A Quantitative Validation Method of Kriging Metamodel for Injection Mechanism Based on Bayesian Statistical Inference

metals

Article A Quantitative Validation Method of Kriging Metamodel for Injection Mechanism Based on Bayesian Statistical Inference

Dongdong You 1,2,3,* , Xiaocheng Shen 1,3, Yanghui Zhu 1,3, Jianxin Deng 2,* and Fenglei Li 1,3

1 National Engineering Research Center of Near-Net-Shape Forming for Metallic Materials, South China University of Technology, Guangzhou 510640, China; [email protected] (X.S.); [email protected] (Y.Z.); ﬂ[email protected] (F.L.) 2 Guangxi Key Lab of Manufacturing System and Advanced Manufacturing Technology, Guangxi University, Nanning 530003, China 3 Guangdong Key Laboratory for Advanced Metallic Materials processing, South China University of Technology, Guangzhou 510640, China * Correspondence: [email protected] (D.Y.); [email protected] (J.D.); Tel.: +86-20-8711-2933 (D.Y.); +86-137-0788-9023 (J.D.) Received: 2 April 2019; Accepted: 24 April 2019; Published: 27 April 2019

Abstract: A Bayesian framework-based approach is proposed for the quantitative validation and calibration of the kriging metamodel established by simulation and experimental training samples of the injection mechanism in squeeze casting. The temperature data uncertainty and non-normal distribution are considered in the approach. The normality of the sample data is tested by the Anderson–Darling method. The test results show that the original difference data require transformation for Bayesian testing due to the non-normal distribution. The Box–Cox method is employed for the non-normal transformation. The hypothesis test results of the calibrated kriging model are more reliable after data transformation. The reliability of the kriging metamodel is quantitatively assessed by the calculated Bayes factor and confidence. The Bayesian factor and the confidence level results indicate that the kriging model demonstrates improved accuracy and is acceptable after data transformation. The influence of the threshold ε on both the non-normally and normally distributed data in the model is quantitatively evaluated. The threshold ε has a greater influence and higher sensitivity when applied to the normal data results, based on the rapid increase within a small range of the Bayes factors and confidence levels.

Keywords: squeeze casting; bayesian inference; uncertainty; kriging metamodel

1. Introduction For the injection mechanism, which is a signiﬁcant component of a squeeze casting machine, uncertainty analysis is very important for the optimal design of mechanical structures and injection processes. The metamodel method is eﬀective and feasible for data analysis and prediction. In our previous study [1,2], an uncertainty analysis based on a kriging metamodel of the injection mechanism in squeeze casting and a literature review were performed. To evaluate and improve the prediction accuracy with uncertainty, the proposed kriging model needs quantitative validation and calibration. This study focuses on the quantitative method that considers data non-normality and uncertainty for the validation and calibration of kriging metamodels. Model validation employs a reasonable validation criterion established by probability and statistics theory, fuzzy mathematics, interval estimation methods, and other information theories to evaluate the prediction model and experimental model, considering data uncertainty, to obtain

