A novel ensemble convex hull-based classification model for bevel gearbox fault diagnosis

Geometric learning models originated from vector space geometry. They all aim to describe a set with the linear combination of specific vectors. By imposing restrictions on the weights of the combinations, different geometric models such as CH, AH, and HD are formed. The CH model requires that the sum of the combined weights is 1, and each weight is non negative. When applied to pattern recognition, the CH model can be understood as the smallest representative element of all samples in the feature space.

Suppose the sample set of one class is $S = \{ {s_i}\} ,i = 1, \ldots ,n$, where ${s_i} \in {R^n}$, the CH of $S$ is:

$$\begin{align}{\text{CH}}(S) = \left\{ {\sum\limits_{i = 1}^n {{c_i}} {s_i}|\sum\limits_{i = 1}^n {{c_i}} = 1,0 \leqslant {c_i} \leqslant 1} \right\},\end{align} \tag{ 1 }$$

where $n$ is the number of samples and ${c_i}$ denotes the combination coefficient of the $i{\text{th}}$ sample.

From the definition of CH, we can find that the shape of the CH model depends on the combination coefficient and the distribution of samples in the feature space. That means we can control the shape of the CH by adjusting the combination coefficients. This is very convenient for converting the CH boundary with prior knowledge to obtain different decision functions. This is why CH has the advantage of clear geometric meaning.

Take the binary classification as an example, the essential idea of the CH classification model is to find the maximal margin hyperplane between two CHs, $f = w \bullet s + b$, where $w$ and $b$ denote the normal vector and the bias respectively. Therefore, the problem is transformed into solving the nearest point pairs of two CHs. Assume that the positive sample set is ${S_ + } = \{ {s_{i + }}\} ,i = 1, \ldots ,{n_ + }$, the negative sample set is ${S_ - } = \{ {s_{i - }}\} ,i = 1, \cdots {n_ - }$, then the two CH models can be expressed as:

$$\begin{align}{\text{CH}}({S_ + }) = \left\{ {\sum\limits_{i + = 1}^{n + } {{c_{i + }}{s_{i + }}|\sum\limits_{i + = 1}^{n + } {{c_{i + }} = 1,0 \leqslant {c_{i + }} \leqslant 1} } } \right\},\end{align} \tag{ 2 }$$

$$\begin{align}{\text{CH}}({S_ - }) = \left\{ {\sum\limits_{i - = 1}^{n - } {{c_{i - }}{s_{i - }}|\sum\limits_{i - = 1}^{n - } {{c_{i - }} = 1,0 \leqslant {c_{i - }} \leqslant 1} } } \right\}.\end{align} \tag{ 3 }$$

The objective function is:

$$\begin{align} &\mathop {\min }\limits_c \left\| {\sum\limits_{i = 1}^{n + } {{c_{i + }}{s_{i + }} - \sum\limits_{i = 1}^{n - } {{c_{i - }}{s_{i - }}} } } \right\| \nonumber \\ &s.t.{\text{ }}\sum\limits_{i = 1}^{n + } {{c_{i + }}} = 1,0 \leqslant {c_{i + }} \leqslant 1,i = 1, \ldots ,{n_ + } \nonumber\\ &\quad\; {\text{ }}\sum\limits_{i = 1}^{n - } {{c_{i - }}} = 1,0 \leqslant {c_{i - }} \leqslant 1,i = 1, \ldots ,{n_ - } . \end{align} \tag{ 4 }$$

It can be transformed into a quadratic programming (QP) problem, the optimal normal vector ${w^*}$ and bias constant ${b^*}$ can be expressed by the solution of QP problem $(c_{1 + }^*, \ldots ,c_{n + }^*, \ldots ,c_{1 - }^*, \ldots ,c_{n - }^*)$:

$$\begin{align}\left\{ {\begin{array}{l} {{w^*} = \sum\limits_{i = 1}^{n + } {c_{i + }^*{s_{i + }} - \sum\limits_{i = 1}^{n - } {c_{i - }^*{s_{i - }}} }} \\ {{b^*} = - \frac{1}{2}\left( {\sum\limits_{i = 1}^{n + } {\sum\limits_{j = 1}^{n + } {c_i^*c_j^*\left\langle {{s_{i + }},{s_{i + }}} \right\rangle - } } \sum\limits_{i = 1}^{n - } {\sum\limits_{j = 1}^{n - } {c_i^*c_j^*\left\langle {{s_{i - }},{s_{i - }}} \right\rangle } } } \right)} \end{array}} \right.,\end{align} \tag{ 5 }$$

where $\left\langle {{s_i},{s_i}} \right\rangle$ is the inner product of two eigenvectors, for a given test sample $s$, the prediction function of the CH model can be defined as:

$$\begin{align}f(s) = {\text{sign}}\left\{ {\langle {w^*},s\rangle + {b^*}} \right\}.\end{align} \tag{ 6 }$$

Combining the predictions from several models has proved to be an effective approach for increasing the performance of a single model, which is known as ensemble learning. Ensemble learning completes the classification task by building multiple sub-classifiers with differences and certain accuracy, and uses some strategies to integrate the sub-classifiers. Hansen and Salamon have proved that if the accuracy of each classifier is above 0.5 and the classifiers are independent, the accuracies by average voting will approach 1 [29]. The reasons for the success of ensemble learning can be concluded as statistical, computational and representation learning, bias-variance decomposition and strength-correlation [30–32].

