Research on Improved Deep Convolutional Generative Adversarial Networks for Insufﬁcient Samples of Gas Turbine Rotor System Fault Diagnosis

: In gas turbine rotor systems, an intelligent data-driven fault diagnosis method is an important means to monitor the health status of the gas turbine, and it is necessary to obtain sufficient fault data to train the intelligent diagnosis model. In the actual operation of a gas turbine, the collected gas turbine fault data are limited, and the small and imbalanced fault samples seriously affect the accuracy of the fault diagnosis method. Focusing on the imbalance of gas turbine fault data, an Improved Deep Convolutional Generative Adversarial Network (Improved DCGAN) suitable for gas turbine signals is proposed here, and a structural optimization of the generator and a gradient penalty improvement in the loss function are introduced to generate effective fault data and improve the classification accuracy. The experimental results of the gas turbine test bench demonstrate that the proposed method can generate effective fault samples as a supplementary set of fault samples to balance the dataset, effectively improve the fault classification and diagnosis performance of gas turbine rotors in the case of small samples, and provide an effective method for gas turbine fault diagnosis.


Introduction
As internal combustion type rotating power machinery, gas turbines are the core components of power equipment in industrial fields, and have rapidly developed and been widely used in other fields such as aerospace, industrial power generation, navigation, and land transportation. The reliability and safety of gas turbines are receiving more and more attention. Due to various unavoidable factors, the gas turbine has various failures, and the predetermined function is reduced or even completely lost. As a result, serious or even catastrophic accidents occur, and huge economic losses are felt by enterprises and even industries [1].
Common gas turbine faults are divided into two categories. The first is related to aerodynamics or performance-based faults, such as thermal distortion, compressor erosion and corrosion, compressor fouling, turbine fouling, turbine erosion and corrosion, blade rubbing, etc. Gas Path Analysis (GPA) is an effective technical means for forecasting these faults [2]. At present, data-driven machine learning has been used for gas path diagnosis based on fault sample sets, such as Artificial Neural Network (ANN) [3], Support Vector Machine (SVM) [4], Bayesian network [5], and fuzzy logic [6]. The second type of common gas turbine faults is related to mechanical properties, such as shaft misalignment, rotor dynamic imbalance, bearing defects, and oil film instability [7]. For such faults, there (1) The Improved DCGAN is proposed, and the generator is optimized by adding a onedimensional Deep Convolutional Neural Network With a Wide First-Layer Kernel (WDCNN) to the deconvolution layer of the generator, to build a generator containing the gas turbine vibration data features and generate samples with higher quality and similarity to original samples. (2) The loss function of the generator and discriminator are optimized, and a gradient penalty is adopted to solve the problem of gradient disappearance and improve the applicability of the method. The optimized penalty factor can be adjusted according to the actual situation, which improves the adaptability of the method. (3) The generated effective fault samples are added to the fault dataset, and the proposed method is validated on the gas turbine rotor system fault data to solve the problem of insufficient gas turbine rotor system fault samples and provide new ideas for gas turbine rotor system fault diagnosis.
The remainder of this paper proceeds as follows. Section 1 is the introduction, Section 2 presents theoretical method, Section 3 is the proposed fault diagnosis method, Section 4 is the experiment and results, and Section 5 concludes the paper.

Deep Convolutional Generative Adversarial Network
The Generative Adversarial Network is a machine learning architecture proposed by Ian Goodflow in 2014, and it is one of the most promising methods of unsupervised learning in recent years. Due to its outstanding data generative capabilities, Generative Adversarial Networks have been fully applied in many fields, especially in image processing, text generation, audio processing, data enhancement, and sample prediction. The main idea of GAN is derived from game theory, in which a discriminative model and a generative model are set up to compete with each other to improve the learning effect and obtain high-dimensional and complex data distribution of real samples. Its network structure is shown in Figure 1: samples are combined with original samples to form feasibility samples and supplement the gas turbine rotor fault dataset, and then the fault samples are adopted to build up the fault diagnosis model. Finally, gas turbine test bench experiments are carried out for fault classification and to verify the proposed method. The main contributions can be summarized as follows: (1) The Improved DCGAN is proposed, and the generator is optimized by adding a onedimensional Deep Convolutional Neural Network With a Wide First-Layer Kernel (WDCNN) to the deconvolution layer of the generator, to build a generator containing the gas turbine vibration data features and generate samples with higher quality and similarity to original samples. (2) The loss function of the generator and discriminator are optimized, and a gradient penalty is adopted to solve the problem of gradient disappearance and improve the applicability of the method. The optimized penalty factor can be adjusted according to the actual situation, which improves the adaptability of the method. (3) The generated effective fault samples are added to the fault dataset, and the proposed method is validated on the gas turbine rotor system fault data to solve the problem of insufficient gas turbine rotor system fault samples and provide new ideas for gas turbine rotor system fault diagnosis.
The remainder of this paper proceeds as follows. Section 1 is the introduction, Section 2 presents theoretical method, Section 3 is the proposed fault diagnosis method, Section 4 is the experiment and results, and Section 5 concludes the paper.

