The Scree Test and the Number of Factors: a Dynamic Graphics Approach

2015 ◽  
Vol 18 ◽  
Author(s):  
Rubén Daniel Ledesma ◽  
Pedro Valero-Mora ◽  
Guillermo Macbeth
Keyword(s):  
Principal Component Analysis ◽  
Principal Component ◽  
Real Data ◽  
Component Analysis ◽  
Psychological Research ◽  
Scree Plot ◽  
Dynamic Graphics ◽  
Number Of Factors ◽  
Interactive Data ◽  
Scree Test

AbstractExploratory Factor Analysis and Principal Component Analysis are two data analysis methods that are commonly used in psychological research. When applying these techniques, it is important to determine how many factors to retain. This decision is sometimes based on a visual inspection of the Scree plot. However, the Scree plot may at times be ambiguous and open to interpretation. This paper aims to explore a number of graphical and computational improvements to the Scree plot in order to make it more valid and informative. These enhancements are based on dynamic and interactive data visualization tools, and range from adding Parallel Analysis results to "linking" the Scree plot with other graphics, such as factor-loadings plots. To illustrate our proposed improvements, we introduce and describe an example based on real data on which a principal component analysis is appropriate. We hope to provide better graphical tools to help researchers determine the number of factors to retain.

On Reliability and Number of Principal Components: Joinder with Cliff and Kaiser

1992 ◽  
Vol 75 (3) ◽  
pp. 929-930 ◽  
Author(s):  
Oliver C. S. Tzeng
Keyword(s):  
Principal Component Analysis ◽  
Principal Components ◽  
Correlation Matrix ◽  
Principal Component ◽  
Component Analysis ◽  
Number Of Factors ◽  
Number Of Principal Components ◽  
Scree Test

This note summarizes my remarks on the application of reliability of the principal component and the eigenvalue-greater-than-1 rule for determining the number of factors in principal component analysis of a correlation matrix. Due to the unpredictability and uselessness of the reliability approach and the Kaiser-Guttman rule, research workers are encouraged to use other methods such as the scree test.

scholarly journals Criteria for choosing the number of dimensions in a principal component analysis: An empirical assessment

2020 ◽  
Author(s):  
Renata Silva ◽  
Daniel Oliveira ◽  
Davi Pereira Santos ◽  
Lucio F.D. Santos ◽  
Rodrigo Erthal Wilson ◽  
...  
Keyword(s):  
Machine Learning ◽  
Principal Component Analysis ◽  
Hypothesis Test ◽  
Feature Learning ◽  
Principal Component ◽  
Component Analysis ◽  
Scree Plot ◽  
Open Issue ◽  
Chained Tasks ◽  
High Dimensional Datasets

Principal component analysis (PCA) is an efficient model for the optimization problem of finding d' axes of a subspace Rd' ⊆ Rd so that the mean squared distances from a given set R of points to the axes are minimal. Despite being steadily employed since 1901 in different scenarios, e.g., mechanics, PCA has become an important link in machine learning chained tasks, such as feature learning and AutoML designs. A frequent yet open issue that arises from supervised-based problems is how many PCA axes are required for the performance of machine learning constructs to be tuned. Accordingly, we investigate the behavior of six independent and uncoupled criteria for estimating the number of PCA axes, namely Scree-Plot %, Scree Plot Gap, Kaiser-Guttman, Broken-Stick, p-Score, and 2D. In total, we evaluate the performance of those approaches in 20 high dimensional datasets by using (i) four different classifiers, and (ii) a hypothesis test upon the reported F-Measures. Results indicate Broken-Stick and Scree-Plot % criteria consistently outperformed the competitors regarding supervised-based tasks, whereas estimators Kaiser-Guttman and Scree-Plot Gap delivered poor performances in the same scenarios.

scholarly journals Selecting the number of factors in principal component analysis by permutation testing-Numerical and practical aspects

2017 ◽  
Vol 31 (12) ◽  
pp. e2937 ◽  
Author(s):  
Raffaele Vitale ◽  
Johan A. Westerhuis ◽  
Tormod Naes ◽  
Age K. Smilde ◽  
Onno E. de Noord ◽  
...  
Keyword(s):  
Principal Component Analysis ◽  
Principal Component ◽  
Component Analysis ◽  
Permutation Testing ◽  
Number Of Factors

scholarly journals Microsaccade characterization using the continuous wavelet transform and principal component analysis

2010 ◽  
Vol 3 (5) ◽  
Author(s):  
Mario Bettenbühl ◽  
Claudia Paladini ◽  
Konstantin Mergenthaler ◽  
Reinhold Kliegl ◽  
Ralf Engbert ◽  
...  
Keyword(s):  
Principal Component Analysis ◽  
Wavelet Transform ◽  
Eye Movements ◽  
Continuous Wavelet Transform ◽  
Principal Component ◽  
Real Data ◽  
Component Analysis ◽  
Continuous Wavelet ◽  
Fixational Eye Movements ◽  
Definition Of

During visual fixation on a target, humans perform miniature (or fixational) eye movements consisting of three components, i.e., tremor, drift, and microsaccades. Microsaccades are high velocity components with small amplitudes within fixational eye movements. However, microsaccade shapes and statistical properties vary between individual observers. Here we show that microsaccades can be formally represented with two significant shapes which we identfied using the mathematical definition of singularities for the detection of the former in real data with the continuous wavelet transform. For character-ization and model selection, we carried out a principal component analysis, which identified a step shape with an overshoot as first and a bump which regulates the overshoot as second component. We conclude that microsaccades are singular events with an overshoot component which can be detected by the continuous wavelet transform.

