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Dimension Reduction Breaking the Curse of Dimensionality
Published in Chong Ho Alex Yu, Data Mining and Exploration, 2022
In this example, the sample size is large and therefore the component structure tends to be stable. However, if the sample size is smaller and there are many variables, parallel analysis (Horn 1965) is recommended for verifying the result of PCA. Indeed, numerous studies have confirmed that by far parallel analysis is the most accurate method for extracting components or factors (Buja and Eyubuglu 1992; Glorfeld 1995; Humphreys and Montanelli 1975; Zwick and Velicer 1986). The logic of parallel analysis resembles that of resampling, which had been discussed in Chapter 7. In parallel analysis, the number of components extracted should have eigenvalues greater than those in a random matrix. To be more specific, the algorithm generates a set of random data correlation matrices by bootstrapping the data set (resampling with replacement), and then the average eigenvalues and the 95th percentile eigenvalues are computed. Next, the observed eigenvalues are compared against the re-sampled eigenvalues, and only components with observed eigenvalues greater than those from the resampling are retained. The resampled result functions as an empirical sampling distribution, against which the observed is compared. The rationale of using the 95th percentile of the resampled data eigenvalues is that this is analogous to setting the value of alpha to .05 in hypothesis testing (Cho et al. 2009). The scree plot in Figure 8.4 shows an example. A scree plot is a line plot that shows the eigenvalues on the y-axis against the number of factors or principal components on the x-axis. In this example, the line with diamond points shows the eigenvalue associated with each component yielded by PCA, whereas the lines with squares and triangles result from parallel analysis. This author suggests that only two or three components should be retained because they are above the random results. Parallel analysis can be run in SAS, SPSS, Matlab, or R using the programs developed by O’Connor (2000). These programs can be downloaded from http://si/oconnor-psych.ok.ubc.ca/nfactors/nfactors.html.
Principal Component Analysis
Published in Jhareswar Maiti, Multivariate Statistical Modeling in Engineering and Management, 2023
Scree plot is a graphical technique (Cattell, 1966) which plots eigenvalues (λj) along the ordinate (Y-axis) and the PC numbers along the abscissa (X-axis) as shown in Figure 11.7.
Evaluation of groundwater quality using multivariate, parametric and non-parametric statistics, and GWQI in Ibadan, Nigeria
Published in Water Science, 2023
A Scree plot is a basic linear graph that illustrates the proportion of total variation explained or represented by each element in the data. The factors are arranged, and therefore given a number label, in decreasing order of contribution to total variance. A scree plot is a graph of eigenvalues in ascending order of magnitude. It demonstrates a clear distinction between the slope angle of the strong eigenvalues and the progressive falling off of the remaining components. The five extracted components (eigenvalues >1) appropriately represented the aggregate dimensions of the data set and compensated for 69.894% of the total variance in the current research, whereas the other six factors (eigenvalues <1) accounted for just 30.106% of the total variance (Figure 4).
Aberrant driving behaviors as mediators in the relationship between driving anger patterns and crashes among taxi drivers: An investigation in a complex cultural context
Published in Traffic Injury Prevention, 2023
Zahid Hussain, Qinaat Hussain, Abdrabo Soliman, Semira Mohammed, Wondwesen Girma Mamo, Wael K. M. Alhajyaseen
For analysis, such as descriptive statistics and exploratory factor analysis, the Statistical Package for the Social Sciences (SPSS 27.0) was employed. Exploratory factor analysis (EFA) was conducted to analyze the factor structure of the DBQ and DAS among professional taxi drivers from various cultural backgrounds in the state of Qatar, utilizing principal component analysis (PCA) with the varimax rotation approach. The Kaiser criterion of eigenvalues > 1.0 and Cattell Scree plot were used to compute the number of construct. The scree plot is a graphical representation of eigenvalues (Cattell 1966). They typically rank eigenvalues in descending order from highest to lowest. The clear break “elbow” indicates the number of components the analysis should yield. Using the limitation criteria suggested by Tabachnick and Fidell (2007), items with factor loadings of at least 0.45 are appropriate indicators of the underlying construct.
Knowledge transfer in institutionalised supplier development and operational performance: the mediating role of absorptive capacity
Published in Cogent Engineering, 2021
We assessed the internal consistency and unidimensionality of all the multi-item constructs by means of exploratory factor analysis with varimax rotation using SPSS vol. 23. All items demonstrated high loadings exceeding the 0.4 cut-off point on the intended factors (Field, 2009; Joseph et al., 2014). The results show that the Kaiser–Meyer–Olkin verified the measure of sampling adequacy for the analysis; KMO = .870 was far above the acceptable minimum limit of 0.5 (Field, 2009; Joseph et al., 2014). Moreover, Bartlett’s test of sphericity χ2 (406) = 3606.102, p < .001 indicates that the correlation matrix is not an identity matrix; therefore, the dataset was suitable for factor analysis. Additionally, an initial analysis was run to obtain eigenvalues for each component in the data, which were compared with the scree plot inflexions. The varimax rotation produced a clear structure of factor loadings on a particular component, and factors with loadings above 0.4 were retained as recommended (Hair et al., 2014). Six components were retained, which had eigenvalues over Kaiser’s criterion of 1 and, in combination, explained 71.855 per cent of the variance.