Simultaneous Linear Prediction

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Abstract
Given a set of items (predictors) suppose one wishes to predict another set of items (predictands) in asimultaneous way. Such a situation may occur when the predictands are different measurable aspects of the same phenomenon. Alternatively one might wish to predict the success of an event (say a successfully performed task) which has many correlated or uncorrelated failure modes (say a set of possible mental or physical disabilities each of them by itself precluding the achievement of the said task.) In such a case a unidimensional prediction is of value only if prediction is simultaneous for all possible failure modes. A linear summarization of the predictors is suggested, which is unique and has “maximum” predictability value for all predictands simultaneously. Other summarizations or scores are found that give “maximum” explanation of residual measures on the predictands and that are uncorrelated. The set of those simultaneous linear predictions is compared to the set of the individual multiple regression predictions as used, for instance, in the same context by Horst [4] for each predictand given the original predictors. We suggest that this technique can be applied in particular to the summarization of a subset of items when the whole set of items constitutes the set of predictands.

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