Classification with discriminant analysis

In discriminant analyses which were conducted separately for the three major axes, the existences of the variables fitting into multiple normal distribution, the variance-covariance matrices being equal and not having multiple connection problems were tested. The data used in the study showed multivariate normal distribution. The variance-covariance matrices were tested with Box-M test. Quadratic discriminant function was used since the variance-covariance matrices were not equal (Tabachnick & Fidell, 2007). In order to test whether there were multi linear connection problems, correlation matrix consisting of the correlation coefficients between variables were calculated (Alpar, 2017). The correlation coefficient of hazelnut surface area was found to be well above 0.90; therefore, this variable was excluded from the independent variables (Alpar, 2017). In discriminant analysis, the most effective variables for separating the varieties were found to be sphericity, grain volume, geometric mean diameter and colour parameters, while the least effective ones were found to be deformation, rupture energy, rupture force and grain weight variables.

For all three main axes, the significance level of independent variable groups was checked with equality of group means test. Wilks’ Lambda and F values were considered for these tests. Smaller Wilks’ Lambda values and greater F values indicate that the contribution of the independent variables to the dependent variables is higher. All of the independent variables were found to be statistically significant (p<0.05) (Güzeller, 2016).

 

Classification with discriminant analysis for X axis (DAX)

Fig. 4 shows the basic elements of the architecture combing the field operation with a cloud service. We chose an architecture, which setup was open for a cloud environment. This environment, called the backend, was doing data processing and could be separated into a dataand a function-backend. It also offered a function for image interpretation as a service through a standardized interface. The service was used by network-access, sending requests including the specific parameters of the actual situation of operation. It was suitable for the objective to integrate innovations and knowledge into the environment with an importable function. The function was offered as a service; therefore, we used the term of FaaS.

Eigenvalues of discriminant functions, canonical correlation and Wilks’ Lambda values were calculated. Eigenvalues of the functions were found to range between 0.014 and 11.474. Eigenvalues greater than 0.40 can be considered to be good (Kalaycı, 2018). Greater eigenvalue expresses that more of the variance in the dependent variable can be explained by the function. The first function explained 57.5% of the total variance. Canonical correlation values of the functions were between 0.118 and 0.959 and the square of these values ranged between 0.014 and 0.92. The square of canonical correlation value shows the variance that the function can explain in the dependent variable. Wilks’ Lambda values, which show the part of the functions that cannot be explained by the differences between groups of the total variance, were between 0.001 and 0.986. These values indicate the significance of the eigenvalue statistics for each discriminant function (Kalaycı, 2018).

In order to evaluate the accurate classification of discriminant analysis, that is, the success of the analysis, the classification table of training and test sets was obtained (Table 3). According to the classification results obtained with DAX model, the discriminant function that was chosen for the training set classified 397 of the 426 samples accurately into their varieties, while 29 samples were classified inaccurately. The general accurate classification rate of DAX model was found as 93.2%. On the other hand, DAX model classified 51 of the 55 samples in test set to their varieties accurately, while 4 samples were classified inaccurately. Its general accurate classification rate was found to be 92.7%.

 

 

Table 3. Classification results obtained with DAX, DAY, and DAZ models

y1:’İncekara’, y2:’Kalınkara’, y3:’Kan’, y4:’Kuş’, y5:’Okay28’, y6:’Palaz’, y7:’Sivri’, y8:’Tombul’, y9:’Uzunmusa’, 10:’Yassı Badem’, y11:’Yuvarlak Badem’.

 

 

Table 4. Classification results obtained with ANNX, ANNY, and ANNZ models

y1:’İncekara’, y2:’Kalınkara’, y3:’Kan’, y4:’Kuş’, y5:’Okay28’, y6:’Palaz’, y7:’Sivri’, y8:’Tombul’, y9:’Uzunmusa’, 10:’Yassı Badem’, y11:’Yuvarlak Badem’.

 

 

Classification with discriminant analysis on Y axis (DAY)

Eigenvalues of the discriminant functions were found to range between 0.005 and 12.397. The first function explained 54.6% of the total variance. Canonical correlation values of the functions were between 0.071 and 0.962 and the square of these values ranged between 0.005 and 0.93. Wilks’ Lambda values were between 0.0003 and 0.995 (Kalaycı, 2018).

According to the classification results obtained with DAY model, the discriminant function that was chosen for the training set classified 403 of the 421 samples accurately into their varieties, while 18 samples were classified inaccurately. The general accurate classification rate of DAY model was found as 95.7%. On the other hand, DAY model classified 51 of the 55 samples in test set to their varieties accurately, while 4 samples were classified inaccurately. Its general accurate classification rate was found to be 92.7%. (Table 3).

 

Classification with discriminant analysis on Z axis (DAZ)

Eigenvalues of the discriminant functions were found to range between 0.007 and 11.910. The first function explained 57.7% of the total variance. Canonical correlation values of the functions were between 0.081 and 0.960 and the square of these values ranged between 0.006 and 0.92. Wilks’ Lambda values, which show the part of the functions that cannot be explained by the differences between groups of the total variance, were between 0.001 and 0.993 (Kalaycı, 2018).