Deep Convolutional Generative Adversarial Network
The Generative Adversarial Network is a machine learning architecture proposed by Ian Goodflow in 2014, and it is one of the most promising methods of unsupervised learning in recent years. Due to its outstanding data generative capabilities, Generative Adversarial Networks have been fully applied in many fields, especially in image processing, text generation, audio processing, data enhancement, and sample prediction. The main idea of GAN is derived from game theory, in which a discriminative model and a generative model are set up to compete with each other to improve the learning effect and obtain high-dimensional and complex data distribution of real samples. Its network structure is shown in Figure 1 The main structure of GAN includes a generator and a discriminator [37,38]. The generator is used to generate real samples, and the discriminator is used to determine real samples and fake samples. The generator G inputs a random noise vector z (usually uniform or normal distribution) and maps it to a new multi-dimensional data space to achieve a generated fake sample () Gz . The discriminator D is used to perform classification and to calculate the probability that the testing sample is a real sample. In the training, the generator G tries to learn the data distribution of the real sample to generate a fake sample, and the discriminator D tries to distinguish the generated fake sample from the real sample. The objective function of the generative adversarial network is as follows: The main structure of GAN includes a generator and a discriminator [37,38]. The generator is used to generate real samples, and the discriminator is used to determine real samples and fake samples. The generator G inputs a random noise vector z (usually uniform or normal distribution) and maps it to a new multi-dimensional data space to achieve a generated fake sample G(z). The discriminator D is used to perform classification and to calculate the probability that the testing sample is a real sample. In the training, the generator G tries to learn the data distribution of the real sample to generate a fake sample, and the discriminator D tries to distinguish the generated fake sample from the real sample. The objective function of the generative adversarial network is as follows: During the training process of the GAN network, the objective function of D(x) is: The parameters are optimized by Equation (2) to improve D(x). The objective function of G is: minV Among them, x represents the data sample, P z (z) represents the data distribution of the input noise, G(z) represents the sample generated by the input noise, D(x) represents the probability that sample x determined by the discriminator is the real sample instead of the generated sample. E x∼P data (x) [log D(x)] represents the probability of D(x) obtaining real data; E x∼P z (z) [log(1 − D(G(z)))] represents the probability of D(x) discriminating real data.
During the training process, alternate iterations are performed. Firstly, the network G is fixed, and the discriminator network D is trained so that the labels of the training samples are classified correctly with the greatest probability, with maximized log D(x) and log(1 − D(G(z))). Then, the network D is fixed, and the network G is trained to minimize log(1 − D(G(z))). The training of the generator network puts D(G(z)) close to 1, so that the generator loss will be the smallest. The goal of training the discriminator network is to distinguish the real data and the generated data; that is, the discriminator output of the real data is expected to be close to 1, and the output of the generated data D(G(z)) is close to 0. Finally, the training reaches a balance, and G can estimate the distribution of sample data.
The focus of GAN is to find the Nash equilibrium point, and the goal of the gradient descent method in the neural network is to find the smallest loss function, so in model training, there may be situations where the model cannot converge or the convergence is unstable. Moreover, there may be a pattern collapse phenomenon in GAN; that is, the generated data sample pattern is singular, and it does not cover all categories of real data. Therefore, how to ensure stable training of GAN has always been a difficult topic [33,39].
With the continuous intensive research in the field of image and sample generation by the Generative Adversarial Network, many excellent image and sample generation algorithms have emerged. Radford et al. [40] proposed a Deep Convolutional Generative Adversarial Network. By improving the neural network structure in the GAN model, the deconvolution layer innovatively replaced the fully connected layer of the generator in the basic GAN network, thus achieving excellent performance in image generation tasks. As shown in Figure 2, the powerful feature extraction capability of CNN in DCGAN is adopted to improve the learning effect of the generative network. By use of Batch Normalization (BN) in the layer, the generator can learn stably, so that the model can learn better data sample distribution and more stably generate high-quality samples.
DCGAN improves the GAN structure, and greatly improves the stability of GAN training and the quality of generated data [41,42]. Its main improvements are as follows: Convolutional Neural Network is used instead of the original GAN generator and discriminator network structure; in the generator network, micro-step convolution is used instead of pooling layers and fully connected layers; and in the discriminator network, step-size convolution is used instead of pooling layers and fully connected layers. Batch normalization unified operations are added to both the generator network and the discriminator network, and in order to prevent the oscillation and instability of CNN, batch normalization unified operation is not added to the output layer of the generator network or the input layer of the discriminator network. The tanh activation function is used in the output layer of the generator network; the ReLU activation function is used in all rest layers. In the discriminator network, the LeakReLU activation function is used instead of the ReLU activation function to solve the vanishing gradient problem when the discriminator network returns to the generator network. DCGAN greatly improves the stability of GAN training and the quality of the generated results. normalization unified operation is not added to the output layer of the generator network or the input layer of the discriminator network. The tanh activation function is used in the output layer of the generator network; the ReLU activation function is used in all rest layers. In the discriminator network, the LeakReLU activation function is used instead of the ReLU activation function to solve the vanishing gradient problem when the discriminator network returns to the generator network. DCGAN greatly improves the stability of GAN training and the quality of the generated results.

Convolutional Neural Network (CNN)
The Convolutional Neural Network is a typical deep neural network. It reduces the complexity of the network model through the weight sharing method and improves the efficiency of model training. The advantage of this network is that the network model can be trained with input images or signals, eliminating the need for feature extraction. The Convolutional Neural Network consists of an input layer, convolution layer, pooling layer, fully connected layer, and output layer. The input layer realizes the input of the original dataset, the feature is obtained through convolution calculation of the convolution layer, and the pooling layer performs down-sampling operation on the feature image to reduce the data dimension. The number of convolutional layers and down-sampling layers is often determined according to actual conditions. The fully connected layer, just as the neural network, mainly implements data mapping [43,44].
The idea of the convolutional layer is to use convolution kernels to extract local features, which is similar to an automatic feature extraction, which has been widely used in feature extraction, fault diagnosis, and so on. For time series vibration signals, the one dimension Deep Convolutional Neural Network with a Wide First-Layer Kernel (WDCNN) has strong processing capabilities, which can eliminate complex feature extraction and directly perform operations on the sequence. Its main feature is that it has the first layer of a large convolution core and a multi-layer small convolution kernel with the size 3 × 1. It can provide a faster processing speed and it is suitable for real-time fault diagnosis. In the proposed method, the one-dimensional Deep Convolutional Neural Network with a Wide First-Layer Kernel is used for vibration signals' feature extraction, and it is used as the coding part of the new generator.
The detailed settings of the proposed network structure are shown in Table 1. Vibration signals are used in the input layer. Different kernel sizes, numbers, and strides have

Convolutional Neural Network (CNN)
The Convolutional Neural Network is a typical deep neural network. It reduces the complexity of the network model through the weight sharing method and improves the efficiency of model training. The advantage of this network is that the network model can be trained with input images or signals, eliminating the need for feature extraction. The Convolutional Neural Network consists of an input layer, convolution layer, pooling layer, fully connected layer, and output layer. The input layer realizes the input of the original dataset, the feature is obtained through convolution calculation of the convolution layer, and the pooling layer performs down-sampling operation on the feature image to reduce the data dimension. The number of convolutional layers and down-sampling layers is often determined according to actual conditions. The fully connected layer, just as the neural network, mainly implements data mapping [43,44].
The idea of the convolutional layer is to use convolution kernels to extract local features, which is similar to an automatic feature extraction, which has been widely used in feature extraction, fault diagnosis, and so on. For time series vibration signals, the one dimension Deep Convolutional Neural Network with a Wide First-Layer Kernel (WDCNN) has strong processing capabilities, which can eliminate complex feature extraction and directly perform operations on the sequence. Its main feature is that it has the first layer of a large convolution core and a multi-layer small convolution kernel with the size 3 × 1. It can provide a faster processing speed and it is suitable for real-time fault diagnosis. In the proposed method, the one-dimensional Deep Convolutional Neural Network with a Wide First-Layer Kernel is used for vibration signals' feature extraction, and it is used as the coding part of the new generator.
The detailed settings of the proposed network structure are shown in Table 1. Vibration signals are used in the input layer. Different kernel sizes, numbers, and strides have been selected to perform the one-dimension convolution operation with an appropriate number. For Conv 1, the kernel size, number, and stride are 32, 20, and 8, respectively. The BN and ReLU activation function are adopted after the Conv layer, and the pooling layers are used to perform the pooling operation. The dropout operation is used after FC1 and FC2, and four different classifications can be recognized.