Metals 2019, 9, 493; doi:10.3390/met9050493 www.mdpi.com/journal/metals Metals 2019, 9, 493 2 of 15 a reliable prediction model and quantitatively infer the confidence of the predicted results of the total structural response. During the past ten years, model validation and calibration methods that consider uncertainties have increasingly attracted attention. Commonly used approaches, such as parameter estimation, model parameter correction, and system identification, update computer models to improve the consistency of model output. Oberkampf and Trucano [3] investigated model calibration and validation benchmarks. The design and use method of the code and validation metrics based on experimental data uncertainty was recommended. Chen et al. [4] presented a generic model validation methodology that replaced parts of the experimental evaluation with the model uncertainty analysis and substantially reduced the number of physical tests. Beek et al. [5] proposed a heuristics-enhanced model fusion method that updates the estimated input parameters by reducing information about available points to calibrate a kriging metamodel and accurately obtain the system responses. Recent studies have progressed to the use of statistical analysis methods, such as a hypothesis testing and Bayesian statistical inference, to measure and verify the consistency of the probability distribution between model prediction output and reference data. Oberkampf and Barone [6] conducted a quantitative model validation with uncertainty by developing a statistical hypothesis testing-based method. Hypothesis testing includes the classic method and the Bayesian method. Previous studies have discussed the differences between the two approaches (e.g., [7]). Compared with the classical hypothesis test, the Bayesian approach seems relatively simple and focuses on model acceptance. Selecting test statistics and determining the sampling distribution of statistics is not required. Furthermore, the significance level is not needed in advance, and its rejection domain does not have to be determined. Thus, accurate and stable test results can be obtained. A remarkable feature of the Bayesian approach is the introduction of prior information for Bayesian utilization. In decision-making, prior distribution, which is formed by the quantification of prior information, total information, and samples are integrated by a Bayesian formula to obtain the posterior distribution. Then, the statistical inference based on a posterior distribution may be used to validate the results. Rebba and Mahadevan [8] developed aggregate validation metrics that are based on classical and Bayesian hypothesis testing by multivariate analysis and applied experimental data to validate computational models with uncertainties. Jiang and Mahadevan [9] derived a formula to calculate the Bayes factor and confidence for interval hypothesis testing to improve the total reliability evaluation of the model. A method based on Bayesian inference and factor analysis is applied for the quantitative evaluation of the model confidence in dynamic systems at various levels [10]. They employ the proposed method and a power density pseudospectrum approach for system identification and neural network model validation [11]. In recent years, the Bayesian method has been demonstrated to be effective and feasible in validating and calibrating the kriging model. Arendt et al. [12,13] developed the modular Bayesian approach, which is a form of preposterior analysis using a Gaussian process, and calculated the preposterior covariance matrix of the calibration parameters to predict the degree of identifiability of the systems to be calibrated. Han et al. [14] introduced a statistical methodology that is based on a hierarchical Bayesian model and its application in a biomechanical engineering example, where tuning parameters and calibration parameters were simultaneously determined. Pronzato and Rendas [15] applied a Bayesian local kriging metamodel approach to replace the computationally expensive simulation codes with localized covariance functions and construct interpolators/predictors of random fields. Bachoc et al. [16] proposed a global statistical approach based on a universal kriging model that was calibrated by classical statistical inference to improve the predictions of the underlying physical system, including an uncertainty quantification, and determine the calibration parameters in a Bayesian framework. Jensen et al. [17] explored the feasibility of integrating an adaptive kriging metamodel into a finite element model updating formulation, in which a Bayesian model updating approach based on stochastic simulation was adopted using dynamic response data. De Baar et al. [18] derived gradient-enhanced kriging by the Bayesian theorem for uncertainty quantification, and in the Bayesian framework, improved the kriging process by increasing the posterior covariance when Metals 2019, 9, 493 3 of 15 making maximum likelihood estimates of the hyperparameters. Boojari et al. [19] developed a fixed rank kriging model to simultaneously handle skewness and nonhomogeneity and adopted a Bayesian framework for posterior distribution sampling and spatial prediction. Zhang et al. [20] adopted a kriging model to approximate the inaccessible point in one dimension and combined multiple predictions using Bayesian inference to improve the precision of function estimation. Angelikopoulos et al. [21] drastically reduced the computational cost of uni-modal posterior probability distribution functions in large-scale structural dynamics simulation by combining adaptive kriging with the Bayesian inference of models. Plumlee [22] presented a Bayesian calibration method that considers the orthogonality between the kriging model gradient and the prior distribution on the model bias to mitigate the suboptimally broad problem of the posterior of the calibration parameter and model bias. In statistical analysis and prediction, the interval hypothesis test method requires that the analysis and calculation data follow a normal distribution. An error in normal distribution assumption is usually made when employing Bayesian inference for computational model validation (e.g., [8]). In practical applications, however, the model validity is questioned as the assumption often does not conform to reality. Jiang et al. [23] investigated the influence of a normality assumption on the precision of model quantitative validation with data uncertainty using a Bayesian probabilistic approach. Testing the normality of the data and converting the non-normality data by an appropriate transformation method are necessary for model validation. Various approaches for non-normality data transformation, including the Rosenblatt transformation, Nataf transformation, power and modulus transformations, and Pericchi’s Bayesian method, were discussed by the research of Rebba and Mahadevan [24]. Each transformation technique may be selected according to different specific research questions. Box–Cox transformation [25], which is a power transformation model that is based on the maximum likelihood method, enable linear regression models to satisfy the linearity, homogeneity of variance, independence, and normality without losing information. Jiang et al. [23] converted non-normality data using the Box–Cox method to promote the model accuracy. Bhardwaj et al. [26] utilized a response surface methodology to develop a prediction model and employed the Box–Cox transformation to pursue the non-normality transformation for the surface roughness prediction in the turning of AISI 1019 steel. Currently, the Bayesian kriging method is extensively employed in some fields, including mechanical design, structural optimization, and fluent analysis. Choi et al. [27] addressed the epistemic uncertainty in a structural reliability analysis using the Bayesian kriging approach, which treats the probability as a random variable and establishes the probability density function (PDF) of the limit state function with favorable accuracy. Romero et al. [28] applied a kriging interpolator that includes Monte Carlo computation of Bayesian inference and sensitivity analysis to predict the values of the elastic elements that correspond to an alternative beam T-junction model for optimizing the behaviors of the upper structures of buses and coaches. Belligoli et al. [29] employed a Bayesian calibration of a computational fluid dynamics (CFD) model and the kriging metamodel of a real process to quantify the epistemic uncertainty of ultrasonic flow meter measurements in non-ideal flow conditions. Zelaia et al. [30] developed a kriging surrogate model and conducted an inverse solution using a Bayesian framework to yield robust values for the properties and quantify the property uncertainty in the characterization of mechanical behaviors by instrumented indentation. Sen et al. [31] developed surrogate models based on a modified Bayesian kriging method using the computed pseudo-turbulent stresses of mesoscale simulations to quantify and evaluate the effect of velocity fluctuations for problems of shock–particle interactions. Kim et al. [32] addressed a Bayesian statistical procedure to calibrate the key parameters unknown prior in the model and test the data uncertainty for the analysis problem of piston insertion into the housing in a pyrotechnically actuated device, where the approach was employed. Im and Park [33] presented a particle swarm optimization procedure based on surrogate models and employed Bayesian statistics to obtain more reliable results for the structure optimization of a hub sleeve. Jo et al. [34] developed two adaptive variable-fidelity kriging surrogate models and integrated them with Bayesian-based and difference-based dynamic fidelity indicators, which were Metals 2019, 9, 493 4 of 15 formulated as probabilistic model validation metrics to quantify model-form uncertainties into an efficient global optimization design framework for the design problems. In the material processing field, the Bayesian kriging method has been gradually applied. Jin et al. [35] proposed a Bayesian model calibration method based on likelihood estimation to address the experimental data uncertainties, including inherent error and insufficient data in the material parameter estimation of solder alloy, and accurately predict fatigue failure. Karandikar et al. [36] applied Bayesian inference to estimate the model constants of the Taylor tool life equation for milling and turning operations while considering the uncertainty and predict the tool life by a PDF. Nannapaneni et al. [37] employed Bayesian networks to integrate and quantify the multi-source uncertainties in the performance prediction of injection and welding processes. Kikuchi et al. [38] described a Bayesian kriging optimization method to effectively and accurately determine the grain boundary atomic structures of complex metal oxide materials. Zelaia and Melkote [39] created a Gaussian surrogate model to replace the finite element simulation during the machining process and utilized a Bayesian inference-based calibration method to capture the uncertainty of the Johnson–Cook parameters in experimental orthogonal cutting tests. In this paper, the kriging metamodel with non-normal data is quantitatively calibrated and validated by the proposed Bayesian interval hypothesis testing method. To improve the accuracy of the model evaluation, the Anderson–Darling (A–D) goodness-of-fit approach is introduced for the normality test of sample data, and the non-normal data are transformed via the Box–Cox method. The Bayes factor and confidence are calculated to quantitatively assess the reliability of kriging metamodels established by simulation and experimental training samples, and a sensitivity analysis is performed by changing the threshold ε value to evaluate the influence on model precision.

