A New Modelling and Feature Extraction Method Based on Complex Network and Its Application inMachine Fault Diagnosis

.e application of the existing complex network in fault diagnosis is usually modelled based on the time domain, resulting in the loss of sign frequency-domain features, and the extracted topology features of network are toomacroscopic and insensitive to local changes within the network. .is paper proposes a new method of local feature extraction based on frequency complex network (FCN) decomposition and builds a new complex network structure feature on this basis, namely, subnetwork average degree. .e variation law of signals in frequency domain is obtained with the aid of the structural features of complex network. .e local features that are sensitive to local changes of the network are applied to characterize the whole network, with flexible application and without limitation in mechanism. .e average degree of subnetwork could be regarded as feature parameters for rolling bearing fault diagnosis and degradation state recognition. Analysis on the experimental data and bearing life cycle data shows that the method proposed in this paper is effective, revealing that the extracted features have effective separability and high accuracy in fault recognition and the degradation detection of the life cycle of rolling bearings combined with neural networks. Moreover, the proposed method has reference value for the processing and recognition of other nonstationary signals.


Introduction
Rotating machinery is widely applied in large-scale petroleum, chemical, electric power, metallurgy, and other industries, serving as the core equipment. Its fault will not only affect the operation of the machine itself but also cause loss in follow-up production [1]. erefore, it is of great practical significance to conduct fault diagnosis for rotary machinery. e nonstationary nature of vibration signals of a rotating machine often leads to difficulty in accurately identifying faults. Time-frequency analysis and wavelet, both effective methods to process nonlinear and nonstationary signals [2], still have different degrees of defects. For example, the selection of wavelet basis in wavelet analysis directly affects the analysis result, and the basis function, without adaptive feature, cannot be changed once selected in the analysis [3].
Complex network, a new method to analyze nonlinear and nonstationary signals, can transform the wave features of signals into the topological features of the network, which overcomes the shortcomings of traditional time-frequency analysis [4,5]. It is a simple, intuitive, universal, topological, and strong robust network structure [6,7]. Visible network construction method proposed by Lacasa and Toral can process time series of larger amount of data [8]. Ming et al. presented a new method to construct directed and weighted complex network based on time series symbolic pattern with sliding window method [9]. Recursive complex network was applied by Sun et al. to extract network features of rolling bearings and diagnose faults [10]. Although complex network has achieved good results in nonlinear and nonstationary signal analysis, current diagnostic methods of complex network is usually based directly on time domain, ignoring frequency-domain features, and the extracted features have only global features, without the reflection of local structures of networks, which is difficult to ensure the accurate extraction of fault information. erefore, a new extraction method based on frequency complex network (FCN) model construction and subnetwork average degree is proposed in this paper and is applied in the bearing fault diagnosis, the monitoring and early warning of rolling bearings and the recognition of degradation state. e variation law of fault signal in frequency domain, obtained with the structural features of complex network, is transformed into the statistical features of the complex network structure, without limitation in fault mechanism. e verification results of fault signal of rolling bearings prove that this method is effective in extracting information that represents different fault features of the machine and can achieve accurate differentiation, providing a new effective approach for fault diagnosis of rotary machinery.

Recursive Algorithm.
Based on its inherent characteristics, recursion can be used to show the similar properties in some specific states. e recursive graph principle proposed by Eckmann and others is applied in this paper to visualize recursive graphs using recursive matrixes to characterize the recursive nature of the system [11]. e recursive matrix uses phase space reconstruction to reconstruct a one-dimensional sequence into a high dimensional space. e specific algorithm is as follows: (1) e power system reconstructed by one-dimensional series x i from the time delay τ, and the embedded vector of the m-dimensional phase space is as follows: (2) e distance between the reconstructed phase space i and j is calculated (3) Recursive value is calculated where H(g) is Heaviside function, ‖g‖ is Euclidean norm, and ε is reference threshold value. k i , the degree of a node, is defined as the number of other nodes connected to the node, which reflects the importance of this node. e bigger the degree of a node, the more important the node is [12]. e average degree of a complex network with the number of nodes n is denoted as p which is the mean of the degree of all nodes in the network k i , i.e.,

Construction of Subnetwork
where n is the total number of nodes in the network. e new complex network structure feature (subnetwork average degree) is extracted based on frequency complex network (FCN) decomposition, which reflects degree distribution of different regions in the network. e value of node degree in every region, the arrangement of node degrees, and the location of important node are all correlated. e network decomposition is to evenly divide the established complex network model into several subnetworks on the basis of the order of nodes (the edges of subnetworks are reserved during decomposing), and then, the average degree of each subnetwork can be obtained with formula (4) to generate average degree features of subnetworks, as shown in formula (5), namely, wherein, when determining the number m of subnetworks, m � 2 l is generally selected and should meet the requirement 2 ≤ m ≤ n.

