Application of power-law committee machine to combine five machine learning algorithms for enhanced oil recovery … – Nature.com
This study combines the predictions of five machine learning models by means of the PLCM method to increase the generalization of the model in the context of EOR screening. This study not only assesses the individual machine learning methods in predicting the most suitable EOR techniques, but also takes benefit from the PLCM method optimized by the PSO to increase the prediction accuracy, for the first time in the context of EOR screening. In this manner, the predictive tool is not limited to only one data-driven model, but also takes advantage of the strength points of different types of machine learning algorithms. Figure1 shows the flowchart of this study. First, the required dataset to build and evaluate the utilized models is collected. Then, the data is preprocessed, which includes encoding the textual data into numeric values and normalizing the variables into [0,1]. Then, the individual machine learning models are trained. The hyperparameters of the models are tuned using a grid search with fivefold cross-validation. After training the individual models, their outputs are combined using the PLCM method optimized by the PSO algorithm. Then, the performance of the utilized methods is compared in terms of quantitative and visual evaluation metrics. The metrics, including the accuracy, precision, recall, F1-score, confusion matrix, precision-recall curve, and Receiver Operating Characteristic (ROC) curve to analyze their ability to handle the class imbalance issue. In the end, a feature importance analysis is conducted to find out the most influential input variables on the prediction of suitable EOR techniques. Another specialty of this study is that it uses a more comprehensive dataset than those in the previous studies, which increases the generalization of the developed model.
General flowchart of the study.
In this study, a dataset including 2563 EOR projects (available in Supplementary Information) from 23 different countries applied to sandstone, carbonate, limestone, dolomite, unconsolidated sandstone, and conglomerate reservoirs was collected from the literature5,20,21,22,23,24,25,26,27 to develop the screening methods. The utilized variables include the formation type, porosity (%), permeability (mD), depth (ft), viscosity (cP), oil gravity (API), temperature (F), and the production mechanism before conducting EOR. The EOR techniques include injection of steam, hydrocarbon miscible, hydrocarbon immiscible, CO2 miscible, CO2 immiscible, carbonated water, low-salinity water, CO2 foam, nitrogen miscible, nitrogen immiscible, micellar polymer, surfactant/polymer, surfactant, cyclic steam drive, steam-assisted gas drive (SAGD), liquefied petroleum gas (LPG) miscible, in-situ combustion, polymer, alkaline/surfactant/polymer (ASP), hot water, microbial, air-foam, hydrocarbon miscible-WAG, and hydrocarbon immiscible-WAG. Table 2 reports the statistical analysis of the variables. Since formation is a categorical feature, it was converted to numerical values. Among fifteen different formation types, sandstone, carbonate, and dolomite are the most prevalent formation types with 45%, 10%, and 10% of the total data, respectively. To assess the accuracy of the developed models on unseen data, 85% of the data was used for training and the remaining 15% was used as blind test cases, and fivefold cross-validation is used for hyperparameter tuning. It is common to divide the dataset with a ratio of 70:15:15 as training, validation, and testing subsets. The validation subset is commonly used for tuning the hyperparameters of the models. Nonetheless, in the current study, 5-Fold cross validation was used to tune the hyperparameters, which does not require putting aside a portion of the data for validation. In this technique, the training subset is divided into K (5 in this study) non-overlapping folds. Then, the model is trained and validated K times with the fixed hyperparameters. One of the folds is used for validation and the others for training. Finally, the validation score is calculated as the average of scores over K repetitions. This is repeated for all configurations of the hyperparameters and the set of hyperparameters with the highest cross-validation score is selected. Thereby, as we did not need a separate validation subset, all samples, except for the testing subset, were used for training (85%).
One of the crucial steps before moving to model development is data preprocessing. One type of preprocessing is to encode textual values to numerical values, which is called label encoding. For example, the formation type, previous production mechanism, and EOR techniques are textual features, which were encoded as numbers. Another preprocessing step is scaling the data into similar intervals since the scale of the features differ significantly. For example, viscosity is in the order of 106, while porosity is in the order of tens. In this study, the features were normalized into [0,1] interval using ((X - X_{min } )/(X_{max } - X_{min } )), where (X_{min }) and (X_{max }) are the minimum and maximum of the features in the training subset.
