Bearing fault diagnosis via kernel matrix construction based support vector machine

2.1. Continuous wavelet transform

CWT aims to measure a local similarity between wavelet $\psi \left(t\right)$ at scale $s$ position $\tau$ and signal $f\left(t\right)$. The wavelet coefficient $c\left(s,\tau \right)$ can be defined by Eq. (1):

By shifting $\psi \left(t\right)$ in time and scaling $\psi \left(t\right)$, a wavelet coefficient matrix $\mathbf{C}$ can be created which is viewed as a time-frequency space as Eq. (2) and represents the dynamic characteristics of the signal $f\left(t\right)$:

where $c_{m,n}$ is the coefficient at the $m$th scale and at the $n$th data point of a sample signal.

2.2. Singular value decomposition

SVD is used to decompose the wavelet coefficient matrix $\mathbf{C}$. Assuming matrix $\mathbf{C}$ with the size of $m\times n$, the SVD results can be expressed by Eq. (3):

where $\mathbf{A}$ and $\mathbf{B}$ are orthogonal matrices of $m\times m$ and $n\times n$, respectively. $\mathrm{\Lambda }$ is an $m\times n$ diagonal nonnegative matrix. The diagonal elements in $\mathbf{\Lambda }$ are called singular values (SVs) of $\mathbf{C}$, which are only determined by matrix $\mathbf{C}$ itself and denote the natures of matrix $\mathbf{C}$, namely, the characteristics of a sample signal. Given $m<n$, Eq. (3) can be illustrated in details as Eq. (4):

SVs constitute vector $\mathbf{x}$ described as Eq. (5). $\mathbf{x}$ also denotes the feature vector extracted from a sample signal:

3.1. Kernel matrix pattern based SVM

A training set ${\mathbf{S}}_1$ and a test set ${\mathbf{S}}_2$ are given as below:

Assume $\mathbf{\phi }\left(\mathbf{x}\right)$ is a image of point $\mathbf{x}$ mapped into a high dimensional feature space $\mathbf{F}$ and all the sample images can be separated by a hyper-plane as Eq. (8):

The hyper-plane is determined to solve the following optimization problem:

It is equivalent to solving a constrained convex quadratic programming optimization problem:

$\mathbf{K}$ is named training kernel matrix which is a $l\times l$ symmetric matrix with $k_{ij}=\kappa \left({\mathbf{x}}_i,{\mathbf{x}}_j\right)$, the inner product between the images of two training samples in space $\mathbf{F}$. $l$ is the number of training samples, $\mathbf{e}$ is a column vector with $e_i=1$, $\mathbf{\alpha }={\left({\alpha }_1,{\alpha }_2,\dots ,{\alpha }_l\right)}^T$ is a Lagrange multiplier vector, $\mathbf{\Lambda }$ is an $l\times l$ diagonal matrix with ${\mathrm{\Lambda }}_{ii}=y_i$, $c$ is error penalty constant, $y_i$ is the $i$th sample class label.

By maximizing $\mathbf{W}\left(\alpha \right)$, the optimized ${\mathbf{\alpha }}^{*}$ can be obtained. Thus, the optimized $b^{*}$ can be computed using the following equation:

where ${\mathbf{K}}^i$ is the $i$th column vector of $\mathbf{K}$.

Hence, the pattern function of SVM to predict the class of unseen sample ${\mathbf{x}}_v$ can be written as:

and:

${\mathbf{K}}_t$ is named prediction kernel matrix of $l\times p$ with $k_{iv}={\kappa }_t\left({\mathbf{X}}_i,{\mathbf{X}}_v\right)$, the inner product between the images of a training sample and a test sample in space $\mathbf{F}$. $p$ is the number of the test samples, ${\mathbf{K}}_t^v$ is the $v$th column vector of ${\mathbf{K}}_t$.

According to Eq. (15), the result of pattern analysis just depends on kernel matrices, so it is feasible for SVM to solve nonlinear classification problems by developing appropriate kernel matrices.

3.2. Kernel matrix construction

A novel method on kernel matrix construction (KMC) is presented to solve nonlinear classification problems using pattern analysis based SVM.

To our best knowledge, this KMC based method has not been studied in the field of machinery fault diagnosis. The specific procedure of KMC is stated below and illustrated in Fig. 2.