Maximum Mean Discrepancy (MMD)
The MMD algorithm is a loss function widely used in transfer learning. It is mainly used to measure the distance of the distribution of two different but related sample data and observe the distribution differences in the two types of sample data [45]. In the proposed DCGAN method, MMD is used to measure the similarity between generated samples and original samples. The smaller the MMD, the better the similarity between the generated sample and the original sample, and the higher the quality of the generated samples. MMD mainly uses high-dimensional mapping functions to find the expected difference in RHKS of the two samples to determine the similarity [46]. Assume the samples obey the p distribution D s = {x 1 , x 2 , · · · , x n } and the samples obey the q distribution D m = {y 1 , y 2 , · · · , y m }, function φ : χ → H maps samples to high-dimensional RKHS, where χ is the feature space of the sample. The distribution of different samples in RKHS is effectively matched by means of kernel mean embedding. The unbiased estimation of the kernel average embedding is obtained by calculating the expected value E p [φ(x i )] and E q [φ(y i )] of the sample mapped to the RKHS. φ(x i ) is the representation of sample x i mapped to RKHS through high-dimensional functions. MMD is expressed as follows: RKHS is a complete high-dimensional inner product space. Dot product operations of φ(x i ) and φ(y j ) can be calculated with kernel function. Generally, the Radial Basic Function (RBF) is chosen to represent the infinite dimensions Gaussian kernel: where σ represents the bandwidth of the Gaussian kernel. We rewrite Equation (5) by Equation (4): It can be seen from Equation (6) that when p = q, MMD 2 (p, q) and when p = q, it needs to find the mapping function k(·) to minimize the MMD 2 (p, q), so that the probability distributions of the two samples in RKHS are more similar under the representation φ(·) to reduce the distribution difference in RKHS. In the Improved DCGAN method, new fault samples are generated by the Generative Adversarial Network, and MMD is used to measure the similarity between generated samples and real samples. The smaller the MMD, the better the similarity between the generated sample and the original sample.

The Improved DCGAN
In traditional GAN, random noises are generally used as signal sources to enter into the generator G, and new signals and samples are generated after processing by the network. In order to make the generated signal consistent with the real signal, a discriminator D is trained and used to identify whether the signal is real data or fake data generated by the generator. The random noise generated each time during the operation is a special code that only a group of generators can understand, and it determines how the generated result should be represented. The result generated by using random data as input barely has a relationship with the original signals, leading to low similarity between the signals generated by the generator and original sample signals and long training time of the network.
Each convolutional layer in a Convolutional Neural Network consists of several convolutional units, and the parameters of each convolutional unit are optimized through the back-propagation algorithm. The purpose of the convolution operation is to extract different features of the input. The first convolution layer may only extract some low-level features such as edges, lines, and corners. The network with more layers can iteratively extract more complex features from the low-level features. The transposed convolutional layer can synthesize pictures or signals based on features, and has been used to generate adversarial networks such as DCGAN.
If the advantages of the convolutional layer and the transposed convolutional layer in CNN can be combined, a new generator can be formed. Therefore, in order to obtain the generated samples with higher similarity to the original samples, an Improved DCGAN method is proposed. Figure 3 is shown as the structure of Improved DCGAN. On the basis of DCGAN, the trained WDCNN network is introduced to the generator. The features of input fault signals are obtained through WDCNN network training, so the input contains fault features and the generated samples similar to the original samples are easier to obtain through the generator. The discriminator examines patches of the input samples and determines the probability of the samples being real or fake. As can be observed, the discriminator has an input layer of 32 × 1. Three hidden layers are employed with LeakReLU as the activation function. Finally, the output layer has a dimension of 256 × 1, which is fully connected with a sigmoid activation function for the real and fake data classification. The kernel size for the The generator of the Improved DCGAN network consists of three parts: an encoder network, the reshape network, and a decoder network. The encoder network extracts the feature representation of the input signals, and WDCNNs are used as convolutional layers to construct the encoder. The reshape network consists of the project and reshape layer, which upscales the input using a fully connected operation and reshapes the output to the specified size. The decoder network consists of transposed convolutional layers, and takes the transformed features and outputs the final generated signals as samples.
The discriminator examines patches of the input samples and determines the probability of the samples being real or fake. As can be observed, the discriminator has an input layer of 32 × 1. Three hidden layers are employed with LeakReLU as the activation function. Finally, the output layer has a dimension of 256 × 1, which is fully connected with a sigmoid activation function for the real and fake data classification. The kernel size for the CNN is 3 × 1 in all its layers with stride of 2 for all the hidden layers except by the output with stride of 1. The structure parameters and hyper-parameters of Improved DCGAN are shown in Table 2. Table 2. Structure parameters and hyper-parameters of Improved DCGAN.

