**2.1. Methodology**

Fractal dimension is considered right from its invention [21] to be a good parameter to characterize time sequences of values of natural variables. And a simple, fast and accurate method for calculating the fractal dimension of data's time sequences was presented by Sy-Sang liaw and Feng-Yuan Chiu [20]. This method considers that a time sequence of 2*<sup>M</sup>* + 1 values is separated by a constant time interval which is well fitted by a fractal function *f*(*t*) in the period [0, *T*]. Then, calculating the fractal dimension *D* of *f*(*t*) by using the known values of *f*(*t*) at *tj* = *jT*/2*M*. To achieve this aim, Liaw and Chiu first defined *Lk*(*f*), the piecewise linear interpolation of level *k*(*k* = 0, 1, 2, ..., *M*), to *f*(*t*) as the union of the line segments connecting the points [*tj*, *f*(*tj*)] and [*tj*+1, *f*(*tj*<sup>+</sup>1)], where *tj* = *jT*/2*k*, *j* = 0, 1, 2, ..., 2*<sup>k</sup>* (see Figure 1). And then they checked out how poor the interpolation function *Lk*(*f*) is relative to the next level of interpolation *Lk*<sup>+</sup>1(*f*). The error of *Lk*(*f*) is defined as the sum of the absolute value of the differences of *Lk*(*f*) and *Lk*<sup>+</sup>1(*f*) at all *tj* <sup>=</sup> *jT*/2*k*+<sup>1</sup> <sup>≡</sup> *<sup>j</sup>εk*:

$$\Delta\_{k} \equiv \sum\_{j=0}^{2^{k+1}} |L\_{k+1}(f(t\_j)) - L\_k(f(t\_j))| = \sum\_{j=odd}^{2^{k+1}} |f(t\_j) - \frac{f(t\_j - \varepsilon\_k) + f(t\_j + \varepsilon\_k)}{2}| \quad \mathfrak{t}\_j = j\varepsilon\_k \tag{1}$$

Liaw and Chiu [20] found that the value Δ*<sup>k</sup>* is proportional to (*εk*)1−*<sup>D</sup>* when *k* is large enough.

**Figure 1.** Piecewise interpolation *Lk* (*f*) to a function *f*(*t*) (grey) at level 0 (dotted), 1 (dashed), and 2 (solid). Δ*<sup>k</sup>* (thick solid) denotes the error of the *k*th level interpolation with respect to the *k* + 1 level [20]

Thus, the fractal dimension *D* of *f*(*t*) can be obtained from the slope *s* of the log-plot of Δ*<sup>k</sup>* with respect to the level *k* by *D* = 1 + *s*/*log*2 for large enough *k* values.

In this bearing fault diagnosis method, raw vibration signal will be seen as a time sequences of data. Raw vibration signal is often heavily clouded by various noises due to the compounded effect of other machine elements' interferences and background noises presenting in the measuring device [2]. So, EMD is used to analysis raw vibration signal to filter noise before extracting its fractal feature. As discussed by Huang et al. [10], the EMD method is designated to deal with non-stationary and nonlinear signals. This method is based on the simple assumption that any data consists of different simple intrinsic modes of oscillations. Using the EMD method, complicated signals can be decomposed in a finite set of intrinsic mode functions (IMFs). Each IMF should meet the following two conditions: (1) in the whole data set of a signal, the number of extreme and the number of zero crossings must either equal or differ at most by one, and (2) at any time point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.

Assume *x*(*t*) is a vibration signal, and its empirical mode decomposition process can be described by following steps:

Step 1. Initialize: *r*0(*t*) = *x*(*t*), *i* = 1.

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Multi-sensor information fusion is an emerging interdisciplinary beginning in the military field, and it has already been successfully applied in many different areas. In the field of industrial equipment fault diagnosis, multi-source information fusion technology application is still in its early stage. Multi-sensor information fusion is divided into three levels: sensor level, feature level and decision level. And multiple classifier ensemble approach belongs to decision level information fusion. In the recent years, the use of multiple classifiers has gained a lot of attention and researches have continuously showed the benefits of using multiple classifiers to solve complex problems [4]. In contrast, the feature-level fusion has not probably

