Global-local based wavelet functional principal component analysis for fault detection and diagnosis in batch processes

Wavelet functional principal component analysis for batch process monitoring

Application of functional data analysis to explore movements: walking, running and jumping - A systematic review

Trends in ecologically meaningful water quality variates are important for establishing changes in water quality. Linear assessments of the trend at different monitoring sites are relatively easy to compare but may be misleading as they can poorly represent the nature of the trend. Non-linear trends, on the other hand, more faithfully capture the trend but have been much more difficult to compare between sites. In this article we investigate the ability of functional data analysis to explore between site differences in non-linear water quality trends. Functional data analysis is briefly reviewed, concentrating in particular on the techniques of functional principal component analysis, functional linear modelling and functional cluster analysis. These techniques are then applied to some nutrient and sediment trend data obtained from a number of monitoring sites in three dams in South East Queensland, Australia. Functional data analysis is found to be a valuable exploratory tool for providing insight into the between-site, depth and dam differences. Copyright © 2005 John Wiley & Sons, Ltd.

Different operation phases in batch processes cover distinguishing behaviors, so establishing statistical models for each identical phase become an effective way for batch monitoring. In this paper, a new adaptive phase partition and online fault detection method is proposed, which can track the phase’s transition by time sequence and has less reliance on parameters’ selection. The discussion and analysis of this proposed method follows. In this proposed method, the information contained in every sample time will be evaluated, and the change tendency of feature is demonstrated on a batch prospect. Then, two control bounds are designed for the feature tendency, the stable, and the transitional phases that have a different feature level and play certainly roles in process operation, will be identified automatically. For online monitoring, the new fault detection strategy is composed of modeling the PCA and PLS statistical methods for each identified phase, three statistics are established to ensure the data-decomposing reliable. The proposed method is applied to the industrial penicillin fermentation process, and the experimental result shows better performance in phase partition and fault detection.

Functional data is ubiquitous in many domains such as healthcare, social media, manufacturing process, sensor networks, etc. Functional data analysis involves the analysis of data which is treated as infinite-dimensional continuous functions rather than discrete, finite-dimensional vectors. In this paper, we propose a novel function-on-function regression model based on mode-sparsity regularization. The main idea is to represent the regression coefficient function between predictor and response as the double expansion of basis functions, and then use mode-sparsity constraint to automatically filter out the irrelevant basis functions for both predictors and responses. The mode-sparsity regularization covers a wide spectrum of sparse models for function-on-function regression. The resulting optimization problem is challenging due to the non-smooth property of the mode-sparsity. We develop an efficient and convergence-guaranteed algorithm to solve the problem. The effectiveness of the proposed approach is verified on benchmark functional data sets in various domains.

Nonlinear fault detection for batch processes via improved chordal kernel tensor locality preserving projections

This research implements two important main fields of the study which is functional data analysis and extreme value theory. The aims of this study are to convert the extreme rainfall data into functions and to propose method development for functional data analysis in extreme value theory. Functional data analysis is one of the techniques to convert or transform discrete observations into functions or curves. The daily rainfall data in Petaling Jaya from 1974 to 2003 will be tested on extreme value theory approach which is block maxima in order to practice the functional data analysis technique. The least squares and akaike criteria information method are used to smooth the fourier basis of the functional data analysis. A descriptive statistics which represents the behavior of a rainfall data for Petaling Jaya is analyzed and illustrated by a smooth curve from functional data analysis. This study indicates that there is an increase in rates of the rainfall in Petaling Jaya for the recent period compared to the last 30 years data both in the mean and standard deviation values.

Functional data analysis (FDA) is a relatively new branch of statistics devoted to describing and modelling data that are complete functions. Many relevant aspects of musical performance and perception can be understood and quantified as dynamic processes evolving as functions of time. In this paper, we show that FDA is a statistical methodology well suited for research into the field of quantitative musical performance analysis. To demonstrate this suitability, we consider tempo data for 28 performances of Schumann's Träumerei and analyse them by means of functional principal component analysis (one of the most powerful descriptive tools included in FDA). Specifically, we investigate the commonalities and differences between different performances regarding (expressive) timing, and we cluster similar performances together. We conclude that musical data considered as functional data reveal performance structures that might otherwise go unnoticed.

