁵⁵ is employed to select correlation parameters for model inputs, effectively avoiding the crucial information loss of nonlinear correlation parameters. The related calculation process of MIC is shown in Eqs. (29) and (30).

$$MI(\hbar ,\psi ) = \sum\limits_{\hbar ,\psi } {P(\hbar ,\psi )} \log_{2} \frac{P(\hbar ,\psi )}{{P(\hbar )P(\psi )}}$$

(29)

$$MIC(\hbar ,\psi ) = \mathop {\max }\limits_{\hbar \psi < B(n)} \frac{MI(\hbar ,\psi )}{{\log_{2} (\min \{ |\hbar |,|\psi |\} )}}$$

(30)

where MI(ℏ,ψ) and P(ℏ,ψ), and P(ℏ) and P(ψ) are the mutual information and joint probability density, and the boundary probability densities of ℏ and ψ, respectively. Table 3 lists the 15 flight parameters with strong correlation with navvn according to the MIC correlation analysis results. The correlation determination threshold is taken as 0.6.

Table 3 Flight parameters with correlation to Navvn.

Experimental setup

The experimental datasets are divided into training and testing sets in an 8:2 ratio. The Flight97-104 dataset contains the training set of the Flight97 dataset and the entire Flight104 dataset. The experimental setup includes a Windows 10 (64-bit) operating system, an AMD Ryzen 5 3600 6-Core @3.6 GHz processor, and TensorFlow 2.12.0. In particular, the optimal hyperparameter selection is determined using a grid search method during the pre-training stage. The search ranges for the learning rate are set to [0.001, 0.0015, 0.002], for the dropout rate to [0.1, 0.2, 0.3], and for the batch size to [32, 64, 128]. The number of epochs is set to 100.

Evaluation metrics

This paper uses the true positive rate (TPR), false positive rate (FPR), and accuracy (ACC) rate as the evaluation metrics for AD, and the mean absolute error (MAE) and root mean square error (RMSE) as the evaluation metrics for data recovery, as shown in Eqs. (31)–(35)^56,57, respectively.

$$TPR = \frac{TP}{{TP + FN}} \times 100\%$$

(31)

$$FPR = \frac{FP}{{TN + FP}} \times 100\%$$

(32)

$$ACC = \frac{TP + TN}{{TP + TN + FP + FN}} \times 100\%$$

(33)

$$MAE = \frac{1}{L}\sum\limits_{i = 1}^{L} {|y_{i} - \hat{y}_{i} |}$$

(34)

$$RMSE = \sqrt {\frac{1}{L}\sum\limits_{i = 1}^{L} {(y_{i} - \hat{y}_{i} )^{2} } }$$

(35)

where TP and TN, and FP and FN are correctly identified normal and abnormal data, and incorrectly identified normal and abnormal data, respectively. L is the parameter length. ${y}_{i}$ and ${\widehat{y}}_{i}$ are the raw data without anomalies and the inverse normalized data at point i, respectively. The smaller values of FPR, MAE, and RMSE and the higher values of TPR and ACC indicate better model performance.

AD and recovery results

Performance of direct training with different domains

AD

• 1. AD

  To verify the effectiveness of 1DCB-PSRS as the pre-trained framework, the regression models based on LSTM-RF²⁶, STC-LSTM¹⁸, 1D CNN, BiLSTM, temporal convolutional network (TCN), and support vector regression (SVR) are used for the baseline methods. Figure 8 shows the bias AD results of these methods on all datasets. 1DCB-PSRS obtains the highest ACC values for different datasets. For the Flight97 Flight98, Flight104, and Flight97-104 datasets, 1DCB-PSRS achieves the ACC values of 95.74%, 96.47%, 97.46%, and 95.87% respectively. Meanwhile, 1DCB-PSRS reaches TPR values of 93.76%, 96.09%, 96.44%, and 94.90% for the Flight97, Flight98, Flight104, and Flight97-104 datasets, respectively. In addition to ACC and TPR values, 1DCB-PSRS also exhibits relatively low FPR values, varying from 0.30% to 2.70% on different datasets. In contrast, BiLSTM and 1D CNN have higher FPR values in most datasets, which means these methods identify more abnormal samples as normal. For example, the FPR values of BiLSTM for all datasets are 11.00%, 24.20%, 6.40%, and 29.30%, respectively. 1D CNN achieves FPR values of 43.50%, 21.40%, and 23.50% for the Flight97, Flight98, and Flight97-104 datasets, respectively. Similarly, STC-LSTM also has high false detection rates on different datasets. This may be due not only to the insufficient feature extraction of STC-LSTM, but also to a certain extent to the influence of random noise. In contrast, the performance of LSTM-RF on different datasets is improved by residual smoothing compared STC-LSTM. However, this improvement is still relatively limited. For example, for the Flight98 dataset, LSTM-RF achieves an FPR value of 19.60%. The AD performance of TCN is relatively poor across different datasets. Specifically, for the Flight97 dataset, the TPR and ACC values of TCN are 76.27% and 84.36%, respectively; for the Flight98 dataset, the FPR value of TCN is as high as 21.10%; and for the Flight97-104 dataset, the FPR value of TCN reaches 31.20%. For different datasets, although SVR can correctly detect most or all abnormal samples, its overall performance is still lower than 1DCB-PSRS in terms of TPR and ACC values. Specifically, for different datasets, the TPR and ACC values of SVR are both below 87.00% and 91.00%, respectively.

