Fuzzy Principal Component Analysis Research Articles

잡음 민감성이 향상된 주성분 분석 기법의 비선형 변형

Principal component analysis (PCA) is a well-known method for dimensionality reduction and feature extraction while maintaining most of the variation in data. Although PCA has been applied in many areas successfully, it is sensitive to outliers and only valid for Gaussian distributions. Several variants of PCA have been proposed to resolve noise sensitivity and, among the variants, improved robust fuzzy PCA (RF-PCA2) demonstrated promising results. RF-PCA, however, is still a linear algorithm that cannot accommodate non-Gaussian distributions. In this paper, a non-linear algorithm that combines RF-PCA2 and kernel PCA (K-PCA), called improved robust kernel fuzzy PCA (RKF-PCA2), is introduced. The kernel methods make it to accommodate non-Gaussian distributions. RKF-PCA2 inherits noise robustness from RF-PCA2 and non-linearity from K-PCA. RKF-PCA2 outperforms previous methods in handling non-Gaussian distributions in a noise robust way. Experimental results also support this.

잡음 민감성이 개선된 변형 퍼지 주성분 분석 기법

주성분 분석(PCA)은 차원 축소와 특징 추출을 위해 널리 사용되는 기법 중의 하나이지만 자승 오류의 사용으로 인해 잡음에 민감한 단점이 있다. 이러한 잡음 민감성을 개선하기 위해 다양한 방법이 소개되었고 그 중 improved robust fuzzy PCA(RF-PCA2)는 퍼지 소속도를 이용한 반복적 최적화 기법으로 다른 방법에 비해 우수한 성능을 보였다. 하지만 RF-PCA2 역시 국부적인 최적해에 빠질 수 있으며 그 원인 중 하나는 RF-PCA2 알고리듬이 소속도를 균일한 값으로 초기화시키기 때문이다. 또한 퍼지 소속도를 사용하고 있지만 여전히 목적함수가 자승 오류 최소화에 기초하고 있다는 사실도 그 원인이 된다. 이 논문에서는 RF-PCA2의 이러한 문제점을 개선한 RF-PCA3를 제안한다. 제안하는 알고리듬은 RF-PCA2의 목적 함수를 바탕으로 하고 있다. 여기에 PCA의 목적 함수를 추가하고 초기 소속도 값을 데이터의 분포로부터 계산함으로써 전역 최적해에 가까운 해를 얻을 수 있는 가능성을 높여준다. 이러한 사실들은 실험 결과를 통해 확인할 수 있다. Principal component analysis (PCA) is a well-known method for dimensionality reduction and feature extraction. Although PCA has been applied in many areas successfully, it is sensitive to outliers due to the use of sum-square-error. Several variants of PCA have been proposed to resolve the noise sensitivity and, among the variants, improved robust fuzzy PCA (RF-PCA2) demonstrated promising results. RF-PCA2, however, still can fall into a local optimum due to equal initial membership values for all data points. Another reason comes from the fact that RF-PCA2 is based on sum-square-error although fuzzy memberships are incorporated. In this paper, a variant of RF-PCA2 called RF-PCA3 is proposed. The proposed algorithm is based on the objective function of RF-PCA2. RF-PCA3 augments RF-PCA2 with the objective function of PCA and initial membership calculation using data distribution, which make RF-PCA3 to have more chance to converge on a better solution than that of RF-PCA2. RF-PCA3 outperforms RF-PCA2, which is demonstrated by experimental results.

Using Fuzzy Theory and Principal Component Analysis for Water Shortage Risk Assessment in Beijing, China

Using Fuzzy Theory and Principal Component Analysis for Water Shortage Risk Assessment in Beijing, China

A spatially weighted principal component analysis for multi-element geochemical data for mapping locations of felsic intrusions in the Gejiu mineral district of Yunnan, China