Analysis of the Advantages of Subnetwork Average
Degree. Compared with the existing average degree features, the average degree of subnetworks can reflect the local features of the network in space with more information, which is more effective in fault recognition. e networks (a) and (b) in Figure 1, both six-node network models, have the same nodes with different connections between nodes. e number of connected edges of the nodes indicates that the important node of the network (a) is 2 and that of network (b) is 6. e degree of each node in the network needs to be calculated before the analysis on the degree of network node, with the calculation results shown in Table 1. If the network is characterized by the average value of degree of all nodes, the eigenvalues of the two networks are 2.67 with formula (4), hence no difference between the network (a) and the network (b).
However, if the average degree of subnetworks is extracted, the difference between the two networks can be reflected. e network is evenly divided into two subnetworks, with subnetwork 1 as the connection of nodes 1, 2, and 3 and subnetwork 2 as the connection of nodes 4, 5, and 6. e average degree of subnetworks is generated and a twodimensional feature vector is constructed. erefore, the feature vector of the network (a) is [2.33, 3] and that of the network (b) is [3, 2.33]. It is not difficult to find out that the average degree of subnetworks can mirror the difference between two networks in node connection, which can more accurately reflect the structural features of the network models.

An Extraction Method Based on FCN Subnetwork Average
Degree. In view of the nonstationary nature of vibration signals of rotating machinery and the abundant information contained in frequency domain [13], a new subnetwork average degree feature is proposed in this paper based on frequency complex network (FCN) decomposition. e fault diagnosis process based on subnetwork average degree is shown in Figure 2. e processing steps are as follows: (1) Rotation frequency is calculated as f r � V r /60. Frequency bandwidth m is chosen as an integer multiple of f r or f r. e frequency spectrum Y � y i i�1,2,...,n is obtained from the one-dimensional time series X using Fourier transform. e amplitudes are then squared point by point, added in the frequency band with bandwidth m and arranged in order to obtain the required band energy spectrum [14], denoted as Wherein, E k represents the energy of the K th frequency band, with the formula (2) On the basis of the recursive algorithm, the recursive matrix R reflecting self-similarity of energy spectrum can be obtained. e recursive matrix is regarded as an adjacency matrix to construct a complex network model. e average degree of subnetworks p is extracted as the feature parameters with formula (4) and (5).
(3) In all samples, half of the data are randomly selected to form the training set, and the remaining half are test set. e radical basis function (RBF) neural network [15] is trained with the training sample feature vector set V T to obtain the network model of the system. en, the network is tested and verified with the testing sample feature vector set V C to identify the types of the testing sample.

Performance Degradation Assessment Method Based on Network Structural Features.
Targeting the problems of insensitivity of the traditional performance degradation index to early weak faults and the low accuracy of degradation state recognition [16], this paper proposes a new method to assess the bearing performance degradation based on the structural features of frequency complex network, with the processing flow shown in Figure 3. e basic steps of this method are as follows: (1) Extraction of performance degradation index: the bearing vibration signal is transformed into the frequency band energy spectrum, denoted as E � [E 1 , E 2 , . . . , E k , . . . , E n ], where E k represents the energy of the k th frequency band, with the formula being E k � j+m i�j y 2 i (6). e recursive matrix R is obtained according to the self-similarity of the energy spectrum in different frequency bands. e recursive matrix is regarded as an adjacency matrix to construct a complex network, and the network average degree p is extracted as the bearing performance degradation index with formula (4).
(2) State monitoring: the Layda rule [17] is applied to determine the threshold on the basis of the average degree change curve. When the absolute value of the difference between the mathematical expectation of the data point and its set is greater than the standard deviation of the data set of 3 times, namely, |x i − x| > 3σ, the data point is determined as abnormal data. When the number of abnormal data consecutively exceeds the number of safety times, a fault alert is automatically generated.   Shock and Vibration 3 (3) Degradation state division: based on the evolution law of the performance degradation index p, the key data mutation points are determined. e degradation states of the bearing performance are approximately divided into four operating states: normal, minor fault, deepening of the fault and gradual thinning of the sharp part, and serious fault to failure. (4) Determination of the training set and testing set: the samples in each degradation state are divided into two sample sets according to the parity of the serial number. One sample set is selected as the training sample set, and the other sample set is selected as the testing sample set.  (5). e degradation feature vector set of the training set V T and the testing set V C is then generated. (6) Recognition of degradation state: the number of state categories is set as 4, and the RBF neural network is trained with the training sample feature vector set V T to obtain the network model of the system. en, the network is tested and verified with the testing sample feature vector set V C to identify the degradation state of the testing sample.