Ensemble learning can be proved by the Hoeffding inequality. Take the binary classification as an example; $y \in \left\{ { - 1, + 1} \right\}$ denotes the label. $f$ stands for the actual function and ${h_i}$ means the sub-classifier. The error probability of the sub-classifier is $\varepsilon$, $P({h_i}(x) \ne f(x)) = \varepsilon$, and there are T sub-classifiers. Use a simple voting method to integrate sub-classifiers. The calculation formula is given by equation (7). Suppose that if more than half of the sub-classifiers classify correctly, the result of ensemble is correct.

$$\begin{align}H(x) = {\text{sign}}\left( {\mathop \sum \limits_{i = 1}^T {h_i}(x)} \right).\end{align} \tag{ 7 }$$

Assuming that each sub-classifier is independent, the ensemble error probability can be given as follows:

$$\begin{align}P(H(x) \ne f(x)) &= \sum\limits_{j = 0}^{\left\lfloor {T/2} \right\rfloor } {\left( {\begin{array}{*{20}{c}} T \\ j \end{array}} \right)} {(1 - \varepsilon )^j}{\varepsilon ^{T - j}}\nonumber\\& \leqslant \exp \left( { - \frac{1}{2}T{{(1 - 2\varepsilon )}^2}} \right).\end{align} \tag{ 8 }$$

According to equation (8), the ensemble error probability will decrease exponentially with the increase of the number of sub-classifiers.

Since the CH model has the advantages of clear geometric meaning and is easy to deform, a method is proposed to generate differential sub-classifiers, which is by deforming the original CH. Considering the requirements of differential deformation and the clustering characteristics of samples in the feature space, the distance-based contraction factor and k-nearest neighbor density-based contraction factor are introduced to deform the original CH model. Figure 1 depicts the process of deforming the original CH model, where $\lambda$ is the distance-density-balance constant. The detailed process is depicted as follows.

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Step 1: Shrink the sample point ${{\text{s}}_i}$ towards the barycenter of the CH ${s_{{\text{cent}}}}$. ${\hat s_i}$ denotes the sample after shrinking, which is defined by equation (9):

$$\begin{align}{\hat s_i} = {s_{{\text{cent}}}} + {\mu _i}({s_i} - {s_{{\text{cent}}}}),\end{align} \tag{ 9 }$$

where ${s_{{\text{cent}}}} = \frac{1}{n}\mathop \sum \limits_{i = 1}^n {s_i}$, and ${\mu _i}$ indicate the degree of shrinkage. It is determined by its distance from the barycenter and its local density. That is, the farther the distance is, and the lower the local density is, the greater the degree of shrinkage. The ${\mu _i}$ is defined by equation (10):

$$\begin{align}{\mu _i} = \lambda {\mu _{i\_{\text{dis}}}} + (1 - \lambda ){\mu _{i\_{\text{den}}}},\end{align} \tag{ 10 }$$

where $\lambda$ represents the distance-density-balance constant. The constant gives the tradeoff between distance and density. The distance-based contraction factor ${\mu _{i\_{\text{dis}}}}$ and k-nearest neighbor density-based contraction factor ${\mu _{i\_{\text{den}}}}$ are defined as follows:

$$\begin{align}\left\{ \begin{gathered} {\mu _{i\_{\text{dis}}}} = \frac{2}{{1 + \exp (0.5.\left\| {{s_i} - {s_{{\text{cent}}}}} \right\|)}} \\ {\mu _{i\_{\text{den}}}} = \frac{2}{{1 + \exp \left(0.5.\left\| {\frac{{\varphi ({s_i},{s_{i\_k}})}}{{\max (\varphi (s,{s_k}))}}} \right\|\right)}} \\ \end{gathered} \right..\end{align} \tag{ 11 }$$

The function $\varphi ({s_i},{s_{i\_k}})$ represents the average distance from the sample point ${s_i}$ to the $k$ nearest neighbor points of ${s_i}$, and the function $\max (\varphi (s,{s_k}))$ represents the maximum of the average distances.

