Functional data analysis

Meyer, S. L., Data Analysis for Scientists and Engineers, Wiley, New York, 1975. (This is a more detailed manual on statistics and contains very useful collections of tables, distribution functions, data analysis procedures, and graphical techniques.)... 

Stability of baseline pulmonary function Data analysis Defining the lung edge Defining regions of interest within the lung Correction for tissue attenuation of radioactivity Calculation of dose deposited Expression of results... 

Wavelets in Parsimonious Functional Data Analysis Models... 

In this chapter, compression is achieved by assuming that the data profiles can be approximated by a linear combination of smooth basis functions. The bases used originate from the fast wavelet transform. The idea that data sets are really functions rather than discrete vectors is the main focus of functional data analysis [12-15] which forms the foundation for the generation of parsimonious models. 

J.O. Ramsay and B.W. Silverman, Functional Data Analysis, Springer series in statistics. Springer, New York, (1997). 

Variability analysis by Statistical Control Process and Functional Data Analysis. Case of study applied to power system harmonics... 

Keywords Harmonics, variability, outlier, statistical process control, functional data analysis. 

Abstract. Functional data appear in a multitude of industrial applications and processes. However, in many cases at present, such data continue to be studied from the conventional standpoint based on Statistical Process Control (SPC), losing the capacity of analyzing different aspects over the time. In this study is presented a Statistical Control Process based on functional data analysis to identify outliers or special causes of variability of harmonics appearing in power systems which can negatively impact on quality of electricity supply. The results obtained from the functional approach are compared with those obtained with conventional Statistical Process Control that has been done firstly. 

In classic statistical terminology, an outlier is defined as an observation that by being atypical or erroneous has a very different behavior compared to the other distribution data of which it is part, giving an approximate idea of the variability of the distribution. In the analysis of harmonics, when the variability and its causes are being analyzed, the charts offered by the software of the majority of measuring equipment not allow analysis of the variability. In this study, two different techniques are proposed for the above purpose in the following sections the conventional SPC analysis and the SCP based on functional data analysis. Previously of SPC techniques, it can be done a Box Plots analysis to determine the presence of outliers, but this Box Plots evaluation does not give any information about their causes and do not allow the traceability. 

Control graphs and the concept of rational subgroups can be used successfully in the search and elimination of outliers in harmonics present in electrical systems, provided that the data set follows a normal distribution. When data do not follow a normal distribution. Functional Data Analysis can be utilized effectively in the detection of outliers, also contributing major advantages in the detection of specific variability compared to traditional techniques such as Statistical Control Process. The functional approach greatly enhances the capacity for analysis facilitating massive and systematic analysis of the data. 

J. Sancho, J.J. Pastor, J. Martinez, M.A. Garcia. Evaluation of Harmonic Variability in Electrical Power Systems through Statistical Control of Quality and Functional Data Analysis. Procedia Engineering 63, 2013, pp. 295-302. 

The CMF approach can offer an alternative solution to this problem. Instead of using discrete sets of representative conformations, one can consider for each molecule an infinite number of conformations organized into a continuous manifold, so-called corrformational space . This provides the ability to apply functional data analysis not only to molecular fields but also to molecular geometry in a consistent way. Such corrformational space can be described by means of some probability density function pdf) in 3N-dimensional Euclidean space, where N is the number of atoms in the molecule under study. Having applied several approximations from the arsenal of statistical physics, one can obtain the following expression for calculating atomic kernels instead of Eq. (13.6) ... 

FIGURE 18.3 Example of curved data used for functional data analysis. 

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