Fig. 2Flow chart of training kernel matrix K construction

Step 1: Provide training set ${\mathbf{S}}_1=\left\{\left({\mathbf{x}}_{1,}{y}_1\right),\dots ,\left({\mathbf{x}}_{l,}{y}_l\right)\right\}$ and test set ${\mathbf{S}}_2=\left({\mathbf{x}}_{l+1,}{y}_{}\left(l+1\right)\right),\dots ,\left({\mathbf{x}}_{l+p,}{y}_{l+p}\right)$. Suppose there exists $r$ classes of samples in ${\mathbf{S}}_1$. ${\mathbf{S}}_1$ is used to construct training kernel matrix $\mathbf{K}$ of $l\times l$. ${\mathbf{S}}_1$ and ${\mathbf{S}}_2$ are used for predicting matrix ${\mathbf{K}}_t$ of $l\times p$. Let ${\mathbf{T}}_1$ be a matrix of $l\times l$, ${\mathbf{T}}_2$ of $l\times p$. Initialize ${\mathbf{T}}_1\left({\mathbf{T}}_2\right)$, $\mathbf{K}\mathbf{}\left({\mathbf{K}}_t\right)=$ 0.

Step 2: Produce distance matrix ${\mathbf{D}}_1$$\left({\mathbf{D}}_2\right)$ by computing pairwise distance of samples using Eq. (17). Thus, ${\mathbf{D}}_1$ about pairwise distance of training samples and ${\mathbf{D}}_2$ about pairwise distance between training and test samples are shown as Eq. (18):

$d_{ij}$ denotes the distance of the $i$th training sample and the $j$th training sample in ${\mathbf{D}}_1$ and the distance of the $i$th training sample and the $j$th test sample in ${\mathbf{D}}_2$.

Step 3: Find the $k$ closest neighbors distribution of each sample. The $k$ closest neighbors of each sample are the $k$ least numbers in each column of ${\mathbf{D}}_1$$\left({\mathbf{D}}_2\right)$. Set 1 to the elements in ${\mathbf{T}}_1$$\left({\mathbf{T}}_2\right)$ that have the same locations of the $k$ least numbers in ${\mathbf{D}}_1$$\left({\mathbf{D}}_2\right)$. The rows of ${\mathbf{T}}_1$$\left({\mathbf{T}}_2\right)$ is divided into $r$ blocks, its blocks and columns stand for classes and samples, respectively. Eq. (19) shows the k closest neighbors distribution in different classes by setting 1:

Step 4: Classify using majority vote among the $k$ neighbors. If a sample has the majority of $k$ neighbors within one block, the sample belongs to the block related class. Set 1 to the column within the block, 0 to the rest of that column. For example, if ${\mathbf{x}}_i$$\left({\mathbf{x}}_j\right)$ belongs to the 1st class, ${\mathbf{T}}_1$$\left({\mathbf{T}}_2\right)$ is revised as Eq. (20):

Step 5: Compress multi-classes of ${\mathbf{T}}_1$$\left({\mathbf{T}}_2\right)$ into two classes. The 1st class remains unchangeable and the other classes merge into the 2nd class. Where a sample is 0(1) in the 1st class must be 1(0) in the 2nd class. The updated ${\mathbf{T}}_1$ and ${\mathbf{T}}_2$ are shown as Eq. (21):

Step 6: Select the 1st row of ${\mathbf{T}}_1$$\left({\mathbf{T}}_2\right)$ as a row matrix ${\mathbf{R}}_1$$\left({\mathbf{R}}_2\right)$. ${\mathbf{R}}_1$ reveals training samples class, ${\mathbf{R}}_2$ describes test samples class:

The training kernel matrix $\mathbf{K}$ can be constructed based on ${\mathbf{R}}_1$, it is an $l\times l$ symmetric matrix with diagonal element 1 as Eq. (23). $\mathbf{K}$ reflects the similarity among training samples. The prediction matrix ${\mathbf{K}}_t$ with $l\times p$ can be likewise established according to ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$. ${\mathbf{K}}_t$exhibits the similarity between training and test samples. In $\mathbf{K}$$\left({\mathbf{K}}_t\right)$ “1” means the maximum similarity between corresponding samples and “0” means no similarity:

Step 7: Increase $k=k+1$ and repeat from Step 3 to Step 6 till k exceeds the upper. The upper should be given to a medium value to save computing time.

