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AbstractIn numerical analysis, sparse grids are point configurations used in stochastic finite element approximation, numerical integration and interpolation. This paper is concerned with the construction of polynomial interpolator models in sparse grids. Our proposal stems from the fact that a sparse grid is an echelon design with a hierarchical structure that identifies a single model. We then formulate the model and show that it can be written using inclusion–exclusion formulæ. At this point, we deploy efficient methodologies from the algebraic literature that can simplify considerably the computations. The methodology uses Betti numbers to reduce the number of terms in the inclusion–exclusion while achieving the same result as with exhaustive formulæ.

PROCEDURES FOR EMPIRICAL TESTS OF POLITICAL THEORY SHOULD BE DESIGNED TO EVALUATE THE SUCCESS OF THE SPECIFIC PREDICTION BEING TESTED. THIS PAPAER INTRODUCES(1) A FORMAL LANGUAGE, "PREDICTION LOGIC"; (2)A POPULATION MEASURE, (3) PARTIAL MEASURES FOR THE MULTIVARIATE CASE; (4)BIVARIATE STATISTICAL INFERENCE MEASURES WHEN THE DATA ARISE FROM A SAMPLE RATHER THAN A POPULATION.