# A Review on Data-Driven Condition Monitoring of Industrial Equipment
## 2 Data-Driven Condition Monitoring of Industrial Equipment
### Fault Diagnosis and Prediction
#### 2.2.1 Method Based on Shallow Machine Learning
From the above two theorems, if the number of hidden layer neurons \(\mathrm{l}\) and the number of training samples \(\mathrm{N}\) are equal, then the network can approximate the samples with zero error for any \(\mathrm{w}\), \(\mathrm{b}\). Thus, training for the SLFN model is also equivalent to solving the Equation (25) for the least-and-multiply solution \(\hat{\mathrm{v}}\). \[\left\|\mathrm{H}\Big{(}\hat{\mathrm{w}}_{1},\cdots,\hat{\mathrm{w}}_{\mathrm{ M}},\hat{\mathrm{b}}_{1},\cdots,\hat{\mathrm{b}}_{\mathrm{M}}\Big{)}\hat{ \mathrm{v}}-\mathrm{Y}\right\|=\min_{\mathrm{v}}\left\|\mathrm{H}(\mathrm{w}_{ 1},\cdots,\mathrm{w}_{\mathrm{M}},\mathrm{b}_{1},\cdots,\mathrm{b}_{\mathrm{M }})\mathrm{v}-\mathrm{Y}\right\| \tag{26}\]  
However, when \(\mathrm{N}\) is large, an excessive number of hidden layers implies a larger computational effort, so the size of the hidden layers can be reduced to \(\mathrm{M}\) so that the training error can be approximated by an arbitrarily small \(\varepsilon\). In this case, the Moore-Penrose generalized inverse matrix \(\mathrm{H}^{+}\) of \(\mathrm{H}\) can be used to solve for Equation (26). \[\hat{\mathrm{v}}=\mathrm{H}^{+}\mathrm{Y} \tag{27}\]  
The most common method currently used to solve the generalized inverse matrix \(\mathrm{H}^{+}\) is the singular value decomposition method, regardless of whether \(\mathrm{H}^{\mathrm{T}}\mathrm{H}\) is a singular or non-singular matrix, which is solved by the following equation. \[\hat{\mathrm{v}}=\Big{(}\mathrm{H}^{\mathrm{T}}\mathrm{H}\Big{)}^{-1}\mathrm{H }^{\mathrm{T}}\mathrm{Y} \tag{28}\]  
In contrast to traditional single hidden layer feedforward neural networks, ELM randomly selects weights and biases in the hidden layer and then calculates the weights of the output layer by a regularized linear regression method. Even though the weights of the hidden layer are randomly generated, ELM maintains the universal approximation capability of SLFN. Therefore, from the perspective of learning efficiency, ELM networks are not only simple to operate, but also learn faster and have better global search capability, which can overcome the problem of traditional neural networks falling into local optima, without generating overfitting or inappropriate learning rates, and have better generalization capability. Compared to standard ANN and SVM algorithms, the rapid training speed and excellent learning efficiency of ELM have brought it into focus for equipment condition monitoring [62; 91; 122; 123; 124; 125]. In [126], the RMS values of the voltage and current characteristics of the three-phase induction motor are used as ELM model inputs. Six external faults of the three-phase induction motor were discovered, with ELM providing faster results and reducing the computational load more effectively than SVM and MLP. In [42], the Fourier spectrum features of eight manually collected motor pump vibration signals are utilized as input to an ELM model to categorize motor pump faults such as misalignment, unbalance, and rubbing. When compared to other statistical classifiers (K-Nearest-Neighbor, SVM, etc. ), the generalization and classification accuracy of the ELM were shown to be better to those of the statistical classifiers. In [127], [PERSON] et al. extracted information from the vibration signal of a faulty hydraulic pump using three different signal processing techniques. The retrieved signal features that are most sensitive to the fault are utilized to train the classifier model. Comparing ELM, BP, and SVM classifiers revealed that ELM has benefits in terms of training speed and generalization and is easier in terms of model construction and parameter selection than the other two approaches. In the sphere of condition monitoring, the combination of ELM and signal pre-processing techniques has also garnered considerable interest. By integrating ELM with a depth wavelet autoencoder in [128], ELM was able to classify bearing faults with better accuracy.