So far, a series of different deformed CHs can be produced according to different $\lambda$, resulting in a series of sub-classifiers. In fact, we adopt a different way to build the subspaces of the original dataset from the traditional down-sampling method. We leverage the balanced-distance-density constant $\lambda$ to obtain different subspaces (deformed CHs). Considering the sparsity of samples in the feature space, the subspaces will change a lot with the change of $\lambda$. We can ensure that we will obtain differential sub-classifiers by setting different $\lambda$ especially in case of outliers and noise, so we do not need too many sub-classifiers. In this paper, considering the trade-off between cost and performance, we set $\lambda = \{ 0,0.5,1\}$, $\lambda = 0$ means the shrinkage only according to the density, $\lambda = 0.5$ means the shrinkage comprehensively considering the distance and density, and $\lambda = 1$ means the shrinkage only according to the distance. In this way, it can ensure that there are great differences among the deformed CHs. Besides, it can suppress the interference of noise and outliers to obtain a more reliable data boundary description. In summary, this strategy can guarantee that the sub-classifiers are different and with strong robustness. Therefore, there are finally four sub-classifiers to ensemble plus the original CH.

Step 2: Swell the deformed CHs to eliminate the possible influence of unbalanced samples. Expansion factor ${\phi _s}$ is given by equation (12):

$$\begin{align}{\phi _s} = \frac{1}{{\frac{1}{n}\mathop \sum \limits_{i = 1}^n {\mu _i}}}.\end{align} \tag{ 12 }$$

Step 3: Construct the sub-classifiers. Taking the binary classification as an example, the deformed CH models can be expressed by equations (13) and (14):

$$\begin{align}{\text{CH}}({\hat S_ + }) &= \left\{ \sum\limits_{i + = 1}^{n + } {{\hat c}_{i + }}{{\hat s}_{i + }}|\sum\limits_{i + = 1}^{n + } {{\hat c}_{i + }} = 1,\frac{{1 - {\phi _{\hat s + }}}}{{{n_ + }}} \leqslant {{\hat c}_{i + }}\right.\nonumber\\&\qquad \left. \leqslant \frac{{1 - {\phi _{\hat s + }}}}{{{n_ + }}} + {\phi _{\hat s + }} \right\},\end{align} \tag{ 13 }$$

$$\begin{align}{\text{CH}}({\hat S_ - }) &= \left\{ \sum\limits_{i - = 1}^{n - } {{\hat c}_{i - }}{{\hat s}_{i - }}|\sum\limits_{i - = 1}^{n - } {{\hat c}_{i - }} = 1,\frac{{1 - {\phi _{\hat s - }}}}{{{n_ - }}} \leqslant {{\hat c}_{i - }}\right.\nonumber\\&\qquad \left.\leqslant \frac{{1 - {\phi _{\hat s - }}}}{{{n_ - }}} + {\phi _{\hat s - }} \right\}.\end{align} \tag{ 14 }$$

The objective function of the corresponding sub-classifier is given in equation (15):

$$\begin{align} &\mathop {\min }\limits_{\hat c} \left\| {\sum\limits_{i = 1}^{n + } {{{\hat c}_{i + }}{{\hat s}_{i + }} - \sum\limits_{i = 1}^{n - } {{{\hat c}_{i - }}{{\hat s}_{i - }}} } } \right\| \nonumber \\ &s.t.{\text{ }}\sum\limits_{i + = 1}^{n + } {{{\hat c}_{i + }} = 1,\frac{{1 - {\phi _{\hat s + }}}}{{{n_ + }}} \leqslant {{\hat c}_{i + }} \leqslant \frac{{1 - {\phi _{\hat s + }}}}{{{n_ + }}} + {\phi _{\hat s + }}} ,\nonumber \\ &~~\quad \quad\quad\quad i = 1, \ldots ,{n_ + }\nonumber \\ &~\quad {\text{ }}\sum\limits_{i - = 1}^{n - } {{{\hat c}_{i - }} = 1,\frac{{1 - {\phi _{\hat s - }}}}{{{n_ - }}} \leqslant {{\hat c}_{i - }} \leqslant \frac{{1 - {\phi _{\hat s - }}}}{{{n_ - }}} + {\phi _{\hat s - }}} ,\nonumber \\ &~~\quad\quad\quad\quad i = 1, \ldots ,{n_ - } .\end{align} \tag{ 15 }$$

Formula (15) can be expanded into formula (16), and the problem can be converted into a QP problem:

$$\begin{align} &\mathop {\min }\limits_{\hat c} \sum\limits_{j = 1}^{n + } {\sum\limits_{i = 1}^{n + } {{{\hat c}_{i + }}{{\hat c}_{j + }}{{\hat s}_{i + }}{{\hat s}_{j + }}} } - 2\sum\limits_{i = 1}^{n + } \sum\limits_{j = 1}^{n - } {{\hat c}_{i + }}{{\hat c}_{j - }}{{\hat s}_{i + }}{{\hat s}_{j - }} \nonumber \\ &\quad + \sum\limits_{j = 1}^{n - } {\sum\limits_{i = 1}^{n - } {{{\hat c}_{i - }}{{\hat c}_{j - }}{{\hat s}_{i - }}{{\hat s}_{j - }}} } \nonumber\\& s.t.{\text{ }}\sum\limits_{i + = 1}^{n + } {{\hat c}_{i + }} = 1,\frac{{1 - {\phi _{\hat s + }}}}{{{n_ + }}} \leqslant {{\hat c}_{i + }} \leqslant \frac{{1 - {\phi _{\hat s + }}}}{{{n_ + }}} + {\phi _{\hat s + }} ,\nonumber \\ &~~\quad \quad\quad\quad i = 1, \ldots ,{n_ + } \nonumber \\ & ~\quad{\text{ }}\sum\limits_{i - = 1}^{n - } {{\hat c}_{i - }} = 1,\frac{{1 - {\phi _{\hat s - }}}}{{{n_ - }}} \leqslant {{\hat c}_{i - }} \leqslant \frac{{1 - {\phi _{\hat s - }}}}{{{n_ - }}} + {\phi _{\hat s - }} ,\nonumber \\ &~~\quad \quad\quad\quad i = 1, \ldots ,{n_ - } . \end{align} \tag{ 16 }$$

To solve equation (16), we use the Lagrange multiplier approach to solve the standard convex QP (quadratic programming) problem. Equation (16) can be transformed to a standard QP problem as follows:

$$\begin{align} & \mathop {\min }\limits_x \frac{1}{2}{x^T}Hx \nonumber \\ &s.t.{\text{ }}Aeq * x = beq \nonumber\\ &\quad~~ lb \leqslant x \leqslant ub{\text{ }} \end{align} \tag{ 17 }$$

where $x = {[{\hat c_{1 + }},{\hat c_{2 + }}\ldots {\hat c_{n + }}\ldots {\hat c_{1 - }},{\hat c_{2 - }}\ldots {\hat c_{n - }}]^T}$, $H = \left[ {\begin{array}{*{20}{l}} {{K_{11}}}\;\;\;{ - {K_{12}}} \\ { - {K_{12}}}\;\;\;{{K_{22}}} \end{array}} \right]$, which represents the hessian matrix, ${K_{11}} = {\left[ {{{\hat s}_{1 + }},{{\hat s}_{2 + }}\ldots {{\hat s}_{n + }}} \right]^T} * \left[ {{{\hat s}_{1 + }},{{\hat s}_{2 + }}\ldots {{\hat s}_{n + }}} \right]{\text{, }}{K_{12}} = {\left[ {{{\hat s}_{1 + }},{{\hat s}_{2 + }}\ldots {{\hat s}_{n + }}} \right]^T}$ $* \left[ {{{\hat s}_{1 - }},{{\hat s}_{2 - }}\ldots {{\hat s}_{n - }}} \right]{\text{, }}{K_{22}} = {\left[ {{{\hat s}_{1 - }},{{\hat s}_{2 - }}\ldots {{\hat s}_{n - }}} \right]^T} * \left[ {{{\hat s}_{1 - }},{{\hat s}_{2 - }}\ldots {{\hat s}_{n - }}} \right]$ $Aeq = {\left[ {\begin{array}{*{20}{l}} {1,1\ldots 1}\;\;\;{0,0\ldots 0} \\ {0,0\ldots 0}\;\;\;{1,1\ldots 1} \end{array}} \right]_{2 \times ((n + ) + (n - ))}}$, $Beq = {\left[ {1,1} \right]^T}$. The hessian matrix $H$ is a symmetric matrix, which means that $H$ is a positive semidefinite matrix and the QP problem has the global optimal solution. So we can leverage the Karush–Kuhn–Tucker conditions of the Lagrange multiplier approach to obtain the solution. Then, we can obtain the optimal solution $(\hat c_{1 + }^*, \ldots ,\hat c_{n + }^*, \ldots ,\hat c_{1 - }^*, \ldots ,\hat c_{n - }^*)$ by solving formula (17), and the optimal hyperplane of the sub-classifier is expressed as $f(s) = {\text{sign}}\left\{ {\langle {{\hat w}^*},s\rangle + {{\hat b}^*}} \right\}$. The normal vector ${\hat w^*}$ and bias constant ${\hat b^*}$ are given by equation (18):

$$\begin{align}\left\{ {\begin{array}{l} {{{\hat w}^*} = \sum\limits_{i = 1}^{n + } {\hat c_{i + }^*{{\hat s}_{i + }} - \sum\limits_{i = 1}^{n - } {c_{i - }^*{{\hat s}_{i - }}} }} \\ {{{\hat b}^*} = - \frac{1}{2}\left( {\sum\limits_{i = 1}^{n + } {\sum\limits_{j = 1}^{n + } {\hat c_i^*\hat c_j^*\left\langle {{{\hat s}_{i + }},{{\hat s}_{i + }}} \right\rangle - } } \sum\limits_{i = 1}^{n - } {\sum\limits_{j = 1}^{n - } {\hat c_i^*\hat c_j^*\left\langle {{{\hat s}_{i - }},{{\hat s}_{i - }}} \right\rangle } } } \right)} \end{array}} \right. .\end{align} \tag{ 18 }$$