By using information fusion theory, this chapter will introduce some bearing fault diagnosis approaches. And these methods can divide into two categories: fault diagnosis based on feature-level fusion [11] and fault diagnosis based on decision-level fusion [14]. In the proposed fusion methods for bearing fault diagnosis, some intelligent algorithms are used for feature dimension reduction or pattern recognition. The feature-level fusion approach for bearing fault diagnosis is using gene expression programming (GEP), while the decision-level fusion approach using multiple classifier ensemble method. And the decision-level fusion approach is based on the new bearing fault diagnosis method [12] which uses empirical mode

**2. Bearing fault diagnosis using fractal feature parameter classification**

Faulty and normal machine conditions are always treated as classification problems based on learning pattern from empirical data modeling in complex mechanical processes and systems [31]. In this approach, a general framework for applying classification methods to fault diagnosis problems includes two steps: representative feature extraction and pattern classification. Feature extraction is a mapping process from the measured signal space to the feature space. Representative features which demonstrate the information of fault are extracted from the feature space. Pattern classification is the process of classifying the extracted features into different categories by geometric, statistic, neural or fuzzy classifiers. And recently, the development of artificial intelligence techniques has led to their application in fault diagnosis area. Meanwhile, artificial neural networks (ANNs) and support vector machines (SVMs) have been successfully applied to the intelligent fault diagnosis of

In practice, the classical approach is not always reliable when the extracted features are contaminated by noise. And most intelligent fault diagnosis approaches are complex, especially in solving multiple fault diagnosis problems. In this section, a novel, simple, fast and reliable intelligent method for solving multiple fault diagnosis problem will be proposed.

Fractal dimension is considered right from its invention [21] to be a good parameter to characterize time sequences of values of natural variables. And a simple, fast and accurate method for calculating the fractal dimension of data's time sequences was presented by Sy-Sang liaw and Feng-Yuan Chiu [20]. This method considers that a time sequence of 2*<sup>M</sup>* + 1 values is separated by a constant time interval which is well fitted by a fractal function *f*(*t*) in

And this approach is based on EMD and fractal feature parameter extraction.

received the amount of attention it deserves [32].

mechanical equipment [27].

**2.1. Methodology**

decomposition (EMD) and fractal feature parameter classification.

Step 2. Extract the *i*-th intrinsic mode function (IMF) *ci*(*t*): Step 2.1. Initialize: *<sup>h</sup>*0(*t*) = *ri*−1(*t*), *<sup>j</sup>* = 1.

Step 2.2. Determine all the maximal values, minimal value points of *hj*−1(*t*) and fit all extreme points into the upper and lower envelope of the original signal with the cubic spline line.

Step 2.3. Determine the mean value of the upper and lower envelope of *hj*−1(*t*), designated as *mj*−1(*t*).

Step 2.4. Calculate the difference between *hj*−1(*t*) and *mj*−1(*t*), *hj*(*t*): *hj*(*t*) = *hj*−1(*t*) − *mj*−1(*t*).

Step 2.5. If *hj*(*t*) satisfies the conditions of IMF, then it is designated as *ci*(*t*) = *hj*(*t*). Otherwise, update the value of *j*: *j* = *j* + 1, and return to Step 2.2.

Step 3. Get the remaining signal: *ri*(*t*) = *ri*−1(*t*) − *ci*(*t*), after decomposing the *<sup>i</sup>*-th IMF.

Step 4. When *ci*(*t*) or *ri*(*t*) satisfies the given termination condition, the cycle is ended. Designate the final remaining signal as *rn*(*t*) (*n* = *i*). Otherwise, update the value of *i*: *i* = *i* + 1, and return to Step 2.

Finally, raw vibration signal can be decomposed into *n* IMFs: *ci*(*t*), *i* = 1, ..., *n* and one residue function *rn*(*t*):

$$\mathbf{x}(t) = \sum\_{i=1}^{n} c\_i(t) + r\_n(t) \tag{2}$$

**Figure 2.** The resulting empirical mode decomposition components and fractal features from the inner

Operating Sample *p*(*c*1) *p*(*c*2) *p*(*c*3) *p*(*c*4) *p*(*c*5) *p*(*cr*)

