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Modelling Procedures
Published in Vanesa Magar, Sediment Transport and Morphodynamics Modelling for Coasts and Shallow Environments, 2020
with the normalisation factors now incorporated in cp. From the properties of real symmetric matrices, one can deduce some of the properties of the data: The trace of A is equal to the mean-square value of the data or the energy.Each eigenvalue λp represents the relative contribution of mode p to the total variability.The matrix can be arranged so that λi are in decreasing order with increasing i, so that e1 accounts for most of the mean-square value, e2 for most of the remaining mean-square value, and so on.In general, the first five modes capture more than 90% of the total energy, unless the system has a lot of red noise (Vautard & Ghil 1989).The shape functions can be interpreted as ‘modes’ of variation, as in Fourier analysis.The number of maxima and minima in an eigenfunction increases with the order of the eigenfunction.
Basic prediction methods in marine sciences
Published in David R. Green, Jeffrey L. Payne, Marine and Coastal Resource Management, 2017
A key objective of empirical modelling is to understand the structure of data variability, both in time (within x or within each of m components of x) and/or between m components of x. If these components correspond to variable(s) observed at various locations, the problem fits spatio-temporal modelling. There are two modes of variation: deterministic and stochastic. Deterministic variations are regular and can be described by simple functions, such as polynomials and harmonic functions. In contrast, stochastic variations are irregular and may be modelled by stochastic processes, the structure of which may vary from simple stochastic models to the complicated ones. Importantly, the two modes are usually superimposed on each other and reveal dissimilar importance for the specific time series (Figure 8.2). In general, we can decompose a time series x into a deterministic component d(x) and a stochastic component s(x), known also as residuals, in the following way: x=d(x)+s(x).
A statistical shape model of the tibia-fibula complex: sexual dimorphism and effects of age on reconstruction accuracy from anatomical landmarks
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2022
Olivia L. Bruce, Michael Baggaley, Lauren Welte, Michael J. Rainbow, W. Brent Edwards
A principal component analysis (PCA) was applied to the registered data to obtain the average shape and modes of variation (i.e., principal components) for the sample. An analysis described by Mei et al. (2008) evaluating bootstrap stability on mode direction and comparison with noise was used to determine the number of principal components to retain. Eight principal components accounting for 96.2% of the total variance in the model were ultimately retained. Scores for each retained principal component were compared using unpaired t-tests to determine if and how size and shape differed between sexes (SPSS v.26, IBM, NY, USA, α = 0.05). Centroid size, the square root of the sum of squared Euclidean distances of all points in a shape from the centroid of the shape, was calculated. Pearson correlations were used to evaluate whether principal component scores were correlated with size.
Integration of cortical thickness data in a statistical shape model of the scapula
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2020
Jonathan Pitocchi, Roel Wirix-Speetjens, G. Harry van Lenthe, María Ángeles Pérez
Statistical shape models (SSMs) provide a valuable way to describe shape variability within a training dataset. Since their introduction (Cootes and Taylor 2001), these models have been used for multiple applications: to automatically segment bone structures (Lamecker et al. 2004; Ma et al. 2017), to study the shapes of human anatomy (Sarkalkan et al. 2014; Salhi et al. 2017; Sintini et al. 2018; Casier et al. 2018), to virtually reconstruct large bone defects (Vanden Berghe et al. 2017; Poltaretskyi et al. 2017; Plessers et al. 2018; Abler et al. 2018), to build 3D models starting from 2D information (Grassi et al. 2017; Mutsvangwa et al. 2017). The main concept behind SSM techniques is to perform principal component analysis (PCA) on corresponding landmarks derived from the dataset objects and to extract the main modes of variation. Thus, each subject in the training dataset can be described by a linear combination of principal components (PCs), corresponding to these modes of variation. Moreover, new instances, representative of the population, can be generated by varying the PC (Cootes and Taylor 2001).
Statistical kinematic modelling: concepts and model validity
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2022
Kate Duquesne, Pavel Galibarov, Jose-de-Jesus Salazar-Torres, Emmanuel Audenaert
Another appealing alternative is Functional Data Analysis (FDA). In FDA, time-series are treated as functions (Ramsay et al. 2009a). Functional Principal Component Analysis (FPCA) is then used as the equivalent of PCA in the functional domain (Warmenhoven et al. 2021) and can be used to examine multiple variables simultaneously (Happ and Greven 2018). To indicate the primary modes of variation, Multivariate Functional Principal Component Analysis (MFPCA) uses eigenfunctions, whereas PCA and PPA use eigenvectors (Ramsay et al. 2009a).