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Analysis of Dynamic Data
Published in Shyama Prasad Mukherjee, A Guide to Research Methodology, 2019
In the second situation, the regression coefficients cannot be estimated by the ordinary least-squares method since the number of variables far exceeds the number of data points (units). One can use the principal-component-based partial least-squares approach or refer to functional principal-component analysis (FPCA). FPCA is an important dimension-reduction tool and in sparse data situations can be used to impute functional data. Penalized splines have also been used to estimate regression coefficients. Classic functional regression models are linear, though non-linear models have also been investigated. This type of non-parametric functional regression has been discussed in the literature (Ferraty and Vieu, 2006). In fact, functional regression is an active area of research and includes combinations of (1) functional responses with functional covariates, (2) vector responses with functional covariates and (3) functional responses with vector covariates.
A data-centric bottom-up model for generation of stochastic internal load profiles based on space-use type
Published in Journal of Building Performance Simulation, 2019
R. M. Ward, R. Choudhary, Y. Heo, J. A. D. Aston
In our analysis, we consider each day of data per space-use type to be a data sample, in the form of 24 hourly values of electricity consumption attributable to plug loads, and a function is fitted to each data sample. A particularly useful aspect of FDA is functional Principal Component Analysis (fPCA). Similar to principal component analysis of discrete data, fPCA serves to identify the fundamental principal components of the data – in this case themselves functions – which can be used to generate the original data from the mean function together with a weighted sum of those components. A function , constructed from a mean and i principal components with weightings , is simply represented using the equation: In this approach, the mean and principal component functions are the same across all the data samples. The only parameters that change from one sample to another are the weightings, or scores, . This means that if we can find a set of principal components that govern the behaviour across all space-use types, the difference between the space-use types lies in the distribution of the scores for each principal component.
Image-Based Prognostics Using Penalized Tensor Regression
Published in Technometrics, 2019
Xiaolei Fang, Kamran Paynabar, Nagi Gebraeel
The third baseline model, which we designated as “FPCA,” uses functional principal components analysis (FPCA) to model the overall image intensity. To be specific, we first transform the degradation image stream of each system into a time-series signal by taking the average intensity of each observed image. Next, FPCA is applied to the time-series signals to extract features. FPCA is a popular functional data analysis technique that identifies the important sources of patterns and variations among functional data (time-series signals in our case) (Ramsay and Silverman 2005). The time-series signals are projected to a low-dimensional feature space spanned by the eigen-functions of the signals’ covariance function and provides fused features called FPC-scores. Finally, FPC-scores are regressed against the TTFs by using LLS regression. More details about this FPCA prognostic model can be found in (Fang, Zhou, and Gebraeel 2015).
Multiple profiles sensor-based monitoring and anomaly detection
Published in Journal of Quality Technology, 2018
Chen Zhang, Hao Yan, Seungho Lee, Jianjun Shi
A straightforward monitoring statistic for is to apply the traditional parametric or nonparametric SPC methods. However, in many situations the total number of grid time points (i.e., ) is much larger than the number of IC reference samples, . We thus suffer from the curse of dimensionality. Hence it is usually important to perform dimension reduction before the monitoring step. One fundamental technique is to use functional PCA (FPCA) to extract a few major and typical features from the functional data. FPCA has been applied in univariate Phase I profile monitoring in Yu, Zou, and Wang (2012). Recently, Paynabar et al. (2016) extended FPCA to multichannel profiles as MFPCA by appropriately addressing their cross-correlations.