Extraction of FCN Subnetwork Average Degree of Bearing
Condition. In this paper, the vibration signals of rolling bearings of different fault types are applied to verify the effectiveness of the proposed features. A test rig as shown in Figure 4 is used to collect fault signals of the rolling bearing. e type of rolling bearing used in the experiment is 6308, with 8 rolling elements. Vibration signals are obtained by an acceleration transducer installed on the bearing seat, with a speed of 1050 r/min and a sampling frequency of 10240 Hz.
ere are four types of rolling bearing states, namely, rolling element fault, inner ring fault, outer ring fault, and normal state. Twenty samples are collected for each type of state, hence 80 sets of data in total.
e experiment data of the rolling bearing are processed with the FCN subnetwork average degree extraction method. Firstly, the energy spectrum of the frequency band is established based on time series. To keep the periodic information of the data as much as possible, the frequency bandwidth is chosen as an integral multiple of the bearing rotation frequency or the rotation frequency. e frequency bandwidth m used in this paper is 4 times the rotation frequency.
e rotation speed of the bearing during the experiment is 1050 r/min, and the frequency bandwidth is determined as 70 Hz after calculation.
Based on the energy spectrum of frequency band, a frequency complex network (FCN) model is constructed with the application of recursive algorithm. e adjacency matrix corresponding to the complex network model is the same as recursive matrix. en, the network is visualized, and the model displayed is a complex network model, unweighted and undirected. e display layout is set as FR (Fruchterman-Reingold); the size of the network nodes is determined by the degree of the nodes. e larger the value of the degree, the larger the size of the nodes and the darker the color. e frequency complex network (FCN) model corresponding to different states of the rolling bearing is shown in Figure 5.
It can be seen from Figure 5 that the complex network shape corresponding to rolling element fault and inner ring fault is similar, but there is still a big difference in the attributes of the nodes and edges. Outer ring faults correspond to complex network with sparse edges and prominent nodes.
e normal corresponding complex network is close to a fully connected network. erefore, different network models under different states of the rolling bearing have different network structures, which lays a good foundation for the extraction of network topological features.  e FCN model of the rolling bearing in different states is evenly decomposed into 8 subnetworks, and the average degree of nodes in each subnetwork is extracted to form an eight-dimensional feature vector, and the feature distribution diagram of different faults of the rolling bearing is obtained as shown in Figure 6. Wherein, Figure 6(a) is a three-dimensional graph of degree distribution of subnetwork nodes.
ere are in total 80 samples, and 8-dimensional features are extracted from each sample. Figure 6(b) is a platform of feature distribution, with different colors representing the amplitudes of sample features. e horizontal axis represents features, and the vertical axis represents samples, with 1-20 groups being rolling element faults, 21-40 groups being inner ring faults, 41-60 groups being outer ring faults, and 61-80 being normal.
One sample is chosen from each feature set of the rolling element faults, the inner ring faults, the outer ring faults, and the normal state to draw a bar graph, as shown in Figure 7. It is not difficult to find out that the features of four states (the rolling element faults, the inner ring faults, the outer ring faults, and the normal) differ greatly from each other, especially the normal state and the outer ring fault state.

Shock and Vibration
For the rolling element faults and the inner ring faults, the features of the 1 st and the 6 th dimension have great di erence.

