• Journal of the Korean Data and Information Science Society
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• v.25 no.2
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• pp.423-429
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• 2014

This paper discusses a method for getting Type I sums of squares by projections under a two-way fixed-effects model when variances of errors are not equal. The method of weighted least squares is used to estimate the parameters of the assumed model. The model is fitted to the data in a sequential manner by using the model comparison technique. The vector space generated by the model matrix can be composed of orthogonal vector subspaces spanned by submatrices consisting of column vectors related to the parameters. It is discussed how to get the Type I sums of squares by using the projections into the orthogonal vector subspaces.

• Journal of the Korean Data and Information Science Society
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• v.25 no.4
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• pp.799-805
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• 2014

This paper deals with a method for getting the Type III sums of squares on the basis of projections under the assumption of two-way fixed effects model. For unbalanced data in general total sum of squares is not equal to the sum of componentwise Type III sums of squares. There are some differencies between two quantities. The suggested method using projections can detect where the differences occur and how much they are different. The traditional ANOVA method could not explain clearly the differences. It also discusses how eigenvectors and eigenvalues of the projection matrices can be used to get the Type III sums of squares.

• Investigative Magnetic Resonance Imaging
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• v.9 no.1
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• pp.36-42
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• 2005

Purpose : To develop a novel approach to calculate the sensitivity profiles of the phased array coil for use in non-uniform intensity correction. Materials and Methods : The proposed intensity correction method estimates the sensitivity profile of the coil to extract intensity variations that represent the scanned image. The sensitivity profile is estimated by fitting a non-linear curve to various angles of projections through the imaged object in order to eliminate the high-frequency image content. Filtered back projection is then used to compute the estimates of the sensitivity profile of each coil. The method was applied both to phantom and brain images from 8-channel phased-array coil and 4-channel phased-array coil, respectively. Results : Intensity-corrected images from the proposed method have more uniform intensity than those from the commonly used 'sum-of-squares' approach. By using the proposed correction method, the intensity variation was reduced to $6.1\%$ from $13.1\%$, acquired from the 'sum-of-squares'. Conclusion : The proposed method is more effective at correcting the intensity non-uniformity of the phased-array surface-coil images than the conventional 'sum-of-squares' method.

• Communications for Statistical Applications and Methods
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• v.13 no.2
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• pp.233-241
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• 2006

Classical principal component analysis (PCA) can be formulated as finding the linear subspace that best accommodates multidimensional data points in the sense that the sum of squared residual distances is minimized. As alternatives to such LS (least squares) fitting approach, we produce LMS (least median of squares) and LTS (least trimmed squares)-type PCA by minimizing the median of squared residual distances and the trimmed sum of squares, in a similar fashion to Rousseeuw (1984)'s alternative approaches to LS linear regression. Proposed methods adopt the data-driven optimization algorithm of Croux and Ruiz-Gazen (1996, 2005) that is conceptually simple and computationally practical. Numerical examples are given.

• Communications for Statistical Applications and Methods
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• v.27 no.5
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• pp.523-533
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• 2020

This paper suggests three available methods for finding nonnegative estimates of variance components of the random effects in mixed models. The three proposed methods based on the concepts of projections are called projection method I, II, and III. Each method derives sums of squares uniquely based on its own method of projections. All the sums of squares in quadratic forms are calculated as the squared lengths of projections of an observation vector; therefore, there is discussion on the decomposition of the observation vector into the sum of orthogonal projections for establishing a projection model. The projection model in matrix form is constructed by ascertaining the orthogonal projections defined on vector subspaces. Nonnegative estimates are then obtained by the projection model where all the coefficient matrices of the effects in the model are orthogonal to each other. Each method provides its own system of linear equations in a different way for the estimation of variance components; however, the estimates are given as the same regardless of the methods, whichever is used. Hartley's synthesis is used as a method for finding the coefficients of variance components.

• Journal of the Korean Data and Information Science Society
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• v.26 no.1
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• pp.31-39
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• 2015

This paper deals with a method for estimating variance components on the basis of projections under the assumption of random effects model. It discusses how to use projections for getting sums of squares to estimate variance components. The use of projections makes the vector subspace generated by the model matrix to be decomposed into subspaces that are orthogonal each other. To partition the vector space by the model matrix stepwise procedure is used. It is shown that the suggested method is useful for obtaining Type I sum of squares requisite for the ANOVA method.

• The Korean Journal of Applied Statistics
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• v.28 no.2
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• pp.151-158
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• 2015

This paper discusses how to use projections for the analysis of data from balanced incomplete block designs. A model is suggested as a matrix form for the interblock analysis. A second set of treatment effects can be found by projections from the suggested interblock model. The variance and covariance matrix of two estimated vectors of treatment effects is derived. The uncorrelation of two estimated vectors can be verified from their covaraince structure. The fitting constants method is employed for the calculation of block sum of squares adjusted for treatment effects.