ANN is a learning algorithm that is inspired by the human brain. ANN can figure out the relationship between the inputs and outputs without the need for complex mathematical or computational methods. Among the various types of ANN, the Multilayer Perceptron (MLP-ANN) stands out as the most commonly used28,29,30. The MLP includes three layers, namely input, hidden, and output layers31,32, as illustrated in Fig.2. As shown, each layer consists of computational units known as neurons. The number of neurons in the input and output layers is the same as the dimension of the input and output variables, respectively. The number of hidden layers and their size should be determined by trial and error. Each neuron is connected to all neurons of the previous layers, which represents a unique linear combination of the data coming in from previous layer. The linear combination takes place using a set of weights. For example, (W_{xh}) represents the set of weights mapping the inputs to the hidden layers, and (W_{ho}) represents the set of weights mapping the hidden neurons to the output layer. Another critical aspect of an ANN model is the activation function, which receives the results of the linear combination, known as activations, and determines the activation of each neuron. Including hidden layers with non-linear activation functions in an ANN empowers it to capture non-linear dependencies. The weights are learned during the training phase of the model, which is the ultimate goal of the training process. Using these weights, the outputs, represented by (hat{y}), are calculated by the feed-forward process as below.
$$hat{y} = fleft( {mathop sum limits_{i = 1} W_{ij} x_{i} + b_{j} } right),$$
(1)
where f isthe activation function; (b_{j}) is the hidden layer bias; (x_{i}) is theinput for the ith variable; and, (W_{ij}) is theconnection weight between the ith input and jth neuron.
Schematic structure of an ANN.
The learning process in an ANN is actually adjusting the weights and biases in the hidden layers using the backpropagation algorithm to minimize the loss function between the predicted and actual values28,33. In a multiclass classification problem, the outputs are converted to one-hot encoded vectors, where all elements of the vectors are zeros except for the element corresponding to that specific sample class. To handle multiclass classification, the categorical cross entropy is used as the loss function, which is defined as follows.
$$CCEleft( W right) = mathop sum limits_{i = 1}^{C - 1} y_{i} log left( {hat{y}_{i} } right),$$
(2)
where y denotes the vector of actual outputs and C is the number of classes. Each output in a multiclass problem is a vector of probabilities for each class. The probabilities are calculated using the Softmax activation function. To minimize the loss function, the gradient of the loss with respect to the weights and biases must be calculated and back propagated to all layers to update the weights. Given the gradient of the loss function, the weights can be updated as follows.
$$W^{t + 1} = W^{t} - eta nabla_{W} CCE,$$
(3)
where (W^{t + 1}) and (W^{t}) are the new and current weights, (eta) is the learning rate, and (nabla_{W} CCE) is the gradient of the loss function calculated by an optimization algorithm, such as Adam, Stochastic Gradient Descent (SGD), RMSprop, Adagrad, Momentum, Nestrov and Accelerated Gradient34,35.
ANNs offer a variety of hyperparameters that can be tuned to optimize the models performance. It includes options for controlling model structure, learning rates, and regularization. Furthermore, ANNs incorporate class weights into the loss function, addressing the problem of class-imbalance, which is useful for the problem understudy. It also supports multiclass classification. Accordingly, one of the utilized methods in this study is the ANN.
According to the explanations, the control parameters of the ANN are the number of hidden layers, number of neurons in the hidden layers, activation functions, the optimizer, and learning rate, which should be fine-tuned to achieve a satisfactory performance.