Model
Network Layer Kernel Size Stride Activation Function

Loss Function Optimization Based on Gradient Penalty
In the process of Generative Adversarial Network training, the loss function is the key to the quality of network learning. If the loss function is incorrect, it will lead to training instability and even exploding gradient problems, and difficulty to train the correct model. The loss function in GAN is used still in DCGAN continuously. Based on DCGAN, an improved method with a gradient penalty is adopted in the proposed method, in which the original loss function in GAN is abandoned and an additional gradient penalty is added to the loss function of the discriminator to solve the vanishing gradient problem and the exploding gradient problem in training. The sparse gradient penalty algorithm is used as the loss function in the process of model training, the negative gradient direction is used as the search direction, and the minimum value is solved along the gradient descent direction, so as to approach the minimum deviation in a recursive way. In the training process, after each forward propagation, the real values of loss value and output value will be obtained. The smaller the loss value, the better the model.
Firstly, the samples x r and x g are obtained in the real sample space P r and the generated sample space P g , respectively, and thenx is obtained by random interpolation between the real sample x r and the generated sample x g , as shown in Equations (7) and (8): where ε represents the random sampling of the samples. According to the joint space samplingx, the discriminator output D(x) is made, and then the joint derivation is carried out.
where i represents the number of batch size samples. Equation (9) is used for the square difference processing to the joint derivative grad. In order to adapt to the sample characteristics, the joint space sample loss function loss eps is finally obtained after the square difference. The gradient penalty term is shown in Equation (10): where ∇ x D(x) represents the gradient of the discriminator output value in the X direction; E x∼x represents the mathematical expectation when the input of the discriminator is a random interpolation.
where N grad represents the row number of the matrix grad; λ represents the gradient penalty parameter. The original loss function of the discriminator is shown in Equation (12): where E x∼P g [D( x)] represents the mathematical expectation when the inputs of discriminator are generated samples; E x∼P r [D(x)] represents the mathematical expectation when the inputs of the discriminator are real samples.
In the discrimination of samples, the goal of the discriminator loss function is to reduce the gap between the predicted value of the model and the real value as much as possible. The differences between the loss functions of the discriminator discriminating real and fake samples are weighted and the final discriminator loss function is obtained.
In the loss function of the generator, since the problem solved by the generator is different from that of the discriminator, the loss in the model is mainly caused by convolution processing. The loss caused by joint sampling samples is calculated according to the gradient descent method, and the loss function formula is as follows: The final generator loss function G loss is as follows: where β represents the gradient penalty parameter of the generator, which can be used to adjust the intensity of gradient penalty. Compared with DCGAN, the new gradient penalty term will make the overall training of the Deep Convolution Generative Adversarial Network with a gradient penalty more stable, and the convergence and training speed faster.

The Proposed Gas Turbine Rotor System Fault Diagnosis Method
In traditional Generative Adversarial Networks, random data are used as the encoding source to generate samples. In order to improve network training stability and the generated signals' quality, the inputs introduce fault features in relation to the original signals. According to the Improved DCGAN structure and optimized loss function, an improved method is proposed, which can be applied to gas turbine rotor fault diagnosis under different unknown conditions with small labeled samples, with the purpose of generating effective sample data and supplying a dataset for fault diagnosis. Figure 4 is a flow chart of the proposed gas turbine rotor fault diagnosis method. The whole process is summarized as follows: Appl

Experiment and Data Acquisition
To further verify the performance of the proposed method, the gas turbine rotor system fault experiment from the test bench is adopted. The rotor system of a gas turbine is shown in Figure 5, which is mainly composed of a compressor, turbine, rotating shaft, and so on. The low-pressure turbine is connected to the low-pressure compressor through the low-pressure rotating shaft, and the high-pressure turbine is connected to the high-pressure compressor through the high-pressure rotating shaft sleeved on the low-pressure rotating shaft. Due to the complex and high-temperature internal environment of gas tur- Firstly, accelerometers installed on different parts of the gas turbine are adopted to collect vibration data, and then the obtained signals are preprocessed by time domain and frequency domain analysis. Combining the analysis with expert diagnosis, labeled samples are obtained and the data are divided into the training set and testing set.
Secondly, the fault samples are adopted as the input of the proposed DCGAN network to generate fault samples. The one-dimensional CNN network suitable for vibration signal feature extraction is constructed, and the network and parameters are used for the generator of the Improved DCGAN network, and a new generator for feature extraction of the actual sample data is formed to continually generate new associated samples.
Thirdly, the sample maximum mean difference (MMD) method is used to determine the mean difference between the generated signal and the real signal, and combine the generated data that meet the requirements with the original small sample data to form a new dataset.
Lastly, fault classification and diagnosis are performed on the generated new sample set, and comparisons of results and accuracy analyses are performed to verify the method.

