# Modified Local Linear Embedding Algorithm for Rolling Element Bearing Fault Diagnosis
## 4 Weight Construction Based on L1-Norm  
### The Effect of Noise on Weight Construction  
In the process of dimensionality reduction, except selecting reasonable and effective neighborhood, an accurate local linear weight construction is also the key factor affecting the dimensionality reduction effect [20]. The construction of weight matrix is the main basis for getting the mapping from high-dimensional space to low-dimensional space of the manifold structure. If the weight construction is not accurate, it will lead to the deviation of dimensionality reduction and even cause distortions.  
Due to complex coupling among the modules on industrial production process, the mechanical fault data can reflect the fault information, but at the same time it often doped with noise. When LLE algorithm is used to reduce the dimension of fault data, noise points can lead to an inaccuracy of weight construction, which will affect the dimensionality reduction and reduce the accuracy of fault diagnosis. There is a detailed analysis of noise effect [21].  
Let \(x_{0}\) represents the sample point \(x_{i}\), U(x\({}_{i}\)) represents the neighborhood of \(x_{0}\). Suppose \(x_{1}\), \(x_{2},\ldots,\ x_{k}\in U(\ x_{0})\), and \(x_{0}=\sum_{i=1}^{k}w_{i}x_{i}\), \(\sum_{i=1}^{k}w_{i}=1\). We define the data points which contaminated by noise as \(\widetilde{x}_{i}=x_{i}+\varepsilon_{i}\ (i=0,1,2,\ldots,k)\), where \(\varepsilon_{i}\ (i=0,1,2,\ldots,k)\) indicates the noise of each neighborhood point. According to LLE algorithm, after noise pollution, there are data points \(\widetilde{x}_{0}=\sum_{i=1}^{k}\widetilde{w}_{i}\widetilde{x}_{i}\), \(\sum_{i=1}^{k}\widetilde{w}_{i}=1\), and \(\widetilde{x}_{i}\in U(\widetilde{x}_{0})\). \(\widetilde{w}_{i}\) represents the elements in local reconstruction matrix of the contaminated data points.  
Then, when the neighborhood points are without noise pollution, its reconstructed weights can be expressed as following:  
\[\left\{\begin{array}{c}X_{0}=(\ x_{1},\ x_{2},\ldots,\ x_{k})\\ W_{0}=(\ w_{1},\ w_{2},\ldots,\ w_{k})^{T}\end{array}\right. \tag{7}\]  
When the polluted neighborhood points and its reconstructed weights can be expressed as following:  
\[\left\{\begin{array}{c}\widetilde{X_{0}}=(\widetilde{x_{1}},\ \widetilde{x_{2}},\ldots,\ \widetilde{x_{k}})\\ \widetilde{W_{0}}=(\widetilde{w_{1}},\ \widetilde{w_{2}},\ldots,\ \widetilde{w_{k}})^{T} \end{array}\right. \tag{8}\]  
Figure 1: The flow diagram of neighborhood selection algorithm in M-Local Linear Embedding (LLE) algorithm.  
Thus, there are  
\[\left\{\begin{array}{l}\mathrm{x}_{0}=X_{0}W_{0}\\ \widetilde{x_{0}}=\widetilde{X_{0}}\widetilde{W_{0}}\end{array}\right. \tag{9}\]  
There is a theorem: If it is independent of noise points, different dimensions, and each \(\widetilde{W_{0}}\). The mean value of each noise point is 0, and they have the same variance [21]. The error of reconstructed weight matrix for the data point is \(\delta W_{0}=\widetilde{W_{0}}-W_{0}\). The following estimate is made:  
\[E||\delta W_{0}||^{2}\leq\frac{k(k+1)\sigma^{2}}{\lambda_{min}I}E||\widetilde{ W_{0}}||^{2} \tag{10}\]  
Among them, \(||\cdot||\) is the Euclidean norm, \(\sigma^{2}=\sum_{i=1}^{\mathrm{D}}\sigma_{i}^{2}\), \(\sigma_{i}^{2}=Var(\varepsilon_{i})\), \((i=0,\,1,\,2,\ldots,D)\), and \(\varepsilon_{i}\) represents the \(k\)-th component of \(\varepsilon\), \(l=rank(X_{0})\), \(\lambda_{min}\) is the minimum nonzero eigenvalue of \(X_{0}{}^{T}X_{0}\). For detailed proof, please refer to document [20].  
It can be seen from the above theorem, the main factors affecting the robustness of LLE algorithm are: noise distribution \(\sigma^{2}\), neighborhood selection \(k\), and the energy of weight matrix \(||\widetilde{W_{0}}||\). Therefore, in order to enhance the ability of LLE algorithm to resist noise, besides the improvement of neighborhood construction, we should also reduce the weight matrix energy. Only by reducing the error of the local linear reconstruction weight matrix, can we overcome the sensitivity of LLE algorithm to noise.