An inherent feature of batch processes is multiple operational phases. So many different phased-based monitoring methods had been proposed in order to monitor batch processes with multiple phases. The key of those methods is how to divide the phases of batch processes. At present, the most of the existing phase-division methods segment the batch process into several phases, where the same phase for difference batch run has equal duration. However, due to production conditions or other factors, the every phase's duration for different batch run is different. To solve this problem, a novel phase division method based on Fisher Discriminant Analysis (FDA) and Dissimilarity Index (DISSIM) is proposed in this paper. In DISSM-FDA algorithm, DISSIM is used to determine start and end points for each phase and rough division results are given. Then, the unequal phase division algorithm is advanced based on FDA that used to extract the phase characteristics. Application results to an industrial typical multi-phase streptomycin fermentation process show the effectiveness of the proposed method.

Compared with the normal process, the statistical values or the relationships between process variables and the quality that represent process situations would change in an abnormal batch. A novel method named quality relevant fault detection based on statistical pattern and regression coefficients (SPRC) is proposed for the batch process. Firstly, the statistical patterns of the process data, such as mean value and SD, are computed to quantify process characteristics. The regression model is built via linear methods, such as multiple linear regression, least absolute shrinkage and selection operator, to describe the linear relationship. The mutual information between the quality and the process parameters is used to characterize the nonlinear relationship. In this way, statistical patterns and regression coefficients which express batch information constitute the two‐way matrix. Then, the matrix is dimensionally reduced by related method, such as principal component analysis. The relationships hidden in batch process could be sought by SPRC. Finally, two cases, penicillin fermentation and steel hot rolling, are used to validate the feasibility and advantages of the proposed method.

In the chemical and biochemical industries, batch processes play an important role in producing high-quality and value-added specialty products. The variables measured in such processes often have periodic trajectories with varying batch duration and fixed operational stages. The measured value of a process variable provides only part of the information about process conditions, while the process monitoring and operation activities mostly rely on the trends and trajectories of these variables. It is necessary to construct a trend-monitoring statistic to detect an abnormality in a batch process variable trajectory to provide operation decision support and guarantee product quality. In this paper, a new trend cumulative sum (CUSUM) chart is introduced as a way to detect the deviation of batch process variable trends, which may be caused by abnormal operations. This method relies on the extraction of variable trend features as a combination of derivatives through a functional description of variable trajectories. This then leads to the construction of an individual trend CUSUM chart and a multiple-trend CUSUM chart. The multiple-trend CUSUM chart, the sum of the trend CUSUM charts of all process variables, aims to improve the monitoring efficiency by combining the trend deviations of all variables in a single chart. The potential of these novel univariate control charts is demonstrated using the batch manufacture of polypropylene (PP). When several simulated and actual batches are studied, the results show that the trend CUSUM chart is capable of capturing trend abnormalities. We also show, by comparison, that alarms can be triggered in the trend CUSUM chart before the measured process variables exceed the control limits of their Shewhart individuals charts, allowing for the corrections to be made at an early stage of an abnormal situation.

Data‐driven soft sensor models have been extensively utilized in industrial processes. Batch processes are usually employed to manufacture low‐volume and high value‐added products in chemical, materials, and pharmaceutical industries. The most distinctive features of batch process lie in nonlinear, repetition, and slow time varying characteristics. In this paper, a data‐driven soft sensor modelling method based on linear slow feature analysis (LSFA) and least squares support vector regression (LSSVR) is proposed. In this method, LSFA was used to effectively capture the driving force behind the data features that change as slowly as possible. Then, a LSSVR model was constructed between the extracted slow feature variables and quality variables. Finally, a numerical example, industrial penicillin fermentation processes, and cobalt oxalate synthesis process were utilized to confirm the prediction accuracy and model reliability of the proposed approach.

