Instar determination by constrained gaussian mixture models according to Dyar's rule

Despite its importance in ecological studies and pest controls, the lack of knowledge of the life cycle and the ambiguity of data challenge the accurate determination of insect nymphs regarding many insect species. Finite mixture models are often utilized to classify instars without knowing the instar number. This study derives parsimonious gaussian mixture models using parameter constraints motivated by Dyars rule. Dyars rule explains the growth pattern of larvae and nymphs of insects by assuming a constant ratio of head capsule width for every two adjacent development stages. Accordingly, every mean value of log-transformed data in each instar stage is considered a linear function, where two Dyar constants are an intercept and a slope for the instar stages, respectively, to infer the instar stage of samples. The common variance for every instar stage regarding log-transformed data can be assumed in a mixture model, as well. If valid, these assumptions will allow an efficient estimation of the model by reducing free parameters. As a result, four model hypotheses are proposed for each assumption of instar counts depending on whether these two parameter constraints are applied. After model estimation, the proposed method uses the ICL criterion to choose the optimal counts of nymphal stages, and parametric bootstrap LR tests are applied to decide the most efficient model regarding parameter constraints. The proposed method could attain the correct model settings during the simulation study. This study also discusses the interpretation of the results of real insect data sets that concord with Dyars rule or not.