Research On Weighted Efficiency Optimal Design For Experiments With Mixtures

The experiment is an effective way to obtain data and an essential means for exploring and discovering the relationship,law,or nature between objectives.Scientific experimental design effectively obtains experimental information and statistical laws with as few numbers of experiments as possible.Optimal mixture experimental design is an essential topical research problem that applies optimal design theory to mixture experiments.Its purpose is to propose the optimality criterion for measuring the quality of different experimental designs from the statistical point of view,and to obtain the optimal experimental design for a given mixture model.The design proposed based on optimal design theory will be optimal if the form of the underlying mixture model is known.However,the optimal design may not provide the best experimental design or may have poor performance in parameter estimation or response prediction variance if the form of the mixture model is unknown at the design stage or there are deviations from the true model.To improve the theory and methodology of optimal mixture experimental design,this thesis proposes a weighted efficiency optimal design from the point of view of design efficiency for the cases where the form of the mixture model is known or unknown.On the other hand,a lattice point partition design is proposed from the perspective of uniformity of the distribution of experimental points for the case where the model is unknown.First,a particular class of permutation matrices for optimal mixture experiments is defined,and gives a series of related properties of the permutation matrix.Meanwhile,the R-optimal designs of the q-component Becker models H2 and H3 are investigated from the perspective of saturated design,and the equivalence of the R-optimal designs of the models H2 and H3 under the weighted simplex centroid design is discussed,but the saturated R-optimal allocations do not satisfy the general equivalence theorem.This thesis constructs a class of non-saturated R-optimal designs by increasing the number of experimental points and providing specific numerical results.It is verified by examples that this design has less efficiency loss compared to the A-optimal design and the D-optimal design.Secondly,the basic theory of weighted R-efficiency optimal design is studied,mainly including the weighted R-efficiency optimal design method and the general equivalence theorem.In addition,the construction method of the weighted Refficiency optimal design for the first-order Scheffe and second-order Kronecker models is discussed.It is also verified by numerical simulations that the weighted R-efficiency optimal design has less efficiency loss and is more robust.Furthermore,two different methods for constructing weighted efficient optimal designs based on design efficiency are investigated.The methods for constructing the weighted efficiency optimal designs with D-and I-optimality and with R-and I-optimality are presented,which include the corresponding criterion functions,general equivalence theorems,and search algorithms for determining the optimal weighting parameters.Meanwhile,two types of weighted efficiency optimal designs are constructed for the mixture experimental problem without and with additional constraints,respectively.Comparing the two types of weighted efficiency optimal designs,D-,R-and I-optimal designs from the three different aspects,that is,the efficiency of the designs,the variation of the support points,and the prediction variance on the simplex,we find that the constructed weighted efficiency optimal designs have less efficiency loss and the prediction variance is relatively small.Finally,a class of lattice point partition designs is constructed for the case of the unknown model from the perspective of uniformity of design point dispersion,and the lattice point partition methods are given for three different mixture constraint regions.In addition,the analytical expressions of mean square error discrepancy(MSED)and maximum distance discrepancy(MD)are given under the standard sub-simplexes,and a class of lattice point partition designs is constructed based on the vertices of the sub-simplexes.It is also shown that the validity of the design by examples,and which has smaller overall response prediction variance and less of efficiency loss.