  Fig. 8

  

  The bias AD results of 1DCB-PSRS, 1D CNN, BiLSTM, LSTM-RF, STC-LSTM, TCN, and SVR.

  Figure 9 illustrates the drift AD results of the above methods on all datasets. The overall performance of 1DCB-PSRS is still superior to baseline methods. For different datasets, the ACC, TPR, and FPR values of 1DCB-PSRS vary from 94.11% to 98.68%, 91.35% to 97.94%, and 0.00% to 2.00%, respectively. The TPR values of BiLSTM for the Flight97 and Flight98 datasets are 92.35% and 100%, respectively, STC-LSTM for the Flight97 dataset is 96.59%, 1D CNN for the Flight97-104 dataset is 99.29%, and TCN for the Flight97-104 dataset is 96.29%, which are better than that of 1DCB-PSRS. The FPR values of SVR for the Flight98 and Flight97-104 dataset are all 0.00%, which are also better than that of 1DCB-PSRS. However, these methods fail to strike a better balance between accuracy and false detection rate. In contrast, 1DCB-PSRS demonstrates more robustness on different datasets and can maintain high accuracy and low false detection rates.

  Fig. 9

  

  The drift AD results of 1DCB-PSRS, 1D CNN, BiLSTM, LSTM-RF, STC-LSTM, TCN, and SVR.

  Table 4 shows the average values of AD performance metrics for each method on all datasets, denoted by $\overline{\text{ACC} }$, $\overline{\text{TPR} }$, and $\overline{\text{FPR} }$, respectively. Although the $\overline{\text{FPR} }$ value of 1DCB-PSRS for drift anomaly is 1.15%, which is higher than that of SVR, its $\overline{\text{FPR} }$ value for bias anomaly is 0.90%, significantly lower than that of SVR and other baseline methods. In addition, compared to the baseline methods, 1DCB-PSRS demonstrates the best average performance metrics in $\overline{\text{ACC} }$ and $\overline{\text{TPR} }$ values, achieving 96.59% and 95.43% for bias anomaly, and 96.38% and 95.30% for drift anomaly, respectively. The average performance metrics of LSTM-RF are also relatively good with $\overline{\text{ACC} }$, $\overline{\text{TPR} }$, and $\overline{\text{FPR} }$ values of 93.08%, 91.69%, and 4.15%, and 93.30%, 92.49%, and 5.30%, respectively, for bias and drift anomalies. Although the $\overline{\text{TPR} }$ values of BiLSTM for bias and drift anomalies are all 94.17%, second only to 1DCB-PSRS, its $\overline{\text{FPR} }$ values are equally high at 21.98% and 17.73%, respectively. The $\overline{\text{FPR} }$ values of 1D CNN, STC-LSTM, and TCN are also relatively high, ranging from 11.38% to 22.18%. SVR exhibits relatively poor performance in $\overline{\text{ACC} }$ and $\overline{\text{TPR} }$ values, with values of 88.41% and 83.97% for bias anomaly, and 89.33% and 84.26% for drift anomaly, respectively.

  Table 4 The average AD performance of the above methods.

• 2. Data recovery.

