# Optimization of the ANNs Predictive Capability Using the Taguchi Approach: A Case Study

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## Abstract

**:**

_{12}orthogonal array, turning experiments were conducted to identify the best parametric set of an ANNs design, considering different combinations of sample number, scaling, training rate, activation functions, number of hidden layers, and epochs. The composite desirability value for the multi-response variables has been obtained through the desirability function analysis (DFA). The parameters’ optimum levels have been identified using this methodology.

## 1. Introduction

_{i}are multiplied by the weights w

_{i}, with w

_{0}as bias. Bias shifts the activation function by adding a constant to the input to better fit the data prediction. Bias in neural networks can be thought of as analogous to the role of a constant in a linear function, whereby the constant value transposes the line: without w

_{0}, the line would go through the origin (0, 0), and the fit could be poor. If the weighted sum of the inputs “a” overcomes a threshold value T, neurons release its output y, which is a function of (a-T). In other words, the arriving signals (inputs), multiplied by the connection weights, are first summed and then passed through an activation function (θ) to produce the output [6]. A unit feeds its output to all the nodes on the next layer, but there is no feedback to the previous layer (feed-forward network). Weights represent the system memory and indicate the importance of each input neuron in the output generation processing. The activation function is used to introduce non-linearity in the network’s modeling capabilities [7]. Activations are typically a number within the range of 0 to 1, and the weight is a double, e.g., 2.2, −1.2, 0.4.

## 2. Materials and Methods

#### 2.1. The Multi-Layer Perceptron (MLP) Model

#### 2.1.1. Neurons and Layers

#### 2.1.2. Activation Function

#### 2.1.3. The Training Process and Pre-Treatment of Data

- Training phase, to find the weights that represent the relationship between ANN inputs and outputs.
- Testing phase, to optimize the weights and to estimate model accuracy by error indicators.

#### 2.1.4. The Training Cycles and Network Performance

- (a)
- Coefficient of Determination (R
^{2})

- (b)
- Mean Absolute Error (MAE)

- (c)
- Root Mean Squared Error (RMSE)

^{2}provides the variability measure of the data reproduced in the model. As this test does not give the model’s accuracy, other statistical parameters have to be reported. MAE and RMSE measure residual errors, which provide a global idea of the difference between the observed and modeled values.

^{2}are often used to select the “better” neural network [43].

#### 2.1.5. The Taguchi Design of Experiments Method

#### 2.2. A Case Study: Use of Artificial Neural Networks to Evaluate Organic and Inorganic Contamination in Agricultural Soils

^{2}is 0.87, and the RMSE and MAE values in the training and test set are of the same order. Therefore, the MLP model provides reliable predictions.

#### The Taguchi Design Steps to Optimize the Predictive Xapability of ANNs

## 3. Results

#### 3.1. Step 1. Identify the Main Function and Its Side Effects

#### 3.2. Step 2. Identify the Objective Function and Type of Response(s) to Be Optimized

^{2}, one of the three performance measures (§ 2.1.4), was chosen as the objective function or response variable to be maximized. The target value of R

^{2}has been set to 0.70 or more. Taguchi recommends using a loss function to quantify a design’s quality, defining the loss of quality as a cost that increases quadratically with the deviation from the target value. Usually, the quality loss function considers three cases: nominal-the-best, smaller-the-better, and larger-the-better [51], corresponding to three types of continuous measurable responses to be optimized [52]:

^{2}qualifies as an LTB response.

#### 3.3. Step 3. Identify the Control Factors and Their Levels

- Number of samples. The experiment can detect the minimum number of sample units sufficient for the network to learn. In a 64-sample database, three levels of this factor have been fixed: 10, 30, and 50 units.
- Input scaling rule. Two levels are corresponding to the different criteria to scale input data (§ 2.3):

- (a)
- normalization, using the formula$${p}_{i}=\frac{{x}_{i}-\mathrm{min}\left(x\right)}{\mathrm{max}\left(x\right)-\mathrm{min}\left(x\right)}$$
- (b)
- standardization, using the formula$${z}_{i}=\frac{{x}_{i}-\mathsf{\mu}}{\sigma}$$

- Training rate (%): three levels of percentage have been considered for computing the size of the training set (test set): 60% (40%), 70% (30%), and 80% (20%).
- Activation function of hidden and output nodes: as mentioned above (§ 2.2), two levels have been chosen for the activation function of the hidden nodes (sigmoid and hyperbolic tangent), and three levels for the activation function of output nodes (identity function, sigmoid, and hyperbolic tangent).
- Number of hidden layers: to determine if a deep network has better predictive performance, two levels of this factor have been considered: one or two hidden layers, as allowed by Neural Network function in IBM-SPSS.
- Epochs. The training process duration has been set to three levels: 10, 10,000, and 60,000 epochs.