Feature Separability.
e principal component analysis (PCA) algorithm [18] is applied in the dimensional reduction and visualization of average degree features of subnetworks to obtain the feature dimensional reduction graph as shown in Figure 8. It is not di cult to nd out that the four samples of the same type of rolling bearing have higher degree of clustering, and there is a clear degree of di erentiation with no overlap between di erent types of samples, which proves that the feature categories proposed in this paper have strong separability. Wherein, the sample clustering degrees of the outer ring fault and the normal sate are the highest and farthest from other sample types, which is consistent with the conclusion drawn from the subnetwork average degree feature diagram above. e existing complex network methods are mostly modelled based on the time domain, such as limited penetrable visibility graph method [19], and a topological feature parameter of the network is extracted to represent the entire network. Features such as the average path length, clustering coe cient, and degree, calculated on the basis of    the entire network, are not sensitive to local changes within the network and hence cannot reflect the microscopic feature of the network. Figure 9 shows the average path length, clustering coefficient, and degree based on limited penetrable visibility graph method of the four states of the rolling bearing, respectively. e horizontal axis represents samples, and there are 20 samples for each state. e vertical axis represents the network topological features corresponding to the samples. It can be seen that the ability of the three features of the average path length, the clustering coefficient, and the degree to distinct different types of samples of rolling bearing is poor. Only the normal state is significantly different from the three faults states which overlap with each other, resulting in difficulty to accurately distinguish with one other. erefore, compared with the features proposed in this paper, the existing network topological feature categories have poor separability and cannot be applied to subsequent diagnosis and recognition. e classification separability measure, a distance-based feature evaluation function, is defined as the ratio of interclass divergence and interclass divergence in a sample, which can evaluate the effectiveness of a feature subset. e smaller the value, the better the separability. For the same classifier, the good interclass separability can significantly improve the identification rate of the classifier; and more confounding and overlapping feature information makes pattern recognition very difficult for classifiers. A quantitative comparative analysis is made to network topological feature vector LCD, the combination of the average path length L, the clustering coefficient C, the degree D, and subnetwork average degree feature (reduced to three dimensions) proposed in this paper, with the comparison results shown in Table 2 which indicates that the classification separability measure of the proposed features is smaller than that of commonly used features LCD. is proves that the features of this paper are superior to the common network topological features and are consistent with the above qualitative analysis results.