scholarly journals Kernel Principal Component Analysis Allowing Sparse Representation and Sample Selection

2019 ◽  
Vol 13 (1) ◽  
pp. 9-20 ◽  
Author(s):  
Duo Wang ◽  
Toshihisa Tanaka
Keyword(s):  
Principal Component Analysis ◽  
Sparse Representation ◽  
Sample Selection ◽  
Principal Component ◽  
Real Data ◽  
Component Analysis ◽  
Kernel Functions ◽  
Kernel Principal Component Analysis ◽  
Alternating Direction ◽  
Sparse Kernel

Kernel principal component analysis (KPCA) is a kernelized version of principal component analysis (PCA). A kernel principal component is a superposition of kernel functions. Due to the number of kernel functions equals the number of samples, each component is not a sparse representation. Our purpose is to sparsify coefficients expressing in linear combination of kernel functions, two types of sparse kernel principal component are proposed in this paper. The method for solving sparse problem comprises two steps: (a) we start with the Pythagorean theorem and derive an explicit regression expression of KPCA and (b) two types of regularization $l_1$-norm or $l_{2,1}$-norm are added into the regression expression in order to obtain two different sparsity form, respectively. As the proposed objective function is different from elastic net-based sparse PCA (SPCA), the SPCA method cannot be directly applied to the proposed cost function. We show that the sparse representations are obtained in its iterative optimization by conducting an alternating direction method of multipliers. Experiments on toy examples and real data confirm the performance and effectiveness of the proposed method.

Principal Component Analysis Using the Factor Procedure

2010 ◽  
pp. 171-193
Author(s):  
Sean Eom
Keyword(s):  
Principal Component Analysis ◽  
Factor Analysis ◽  
Principal Component ◽  
Component Analysis ◽  
Statistical Techniques ◽  
Data Input ◽  
Number Of Factors ◽  
Rotation Method ◽  
Definition Of

This chapter describes the factor procedure. The first section of the chapter begins with the definition of factor analysis. This is the statistical techniques whose common objective is to represent a set of variables in terms of a smaller number of hypothetical variables (factor). ACA uses principal component analysis to group authors into several catagories with similar lines of research. We also present many different approaches of preparing datasets including manual data inputs, in-file statement, and permanent datasets. We discuss each of the key SAS statements including DATA, INPUT, CARDS, PROC, and RUN. In addition, we examine several options statements to specify the followings: method for extracting factors; number of factors, rotation method, and displaying output options.

scholarly journals Principal Component Analysis (PCA) for Powder Diffraction Data: Towards Unblinded Applications

Crystals ◽  
2020 ◽  
Vol 10 (7) ◽  
pp. 581
Author(s):  
Dmitry Chernyshov ◽  
Iurii Dovgaliuk ◽  
Vadim Dyadkin ◽  
Wouter van Beek
Keyword(s):  
Principal Component Analysis ◽  
Diffraction Data ◽  
Site Occupancy ◽  
Principal Component ◽  
Real Data ◽  
Component Analysis ◽  
Gas Uptake ◽  
Evolution Characteristics ◽  
Isothermal Gas ◽  
External Stimuli

We analyze the application of Principal Component Analysis (PCA) for untangling the main contributions to changing diffracted intensities upon variation of site occupancy and lattice dimensions induced by external stimuli. The information content of the PCA output consists of certain functions of Bragg angles (loadings) and their evolution characteristics that depend on external variables like pressure or temperature (scores). The physical meaning of the PCA output is to date not well understood. Therefore, in this paper, the intensity contributions are first derived analytically, then compared with the PCA components for model data; finally PCA is applied for the real data on isothermal gas uptake by nanoporous framework γ –Mg(BH 4 ) 2 . We show that, in close agreement with previous analysis of modulation diffraction, the variation of intensity of Bragg lines and the displacements of their positions results in a series of PCA components. Every PCA extracted component may be a mixture of terms carrying information on the average structure, active sub-structure, and their cross-term. The rotational ambiguities, that are an inherently part of PCA extraction, are at the origin of the mixing. For the experimental case considered in the paper, the extraction of the physically meaningful loadings and scores can only be achieved with a rotational correction. Finally, practical recommendations for non-blind applications, i.e., what boundary conditions to apply for the the rotational correction, of PCA for diffraction data are given.

Constrained Principal Component Analysis: Various Applications

2002 ◽  
Vol 27 (2) ◽  
pp. 105-145 ◽  
Author(s):  
Michael A. Hunter ◽  
Yoshio Takane
Keyword(s):  
Principal Component Analysis ◽  
Bootstrap Method ◽  
Principal Component ◽  
Component Analysis ◽  
Psychological Research ◽  
Unified Framework ◽  
Dependent Variables ◽  
Criterion Variables ◽  
Order Structures ◽  
The Bootstrap Method

Constrained Principal Component Analysis (CPCA) is a method for structural analysis of multivariate data. It combines regression analysis and principal component analysis into a unified framework. This article provides example applications of CPCA that illustrate the method in a variety of contexts common to psychological research. We begin with a straightforward situation in which the structure of a set of criterion variables is explored using a set of predictor variables as row (subjects) constraints. We then illustrate the use of CPCA using constraints on the columns of a set of dependent variables. Two new analyses, decompositions into finer components and fitting higher order structures, are presented next, followed by an illustration of CPCA on contingency tables, and CPCA of residuals that includes assessing reliability using the bootstrap method.

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