Normal 1 62.5909 26.8249 8.9951 6.6883 1.7284 0.6491

Outer race 1 442.1013 45.3849 17.8250 5.9114 3.0012 0.6935

Inner race 1 487.9925 151.8692 59.1485 14.1692 4.6696 1.5808

**Table 1.** Fractal feature parameters of different operating condition samples (defect size: 0.007inches)

and a torque sensor. The bearings are installed in a motor driven mechanical system. The dynamometer is under control so that desired torque load levels can be achieved. Vibration data is collected using accelerometer, which is attached to the housing with magnetic bases. Accelerometer is placed at the 12 o'clock position at the driven end of the motor housing. In machine condition monitoring, an accelerometer can provide rich information about conditions of several machine components. For example, the measured data from the accelerometer in this experiment is a mixture of signals reflecting conditions of the bearing inner race, outer race and rolling elements. The vibration data are collected by a 16 channel

Ball 1 295.3898 32.5473 19.6932 4.1315 2.0519 0.3888

2 67.8337 25.7203 8.9875 5.9469 2.8435 0.4777

Bearing Fault Diagnosis Using Information Fusion and Intelligent Algorithms 119

2 479.2442 44.4325 19.1783 6.6492 3.4410 0.4973

2 511.5880 149.7561 68.9864 28.0217 5.8445 2.6485

2 270.6188 33.8913 20.9217 4.4377 2.3469 0.3063

race fault signal sample

DAT recorder with 12,000 Hz.

condition index

In this work, representative feature is fractal feature parameter extracting from each IMF. Because the method of fractal dimensions of time sequences needs *k* to be large enough, we use fractal feature parameter. And fractal feature parameter of each IMF will be calculated as Equation 3 shows. It is easy to know that the IMF's numbers of different raw vibration signal samples are different. And in the vibration signal examination, we find that the rich operating condition information is inside the front IMFs. So, we can integrate the residual IMFs into a component. In this new method, a parameter *L* is set to denote the number of IMF using to extract representative feature. And the *L*-th IMF will be re-denoted as *cr*(*t*) whose calculation form as Equation 4. Then, the feature set of each raw signal has *L* fractal feature parameters. For example, we set the value of parameter *L* as *L* = 6. Figure 2 summarizes all the IMFs and fractal features obtained from a bearing inner race fault signal sample. Table 1 presents the fractal feature parameters of IMFs of different operating condition vibration signal samples. And from Table 1, it is clear that fractal feature parameter sets of the same operating condition are similar, and it is easy to distinguish different operating conditions of fractal feature parameter.

$$p = \sum\_{k=0}^{M-1} \Delta\_k \tag{3}$$

$$c\_r(t) = \sum\_{i=L}^{n} c\_i(t) \tag{4}$$

#### **2.2. Results and discussion**

By using fractal feature parameter classification, bearing fault diagnosis method is applied to the bearing fault signal analysis from the Case Western Reserve University website [3]. The ball bearings are installed in a motor driven mechanical system, as shown in Figure 3. By a self-aligning coupling, a three-phase induction motor is connected to a dynamometer

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spline line.

*mj*−1(*t*).

function *rn*(*t*):

fractal feature parameter.

**2.2. Results and discussion**

designated as *mj*−1(*t*).

*i* = *i* + 1, and return to Step 2.

Step 2.2. Determine all the maximal values, minimal value points of *hj*−1(*t*) and fit all extreme points into the upper and lower envelope of the original signal with the cubic

Step 2.3. Determine the mean value of the upper and lower envelope of *hj*−1(*t*),

Step 2.4. Calculate the difference between *hj*−1(*t*) and *mj*−1(*t*), *hj*(*t*): *hj*(*t*) = *hj*−1(*t*) −

Step 2.5. If *hj*(*t*) satisfies the conditions of IMF, then it is designated as *ci*(*t*) = *hj*(*t*).

Step 3. Get the remaining signal: *ri*(*t*) = *ri*−1(*t*) − *ci*(*t*), after decomposing the *<sup>i</sup>*-th IMF. Step 4. When *ci*(*t*) or *ri*(*t*) satisfies the given termination condition, the cycle is ended. Designate the final remaining signal as *rn*(*t*) (*n* = *i*). Otherwise, update the value of *i*:

Finally, raw vibration signal can be decomposed into *n* IMFs: *ci*(*t*), *i* = 1, ..., *n* and one residue

