Shock and Vibration

Volume 2016, Article ID 1212457, 14 pages

http://dx.doi.org/10.1155/2016/1212457

## Multisensor Fused Fault Diagnosis for Rotation Machinery Based on Supervised Second-Order Tensor Locality Preserving Projection and Weighted -Nearest Neighbor Classifier under Assembled Matrix Distance Metric

^{1}School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China^{2}Department of Mechanical and Dynamic Engineering, Harbin University of Science and Technology, Harbin 150080, China

Received 15 June 2016; Revised 21 October 2016; Accepted 24 October 2016

Academic Editor: Fiorenzo A. Fazzolari

Copyright © 2016 Fen Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to sufficiently capture the useful fault-related information available in the multiple vibration sensors used in rotation machinery, while concurrently avoiding the introduction of the limitation of dimensionality, a new fault diagnosis method for rotation machinery based on supervised second-order tensor locality preserving projection (SSTLPP) and weighted* k*-nearest neighbor classifier (WKNNC) with an assembled matrix distance metric (AMDM) is presented. Second-order tensor representation of multisensor fused conditional features is employed to replace the prevailing vector description of features from a single sensor. Then, an SSTLPP algorithm under AMDM (SSTLPP-AMDM) is presented to realize dimensional reduction of original high-dimensional feature tensor. Compared with classical second-order tensor locality preserving projection (STLPP), the SSTLPP-AMDM algorithm not only considers both local neighbor information and class label information but also replaces the existing Frobenius distance measure with AMDM for construction of the similarity weighting matrix. Finally, the obtained low-dimensional feature tensor is input into WKNNC with AMDM to implement the fault diagnosis of the rotation machinery. A fault diagnosis experiment is performed for a gearbox which demonstrates that the second-order tensor formed multisensor fused fault data has good results for multisensor fusion fault diagnosis and the formulated fault diagnosis method can effectively improve diagnostic accuracy.

#### 1. Introduction

As one of the most common mechanical equipment classes, rotation machinery occupies an important role in industrial applications such as manufacturing, metallurgy, energy, and transportation. Due to tough working environments, similar materials, and structural properties, rotation machinery can be subject to malfunctions or failures. This can significantly decrease machinery service performance including manufacturing quality and operation safety and cause machinery to break down, which may lead to serious catastrophes [1]. Accordingly, research into fault diagnosis of rotation machinery has attracted considerable attention by researchers in related domains in recent years. The vibration signals collected from velocity or accelerator sensors located in machinery housing are generally regarded as the foundation of fault diagnostic procedures. However, most existing studies on fault diagnosis of rotation machinery have empirically or experimentally focused on analyzing single sensor signals [2–4], and the remaining studies have performed multisensor fused fault diagnosis through complex fusion algorithms such as blind source separation (BSS) [5] and D-S evidence theory. The single-sensor-based fault diagnosis methods belonging to the former category of studies generally lead to loss of valuable information available from multiple sensors, and the multisensor fused diagnosis methods appearing in the latter category of studies are tend to cause a high computational load. To tackle these issues, this paper presents a second-order tensor representation of fault samples including fault feature dimensions and sensor locations dimensions, which is used in an efficient multisensor fused fault diagnosis framework.