2. Bayesian Statistics Inference Methodology

2.1. Bayesian Statistics Inference Modeling Temperature is treated as a single output variable in this study; thus, for the single output model, let ye and yp represent the experimental result and the prediction result, respectively. Then, two simple hypotheses—a null hypothesis (H0 : ye = yp) and an alternative hypothesis (H1 : ye , yp)—can be defined for a certain point. If the null hypothesis is valid, the model is accepted, whereas the model is rejected when the alternative hypothesis is valid. In Bayesian interval hypothesis testing, let d = ye yp represent the difference between the experimental result and the prediction result of − the model. The null hypothesis can be expressed as H : d ε, and the alternative hypothesis can be 0 | | ≤ expressed as H : d > ε, where ε is the present threshold. 1 | | In Bayesian hypothesis testing, each prediction difference d is assumed to have a PDF or likelihood function, namely, d H L(d H ) and d H L(d H ). When the prior distribution of d is unknown, | 0 ∼ | 0 | 1 ∼ | 1 the Gaussian distribution is adopted followed by further updating based on the Bayesian approach. Usually, we make the following assumptions: 2 (1) Prediction difference d = d , d , ... , dn follows the normal distribution N µ, σ with the { 1 2 } variance σ2, which is estimated from experimental or simulation data, where n is the number of sample points; (2) Prior PDF f (µ) from the null hypothesis and alternative hypothesis follows the normal distribution N ρ, τ2 . In the prior PDF, let the mean ρ = 0 and the variance τ2 = σ2 [40,41], i.e., assume that the prior information equals the observation values. To reflect the extent to which sample information and prior distribution support the null hypothesis, the Bayes factor is introduced, i.e., likelihood ratio in the interval hypothesis test. Based on the Bayesian Metals 2019, 9, 493 5 of 15 theorem and given validation data, the expression of the interval-based Bayes factor is obtained as follows [9]: R ε Metals 2019, 9, x FOR PEER REVIEWf ( Data H ) L(d µ) f (µ)dµ 5 of 16 0 ε | Bi = | = R ε − R (1) f (Data H1) − L(d µ) f (µ)dµ + ∞ L(d µ) f (µ)dµ According to the Bayesian theorem,| 𝑓(𝜇|𝒅) ∝ 𝐿(𝒅|𝜇)𝑓(𝜇) εand let 𝐾= 𝑓(𝜇|𝒅) d𝜇; Equation −∞ | | R ε (1) can be simplified as f ( d) L(d ) f ( ) K = f ( d) According to the Bayesian theorem, µ µ µ and let ε µ dµ; Equation (1) | ∝ | − | can be simplified as 𝑓 (𝜇|𝑑) 𝑑𝜇 Rε 𝐾 𝐵 = f (µ d)dµ = (2) 𝑓 (𝜇|𝑑)ε 𝑑| 𝜇+ 𝑓 (𝜇|𝑑) 𝑑𝜇 1−𝐾K Bi = Rε − R = (2) − f (µ d)dµ + ∞ f (µ d)dµ 1 K ε − where K is calculated as −∞ | | where K is calculated as ! ! λ𝜆2 −𝜇µ0 λ𝜆1 −𝜇µ0 K𝐾=𝛷= − −𝛷 − (3) Φ 𝜎 Φ 𝜎 (3) σ0 − σ0 𝛷(⋅) 𝜆 =−𝜀 𝑛𝜏2+𝜎2 𝜆 =𝜀 𝑛𝜏 +𝜎2 2 wherewhere Φ( ) followsfollows aa standardstandard normalnormal distributiondistribution withwith λ1 = ε√√nτ + σ , λ2 = √ε √nτ + σ, , _ 2 2 2· 2 2 2 2 − 𝜎σ0 =𝜎= σ 𝜏τ ,, and and µ𝜇0 =(𝑛𝑒= neτ̄𝜏 ++𝜌𝜎ρσ )//√𝑛𝜏nτ +𝜎+ σ . FigureFigure 1a1a demonstrates the geometric meaning forfor twotwo hypotheses,hypotheses, wherewhere ε𝜀 = -𝜀 εandand 𝜀ε =𝜀= .ε . 1 − 2 InIn the the figure, figure, KK isis the the shaded shaded part part in in the the interval interval ((𝜀ε,,𝜀ε).. The The variables variables KK andand 1- 1 KK representrepresent the the area area 1 2 − ofof the the posterior posterior probability probability density density with with the the null null hypothesis hypothesis and and the the area area of of the the posterior posterior probability probability densitydensity with with the the alternative alternative hypothesis, hypothesis, respectively. respectively. The The Bayes Bayes factor factor represents represents the the area area ratio ratio of of the the twotwo hypotheses. hypotheses. Figure Figure 1b1b shows the Bayes factorfactor correspondingcorresponding toto didifferentfferent probabilityprobability densitiesdensities in in a afixed fixed interval. interval. WhenWhen thethe meanmean value is constant, th thee smaller the the variance variance is, is, the the steeper steeper the the function function curvecurve is is and and the the larger larger the the KK valuevalue is. When When the validation data support thethe modelmodel ((d|𝑑| ≤𝜀ε),), the the area area of | | ≤ ofthe the posterior posterior probability probability density density will will increase increase with with the the null null hypothesis, hypothesis, i.e., i.e., the the area area of the of the shaded shaded part partin Figure in Figure1b will 1b increase;will increase; otherwise, otherwise, the area the area will decrease.will decrease.

ε2 f(μ|d) B >1 f(μ|d) K= f(μ|d)dμ 3 ∫ε1

B=B=K/(1-K) B1<1 B2=1 K

ε1 μ' ε2 μ ε1 μ' ε2 μ

(a) (b)

