Vittadello Sean T, Stumpf Michael P H
School of BioSciences and School of Mathematics and Statistics, The University of Melbourne, Parkville, Victoria 3010, Australia.
R Soc Open Sci. 2021 Oct 13;8(10):211361. doi: 10.1098/rsos.211361. eCollection 2021 Oct.
In many scientific and technological contexts, we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models . Here, we develop and illustrate two such approaches that allow us to compare model structures in a systematic way by representing models as simplicial complexes. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent model structure, from which the model distances are also obtained. We then expand on this measure of model distance to study the concept of model equivalence to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.
在许多科技背景下，我们对合适的数学模型的结构和细节了解甚少。因此，我们常常需要比较不同的模型。利用现有的数据，我们可以使用形式化的统计模型选择方法来比较和对比不同数学模型描述此类数据的能力。然而，目前缺乏比较不同模型的严格方法。在此，我们开发并阐述了两种这样的方法，通过将模型表示为单纯复形，使我们能够以系统的方式比较模型结构。利用单纯代数拓扑中成熟的概念，我们基于模型的单纯表示定义了模型之间的距离。采用具有平坦滤过的持久同调为模型提供了作为持久区间的替代表示，持久区间代表模型结构，模型距离也由此得出。然后，我们扩展这种模型距离度量来研究模型等价的概念，以确定模型的概念相似性。我们应用模型比较方法来证明发育生物学中位置信息模型和图灵模式模型之间的等价性，这对于两类先前被认为不相关的模型而言是一项新的发现。