So far, the work of constructing the sub-classifier is completed. Because equation (18) only involves the inner product operation between the feature vectors, the original feature space can be mapped to the high-dimensional Hilbert space by introducing a kernel trick. The maximum margin hyperplane can be divided in the high-dimensional space to solve the linearly inseparable problem. In this paper, the Gaussian kernel function is used:

$$\begin{align}\kappa ({s_i},{s_j}) = \exp \left( { - \frac{{{{\left\| {{s_i} - {s_j}} \right\|}^2}}}{{2{\sigma ^2}}}} \right),\end{align} \tag{ 19 }$$

where $\sigma > 0$, represents the bandwidth of the Gaussian kernel. For multi-classification problems, the 'one-against-one' strategy is adopted to extend to a multi-classifier.

We have obtained the set of sub-classifiers with differences, $\left\{ {{f_m}} \right\},m = 1, \ldots ,M$. The output of EnCH will be obtained by utilizing a bagging algorithm, which integrates the sub-classifiers by weight. The final decision function of the EnCH is defined by equation (20):

$$\begin{align}F(s) = {\text{sign}}\left(\mathop \sum \limits_{m = 1}^M {\varepsilon _m} * {f_m}(s)\right),\end{align} \tag{ 20 }$$

where ${\varepsilon _m}$ is the weight of the $m{\text{th}}$ sub-classifier, and ${f_m}(s)$ represents the prediction results of the $m{\text{th}}$ sub-classifier. The basic structure of EnCH is presented in figure 2. The complete algorithm of EnCH is presented in algorithm 1.

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Algorithm 1. EnCH classification model.
Input: Training set $S:\{ {s_i},{y_i}\} _{i = 1}^n,{y_i} \in \{ - 1, + 1\}$, validation set $V:\{ {v_i},{y_i}\} _{i = 1}^n,{y_i} \in \{ - 1, + 1\}$ and parameter set $\left\{ {k = 3,\;\lambda = \left[ {0,0.5,1} \right]} \right\}$.
Output: Decision functions of the EnCH classification model.
Procedure:
1. Normalize the data set;
2. Get M groups of new samples with different degrees of shrinkage, by equations (8), (9);
3. Get M groups of deformed CHs, by equations (7), (10) – (12);
4. Construct the classification hyperplane between the deformed CHs of different classes in each group, and get M sub-classifiers ${f_m},m = 1,2, \ldots ,M$, according to equations (13) – (15);
5. Get the weight ${\varepsilon _m}$ on the validation set, integrate all sub-classifiers, and the final decision function is $F(s) = {\text{sign}}\left(\mathop \sum \nolimits_{m = 1}^M {\varepsilon _m} * {f_m}(s)\right)$.

When the fault modes of a bevel gear box are different, the time-domain waveform and frequency-domain waveform of the vibration signal will change, and the corresponding time-domain statistical parameters and frequency-domain statistical parameters will also change. Therefore, taking time-domain statistical parameters and frequency-domain statistical parameters as the feature vectors has been widely used in the field of fault diagnosis. In this paper, 12 time-domain statistical parameters (mean, root mean square, square root amplitude, mean amplitude, maximum peak, standard deviation, skewness, kurtosis, crest factor, clearance factor, shape factor, and impulse factor) and eight frequency-domain statistical parameters (spectral amplitude mean, spectral amplitude standard deviation, spectral standard deviation frequency, spectral amplitude skewness, spectral frequency skewness, spectral amplitude kurtosis, spectral frequency kurtosis, and spectral gravity frequency) are selected to form the feature vectors of samples. Specific calculation formulas of these parameters refer to literature [16].

The reason to select these features is because they can reflect the characteristics of time domain waveforms and frequency domain waveforms more comprehensively. Failure modes of the machines are closely related to these features. For instance, skewness and kurtosis can reflect the degree of signal deviation from the normal distribution, which are sensitive to cracks and spalling failures. The crest factor, clearance factor, shape factor and impulse factor are non-dimensional parameters that keep stable under different working conditions. The case in the frequency domain is similar to the case of the time domain.

The fault diagnosis procedure based on EnCH is graphically presented in figure 3, and the main steps are as follows:

• (a)  
  Signal acquisition and processing: use the sensor to obtain vibration signals of the bevel gear box under different fault modes.
• (b)  
  Feature extraction in the time domain and frequency domain: analyze the vibration signal in the time domain and frequency-domain to extract the corresponding time-domain statistical parameters and the corresponding frequency-domain statistical parameters to form the feature vectors.
• (c)  
  Division of the dataset: divide the data set into a training set and a test set with a certain proportion, and form the original CH according to the training set.
• (d)  
  Training of the EnCH: according to the original CH, the distance-based contraction factor and k-nearest neighbor density-based contraction factor of each sample are calculated to form a series of deformed CHs. The deformed CHs are then trained to obtain a series of sub-classifiers. Finally, the sub-classifiers are integrated.
• (e)  
  Validation of the effect: input the test samples into the trained EnCH model to validate the effect.