#### 3.4. Step 4. Select a Suitable Orthogonal Array and Construct the Experiment Matrix

^{k}runs. Thus, for seven factors at two or three levels, it would require many experiments (2

^{7}or 3

^{7}) to be carried out, and too many observations to be economically viable, as stated above (§ 3).

_{12}(Table 3):

#### 3.5. Step 5. Conduct the Experiments

_{12}, each experiment has been conducted two times (24 runs in total), corresponding to 12 MLP models whose R

^{2}, RMSE, and MAE in the training and test set have been calculated in each run. Table 4 shows the measured values of response variable R

^{2}, the mean of R

^{2}in each run, and the S/N ratio obtained from each trial’s different networks.

#### 3.6. Step 6. Predict the Optimum MLP Model and Its Performance

^{2}calculated in each run and the signal/noise ratio (S/N Ratio). A standard approach for multi-response optimization is to identify one of the response variables as primary considering it as an objective function and other responses as constraints [54]. Several methods of multiple response optimization have been proposed in the literature. Among them, the utilization of the desirability function is the most efficient approach [55]. In the desirability function analysis (DFA), each response is transformed into a desirability value d

_{i}, and the total desirability function D, which is the geometric mean of the single d

_{i}, is optimized [56]. Desirability D is an objective function that ranges from 0 to 1. If the response is on target, the desirability value will be equal to 1, and DFA will not maximize the desirability value. When the response falls within the tolerance range but not on the desired value, the corresponding desirability will be between 0 and 1 [57]. As the response approaches the target, the desirability value will become closer to 1.

- Number of samples: at least 50 samples are required to obtain an optimal model. Thus, a small number of units could cause unbiased predictions.
- Input scaling rule: the normalization of input variables produces better results than the standardization rule.
- Training rate (%): in an optimal MLP model, the training set must consider 70% of database units. Thus, the test set represents the remaining 30%.
- Activation function of hidden and output nodes: the best activation function is the hyperbolic tangent for both hidden and output nodes.
- Number of hidden layers: according to Taguchi’s design, a deep network is not the best solution for this analysis; one hidden layer has been more than enough to optimize the forecasts.
- Epochs: the model accuracy has been determined in 10,000 epochs.

#### 3.7. Step 7. Conduct the Verification Experiment

_{def1}, MLP

_{def2}, and MLP

_{opt}) are summarized as follows (Table 6).

_{def1}and MLP

_{def2}, the best solution is to consider a higher number of samples increasing performance in case of an arbitrary choice of architecture parametric set. However, the optimal model MLP

_{opt}, in which the original dataset has been normalized, has produced, with a longer training time, more reliable predictions than MLP

_{def1}: the RMSE and MAE values (in training and test sets) are lower, and R

^{2}is 0.93.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**A Three-layer Multi-Layer Perceptron (MLP), with m input nodes (IN), h hidden nodes (H), and t output nodes (OUT). The term a

_{0}

^{(J)}is the bias, for J = IN, H

**Figure 3.**MLP feed-forward network 10-3-1. Input variables have been standardized. The training rate set was 70.3%. Activation functions were hyperbolic tangent for hidden nodes and identity function for output nodes. The prediction error has been minimized across ten epochs. The RMSE and MAE values in the training and test set are of the same order, even if not remarkably low, and R

^{2}is 0.87. The network provides reliable predictions.

