The Weibull distribution is regarded as among the finest in the family of failure distributions. One of the most commonly used parameters of the Weibull distribution (WD) is the ordinary least squares (OLS) technique, which is useful in reliability and lifetime modeling. In this study, we propose an approach based on the ordinary least squares and the multilayer perceptron (MLP) neural network called the OLSMLP that is based on the resilience of the OLS method. The MLP solves the problem of heteroscedasticity that distorts the estimation of the parameters of the WD due to the presence of outliers, and eases the difficulty of determining weights in case of the weighted least square (WLS). Another method is proposed by incorporating a weight into the general entropy (GE) loss function to estimate the parameters of the WD to obtain a modified loss function (WGE). Furthermore, a Monte Carlo simulation is performed to examine the performance of the proposed OLSMLP method in comparison with approximate Bayesian estimation (BLWGE) by using a weighted GE loss function. The results of the simulation showed that the two proposed methods produced good estimates even for small sample sizes. In addition, the techniques proposed here are typically the preferred options when estimating parameters compared with other available methods, in terms of the mean squared error and requirements related to time.

The parameters of the Weibull distribution are widely used in reliability studies and many engineering applications, such as the lifetime analysis of material strength [

The form of the probability density function (PDF) of two parameters of WD is given by:

The cumulative distribution function (CDF) and the survival function

Several approaches to estimating the parameters of the WD have been proposed [

Manual approaches include the ordinary least squares [

In addition to computational methods, many studies in the literature have attempted to use the neural network (NN) to anticipate the parameters of the WD in many areas, such as the method developed by Jesus that applies the Weibull and ANN analysis to anticipate the shelf life and acidity of vacuum-packed fresh cheese [

Recently, a few methods have been attempted to combine the robustness of the ANN and some of the above statistical methods. Maria modeled the distribution of tree diameters using the OLS and the ANN [

In the proposed method, we solve the problem whereby the reliability of the OLS method is compromised by outliers through the introduction of a pre-trained neural network after the linearization of the CDF. The remaining sections of this paper are organized as follows: Section 2 provides a review of different numerical and graphical methods for estimating the parameters of the WD, such as the MLE, OLS, WLS, and BLGE. In Section 3 we present the proposed methods. To evaluate their appropriateness in comparison with competing methods, the relevant performance metrics are covered in Section 4. The results are discussed in Section 5. Finally, the conclusions of this study are provided in Section 6.

The most commonly used approaches to estimate the parameters

Let the set

The partial derivatives of the equation for

The MLE estimator

The parameter

To estimate the parameters of the WD, the OLS method is extensively used in mathematics and engineering problems [

Let

Let

The estimates

Therefore, the estimates

The estimates

In the WLS estimate, the parameters

The biggest challenge in the application of the WLS is in finding the weights

Hence, the weights can be written as follows:

Minimizing

In this section, the approximate Bayesian estimator under a GE loss function of the parameters

The parameters

Moreover, it can be asymptotically estimated by:

For the two-parameter case

The functions in

To apply the Lindley model of

The elements

The general entropy loss function

The BLGE of

In the same way, the BLGE of

In the following sections, we describe the proposed BLWGE and OLSMLP methods.

The WGE loss function was proposed as dependent on the weighted loss GE function as follows:

Based on the posterior distribution of the parameter

Thus, we can find that

Consequently, the BLWGE of parameter

We note that the GE is a special case of the WGE when

Based on the WGE and by using

Thus, the BLWGE

Similarly, the BLWGE

Thus, the weighted Bayes estimator for the shape parameter

As previous studies have shown [

We now describe the proposed method, which is divided into two main parts: the linearization of the CDF, and the application of a feedforward network with backpropagation to estimate the values of

The OLS method takes the CDF defined in

Therefore, instead of using the slope and the intercept, we propose applying Algorithm 1 as described below.

The steps used to evaluate the parameters of the WD from the input csv file are described by Algorithm 1.

Normalization is an essential preprocessing tool for a neural network [

To estimate the parameters of the WD, we propose using a multilayer perceptron (MLP), which is a feedforward network with backpropagation [

Various criteria have been proposed in the literature to fix the number of hidden neurons [

The hyperbolic tangent activation function (

The objective of our neural network is a model that performs well on the data used in both the training and the test datasets. For this reason, we add a well-known regularization layer as described in the next section.