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The vibration signal dataset used in this paper is collected from a drivetrain dynamic simulator test bench in our laboratory. The test bench is shown in figure 4. The bevel gear pair uses a gear with 18 teeth as the input end and a gear with 36 teeth as the output end. Before the experiment, the electric discharge wire-cutting (EDM) technology was used to cut the driving bevel gear at different depths to simulate five different gear crack degrees. During the experiment, the input-axis speed is set to 1500 r min⁻¹ and the brake load is set to 4 N m⁻¹ under each fault condition. The sampling frequency is set to 4096 Hz.

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The vibration signal dataset obtained contains six different modes, including normal mode and five fault modes. Each mode contains 60 samples, and each sample contains 4096 sampling points. We then extract the features of each sample from the time domain and frequency domain respectively, to constitute the feature vector containing 20 dimensions, and get the dataset for experiment. Specific information about the experimental dataset is summarized in table 1. Figure 5 displays the time-domain waveforms of the vibration signals under six different fault modes.

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Experiment 1 is used to verify the following three points: (a) the proposed model can improve the generalization of the sub-CH classification model; (b) the proposed model is superior to other geometric learning models; and (c) the deep learning methods will suffer performance degradation when they are facing small sample tasks.

Four typical geometric learning models, SVM, CH, AH, and HD, are selected for comparative experiments. In the experiment, 18 samples are randomly selected from the 60 samples contained in each fault mode as the training dataset, and the rest as the test dataset. All of the models used for comparison adopt five-fold cross-validation to select their optimal parameters, and the candidate range of the gaussian kernel width $\sigma$ and the penalty parameters $\xi$ of SVM are set as $\sigma \in \left\{ {{2^{ - 3}},{2^{ - 2.5}},{2^{ - 2}}, \ldots ,{2^{10}}} \right\}$and $\xi \in \left\{ {0,5,10, \ldots ,50} \right\}$ respectively. The optimal parameters of all models are displayed in table 2, and all models involved in the experiment are run ten times using random selected training data with the optimal parameters to avoid contingency in the experiment.

Table 2. The optimal hyperparameters of all models.

	HD	AH	SVM	CH	EnCH
Gaussian kernel width ($\sigma$)	21.5	25	22.5	2	${\sigma _1}$ = 24, ${\sigma _2}$ = 24.5, ${\sigma _3}$ = 24, ${\sigma _4}$ = 2
Penalty parameters ($\xi$)	—	—	20	—	—

First, according to the selection strategy for the distance-density-balance constant $\lambda$ in section 2.1, we get the EnCH classification model with four sub-classifiers. Table 3 shows the average classification accuracy for the ten experiments for each sub-classifier, as well as the average classification accuracy and ensemble classification accuracy of four sub-classifiers under the optimal parameters. We find that the ensemble result is better than the accuracy of each sub-classifier and their average accuracy, indicating that the proposed model has the ability to improve the generalization of the original CH model. To further illustrate the difference among the sub-classifiers, taking the first trial as an example, the prediction labels of the test dataset under each sub-classifier are shown in figure 6, where the accuracies of sub-classifier 1, sub-classifier 2, sub-classifier 3 and sub-classifier 4 are 98.01%, 98.41%, 97.62% and 97.22% respectively, while the final ensemble accuracy is 98.81%. It is obvious that the accuracy of the sub-classifier 3 is close to the ensemble classification accuracy. But it is worth nothing that the sub-classifier 3 does not correspond to the original CH classifier (from table 4, we can see the accuracy of the original CH is 96.15%), it corresponds to the deformed CH classifier. So it still reflects that the proposed method can improve the performance of the base model. That is to say, although the accuracies of sub-classifiers may be close to the ensemble accuracy sometimes, it is unpredictable which sub-classifier will obtain excellent results, so the use of ensemble learning can improve the stability of the model and always obtains a better result.

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However, we can also see that when the dataset is pure, because each sub-classifier has achieved high accuracy, the difference among sub-classifiers is not obvious. Subsequent experiments (sections 4.3 and 4.4) find that the worse the dataset conditions are, the better the performance of the proposed model will be.

The classification accuracies of all learning models in every trial are shown in figure 7, and table 4 shows the average accuracy of the experiments repeated ten times for each model. The experimental results prove that each geometric learning model achieves relatively high classification accuracy under the condition of few samples, but the performance of the proposed classification model is more stable. And the proposed model obtains the highest classification accuracy except for the fourth trial, which can verify the superiority of the proposed model.