  Table 5 presents the data recovery results of the above methods across all datasets. For different anomaly types of the Flight97, Flight98, and Flight104 datasets, 1DCB-PSRS obtains the smallest MAE and RMSE values. Specifically, for bias and drift anomalies of the Flight97 dataset, the MAE and RMSE values of 1DCB-PSRS are 1.3460 and 1.4298, and 1.7795 and 1.9103, respectively. For bias and drift anomalies of the Flight98 dataset, the MAE and RMSE values of 1DCB-PSRS are 0.5475 and 0.6234, and 0.7922 and 0.9071, respectively. For bias and drift anomalies of the Flight104 dataset, the MAE and RMSE values of 1DCB-PSRS are 1.0323 and 0.9806, and 1.3077 and 1.2106, respectively. Although the RMSE value of 1DCB-PSR for bias anomaly of Flight97-104 dataset is 1.3544, slightly higher than that of LSTM-RF, its MAE value is the lowest compared to the baseline methods, at 0.9684. For drift anomaly of the Flight97-104 dataset, the MAE and RMSE values of 1DCB-PSR are the lowest, at 0.9289 and 1.2751, respectively. This benefits from 1DCB-PSRS fully leveraging the advantages of 1D CNN and BiLSTM in time-series data modeling, enabling the comprehensive extraction of spatiotemporal features from flight data, thereby achieving more accurate predictions. In contrast, the data recovery performance of 1D CNN, BiLSTM, LSTM-RF, STC-LSTM, and TCN is relatively inferior, with MAE and RMSE values ranging from 0.6013 to 1.8954 and from 0.8707 to 2.3609, respectively, for different anomaly types of the Flight97, Flight98, Flight104, and Flight97-104 datasets. This is because 1D CNN, BiLSTM, LSTM-RF, STC-LSTM, and TCN either focus on local feature extraction or temporal feature modeling, limiting their representation capability for complex flight data. Due to SVR focusing only on spatial feature modeling while ignoring temporal characteristics, its data recovery performance is the worst, with MAE and RMSE values for bias and drift anomalies across different datasets ranging from 1.9596 to 3.5461 and from 2.8096 to 4.5183, respectively.

  Table 5 Data recovery results for 1DCB-PSR, 1D CNN, BiLSTM, LSTM-RF, STC-LSTM, TCN, and SVR in all datasets.

Performance of transfer learning with limited training or fine-tuning samples

The effectiveness of 1DCB-PSRS has been verified in the above experiments. Therefore, based on 1DCB-PSRS as the pre-trained framework, the feasibility and effectiveness of the knowledge transfer method based on multi-source data fusion is further explored in the target domain with limited samples. Specifically, the first 10% and the first 20% data are used as the training or fine-tuning samples and the last 20% are used as the testing samples in the target domain Flight98 dataset. This helps to provide a more in-depth understanding of the knowledge transfer-based approach’s performance in the target domain with limited samples. In this subsection, four groups of experiments are set up, which are described as follows:

• Direct training with limited samples of the target domain Flight98 dataset, denoted by E1.

• Transfer from the source domain Flight97 dataset to the target domain Flight98 dataset, denoted by E2.

• Transfer from the source domain Flight104 dataset to the target domain Flight98 dataset, denoted by E3.

• Transfer from the source domain Flight97-104 dataset to the target domain Flight98 dataset, denoted by E4.

• 1. AD

  Figure 10 displays the AD results of E1, E2, E3, and E4. For bias and drift anomalies, the ACC, TPR, and FPR values of E1 with 10% training samples are 28.72%, 13.41%, and 37.60%, respectively. Although the ACC and TPR values of E1 with 20% training samples are significantly improved, its FPR values are as high as 74.20% to 75.40%, respectively, for bias and drift anomalies. The reason for E1’s poor performance is model underfitting due to insufficient training samples. The performances of E2, E3, and E4 are substantially improved compared to E1 with 10% fine-tuning samples. The overall AD performances of E2, E3, and E4 with 20% fine-tuning samples are also better than that of E1 with 20% training samples. This indicates that for the tasks in this paper, it is feasible to enhance the performance of the target task with limited samples by transferring the source task knowledge to the target task. E3 shows the lowest FPR values under 10% and 20% fine-tuning samples, ranging only from 1.10% to 1.60%, slightly better than E4. However, in terms of overall performance, E4 outperforms E1, E2, and E3 in both 10% and 20% fine-tuning samples. In particular, E4 obtains the highest ACC and TPR values of 80.25% and 85.75%, and 73.95% and 80.36%, respectively, for bias and drift anomalies with 20% fine-tuning samples. Meanwhile, E4 has relatively lower FPR values, ranging only from 2.30% to 5.80%. In addition, there is no significant improvement in the overall performance of E2 and E3 with 20% fine-tuning samples compared to 10% fine-tuning samples. This may be due to the large difference in data distribution between a single source domain and the target domain. This difference may result in the model that cannot fit the data of the target task well.

  Fig. 10

  

  The AD results of E1, E2, E3, and E4.

