The Biplot as a Diagnostic Tool for Models of Two-Way Tables

Bradu, Dan; Gabriel, K. Ruben

doi:10.2307/1268161

Cited by 69 publications

(96 citation statements)

References 16 publications

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“…This approach is an alternating least square (ALS) algorithm that works iteratively and starts with the leading eigentriple λ 1 , u 1 , and v 1 . It starts with an initial guess of u 1 . Then a least square regression is applied to every column of M by using u 1 to get λ 1 v 1 .…”

Section: Alternating Approachmentioning

confidence: 99%

“…Then a least square regression is applied to every column of M by using u 1 to get λ 1 v 1 . At the end of the regression cycle of columns, scaling the vector estimate to unit length can give v 1 , and the scale factor is λ 1 . After getting the estimation of v 1 , the least square regression of every row of M is performed by using v 1 to refine u 1 .…”

Section: Alternating Approachmentioning

confidence: 99%

“…After getting the estimation of v 1 , the least square regression of every row of M is performed by using v 1 to refine u 1 . Then, the estimation is normalized to get the new u 1 . These steps are repeated until convergence.…”

Section: Alternating Approachmentioning

confidence: 99%

“…SVD has been applied in many methods, such as biplot [1], correspondence analysis [2], and principal component analysis. There are also many SVD applications, for example, image compression, gene data analysis, etc.…”

Section: Introductionmentioning

confidence: 99%

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Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance

Wang

2017

Data

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Abstract:The Singular Value Decomposition (SVD) is a fundamental algorithm used to understand the structure of data by providing insight into the relationship between the row and column factors. SVD aims to approximate a rectangular data matrix, given some rank restriction, especially lower rank approximation. In practical data analysis, however, outliers and missing values maybe exist that restrict the performance of SVD, because SVD is a least squares method that is sensitive to errors in the data matrix. This paper proposes a robust SVD algorithm by applying an adjustable robust estimator. Through adjusting the tuning parameter in the algorithm, the method can be both robust and efficient. Moreover, a sequential robust SVD algorithm is proposed in order to decrease the computation volume in sequential and streaming data. The advantages of the proposed algorithms are proved with a financial application.

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Section: Alternating Approachmentioning

confidence: 99%

Section: Alternating Approachmentioning

confidence: 99%

Section: Alternating Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance

Wang

2017

Data

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“…Approximation of matrices by other matrices of lower rank plays a useful role in fitting models to data (Mandel [15], [16]; Bradu and Gabriel [1]), in graphical representation of data by means of biplots (Gabriel [4], [5]), in principal component analysis (Whittle [24]) and in other mutivariate techniques. (In fact the underlying approach of S. N. Roy and his students [19], [20], has been that of studying the rank one approximation of the data matrix).…”

Section: Introductionmentioning

confidence: 99%

Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights

1979

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Evaluation of the predictive power of biplot axes to automate the construction and layout of biplots based on the accuracy of direct readings from common outputs of multivariate analyses: 1. application to principal component analysis

Alves

2012

Journal of Chemometrics

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Predictive biplots, as developed by J.C. Gower and coworkers, can be a very useful tool to aid the interpretation of the outcomes of multivariate analyses. This paper covers a statistical methodology that enables the automation of the construction of predictive biplots, as well as an R function, AutoBiplots.PCA( ), which applies the methodology to principal components analysis. A case study based on the sensory analysis of coffees is used to illustrate the methodology as well as the outputs of the R function. The method relies on the definition of a variable's mean standard predictive error, mspe, as the degree of accuracy in the process of predicting the original values from the biplots, which is compared with a predefined tolerance value (T axis ) to decide if the correspondent biplot axis is drawn in the biplot. Standard predictive errors, spe, are calculated for each unit in relation to each biplot axis in each two-dimensional plot and are compared with a predefined tolerance value (T units ) to decide which units shall be faced as outliers. The R function automates the process, enabling the user to decide on the degree of precision of the actual analysis. Besides providing a solution for the automatic production of predictive biplots, the methodology offers new insights for the interpretation of multivariate analyses outputs on the basis of a sound principle, the degree of precision of the analysis. This provides an automatic way for the selection of variables that explain latent dimensions and also helps in deciding on the number of important latent dimensions for model developments.

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The Biplot as a Diagnostic Tool for Models of Two-Way Tables

Cited by 69 publications

References 16 publications

Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance

Adjustable Robust Singular Value Decomposition: Design, Analysis and Application to Finance

Lower Rank Approximation of Matrices by Least Squares With Any Choice of Weights

Evaluation of the predictive power of biplot axes to automate the construction and layout of biplots based on the accuracy of direct readings from common outputs of multivariate analyses: 1. application to principal component analysis

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