Step 8: Take the average of the matrices $\mathbf{K}$$\left({\mathbf{K}}_t\right)$. A number of $\mathbf{K}$$\left({\mathbf{K}}_t\right)$ would be produced with the closest neighbor $k$ changing from the lower to the upper. Average these matrices to get better intrinsic relations among samples. The averaged $\mathbf{K}$$\left({\mathbf{K}}_t\right)$ is applied to the pattern function for classification.

4.1. Experimental setup and vibration data

The experiment data about faulty bearings is taken from the Case Western Reserve University Bearing Data Center. The vibration data has been widely utilized as a standard dataset for REB diagnosis. As shown in Fig. 4, the test stand consists of a 2 hp motor (left), a torque transducer/encoder (center), a dynamometer (right), and control electronics. The test bearings support the motor shaft. Motor bearings were seeded with faults using electro-discharge machining. Faults ranging from 0.007 inches in diameter to 0.021 inches in diameter were introduced separately at the inner raceway, rolling element and outer raceway. Faulted bearings were reinstalled into the test motor and vibration data was recorded for motor loads of 0 to 3 horsepower (motor speeds of 1797 to 1720 RPM). Bearing Information is shown as Table 1 and Table 2. Vibration signal was collected using accelerometers, which were attached to the drive end of the motor housing with magnetic bases. Then vibration signal was digitalized through a 16 channel DAT recorder. Digital data was collected at 48.000 samples per second for drive end bearing faults and post processed in a MATLAB environment. Speed and horsepower data were collected using the torque transducer/encoder and were recorded.

In this experiment, the vibration data of the drive end bearing are chosen to perform location and severity identification of bearing fault. The sampling frequency is 48 kHz and each sample contains 2048 data points. Four different bearing conditions, i.e. healthy state, outer race fault, inner race fault and ball fault are observed for fault location recognition using KMCSVM. In addition, four types of fault severities (healthy, 0.007, 0.014 inch and 0.021 inch) are also considered to assess KMCSVM classification performance.

4.2. Feature extraction

Referring to wavelet selection criterion in subsection “Wavelet selection” presented in [37], the energy to entropy ratios about six different wavelets including the Shannon, Gaussian, Complex Morlet, Daubechies, Meyer and Morlet are plotted in Fig. 5. due to the maximum energy to entropy ratio, the Shannon wavelet is selected as the best mother wavelet to perform continuous wavelet transform. The feature vectors are calculated from the coefficient matrices using SVD.

Fig. 4Rolling element bearing test rig

Fig. 5Energy to entropy ratios of datasets using wavelets

4.3. Classification of bearing conditions

The performance of KMCSVM is evaluated by identifying bearing fault location and fault severity, and compared with other kernel pattern recognition methods like CTKSVM, L-SVM and RBFSVM that have been studied in the previous work [37]. CTKSVM is a SVM based on the classification tree kernel which is constructed using fuzzy pruning strategy and tree ensemble learning algorithm to improve the diagnostic capability of REB fault. L-SVM makes use of classical linear kernel as well as RBFSVM with radial basis function to diagnose REB fault. Both five-fold cross validation and independent test are conducted to obtain the classification accuracy of these SVM classifiers. To discover the true fault from the possible multi-faults, SVM classifiers are trained in a tournament of one against others by setting one class as +1 and others as –1, and continuous to detect unknown sample in the same manner.

4.3.1. Identification of fault location

Fault location recognition strives to distinguish four different bearing conditions, i.e. healthy state, outer race fault, inner race fault and ball fault. Table 3 lists 12 datasets with various loading, fault size and shaft speed for analysis. There are 48 samples for each state, thus total 192 samples for all states in each dataset shown as Table 4.

The groups of sample sets are allocated in the way that satisfies the tournament of training and test using five-fold cross validation and independent test as described in Table 5.

