From the literature three types of predictors for factor scores are available. These are characterized by the constraints: linear, linear conditionally unbiased, and linear correlation preserving. Each of these constraints generates a class of predictors. Best predictors are defined in terms of L?wner's partial matrix order applied to matrices of mean square error of prediction. It is shown that within the first two classes a best predictor exists and that it does not exist in the third. Copyright ? 1996 by Marcel Dekker, Inc.
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Anderson and Rubin and McDonald have proposed a correlation-preserving method of factor scores prediction which minimizes the trace of a residual covariance matrix for variables. Green has proposed a correlation-preserving method which minimizes the trace of a residual covariance matrix for factors. Krijnen, Wansbeek and Ten Berge have proposed minimizing the determinant rather than the trace of the latter covariance matrix, and offered an iterative procedure to that effect. In the present paper it is shown that the iterative procedure can be replaced by a closed-form solution. When all unique variances are strictly positive, this solution is the same as McDonald's. The solution coincides with Green's solution in certain special cases, for instance, when the factors are orthogonal.
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The assumptions of the model for factor analysis do not exclude a class of indeterminate covariances between factors and error variables (Grayson, 2003). The construction of all factors of the model for factor analysis is generalized to incorporate indeterminate factor-error covariances. A necessary and sufficient condition is given for indeterminate factor-error covariances to be arbitrarily small, for mean square convergence of the regression predictor of factor scores, and for the existence of a unique determinate factor and error variable. The determinate factor and error variable are uncorrelated and satisfy the defining assumptions of factor analysis. Several examples are given to illustrate the results.
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Since the 1970s, Caribbean reefs have transitioned from coral-dominated to algal-dominated ecosystems. The prevalence of algae reduces coral recruitment, rendering the reefs unable to recover from additional disturbances and jeopardizing crucial ecosystem services, including coastal protection, fisheries, and tourism. One of the main factors to the proliferation of algae is the scarcity of grazers, which is a result of overfishing and disease outbreaks. While fishing supports livelihoods, enhances local food security, and is an integral part of the Caribbean communities' culture, it remains a significant threat to coral reefs. Consequently, the Nature and Environmental Policy Plan (NEPP) 2020-2030, outlining conservation and restoration priorities in the Caribbean Netherlands, underscores the necessity of an integrated approach to tackle the complex challenges of coral reef restoration and fisheries development. The Saba government, and nature management organizations of Bonaire, St. Eustatius, and Saba are implementing the NEPP. Together with University of Applied Sciences Van Hall Larenstein, Wageningen University and WWF, they aim to identify novel species of native invertebrate grazers with the dual purpose of reef restoration and fisheries diversification. The Caribbean king crab (Maguimithrax spinosissimus), the West Indian sea egg (Tripneustes ventricosus), and the West Indian top shell (Cittarium pica) have been identified as potential candidates. Despite their preference to graze on macroalgae, their current densities are inadequate. Population enhancement of these species holds promise for reducing algae, promoting biodiversity, and simultaneously supporting small-scale fisheries. However, there is limited knowledge regarding the ecological effects and socio-economic potential of these grazers. The ReefGrazers project aims to assess the current densities of these herbivores around the BES islands, analyze their impacts on the reef, and evaluate their retention post-restocking. Socio-economic research will quantify current small-scale fishing practices, while market analysis will help assess the potential for the development of these novel resources as sustainable fisheries.