Recognition Results.
In order to verify the effectiveness of the subnetwork average degree feature in fault diagnosis of rolling bearings, a radial basis function (RBF) neural network classifier [16] is used to conduct the classification test. Ten training samples and 10 testing samples are selected in each state, hence 40 training samples and 40 testing samples in total. For the 40 training samples of the four types, 40 16dimensional feature vectors are generated with the above method.
e 40 subnetwork average degree features are trained and classified with RBF neural network, and the different fault classification charts of the rolling bearing are obtained as shown in Figure 10, wherein the first type is the rolling element faults, the second type is the inner ring faults, the third type is the outer ring faults, and the fourth type is the normal state. It is not difficult to find out that when the subnetwork average degree feature is input into the RBF neural network, the recognition accuracy rate can reach 100%.
Time-domain feature parameters, easy to calculate and clear in physical meanings, have been applied in fault detection and state analysis for various types of machinery. In this paper, the following 14 types of time-domain statistical features are applied, namely, mean, standard deviation, variance, skewness, kurtosis, peak-to-peak value, square root amplitude, average amplitude, rms amplitude, extreme value, waveform index, peak index, pulse index, and margin index. e above parameters constitute the feature vectors representing the state of the bearing, which are compared with the features mentioned herein in the RBF recognition accuracy rate. Similarly, 10 training samples are selected for each state, and the remaining 10 groups of samples are identified using the RBF neural network as a classifier, with the recognition results shown in Figure 11. It is not difficult to find out that when the time-domain statistical parameters are input into the RBF neural network, the recognition accuracy rate can reach 97.5%, which indicates that better separability features are very important for subsequent accurate recognition.
In order to more objectively compare the recognition results, 10 groups of samples are randomly selected from each fault set as training samples, and RBF neural network is used as the classifier to identify the remaining 10 groups of samples.
e time-domain feature parameters mentioned above and the features in this paper are independently used. e recognition results in Table 3 are the mean and variance of the accuracy rate of each algorithm after running 10 times independently in MATLAB.
e results indicate that the mean value of the feature recognition accuracy rate in this paper is high, and the variance is small, which proves that the features in this paper are not only good in classification separability, but also relatively stable, with no significant fluctuation in the recognition accuracy for different training data in the same data set. e vibration data are collected once every 10 minutes, and the data length is 20480 points. In this paper, the data of bearing 1 in experiment 2 are selected, with a total of 984 samples during the experiment, and bearing 1 is seriously damaged [20]. e method proposed in this paper is applied to extract the performance degradation index. e corresponding band energy spectrum of each sample is obtained. In order to retain the periodic information of the data as much as possible, the frequency bandwidth m used in this paper is the bearing rotation frequency. e 500 th , 533 rd , 600 th , 702 nd , 823 rd , and 950 th samples are successively selected to draw their band energy spectrum, as shown in Figure 12. It is clear that the spectral peaks of the energy spectrum in the fault frequency band are more and more prominent with the deepening of the fault degree. Since the intervals between the spectral peaks are the same, the self-similar features of the energy spectrum are more and more obvious. e frequency band energy spectrum is regarded as a one-dimensional sequence, and a recursive algorithm is applied to extract the self-similar features of the sequence. By setting the delay time as τ 6 and embedding dimension as s 2, a recursive matrix corresponding to the energy spectrum sequence can be obtained. Donner et al. found that the recursive matrix can be regarded directly as the adjacency matrix of the network, the sequence is described in the form of a network, and self-similar feature inside the sequence is re ected by the structural features of the network. A recursive algorithm is applied to transform each sample into a complex network model represented by points and edges. e average degree features of each network model are then extracted. e average degree change curve of the entire experimental sample is shown in Figure 13, and the rst half of the curve is comparatively steady, which means that the bearing is in normal state. e curve suddenly drops at the 533 rd point, and then it comes into a more stable  Shock and Vibration decline phase, which means that the bearing is experiencing early weak faults, and the fault degree gradually deepens. e curve drops sharply at the 702 nd point and then slowly increases, indicating that the bearing may have a partial fault and spall. e sharp part of the bearing is gradually attened. e curve falls sharply again at the 823 rd point and became more and more unstable, indicating that the bearing is in a state of serious degradation until it completely fails. It is not di cult to nd out that the average degree feature is sensitive to the early faults of the rolling bearing. With the degradation of the bearing, the feature value gradually decreases and the uctuation amplitude gradually increases.

State Monitoring and Warning.
e above analysis indicates that the average degree feature tends to gradually decrease with the performance degradation of the rolling bearing. erefore, a threshold can be set, and when the average degree exceeds the threshold, a fault alarm is given. e threshold is determined with the Layda rule in this paper. When the absolute value of the di erence between the data point of the average degree and the mathematical expectation of the set is greater than 3 standard deviations, that is |x i − x| > 3σ, the data point is identi ed as abnormal. In order to avoid accidental errors, the situation that abnormality occurs to the average degree in ve consecutive times is de ned as device abnormal, and thereby a fault alarm is given. As shown in Figure 14, an alarm of abnormal data is given in the 533 rd set of data with the method proposed in this paper, indicating that the average degree has exceeded the threshold range ve times successively, which proves that this method is sensitive to the early fault of the bearing and can detect the exact time of the early fault, so as to succeed in the monitoring and early warning of the bearing.