Experiment and Data Acquisition
To further verify the performance of the proposed method, the gas turbine rotor system fault experiment from the test bench is adopted. The rotor system of a gas turbine is shown in Figure 5, which is mainly composed of a compressor, turbine, rotating shaft, and so on. The low-pressure turbine is connected to the low-pressure compressor through the low-pressure rotating shaft, and the high-pressure turbine is connected to the high-pressure compressor through the high-pressure rotating shaft sleeved on the low-pressure rotating shaft. Due to the complex and high-temperature internal environment of gas turbines, the whole machine vibration measurement is often used in the engineering application. Through multiple experiments and analyses at different positions and angles, as shown in Figure 5, a speed sensor (sensor 1) is installed on the radial position at the front end of the low-pressure compressor casing as the front measuring point, a speed sensor (sensor 2) is installed at the radial position of the casing between the high-pressure compressor and the combustion chamber as the back measuring point, and the rotor speed sensor for measuring high-and low-pressure rotor speed is installed inside the gas engine. The numbers 1-5 in the Figure 5 are the support points for high-and low-pressure rotor, among which, 1, 2 and 5 are the low-pressure rotor support points, and 3 and 4 are the high-pressure rotor support points. The sampling frequency is 6000 Hz, and the experiment time lasts 8000 s. In the actual operation of the gas turbine rotor system, the time domain statistics in the vibration signals will change with the change in the working conditions and state of the gas turbine rotor system. Therefore, the time domain analysis of gas turbine rotor system vibration signals can be used to preliminarily observe and judge the state and trend of the rotor system, and further for the gas turbine rotor system fault diagnosis and analysis. The vibration signals collected by the gas turbine sensor are shown in Figure 6, and the effective values of gas turbine vibration signals in the time domain can be obtained. From Figure  6, it can be seen that the amplitude of the front measuring point vibration data is large and exceeds the limit in some time periods, and by analysis, it can be concluded that faults The sampling frequency is 6000 Hz, and the experiment time lasts 8000 s. In the actual operation of the gas turbine rotor system, the time domain statistics in the vibration signals will change with the change in the working conditions and state of the gas turbine rotor system. Therefore, the time domain analysis of gas turbine rotor system vibration signals can be used to preliminarily observe and judge the state and trend of the rotor system, and further for the gas turbine rotor system fault diagnosis and analysis. The vibration signals collected by the gas turbine sensor are shown in Figure 6, and the effective values of gas turbine vibration signals in the time domain can be obtained. From Figure 6, it can be seen that the amplitude of the front measuring point vibration data is large and exceeds the limit in some time periods, and by analysis, it can be concluded that faults may occur. For further analysis, the root mean square (RMS) values of vibration signals are calculated and shown in Figure 7. The sampling frequency is 6000 Hz, and the experiment time lasts 8000 s. In the actual operation of the gas turbine rotor system, the time domain statistics in the vibration signals will change with the change in the working conditions and state of the gas turbine rotor system. Therefore, the time domain analysis of gas turbine rotor system vibration signals can be used to preliminarily observe and judge the state and trend of the rotor system, and further for the gas turbine rotor system fault diagnosis and analysis. The vibration signals collected by the gas turbine sensor are shown in Figure 6, and the effective values of gas turbine vibration signals in the time domain can be obtained. From Figure  6, it can be seen that the amplitude of the front measuring point vibration data is large and exceeds the limit in some time periods, and by analysis, it can be concluded that faults may occur. For further analysis, the root mean square (RMS) values of vibration signals are calculated and shown in Figure 7. In Figure 7, the high-pressure rotor speed changes continuously under multiple working conditions; the dotted line is the change in the high-pressure rotor frequency, the green line is the vibration limit value, the red line is the RMS for the front measuring point, and the blue line is the RMS for the back measuring point. It can be found from the gas turbine sensor data analysis that the speed was constantly changing in various working conditions during the experiment. The effective value of the front measuring point vibration signals and the effective value of the back measuring point vibration signals change with the working conditions. Among them, the effective value of the front measuring  Through further analysis, the pre-measurement vibration data obtained from the g turbine test bench experiment can be divided into four classes, normal state, air flow e citation, unbalanced, and misalignment, and these test data will be used as the dataset fault diagnosis. The vibration signals of different faults for gas turbines rotor system a shown in Figure 8. In Figure 7, the high-pressure rotor speed changes continuously under multiple working conditions; the dotted line is the change in the high-pressure rotor frequency, the green line is the vibration limit value, the red line is the RMS for the front measuring point, and the blue line is the RMS for the back measuring point. It can be found from the gas turbine sensor data analysis that the speed was constantly changing in various working conditions during the experiment. The effective value of the front measuring point vibration signals and the effective value of the back measuring point vibration signals change with the working conditions. Among them, the effective value of the front measuring point vibration signals has the phenomenon of vibrations exceeding the limit for several time periods, and the effective value of the back measuring point vibration signals has never exceeded the limit. It can be found that in multiple periods of time, due to the structure and assembly of the gas turbine, the gas turbine stalls due to the rotation of the high-pressure compressor during the speed-up condition, resulting in air flow excitation, further causing temporary unbalance of the high-pressure rotor, resulting in vibration overrun.
Through further analysis, the pre-measurement vibration data obtained from the gas turbine test bench experiment can be divided into four classes, normal state, air flow excitation, unbalanced, and misalignment, and these test data will be used as the dataset for fault diagnosis. The vibration signals of different faults for gas turbines rotor system are shown in Figure 8. Through further analysis, the pre-measurement vibration data obtained from the gas turbine test bench experiment can be divided into four classes, normal state, air flow excitation, unbalanced, and misalignment, and these test data will be used as the dataset for fault diagnosis. The vibration signals of different faults for gas turbines rotor system are shown in Figure 8. The time-frequency analyses of gas turbine rotor system vibration signals under misalignment and unbalanced states are shown in Figure 9. It can be seen from Figure 9a that the abnormal frequency (131.5 Hz) is about 1.47 times the low-voltage rotating frequency (89.1 Hz), and the frequency is not fixed. After preliminary judgment, it is determined as faults by rotor misalignment. In addition to the abnormal frequency associated with excessive vibration, the double frequency component 101 Hz of low-pressure rotor frequency 52 Hz appears when the high speed drops to slow speed, as shown in Figure 9b. By analysis, this may be caused by the rotor unbalanced. After discussion and consultation with experts, it was preliminarily determined that the abnormal frequency and the reasons for vibration exceeded the standard. Through examining actual faults and machine maintenance, the fault type was determined, and the label was given as the experimental sample.
The rotor misalignment could cause the radial double frequency vibration, and axial vibration of the rotor could cause the fatigue damage of the shaft, as shown in Figure 10. If the rotor misalignment lasts for a long time, it will aggravate and cause other more serious faults; thus, it must be effectively identified. The gas turbine rotor system unbalance will directly lead to excessive vibration of the whole machine, which will cause a series of deformed or even damaged elements. The deformation caused by vibrations will produce rub impact, leading to fatigue damage of the rotor or other parts, such as the fatigue crack shown in Figure 11. faults by rotor misalignment. In addition to the abnormal frequency associated with e cessive vibration, the double frequency component 101 Hz of low-pressure rotor fr quency 52 Hz appears when the high speed drops to slow speed, as shown in Figure 9 By analysis, this may be caused by the rotor unbalanced. After discussion and consult tion with experts, it was preliminarily determined that the abnormal frequency and t reasons for vibration exceeded the standard. Through examining actual faults and m chine maintenance, the fault type was determined, and the label was given as the expe mental sample.   The rotor misalignment could cause the radial double frequency vibration, and axial vibration of the rotor could cause the fatigue damage of the shaft, as shown in Figure 10. If the rotor misalignment lasts for a long time, it will aggravate and cause other more serious faults; thus, it must be effectively identified. The gas turbine rotor system unbalance will directly lead to excessive vibration of the whole machine, which will cause a series of deformed or even damaged elements. The deformation caused by vibrations will produce rub impact, leading to fatigue damage of the rotor or other parts, such as the fatigue crack shown in Figure 11.

The Training of Improved DCGAN
Because the labeling of fault samples requires a lot of experimental and expert experience, the fault samples are not enough. Therefore, in order to improve the accuracy of fault diagnosis, the proposed method is used to generate fault samples with very high similarity with the original samples to supplement all kinds of samples and make up for the imbalance of samples. Figure 12 is the flow chart of gas turbine fault data generation and fault diagnosis of gas turbine rotor systems. The original gas turbine sample is labeled and used as the training dataset; the length of each fault sample is 1024. Then, the labeled samples are taken as the input to Improved DCGAN to achieve the generated samples. The number of samples in the training dataset including generated samples is 1000 for each class. The number of samples in the original training dataset, the training dataset including the generated samples, and the testing dataset are shown in Table 3.