In this work, representative feature is fractal feature parameter extracting from each IMF. Because the method of fractal dimensions of time sequences needs *k* to be large enough, we use fractal feature parameter. And fractal feature parameter of each IMF will be calculated as Equation 3 shows. It is easy to know that the IMF's numbers of different raw vibration signal samples are different. And in the vibration signal examination, we find that the rich operating condition information is inside the front IMFs. So, we can integrate the residual IMFs into a component. In this new method, a parameter *L* is set to denote the number of IMF using to extract representative feature. And the *L*-th IMF will be re-denoted as *cr*(*t*) whose calculation form as Equation 4. Then, the feature set of each raw signal has *L* fractal feature parameters. For example, we set the value of parameter *L* as *L* = 6. Figure 2 summarizes all the IMFs and fractal features obtained from a bearing inner race fault signal sample. Table 1 presents the fractal feature parameters of IMFs of different operating condition vibration signal samples. And from Table 1, it is clear that fractal feature parameter sets of the same operating condition are similar, and it is easy to distinguish different operating conditions of

*ci*(*t*) + *rn*(*t*) (2)

Δ*<sup>k</sup>* (3)

*ci*(*t*) (4)

*n* ∑ *i*=1

*p* =

*cr*(*t*) =

*M*−1 ∑ *k*=0

> *n* ∑ *i*=*L*

By using fractal feature parameter classification, bearing fault diagnosis method is applied to the bearing fault signal analysis from the Case Western Reserve University website [3]. The ball bearings are installed in a motor driven mechanical system, as shown in Figure 3. By a self-aligning coupling, a three-phase induction motor is connected to a dynamometer

Otherwise, update the value of *j*: *j* = *j* + 1, and return to Step 2.2.

*x*(*t*) =

**Figure 2.** The resulting empirical mode decomposition components and fractal features from the inner race fault signal sample


**Table 1.** Fractal feature parameters of different operating condition samples (defect size: 0.007inches)

and a torque sensor. The bearings are installed in a motor driven mechanical system. The dynamometer is under control so that desired torque load levels can be achieved. Vibration data is collected using accelerometer, which is attached to the housing with magnetic bases. Accelerometer is placed at the 12 o'clock position at the driven end of the motor housing. In machine condition monitoring, an accelerometer can provide rich information about conditions of several machine components. For example, the measured data from the accelerometer in this experiment is a mixture of signals reflecting conditions of the bearing inner race, outer race and rolling elements. The vibration data are collected by a 16 channel DAT recorder with 12,000 Hz.

sizes of 0.007, 0.014 and 0.021 inches under four different loads. It covers four different operating conditions, too. Each class data subset has been partitioned into two equal halves, one partition is used for training, while the other for testing. The purpose of data set C is to

In order to evaluate the classification performance of the fractal feature parameter of IMF, orthogonal quadratic discriminant function (OQDF-E) [9] is used to train and test on three data sets showed in Table 2. Table 3 gives the classification performance on various data sets. The new bearing fault diagnosis method can get good decision accuracy as Table 3 shows. Table 4 extends the analysis of results and shows the classification performance between normal and fault operating condition. From Table 4, we can see that the new method using

> Data set A B C Train accuracy (%) 100 100 78.25 Test accuracy (%) 100 78.13 80

Operating condition Normal Fault Normal Fault Normal Fault Test accuracy (%) 100 100 100 100 100 100

In above section, we have proposed a simple, fast and good performance fault diagnosis approach. This approach is based on single sensor source and using individual classifier. It can obtain high accuracy on the multiple fault types recognition problems under the same fault degree. But when under multiple fault degrees, it declines in performance. To deal with this problem, this section will introduce a new method based on decision-level fusion for bearing fault diagnosis. The new fusion method includes four stages. These four stages are vibration signal acquisition and decomposition, fractal feature parameter extraction, single data source fault diagnosis and decision-level fusion for fault diagnosis. The first three stages are the same with the method described in the above section. So, we only state the last step in

Given a specific pattern recognition problem, different classifier has different classification performance. Very satisfactory results can not always be got if we simply conduct a study on a single classifier to improve its classification accuracy. Multiple classifier system (MCS) can overcome limitations of individual classifier and enhance classification accuracy. The techniques of combining the outputs of several classifiers have been applied to a wide range of real problems and it has been shown that MCSs outperform the traditional approach of

The most often used classifiers combination approaches in MCS include the majority voting [30], the weighted combination (weighted averaging) [18], the probabilistic schemes [16, 17],

Data set A Data set B Data set C

Bearing Fault Diagnosis Using Information Fusion and Intelligent Algorithms 121

test the reliability of the novel approach in identifying the various grades of fault.

fractal feature parameter can get perfect performance in fault detection.