CatBoost is a gradient-boosting tree construction method36, which makes use of both symmetric and non-symmetric construction methods. In CatBoost, a tree is learned at each iteration with the aim of reducing the error made by previous trees. Figure3 shows the process of CatBoost tree building. In this figure, the orange and blue circles represent a dataset with two classes. The process starts with a simple initial model, assigning the average of the entire dataset to a single leaf node. Then, the misclassified samples (enlarged circles in Fig.3) are identified and new trees are added based on the gradient boosting approach. Afterward, the predictions are updated to the combination of the predictions made by all trees. By adding new trees at each iteration, the number of misclassified samples decreases. Adding the trees continues until either the minimum number of samples required for splits or the maximum depth of the trees is reached. For categorical features, the CatBoost algorithm employs a symmetric splitting method for each feature. Then, based on the type of the feature, it chooses one of the split methods for each feature to create a new branch for each category37.
Schematic of the CatBoost tree construction.
Considering a training dataset with (N) samples, where (X) is the matrix of inputs ((x_{1} ,; ldots ,;x_{N})) and (y) is the vector of outputs ((y_{1} ,; ldots ,;y_{N})), the goal is to find a mapping function, (f(X)), from the inputs to the outputs. Here, (f(X)) is the boosted trees. Just like the ANN, the CatBoost needs a loss function ((L(f))) to be minimized to perform the optimal tree building strategy.
Now, the learning process entails minimizing the (L(f)).
$$f^{*} (X) = arg ;mathop {min }limits_{f} L;(f) = arg ;mathop {min }limits_{f} mathop sum limits_{i = 1}^{N} L;(y_{i} ,;hat{y}_{i} ),$$
(4)
If the algorithm entails M gradient boosting steps, a new estimator hm can be added to the model.
$$f_{m + 1} ;(x_{i} ) = f_{m} ;(x_{i} ) + h_{m} ;(x_{i} ),$$
(5)
where (f_{m + 1} ;(x_{i} )) is the new model, and (h_{m} ;(x_{i} )) is the newly added estimator. The new estimator is determined by employing the gradient boosting algorithm, where the steepest descent obtains (h_{m} = - ;alpha_{m} g_{m}) where (alpha_{m}) is the step length and (g_{m}) is the gradient of the loss function.
Now, the addition of a new tree/estimator can be accomplished by
$$f_{m + 1} (x) = f_{m} (x) + left( {arg mathop {min }limits_{{h_{m} in H}} left[ {mathop sum limits_{i = 1}^{N} Lleft( {y_{i} , ;f_{m} (x_{i} ) + h_{m} (x_{i} ) } right)} right]} right);(x),$$
(6)
$$f_{m + 1} (x) = f_{m} (x) - alpha_{m} g_{m} .$$
(7)
By taking benefit from the gradient boosting approach, the ensemble of decision trees built by the CatBoost algorithm often leads to a high prediction accuracy. The CatBoost also uses a strategy known as ordered boosting to improve the efficacy of its gradient-boosting process. In this type of boosting, a specific order is used to train the trees, which is determined by their feature importance. This prioritizes the most informative features, resulting in more accurate models38. The algorithm offers a wide range of regularization methods, such as depth regularization and feature combinations, which helps prevent overfitting. This is specifically useful when dealing with complex datasets.
The CatBoost offers a range of control parameters to optimize the structure of the model. These parameters include the number of estimators, maximum depth of the trees, maximum number of leaves, and regularization coefficients. These control parameters are optimized in this study to obtain the best performance from the model.
KNN is a non-parametric learning algorithm proposed by Fix and Hodges39. This algorithm does not have a training step and determines the output of a sample based on the output of the neighboring samples10. The number of neighbors is denoted by K. With K=1, the label of the sample is as of the nearest sample. As the name of this algorithm implies, the K nearest neighbors are found based on the distance between the query sample and all samples in the dataset. Euclidean, Minkowski, Chebyshev, and Manhattan distances are some common distance measures. The Minkowski distance is a generalization of the Euclidean and the Manhattan distance with (p = 2) and (p = 1), respectively. p is the penalty term in Lp norm, which can be a positive integer. The distance between the samples greatly depends on the scale of the features. Therefore, feature scaling is of great importance40. After finding the K nearest samples to the new sample (query), its label is determined using Eq.(8).
