Modeling Fruit Growth of ‘Triunfo’ Pear Grown in North Espírito Santo State

The objective of this study was to obtain mathematical models to estimate non-destructively the fruit mass of pear cv. ‘Triunfo’. To this end, 128 fruits from all developmental stages collected at three different times were used. Fruits were measured for maximum length (L), maximum width (W) and observed mass (OM). For the adjustment, with a sample of 100 fruits, the models first degree linear, quadratic and power were tested, in which the OM was used as the dependent variable in function of L and W. From a sample of 28 fruits, separated for this purpose the equations were validated. Thus, it indicates an equation of the quadratic model represented by EM = 36.020218 – 3.067232(W) + 0.082568(W) using from the measurement of the largest fruit width (W), as the most accurate to estimate the fruit mass of pear cv. ‘Triunfo’.


Introduction
Belonging to the Rosaceae family, the pear (Pyrus communis L.) is the third most consumed temperate fruit in Brazil, with a production of 22,108 tonnes in 2017, with standing out the states of Rio Grande do Sul (12,839 t), Santa Catarina (7,334 t), Paraná (1,341 t), São Paulo (300 t) and Minas Gerais (294 t) (IBGE, 2018). However, Brazil is unable to meet all the demand for supplying the national market, making it necessary to import a large part of the fruits consumed in the country (Oliveira et al., 2015). In part, the insufficient Brazilian production is linked to the lack of studies that seek to adapt new cultivars to regions with productive potential (Fachinello et al., 2011).
The pear tree is a plant of the temperate climate, which requires temperatures of 7.2 o C or less for a good vegetative and reproductive development (Oliveira et al., 2015). There are cultivars that require a high amount of cold hours (700 to 1200 h) and there are those with less accumulated cold hours (less than 500 h), which are more interesting for Brazilian producers (Nakasu & Faoro, 2003).
Among the cultivars with little demand for accumulated cold hours, we can highlight the cv. 'Triunfo' which presents large fruits, with an oblong shape, thick film, green color, and peel scores, the pulp is firm, granulated, with an acidic sweet flavor and of regular quality (Nakasu et al., 2007). This cultivar has been shown to have great potential for the tropical and subtropical regions of Brazil (Bettiol Neto et al., 2014;Oliveira et al., 2015). However, there are few studies that prove its productive effectiveness (Bettiol Neto et al., 2014).
In relation to the knowledge of the fruit growth patterns, this practice facilitates the decision making of the cultural treatments, allowing the adoption of appropriate techniques in the management, harvesting and post-harvesting of the fruits, in addition to indicating maturation and quality (Muianga et al., 2016). The validation was done based on 28 fruits, where the values of L and W were replaced in the proposed equations for modeling, thus obtaining the estimated mass (EM, in g). A simple linear regression model (EM = β 0 + β 1 x) was fitted for each model, where EM was the dependent variable as a function of OM as an independent variable. For each of the simple regression adjustments, the hypotheses H 0 : β 0 = 0 versus H a : β 0 ≠ 0 and H0: β 1 = 1 versus Ha: β 1 ≠ 1 were tested, using the Student's t test at 5% probability. Also determined for all equations the mean absolute error ( The best equation was defined by the following criteria: linear coefficient (β 0 ) not different from zero; slope (β 1 ) not different from one; MAE and RMSE closer to zero, and Willmott's d index (Willmott, 1981) closer to one. Statistical analyzes were performed with the aid of the R software (R Core Team, 2018), using the data package ExpDes.pt version 1.2 (Ferreira et al., 2018).

Results and Discussion
The equations resulting from the measures of the length and width of the fruits are shown in Table 1. The width, in general, presented a greater relationship with the mass of the fruits in all the models studied. The best adjustment in relation to the width may be associated with the piriform characteristic of the fruits, where there is a greater accumulation of mass in the medial-basal region, favoring the relationship between these characteristics. However, the quadratic and power model equations showed coefficient of variation values above 0.90, which according to Rojas-Lara et al. (2008) is indicative of a good correlation between the variables under study. Table 1. First degree linear model equations, quadratic and power adjusted using the observed mass (OM) in function of length (L) and width (W) and coefficient of determination (R 2 ), to estimate the mass of pear fruits cv. 'Triunfo' Based on the validation criteria (Table 2), only the quadratic model and the power model, obtained from the largest fruit width, had results not different from zero and not different from one by the Student's t test (p < 0.05) for the linear coefficients (β 0 ) and slope (β 1 ) respectively ( Table 2). The non-significance of these parameters is an excellent indicator of the accuracy of these equations, since when the observed mass is zero, the mass estimated by the model will also be zero and as there is a gradual increase in the observed mass, the estimated mass will follow the same behavior. However, the quadratic model equation presented mean absolute error (MAE) and root mean square error (RMSE) values closer to zero and d index value closer to one, in addition, this equation had a value of determination coefficient (R 2 ) slightly higher than the power model equation, being the same equation with higher values of R 2 in the modeling adjustment, thus, this model was the one that best met the statistical criteria established in this study.

Conclusion
The pear fruits of the cultivar 'Triunfo', can have their mass estimated through their greater width (W) by the quadratic model equation represented by EM = 36.020218 -3.067232(W) + 0.082568(W) 2 , in a simple, precise and non-destructive way.