**3. Decision-level fusion for bearing fault diagnosis**

**Table 3.** Classification performance

**Table 4.** Fault detection performance

this section.

**3.1. Methodology**

using a single high-performance classifier [26].

**Figure 3.** Schematic diagram of the experimental setup

As we referred in Figure 3, in the mechanical system, single point faults were introduced to the test bearings using electro-discharge machining with fault diameters of 7 mils, 14 mils and 21 mils. Each bearing was tested under four different loads (0, 1, 2 and 3 hp). Three bearing data sets (A-C) were obtained from the experimental system under the four different operating conditions: (1) under normal condition, (2) with outer race fault, (3) with inner race fault, (4) with ball fault. The detailed descriptions of the three data sets are shown in Table 2.


**Table 2.** Description of three data sets

Data set A is formed by 320 samples. These samples include 4 different operating information under 4 conditions (0, 1, 2 and 3 hp), and among which the fault defect size is 0.007 inches. Every operating condition has 80 data samples. The whole data set is divided into 2 parts: 160 samples for training and 160 for testing. So, the task can be viewed as a four-class classification aimed at 4 different operating conditions. Data set B also contains 320 data samples. The training samples are including samples with 0.007 inches fault defect, while testing samples 0.021 inches fault defect. Data set B is used to further investigate the performance of fault diagnosis scheme. Data set C comprises 800 data samples including three different defect sizes of 0.007, 0.014 and 0.021 inches under four different loads. It covers four different operating conditions, too. Each class data subset has been partitioned into two equal halves, one partition is used for training, while the other for testing. The purpose of data set C is to test the reliability of the novel approach in identifying the various grades of fault.

In order to evaluate the classification performance of the fractal feature parameter of IMF, orthogonal quadratic discriminant function (OQDF-E) [9] is used to train and test on three data sets showed in Table 2. Table 3 gives the classification performance on various data sets. The new bearing fault diagnosis method can get good decision accuracy as Table 3 shows. Table 4 extends the analysis of results and shows the classification performance between normal and fault operating condition. From Table 4, we can see that the new method using fractal feature parameter can get perfect performance in fault detection.


**Table 3.** Classification performance

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As we referred in Figure 3, in the mechanical system, single point faults were introduced to the test bearings using electro-discharge machining with fault diameters of 7 mils, 14 mils and 21 mils. Each bearing was tested under four different loads (0, 1, 2 and 3 hp). Three bearing data sets (A-C) were obtained from the experimental system under the four different operating conditions: (1) under normal condition, (2) with outer race fault, (3) with inner race fault, (4) with ball fault. The detailed descriptions of the three data sets are shown in Table 2. Data The number of The number of Defect size(inches) Operating Class set training sample testing sample (training/testing) condition label A 40 40 0/0 Normal 1

> 40 40 0.007/0.007 Outer race fault 2 40 40 0.007/0.007 Inner race fault 3 40 40 0.007/0.007 Ball fault 4

> 40 40 0.007/0.021 Outer race fault 2 40 40 0.007/0.021 Inner race fault 3 40 40 0.007/0.021 Ball fault 4

> 40 0.007/0.007 Outer race fault 2 40 0.007/0.007 Inner race fault 3 40 0.007/0.007 Ball fault 4 40 0.014/0.014 Outer race fault 5 40 0.014/0.014 Inner race fault 6 40 0.014/0.014 Ball fault 7 40 0.021/0.021 Outer race fault 8 40 0.021/0.021 Inner race fault 9 40 0.021/0.021 Ball fault 10

B 40 40 0/0 Normal 1

C 40 40 0/0 Normal 1

Data set A is formed by 320 samples. These samples include 4 different operating information under 4 conditions (0, 1, 2 and 3 hp), and among which the fault defect size is 0.007 inches. Every operating condition has 80 data samples. The whole data set is divided into 2 parts: 160 samples for training and 160 for testing. So, the task can be viewed as a four-class classification aimed at 4 different operating conditions. Data set B also contains 320 data samples. The training samples are including samples with 0.007 inches fault defect, while testing samples 0.021 inches fault defect. Data set B is used to further investigate the performance of fault diagnosis scheme. Data set C comprises 800 data samples including three different defect

**Figure 3.** Schematic diagram of the experimental setup

**Table 2.** Description of three data sets


**Table 4.** Fault detection performance
