Class Definition and Mixture Class Definition by Means of Construction of Convex Hull Boundaries: Application to Analysis for Animal Fat Adulteration

1989 ◽  
Vol 72 (1) ◽  
pp. 41-47
Author(s):  
Anne Thielemans ◽  
Hubert De Brabander ◽  
Desire L Massart
Keyword(s):  
Principal Component Analysis ◽  
Convex Hull ◽  
Principal Component ◽  
Component Analysis ◽  
Animal Fat ◽  
Class Definition ◽  
Component Plane ◽  
Mixture Class ◽  
Simulated Test ◽  
Definition Of

Abstract A visual classification technique based on the construction of convex hull boundaries in combination with a principal component analysis is described. This combined technique was evaluated in the situation in which a distinction has to be made between 2 pure animal fat classes and the corresponding mixture class. In the first instance, a principal component analysis is carried out to ensure the 2-dimensional and thus visual aspect of the technique. Convex hulls are then constructed in the 2-dimensional principal component plane to delimit the boundaries of the different classes to be defined. The effectiveness of the constructed hull boundaries in the definition of class-membership was investigated by means of the classification of different simulated test samples. The results show that, at least for the tested applications, the technique is valid, although some false positive classifications occur. The detection of outliers especially seemed to pose problems. Therefore, some propositions are made of how to refine the developed hull technique to enhance the classification results.

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.

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.

CONDITIONS FOR THE APPLICABILITY OF THE PRINCIPAL COMPONENT ANALYSIS TO THE CHARACTERIZATION OF THE 1/f-NOISE

2006 ◽  
Vol 06 (01) ◽  
pp. L17-L28 ◽  
Author(s):  
JOSÉ MANUEL LÓPEZ-ALONSO ◽  
JAVIER ALDA
Keyword(s):  
Principal Component Analysis ◽  
Principal Component ◽  
Random Variable ◽  
Component Analysis ◽  
Optical Systems ◽  
Data Sets ◽  
Regular Time ◽  
Pca Method ◽  
Definition Of

Principal Component Analysis (PCA) has been applied to the characterization of the 1/f-noise. The application of the PCA to the 1/f noise requires the definition of a stochastic multidimensional variable. The components of this variable describe the temporal evolution of the phenomena sampled at regular time intervals. In this paper we analyze the conditions about the number of observations and the dimension of the multidimensional random variable necessary to use the PCA method in a sound manner. We have tested the obtained conditions for simulated and experimental data sets obtained from imaging optical systems. The results can be extended to other fields where this kind of noise is relevant.

Régionalisation du régime des précipitations dans la région des Bois-francs et de l'Estrie par l'analyse en composantes principales

1998 ◽  
Vol 25 (6) ◽  
pp. 1050-1058 ◽  
Author(s):  
T O Siew-Yan-Yu ◽  
J Rousselle ◽  
G Jacques ◽  
V.-T.-V. Nguyen
Keyword(s):  
Principal Component Analysis ◽  
Principal Component ◽  
Component Analysis ◽  
Precipitation Regime ◽  
Regional Patterns ◽  
Air Masses ◽  
Orographic Effect ◽  
High Precipitation ◽  
Definition Of ◽  
Homogeneous Regions

A definition of homogeneous regions in terms of precipitation regime is achieved by the use of principal component analysis (PCA). The method has been shown to be a reliable regionalization tool even though it was applied to a territory showing rather complex physiography and high precipitation variation. Results based on the application of the PCA to the interstation correlation matrix of precipitation have indicated four distinct homogeneous regions. These regional patterns can be explained by the orographic effect and by the circulation of air masses within the study region.Key words: homogeneous regions, rainfall, principal component analysis, orographic effect.

Convexification of Permutation-Invariant Sets and an Application to Sparse Principal Component Analysis

2021 ◽  
Author(s):  
Jinhak Kim ◽  
Mohit Tawarmalani ◽  
Jean-Philippe P. Richard
Keyword(s):  
Principal Component Analysis ◽  
Convex Hull ◽  
Principal Component ◽  
Component Analysis ◽  
Invariant Sets ◽  
Alternative Proof ◽  
Sparse Principal Component Analysis ◽  
Cardinality Constraint ◽  
Semidefinite Programming Relaxation ◽  
Invariant Norm

We develop techniques to convexify a set that is invariant under permutation and/or change of sign of variables and discuss applications of these results. First, we convexify the intersection of the unit ball of a permutation and sign-invariant norm with a cardinality constraint. This gives a nonlinear formulation for the feasible set of sparse principal component analysis (PCA) and an alternative proof of the K-support norm. Second, we characterize the convex hull of sets of matrices defined by constraining their singular values. As a consequence, we generalize an earlier result that characterizes the convex hull of rank-constrained matrices whose spectral norm is below a given threshold. Third, we derive convex and concave envelopes of various permutation-invariant nonlinear functions and their level sets over hypercubes, with congruent bounds on all variables. Finally, we develop new relaxations for the exterior product of sparse vectors. Using these relaxations for sparse PCA, we show that our relaxation closes 98% of the gap left by a classical semidefinite programming relaxation for instances where the covariance matrices are of dimension up to 50 × 50.