Large volumes of feature parameters generated by time-domain, frequency-domain, and time-frequency-domain analysis of vibration signals are commonly integrated into a high- dimensional data set to obtain accurate fault diagnostic results [6]. This high-dimensional feature set can provide more valuable information, but it also increases the computational load and may even trigger dimensionality issues. One approach to address this problem is to apply dimension reduction technology. Compared with classical linear dimensionality reduction methods such as principle component analysis (PCA) [7], linear discriminate analysis (LDA) [8], and multidimensional scaling (MDS) [9], a new technology for discovering intrinsic low-dimensional structure of nonlinear distributed data hidden in high-dimensional space has emerged which is known as manifold learning and has become a current research focus. Representative manifold learning methods include isometric mapping (ISOMAP) [10], locality linear embedding (LLE) [11], Laplacian eigenmaps (LE) [12], and local tangent space alignment (LTSA) [13]. The effectiveness of these basic manifold learning algorithms and their variants for fault diagnosis of rotation machinery has been validated frequently by a large number of studies. For instance, Li et al. [14]. proposed a fault diagnosis method using dimension reduction with linear local tangent space alignment (LLTSA). Ding et al. [15] developed a fusion feature extraction method based on locality preserving projection (LPP) for rolling element bearing fault classification. Additionally, an envelope manifold demodulation method was investigated for planetary gear fault detection in [16]. It should be observed that the input sample for these methods is generally represented by a vector with a high-dimensional feature space. It is obvious that these manifold learning algorithms are not suitable when a multisensor fused faulty sample is represented as a second-order tensor, namely, a matrix. Furthermore, tensor representation based manifold learning methods have received little investigation for fault diagnosis. Fortunately, there are several second-order or higher-order tensor extended manifold algorithms, such as second-order tensor locality preserving projection (STLPP) [17], tensor neighborhood preserving embedding (TNPE) [18], a tensor version of discriminant locality linear embedding (DLLE/T) [19], and tensor PCA [20]. These algorithms have been progressively applied in the areas of two-dimensional or higher-dimensional image classification, computer vision, and pattern recognition and offer a feasible solution for tensor-represented fault diagnosis. Out of the methods mentioned above, STLPP possesses the ability to discover intrinsic local geometric and topological properties of a manifold embedded in a second-order tensor space, on the basis of inherited strengths of LPP. However, it has been found that there are several limitations of the STLPP algorithm. For instance, STLPP is an unsupervised method for dimension reduction and thus does not consider discriminant information which is useful for fault classification. Secondly, the similarity with second-order tensor formed samples in traditional STLPP has been computed using the Frobenius distance measure [17] which is the same as the Euclidean distance of the vectorized version of matrix formed samples, so it may still cause a loss of spatial locality information. To tackle these problems, this paper introduces the concept of supervision into the framework of a traditional STLPP and employs an assembled matrix distance metric (AMDM) which has been successfully utilized in 2DPCA [21] into the construction of a similarity weighting matrix to obtain better matching between two second-order tensor formed faulty samples.

To further improve the accuracy and the efficiency of fault diagnosis, intelligent classification methods are considered as an indispensable component in the diagnostic procedure. These methods include artificial neural networks (ANN) [22], support vector machines (SVM) [23], and fuzzy-based systems [24] as well as Bayesian based classifiers [25]. Compared with these methods, the* k*-nearest neighbor classifier (KNNC) ranks* k* neighbors of testing samples from training samples and uses the class labels of similarity neighbors to classify input test samples by evaluating the similarity between samples in the feature space [26, 27]. The KNNC method has many benefits, including a lower calculation requirement, quicker speed, and higher pattern recognition accuracy [28]. Therefore, it is considered to be the simplest tool for faulty pattern recognition. There are some existing shortcomings for traditional KNNC which classifies the sample labels using unified weights, and thus the weighted k-nearest neighbor classifier (WKNNC) was developed which assigns different weights to nearest neighbors to represent the impact of each neighbor on each unknown sample. Therefore, this paper uses the WKNNC to establish the relationships between features of samples and conditional classifications. Additionally, the AMDM mentioned above is also employed for the similarity evaluation of low-dimensional second-order formed samples after SSTLPP based dimension reduction in WKNNC.

The remainder of this paper is organized as follows. The proposed supervised second-order tensor locality preserving projection based on assembled matrix distance metric (SSTLPP-AMDM) algorithm is discussed in detail in Section 2. The weighted* k*-nearest neighbor classifier with an assembled matrix distance metric (WKNNC-AMDM) is described in Section 3. Section 4 provides the overall framework for the proposed multisensors fused fault diagnosis. In Section 5, a fault diagnosis experiment is performed for a gearbox to validate the proposed method. Finally, the conclusions are given in Section 6.