Table 3Description of 12 datasets on fault locations

Dataset	1	2	3	4	5	6	7	8	9	10	11	12
Fault (inch)	0.007	0.007	0.007	0.007	0.014	0.014	0.014	0.014	0.021	0.021	0.021	0.021
Load (HP)	0	1	2	3	0	1	2	3	0	1	2	3
Speed (RPM)	1796	1772	1750	1725	1796	1772	1750	1725	1796	1772	1750	1725

Table 4Composition of dataset on fault locations

Fault type	Sample size
H	48
O	48
I	48
B	48
H – healthy, O – outer race defect, I – inner race defect, B – ball defect

Table 5Sample set with different fault locations for training and test

Sample label		5-fold cross validation	Independent test
			Training	Test
1	H vs. (O+I+B)	48: (48 + 48 + 48)	24: (24 + 24 + 24)	24: (24 + 24 + 24)
2	O vs. (I+B)	48: (48 + 48)	24: (24 + 24)	24: (24 + 24)
3	B vs. (O+I)	48: (48 + 48)	24: (24 + 24)	24: (24 + 24)
4	I vs. (O+B)	48: (48 + 48)	24: (24 + 24)	24: (24 + 24)
5	I vs. B	48:48	24: 24	24: 24
6	O vs. I	48:48	24: 24	24: 24
7	O vs. B	48:48	24: 24	24: 24

Fig. 6 illustrates the accuracy of the four classifiers corresponding to the 12 datasets in Table 3 using five-fold cross validation. The classification accuracy of RBFSVM is obviously lowest among all the methods. In eight cases (Fig. 6(b)-(h), Fig. 6(k)), KMCSVM achieves a higher classification accuracy. In three cases (Fig. 6(a), Fig. 6(i), Fig. 6(l)), the classification rates based on KMCSVM, CTKSVM and L-SVM are almost similar to each other. Only in one case (Fig. 6(j)), the classification accuracy of KMCSVM is slightly lower than those of CTKSVM and L-SVM. As a whole, the classification ability increases in the order of RBFSVM, L-SVM, CTKSVM and KMCSVM. Additionally, the classification accuracy of KMCSVM maintains the least fluctuation. It indicates that KMCSVM is insensitive to the changes of sample sets.

The classification accuracy of KMCSVM is observed as the fault size changes under specific loads (0 HP, 1 HP, 2 HP, 3 HP). It can be inferred from Fig. 7 that the accuracy of KMCSVM descends in sequence of fault sizes from 0.007 to 0.021 then to 0.014 inch except that the accuracy alternately occurs between 0.014 and 0.021 inch under 1 HP load as described Fig. 7(b). In the early stage of bearing fault (0.007 inch), the accuracy arrives at 100 %. The accuracy then falls with the growth of bearing fault (0.014 inch). When the fault size further enlarges (0.021 inch), the classification accuracy rises again.

Fig. 6Accuracy of the four classifiers corresponding to 12 datasets

Fig. 8 describes the classification accuracy of KMCSVM with the load variation while fixing the fault size. In Fig. 8(a), the accuracy for fault with 0.007 inch always keeps 100 %. So KMCSVM is robust against the load interference and excellent fault classification performance. From Fig. 8(b) and Fig. 8(c), it demonstrates that the loading disturbances bring the accuracy fluctuations irregularly.

It also can be seen from Table 6 that the average accuracy of KMCSVM, whenever five-fold cross validation or independent test, is the highest (all more than 95.60 %). The corresponding training and test time are summarized in Table 7. For 5 folds cross validation, the computational cost of training KMCSVM is higher than that of the other three methods. The reason is that the construction of training kernel matrix needs more computational time. Once KMCSVM is trained, it has the efficient diagnosis capability with no more than 8.3 s. For independent validation, it takes less time to train (less than 9.97 s) and test (less than 3.02 s) KMCSVM which is very close to other methods. Thereby, KMCSVM displays its outstanding fault diagnosis performance.

Fig. 7Accuracy of KMCSVM with the fault size variation

Fig. 8Accuracy of KMCSVM with the load variation

4.3.2. Identification of fault severity

Fault severity recognition seeks to evaluate REB fault size that influences the machinery health and its lifetime. In Table 8, four types of fault severity conditions are considered to assess KMCSVM classification performance using datasets in Table 9.

The groups of sample sets are provided by means of tournament to identify different fault sizes as described in Table 10.

