Multi-Scale Principle Component Analysis and Mod-MSPCA

As described in the previous section, PCA is a great tool to reduce the dimensionality of the original data matrix by capturing the most important underlying variations and relationships between the variables. This characteristic makes it suitable for application involving multiple sensors. However, by itself, PCA is capable of feature extraction along only one localization, either in time or frequency; it lacks the capabilities to determine events with energy and power spectrums changing in both time and frequency. Another shortcoming to this technique is its limited capacity to reduce the error inherent in the original data matrix which deteriorates the reliability of the produced results. Unlike principal component analysis, wavelet transforms ensure the reduction of error by filtering the original data, and they have the ability to represent deterministic features with a small number of relatively large coefficients. Added to that, wavelets can easily isolate events in both time and frequency. However, they lack the capability to carry multivariable processing that is required by most industrial processes in order to distinguish sensor faults from process faults.

That’s why combining PCA and wavelets in one powerful method, known as Multiscale Principal Component Analysis (MSPCA) introduced by Bakshi [56], harnesses the advantages of each technique simultaneously. With that, MSPCA combines the extraction of deterministic features, done by wavelet transform, with the decorrelation and isolation of linear relationships which is carried by PCA. Figure 2.11 presents an overview of the technique where 𝑔 and ℎ are the low and high pass filters respectively.

Figure 2.11 Schematic Diagram of a Three-Scale MSPCA Technique [9]

To explain the above Figure 2.11, each column in the measurement data matrix 𝑋 undergoes the wavelet decomposition first. For each scale in the resulting decomposition, the wavelet coefficients serve as inputs to the PCA’s covariance, SSCP, or correlation matrix. After that, both of PCA loading and scores matrices, T and P, are calculated for each wavelet coefficient. Finally, MSPCA builds the PCA in-control model (upper limits of 𝒯², 𝒬, and 𝜑) by selecting both wavelet coefficients and number of principal components through either Kaiser or Heuristic method. To sum up, process monitoring using MSPCA results in determining the independent PCs and their limits at each wavelet scale.

Although MSPCA provides many advantages over PCA and wavelet stand-alone techniques, it still lacks the ability to detect quickly and continuously the presence of shifts between operational states, e.g. abnormal to normal process state. In fact, Yoon and MacGregor [57] showed that MSPCA, due to the inherent scale decomposition, continues to flag false alarms even after the process has returned to its normal operations. Added to that, since the wavelet transform down-samples the

adds a final step that combines up-scaling and Reconstruction Based Charts (RBC) which allows to diagnose the faults better. Furthermore, in this modified version, all produced wavelet coefficient are reconstructed using reconstruction filters before going as inputs to the PCA. Figure 2.12 describes this concept in details.

Figure 2.12 Schematic Diagram of a Three-Scale Mod-MSPCA Technique [9]

One common characteristics between MSPCA and Mod-MSPCA is present in the calculation of the final PCA statistical indices. In fact, these last are applied on each individual scale of the wavelet transform. Therefore, the previously defined 𝒯², 𝒬, and 𝜑 in (2.18) ~ (2.21) can be rewritten as:

𝒯_𝑗²= ∑ ^𝜏𝑖

𝜆_𝑖,𝑗

𝐴𝑖=1 , where j = 1,2, ..., J+1 (2.28)

𝒬_𝑗= ∑^𝐴_𝑖=1𝑥̃_𝑖,𝑗² , where j = 1,2, ..., J+1 (2.29)

𝜑_𝑗= ^𝒬𝑗

𝛿_𝑗²+ ^𝒯𝑗

2

𝜏_𝑗² , where j = 1,2, ..., J+1 (2.30)

Where J is the number of levels in the wavelet transform, 𝐴 the number of principal components, and 𝜆_𝑖,𝑗 eigenvalues of the i^th variable’s variance at the j^th level. To recover the overall indices from the individual scales, the following three equations are employed:

𝒯²= ∑ 𝒯_𝑗²

As mentioned before, unlike MSPCA, Mod-MSPCA takes these computed PCA statistics one step further by generating the reconstruction based charts. Dunia and Qin [58] showed that the combination of RBC with PCA results guarantees a higher rate of correct diagnosis. This is the case as faulty variables produce the highest contribution to the reconstructed results. To prove this, they modelled the faults using the equation 𝑥 = 𝑥^∗+ 𝑓, where 𝑥^∗ reflects the normal operation sample vector, and 𝑓 reflects the faulty vector defined as 𝑓 = 𝔣 𝜉_𝑖. The m x 1 vector 𝜉_𝑖 = [0 … 0 1 0 … 0 ]^𝑇, with the i^th element equal to unity, is the fault direction vector. Consequently, the RBC technique aims at reconstructing 𝑥^∗ from every new sample vector x along all possible fault directions. The final minimum 𝑥^∗direction, where 𝑓 contribution is maximum, is used to identify the faulty variable. The RBC formulas combined with PCA’s 𝒯², 𝒬, and 𝜑 statistics are as follows:

𝑅𝐵𝐶_𝑖,𝒯²= ^{(𝑥 𝐷̂ 𝜉𝑖)2} (2.19), and (2.21) respectively. These equations ensure fault isolation through reconstruction-based contribution analysis. And by that, Mod-MSPCA is a powerful tool to detect, identify, and analyze different types of faults.

Yoon and MacGregor [57] highlighted the validity of Mod-MSPCA as a reliable FDD technique using a Monte Carlo simulation with a Continuous Stirred Tank Reactor (CSTR) system. A total of 500 sets of training and testing data were generated to cover different types of faults regarding the inlet temperature measurements. Mod-MSPCA performance was compared with PCA in order to detect and isolate sensor spikes, sensor drifts, sensor biases, and other complex phenomena like scaling effects. In the end, Mod-MSPCA was more effective than regular PCA methods, and it was specifically insightful for faults that are localized in frequency or appear in wavelet scales with small variances.

Lachouri [59] relied on MSPCA stand alone technique to monitor bearing faults. To put this method to the test, he used known recordings of both healthy and faulty bearing’s vibrational signals. The results of this work demonstrated that most faults violated the 95% upper limit threshold for SPE statistics (𝒬) which was used to signal their presence. Besides, Lachouri showed that SPE alone (without 𝒯² statistics) can detect most of machinery faults that are related to bearings.

Wenying [60] combined MSPCA with a powerful classification technique using Support Vector Machines (SVM) to diagnose the health of induction motors. In this method, both 𝒯² 𝑎𝑛𝑑 𝒬 results were fed directly as inputs to the SVM. To test the validity of the method, vibration data was collected from 4 accelerometers mounted on the motor in all 4 directions. The combination of MSPCA and SVM was very accurate in detecting and isolating 4 induced faults: abnormal stators, non-uniform air-gap, unbalanced rotor, and faulty rolling bearings.