Regularization is a technique that can prevent overfitting [

The optimization of deep networks is an active area of research [

To evaluate the proposed methods with respect to other methods, we used two statistical tools, the mean squared error (MSE) and the mean absolute percentage error (MAPE) [

We generated 250,000 random data points from the WD for different parameters and different values of

We used the same dataset for the neural network in the training phase, but applied one sample to each shape/scale pair. This was unlike in the other methods (MLE, OLS, WLS, BLGE, and BLWGE), which used 10,000 samples to estimate the parameters of the WD. This dataset was divided into two subsets. The first subset was used to fit the model, and is referred to as the training dataset; it was characterized by known inputs and outputs. The second subset is referred to as the test dataset, and was used to evaluate the fitted machine learning model and make predictions on the new subset, for which we did not have the expected output. We chose the train–test procedure for our experiments because we guessed that we had a sufficiently large dataset available.

In all experiments, we trained the model with Google Collaboratory (GPU) for 25 epochs. We used the Nadam optimizer with learning rate of

In all experiments, the parameters of the BLWGE and BLGE were empirically determined. The values of the weights

To illustrate how the sample size affects the calculation of the MSE,

From

Scale |
Shape |
||||||
---|---|---|---|---|---|---|---|

1.5 | 0.049387 | 0.056885 | 0.056176 | 0.085488 | 0.096340 | 0.066142 | |

1.75 | 0.034172 | 0.555655 | 0.038032 | 0.088573 | 0.094040 | 0.027932 | |

2 | 0.027430 | 0.982985 | 0.030333 | 0.103455 | 0.095413 | 0.014994 | |

2.5 | 0.043991 | 2.225489 | 0.047444 | 0.310981 | 0.419640 | 0.643763 | |

3.25 | 0.072562 | 0.587675 | 0.078860 | 0.277142 | 0.538829 | 0.904675 | |

4 | 0.110517 | 0.116685 | 0.118746 | 0.269783 | 0.636383 | 1.331979 |

Scale |
Shape |
||||||
---|---|---|---|---|---|---|---|

1.5 | 0.348991 | 0.236889 | 0.209179 | 0.243708 | 0.199329 | 0.042465 | |

1.75 | 0.457484 | 0.314301 | 0.284707 | 0.320195 | 0.263958 | 0.026617 | |

2 | 0.604420 | 0.420835 | 0.376867 | 0.423471 | 0.348965 | 0.178389 | |

2.5 | 2.429318 | 1.643576 | 1.499686 | 1.693472 | 1.387491 | 3.685218 | |

3.25 | 2.407258 | 1.636167 | 1.479661 | 1.679463 | 1.378634 | 1.997234 | |

4 | 2.461472 | 1.658105 | 1.496513 | 1.726041 | 1.418305 | 0.925600 |

From

Parameters of WD |
|||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Methods | Estimated |
Estimated |
Statistical analysis | ||||||||||

MSE | Ranking | MAPE (%) | Ranking | Global rank | Time Computation | ||||||||

MLE | 9.95111 | 2.98894 | 11.05354 | 0.01459097 | 6 | 1 | 0.27084 | 0.0319 | 6 | 1 | 6 | 1 | 2.87 ms |

OLS | 8.23364 | 3.00974 | 7.47628 | 30.15793 | 4 | 6 | 0.24563 | 0.64591 | 5 | 6 | 5 | 6 | 0.9 us |

WLS | 8.01792 | 3.00197 | 6.871836 | 0.01563281 | 3 | 2 | 0.2394 | 0.033138 | 4 | 2 | 4 | 2 | 0.98 us |

BLGE | 9.06909 | 1.8185 | 7.744257 | 1.509358 | 5 | 5 | 0.23293 | 0.39383 | 3 | 5 | 3 | 5 | 2.81 ms |

BLWGE | 8.40374 | 2.09433 | 6.369233 | 0.8291446 | 2 | 4 | 0.22206 | 0.30189 | 2 | 4 | 2 | 4 | 2.77 ms |

OLSMLP | 8.1827 | 2.95646 | 0.424712 | 0.0380619 | 1 | 3 | 0.061342 | 0.051703 | 1 | 3 | 1 | 3 | 17.61 us |

This study proposed a method to estimate the parameters of the WD. This method is based on the OLS graphical method and the MLP neural network. The MLP solves the problems caused by the presence of outliers and eases the difficulty of determining the weights in the WLS method. It yielded acceptable results in simulations, especially in terms of shape estimation. It is also faster than the MLE, BLGE, and BLWGE.

We also proposed a second method (BLWGE), in which we introduced weight to the GE loss function. The results of simulations showed that BLWGE yields good results, especially in terms of shape estimation, compared with the other methods.

This project was supported by the Deanship of Scientific Research at Prince Sattam bin Abdulaziz University under Research Project No. 2020/01/16725.