The Training of Improved DCGAN
Because the labeling of fault samples requires a lot of experimental and expert experience, the fault samples are not enough. Therefore, in order to improve the accuracy of fault diagnosis, the proposed method is used to generate fault samples with very high similarity with the original samples to supplement all kinds of samples and make up for the imbalance of samples. Figure 12 is the flow chart of gas turbine fault data generation and fault diagnosis of gas turbine rotor systems. The original gas turbine sample is labeled and used as the training dataset; the length of each fault sample is 1024. Then, the labeled samples are taken as the input to Improved DCGAN to achieve the generated samples. The number of samples in the training dataset including generated samples is 1000 for each class. The number of samples in the original training dataset, the training dataset including the generated samples, and the testing dataset are shown in Table 3.  Figure 12. Fault data generation and classification of gas turbine rotor system.

Number of Samples in
The training parameters of the proposed Improved DCGAN network model are se as follows: the batch size is set as 11, the learning rate is 0.000001, the number of mode training iterations is 80,000, the generator gradient penalty parameter  is 0.000001, an the discriminator gradient penalty parameter  is 10. The optimization method of th generator and discriminator is the Adam optimization algorithm. In order to ensure tha  The training parameters of the proposed Improved DCGAN network model are set as follows: the batch size is set as 11, the learning rate is 0.000001, the number of model training iterations is 80,000, the generator gradient penalty parameter β is 0.000001, and the discriminator gradient penalty parameter λ is 10. The optimization method of the generator and discriminator is the Adam optimization algorithm. In order to ensure that the generated signals and the real signals have a large degree of similarity and a small error, the MMD should be set as less than 0.125.
In the adversarial training between the generator and discriminator, the generator tries to generate samples that could cheat the discriminator, and the discriminator learns to distinguish between the generated samples and the real signals with training. Figure 13 shows the loss function waveform of the generator and discriminator. In adversarial learning between the generator and discriminator, the generated samples increase with iteration, and the loss function gradually changes. At the beginning, the loss of generator is smooth, and then gradually increases and finally tends to be stable. The loss of the discriminator gradually decreases with the training time, and tends to be stable. After adversarial training, the generator generates higher-quality vibration signals.
It can be seen from Figure 13 that compared with DCGAN, the newly added gradient penalty term in the proposed DCGAN made the overall training of the Deep Convolutional Generative Adversarial Network more stable, and the convergence speed faster.
to distinguish between the generated samples and the real signals with training. Figure 13 shows the loss function waveform of the generator and discriminator. In adversarial learning between the generator and discriminator, the generated samples increase with iteration, and the loss function gradually changes. At the beginning, the loss of generator is smooth, and then gradually increases and finally tends to be stable. The loss of the discriminator gradually decreases with the training time, and tends to be stable. After adversarial training, the generator generates higher-quality vibration signals. It can be seen from Figure 13 that compared with DCGAN, the newly added gradient penalty term in the proposed DCGAN made the overall training of the Deep Convolutional Generative Adversarial Network more stable, and the convergence speed faster.

The Generated Data and Results Comparison
The original signals and generated fault sample signals under four states by the proposed method are demonstrated. Figure 14 shows the generated sample signals of the gas turbine rotor system under the normal state. Figure 15 and Figure 16 show the generated sample signals of the gas turbine rotor system under the airflow excitation state and the unbalanced state. Figure 17 shows the generated sample signals of the gas turbine rotor system under the misalignment state. For comparison, the generated fault sample signals by GAN and DCGAN are also displayed.
Meanwhile, the original signals and generated signals based on GAN, DCGAN, and the Improved DCGAN method are analyzed by frequency domain and time-frequency domain spectrum analysis. In order to better verify the similarity between the generated samples and the original samples, spectrum analysis and Structural Similarity (SSIM) analysis are used for verification.
(1) Spectrum analysis The similarities between the generated samples and the original samples are analyzed by time domain analysis, Fast Fourier Transform (FFT) spectrum analysis, Short-Time Fourier Transform (STFT) spectrum analysis, and Continue Wave Transform (CWT) analysis. It can be seen from the comparison of spectrum signals in normal states in Figure  14 that the spectrum of the newly generated sample signals is consistent with the spectrum of real sample signals. It can be clearly seen from the FFT and CWT diagrams that the samples generated by the proposed method have the highest similarity with the original signals. The generated samples by GAN and DCGAN are similar to the original samples, but not completely equivalent, and there are small deviations at some peaks.