#### 2. Supervised Second-Order Tensor Locality Preserving Projection Based on Assembled Matrix Distance Metric (SSTLPP-AMDM)

##### 2.1. Introduction to Second-Order Tensor Locality Preserving Projection (STLPP)

As the tensor extension of LPP, TLPP is essentially equivalent to finding a linear approximation of the eigenfunctions of the Laplace Beltrami operator in a tensor space. The incipient TLPP which was initially presented by He et al. [17] in 2005 is a second-order case and was reviewed and then extended for a universal* n*-order version by Dai and Yeung [18] in 2008. Since the multisensor fused faulty sample studied in this paper is represented by a second-order tensor form, namely, in a matrix form, the second-order TLPP (STLPP) algorithm is the focus in the following discussion. Given matrix formed samples , the aim of STLPP is to find two transformation matrices and by optimizing the following formulation:where is the Frobenius norm of the matrix; that is, ; denotes the elements of the weight matrix of the nearest neighbor graph , which is equal to when is one of the nearest neighbors of or is one of the nearest neighbors of ; otherwise it is equal to zero. is a diagonal matrix; .

Using a series of mathematical derivations, the optimal values for and are obtained by iteratively computing the generalized eigenvectors of the following formulations:where , , , and .

Finally, the low-dimensional representations of the original data are obtained using .

##### 2.2. Computation of a Supervised Similarity Weighting Matrix Based on AMDM

As described in the previous section, there are a certain number of limitations when using the prevailing computation method for the similarity weighting matrix of the nearest neighbor graph . For instance, the Frobenius distance metric (FDM) used for the similarity evaluation between different second-order tensor formed samples is essentially the Euclidean distance of the vectorized version of the matrix formed samples, and thus it neglects spatial geometrical information of each element in matrix formed samples and thus has poor matching performance for different samples. Additionally, class label information of the training samples is not effectively used in the traditional STLPP, although this information can be helpful for subsequent accurate classification assignment. To address these issues, this paper formulates a novel supervised similarity matrix computation method that decides the similarity between matrix formed samples using an assembled matrix distance metric that takes the classification information into account.

Firstly, for any two arbitrary matrix formed samples and , the distance between the two samples can be measured using the following assembled matrix distance metric (AMDM) [21]:where denotes a variable parameter which strongly affects the representation ability of the defined distance function for subsequent classification assignment. It is obviously that the Frobenius distance metric is a special case of the AMDM with , and the Yang distance metric proposed by Yang et al. in [29] is another special case for . It has also been theoretically and experimentally verified that an assembled matrix distance metric with a lower value of , that is, , outperforms existing Frobenius distance and Yang distance measures in terms of the final classification accuracy. Accordingly, the value of for the employed AMDM is set between 0 and 1, , and its exact value is determined by repeated experiments.

Secondly, by understanding the class label information of the training samples and the AMDM based distances between samples, the proposed supervised similarity weighting matrix based on AMDM can be defined aswhere denotes the element at column and row in the new formulated supervised similarity matrix , which represents the similarity degree of the matrix formed samples and . and are the class labels of samples and , respectively. is the penalty coefficient which is used to characterize the reduction in the similarity degree. Since is one of the nearest neighbors of or is one of the nearest neighbors of , the corresponding class labels are inconsistent, and thus the value of should be set to .

The newly formulated similarity weighting matrix computation equation shown in (4) can be viewed as the combination and extension of the prevailing “0-heat kernel function” and the “0-1” binary mode, in which the former is intimately related to the manifold structure and the latter is regarded as the direct expression of the label information. The properties and corresponding advantages of the supervised similarity weighting matrix based on AMDM can be summarized as follows. (i) A more accurate representation of the matching relationship between matrix formed samples can be achieved using AMDM rather than traditional STLPP, which uses the Frobenius distance metric. (ii) The inclusion of the penalty parameter results in larger differences between 1 and as the assembled matrix distance increases, which allows the interclass and intraclass similarity to be easily distinguished.

##### 2.3. SSTLPP-AMDM Algorithm

This paper proposes a novel supervised second-order tensor locality preserving projection algorithm with the assembled matrix distance metric (SSTLPP-AMDM) that uses the improvements in both the matrix distances computation of samples in the projection space and the similarity weighting matrix computation expression. In contrast to traditional STLPP, the two transformation matrices and that represent both the neighborhood graph structure and the class label information are obtained by solving the following objective function:The distance between two mapped sample points and in the embedded tensor space is measured using the assembled matrix distance metric to achieve a better matching result. The element of the supervised similarity weighting matrix which is computed by (4) is employed to represent the neighboring degree of samples and and considers both the local structure and class information. The diagonal matrix has the ability to characterize the degree of importance of the mapped sample point in the embedded tensor space to represent the original sample point .