  Table 6 shows the difference of each corresponding performance metric for bias and drift anomalies of E1, E2, and E3 compared to E4 with 10% and 20% training or fine-tuning samples. “-” indicates a decrease and “ + ” indicates an increase. For bias and drift anomalies, the ΔACC, ∆TPR, and ∆FPR values of E1 with 10% and 20% training samples have the largest range of variations, which are − 45.91% to − 25.28%, − 50.32% to − 5.18%, and + 27% to + 71.80%, respectively. This is due to the limited training samples, which makes the model underfitting, as described earlier. Although E2’s TPR value with 10% fine-tuning samples increases by 2.27% compared to E4 for bias anomaly, its ACC and FPR values decrease by 12.19% and 15.47%, and increase by 45.00% and 47.80%, respectively, for bias and drift anomalies. The ranges of ∆ACC, ∆TPR, and ∆FPR values for E2 with 20% fine-tuning samples are similarly large, at − 17.03% and − 21.25%, − 9.09% and − 16.22%, and + 34.60% and + 32.30%, respectively. Although E4 has slightly higher FPR values than E3 with 10% and 20% fine-tuning samples, its ΔFPR ranges from only − 4.50% to − 0.80%, which is relatively lower. In contrast, for bias and drift anomalies, the △ACC and △TPR values of E3 with 10% fine-tuning samples are − 7.19% and − 6.75%, and − 11.09% and − 10.27%, respectively. The ΔACC and ΔTPR values of E3 with 20% fine-tuning samples are -9.00% and -11.06%, and -15.18% and -17.31%, respectively, for bias and drift anomalies. The experimental results show that E4 exhibits a substantial overall increase in the TPR and ACC values while maintaining lower FPR values compared to E1, E2, and E3.

  Table 6 The difference of each corresponding performance metric for bias and drift anomalies of E1, E2, and E3 compared to E4 with 10% and 20% training or fine-tuning samples.

• 2. Data recovery

  Figures 11 and 12 display the visualization of data recovery for E1, E2, E3, and E4 with 10% and 20% training or fine-tuning samples, respectively, where the blue curves are the original data without anomaly and the red curves are the predicted data. With 10% training or fine-tuning samples, E1, E2, and E3 exhibit varying degrees of recovery deviations, as indicated by the black circles in Fig. 11. Notably, E1 shows the largest recovery bias with 10% training samples. As the number of training samples increases, the recovery bias of E1 improves significantly. However, the relatively obvious recovery bias of E1 remains, as depicted by the blue circles in Fig. 12. The recovery performances of E2 and E3 do not appear to show significant improvement with 10% and 20% fine-tuning samples, possibly due to the more pronounced difference in feature distribution between the source and target domains, as mentioned earlier. In contrast, through the multi-source fusion strategy, E4 effectively mitigates the negative impact of domain differences on the transfer learning process, leading to superior recovery performance.

  Fig. 11

  

  Data recovery visualization for E1, E2, E3, and E4 with 10% training or fine-tuning samples.

  Fig. 12

  

  Data recovery visualization for E1, E2, E3, and E4 with 20% training or fine-tuning samples.

Table 7 provides the MAE and RMSE values for E1, E2, E3, and E4 with 10% and 20% training or fine-tuning samples. For bias and drift anomalies, the MAE and RMSE values of E1 with 10% training samples are 2.6580 and 2.6797, and 3.1705 and 3.1882, respectively. The MAE and RMSE values of E1 with 20% training samples improve dramatically, with 1.3148 and 1.3905, and 1.5768 and 1.6872 for bias and drift anomaly, respectively. The MAE and RMSE values of E2 and E3 are lower than E1 with 10% fine-tuning samples. This suggests that the target domain can efficiently leverage the knowledge from the single-source domain to alleviate the recovery bias. However, the MAE and RMSE values of E2 and E3 with 20% fine-tuning samples show an increment compared to E1, which may be attributed to the larger differences between the source and target domains, as described earlier. Compared to E1, E2, and E3, E4 with 10% and 20% fine-tuning samples demonstrates the lowest MAE and RMSE values, indicating that multi-source data fusion can provide a more enriched data information, reducing recovery deviation.

Table 7 The MAE and RMSE values for E1, E2, E3, and E4 with 10% and 20% training or fine-tuning samples.

