{'fraction_non_alphanumeric': 0.02113821138211382, 'fraction_numerical': 0.0, 'mean_word_length': 5.124378109452737, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Optimal designs of collapsed Scheffé model

Abstract In mixture experiments, sometimes, there are some so-called collapse phenomena among its mixture components. In view of these collapse phenomena, we present a detailed discussion of the influence thereof on the response models and the structures of the corresponding optimal designs. In this paper, we first propose a set of concepts related to the collapse phenomena, including collapsed mixture models, collapsed Scheffé model, multiple mixture simplex, and direct sum measure. For the collapsed Scheffé model, we first prove a new inequality which is based on positive definite quadratic form. And applying this inequality, we prove that the optimal design of any collapsed Scheffé model can be obtained by combining the optimal designs of all its local models. Meanwhile, we also give out the coefficients of all the designs in the combinations. The relevant optimization criteria discussed in this paper include D-, A- and - optimality criteria. The conclusions of this paper clarify the optimal design structures for the collapsed Scheffé models. Applying these conclusions, a new method for constructing the optimal designs of the collapsed Scheffé models is also given.