Principal component analysis (PCA) is frequently used in geosciences for information extraction. In many applications, masking PCA has been used to create subsets of samples or sub-areas to enhance the effect of the main objects of interest. In this paper we suggest how the representativeness of samples or pixels can be quantified using a fuzzy membership function based on fuzzy set theory. In this new method, the relative importance of pixels or samples can be taken into account using a multivariate statistical method such as PCA. A Fuzzy Masking PCA is proposed and implemented in GeoDAS GIS on the basis of a spatially weighted PCA (SWPCA). This paper introduces the mathematical treatment of the fuzzy masking PCA and follows a case study of identifying the locations of intrusive bodies from geochemical data in the Gejiu mineral district in Yunnan, China. Power-law functions based on the inverse distance from mapped felsic intrusions are applied as weighting functions in FMPCA. The results indicate that fuzzy mask PCA increases the signal-noise ratio of the component representing igneous intrusions and decreases the influence of sedimentary rocks. The areas delineated as potential areas for new intrusions (including buried intrusions) are valuable guides for Sn mineral prospecting.

잡음 민감성이 개선된 퍼지 주성분 분석

주성분 분석(PCA)은 데이터의 차원을 줄이면서 최대의 데이터 변이를 보존하는 기법으로 차원 축소나 피처 추출을 위해 널리 사용되고 있다. 하지만 PCA는 잡음에 민감한 단점이 있으며, 이러한 잡음 민감성을 해결하기 위해 여러 가지 PCA 변형이 제안되었다. 그 중 robust fuzzy PCA(RF-PCA)는 퍼지 소속도를 사용하여 잡음의 영향을 효과적으로 줄일 수 있음이 입증되었다. 하지만 RF-PCA 역시 몇 가지 문제점이 있고, 수렴성이 그 중 하나이다. RF-PCA는 소속도와 주성분을 갱신할 때 서로 다른 목적 함수를 사용하므로 수렴 속도가 느리고 구해지는 해가 국부 최적 해임을 보장하지 않는다. 이 논문에서는 RF-PCA의 문제점을 해결하기 위해 하나의 목적 함수를 이용해 소속도와 주성분을 갱신할 수 있는 방법을 제안한다. 제안한 방법, RF-PCA2는 반복 최적화를 이용함으로써 국부 최적해에 수렴함을 보장하며, RF-PCA에 비해 빠른 수렴 속도를 가지고, 잡음 민감성이 줄어든다. 이러한 사실들은 실험 결과를 통해 확인할 수 있다. Principal component analysis (PCA) is a well-known method for dimension reduction while maintaining most of the variation in data. Although PCA has been applied to many areas successfully, it is sensitive to outliers. Several variants of PCA have been proposed to resolve the problem and, among the variants, robust fuzzy PCA (RF-PCA) demonstrated promising results. RF-PCA uses fuzzy memberships to reduce the noise sensitivity. However, there are also problems in RF-PCA and the convergence property is one of them. RF-PCA uses two different objective functions to update memberships and principal components, which is the main reason of the lack of convergence property. The difference between two functions also slows the convergence and deteriorates the solutions of RF-PCA. In this paper, a variant of RF-PCA, called RF-PCA2, is proposed. RF-PCA2 uses an integrated objective function both for memberships and principal components. By using alternating optimization, RF-PCA2 is guaranteed to converge on a local optimum. Furthermore, RF-PCA2 converges faster than RF-PCA and the solutions found are more similar to the desired solutions than those of RF-PCA. Experimental results also support this.

A material supplier selection model for property developers using Fuzzy Principal Component Analysis

Nowadays, the participants in construction industry are facing pressure to meet the higher customer expectations under a tighter budget. On the other hand, an effective and efficient material supplier selection system has a significant effect in the business success of property developers. The aim of this study is to investigate a selection model based on Fuzzy Principal Component Analysis (PCA) for solving the material supplier selection problem from the perspective of property developers. First, the Triangular Fuzzy Numbers is used to quantify the decision makers' subjective judgments. Second, PCA is employed to compress the data of the selection criteria and eliminating the multicollinearity among them. Third, the linear combined score of PCA (SCORE PCA) is used to rank the suppliers. Four material purchases are used to validate the proposed selection model. The results show that the proposed model can be adopted in construction material supplier selection by the property developers.