scholarly journals Convex Formulations for Fair Principal Component Analysis

2019 ◽  
Vol 33 ◽  
pp. 663-670 ◽  
Author(s):  
Matt Olfat ◽  
Anil Aswani
Keyword(s):  
Principal Component Analysis ◽  
Principal Component ◽  
Component Analysis ◽  
Health Data ◽  
Kernel Pca ◽  
Semidefinite Programs ◽  
Data Points ◽  
Insurance Rates ◽  
Definition Of ◽  
Reduced Data

Though there is a growing literature on fairness for supervised learning, incorporating fairness into unsupervised learning has been less well-studied. This paper studies fairness in the context of principal component analysis (PCA). We first define fairness for dimensionality reduction, and our definition can be interpreted as saying a reduction is fair if information about a protected class (e.g., race or gender) cannot be inferred from the dimensionality-reduced data points. Next, we develop convex optimization formulations that can improve the fairness (with respect to our definition) of PCA and kernel PCA. These formulations are semidefinite programs, and we demonstrate their effectiveness using several datasets. We conclude by showing how our approach can be used to perform a fair (with respect to age) clustering of health data that may be used to set health insurance rates.

scholarly journals Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

2006 ◽  
Vol 38 (2) ◽  
pp. 299-319 ◽  
Author(s):  
Stephan Huckemann ◽  
Herbert Ziezold
Keyword(s):  
Principal Component Analysis ◽  
Riemannian Manifolds ◽  
Principal Component ◽  
Component Analysis ◽  
Intrinsic Metric ◽  
Shape Spaces ◽  
Intrinsic Mean ◽  
Pass Through ◽  
Definition Of ◽  
Euclidean Mean

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

scholarly journals Comparison of dimension reduction methods on fatty acids food source study

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Yifan Chen ◽  
Yusuke Miura ◽  
Toshihiro Sakurai ◽  
Zhen Chen ◽  
Rojeet Shrestha ◽  
...  
Keyword(s):  
Fatty Acids ◽  
Principal Component Analysis ◽  
Factor Analysis ◽  
Principal Component ◽  
Component Analysis ◽  
Food Sources ◽  
Fat Intake ◽  
Plant Oil ◽  
Animal Fat ◽  
Lipid Fractions

AbstractSerum fatty acids (FAs) exist in the four lipid fractions of triglycerides (TGs), phospholipids (PLs), cholesteryl esters (CEs) and free fatty acids (FFAs). Total fatty acids (TFAs) indicate the sum of FAs in them. In this study, four statistical analysis methods, which are independent component analysis (ICA), factor analysis, common principal component analysis (CPCA) and principal component analysis (PCA), were conducted to uncover food sources of FAs among the four lipid fractions (CE, FFA, and TG + PL). Among the methods, ICA provided the most suggestive results. To distinguish the animal fat intake from endogenous fatty acids, FFA variables in ICA and factor analysis were studied. ICA provided more distinct suggestions of FA food sources (endogenous, plant oil intake, animal fat intake, and fish oil intake) than factor analysis. Moreover, ICA was discovered as a new approach to distinguish animal FAs from endogenous FAs, which will have an impact on epidemiological studies. In addition, the correlation coefficients between a published dataset of food FA compositions and the loading values obtained in the present ICA study suggested specific foods as serum FA sources. In conclusion, we found that ICA is a useful tool to uncover food sources of serum FAs.

scholarly journals Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces

2006 ◽  
Vol 38 (02) ◽  
pp. 299-319 ◽  
Author(s):  
Stephan Huckemann ◽  
Herbert Ziezold
Keyword(s):  
Principal Component Analysis ◽  
Riemannian Manifolds ◽  
Principal Component ◽  
Component Analysis ◽  
Intrinsic Metric ◽  
Shape Spaces ◽  
Intrinsic Mean ◽  
Pass Through ◽  
Definition Of ◽  
Euclidean Mean

Classical principal component analysis on manifolds, for example on Kendall's shape spaces, is carried out in the tangent space of a Euclidean mean equipped with a Euclidean metric. We propose a method of principal component analysis for Riemannian manifolds based on geodesics of the intrinsic metric, and provide a numerical implementation in the case of spheres. This method allows us, for example, to compare principal component geodesics of different data samples. In order to determine principal component geodesics, we show that in general, owing to curvature, the principal component geodesics do not pass through the intrinsic mean. As a consequence, means other than the intrinsic mean are considered, allowing for several choices of definition of geodesic variance. In conclusion we apply our method to the space of planar triangular shapes and compare our findings with those of standard Euclidean principal component analysis.

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