Discussion

The main findings of this study can be summarized as follows based on the above experimental results:

First, 1DCB-PSRS performed well in detecting bias and drift anomalies for different datasets, with average ACC values above 96.00%, average TPR values above 95.00%, and average FPR values below 1.20%, demonstrating good robustness and generalization. In addition, 1DCB-PSRS also achieved optimal data recovery performance, with MAE and RMSE values ranging from 0.5475 to 1.4298, and 0.7922 and 1.9103, respectively, for different anomaly types of different datasets. This was made possible by the fact that 1DCB-PSRS used 1D CNN and BiLSTM to construct the prediction model, which was able to fully extract the spatiotemporal features of flight data and realize the accurate prediction of the monitored parameter. Meanwhile, the anomaly detection performance was further improved by using MIC to select correlated flight parameters as model inputs and EWMA to smooth the residuals to reduce the effects of uncorrelated flight parameters and random noise. These results emphasize the significant advantages of 1DCB-PSRS in UAV flight data AD and data recovery, bringing important reference value for practical applications.

Second, using 1DCB-PSRS as the pre-training framework, a series of experiments were conducted with limited samples. E1 exhibited inferior performance primarily due to insufficient training samples, resulting in model underfitting. E2 and E3 mitigated the problem of model underfitting to some extent. Nevertheless, despite an increase in fine-tuning samples, the AD performances of E2 and E3 did not show substantial enhancement. This observation could be attributed to the large difference between the single-source and target domain, thus limiting the performance improvement of E2 and E3. E4 exhibited a noticeable advantage in AD performance over E1, E2, and E3, with higher ACC and TPR values and relatively low FPR values. For data recovery, E4 had the smallest MAE and RMSE values. The experimental results demonstrated that the knowledge transfer method based on multi-source data fusion exhibited remarkable advantages for the task addressed in this study and is expected to be widely used in practical scenarios.

However, it is important to recognize that there are some limitations to this study. First, the effectiveness of the proposed method is validated in this study by using anomaly injection to obtain bias and drift anomaly data. However, other anomaly types such as stuck anomalies and composite anomalies^17,58 are not included in the experimental scope, resulting in the effectiveness of the proposed method in multiple anomaly types that has not yet been fully evaluated. Second, this paper utilizes FPR, TPR, ACC, MAE, and RMSE as AD and data recovery performance metrics, which may make the evaluation of the results less comprehensive. Finally, this paper leverages deep learning to achieve AD and data recovery, but its poor interpretability may hinder the credibility and generalization of the proposed method in practical applications.

Therefore, considering the above limitations, future research will incorporate more evaluation metrics (such as F1 score and t-tests) and anomaly types to comprehensively assess the model performance. Meanwhile, enhancing the model interpretability using methods like SHapley Additive exPlanations (SHAP)⁵⁹ and Local Interpretable Model-agnostic Explanation (LIME)⁶⁰ is also an important direction for future research.

Conclusion and prospects

To improve the AD and recovery performance of UAV flight data with limited samples, a novel data-driven framework 1DCB-PSRS was first proposed. It was based on the designed 1D CNN-BiLSTM prediction model to fully capture the complex spatiotemporal features of flight data while integrating MIC-based parameter selection and EWMA-based residual smoothing to reduce the interference of irrelevant flight parameters and random noise. Then, an innovative knowledge transfer approach based on multi-source data fusion was proposed for UAV flight data AD and recovery with limited samples. It employed 1DCB-PSRS as the pre-training framework to transfer knowledge using data distribution information from multiple source domains, avoiding reliance on a single source domain data. Finally, extensive experiments were conducted on real UAV flight data. With sufficient training data, for the bias and drift anomalies of different datasets, the average ACC values of 1DCB-PSRS were all over 96.00%, the average TPR values were all over 95.00%, and the average FPR values were all below 1.20%. Meanwhile, for different anomaly types of different datasets, 1DCB-PSRS also achieved the best data recovery performance with sufficient data, with MAE and RMSE values ranging from 0.5475 to 1.4298, 0.7922 and 1.9103, respectively. With limited training or fine-tuning samples, E4 achieved a good AD performance. Particularly with 20% fine-tuning samples, E4 showed the highest ACC and TPR values of 80.25% and 85.75%, and 73.95% and 80.36%, respectively, for bias and drift anomalies. For data recovery, E4 consistently has the lowest MAE and RMSE values. The effectiveness of 1DCB-PSRS as the pre-training framework and the knowledge transfer method based on multi-source data fusion was proved according to the experimental results.

In future research, more metrics will be included to comprehensively assess the model performance. Meanwhile, the performance of the proposed method will be validated in more anomaly types. In addition, model interpretability will be actively explored in the future.

Data availability statement

All original data from this study can be accessed through the link https://conservancy.umn.edu. Researchers interested in accessing and analyzing these data can directly obtain resources from the specified repository.