$$hat{f}(x_{q} ) leftarrow {text{arg }};mathop {max }limits_{c in C} mathop sum limits_{i = 1}^{K} delta (c, ;f(x_{i} )), quad delta (a,;b) = 1 quad {text{if}};; a = b.$$
(8)
where (x_{q}) is the new sample, (f(x_{i} )) is the label of the ith neighboring sample, C denotes the number of classes, and (delta (a,;b)) is the Kronecker delta which is 1 if (a = b) and 0 otherwise. An extension to KNN is the distance-weighted KNN, where the inverse of the distances between the samples are used as the weights. In this manner, the prediction for the query sample will be
$$hat{f}(x_{q} ) leftarrow {text{arg }};mathop {max }limits_{c in C} mathop sum limits_{i = 1}^{K} w_{i} delta (c,; f(x_{i} )),quad delta (a,;b) = 1 quad {text{if}} ;;a = b,$$
(9)
where (w_{i}) is the inverse of the distance between the query sample and sample i, (w_{i} = 1/D(x_{q} ,;x_{i} )). Consequently, the closer neighbors will have a higher impact on the predicted label.
One distinctive feature of KNN that sets it apart from other machine learning methods is its ability to handle incomplete observations and noisy data41. This technique enables the identification of significant patterns within noisy data records. Another advantage of KNN is that it does not require any training and building and the model optimization can be done quite quickly. According to the above explanations, the controlling parameters of KNN are the number of neighbors (K), using/not using distance weighting, penalty terms, and the algorithm used to compute the nearest neighbors.
SVM is a binary classification algorithm introduced by Cortes and Vapink42. SVM can be implemented to solve problems with linear or non-linear behavior43,44. However, non-linear data should be mapped into a higher-dimensional space to make it linearly separable. This technique is called the kernel trick. The classification is done by a decision boundary which has the maximum margin from both classes. Figure4 shows the schematic of an SVM classifier for a binary classification task. The margins are constructed by finding the support vectors in each class and drawing the hyperplanes from the support vectors45. The hyperplanes are shown by dashed lines and the decision boundary is drawn between them. In this figure, the green circles represent the positive (+1) and the blue circles represent the negative (1) classes. The circles on the hyperplanes are the support vectors. The decision boundary with the maximum margin from the classes results in the highest generalization.
Schematic of a binary SVM.
By considering the mapping function (emptyset (X)) and inputs (X) and outputs (y), the equation of the decision boundary can be written as follows46:
$$W^{T} emptyset (X) + b = 0,$$
(10)
where W is the weight parameters and b is the bias term. The smallest perpendicular distance between the hyperplanes is known as the margin, which is double the distance between the support vectors and the decision boundary. Assuming that the data is separated by two hyperplanes with margin (beta), after rescaling W and b by (beta /2) in the equality, for each training example we have
$$y_{i} left[ {W^{T} emptyset (x_{i} ) + b} right] ge 1,quad i = left{ {1,;2, ldots ,;M} right}.$$
(11)
For every support vector ((X_{s} , ;y_{s})) the above inequality is an equality. Thereby, the distance between each support vector and the decision boundary, r, is as follows
$$r = frac{{y_{s} (W^{T} X_{s} + b)}}{left| W right|} = frac{1}{left| W right|},$$
(12)
where (left| W right|) is the L2 norm of the weights. Therefore, the margin between the two hyperplanes becomes (frac{2}{left| W right|}). The goal is to maximize (frac{2}{left| W right|}), which is equivalent to minimizing (frac{1}{2}W^{T} W). Consequently, the optimization problem of the SVM is:
$$begin{gathered} arg ;mathop {min }limits_{W,b} frac{1}{2}W^{T} W, hfill \ subject; to ;y_{i} left[ {W^{T} emptyset (x_{i} ) + b} right] ge 1,quad {text{for}};;i = 1,; ldots ,;M. hfill \ end{gathered}$$
(13)