Figure 1. Interval-based Bayes factor. (a) Geometric meaning of Bayes factor; (b) Bayes factors B of Figure 1. Interval-based Bayes factor. (a) Geometric meaning of Bayes factor; (b) Bayes factors 𝐵 iof differentdifferent probability probability density density functions functions (PDFs). (PDFs). 2.2. Assessment of Model Confidence 2.2. Assessment of Model Confidence The reasonable initial interval threshold ε, which determines the value of K in Equation (2) should The reasonable initial interval threshold 𝜀, which determines the value of K in Equation (2) be established before model validation. When the threshold tends to be infinity, the K value is infinitely should be established before model validation. When the threshold tends to be infinity, the K value close to 1, and the uncertainty of the Bayes factor will increase, which indicates that the model is is infinitely close to 1, and the uncertainty of the Bayes factor will increase, which indicates that the always acceptable. Conversely, when the threshold tends to zero, the K value will also approach an model is always acceptable. Conversely, when the threshold tends to zero, the K value will also infinitesimal, and the Bayes factor will approach zero, which indicates that the data will not support the approach an infinitesimal, and the Bayes factor will approach zero, which indicates that the data will null hypothesis, namely, reject the model. A sensitivity analysis is performed to quantitatively study not support the null hypothesis, namely, reject the model. A sensitivity analysis is performed to the influence of the various threshold ε value on the model reliability. To facilitate the comparison of quantitatively study the influence of the various threshold 𝜀 value on the model reliability. To values in a larger range, the non-negative Bayes factor is usually transformed into logarithmic form, facilitate the comparison of values in a larger range, the non-negative Bayes factor is usually i.e., bi = ln(Bi). transformed into logarithmic form, i.e., 𝑏 =𝑙𝑛(𝐵). Kass and Raftery [41] argue that the logarithmic Bayes factor between 0 and 1 is insufficient to support the null hypothesis, a factor between 3 and 5 is strong evidence, and a factor greater than 5 is very strong evidence. On the same order of magnitude, a negative 𝑏 is considered to reject the null hypothesis 𝐻 and support the alternative hypothesis 𝐻.

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Kass and Raftery [41] argue that the logarithmic Bayes factor between 0 and 1 is insuﬃcient to support the null hypothesis, a factor between 3 and 5 is strong evidence, and a factor greater than 5 is very strong evidence. On the same order of magnitude, a negative bi is considered to reject the null hypothesis H0 and support the alternative hypothesis H1. The prior probabilities of two hypotheses are assumed to be Pr(H ) = π and Pr(H ) = 1 π , 0 0 1 − 0 respectively. According to the Bayesian theorem, the model conﬁdence can be calculated by the validation data and expressed as follows:

B π κ = Pr(H Data) = i 0 (4) 0| B π + 1 π i 0 − 0 From Equation (4), B approaching indicates 100% conﬁdence and that the model will be i ∞ accepted. Conversely, Bi approaching 0 indicates 0% conﬁdence and that the model will be rejected. Additionally, the prior information in Equation (4) includes expert knowledge about model accuracy. Empirical knowledge is not available for reference in this study, which indicates that prior knowledge of the hypotheses is not available; thus, the unbiased assumption π0 = 0.5 is employed.

2.3. Normality Test and Power Transformation Normal distribution assumption is the basis of Bayesian hypothesis testing, and the normality test of data, for example, a t test or F test, is necessary in the statistical analysis of actual problems. The total validation data of a non-normal distribution are calculated in vain. Statistical analysis, such as regression analysis and variance analysis, also requires normality testing of data to be performed in advance. The test methods include the Kolmogorov–Smirnov (K–S), Shapiro–Wilk, and Ryan–Joiner methods, and some goodness of fit tests, such as the chi-square, Cramer–von Mises, and Anderson–Darling (A–D) methods [42]. The A–D goodness of fit test [43] is an algorithm based on the empirical cumulative distribution function (ECDF). This test compares the empirical cumulative distribution of the sample data with the expectation distribution when the hypothetical data are normally distributed. If the difference is too large, the hypothesis of the total normal distribution will be denied. This method is suitable for small and large samples. The A–D test strengthens the consideration of tail data, which causes a correction of the K–S test with higher sensitivity. Therefore, the A–D test is employed to test the normality of the errors between the prediction and validation data of the model, i.e., null hypothesis H0: error data follow a normal distribution; alternative hypothesis H1: error data does not follow a normal distribution. This method employs A–D statistics to measure the extent to which data is subject to a particular distribution. For specific data sets and distributions, the smaller the A–D value, the better the distribution fits the data. The corresponding P value obtained via the A–D value can be used to verify whether the data are derived from the selected distribution. Comparing the P value with the present value α (usually 0.05 or 0.10), if the P value is larger, the null hypothesis H0 will be accepted and the validation data are normal; otherwise, H0 will be rejected. Normal transformation of non-normal data is very important for model validation. The power transformation method is used to transform data that do not follow a normal distribution. As a data preprocessing technique, the Box–Cox transformation performs normal pre-ranking transformation of the original data without strict requirements and assumptions. Information processing is more accurate and easier to implement, and the strict closed interval of the response distribution is not required; thus, the Box–Cox approach is extensively employed [44]. Let d = (d1, d2, ... , dn), where d1, d2, ... , dn represents the original data of a single output variable with n points. The Box–Cox transformation is expressed as [45] ( [(c + d)λ 1]/λ λ , 0 T = − (5) d ln(c + d) λ = 0 Metals 2019, 9, 493 7 of 15

where Td represents the transformed data; c is a constant that is utilized to make (c + d) values positive when some d values are negative, as the Box–Cox transformation cannot be employed for negative values. λ is the transform coeﬃcient, and the main task of data transformation is to obtain an appropriate estimation of the transform parameters. Generally, the estimation can be obtained by the following maximum likelihood estimation method: Deﬁne the likelihood function as

n n X L(λ) = ln(σ2) + (λ 1) ln(d ) (6) − 2 − i i=1 where σ2 is the variance of transformed data. The optimal solution of λ can be obtained by solving the unconstrained nonlinear optimization problem of Equation (6). The optimization methods of the parameter solution include the steepest descent method, genetic algorithm, and then Newton–Raphson method.

3. Results and Discussion The Bayesian statistical inference model is used to analyze the test sample data and quantitatively verify the kriging metamodel constructed from simulation and experimental training samples in previous research [2]. Orig_Krig_Test is the raw data d1 in the initial model; Orig_Cal_ Test is the 2 raw data d in the calibration model; Trans_Krig_Test is the transformed data Td1 in the initial model; and Trans_Cal_Test is the transformed data Td2 in the calibration model.

