Lasso Model for Morphological Covariation Patterns between Colossoma macropomum and Piaractus orinoquensis × Colossoma macropomum Hybrid

Manuel Milla Pino, Danny Villegas Rivas*, César Osorio Carrera, Nancy Carruitero Avila, Teresita Merino Salazar, Henry Díaz Merino, Carola Calvo Gastañaduy, Ricardo Shimabuku Ysa, Juan De La Cruz Lozado, River Chávez Santos, Dora Calvo Gastañaduy, Lillet Villavicencio Palacios

Abstract

Currently, great importance has been given to the study of external morphology, especially in fish, when it is used as a means of identifying hybrids. This paper considers a LASSO model based on the truss protocol to compare morphological covarion patterns between specimens of C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀). In this study, 25 specimens of C. macropomum and 20 specimens of the hybrid P. orinoquensis (♂) × C. macropomum (♀), were analyzed, respectively. The method "Truss protocol" or "trusses" was used. LASSO model achieved to reduce the mean squared error. The final model obtained contains only seven covariates. LASSO model fitted on the morphological covariation patterns between specimens of C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀) showed a good fit and allowed to correctly classify most of the specimens. Differences were observed in the area of the head and the anterior part of the fish evidenced in covariates associated with hydrodynamic abilities and with foraging.

Keywords: Morphometry, Truss protocol, Fishes, Lambda, Shrinkage regression

Introduction

The family Characidae is the most diverse family of freshwater fish species in South America (Diachkova et al., 2019; Nurmayanti et al., 2019). The implementation of morphometric analysis in some species provides scientific knowledge that helps genetic improvement. The morphological characters are physical evidence of the expression of the genotype. Therefore, the differences between specific body characteristics can become very important to establish patterns of differentiation and inheritance (Lazzarotto et al., 2017). In continental fish, the morphometric characteristics referring to the anatomical shape have been used to evaluate the productive response in rearing both in natural environments and in captivity. Currently, there are more modern and precise morphometric analysis techniques, such as geometric morphometry (Bookstein et al., 1985), which together with multivariate statistical analysis and means of direct visualization, constitute one of the most useful tools to describe the biological form and its changes.

Generally, these techniques are based on a set of measured distances between identifiable points on the organisms. In most cases, the measurements (distances between homologous points) present a high correlation, which is exploited in the models that are frequently used to compare between species. However, a variable selection model has desirable requirements: accurate predictions and interpretable models, and stability, that is, small changes in the data should not cause large changes in the predictors used. Traditional methods of variable selection, such as ridge regression, all subsets regression, or stepwise regression, fail one or more of the above requirements. The LASSO regression models (Hastie et al., 2015) are based on the multiple linear models and seek to achieve its “regularization”. Although LASSO works successfully on many occasions, it has some limitations, which can be solved with the model known as Elastic Net (Ramos, 2018). In this sense, this paper considers the LASSO model based on the truss protocol to compare morphological covarion patterns between specimens of C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀) when p > n, that is, we have more variables than observations using lars and glmnet package in R.

Materials and Methods

Morphological Covariation Patterns between C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀)

In this study, 25 adult specimens of C. macropomum and 20 adult specimens of the hybrid P. orinoquensis (♂) × C. macropomum (♀) with an average weight of 600g, from artificial ponds of a fish farm in Portuguesa state, Venezuela, were analyzed. Within the sample of each species, there are mixed male and female individuals. The method "Truss protocol" or "trusses" (Strauss and Bookstein, 1982) was used, which achieves an exhaustive reconstruction of the shape from the distances between the homologous anatomical landmarks (Table 1 and Figure 1). The distances connecting these landmarks form a series of continuous quadrilaterals with their respective internal diagonals (Figure 1), which allows detecting differences in shape in the vertical, horizontal, and oblique directions. The limitation of this study is the number of measures necessary to achieve better efficiency in estimating parameters related to the morphology of these species.

Table 1. Truss Measurements from C. macropomum and the Hybrid P. orinoquensis (♂) × C. macropomum (♀) Specimens

The morphological covariation patterns between specimens of C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀) were studied using LASSO models in the R package (Team, 2020).

The LASSO Method

This method combines a regression model with a procedure for contracting some parameters towards zero and selecting variables, by imposing a restriction or penalty on the regression coefficients.

Below is a formulation of Lasso as an optimization problem (for details see Ramos, 2018):

Suppose we have the data , where  t are the predictor variables and  are the responses. We can consider that the  are standardized, that is,

or in other words, they have zero mean and variance 1. If the previous condition is not verified, it is enough to classify the variables as part of the preprocessing.

