# Modified Local Linear Embedding Algorithm for Rolling Element Bearing Fault Diagnosis
## 1 Introduction  
As the critical component of mechanical equipment, once rolling element bearing malfunctions in the process of industrial production, it will bring cascading failures and even cause the system to collapse. So, if the rolling element bearing is broken, it will result in huge economic losses or even a major security accident [1]. Therefore, accurate fault diagnosis of rolling element bearing is an important guarantee for the reliability and effectiveness of mechanical equipment. With the complexity of industrial machinery structure and the diversity of modules, fault signals of rolling element bearing often show non-stationary, non-linear and high-dimensional characteristics [2], and noise is often doped among the collected signals. In recent years, manifold learning has been applied to mechanical fault diagnosis field because of its good ability in nonlinear dimensionality reduction [3]. The representative methods of manifold learning are listed as following: the Local Linear Embedding (LLE) algorithm proposed by [PERSON] [4], Isometric Feature Mapping (ISOMAP) algorithm put forward by [PERSON] et al. [5], and Local Tangent Space Alignment (LTSA) algorithm advanced by [PERSON], [PERSON] et al. [6]. The LLEalgorithm is a classical and effective nonlinear dimensionality reduction method, which can learn a high dimensional manifold structure in arbitrary dimension. Furthermore, due to the small computational complexity of LLE algorithm, it is simple and easy to implement. LLE algorithm defines that each sample point and its neighborhood points are located in a local linear manifold surface, so each sample point can be represented by neighborhood points [7]. In fault diagnosis field, LLE algorithm can effectively extract the sensitive features of sample data and filter redundant information, so it can reduce the dimension of original feature space. [PERSON], [PERSON] et al. used LLE to reduce the dimension of observation, the results showed that the precision of fault diagnosis model of rotating machinery platform was effective [8]. Besides, [PERSON], [PERSON] et al. proposed a rolling bearing fault diagnosis method based on LLE and least squares support vector machine, and it can effectively identify the nonlinear fault features existing in high dimension space [9]. However, LLE algorithm constructs low dimensional manifold through the weighted method to keep the local geometric relationship of original manifold. Since the local linear definition of LLE algorithm, it causes that LLE algorithm is susceptible to the interference of noise. In the past decade, some algorithms have been proposed to improve the robustness of LLE algorithm. [PERSON] and [PERSON] proposed a robust LLE algorithm based on the robust PCA algorithm [10], this method reduced the interference of outliers effectively by adjusting the distribution deviation in the statistical process. [PERSON] et al. put forward a robust LLE algorithm based on penalty function, which combined L1 norm with L2 norm in weight function to improve the anti-noise ability [11]. [PERSON], [PERSON] and [PERSON], [PERSON] adopted L1 norm to construct the weight matrix to improve the ability to resist noise in original LLE algorithm [12]. In order to improve the stability of LLE algorithm under strong noise, [PERSON], [PERSON] et al. proposed an improved algorithm based on sparse constraint. In the calculation process of this method, the L1 norm is added to penalty constraint, which can make the optimal reconstruction weight matrix become sparser [13]. [PERSON] et al. constructed reasonable weight by using different neighborhood sizes, thus it can reduce the effect of interference points in dimensionality reduction [14]. The above methods are only adopted a separate improvement to enhance the robustness of LLE algorithm, and only use artificial data sets to simulate.