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In addition, in order to test the stability of the performance of the proposed model, different proportions (0.1, 0.2, 0.3, 0.4, 0.5) of training samples are set for training. The average accuracies of the ten repeated experiments with randomly selected training data in every trial are shown in figure 8. The results show that the classification accuracy of different models is improved with the increase of the number of training samples. The performance of SVM with few samples is the worst, but it is improved significantly with the increase of the proportion of training samples. The stability of the proposed model is the best, and the highest accuracies are obtained under different training sample proportions. It can be predicted that the deformation operation to the original CH makes the boundary estimation of the dataset in the feature space more reliable, so the performance is more stable.

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To further prove the necessity and superiority of the proposed method compared with deep learning methods under few samples conditions, we use the convolutional neural network (CNN) for comparison experiments. The structure and parameters of the CNN are presented in table 5. Note that the Batch Normalization is selected for the necessary modules to avoid the overfitting. We choose 16 as the training batch size. The SGD optimizer and CrossEntropyLoss are selected. The learning rate is set as 0.001.

First, we set the proportion of the training set as 0.3. The change of error and accuracy of the training and test during the process of training are shown in figure 9. We can see that the phenomenon of over fitting is very serious. When the training loss is close to 0 and the training accuracy is close to 1, the test loss and accuracy have poor performance. The best diagnostic accuracy is only 76.85%, which is far lower than the proposed method and it takes longer to train. Therefore, this can illustrate that deep learning models may create a performance degradation problem when facing few samples tasks.

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To observe the change of performance with the change of proportion of the training set, we improve the training set proportion to 0.5. Figure 10 shows the training result. We can see the performance of the CNN model has not improved significantly. Overfitting still exists, and the best accuracy is only improved to 81.11%. That is because the 0.5 proportion (30 training samples) is still not enough for a deep learning model.

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Because it is inevitable that the vibration signal is mixed with noise during the process of collection, which will deform the boundary of the dataset in the feature space, the decision of the maximal margin hyperplane of the geometric learning model will be interfered with, and the diagnosis accuracy will decrease. In the proposed EnCH classification model, each sub-classifier benefits from the shrinking to the original CH model based on two aspects of distance and density. Theoretically, the deformed sub-CH model can better represent the essential distribution of the dataset, so it will have a good suppression effect on the interference of noise. Therefore, experiment 2 is carried out to prove the following conclusions: (a) the EnCH classification model is robust to noise and has better performance than other geometric learning models; and (b) the difference among sub-classifiers is more obvious and the corresponding ensemble performance is better than experiment 1 under the condition with noise.

In the experiment, Gaussian white noise with different signal-to-noise ratio (SNR) is used to simulate the noise mixed in an actual collected signal. First, we add white noise with SNR = −6 ∼ 2 dB to the original vibration signal, and then calculate the features in the time domain and frequency domain. Finally, we get the dataset with different degrees of noise. The following experiments are carried out with different degrees of noisy dataset, and the proportion of the training dataset and the selection of parameters are the same as those in experiment 1. Figure 11 shows the time-domain waveform of the vibration signal (1.0 mm crack in the drive gear) when the noise intensity is −6 dB, −4 dB, −2 dB, 0 dB and 2 dB respectively and the original signal.

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The test results are presented in figure 12. We can see the performances of SVM, CH, AH and HD models are poor under strong noise interference, illustrating that, for the geometric classification model, noise does affect the decision of the hyperplane. Due to AH's loose estimation of the sample distribution boundary, the influence of noise on AH decreases gradually with the reduction of noise level, and its classification accuracy improves rapidly. Although the Gaussian white noise produces a great change on the CH's boundary, the shape of the CH, which is the key in the decision of the hyperplane, changes little. Therefore, the CH model is more resistant to noise interference than other geometric learning models, and its performance is also better. As an improvement of the original CH model, the EnCH classification model not only retains the advantages of the original CH model, but also significantly improves the generalization ability, at the same time, it has stronger noise-robustness and higher classification accuracy compared with other geometric learning models.

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When the noise intensity is −6 dB, taking the first trial as an example, the confusion matrices of the classification results of each sub-classifier and the confusion matrix of the EnCH classifier are shown in figure 13. The classification accuracies of sub-classifier 1, sub classifier 2, sub classifier 3, sub classifier 4 and EnCH classifier are 88.49%, 87.69%, 87.69%, 88.09% and 91.27% respectively. We can see that the difference among sub-classifiers is more obvious and the generalization ability has more significant improvement than experiment 1.

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To further demonstrate that the proposed method has the ability to suppress the influence of noise, we give the distribution of the samples in the feature space before and after the deformation of CH. The noise of SNR = −6 db is added to the original data. The result is given in figure 14. As the result shows, the deformed CHs obtain different distributions of the samples, and CH1 and CH2 are obviously has better clustering result compared with original CH.