Nonetheless, to increase the generalization of the model and avoid overfitting, slack variables ((xi))46,47 are used (see Fig.3), which allow the model to have some miss-classified samples during training. This approach is known as the soft margin approach. Now, the optimization problem becomes
$$begin{gathered} arg ;mathop {min }limits_{W,b} left( {frac{1}{2}W^{T} W + cmathop sum limits_{i} xi_{i} } right), hfill \ subject; to; y_{i} left[ {W^{T} emptyset (x_{i} ) + b} right] ge 1 - xi_{i} ,quad {text{for}};;i = 1,; ldots ,;M. hfill \ end{gathered}$$
(14)
where c is a regularization factor that controls the weight of the slack variables in the loss function. Equation(14) is a dual optimization problem, which is solved using the Lagrange approach. The Lagrange approach converts a dual-optimization problem to a standard one by incorporating the equality and inequality constraints to the loss function. Thereby, Eq.(14) becomes
$$begin{gathered} L(W,;b,;alpha ) = frac{1}{2}W^{T} W - mathop sum limits_{i = 1}^{M} alpha_{i} left[ {y_{i} left( {W^{T} emptyset (X_{i} ) + b} right) - 1} right], hfill \ subject; to ;;0 le alpha_{i} le c,quad i = 1,; ldots ,;M. hfill \ end{gathered}$$
(15)
where (alpha_{i})s are Lagrange multipliers. To minimize the above loss function, its derivatives with respect to W and b are set equal to zero. By doing this, we obtain (W = sumnolimits_{i = 1}^{M} {alpha_{i} y_{i} emptyset (X_{i} )}) and (sumnolimits_{i = 1}^{M} {alpha_{i} y_{i} = 0}). Plugging these back into the Lagrange gives the dual formulation.
$$begin{gathered} arg ;mathop {max }limits_{alpha } - frac{1}{2}mathop sum limits_{i,j = 1}^{M} alpha_{i} alpha_{j} y_{i} y_{j} emptyset (X_{i} )emptyset (X_{j} ) + mathop sum limits_{i = 1}^{M} alpha_{i} , hfill \ subject;; to; mathop sum limits_{i = 1}^{M} alpha_{i} y_{i} = 0, ;;0 le alpha_{i} le c, ;;i = 1,; ldots ,;M. hfill \ end{gathered}$$
(16)
Equation(16) is solved using a Quadratic Programming solver to obtain the Lagrange multipliers (alpha_{i}). (alpha_{i}) is non-zero only for the support vectors. Parameter b does not appear in the dual formulation, so it is determined separately from the initial constraints. Calculating (emptyset (X_{i} )emptyset (X_{j} )) is computationally expensive since it requires two mapping operations and one multiplication, especially if the data is high-dimensional. To tackle this problem, the Kernel trick is introduced, where (emptyset (X_{i} )emptyset (X_{j} )) is represented as a kernel function (K(X_{i} ,;X_{j} )) based on the Mercers Theorem48. Finally, after determining the Lagrange multipliers, the prediction for a new sample z is calculated as follows
$$y = signleft( {mathop sum limits_{i = 1}^{n} alpha_{i} y_{i} K(X_{i,} z) + b} right).$$
(17)
The kernel function should be determined by trial and error. Some of the commonly used kernels are the linear, polynomial, and radial basis function (RBF) kernels.
SVM is one of the most successful machine learning algorithms in hand-written digit recognition49,50. SVMs can handle high-dimensional data, making them suitable for tasks with a large number of features. Because of taking benefit from the maximum margin theory and slack variables, SVMs are resistant to overfitting. One special feature of the SVMs, making them different than other artificial intelligence tools, is the kernel trick that enables SVMs to solve different kinds of non-linear classification problems. The convex nature of the loss function of the SVM leads to a convex optimization problem, which ensures converging to a global optimum. Finally, memory efficiency due to using only support vectors to construct the model and ability to handle class-imbalance by incorporating the class weights to the loss function are two other advantages of the SVMs making them suitable for the EOR screening problem in this study.
According to above explanations, some of the most important control parameters of the SVM are the kernel function, regularization factor (c), the degree of polynomial kernels, the intercept of polynomial kernels (coef0), and class weights. Class weights are used to tackle the class-imbalance issue by giving larger weights to rare classes in calculating the loss function.
