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Principal Component Analysis
Published in Nong Ye, Data Mining, 2013
Principal component analysis explains the variance–covariance matrix of variables. Given a vector of variables x′ = [x1, …, xp] with the variance–covariance matrix ∑, the following is a linear combination of these variables: ()yi=a′ix=ai1x1+ai2x2+⋯+aipxp.
Principal Component Analysis
Published in Mohssen Mohammed, Muhammad Badruddin Khan, Eihab Bashier Mohammed Bashier, Machine Learning, 2016
Mohssen Mohammed, Muhammad Badruddin Khan, Eihab Bashier Mohammed Bashier
When collecting real data, usually the random variables that represent the data attributes are expected to be highly correlated. The correlations between these random variables can always be seen in the covariance matrix. The variances of the random variables are found in the diagonal of the covariance matrix. The sum of the variances (diagonal elements of the covariance matrix) gives the overall variability. In Table 12.1, we show the covariance matrix of the “ecoli” data consisting of seven attributes.
Identification of Stochastic Systems
Published in William Delaney, Erminia Vaccari, Dynamic Models and Discrete Event Simulation, 2020
William Delaney, Erminia Vaccari
The mean y and the covariance cov(Yt,Yt+h) provide sufficient information to determine all the statistical properties of a Gaussian process. In general, the covariance matrix can be large and can have small hard-to-estimate terms. As pointed out in Section VI.D of Chapter 3, which could be reread fruitfully at this point, such difficulties necessitate combining the Gaussian hypothesis with others, such as station -arity, independence, the Markov property, and so on.
A methodological approach to assess the territorial vulnerability in terms of people and road characteristics
Published in Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2022
Roberta Maletta, Giuseppe Mendicino
The first step of performing PCA is to assess data suitability (reliability and adequacy). A fundamental step to calculating the principal component vectors is the computation of the covariance matrix , which is a symmetric matrix with variances on its diagonal and covariances off-diagonal. A covariance matrix expresses the correlation between the different variables in the data set. It is essential to identify dependent variables because they contain biased and redundant information, which reduce the overall performance of the model. Thus, PCA requires the evaluation of the eigenvectors and eigenvalues of the covariance matrix. The eigenvalues of are roots of the characteristic equation:where is the eigenvalue associated with the eigenvector of covariance matrix. Rearranging the above equation, we get .
An orifice shape-based reduced order model of patient-specific mitral valve regurgitation
Published in Engineering Applications of Computational Fluid Mechanics, 2021
J. Franz, K. Czechowicz, I. Waechter-Stehle, F. Hellmeier, F. Razafindrazaka, M. Kelm, J. Kempfert, A. Meyer, G. Archer, J. Weese, R. Hose, T. Kuehne, L. Goubergrits
For the definition of the orientation and planarity indices, we approximated 2D projections of the 3D orifice surfaces onto two different planes: (i) the best-fitting valve plane with respect to the whole mitral valve geometry, and (ii) the best-fitting orifice plane, on which the projected orifice area was maximized. To determine these two best-fitting planes, we first computed the covariance matrix of the 3D point cloud defining (i) the mitral valve geometry and (ii) the 3D orifice surface, respectively. Using principal component analysis, the covariance matrix was decomposed into its eigenvectors and eigenvalues. The normal vector n of the best-fitting plane was determined as the cross product of the two eigenvectors that resulted in the maximum area projected onto a plane normal to n. The 2D projected areas of the 3D orifices onto the best-fitting planes were then approximated by alpha shapes with a manually adjusted alpha radius for each orifice.
Monitoring exhaust air temperature and detecting faults during milk spray drying using data-driven framework
Published in Drying Technology, 2023
The next important step in the PCA is to obtain the covariance matrix. This is carried out using expression: where & and & are the mean of the sample data, is the number of data points, observations in data set and is observations in data set y. The covariance matrix obtained is shown in the form of heat map as shown in Figure 3. The aim of this step is to understand how the variables of the input data set are varying from the mean with respect to each other, or in other words, to see if there is any relationship between them. The Eigen values of the covariance matrix are obtained and are used for deciding the contribution of each principal component in the data. This contribution of the information through each of the principal components is shown in Figure 4. It is noticed that none of the principal components carry the information which is more than 50%. Therefore, it is concluded that PCA is not suitable for developing the predictive model to obtain the performance of spray dryer. To confirm this further, the linear regression was carried out by using the principal components which carries maximum information in descending order. The predictive model is first trained and its minimum RMSE is 0.462 (Table 6). The results of this regression for testing are documented for all the combination in Table 3 in terms of RMSE values. It is observed that the minimum RMSE value is obtained for first 14 principal components beyond which it remains constant at 0.453. This RMSE value is itself very high and hence it is further confirmed that the regression using principal components fails to predict the performance of spray dryer. It is therefore proposed to use variable ranking method for deciding the parameters which affects the cyclone exit temperature in the order of its dominance.