3.1. Normality Test Analysis The A–D test method is used in the normality test of the corresponding sample data before the Bayesian hypothesis test is performed. Figure2 displays a probability plot of the corresponding error values, where the confidence level α = 0.05. Figure2a shows a normality test of 70 test sample data of the kriging model, in which the abscissa represents the difference value. The difference mean of the predicted value and the sample data is represented by the last two digits of the decimal, and the variance is 0.1465, which indicates the small total difference and the relatively concentrated and low discrete degree difference data. However, the drawing points do not completely coincide with the fitted line and the value P = 0.008 < 0.05; thus, these sample data can only be regarded as an approximate normal distribution. Figure2b shows a normal test of the di fference values of 118 test sample data for the calibrated kriging model. The difference means and standard deviation of the predicted value and the sample value are smaller in the calibrated model, which indicates that the difference data are more concentrated and the discrete degree is lower. As shown in Figure2b, most of the data points of the distribution fall outside the three normal fitting curves, and the graphs are not well fitted with P values less than 0.005, which indicates that the 118 sample data do not follow a normal distribution. Therefore, non-normal data transformation should be performed to obtain a set of data that approximately follow a normal distribution, and then the Bayesian hypothesis test can be performed. Metals 2019, 9, x FOR PEER REVIEW 8 of 16

Metals 2019, 9, x FOR PEER REVIEW 8 of 16 Metals 2019, 9, 493 8 of 15 -1.0 -0.5 0.0 0.5 1.0

99.9 Orig_Krig_Test -1.0 -0.5 Orig0.0_Cal_Test0.5 1.0 Orig_Krig_Test Orig_Krig_Test Orig_Cal_Test 99.9 Mean -0.03177 Orig_Krig_Test 99 STDEV 0.3827 Mean -0.03177 (a) (b) N 70 99 STDEV 0.3827 95 AD 1.061 (a) (b) N 70 90 P 0.008 95 AD 1.061 Orig_Cal_Test 80 P 0.008 90 Mean 0.02756 70 Orig_Cal_Test 80 STDEV 0.3008 60 Mean 0.02756 N 118 7050 STDEV 0.3008 60 AD 4.938 40 N 118 5030 P ＜0.005 40 AD 4.938 20

Probability Percent 30 P ＜0.005 2010 Probability Percent 5 10 5 1

1 0.1 -1 0 1 0.1 -1 0 1 Figure 2. Anderson–Darling (A–D) tests of the original difference data: (a) initial and (b) calibrated Figure 2. Anderson–Darling (A–D) tests of the original diﬀerence data: (a) initial and (b) calibrated kriging model. Figurekriging 2. model. Anderson–Darling (A–D) tests of the original difference data: (a) initial and (b) calibrated kriging model. 3.2.3.2. Box–CoxBox–Cox Transformation Transformation Results Results 3.2. Box–Cox Transformation Results AsAs both both sets sets of of sample sample data data contain contain negative negative values, values, the the constant constantc is c introduced is introduced to ensure to ensure that that the valuethe Asvalue of both(c +of setsd )(𝒄+𝒅is of positive. sample) is positive. data Let c _containKrig Let= negative𝑐1.1837 _ 𝐾 , c 𝑟𝑖𝑔_values,Cal = 0.4203 the= 1.1837 , 𝑐 constant _( 𝐶= 𝑎𝑙µ +c is min= introduced 0.4203(=) and|𝜇| + add| 𝑚𝑖𝑛to them ensure|) and to eachthat add | | | | theelementthem value to of eachof the (𝒄+𝒅 krigingelement) is model ofpositive. the and kriging calibrationLet 𝑐 model _ 𝐾 model. 𝑟𝑖𝑔and calibration The = 1.1837 , 𝑐 nonlinear _ 𝐶 model. 𝑎𝑙 optimization =The 0.4203(= nonlinear|𝜇| + problem|𝑚𝑖𝑛 |optimization) and is solved add themaccordingproblem to each is to solved Equationelement according (6)of tothe obtain tokriging Equation the model optimal (6) toand solution.obtain calibration the The optimal solutionmodel. solution. processThe nonlinear The is shownsolution optimization in process Figure3 ,is problemandshownλ_Krig inis Figuresolved= 1.31 3,according and andλ _𝜆_𝐾𝑟𝑖𝑔=1.31Cal to= Equation0.15 are estimated. and(6) to 𝜆_𝐶𝑎𝑙=0.15 obtain the optimal are estimated. solution. The solution process is shown in Figure 3, and 𝜆_𝐾𝑟𝑖𝑔=1.31 and 𝜆_𝐶𝑎𝑙=0.15 are estimated. Cofidence level lower limit Cofidence level upper limit Cofidence level lower limit Cofidence level upper limit 7 1.0 Cofidence level lower limit CofidenceLambda level upper limit Cofidence level lower limit CofidenceLambda level upper limit 76 （CL 95.0% ） 1.00.9 （CL 95.0%） Lambda EstimateLambda 1.31 Estimate 0.15 （CL 95.0% ） 0.90.8 （CL 95.0%） 65 CL lowe r limit 0.96 CL lowe r limit -0.04 Estimate 1.31 CLEstimate upper limit 0.33 0.15 CL upper limit 1.75 0.80.7 54 CL lowe r limit 0.96 CL lowe r limit -0.04 CL upper limit 0.33 CL uppe r limit 1.75 0.70.6 STDEV

4 STDEV 3 0.5 0.6

2 STDEV

STDEV 3 0.4 0.5 1 0.3 2 0.4 Limit Limit 0 0.2 1 -2 -1 0 1 2 3 4 5 0.3 -1 0 1 2 3 Lambda Limit Lambda Limit 0 0.2 -2 -1 0 1 2 3 4 5 -1 0 1 2 3 Lambda(a) Lambda(b)

FigureFigure 3. 3.Maximum Maximumlikelihood likelihood estimationestimation ofofλ 𝜆:(: (aa)) KrigingKriging model’smodel’s estimationestimation ofofλ 𝜆;(; b(b)) Calibrated Calibrated krigingkriging model’s model’s estimation estimation(a) of ofλ .𝜆. (b) Figure 3. Maximum likelihood estimation of 𝜆: (a) Kriging model’s estimation of 𝜆; (b) Calibrated The Box–Cox converted data is obtained according to Equation (5). Figures4 and5 show the krigingThe Box model’s–Cox estimationconverted of data 𝜆. is obtained according to Equation (5). Figure 4 and Figure 5 show originalthe original data data of the of kriging the kriging model model and calibration and calibration model model and the and A–D the normality A–D normality test probability test probability maps of the converted sample data, respectively. Figure4a shows the original data, and Figure4b displays