RKF-PCA: Robust kernel fuzzy PCA

Principal component analysis (PCA) is a mathematical method that reduces the dimensionality of the data while retaining most of the variation in the data. Although PCA has been applied in many areas successfully, it suffers from sensitivity to noise and is limited to linear principal components. The noise sensitivity problem comes from the least-squares measure used in PCA and the limitation to linear components originates from the fact that PCA uses an affine transform defined by eigenvectors of the covariance matrix and the mean of the data. In this paper, a robust kernel PCA method that extends the kernel PCA and uses fuzzy memberships is introduced to tackle the two problems simultaneously. We first introduce an iterative method to find robust principal components, called Robust Fuzzy PCA (RF-PCA), which has a connection with robust statistics and entropy regularization. The RF-PCA method is then extended to a non-linear one, Robust Kernel Fuzzy PCA (RKF-PCA), using kernels. The modified kernel used in the RKF-PCA satisfies the Mercer's condition, which means that the derivation of the K-PCA is also valid for the RKF-PCA. Formal analyses and experimental results suggest that the RKF-PCA is an efficient non-linear dimension reduction method and is more noise-robust than the original kernel PCA.

Evaluation of the impacts of homocysteine and hypoxia on vascular endothelial function based on the profiling of neuro-endocrine-immunity network in rats

Homocysteine (Hcy) and hypoxia can induce vascular endothelial dysfunction. The objective of this study was to compare the three models of rat vascular endothelial dysfunction which were established by Hcy, hypoxia and Hcy plus hypoxia, respectively. The assessment of models referred to vasoactive substances and neuro-endocrine-immunity network (NEI network). Fuzzy cluster analysis (FCA) and principal components analysis (PCA) were performed to process multivariate data from NEI network (containing 17 indices). The vascular endothelial function had all been impacted in the three models, while the levels of endothelin (ET) and von Willebrand factor (vWF) in hypoxia group had no significant differences when compared with the control. In the aspect of NEI network, the expression of neurotransmitter was disturbed in the Hcy group, while the systems of endocrine and immunity were out of balance in the hypoxia group, and the dysregulation of the whole NEI network was observed in the Hcy plus hypoxia group. The relationship between endothelial function and NEI network revealed that the levels of NO, vWF and soluble intercellular adhesion molecule-1 (sICAM-1) were associated with the expression of endocrine and immunity system, while the levels of ET and sICAM-1 were associated with the expression of neurotransmitters. The results indicated that some neurotransmitters may play a central role in Hcy induced endothelial dysfunction, while some markers of endocrine and immune systems may wide influence hypoxia induced endothelial dysfunction.

Classical and fuzzy principal component analysis of some environmental samples concerning the pollution with heavy metals

In this paper two robust fuzzy principal component algorithms are explained and their robustness and efficiency are demonstrated. The efficiency of the applied procedures was illustrated on two data sets. The first one contains 125 soil samples (Northern Romania) and 13 ion concentrations (lead, copper, manganese, zinc, nicker, cobalt, chromium, cadmium, calcium, magnesium, potassium, iron and aluminum) and the second data set consists of 234 differently polluted sampling locations (East Germany) characterized by four variables: soil lead content, plant lead content, traffic density, and distance from the road. The fuzzy principal component algorithms achieved better results mainly because they are more compressible than classical PCA and very robust to outliers. Considering, for example, a three component model for the first data set, fuzzy principal component analysis-first component (FPCA-1) accounts for 73.82% of the total variance and fuzzy principal component analysis-all components (FPCA-o) 85.44%; PCA accounts only for 68.22%. The first two principal components explain 63.11% of the total variance in the case of FPCA-1 and 77.16% in the case of FPCA-o as compared to only 57.43% for PCA. As a direct consequence, PCA showed only a partial separation of samples onto the plane or in the space described by different combination of two or three principal components, whereas a much sharper differentiation of the samples, from their origin and location point of view, is observed when FPCAs are applied.