The optimal transformation matrices and are solved in a similar way to traditional STLPP by applying an iterative scheme. The specific implementation process can be described as follows. Firstly, an initial matrix is set as an identity matrix and the first iterative solution of is then obtained by solving the generalized eigenvector problem shown in (6). Secondly, is updated by solving the generalized eigenvector problem shown in (7). By iteratively computing the generalized eigenvectors of (6) and (7) for a predefined number of repetitions, the optimal transformation matrices and are obtained. Finally, the second-order low-dimensional projection of the original second-order high-dimensional sample is obtained.

In summary, there are two main advantages to the newly proposed SSTLPP-AMDM. () The local structure information and the class information act cooperatively in the computation of the similarity weighting matrix, and thus the supervised similarity weighting matrix proposed in this paper outperforms other prevailing similarity weighting matrix computation methods in terms of representation of the similarity degree between samples. () The application of AMDM to measure the distance between both the sample points in the original second-order tensor space and the mapped sample points in the embedded second-order tensor space ensures that the measured samples have a better matching performance than the existing Frobenius distance measure. Therefore, the SSTLPP-AMDM algorithm has superior classification and dimension reduction characteristics than traditional STLPP.

#### 3. Weighted -Nearest Neighbor Classifier with Assembled Matrix Distance Metric (WKNNC-AMDM)

As stated above, the KNNC method proposed by Cover and Hart in 1967 [28] is regarded by many as the simplest pattern classification algorithm. Due to its advantages of a lower calculation requirement, quicker speed, and higher identification accuracy, KNNC has been widely applied to various types of pattern recognition problems, especially fault diagnosis issues. The main KNNC concept is described in the following two steps.

*Step 1. *For a given unknown labeled sample ,* k* similar samples in the training sample set are searched to construct a neighbor set .

*Step 2. *A maximum voting rule is used on all samples in to obtain the class that belongs to.

The above description shows that there are two focus points to KNNC: a similarity measurement method between samples and the establishment of a decision rule. For the first focus point, there have been many similarity measurement methods suggested by previous publications, such as the Euclidean distance, the Manhattan distance, and the cosine angle. However, these vector representations of the data-based metric indexes described above are unsuitable for similarity measurement of the matrix formed data points appearing in this paper. Thus, the AMDM is introduced for the similarity computation of samples in KNNC. It is known that AMDM outperforms common FDM in terms of the similarity presentation between matrix formed samples for classification. Additionally, since selection of neighbors is greatly impacted by the sparsity of the sample distribution, this paper employs a novel assembled matrix distance based on density to efficiently measure the similarity between and its neighbor , using the following formula:

Unlike the classical KNNC voting strategy that uses unified weights for neighbors, in this paper, a weighted voting strategy is used to form the weighted* k*-nearest neighbor classifier (WKNNC), which assigns different weights to each sample in , reflecting the influence each neighbor has on an unknown sample . A new neighbor set is generally reconstructed in ascending order of distance; that is, , and thus the voting weight of sample is computed using the following equation:

Consequently, the class label of an unknown labeled sample can be determined as follows:where denotes the class label of in and is the Di carat function which has the functional value equal to 1 when , and otherwise it is equal to zero.

Additionally, the selection of is an issue that requires attention in the WKNNC algorithm. In this paper the value of is set to , since the classification precision is only just assured when the number of samples in equals , where the number of classes in training set is [30].

#### 4. Overall Framework of the Proposed Fault Diagnostic Method

Based on the preparations above, this paper proposes a novel multisensor fused fault diagnosis method based on SSTLPP-AMDM and WKNNC-AMDM for rotation machinery. The flow chart for the proposed method is shown in Figure 1. There are three main steps to the diagnostic procedure, which will be discussed in detail in this section.