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The Box–Cox converted data is obtained according to Equation (5). Figure 4 and Figure 5 show the original data of the kriging model and calibration model and the A–D normality test probability maps of the converted sample data, respectively. Figure 4a shows the original data, and Figure 4b displays the probability map of the transformed data. Compared with Figure 4a, the data points of transformed data primarily fall within the three normal fitting curves with the A–D value reduced by nearly one-half and the P value 0.069 > 0.05, which indicates that the transformed data follow a normal distribution and can be applied to the Bayesian hypothesis test. Similarly, Figure 5a and b display probability plots of the original sample data and the transformed data. In Figure 5a, the majority of the data points fall outside these three normal fitting curves, which indicates that the original data poorly fits. However, after the Box–Cox transformation, Metalsthe data2019, 9points, 493 almost fall within the fitting curves, the A–D value decreases by an order9 of 15of magnitude in this process, and the P value is 0.264 > 0.05. The transformed data of the two models exhibit a more normal distribution, and the spatial and temporal distribution data of the injection the probability map of the transformed data. Compared with Figure4a, the data points of transformed mechanism in the squeeze casting process can obtain the expected effect with the Box–Cox data primarily fall within the three normal ﬁtting curves with the A–D value reduced by nearly one-half transformation. Subsequently, the Bayesian interval hypothesis test is employed to study the effects and the P value 0.069 > 0.05, which indicates that the transformed data follow a normal distribution of non-normal and normal distribution data in the model. and can be applied to the Bayesian hypothesis test.

-1 0 1

99.9 Orig_Krig_Test Trans_Krig_Test Orig_Krig_Test Mean -0.03177 99 STDEV 0.3827 (a) (b) N 70 95 AD 1.061 90 P 0.008 Trans_Krig_Test 80 Mean 0.1781 70 STDEV 0.3807 60 N 70 50 40 AD 0.688 30 P 0.069 20 Probability Percent 10 5

0.1 -1 0 1

Metals 2019, 9, x FOR PEER REVIEW 10 of 16 Figure 4. A–D tests of original and transformed diﬀerence data in the kriging model: (a) original and (Figureb) transformed 4. A–D tests diﬀerence of original data. and transformed difference data in the kriging model: (a) original and (b) transformed difference data. -3 -2 -1 0 1 Orig_Cal_Test Trans_Cal_Test 99.9 Orig_Cal_Test Mean 0.02756 99 (a) (b) STDEV 0.3008 N118 95 AD 4.938 P<0.005 90 Trans_Cal_Test 80 Mean -0.9120 70 STDEV 0.5732 60 N118 50 AD 0.455 40 P0.264 30

Probability Percent 20

10 5

0.1 -1.0 -0.5 0.0 0.5 1.0 Figure 5. A–D tests of original and transformed difference data in the calibrated model: (a) original Figureand (b )5. transformed A–D tests of di originalfference and data. transformed difference data in the calibrated model: (a) original and (b) transformed difference data. Similarly, Figure5a,b display probability plots of the original sample data and the transformed 3.3.data. Bayesian In Figure Hypothesis5a, the majority Testing of the data points fall outside these three normal fitting curves, which indicatesBoth the that kriging the original model data and poorlythe calibrated fits. However, model ar aftere validated the Box–Cox by Bayesian transformation, hypothesis thetesting. data According to Equation (3) and Equation (4), the Bayes factor and confidence, respectively, of the corresponding model are solved. Table 1 lists the relevant calculation parameters of the Bayesian hypothesis testing, where the threshold 𝜀=𝜎/4. For example, for 70 sample data of the kriging model, the mean 𝜇 = −0.0318 and the variance 𝜎 = 0.1465 are obtained. Considering 𝜌=0, 𝜏=𝜎 = 0.3827, and 𝜀=𝜎/4 = 0.0957 and substituting into Equations (3) and (4), the Bayes factor 𝑏_ = 2.1275 and confidence 𝜅_ = 89.35% can be calculated. The other parameters of the model are obtained by the same approach. When the logarithmic Bayes factor is larger than 5, the null hypothesis is supported. However, the value between 2 and 5 represents powerful evidence [41]. As shown in Table 1, after Box–Cox transformation, the Bayes factor is greater than 5 and the confidence is 100% in both the original kriging model and the calibrated model, which indicates that the null hypothesis is true. For the original difference d, as the sample data of the two models do not follow a normal distribution, the results of the Bayesian hypothesis testing based on the normal distribution may not be reliable. For example, for the original kriging metamodel, the Bayes factor is 2.1275, which indicates that the result supports the null hypothesis to a certain extent but does not fully support it. However, for hypothesis testing of non-normally distributed data, even if its confidence reaches 100% and the Bayes factor is larger than 5, the conclusion that the null hypothesis is true cannot be directly obtained.

Table 1. Bayesian hypothesis testing for the kriging and calibrated models.

Test Data points Mean μ Variance σ2 Bayes factor (ln) Confidence (%) Orig_Krig_Test 70 –0.0318 0.1465 2.1275 89.3543 Trans_Krig_Test 70 0.1781 0.1449 31.4951 100 Orig _Cal_Test 118 0.0276 0.0905 2.3639 91.4037 Trans_Cal_Test 118 –0.9120 0.3286 16.6852 100

Metals 2019, 9, 493 10 of 15 points almost fall within the fitting curves, the A–D value decreases by an order of magnitude in this process, and the P value is 0.264 > 0.05. The transformed data of the two models exhibit a more normal distribution, and the spatial and temporal distribution data of the injection mechanism in the squeeze casting process can obtain the expected effect with the Box–Cox transformation. Subsequently, the Bayesian interval hypothesis test is employed to study the effects of non-normal and normal distribution data in the model.