Since SVM is a binary classifier, to perform multi-class classification, one-to-rest or one-to-one approaches are used. In this study, the one-to-rest approach is used, where (C) SVM models are trained. Each SVM model predicts membership of the samples in one of the C classes.
In the context of machine learning, Random Forest (RF) is an ensemble learning technique that builds a multitude of decision trees during training and combines their outputs to make more accurate and robust predictions51. RF is a supervised learning method, suitable for classification and regression tasks. Each tree in the forest is constructed independently, using a random subset of the features and samples with replacement from the training data52. This randomness adds diversity to the decision-making process, preventing the model from too much focusing on idiosyncrasies in the data. An RF takes a random approach to selecting a subset of input variables/features (controlled by the maximum number of features), and performs the optimal split to divide a node based on a split criterion. Avoiding tree pruning ensures maximal tree growth. As a result, a multitude of trees are constructed, and the model employs a voting mechanism to determine the most prevalent class in a classification task.
Each tree makes its own prediction, and the final decision is determined by the majority voting paradigm. This approach not only enhances the prediction accuracy of the model but also makes it stronger against overfitting. Figure5 shows the schematic of a random forest where n trees are used to make a prediction. Each subset is randomly selected from the dataset and divided into two parts, including the bag and out-of-bag (OOB) parts. The data in each bag is used to build a tree and the data in OOB is used to test that tree. The OOB subset serves as an ongoing and unbiased estimation of the general prediction error, predating the verification of prediction accuracy through the independent testing subset for the aggregated results. When (X) is inputted to the ensemble, each tree provides a separate output ((o_{1} ,; ldots , ;o_{n})). In the end, the ultimate class of the inputs is determined by the same approach given in Eq.(8).
Schematic of the random forest tree construction.
The RF produces competing results to boosting and bagging, without any alteration to the training set. It minimizes the bias by incorporating a random sample predictor before each node segmentation. The RF model can handle high-dimensional data, without need for feature selection. Its implementation in Python is relatively straightforward, boosting training speeds and easy parallelization. Given these advantages, it is becoming increasingly popular among data scientists52,53.
According to the above explanations, the control parameters of a random forest are the split criterion, maximum depth of trees, the number of estimators, and the maximum number of features. These control parameters are fine-tuned to achieve the best performance. There is also another control parameter, which is the minimum number of samples required to split a node, but it is not investigated in this study.
A committee machine is a technique to merge the output of a multitude of predictive models to come up with a single prediction33. The benefit of this technique is to take advantage of the results of different alternatives for modeling a particular problem, instead of using only one model. The individual models are selected in such a way that at least one model from each type of machine learning models is included. Thereby, we can take benefit from the strength points of different types of learning algorithms. By using the PLCM technique, the chance of overfitting can be lowered33. There are two main approaches to combine the output of individual models, namely the static and dynamic approaches. In the static method, a linear combination of the individual outputs is used to get the ultimate output, while the dynamic approach uses a non-linear combination of the outputs. In this study, the dynamic approach with a power-law model is used to accomplish the integration task. Equation(18) shows the power-law model.
$$y = mathop sum limits_{i = 1}^{5} alpha_{i} y_{i}^{{beta_{i} }} ,$$
(18)
where (y) is the ultimate output, (alpha_{i}) and (beta_{i}) are the coefficients that must be optimized to achieve the goal of the power-law committee machine, and (y_{i}) is the output of the (i)-th individual predictive model. In this study, the coefficients of the power-law model ((alpha_{i}) and (beta_{i})) are optimized by the PSO algorithm to achieve a satisfactory integration of the outputs. The PSO is described in the following subsection.