The Generated Data and Results Comparison
The original signals and generated fault sample signals under four states by the proposed method are demonstrated. Figure 14 shows the generated sample signals of the gas turbine rotor system under the normal state. Figures 15 and 16 show the generated sample signals of the gas turbine rotor system under the airflow excitation state and the unbalanced state. Figure 17 shows the generated sample signals of the gas turbine rotor system under the misalignment state. For comparison, the generated fault sample signals by GAN and DCGAN are also displayed.
Appl. Sci. 2022, 12, x FOR PEER REVIEW Figure 15 shows the vibration signals and generated sample signals un excitation from time domain signals, frequency domain, and time-frequency s can be seen from the FFT spectrum diagram of the original signals that there a doubling frequency components in the frequency domain, including funda quency f0, double frequency 2f0, and triple frequency 3f0. Meanwhile, it can b the FFT spectrum diagram of generated samples by the proposed method th quency features are similar to the features of original signals under air flow which proves that the key features of the samples generated by the proposed consistent with those of original samples, and the quality of the generated sam The samples generated by GAN have no obvious multi-frequency component diagram, and the samples generated by DCGAN have no obvious multi-freq ponents in the CWT diagram. By the method, the generator can generate new citation samples with a similar distribution to the original samples, which ar sistent with the original signals. With this method, the training samples can be and the training of the classifier can be enhanced to further improve the gen performance of the model. Figure 16 shows the generated gas turbine rotor fault sample signals un balanced state, it can be seen that the generated samples by the proposed m higher similarity with the original signals than those of GAN and DCGAN shows the generated gas turbine rotor fault sample signals under the misalig generated by GAN, DCGAN, and Improved DCGAN. It can be seen from FFT CWT diagrams that the frequency of the generated samples by Improve emerged with double frequency components, which indicates that the genera bine rotor system unbalanced samples are consistent with real samples from trend and can be used as extended samples of the original samples.       (2) Structural similarity analysis In order to further analyze the accuracy of the generated signals, the S obtained FFT, STFT, and CWT spectrum images can be used for comparativ [47,48].
SSIM evaluates the image quality from three aspects, brightness, contrast, ture, which can be expressed by Equation (16) x y represent the brightness term, contrast term, and structu the image, respectively. The calculation process is shown in Equation (17) to Equ  (1) Spectrum analysis The similarities between the generated samples and the original samples are analyzed by time domain analysis, Fast Fourier Transform (FFT) spectrum analysis, Short-Time Fourier Transform (STFT) spectrum analysis, and Continue Wave Transform (CWT) analysis. It can be seen from the comparison of spectrum signals in normal states in Figure 14 that the spectrum of the newly generated sample signals is consistent with the spectrum of real sample signals. It can be clearly seen from the FFT and CWT diagrams that the samples generated by the proposed method have the highest similarity with the original signals. The generated samples by GAN and DCGAN are similar to the original samples, but not completely equivalent, and there are small deviations at some peaks. Figure 15 shows the vibration signals and generated sample signals under airflow excitation from time domain signals, frequency domain, and time-frequency spectrum. It can be seen from the FFT spectrum diagram of the original signals that there are multiple doubling frequency components in the frequency domain, including fundamental frequency f 0 , double frequency 2f 0 , and triple frequency 3f 0 . Meanwhile, it can be seen from the FFT spectrum diagram of generated samples by the proposed method that these frequency features are similar to the features of original signals under air flow excitation, which proves that the key features of the samples generated by the proposed method are consistent with those of original samples, and the quality of the generated samples is high. The samples generated by GAN have no obvious multi-frequency components in the FFT diagram, and the samples generated by DCGAN have no obvious multi-frequency components in the CWT diagram. By the method, the generator can generate new airflow excitation samples with a similar distribution to the original samples, which are most con-sistent with the original signals. With this method, the training samples can be expanded, and the training of the classifier can be enhanced to further improve the generalization performance of the model. Figure 16 shows the generated gas turbine rotor fault sample signals under the unbalanced state, it can be seen that the generated samples by the proposed method had higher similarity with the original signals than those of GAN and DCGAN. Figure 17 shows the generated gas turbine rotor fault sample signals under the misalignment state generated by GAN, DCGAN, and Improved DCGAN. It can be seen from FFT, STFT, and CWT diagrams that the frequency of the generated samples by Improved DCGAN emerged with double frequency components, which indicates that the generated gas turbine rotor system unbalanced samples are consistent with real samples from the overall trend and can be used as extended samples of the original samples.
(2) Structural similarity analysis In order to further analyze the accuracy of the generated signals, the SSIM of the obtained FFT, STFT, and CWT spectrum images can be used for comparative analysis [47,48].
SSIM evaluates the image quality from three aspects, brightness, contrast, and structure, which can be expressed by Equation (16): where x represents the measured image and y represents the real image. l(x, y), c(x, y), and s(x, y) represent the brightness term, contrast term, and structure term of the image, respectively. The calculation process is shown in Equation (17) to Equation (19): c(x, y) = 2σ x σ y + C 2 s(x, y) = 2σ xy + C 3 σ x σ y + C 3 (19) where µ x and µ y represent the mean value of the measured image and the real image, respectively; σ 2 x and σ 2 y represent the variance between the measured image and the real image, respectively; and σ x σ y represents the covariance between the measured image and the real image. C 1 , C 2 , and C 3 are the constant terms to avoid the denominator being zero, and the relationships between the three are C 1 = (K 1 × L) 2 , C 2 = (K 2 × L) 2 , and C 3 = C 2 /2, in which K 1 = 0.01, K 2 = 0.03, and L = 255. The simplified equation of SSIM is converted to Equation (20): The value range of SSIM is between 0 and 1. The larger the SSIM values, the more similar the two images are, and the better quality of the measured image. According to SSIM, the SSIM of FFT, STFT diagrams, and CWT diagrams of generated signals are calculated and compared with SSIM of the real signal spectrum. The results are shown in Table 4.
From Table 4, it can be concluded that the SSIM of generated sample signals by DCGAN is larger than the SSIM of generated sample signals by GAN, and the structural similarity of the four generated fault sample signals generated by the Improved DCGAN method is strongest, which meets the requirements of samples required for fault diagnosis. The time of generating samples by GAN is 21,280 s, the time of generating samples by DCGAN is 28,120 s, and the time of generating samples by Improved DCGAN is 33,440 s. By combining the powerful feature extraction of CNN with the data generation ability of DCGAN, the Improved DCGAN method was proposed and verified in gas turbine rotor system test experiments, and the results demonstrate that it can generate effective samples and solve the problem of difficulty in extracting gas turbine rotor fault features under the condition of unbalanced and small samples.

Classification Model and Results
After new samples are generated, the original samples and generated fault samples form a new dataset. The new gas turbine sample dataset is divided into a training dataset and a testing dataset. The details of the datasets are shown in Table 3. The original training dataset has 2100 samples for fault diagnosis, the training dataset including generated samples has 4000 samples, and the testing dataset has 2000 samples. In order to examine the superiority of the proposed method, several fault diagnosis methods including Support Vector Machine (SVM), Deep Belief Network (DBN) [49], Two-Dimensional Convolutional Neural Network with Short-Time Fourier Transform of the signals as input (STFT-CNN), WDCNN, and Long Short-Term Memory (LSTM) [50] are adopted for diagnosis classification comparison of the target data. SVM is proposed on the basis of statistical learning theory and used in mechanical fault diagnosis because of its faster training convergence rate and stronger generalization ability. DBN saves the process of manually extracting features and fuses feature extraction and classification together. STFT-CNN adopts the Short-Time Fourier Transform (STFT) of the obtained sample signals as input and the Two-Dimensional Convolutional Neural Network is used for fault classification and analysis. WDCNN is more suitable for one-dimensional vibration signals with high classification accuracy, and the parameter settings of WDCNN are the same as those in Table 1. LSTM has strong adaptability for time series data and is widely used in prediction and classification. Table 5 shows the specific classification results of each fault by different diagnosis methods. Precision refers to the ratio of the number of correctly predicted positive samples to the number of all predicted positive samples. When SVM is used to classify new samples, the precision of normal, airflow excitation, unbalanced, and misalignment states are 92.50%, 94.60%, 90.30%, and 91.30%, respectively, and the classification accuracy is 92.15%. The classification accuracy of DBN and STFT-CNN improved to 98.03% and 98.08%. When WDCNN is used to classify new samples, the classification precision for normal, airflow excitation, unbalanced, and misalignment states are 99.80%, 98.1%, 98.30%, and 98.40%, respectively, and the classification accuracy is 98.65%. From the classification results, it can be seen that WDCNN is suitable for gas turbine rotor fault classification, and WDCNN was thus chosen as the classification model for gas turbine rotor fault classification. The proposed Improved DCGAN model generates higher-quality data, and the new dataset can be used for further fault classification. The WDCNN accurately learns the data distribution and features of different states of the gas turbine, and can provide higher diagnosis accuracy for the gas turbine rotor system.