3.3. Bayesian Hypothesis Testing Both the kriging model and the calibrated model are validated by Bayesian hypothesis testing. According to Equations (3) and (4), the Bayes factor and confidence, respectively, of the corresponding model are solved. Table1 lists the relevant calculation parameters of the Bayesian hypothesis testing, where the threshold ε = σ/4. For example, for 70 sample data of the kriging model, the mean µ = 0.0318 and the variance σ2 = 0.1465 are obtained. Considering ρ = 0, τ = σ = 0.3827, OK − OK OK and ε = σOK/4 = 0.0957 and substituting into Equations (3) and (4), the Bayes factor bi_OK = 2.1275 and confidence κi_OK = 89.35% can be calculated. The other parameters of the model are obtained by the same approach. When the logarithmic Bayes factor is larger than 5, the null hypothesis is supported. However, the value between 2 and 5 represents powerful evidence [41]. As shown in Table1, after Box–Cox transformation, the Bayes factor is greater than 5 and the confidence is 100% in both the original kriging model and the calibrated model, which indicates that the null hypothesis is true. For the original difference d, as the sample data of the two models do not follow a normal distribution, the results of the Bayesian hypothesis testing based on the normal distribution may not be reliable. For example, for the original kriging metamodel, the Bayes factor is 2.1275, which indicates that the result supports the null hypothesis to a certain extent but does not fully support it. However, for hypothesis testing of non-normally distributed data, even if its confidence reaches 100% and the Bayes factor is larger than 5, the conclusion that the null hypothesis is true cannot be directly obtained.

Table 1. Bayesian hypothesis testing for the kriging and calibrated models.

Test Data Points Mean µ Variance σ2 Bayes Factor (ln) Conﬁdence (%) Orig_Krig_Test 70 0.0318 0.1465 2.1275 89.3543 − Trans_Krig_Test 70 0.1781 0.1449 31.4951 100 Orig _Cal_Test 118 0.0276 0.0905 2.3639 91.4037 Trans_Cal_Test 118 0.9120 0.3286 16.6852 100 −

Figure6 shows the e ffect of the Bayesian interval hypothesis test on the non-normal data and normal distribution data in the model. The figure is divided into four graphs (a)–(d) corresponding to Table1. All curves show di fferent increasing trends with an increase in the threshold value. In Figure6a,b, for raw data that do not follow a normal distribution, the corresponding Bayes factors and Bayesian confidence are fairly close, where the Bayes factors are greater than 2, and the confidence is approximately 90%. During the entire process of threshold change, the Bayes factor and confidence values of the calibrated kriging model are slightly higher than those of the initial kriging model. However, for the hypothesis test results of non-normal distribution data, the reliability of the conclusions needs further exploration. Figure6c,d illustrate the Bayes factor and confidence of Box–Cox transformed data. The two parameter values produce results that are opposite to those in the first two figures during the entire process of threshold change. Changes in the Bayes factor and confidence curve of the initial kriging model are faster than those of the calibrated model. Table1 shows the final values. Although the Bayes factor of the initial model is nearly twice that of the calibration model, the Bayes factors of both models are greater than 5. Thus, the null hypothesis is supported, and the accuracy of both models satisfies the requirements. The curve change of the initial kriging model is faster in the graph due to two aspects. The first aspect is that adequate selection of the training sample at the beginning Metals 2019, 9, 493 11 of 15 of the model construction facilitates the precision of the initial kriging model. The second aspect is that the normal distribution level of the transformed sample data of the initial model is worse than that of the calibrated model. Combining Figures4 and5, the normal test P value of the initial model is not large—only 0.069—and slightly larger than the significance level. This value indicates that the probability of following a normal distribution is only 5%. The P value of the calibrated kriging model is 0.264, which is three orders of magnitude greater than the P value of the original data and considerably larger than the P value and significance level of the original kriging model. The reliability of the results, which were produced by the Bayesian hypothesis test approach, is closely related to the normal distribution of the sample data: the higher the probability that the samples follow a normal distribution is, the higher the reliability of the hypothesis test results. Therefore, for both kriging models that support the null hypothesis, the accuracy may satisfy the requirements. Both models can beMetals employed 2019, 9, x FOR forthe PEER next REVIEW study; however, the calibrated kriging model is preferred to some extent.12 of 16

5 100

3 80

1 60

-1 40 Bayes factor(ln) Bayes Bayes Confidence(%)

-3 20

Kriging original model Kriging original model Kriging calibration model Kriging calibration model -5 0 0 30 60 90 120 0306090120 eps scale(%) eps scale(%) (a) (b) 40 100

30 80

20 60

10 40 Bayes factor(ln) Bayes factor(ln) Bayes

0 20 Kriging original model Kriging original model Kriging calibration model Kriging calibration model -10 0 03060901200306090120 eps scale(%) eps scale(%) (c) (d)

Figure 6.6. Bayesian hypothesis testing for the krigingkriging andand calibratedcalibrated models:models: (a) Bayes factor ofof thethe original data; ( (bb)) Bayes Bayes confidence conﬁdence of of the the original original data; data; (c) ( cBayes) Bayes factor factor of the of thetransformed transformed data; data; (d) (Bayesd) Bayes confidence conﬁdence of the of thetransformed transformed data. data.

3.4. Sensitivity Analysis Analysis InIn thethe nullnull hypothesishypothesis andand thethe alternativealternative hypothesis,hypothesis, εɛ isis thethe presentpresent threshold.threshold. The K valuevalue inin Figure1 1 and and Equation Equation (2) (2) depends depends on on the the threshold threshold εɛ of the initial interval. Therefore, the reasonable thresholdthreshold hashas beenbeen determineddetermined beforebefore modelmodel veriﬁcation,verification, andand thethe eeffectﬀect ofof variousvarious εɛ within this range on model reliability was quantitatively explored, as shown in Figures 7 and 8. In the figures, the X axis is the change in the threshold ɛ with the maximum, the left Y-axis represents the Bayesian confidence, which corresponds to the red curves, and the right Y-axis is the Bayes factor that corresponds to the blue curves.

