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THE INTERNET ENCYCLOPAEDIA OF
PERSONAL CONSTRUCT PSYCHOLOGY |
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| Main Page |
| Contents |
| Alphabetical Index |
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| Impressum |
| Singular-value-decomposition |
| This technique is allied to both principal components and correspondence analysis. It was originally devised by Eckart and Young in 1936 (and is sometimes known as Eckart-Young decomposition) who showed that a matrix (such as a repertory grid) can be approximated by another matrix of lower rank (rank being a property of matrices akin to number of dimensions or factors need to account for the data in the matrix). This is the most widely known method of representing both constructs and elements in a single analysis and was originally introduced by Patrick Slater who confusingly promulgated it initially as ‘The Principal Components of a Repertory Grid’ (Slater, 1964) although it is now more commonly known as INGRID. The singular-values are the square roots of the eigenvalues of the cross-products matrix. The procedure produces two sets of eigenvectors, (see principal components for a brief definition of eigenvalues and eigenvectors) one for the rows of the matrix and one for the columns. The row (construct) eigenvectors can be multiplied by the square root of the singular values to give loadings for the constructs which reflect the importance of the ‘factor’ (in terms of variance accounted for), and the column (element) eigenvectors similarly rescaled. The two loading matrices may be then be multiplied together to produce an approximation of the original matrix (grid) which is an advantage over principal components and which only reproduces correlations. Another advantage is that a singular value decomposition can be carried out when there is no variation in a row or column of the grid. Singular-value-decomposition analysis is usually accompanied by some form of pre-scaling of the grid to remove statistical artefacts. For example, INGRID removes the effect of (row) construct means to avoid them influencing the derived loadings. Other approaches suggest double-centring, or removing both element and construct means. |
| References |
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Richard C. Bell
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