Table 6Average accuracy of 4 classifiers using 12 datasets on fault locations

Sample set	5 folds cross validation				Independent validation
	KMCSVM	CTKSVM	L-SVM	RBFSVM	KMCSVM	CTKSVM	L-SVM	RBFSVM
H vs. (O+I+B)	99.87	99.57	97.79	49.01	99.65	99.39	93.75	44.44
O vs. (I+B)	97.57	95.95	92.59	66.38	97.68	95.60	92.59	68.06
B vs. (O+I)	96.99	93.87	89.12	48.15	96.41	93.52	87.50	62.38
I vs. (O+B)	95.72	94.04	90.58	72.34	95.60	94.10	84.49	74.19
I vs. B	96.09	93.84	92.45	66.84	95.66	92.36	92.19	58.16
O vs. I	97.22	96.34	96.09	64.18	97.40	96.35	95.83	56.42
O vs. B	98.61	96.53	92.53	63.37	98.26	96.18	93.39	69.44

Table 7Average training time and test time of 4 classifiers using 12 datasets on fault locations

Sample set	Time (s)	5 folds cross validation				Independent validation
		KMCSVM	CTKSVM	L-SVM	RBFSVM	KMCSVM	CTKSVM	L-SVM	RBFSVM
H vs. (O+I+B)	Train	262.73	12.86	10.75	15.82	9.97	1.90	1.57	4.97
	Test	8.30	0.01	0.01	0	3.02	0	0	0
O vs. (I+B)	Train	108.29	11.94	10.76	12.87	3.96	1.98	1.60	3.11
	Test	5.90	0	0	0	1.25	0	0	0
B vs. (O+I)	Train	109.28	14.42	11.95	14.27	3.93	2.10	1.76	3.16
	Test	5.54	0	0	0	1.25	0	0	0
I vs. (O+B)	Train	107.91	14.31	11.93	13.80	4.14	2.12	1.93	3.35
	Test	5.23	0	0	0	1.28	0	0	0
I vs. B	Train	33.66	9.76	8.45	9.29	1.22	1.55	1.35	1.76
	Test	1.51	0	0	0	0.42	0	0	0
O vs. I	Train	29.03	8.51	7.40	8.30	1.42	1.44	1.28	1.71
	Test	1.39	0	0	0	0.40	0	0	0
O vs. B	Train	28.54	9.53	8.09	8.97	1.31	1.55	1.41	1.83
	Test	1.38	0	0	0	0.40	0	0	0

Table 8Composition of dataset on fault severity

Fault severity	Sample size	Defect size(inch)
H	48	0
S1	48	0.007
S2	48	0.014
S3	48	0.021
H – healthy, S1 – fault with 0.007 inch, S2 – fault with 0.014 inch, S3 – fault with 0.021 inch

Table 9Description of 12 datasets on fault severity

Dataset	1	2	3	4	5	6	7	8	9	10	11	12
Location	O	O	O	O	I	I	I	I	B	B	B	B
Load (HP)	0	1	2	3	0	1	2	3	0	1	2	3
Speed (RPM)	1796	1772	1750	1725	1796	1772	1750	1725	1796	1772	1750	1725

Fig. 9, Fig. 10 and Fig. 11 illustrate the accuracy of KMCSVM tested on the 12 datasets in Table 6 using five-fold cross validation and compared with CTKSVM, L-SVM and RBFSVM. Clearly, RBFSVM contributes to the lowest accuracy. In seven cases (Fig. 9(a)-(d) and Fig. 10(a)-(c)), KMCSVM reaches the highest 100 %. In four cases Fig. 10(d), Fig. 11(a) and Fig. 11(c)-(d)), the accuracy based on KMCSVM are second only to L-SVM. Fig. 11(b) indicates the accuracy of KMCSVM is slightly lower than those of CTKSVM and L-SVM. Consequently, KMCSVM is highly suitable for fault severity recognition of bearing outer race and inner race. Moreover, the accuracy curves of KMCSVM stay little fluctuation. It exhibits good stability of KMCSVM on changes of sample sets and load interference.

Fig. 9Accuracy of the four classifiers for fault severity in bearing outer race

Fig. 10Accuracy of the four classifiers for fault severity in bearing inner race

Table 11 gives the average accuracy of 4 classifiers about REB fault severity recognition. For five-fold cross validation, the classification performance of KMCSVM is slightly lower than L-SVM because KMCSVM is not so well as L-SVM in fault severity recognition of bearing ball. However, KMCSVM is the best one of 4 classifiers which gets the highest accuracy for independent test. The corresponding training and test time are shown in Table 12. The computational cost of training and test KMCSVM is similar to that used for fault locations diagnosis mentioned above.