Results Visualization Comparison
In order to verify the performance of the proposed method, the gas turbine samples are generated by GAN, DCGAN, and Improved DCGAN to form new datasets: dataset with original samples, dataset including generated samples by GAN, dataset including generated samples by DCGAN, and dataset including generated samples by Improved DCGAN. The four datasets are selected to train the classification model and classify the faults with different methods. Table 6 and Figure 18 display the classification accuracy comparisons of different diagnostic methods with the four datasets. When SVM is adopted, the classification accuracies of the dataset with original samples, the dataset including generated samples by GAN, the dataset including generated samples by DCGAN, and the dataset including generated samples by Improved DCGAN are 88.25%, 91.05%, 91.9%, and 92.15%, respectively; when LSTM is used, the classification accuracies of the four datasets are 90.05%, 92.25%, 94.3%, and 96.83%, respectively. When DBN and STFT-CNN are used, the classification results greatly improved. It can be seen that the classification effect by WDCNN performs best under each dataset compared with other methods. Meanwhile, it can be seen that the classification accuracy of the dataset with original samples, the dataset including generated samples by GAN, the dataset including generated samples by DCGAN, and dataset the including generated samples by Improved DCGAN are 92.1%, 96.4%, 97.9%, and 98.65%, respectively. It can be concluded that the classification accuracy of the dataset including generated samples by the Improved DCGAN method obviously increased compared with the dataset including generated samples by GAN and the dataset including generated samples by DCGAN. For the sake of displaying the classification results of the fault datasets by these methods, the visualized t-distributed Stochastic Neighbor Embedding (t-SNE) diagrams [51] are shown in Figure 19. Purple dots represent normal samples, blue dots represent airflow excitation fault samples, green dots are unbalanced fault samples, and yellow dots are misalignment fault samples. It can be seen in Figure 19a that the original samples without generated samples are mostly classified, unbalanced fault samples and misalignment fault samples are gathered together and not well separated, and some airflow excitation faults are misclassified. Figure 19b displays the classification results of the dataset including gen-erated samples by GAN; it can be seen that the classification results improved. Figure 19c displays the classification results of the dataset including generated samples by DCGAN; the four types of samples are effectively separated, although some misalignment fault samples are classified as normal samples. In Figure 19d, the classification results of the proposed method are the best, with most samples classified correctly. method obviously increased compared with the dataset including generated samples by GAN and the dataset including generated samples by DCGAN.  For the sake of displaying the classification results of the fault datasets by these methods, the visualized t-distributed Stochastic Neighbor Embedding (t-SNE) diagrams [51] are shown in Figure 19. Purple dots represent normal samples, blue dots represent airflow excitation fault samples, green dots are unbalanced fault samples, and yellow dots are misalignment fault samples. It can be seen in Figure 19a that the original samples without generated samples are mostly classified, unbalanced fault samples and misalignment fault samples are gathered together and not well separated, and some airflow excitation faults are misclassified. Figure 19b displays the classification results of the dataset including generated samples by GAN; it can be seen that the classification results improved. Figure  19c displays the classification results of the dataset including generated samples by DCGAN; the four types of samples are effectively separated, although some misalignment fault samples are classified as normal samples. In Figure 19d, the classification results of the proposed method are the best, with most samples classified correctly.

Conclusions
In this paper, a data augmentation method based on an improved Deep Convolution Generation Adversarial Network for gas turbine rotor system fault diagnosis is proposed. This method utilizes the powerful feature extraction and data generation ability of GAN to generate effective fault samples for fault diagnosis. The main highlights are summarized as follows.

Conclusions
In this paper, a data augmentation method based on an improved Deep Convolution Generation Adversarial Network for gas turbine rotor system fault diagnosis is proposed. This method utilizes the powerful feature extraction and data generation ability of GAN to generate effective fault samples for fault diagnosis. The main highlights are summarized as follows.
Firstly, based on the Deep Convolution Generation Adversarial Network, an Improved DCGAN method combining the data generation ability of DCGAN with the feature extraction ability of WDCNN was proposed, and a new signal generator more suitable for gas turbine rotor fault signals was constructed.
Secondly, a gradient penalty was adopted to adaptively optimize the loss function according to different gas turbine rotor system faults under unknown conditions.
Finally, the effectiveness of the method was verified by gas turbine rotor system fault experiments. The results indicate that the classification accuracy improved greatly with the proposed method compared to the GAN method and the DCGAN method, and the proposed method can generate effective fault samples to solve imbalances of fault samples.
In the proposed Improved DCGAN method, the improvement of the generator makes the calculation process more complex, as it needs to be calculated by high-performance computers and this consumes more time compared with GAN and DCGAN. In further work, the attention module could be introduced to enhance the extraction of fault features. In addition, the Improved DCGAN network structure optimization could be improved to shorten the consuming time for fault diagnosis.
To sum up, the proposed method effectively supplements fault samples, solves the problem of small samples and unbalanced samples, improves the fault diagnosis accuracy, and provides an effective method and data supply for fault diagnosis of gas turbine rotor systems. Meanwhile, the method makes it more convenient for managers to understand the operation status of gas turbines, formulate the evaluation criteria of maintenance and repair, and save costs associated with suspending detection and maintenance. In the future, different sensor signals such as vibration, temperature, and flow of gas turbines can be combined for multi-source data fusion to further improve the diagnosis accuracy of gas turbines.
Author Contributions: S.L., methodology and writing; H.W., review and formal analysis; X.Z., data curation and field inspection. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the National Natural Science Foundation of China (No. 51975058).

Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study are available from the corresponding author upon reasonable request.