According to the classification results obtained with DAZ model, the discriminant function that was chosen for the training set classified 399 of the 419 samples accurately into their varieties, while 20 samples were classified inaccurately. The general accurate classification rate of DAZ model was found as 95.2%. On the other hand, DAZ model classified 47 of the 53 samples in test set to their varieties accurately, while 6 samples were classified inaccurately. Its general accurate classification rate was found to be 88.7% (Table 3).

 

Classification with artificial neural networks

Since the network structure used in ANN applications is similar for three main axes, they were evaluated together. In the study, the best results were obtained by using feed forward back propagation network structure and Levenberg-Marquardt training algorithm. In addition, hyperbolic sigmoid tangent function was chosen for the hidden layer and softmax function was chosen for the output layer as transfer functions. While the networks were trained, it was ensured that the training was terminated before overfitting occurred by controlling it with the verification set. The success of the network was controlled with the test set. The best training results were obtained

from the networks having 60, 60 and 40 neuron numbers, respectively for X, Y and Z axes.

 

Classification with artificial neural networks on X axis (ANNX)

According to the classification results obtained with ANNX model, 419 of the 426 hazelnut samples were classified accurately into their varieties and 7 samples were classified inaccurately. Its general accurate classification rate was found to be 98.4%. By using the test set, 49 of 55 hazelnut samples were classified accurately into their varieties, while 6 samples were classified inaccurately. Its general accurate classification rate was found to be 89.1%. The samples belonging to ‘İncekara’, ‘Kuş’, ‘Okay28’, ‘Sivri’, ‘Tombul’ and ‘Yuvarlak Badem’ hazelnut varieties were classified accurately (Table 4).

 

Classification with artificial neural networks on Y axis (ANNY

According to the classification results obtained with ANNY model, 418 of the 421 hazelnut samples were classified accurately into their varieties and 3 samples were classified inaccurately. Its general accurate classification rate was found to be 99.3%. By using the test set, 51 of 55 hazelnut samples were classified accurately into their varieties, while 4 samples were classified inaccurately. Its general accurate classification rate was found to be 92.7%. While one sample belonging to ‘Kuş’ variety was classified inaccurately, three samples belonging to ‘Kalınkara’ variety was classified inaccurately. All the other samples were classified accurately into their varieties (Table 4).

 

Classification with artificial neural networks on Z axis (ANNZ)

According to the classification results obtained with ANNZ model, 413 of the 419 hazelnut samples were classified accurately into their varieties and 6 samples were classified inaccurately. Its general accurate classification rate was found to be 98.6%. By using the test set, 46 of 53 hazelnut samples were classified accurately into their varieties, while 7 samples were classified inaccurately. Its general accurate classification rate was found to be 86.8%. While all samples belonging to ‘İncekara’, ‘Kalınkara’, ‘Okay28’, ‘Yassı Badem’ and ‘Yuvarlak Badem’ varieties were classified accurately, one sample belonging to ‘Kan’, ‘Kuş’, ‘Palaz’, ‘Tombul’ and ‘Uzunmusa’ was classified inaccurately and two samples belonging to ‘Sivri’ were classified inaccurately (Table 4).

Variety classification is required for many different purposes such as machine design, evaluation of consumer preference, cultivar identification for cultivar registrations, investigation of heritability of fruit shape traits, analysis of shape abnormalities (Kays, 1999; Cannon & Manos, 2001; Beyer et al., 2002; Brewer et al., 2007; Taner et al., 2021).

Similar results were obtained with both ANN and discriminant analysis models for the variety classification of hazelnuts. The reason for obtaining these similar results with both models can be due to processing data used in this study with multivariate normal distribution analyses before fitting the models. In fact, ANN can be more successful in nonlinear data according to the literature (Ford et al., 2004).

In previous studies, hazelnut cultivars have not been classified by using both ANN and discriminant analysis based on their physical, mechanical, and optical properties. However, one study using discriminant analysis were available in the literature (Menesatti et al., 2008). In that study, four traditional Italian hazelnut cultivars were classified by discriminant analysis. They found the classification success of 77.5%-98.8%. ANN and discriminant analysis methods were used separately or together to classify different agricultural products (Chen et al., 2010; Azizi et al., 2016; Taner et al., 2018).

In conclusion, the most effective variables for the classification of hazelnut varieties were sphericity, grain volume, geometric mean diameter and image processing variables, while the least effective ones were deformation, rupture energy, rupture force and grain weight variables. ANN and discriminant analysis gave close results to each other probably because of that nonlinear data used in this study were normally distributed. ANN and discriminant analysis models can be used successfully for the classification of hazelnut varieties. The classification performance of ANN and discriminant analysis models for the same hazelnut varieties grown in different parts of Turkey and in different climatic conditions can be determined in future studies. In addition, smartphone applications using the models proposed in this study can be developed.