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The kernel-based geometric learning model relies on support vectors to determine the decision function. However, outliers are very likely to be boundary points (support vectors) of the geometric model, which will greatly interfere with the decision of the maximum margin hyperplane, and the corresponding classification accuracy will be greatly reduced. The EnCH classification model considers the clustering characteristics of sample points in the feature space, and introduces the local contraction factor for each sample point. Theoretically, because the outliers are far away from the barycenter of CH and the local density at outliers is low, the degree of shrinkage is greater. This will restrain the influence of outliers on the boundary of CH, and the obtained boundary will be more fitted with the actual distribution boundary, which will improve the classification accuracy. So, experiment 3 is used to prove these hypotheses: (a) the EnCH classification model can automatically identify outliers and give them a large degree of shrinkage; and (b) when the dataset contains outliers, the classification accuracy of the EnCH classification model is also significantly better than other geometric classification models.

For a specific fault mode, the samples in other fault modes are taken as the outliers of this fault mode, and these outliers are used to replace the normal samples to ensure that the number of samples in the training dataset and test dataset does not change. The experiment tests the classification accuracy of all models with the number of outliers being $l(l = 1,2,3,4,5)$. The training dataset proportion of each fault mode is 0.3, that is, 18 training samples, and the rest are set as test samples. The optimal parameters are the same as those in experiment 1. The experiment was repeated 10 times with a randomly selected training dataset.

In order to observe the contraction degree of each sample before training, we recorded the order number of the outliers in the training dataset in table 6 when five outliers are inserted to each fault mode. The contraction degree of all training samples in the first kind of deformed CH is shown in figure 15. It can be found that, except for few exceptions, the outliers all get the relative maximum degree of shrinkage in the 18 training samples. It proves that the proposed model has the ability to identify outliers and give them a large degree of shrinkage. It also verifies that the proposed model can suppress the influence of outliers.

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Table 7 presents the classification accuracy of each model with a different number of outliers. The results show first that the classification accuracy of all classification models in the experiment decreases significantly with the increase of the number of outliers, illustrating that the geometric learning models are sensitive to outliers. Second, that although SVM greatly suppresses the interference of outliers through the penalty parameters, and achieves similar classification accuracy as the proposed model, considering the complex parameter-optimized process, its time cost and calculation cost are huge. At the same time, it is impossible to predict whether the dataset obtained in the actual fault diagnosis process is polluted by outliers, so it is hard to guide the selection of the SVM parameters. The proposed model can adapt to different datasets by using the same parameters. Therefore, the performance of the EnCH classification model is better in general. Third, compared with the original CH model, the generalization performance of the proposed model has been significantly improved. At the same time, compared with other geometric learning models, the proposed model has also achieved higher classification accuracy.

To further demonstrate the effect of the shrinking operation on outliers, we give the distribution of the samples in the feature space before and after the deformation of CH. Before shrinking, 18 outliers are added to the original CH. The result is shown in figure 16, which shows that CH1 and CH3 can obviously confine the outliers to obtain a more realistic distribution.

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The three experiments above prove that the proposed method has better generalization ability and robustness compared with shallow models. However, the necessity and advantages of the method proposed in this paper, compared with other ensemble learning methods, still need to be verified. So, experiment 4 is used to prove these hypotheses: (a) the proposed method can use fewer sub-classifiers to achieve higher diagnostic accuracy compared with the ensemble methods based on the bagging strategy; (b) as strong classifiers, geometric learning models make it hard to obtain differential sub-classifiers, and ensemble learning does not work on geometric learning models if there is no special processing.

The typical ensemble learning algorithm random forest is chosen for comparison to see the relationship between the number of sub classifiers and classification accuracy. In this experiment, the proportion of the training dataset and the hyperparameters of EnCH are the same as those in experiment 1. For the random forest, the number of sub-classifiers (decision trees) is set equal to EnCH (four sub-classifiers) for the sake of fairness. The down-sampling rate is set to 0.8 which has been proved to be optimal by experiments. Different proportions (0.1, 0.2, 0.3, 0.4, 0.5) of training samples are set for training. The average accuracies of the ten repeated experiments are shown in figure 17. The result shows that EnCH obtains better results on different training sample proportions.

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In addition, we illuminate the relationship between the number of sub classifiers and classification accuracy in the general ensemble learning method based on the bagging strategy. Figure 18 shows the variation of classification accuracy of the random forest with increasing number of sub-classifiers in the case where the proportion of training samples is 0.3. We can see that the classification accuracy increases slightly with the increase of the number of sub-classifiers, but the best accuracy is still lower than EnCH.

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To verify the second hypothesis, SVM is selected as the base model of ensemble learning to see the result of ensemble learning based on strong classifiers. In this experiment, the hyperparameters of EnCH and the ensemble SVM are set to the same as in experiment 1. The down-sampling rate is set to 0.8 and the number of sub-classifiers (sub-SVM) is also set equal to EnCH for the sake of fairness. From the result in figure 19, we can see that the classification accuracy is even worse than the single SVM in experiment 1, so there is reason to believe that ensemble learning does not work on geometric learning models if there is no special processing.

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