Rime samples characterization and comparison using classical and fuzzy principal components analysis

Abstract The main objective of this paper is to introduce principal component analysis and two robust fuzzy principal component algorithms as useful tools in characterizing and comparing rime samples collected in different locations in Poland (2004–2007). The efficiency of the applied procedures was illustrated on a data set containing 108 rime samples and concentration of anions, cations, HCHO, as well as pH and conductivity. The fuzzy principal component algorithms achieved better results mainly because they are more compressible than classical PCA and very robust to outliers. For example, a three component model, fuzzy principal component analysis-first component (FPCA-1) accounts for 62.37% of the total variance and fuzzy principal component analysis-orthogonal (FPCA-o) 90.11%; PCA accounts only for 58.30%. The first two principal components explain 51.41% of the total variance in the case of FPCA-1 and 79.59% in the case of FPCA-o as compared to only 47.55% for PCA. As a direct consequence, PCA showed only a partial differentiation of rime samples onto the plane or in the space described by different combination of two or three principal components, whereas a much sharper differentiation of the samples, regarding their origin and location, is observed when FPCAs are applied.

Fuzzy principal component analysis and its Kernel-based model

Principal Component Analysis (PCA) is one of the most important feature extraction methods, and Kernel Principal Component Analysis (KPCA) is a nonlinear extension of PCA based on kernel methods. In real world, each input data may not be fully assigned to one class and it may partially belong to other classes. Based on the theory of fuzzy sets, this paper presents Fuzzy Principal Component Analysis (FPCA) and its nonlinear extension model, i.e., Kernel-based Fuzzy Principal Component Analysis (KFPCA). The experimental results indicate that the proposed algorithms have good performances.

Automatic Identification of Artistic Pigments by Raman Spectroscopy Using Fuzzy Logic and Principal Component Analysis

This work offers an automatic identification system of Raman spectra of artistic pigments. The proposed methodology is based on a fuzzy logic system, and uses principal component analysis to reduce redundancies in data and the correlation operator as an index of similarity between two Raman spectra. Moreover, as sometimes pigments are used in mixtures by artist, the designed system is able to recognize binary mixtures of pigments on the basis of their Raman fingerprints.

Assessment of the impact of a phosphatic fertilizer plant on the adjacent environment using fuzzy logic

AbstractThe impact of a phosphatic fertilizer plant on the adjacent environment was examined. Selected rare earth elements, heavy metals and metalloids were determined in substrates and products, waste by-product, and grass and soil samples. Concentration gradients of elements in grass and soil samples along the southerly and easterly directions were examined and compared with the content of interior soil and grass samples, substrates, and products. Results were compared with available data on soil permissible element concentration levels. Two fuzzy principal component analysis (FPCA) methods for robust estimation of principal components were applied and compared with classical PCA. The efficiency of the new algorithms is illustrated. The investigation explored the impact of the plant on the adjacent environment. The most reliable results, in good agreement with types of samples, were produced using the FPCA-O algorithm

An advanced multivariate statistical approach to study coastal sediment data

AbstractThe present paper deals with the application of classical and fuzzy principal components analysis to a large data set from coastal sediment analysis. Altogether 126 sampling sites from the Atlantic Coast of the USA are considered and at each site 16 chemical parameters are measured. It is found that four latent factors are responsible for the data structure (“natural”, “anthropogenic”, “bioorganic”, and “organic anthropogenic”). Additionally, estimating the scatter plots for factor scores revealed the similarity between the sampling sites. Geographical and urban factors are found to contribute to the sediment chemical composition. It is shown that the use of fuzzy PCA helps for better data interpretation especially in case of outliers.

ファジィクラスタリングを用いた主成分分析における変量選択とデータマイニングへの応用

Knowledge discovery in databases (KDD) or data mining involves fitting models to or determining patterns from high dimensional data sets, and extraction of correlation rules plays an important role. This paper proposes a new approach to knowledge discovery with linear model estimation, in which principal component analysis (PCA) is performed by selecting variables. The proposed algorithm is a hybrid of fuzzy clustering and PCA based on lower rank approximation of data matrix, in which the relative responsibilities of the variables are estimated by using possibilistic constraint for memberships. The proposed algorithm is also enhanced to a local PCA model that can be used for data mining by performing both of linear model estimation and stratified sampling.