Fig. 11Accuracy of the four classifiers for fault severity in bearing ball

Table 10Sample set with different severities for training and test

Sample label		5 folds cross validation	Independent test
			Training	Test
1	H vs. (S1+S2+S3)	48: (48 + 48 + 48)	24: (24 + 24 + 24)	24: (24 + 24 + 24)
2	S1 vs. (S2+S3)	48: (48 + 48)	24: (24 + 24)	24: (24 + 24)
3	S3 vs. (S1+S2)	48: (48 + 48)	24: (24 + 24)	24: (24 + 24)
4	S2 vs. (S1+S3)	48: (48 + 48)	24: (24 + 24)	24: (24 + 24)
5	S2 vs. S3	48:48	24: 24	24: 24
6	S1 vs. S2	48:48	24: 24	24: 24
7	S1 vs. S3	48:48	24: 24	24: 24

Table 11Average accuracy of 4 classifiers using 12 datasets on fault severity

Sample set	5 folds cross validation				Independent validation
	KMCSVM	CTKSVM	L-SVM	RBFSVM	KMCSVM	CTKSVM	L-SVM	RBFSVM
H vs. (S1+S2+S3)	100	99.48	92.54	61.24	99.74	99.13	80.73	83.42
S1 vs. (S2+S3)	98.84	97.28	95.14	58.62	98.73	92.94	83.91	80.44
S3 vs. (S1+S2)	98.15	97.79	99.42	74.17	98.03	96.76	92.13	85.07
S2 vs. (S1+S3)	98.21	96.59	98.67	67.85	97.80	93.12	86.81	82.87
S2 vs. S3	97.83	98.10	98.00	60.68	97.57	96.89	92.19	90.11
S1 vs. S2	99.16	96.70	99.48	57.12	98.79	93.75	88.02	84.03
S1 vs. S3	98.96	98.09	99.48	65.02	99.13	98.96	94.27	80.21

According to the results in the above experiments, KMCSVM earns higher accuracy in diagnosis of fault locations and severities compared to the other three methods. The success of KMCSVM owes to the strategy for the construction of kernel matrix $\mathbf{K}$ and ${\mathbf{K}}_t$. This strategy can effectively suppress irrelevant features and mine the similarity degree of samples. So $\mathbf{K}$ and ${\mathbf{K}}_t$ can express the intra-class compactness and inter-class separation more objectively than CTKSVM. RBFSVM and L-SVM employ fixed kernels that have nothing to do with the analyzed samples, thus fall behind KMCSVM and CTKSVM. Hence, KMCSVM is a competitive method for REB fault diagnosis.

Table 12Average training time and test time of 4 classifiers using 12 datasets on fault severity

Sample set	Time (s)	5 folds cross validation				Independent validation
		KMCSVM	CTKSVM	L-SVM	RBFSVM	KMCSVM	CTKSVM	L-SVM	RBFSVM
H vs. (S1+S2+S3)	Train	164.32	11.75	10.68	15.84	15.58	1.87	1.53	3.76
	Test	5.86	0.01	0	0.01	5.08	0	0	0
S1 vs. (S2+S3)	Train	66.69	11.82	10.56	12.46	6.62	1.91	1.55	2.53
	Test	5.42	0	0.01	0	2.07	0	0	0
S3 vs. (S1+S2)	Train	63.68	11.51	9.60	11.60	6.44	1.74	1.43	2.38
	Test	5.54	0	0	0	2.15	0	0	0
S2 vs. (S1+S3)	Train	61.69	12.16	10.27	12.42	6.49	1.89	1.56	2.47
	Test	5.22	0	0	0	2.16	0	0	0
S2 vs. S3	Train	17.85	8.74	7.38	8.52	2.14	1.52	1.26	1.53
	Test	0.98	0	0	0	0.65	0	0	0
S1 vs. S2	Train	17.30	9.24	7.89	8.97	2.10	1.53	1.30	1.60
	Test	1.03	0	0	0	0.64	0	0	0
S1 vs. S3	Train	17.02	7.78	6.72	7.73	2.30	1.35	1.16	1.45
	Test	0.95	0	0	0	0.67	0	0	0