The knowledge of the urban air quality represents the first step to face air pollution issues. For the last decades many cities can rely on a network of monitoring stations recording concentration values for the main pollutants. This paper focuses on functional principal component analysis (FPCA) to investigate multiple pollutant datasets measured over time at multiple sites within a given urban area. Our purpose is to extend what has been proposed in the literature to data that are multisite and multivariate at the same time. The approach results to be effective to highlight some relevant statistical features of the time series, giving the opportunity to identify significant pollutants and to know the evolution of their variability along time. The paper also deals with missing value issue. As it is known, very long gap sequences can often occur in air quality datasets, due to long time failures not easily solvable or to data coming from a mobile monitoring station. In the considered dataset, large and continuous gaps are imputed by empirical orthogonal function procedure, after denoising raw data by functional data analysis and before performing FPCA, in order to further improve the reconstruction.

Data with spatio-temporal structure can arise in many contexts, therefore a considerable interest in modelling these data has been generated, but the complexity of spatio-temporal models, together with the size of the dataset, results in a challenging task. The modelization is even more complex in presence of multivariate data. Since some modelling problems are more natural to think through in functional terms, even if only a finite number of observations is available, treating the data as functional can be useful (Berrendero et al. in Comput Stat Data Anal 55:2619–2634, 2011). Although in Ramsay and Silverman (Functional data analysis, 2nd edn. Springer, New York, 2005) the case of multivariate functional data is also contemplated, they do not cope with more than one dimension (only time is considered as covariate) in the domain of the considered functions. In estimating functional data through smoothing methods, a proper framework for incorporating space-time structures can be found in the generalized additive models (GAM), while classical dimension reduction techniques for functional data lead to functional principal component analysis (FPCA). In a previous work Ruggieri et al. (J Appl Stat 40(4):795–807, 2013) extended temporal FPCA, that is FPCA on data modelled as functions of the one-dimensional time, to multivariate (more than one variable, in our case pollutants) context. In this paper the computational aspects of FPCA are extended to more than one dimension: space (long, lat) and/or space-time; moreover, multidimensional (spatial, spatio-temporal) FPCA is extended to multivariate case. In order to provide a generalization of FPCA to multidimensional (spatio-temporal) and simultaneously multivariate data, we link GAM models together with the approach proposed by Ramsay and Silverman (2005). The paper describes all the computational details useful to implement this approach, while its effectiveness will be shown by a multivariate spatio-temporal environmental dataset, whose structure is actually very common in literature.

BackgroundA key problem in systems biology is estimating dynamical models of gene regulatory networks. Traditionally, this has been done using regression or other ad-hoc methods when the model is linear. More detailed, realistic modeling studies usually employ nonlinear dynamical models, which lead to computationally difficult parameter estimation problems. Functional data analysis methods, however, offer a means to simplify fitting by transforming the problem from one of matching modeled and observed dynamics to one of matching modeled and observed time derivatives–a regression problem, albeit a nonlinear one.ResultsWe formulate a functional data analysis approach for estimating the parameters of nonlinear dynamical models and evaluate this approach on data from two real systems, the gap gene system of Drosophila melanogaster and the synthetic IRMA network, which was created expressly as a test case for genetic network inference. We also evaluate the approach on simulated data sets generated by the GeneNetWeaver program, the basis for the annual DREAM reverse engineering challenge. We assess the accuracy with which the correct regulatory relationships within the networks are extracted, and consider alternative methods of regularization for the purpose of overfitting avoidance. We also show that the computational efficiency of the functional data analysis approach, and the decomposability of the resulting regression problem, allow us to explicitly enumerate and evaluate all possible regulator combinations for every gene. This gives deeper insight into the the relevance of different regulators or regulator combinations, and lets one check for alternative regulatory explanations.ConclusionsFunctional data analysis is a powerful approach for estimating detailed nonlinear models of gene expression dynamics, allowing efficient and accurate estimation of regulatory architecture.