Kennedy and Eberhart54 introduced the PSO as a population-based optimization algorithm. This algorithm starts solving the problem with random solutions65. Each solution in this algorithm is known as a particle, where a swarm is composed of a multitude of particles. The particles change their position in the solution space by a specified velocity which is updated at each iteration. The particles position determines the solution found by the particle. When the position of the particle changes, a new solution is obtained. The following equations give the updating formulae for the velocity and position of a particle
$$v_{i} (t + 1) = omega v_{i} (t) + c_{1} r_{1} (x_{best,i} (t) - x_{i} (t)) + c_{2} r_{2} (x_{best,g} (t) - x_{i} (t)),$$
(19)
$$x_{i} (t + 1) = x_{i} (t) + v_{i} (t + 1),$$
(20)
where (x_{i}) and (v_{i}) are the position and velocity of particle (i), respectively, (t) is the iteration number, (omega) is the inertia coefficient, (c_{1}) and (c_{2}) are the self-learning and social-learning coefficient, respectively, (r_{1}) and (r_{2}) are two random numbers, (x_{best,i}) is the best solution found by the particle, and (x_{best,g}) is the global best solution. The values of the (x_{best,i}) and (x_{best,g}) are obtained by evaluating the objective function. In this study, the objective function is the negative of prediction accuracy by the PLCM method. The velocity and position of the particles are updated until the algorithm reaches the stopping criterion. The parameters used in Eq.(19) are determined based on the work by Poli et al.56, where (omega ,) (c_{1} ,) and (c_{2}) are set at 0.7298, 1.49618, and 1.49618, respectively.
The PSO is one of the most commonly used optimization algorithms in petroleum engineering57,58,59,60. Among different metaheuristic optimization algorithms, the PSO has shown a better performance compared to the most of other optimization algorithms, such as the genetic algorithm and simulated annealing. The PSO has shown the ability to reach better optimal solutions and faster convergence to similar results than its rivals in many applications61. Thereby, this algorithm is used in this study to optimize the coefficients of the PLCM method.
After describing the tools used in this study, it is necessary to define the evaluation metrics, which are required to evaluate the performance of the proposed method. These metrics include the quantitative and visual indicators that are described in the following subsection.
In this study, quantitative and visual evaluation metrics are used to assess the performance of the proposed method. These metrics include the accuracy, precision, recall, F1-score, confusion matrix, Receiver Operating Characteristic (ROC) curve, and precision-recall curve.
Accuracy is the total number of correct predictions divided by the total number of data points. In binary classification, accuracy is defined as the number of true positives (TP) divided by the number of samples (accuracy = frac{TP}{N}), where N is the total number of data points/samples.
Precision is the portion of positive predictions that are actual positives. Precision focuses on the accuracy of positive predictions. For a binary classification precision is defined as (Precision = frac{TP}{{TP + FP}}), where FP is the number of false positives, which means that the prediction by the model is positive, whereas the actual label of the sample is negative.
Recall gives the portion of the positive samples that are identified as positives. Recall focuses on how well the model captures positive instances. In other words, it is the ratio of true positives to all positive samples in the dataset defined as ({text{Re}} call = frac{TP}{{TP + FN}}), where FN is the number of false negative predictions defined as the samples which are incorrectly classified as negative.
The inverse of the harmonic average of the recall and precision multiplied by 2 is known as F1-Score. F1-Score is defined in Eq.(21).
$$F1{ - }Score = 2frac{PR}{{P + R}},$$
(21)
where P and R are the precision and recall, respectively. A good classifier should have high values of precision and recall, which indicates a high F1-Score.
In multi-class classification, as the problem in this study, each metric is calculated for individual classes and averaged across all classes to obtain a single value. In this manner, each time, one of the classes is considered positive, and other classes are assumed as negative.
In a multiclass problem, the confusion matrix is a (C times C) matrix, where the rows represent the actual class and the columns represent the predicted class of the samples. The values on the main diagonal of the matrix show the number of correct predictions (true positives), and off-diagonal values show the number of incorrect predictions (false positives). The sum of the values on the main diagonal of the matrix divided the total number of samples gives the accuracy, as described above. Also, the diagonal value for each class if divided by the sum of all values in each column gives the class-specific precision, and if divided by the sum of all values in each row gives the class-specific recall.
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