If we denote , the estimate of lasso  is defined as the optimal solution to the optimization problem:

where  is a fitting parameter.

Fixed  that satisfies , optimize in is a differentiable optimization problem in a variable, whose optimality condition is gradient equal to zero.

Prediction and Estimation of the Parameter t

We estimate the prediction error for the LASSO using cross-validation with k-folds.

If we call

where  are the least-squares estimators, and we vary s in a sufficiently small interval, between 0 and 1, for each value of s or respectively of t, we obtain by cross-validation an estimator , of mean square prediction error. We thus determine , the value of t with smaller , and this is the parameter considered.

Algorithms to Find Solutions

Once we have obtained an estimate of t, which we will call , we proceed to solve the optimization problem;

We observe that the previous problem has p variables, since , and a constraint; we can transform this restriction into  linear restrictions:

The previous problem is a convex quadratic optimization problem with  linear constraints. It is possible to obtain an equivalent formulation with a linear number in p of constraints, expanding the number of variables. For this, we make the change:

This problem has  variables since , and 2p + 1 constraints.

The Lars Package in R

Computes the prediction error of cross-validated K-fold mean squared for Forward Stagewise, LASSO, or LARS. For details, see Hastie and Efron (2013).

Results and Discussion

Table 2 and Figure 2 show the fit of the LASSO model on patterns of morphological covariation between C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀)., where the covariates (landmarks distances): eye diameter (X27), anterior edge of the epiphyseal sulcus to the end of the epiphyseal sulcus (X4), posterior edge of the epiphyseal sulcus to the insertion of the pectoral fin (X10), insertion of the pectoral fin to insertion of pelvic fin (X12), posterior edge of the dorsal fin to the fatty fin (X18), anterior edge of the dorsal fin to insertion of pectoral fin (X16) and posterior edge of the dorsal fin to the posterior edge of anal fin (X19) were included in the model, suggesting there are characteristics associated with the morphological covariation patterns that allow differentiation between redundant specimens of C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀). These covariates are associated with morphological covariation patterns that make a difference in the head area and the anterior part of the fish. These covariates are characteristics associated with hydrodynamic abilities and the foraging for food. Figure 3 shows how lasso achieves, using their respective optimal values of λ, to reduce the MSE. The advantage of the final model obtained by lasso is that it is much simpler since it contains only seven covariates. These results coincide with those reported by Perdomo et al. (2017) who compared the morphometry of two continental fish species raised in Trujillo state, Venezuela, and those reported by Villegas et al. (2020a) in a multivariate analysis that allowed a morphometric comparison of a hybrid originated from C. macropomum and P. orinoquensis, and those reported by Villegas et al. (2020b) when studying the redundancy in morphological covariation patterns between C. macropomum and P. orinoquensis. However, the results differ from those indicated by Villegas et al. (2020c) when using a multiple logistic model to study the morphological covariation patterns between the mentioned species. The foregoing reveals what was indicated by Porras-Rivera and Rodríguez-Pulido (2019) and Conte-Grand et al. (2015), who pointed out that external morphology is not always reliable when used as the only means of identification, particularly for hybrid individuals beyond the first generation.

Table 2. LASSO Model Fitted on Morphological Covariation Patterns between C. macropomum and the Hybrid P. orinoquensis (♂) × C. macropomum (♀).

Conclusion

LASSO model achieved, using their respective optimal values of λ, to reduce the mean squared error. The final model obtained by LASSO it was much simpler since it contains only seven covariates. LASSO model fitted on the morphological covariation patterns between specimens of C. macropomum and the hybrid P. orinoquensis (♂) × C. macropomum (♀) showed a good fit and allowed to correctly classify most of the specimens. Differences were observed in the area of the head and the anterior part of the fish between the hybrid and its parent. The morphological differences between these two species were evidenced in covariates associated with hydrodynamic abilities and with foraging. Finally, the results of this research suggest the use of the LASSO model to compare morphological covariation patterns between the hybrid P. orinoquensis (♂) × C. macropomum (♀) and P. orinoquensis when the sample size is less than the number of landmarks (n < p).

Acknowledgments: We thank all the authors for their contribution to this research.

Conflict of interest: Authors have declared that no competing interests exist.

Financial support: The research was financed with own resources

Ethics statement: As per international standards or university standard, ethical approval has been collected and preserved by the authors.

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