Stability of the inverse correlation matrix. Partial ridge regression |
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| Institution: | 1. Department of Applied Mathematics, National Sun Yat-sen University, Taiwan;2. Department of Biostatistics, Columbia University, New York, NY, USA;1. Department of Paediatric # 2, Ivan Horbachevsky Ternopil State Medical University, Ternopil, Ukraine;2. Department of Paediatric and of Paediatric Surgery, Ivan Horbachevsky Ternopil State Medical University, Ternopil, Ukraine;1. Laboratorio de Parasitologia Molecular, Instituto de Pesquisas Biomédicas and Laboratorio de Biologia Parasitaria, Faculdade de Biociências da Pontifícia Universidade Catolica do Rio Grande do Sul (PUCRS), Av Ipiranga 6690, 90690-900, Porto Alegre, RS, Brazil;2. Division of Parasitic Diseases and Malaria, Centers for Disease Control and Prevention, 1600 Clifton Road, MS D-64, Bldg 23, Room 9-440, Atlanta, GA, 30329, USA;3. U.S. Food and Drug Administration, Center for Foods Safety and Applied Nutrition, Office of Applied Research and Safety Assessment, Division of Food and Environmental Microbiology, USA;1. United States Army Institute of Surgical Research, JBSA Fort Sam Houston, Texas;2. San Antonio Military Medical Center, JBSA Fort Sam Houston, Texas;1. Department of Gerontology, The Center for Research and Study of Aging, Faculty of Social Welfare and Health Sciences, University of Haifa, 199 Aba Khoushy Ave. Mount Carmel, Haifa 3498838 Israel;2. Department of Community Mental Health, Faculty of Social Welfare and Health Sciences, University of Haifa; and NATAL Center for the Treatment of Victims of Terror and War, Tel Aviv, Israel;3. NATAL Center for the Treatment of Victims of Terror and War, Tel Aviv, Israel;4. Department of Emergency Medicine, PREPARED Center for Emergency Response Research, Ben Gurion University of the Negev, Beer-Sheba, Israel |
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| Abstract: | In this paper we evaluate the stability of the inverse of a correlation matrix by studying the derivatives of each of its entries with respect to each entry of the correlation matrix. From them we deduce the derivatives of the squared length of the inverse matrix, the variance inflation factors (VIF), and the regression coefficients. To illustrate the procedure, we use a correlation matrix that has already been analyzed by Hoerl and Kennard (1970, Technometrics 12, 69–82), and, by looking at the derivatives of the squared length of the regression vector, we show that the addition of a constant to some of the diagonal entries of the matrix is sufficient for obtaining satisfying estimates of the regression coefficients. This ‘partial ridge regression’ is carried out on the previous matrix and modifies only the coefficients which are perturbed by the collinearity. |
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