Principal component analysis Euclidean distance

For principal component analysis (PCA), the criterion is maximum variance of the scores, providing an optimal representation of the Euclidean distances between the objects. 

It is useful to go back to two-way visualization in principal component analysis to find what really is seen in a plot. A score plot for a two-way PCA model has an orthonormal basis, because the loadings are orthonormal. This can be compared to projecting all points in multidimensional space on a movie screen using a strong light source at a large distance. What is seen in this projection is true Euclidean distance in the reduced space, if both... 

Fluorescence spectra are collected under excitation conditions that are optimized to correlate the emission spectral features with parameters of interest. Principal components analysis (PCA) is further used to extract the desired spectral descriptors from the spectra. The PCA method is used to provide a pattern recognition model that correlates the features of fluorescence spectra with chemical properties, such as polymer molecular weight and the concentration of the formed branched side product, also known as Fries s product, that are in turn related to process conditions. The correlation of variation in these spectral descriptors with variation in the process conditions is obtained by analyzing the PCA scores. The scores are analyzed for their Euclidean distances between different process conditions as a function of catalyst concentration. Reaction variability is similarly assessed by analyzing the variability between groups of scores under identical process conditions. As a result the most appropriate process conditions are those that provide the largest differentiation between materials as a function of catalyst concentration and the smallest variability in materials between replicate polymerization reactions. 

A cluster analysis of the amino acid structures by PCA of the A -matrix is shown in Figure 6.5a note that PCA optimally represents the Euclidean distances. The score plot for the first two principal components (preserving 27.1% and 20.5% of the total variance) shows some clustering of similar structures. Four structure pairs have identical variables 1 (Ala) and 8 (Gly), 5 (Cys) and 13 (Met), 10 (He) and 11 (Leu), and 16 (Ser) and 17 (Thr). Objects with identical variables of course have identical scores, but for a better visibility the pairs have been artificially... 

The following examples demonstrate the usefulness of multivariate methods in the evaluation of field ecological data and laboratory multispecies toxicity tests. In each of the examples, several multivariate techniques were used — generally Euclidean and cosine distances (Figure 11.29), principal components, and nonmetric clustering and association analysis. 

Puchert et al. [20] proposed an extension of the use of PCA with the principal component score distance analysis approach. In their study, they calculated the Euclidean distance between two successive spectra considering the first three factors. A moving block standard deviation was then applied onto the distance terms. It is an extension of fhe work of Storme-Paris ef al., who considered each principal componenf independently [21]. In addition, Puchert et al. used a SIMCA-like approach by developing a successive PCA model with only the most stable spectra in calibration. It allowed them to obtain a better resolution when projecting new samples. 

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