Comparative Analysis.
Time-domain parameters, statistically easy to calculate and clear in physical meanings, have long been applied in state monitoring and fault diagnosis of rolling bearings, among which mean square value C, kurtosis K, etc., are often used as index of bearing performance degradation [21]. e mean square value and kurtosis curves of the entire experimental data are shown in Figures 15 and 16. With the deterioration of bearing faults, the mean square value and kurtosis tend to increase, but both begin to show a signi cant increase at the 702 nd sample, and early faults could not be detected. In addition, the kurtosis uctuates in the normal stage and has poor stability.     e average degree of nodes in each subnetwork is extracted to form the multidimensional degradation feature vector to identify the degradation state of the bearing, that is, the average degree of the subnetwork, p → [p 1 , p 2 , p 3 , . . . , p 16 ]. e average degree change curve of the subnetwork for the entire experimental samples is shown in Figure 17. Since the feature vector is composed of the degree of nodes in di erent areas of the network, there is no redundant information between the values. Hence, it is not necessary to perform dimensionality reduction on the feature vector. As can be seen from Figure 18  In order to further understand the subnetwork average degree feature vectors, principal component analysis (PCA) algorithm is applied to reduce the dimension of the subnetwork average degree feature to obtain the feature reduction dimension shown in Figure 19. In the four degradation states of the rolling bearing, the similar samples have higher degree of aggregation, and there is only partial overlapping between the di erent types of samples, which indicates that the multidimensional degradation feature vectors mentioned in this paper can meet the requirements for distinguishability of di erent states. Wherein, the normal state has the highest degree of aggregation, and the serious degradation to the fault state has the lowest degree, which accords with the basic features of the degradation of the rolling bearing. e radial basis function (RBF) neural network classi er is applied to identify the four states of the rolling bearing life cycle. e rst stage of the bearing is the normal state and contains the rst to 533 rd samples; the second stage is the minor fault, including the 534 th to 702 nd samples; the third stage is deepening of the fault and gradual thinning of the sharp part, including the 703 rd to 823 rd samples; the fourth stage is serious fault to failure, including the 824 th to 984 th samples. e samples in each stage are divided into two sample sets according to the parity of the sequence, with one sample set selected as the training sample set, and the other set is selected as the testing sample set. e training sample feature vector set V T is the training data set of RBF neural network, among which there are 267 training samples in the   rst stage, 84 training samples in the second stage, 61 training samples in the third stage, and 80 training samples in the fourth stage. en, the network is tested and veri ed with the feature vector set V C of the testing sample. e degradation state of the testing sample is identi ed, and the result of the degradation recognition of the rolling bearing is obtained as shown in Figure 17, with a recognition accuracy rate of 93.09%.
It can be seen from Figure 17 that the rst three states of the rolling bearing life cycle have high recognition accuracy, while for the serious fault to failure state, a large number of samples are identi ed as other states, and the recognition accuracy is low. e reason may lie in the unstable performance during the serious fault state, resulting in the uctuation of the extracted features and the low similarity between samples. However, the entire recognition results can basically meet the requirements of the state recognition of bearing degradation.

Comparative Analysis.
Statistical parameter diagnosis, a relatively e ective diagnostic method, has been applied since a long time ago, with the time-domain statistical parameter the most commonly used one. For rolling bearings, the frequency components of the signal, the energy of di erent frequency components, and the peak position of the main frequency spectrum of the energy spectrum will also change when a fault occurs. erefore, the statistical parameters in the frequency domain are often applied to identify the degradation stats of the bearing. In this paper, 14 kinds of time-domain feature parameters [20] and 12 kinds of frequency-domain feature parameters [22] are used to characterize the bearing state and are compared with the features proposed in this paper in RBF recognition rate of bearing degradation. Similarly, the feature vector set V T of the training sample is used as the training data set of the RBF neural network, and then the network is tested and veri ed with the feature vector set V C of the testing sample to identify the degradation state of the testing sample, with the recognition results shown in Table 5. Compared with the time-domain feature parameters and the frequency-domain feature parameters, the average degree of the subnetworks proposed in this paper is more accurate in identifying the degradation state of the bearings.

Conclusion
In order to overcome the de ciencies of existing complex network modelling methods and conventional network topological features, this paper constructs a new complex network model from the perspective of frequency domain and makes application of complex network decomposition to extract the average degree characteristics of subnetworks sensitive to local changes in the network based on the average degree of the bearing condition monitoring and early warning method and the average degree of the subnetwork bearing degradation state recognition method, which are applied in the life cycle of the rolling bearing detection. e results show that the mean and variance of the recognition accuracy of the proposed method are better than those of the time-domain statistics. e average degree of structural features of complex network can e ectively characterize the   degradation process of bearings and is more sensitive to early weaker faults than commonly used time-domain statistical parameters such as kurtosis and RMS values. e degradation recognition method based on average degree of subnetworks can effectively identify the degradation state of bearings, with a recognition rate of 93%, which is higher than the time-domain statistical parameter and the frequency-domain statistical parameter. e method and the degradation features extracted in this paper are sensitive to the early faults and have high accuracy in degradation recognition, which can meet the requirements of the degradation assessment of rolling bearing performance.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.