Factor | Levels | ||
---|---|---|---|

Number of samples | 10 | 30 | 50 |

Input scaling | normalized; standardized | ||

Training rate (%) | 60 | 70 | 80 |

Act. Function H | Sigmoid; Hyperbolic; tangent | ||

Act. Function O | identity | sigmoid | hyp.tangent |

n. Hidden Layers | 1 | - | 2 |

Epochs | 10 | 10,000 | 60,000 |

Factor | Number of Level (n_{i}) | Degrees of Freedom (n_{i}) −1 |
---|---|---|

Mean value | - | 1 |

Number of samples | 3 | 2 |

Input scaling | 2 | 1 |

Training rate (%) | 3 | 2 |

Act. Function H | 2 | 1 |

Act. Function O | 3 | 2 |

n. Hidden Layers | 2 | 1 |

Epochs | 3 | 2 |

OA | Number of Samples | Input Scaling | Training Rate (%) | Act. Function H | Act. Function O | n. Hidden Layers | Epochs |
---|---|---|---|---|---|---|---|

L1 | 10 | norm | 60 | sigm | lin | 1 | 10 |

L2 | 30 | norm | 70 | sigm | hp tg | 1 | 10,000 |

L3 | 50 | norm | 80 | sigm | sigm | 1 | 60,000 |

L4 | 10 | norm | 80 | hp tg | hp tg | 2 | 10,000 |

L5 | 30 | norm | 60 | hp tg | sigm | 2 | 60,000 |

L6 | 50 | norm | 70 | hp tg | lin | 2 | 10 |

L7 | 10 | standard | 80 | sigm | sigm | 2 | 10 |

L8 | 30 | standard | 60 | sigm | lin | 2 | 10,000 |

L9 | 50 | standard | 70 | sigm | hp tg | 2 | 60,000 |

L10 | 10 | standard | 70 | hp tg | lin | 1 | 60,000 |

L11 | 30 | standard | 80 | hp tg | hp tg | 1 | 10 |

L12 | 50 | standard | 60 | hp tg | sigm | 1 | 10,000 |

OA | R^{2}_{1} | R^{2}_{2} | Mean | S/N Ratio |
---|---|---|---|---|

L1 | 0.123 | 0.347 | 0.235 | −15.705 |

L2 | 0.914 | 0.919 | 0.916 | −0.757 |

L3 | 0.914 | 0.845 | 0.879 | −1.135 |

L4 | 0.935 | 0.803 | 0.869 | −1.295 |

L5 | 0.524 | 0.709 | 0.616 | −4.496 |

L6 | 0.793 | 0.91 | 0.851 | −1.458 |

L7 | 0.002 | 0.399 | 0.200 | −50.969 |

L8 | 0.955 | 0.91 | 0.932 | −0.615 |

L9 | 0.912 | 0.899 | 0.905 | −0.863 |

L10 | 0.584 | 0.186 | 0.385 | −12.019 |

L11 | 0.903 | 0.949 | 0.926 | −0.676 |

L12 | 0.86 | 0.549 | 0.704 | −3.683 |

Number of Samples | Input Scaling | Training Rate (%) | Act. Function H | Act. Function O | n. Hidden Layers | Epochs |
---|---|---|---|---|---|---|

50 | norm | 70 | hp tg | hp tg | 1 | 10,000 |

Network Features | MLP_{def1} | MLP_{def2} | MLP_{opt} |
---|---|---|---|

Number of samples | 64 | 50 | 50 |

Scaling | standard | standard | norm |

Training rate | 70 | 70 | 70 |

Act. FunctionH | tg hp | tg hp | tg hp |

Act. FunctionO | lin | lin | tg hp |

Hidden layer | 1 | 1 | 1 |

Epochs | 10 | 10 | 10,000 |

Rsquare | 0.87 | 0.63 | 0.93 |

RMSEtraining | 1.41 | 1.683 | 0.129 |

RMSEtest | 1.788 | 30.319 | 0.025 |

MAEtraining | 0.064 | 0.102 | 0.025 |

MAEtest | 0.65 | 0.488 | 0.691 |

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**MDPI and ACS Style**

Manni, A.; Saviano, G.; Bonelli, M.G.
Optimization of the ANNs Predictive Capability Using the Taguchi Approach: A Case Study. *Mathematics* **2021**, *9*, 766.
https://doi.org/10.3390/math9070766

**AMA Style**

Manni A, Saviano G, Bonelli MG.
Optimization of the ANNs Predictive Capability Using the Taguchi Approach: A Case Study. *Mathematics*. 2021; 9(7):766.
https://doi.org/10.3390/math9070766

**Chicago/Turabian Style**

Manni, Andrea, Giovanna Saviano, and Maria Grazia Bonelli.
2021. "Optimization of the ANNs Predictive Capability Using the Taguchi Approach: A Case Study" *Mathematics* 9, no. 7: 766.
https://doi.org/10.3390/math9070766