Metals 2019, 9, 493 12 of 15 on model reliability was quantitatively explored, as shown in Figures7 and8. In the ﬁgures, the X axis is the change in the threshold ε with the maximum, the left Y-axis represents the Bayesian conﬁdence, Metals 2019, 9, x FOR PEER REVIEW 13 of 16 which corresponds to the red curves, and the right Y-axis is the Bayes factor that corresponds to the Metals 2019, 9, x FOR PEER REVIEW 13 of 16 blue curves. 0 1224364860728496108120 100 35 0 1224364860728496108120 100 Confidence of original data 35 Confidence of transformed data Confidence of original data Confidence of transformed data 80 27 80 27

60 19 60 19

40 11 40 11 factor(ln) Bayes Bayes factor(ln) Bayes Acception Confidence(%)

Acception Confidence(%) 20 3 20 3 Bayes factor of original data Bayes factor of transformedoriginal data data 0 -5 0 1224364860728496108120 Bayes factor of transformed data 0 -5 0 1224364860728496108120eps scale(%) eps scale(%) Figure 7. Effect of various ɛ values for the kriging model with original and transformed data. Figure 7. Eﬀect of various ε values for the kriging model with original and transformed data. Figure 7. Effect of various ɛ values for the kriging model with original and transformed data. 0 1224364860728496108120 100 20 0 1224364860728496108120 100 Confidence of original data 20 Confidence of transformedoriginal data data 80 Confidence of transformed data 14 80 14

60 8 60 8

40 2 40 2 Bayes factor(ln) Acception confidence(%) Acception Bayes factor(ln) 20 -4 Acception confidence(%) Acception 20 Bayes factor of orignal data -4 Bayes factot of transformed data Bayes factor of orignal data 0 Bayes factot of transformed data -10 0 1224364860728496108120 0 -10 0 1224364860728496108120eps scale(%) eps scale(%) Figure 8. Eﬀect of various ε values for the calibrated model with original and transformed data. Figure 8. Effect of various ɛ values for the calibrated model with original and transformed data. Figure 8. Effect of various ɛ values for the calibrated model with original and transformed data. Figure 7 shows the influence of the threshold ɛ on the Bayesian hypothesis test for two sets of sampleFigure data 7 inshows the krigingthe influence model. of Allthe curvesthreshold show ɛ on different the Bayesian increasing hypothesis trends test as forthe two threshold sets of increases.sample data Both in the the Bayes kriging factor model. and theAll confidence curves show gradient different of the increasing transformed trends data as are the greater threshold than thoseincreases. of the Both original the Bayes data, factor which and indicates the confidence a relatively gradient gradual of changethe transformed in the original data aredata. greater When than the those of the original data, which indicates a relatively gradual change in the original data. When the

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Figure7 shows the influence of the threshold ε on the Bayesian hypothesis test for two sets of sample data in the kriging model. All curves show different increasing trends as the threshold increases. Both the Bayes factor and the confidence gradient of the transformed data are greater than those of the original data, which indicates a relatively gradual change in the original data. When the Bayesian confidence of the transformed data exceeds 90%, which is higher than the final confidence of the original data, the growth rate of its Bayes factor gradually accelerates with an increase in ε. The curves in Figure8 illustrate the results of the Bayesian hypothesis testing of the kriging calibration model when four curves intersect at one point. In the interval, the Bayes factor and confidence of the original sample data are slightly larger than those of the transformed data. When this critical point is passed, however, the two curves of the converted data sharply increase. According to the variation in the curves of the two graphs, the increase in ε has some effect on the hypothesis test results of non-normal data but slowly changes with low sensitivity. Conversely, for the sample data that follow a normal distribution, the change in ε has a considerable effect on the results of the hypothesis testing. Both the Bayes factor and confidence can rapidly increase within a small range of ε values. Therefore, the threshold has greater influence and higher sensitivity on normal distribution data.

4. Conclusions Based on a Bayesian framework, a Bayesian statistics inference model for validating a kriging metamodel is established. The Bayesian interval hypothesis test is performed using non-normal spatiotemporal distribution sample data and the normal data transformed by the Box–Cox transformation method to validate the reliability of the kriging metamodel. The metamodel was created by the simulation and experimental training samples for squeeze casting of aluminum alloys in our previous research, in which the temperature data is applicable in the range from 0 to 800 ◦C. The following conclusions are obtained: (1) The Anderson–Darling goodness-of-fit test results show that the original difference data of the injection mechanism, which does not follow or only approximately follows a normal distribution, needs further transformation for Bayesian hypothesis testing. (2) The Box–Cox transformation method successfully converts non-normal data into normal data, where the P value of Trans_Cal_Test is 0.264, which is substantially greater than the P value of 0.069 from the Trans_Krig_Test. For the hypothesis test based on a normal distribution, the hypothesis test results of the calibrated kriging model are more reliable to a certain extent. (3) The Bayesian interval hypothesis test is employed to study the effects of the non-normal data and normal distribution data in the model. For the original data that does not follow a normal distribution, the Bayes factor is greater than 2 and the confidence is approximately 90%. For these hypothesis test results of non-normal data, the reliability of the conclusion needs to be further explored. For the transformed data, the corresponding Bayes factors are greater than 5 and the confidence level is 100%, which indicates that both kriging models support the null hypothesis and that the accuracy satisfies the requirements. The metamodel can be employed in future studies. (4) The change in the threshold has a slow and low sensitivity influence on the hypothesis test results of the non-normal data. Conversely, for the sample data that follow a normal distribution, the influence is distinct. The Bayes factor and confidence can rapidly increase within a small range of values. Thus, the model accuracy can be further improved by controlling the value of the threshold.

Author Contributions: Conceptualization, D.Y. and J.D.; Data curation, X.S. and Y.Z.; Formal analysis, J.D.; Funding acquisition, D.Y.; Investigation, Y.Z.; Methodology, D.Y., and F.L.; Project administration, D.Y.; Software, X.S. and Y.Z.; Validation, D.Y. and F.L.; Visualization, X.S.; Writing—original draft, X.S.; Writing—review and editing, D.Y. Funding: This research was funded by the National Natural Science Foundation of China, grant number 51875209; the Natural Science Foundation of Guangdong Province, grant numbers 2017A030313320 and 2015A030312003; Guangxi Natural Science Foundation of China, grant number 2018GXNSFAA050111; and the Open Fund of Guangxi Key Lab of Manufacturing System and Advanced Manufacturing Technology, grant number 17-259-05S003. Metals 2019, 9, 493 14 of 15

Conﬂicts of Interest: The authors declare no conﬂict of interest. The funders had no role in the design of the study, in the collection, analyses, or interpretation of data, in the writing of the manuscript, or in the decision to publish the results.

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