Wavelet functional principal component analysis for batch process monitoring

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

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.

formula omitted]-type smoothing spline estimators for principal functions

In this work, we propose a novel approach to analyze Epileptic EEG signals using wavelet power spectra and functional principal component analysis. Both continuous and discrete wavelet power spectra are considered. By transforming EEG signals into power spectra, we significantly enhance the functionality of random signals, which makes functional principal component analysis be a suitable technique for further extracting key signal features. We have tested our proposed method using a set of publicly available epileptic EEG. The obtained results demonstrate that wavelet power spectra and functional principal component analysis are promising for feature extraction of epileptic EEG, and they may be useful for epilepsy diagnosis problem.

Hyperspectral image classification using FPCA-based kernel extreme learning machine

Quantitative CT Imaging Analysis for Chronic Lung Allograft Dysfunction in Single Lung Transplant Recipients

We present a functional regression model for diameter prediction. Usually stem form is estimated from a regression model using dbh and height of the sample tree as predictor. With our model additional diameter observations measured at arbitrary locations within the sample tree can be incorporated in the estimation in order to calibrate a standard prediction based on dbh and height. For this purpose, the stem form of a sample tree is modelled as a smooth random function. The observed diameters are assumed as independent realizations from a sample of possible trajectories of the stem contour. The population average of the stem form within a given dbh and height class is estimated with the taper curves applied in the national forest inventory in Germany. Tree deviation from the population average is modelled with the help of a Karhunen–Loeve expansion for the random part of the trajectory. Eigenfunctions and scores of the Karhunen–Loeve expansion are estimated through conditional expectations within the methodological framework of functional principal component analysis (FPCA). In addition to a calibrated estimation of the stem form, FPCA provides asymptotic pointwise or simultaneous confidence intervals for the calibrated diameter predictions. For the application of functional principal component analysis modelling the covariance function of the random process is crucial. The main features of the functional regression model are discussed informally and demonstrated by means of practical examples.

PurposeThe purpose of this paper is to analyze and forecast the Chinese term structure of interest rates using functional principal component analysis (FPCA).Design/methodology/approachThe authors propose an FPCA-K model using FPCA. The forecasting of the yield curve is based on modeling functional principal component (FPC) scores as standard scalar time series models. The authors evaluate the out-of-sample forecast performance using the root mean square and mean absolute errors.FindingsMonthly yield data from January 2002 to December 2016 are used in this paper. The authors find that in the full sample, the first two FPCs account for 98.68 percent of the total variation in the yield curve. The authors then construct an FPCA-K model using the leading principal components. The authors find that the FPCA-K model compares favorably with the functional signal plus noise model, the dynamic Nelson-Siegel models and the random walk model in the out-of-sample forecasting.Practical implicationsThe authors propose a functional approach to analyzing and forecasting the yield curve, which effectively utilizes the smoothness assumption and conveniently addresses the missing-data issue.Originality/valueTo the best knowledge, the authors are the first to use FPCA in the modeling and forecasting of yield curves.

Functional principal component analysis (FPCA) has played an important role in the development of functional time series analysis. This note investigates how FPCA can be used to analyze cointegrated functional time series and proposes a modification of FPCA as a novel statistical tool. Our modified FPCA not only provides an asymptotically more efficient estimator of the cointegrating vectors, but also leads to novel FPCA‐based tests for examining essential properties of cointegrated functional time series.

Functional principal component analysis (FPCA) is a commonly used nonparametric degradation modeling technique. A basic requirement of FPCA is that different units must share the same scale in X-axis. A FPCA-based degradation modeling method has been specially designed for truncated degradation signals whose amplitudes are truncated at the failure threshold. This method formulates the degradation time as a function of state level by axis rotation. Since the degradation signals of different units are truncated at the same failure threshold level, the axis rotation strategy ensures the same scale in X-axis. However, in cases of noisy signals, one state level may correspond to multiple time points, which means that the time function depending on state level does not exist. In addition, this method assumes that the noise term as well as the functional principal component (FPC) scores follow the Gaussian distribution, which restricts the flexibility in real practice. Aiming at the above two issues, this paper develops a new degradation modeling method based on the FPCA. The proposed method formulates the degradation model on the definition of first pass time (FPT), which ensures the existence of the time function depending on state level. The noise term and the FPC scores are not restricted to the Gaussian distribution. A particle filtering based updating algorithm is developed to deal with the non-Gaussian state estimation issue of the FPC scores according to the online condition monitoring signals. The effectiveness of the proposed method is verified using an experiment of milling cutter wear.

In this paper, a novel method based on wavelet transformation of functional data for accurate classification of hyperspectral images is proposed. The motivation of the proposed method is that in the hyperspectral images, the spectral curve of each pixel can be viewed as a function, mathematically. And it can make the best of the abundant spectral information of the original hyperspectral images. Based on this, we have proposed an effective hyperspectral image classification method which mainly used functional principal component analysis (FPCA) before. However, FPCA method focuses on the global features analysis of hyperspectral image data and may not well extract local structural features. In this paper, in order to better reflect the local structural features of hyperspectral images, we make use of the time-frequency localization characteristics of wavelet transformation to decompose hyperspectral images. Then, the sparse wavelet decomposition coefficients are classified by using extreme learning machine (ELM). Experimental results on two popular hyperspectral images show that our method outperforms some state-of-the-art hyperspectral image classification methods.

Functional principal component analysis of H-reflex recruitment curves

Properties of design-based functional principal components analysis

Penalized PCA approaches for B-spline expansions of smooth functional data

The lithium-ion (Li-ion) battery is a promising energy storage technology for electronics, automobiles, and smart grids. Extensive research was conducted in the past to improve the prediction of the remaining capacity of the Li-ion battery. A robust prediction model would improve the battery performance and reliability for forthcoming usage. In the development of a data-driven capacity prediction model of Li-ion batteries, most past studies employed capacity degradation data; however, very few tried using other performance monitoring variables, such as temperature, voltage, and current data, to estimate and predict the battery capacity. In this study, we aimed to develop a data-driven model for predicting the capacity of Li-ion batteries adopting functional principal component analysis (fPCA) applied to functional monitoring data of temperature, voltage, and current observations. The proposed method is demonstrated using the battery monitoring data available in the NASA Ames Prognostics Center of Excellence repository. The main contribution of the study the development of an empirical data-driven model to diagnose the state-of-health (SOH) of Li-ion batteries based on the health monitoring data utilizing fPCA and LASSO regression. The study obtained encouraging battery capacity prediction performance by explaining overall variation through eigenfunctions of available monitored discharge parameters of Li-ion batteries. The result of capacity prediction obtained a root mean square error (RMSE) of 0.009. The proposed data-driven approach performs well for predicting the capacity by employing functional performance measures over the life span of a Li-ion battery.