Principal component analysis versus fuzzy principal component analysis: A case study: the quality of danube water (1985–1996)

Principal component analysis (PCA) is a favorite tool in environmetrics for data compression and information extraction. PCA finds linear combinations of the original measurement variables that describe the significant variations in the data. However, it is well-known that PCA, as with any other multivariate statistical method, is sensitive to outliers, missing data, and poor linear correlation between variables due to poorly distributed variables. As a result data transformations have a large impact upon PCA. In this regard one of the most powerful approach to improve PCA appears to be the fuzzification of the matrix data, thus diminishing the influence of the outliers. In this paper we discuss and apply a robust fuzzy PCA algorithm (FPCA). The efficiency of the new algorithm is illustrated on a data set concerning the water quality of the Danube River for a period of 11 consecutive years. Considering, for example, a two component model, FPCA accounts for 91.7% of the total variance and PCA accounts only for 39.8%. Much more, PCA showed only a partial separation of the variables and no separation of scores (samples) onto the plane described by the first two principal components, whereas a much sharper differentiation of the variables and scores is observed when FPCA is applied.

Robust Fuzzy Principal Component Analysis (FPCA). A Comparative Study Concerning Interaction of Carbon—Hydrogen Bonds with Molybdenum—Oxo Bonds.

Robust Fuzzy Principal Component Analysis (FPCA). A Comparative Study Concerning Interaction of Carbon—Hydrogen Bonds with Molybdenum—Oxo Bonds.

Robust fuzzy principal component analysis (FPCA). A comparative study concerning interaction of carbon-hydrogen bonds with molybdenum-oxo bonds.

Principal component analysis (PCA) is a favorite tool in chemometrics for data compression and information extraction. PCA finds linear combinations of the original measurement variables that describe the significant variations in the data. However, it is well-known that PCA, as with any other multivariate statistical method, is sensitive to outliers, missing data, and poor linear correlation between variables due to poorly distributed variables. As a result data transformations have a large impact upon PCA. In this regard one of the most powerful approaches to improve PCA appears to be the fuzzification of the matrix data, thus diminishing the influence of outliers. In this paper we discuss a robust fuzzy PCA algorithm (FPCA). The new algorithm is illustrated on a data set concerning interaction of carbon-hydrogen bonds with transition metal-oxo bonds in molybdenum complexes. Considering, for example, a two component model, FPCA accounts for 97.20% of the total variance and PCA accounts only for 69.75%.

Fuzzy Principal Component Analysis for Fuzzy Data

Fuzzy Principal Component Analysis for Fuzzy Data

Pattern recognition in geochemical hydrocarbon exploration: A fuzzy approach

For the Swedish Deep Gas Project some 240 soil samples were collected and analyzed for trace metals and ΔC. The data were determined to not be sufficient as anomalous patterns obtained were merely reflecting underlying crystalline or Palaeozoic bedrock. Any possible patterns related to a deep-seated gas source were completely swamped; in addition, glacial transport also presented a problem in interpretation. Therefore, the ARIADNE method was applied to the data set. ARIADNE is a pattern recognition system designed for use in a variety of exploration applications, ranging from geochemical regional surveys to detailed geophysical well logging. The system's core is a fuzzy classifier that can work both on differences in location and dispersion in variable space, either combined or separately. For unsupervised classification, a preprocessor, called NARCISSOS, is used, which, by using fuzzy principal components analysis, extracts a robust background and an appropriate number of anomalous populations. Mean vectors and covariance matrices of all populations are submitted to the ARIADNE classifier. By taking advantage of different patterns emerging by using mean vectors or variance-covariance matrices when classifying in the variable space, the relative influence of transport (e.g., glacial transport) can be estimated and probable source areas also can be established. When ARIADNE was applied to the Deep Gas Project data, two anomalous populations emerged. One was strongly tied, both geographically and chemically, to the Palaeozoic ring structure circumscribing the target area, and the background reflected general chemical features of granitic bedrocks inside and outside of that structure. The second anomaly, however, was not related to any bedrock composition, but rather to structural phenomena in the bedrock. It did not show any signs of glacial dispersions, as the other anomalous group did, and its chemical signture was more or less in concordance with what is found over oil or gas fields in other more “traditional” environments.