Some features of the site may not work correctly. Our result aims at extending the theory of multivariate sub-Gaussian estimators [LM19a] to the quantum setting. Unlike classically, where any univariate estimator can be turned into a multivariate estimator with at most a logarithmic overhead in the dimension, no similar result can be proved in the quantum setting.
Indeed, Heinrich… Expand. View PDF on arXiv. Save to Library Save. Create Alert Alert. Share This Paper. Methods Citations. Figures from this paper. The recurrence matrix, consisting of zeros and ones, can also be used as a basis to extract features that are representative of the dynamic behavior of the time series. This approach is widely referred to as recurrence quantification analysis, and in process engineering, it has mainly been used in the description of electrochemical phenomena and corrosion [53—58], but in principle has general applicability to any dynamic system.
This is also referred to as global, as opposed to local recurrence quantification described in Section 2. The resulting recurrence plot can consequently be treated as an artificial image amenable to analysis by a large variety of algorithms normally applied to textural images, as discussed in more detail in Section 4. Supervised feature extraction 3. Apart from ARIMA models, other models, such as neural networks [60—62], decision trees [63], and just-in-time-learning with PCA [64], have also been proposed.
State space models offer a principled approach for the identification of the subspaces containing the data. Approaches to the monitoring of continuous dynamic process systems. State space models and their variants have been considered by several authors [65—75]. For example, Chen and Liao [62] have used a multilayer perceptron neural network to remove the nonlinear and dynamic characteristics of processes to generate residuals that could be used as input to a PCA model for the construction of simple monitoring charts.
Guh and Shiue [63] have used a decision tree to detect shifts in the multivariate means of process data. Auret and Aldrich [48] have considered the use of random forests in the detection of change points in process systems. In addition, Aldrich and Auret [2] have compared the use of random forests with autoassociative neural networks and singular spectrum analy- sis in a conventional process monitoring framework. The application of deep learning in process monitoring is an emerging area of research that shows particular promising.
This includes the use of stacked autoencoders [76], deep long short term memory LSTM neural networks [77], and convolutional neural networks. Table 2 gives an overview of the feature extraction methods that have been investigated over the last few decades. Case study: Tennessee Eastman process Finally, as an example of the application of a process monitoring scheme incor- porating feature extraction from time series data in a moving window, the following study can be considered.
It is based on the Tennessee Eastman benchmark process widely used in these types of studies. The feature extraction process considered here is an extension of recurrent quantitative analysis discussed in Section 2.
Instead of using thresholded recurrence plots, unthresholded or global recurrence plots are considered, as explained in more detail in below. It captures the dynamic behavior of an actual chemical process, the layout of which is shown in Figure 3. The plant consists of 5 units, namely a reactor, condenser, compressor, stripper and separator, as well as eight components four gaseous reactants A, C, D, E, one inert reactant B, and three liquid products F, G, H [97].
In this instance, the plant- wide control structure suggested by Lyman and Georgakis [99] was used to simu- late the process and to generate data related to varying operating conditions. A total of four data sets were used, that is, one data set associated with NOC and the remaining three associated with three different faults conditions. The TE pro- cess comprises 52 variables, of which 22 are continuous process measurements, 19 are composition measurements, and the remaining 11 are manipulated variables.
These variables are presented in Table 3. Each data set consisted of measure- ments sampled at 3 min intervals. Process flow of Tennessee Eastman benchmark problem. The NOC samples were used to construct an off-line process monitoring model that consisted of a moving window of length b, moving sliding along the time series with a step size s.
The three fault conditions are summarized in Table 4. Fault conditions 3, 9, and 15 are the most difficult to detect, and many fault diagnostic approaches fail to do so reliably. In this case study, the approach previously proposed by Bardinas et al.
The methodology can be briefly summarized as shown in Figure 4. A window of user defined length b slides along the time series A with a user defined step size s, yielding time series segments B , each of which can be represented by a similarity matrix C that is subsequently considered as an image from which features can be extracted via algorithms normally used in multivariate image analysis D.
Description of variables in Tennessee Eastman process. Description faults 3, 9, and 15 in the Tennessee Eastman process. Process monitoring methodology after Bardinas et al. From this matrix, various textural descriptors can be defined. Only four of these were used, as defined by Haralick et al.
The LBP operator is defined for a pixel in the image as a set of binary values obtained by comparing the center pixel intensity with its neighboring pixels. If this is too small, the essential dynamics of the time series would not be captured. On the other hand, if it is too large, it would result in a considerable lag before any change in the process can be detected.
There is also the possibility that transient changes may go undetected altogether. In the case of a moving window, the step size of the moves also needs to be considered. The selection of these two parameters can be done by means of a grid search, and the results of which are shown in Figure 5. Grid search optimization of the window length b and step size s. Figure 6. In Figure 6, principal component score plots of the two optimal feature sets are shown.
The large LBP feature set could not be visualized reliably, as the first two principal components could only capture The variance in the smaller GLCM feature set could be captured with high reliability by the first two principal components of the four features. Also, while thresholding does not preclude the use of a wide range of feature extraction algorithms, recurrent quantification has mostly been applied to dynamic systems based on a set of engineered features that allow some modicum of physical interpretation.
In most diagnostic systems, this is not essential and therefore more predictive feature sets may be constructed. These features could be engineered, as was con- sidered in the case study or they could be learned, by taking advantage of state- of-the-art developments in deep learning. The application of deep learning methods in particular is a highly promising emerging area of research.
Conclusions and future work Data-driven fault diagnosis of dynamic systems has advanced considerably over the last decade or more. In this chapter, the large variety of algorithms currently available has been discussed in terms of a feature extraction problem associated with the data captured by sliding a window across the time series or in some cases making use of a fixed window.
These features could be used in statistical process monitoring frameworks that are well established for steady state systems. In addition, extension of a recent approach to nonlinear time series analysis, namely recurrence quantification analysis, has been considered and shown to be an effective means of monitoring dynamic process systems, such as represented by the Tennessee Eastman benchmark problem in chemical engineering.
As mentioned in Section 4. In future work, the application of convolutional neural networks to extract features from global recurrence plots will be considered. This does not necessarily require a large amount of data, as pretrained networks, such as AlexNet, ResNet, and VGG archi- tectures, and others could possibly be used as is, in what would essentially be a texture analysis problem, similar to the work done by Fu and Aldrich [, ] in the recognition of flotation froth textures, for example.
Conflict of interest The author declares no conflict of interest in this contribution. Licensee IntechOpen. Chemical Engineering Processes. London: Springer; Science. Chinese Journal of Chemical Engineering. Dynamic process fault Chemometrics and Intelligent Laboratory Systems.
Then we say that the complex random vector. The complex Gaussian distribution can be described with 3 parameters: [5]. The characteristic function of complex normal distribution is given by [5]. The circularly-symmetric central complex normal distribution corresponds to the case of zero mean and zero relation matrix, i. The standard complex normal defined in Eq. Thus, the standard complex normal distribution has density. This distribution can be described by density function. Connect and share knowledge within a single location that is structured and easy to search.
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