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Classical and Bayesian predictions applied to Bacillus toxin production

View Article: PubMed Central - PubMed

ABSTRACT

Bacillus thuringiensis is a bacterium with unusual properties that make it useful for pest control in ecoagriculture. It can form a parasporal crystal containing polypeptides (also called delta-endotoxins). These entomopathogenic toxins are made during the stationary phase of the bacterial growth cycle and were initially characterized as an insect pathogen. Nowadays, the use of saturated two-level designs is very popular. This method is especially used in industrial applications where the cost of experiments is expensive. Standard classical approaches are not appropriate to analyze data from saturated designs. It is due to the fact that they only allow to estimate the main factor effects and cannot assure enough freedom degrees to estimate the error variance. In this paper, we propose the use of empirical Bayesian procedures to get inferences for data obtained from saturated designs, inspired from Hadamard matrices. The proposed methodology is illustrated by assuming a dataset to prove the model robustness. The comparison between the two studied mathematical techniques, based on mean square error values (MSE), revealed that Bayesian method (BM) was more accurate than least square method (LSM): for example, the results showed that 2002 and 2000.7 mg/l for experimental and predicted (BM) data were obtained against 2002 and 1991 mg/l for experimental and predicted (LSM) data. This suggested method could be generalized in several application fields in biological sciences.

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MSE variations using least square and Bayesian methods
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Fig1: MSE variations using least square and Bayesian methods

Mentions: Table 3 shows the observed and predicted concentrations of delta-endotoxins. The concentrations ranged from 1502 to 3571 mg l−1. The collected output data are generated from the experimental process using bootstrapping method (Efron 1979). The bootstrap is a type of Monte Carlo method applied based on observed data (Mooney and Duval 1993). The predictive distribution is presented in terms of the difference between the predicted and the observed data (Table 3). One hundred simulated samples of data are used for validation. The stopping criterion for training is a minimum value of the mean square error (MSE). In this study, the minimum value of MSE was taken as 10−6. The MSE was used as the criterion for the training and test data sets to compare the accuracy of the model. The enlarged versions of the simulation output based on Bayesian and linear models are presented in Fig. 1 to illustrate the difference between the mentioned variables. According to the figure, we showed that the majority of the peak points on the Bayesian method curve may indicate that this technique certainly increases goodness of model when compared to least squares method curve, and thus may improve the prediction rate. Differences in residuals between the two models are small when compared pairwise. However, the fact is that Bayesian model residuals are consistently smaller than the linear model residuals. It indicates the degree of robustness of Bayesian method compared to conventional data analysis methods [2002 and 2000.7 mg/l for experimental and predicted (LSM) data vs 2002 and 1991 mg/l for experimental and predicted (BM) data]. In fact, usual approaches such as least square method begin by fitting a model and then optimizing the model to obtain optimal operating settings. These methods do not account for any uncertainty in the parameters or in the form of the model. Bayesian approaches have been proposed recently to account for the uncertainty on the parameters of the model, assuming the model form is identified.Table 3


Classical and Bayesian predictions applied to Bacillus toxin production
MSE variations using least square and Bayesian methods
© Copyright Policy - OpenAccess
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5037111&req=5

Fig1: MSE variations using least square and Bayesian methods
Mentions: Table 3 shows the observed and predicted concentrations of delta-endotoxins. The concentrations ranged from 1502 to 3571 mg l−1. The collected output data are generated from the experimental process using bootstrapping method (Efron 1979). The bootstrap is a type of Monte Carlo method applied based on observed data (Mooney and Duval 1993). The predictive distribution is presented in terms of the difference between the predicted and the observed data (Table 3). One hundred simulated samples of data are used for validation. The stopping criterion for training is a minimum value of the mean square error (MSE). In this study, the minimum value of MSE was taken as 10−6. The MSE was used as the criterion for the training and test data sets to compare the accuracy of the model. The enlarged versions of the simulation output based on Bayesian and linear models are presented in Fig. 1 to illustrate the difference between the mentioned variables. According to the figure, we showed that the majority of the peak points on the Bayesian method curve may indicate that this technique certainly increases goodness of model when compared to least squares method curve, and thus may improve the prediction rate. Differences in residuals between the two models are small when compared pairwise. However, the fact is that Bayesian model residuals are consistently smaller than the linear model residuals. It indicates the degree of robustness of Bayesian method compared to conventional data analysis methods [2002 and 2000.7 mg/l for experimental and predicted (LSM) data vs 2002 and 1991 mg/l for experimental and predicted (BM) data]. In fact, usual approaches such as least square method begin by fitting a model and then optimizing the model to obtain optimal operating settings. These methods do not account for any uncertainty in the parameters or in the form of the model. Bayesian approaches have been proposed recently to account for the uncertainty on the parameters of the model, assuming the model form is identified.Table 3

View Article: PubMed Central - PubMed

ABSTRACT

Bacillus thuringiensis is a bacterium with unusual properties that make it useful for pest control in ecoagriculture. It can form a parasporal crystal containing polypeptides (also called delta-endotoxins). These entomopathogenic toxins are made during the stationary phase of the bacterial growth cycle and were initially characterized as an insect pathogen. Nowadays, the use of saturated two-level designs is very popular. This method is especially used in industrial applications where the cost of experiments is expensive. Standard classical approaches are not appropriate to analyze data from saturated designs. It is due to the fact that they only allow to estimate the main factor effects and cannot assure enough freedom degrees to estimate the error variance. In this paper, we propose the use of empirical Bayesian procedures to get inferences for data obtained from saturated designs, inspired from Hadamard matrices. The proposed methodology is illustrated by assuming a dataset to prove the model robustness. The comparison between the two studied mathematical techniques, based on mean square error values (MSE), revealed that Bayesian method (BM) was more accurate than least square method (LSM): for example, the results showed that 2002 and 2000.7 mg/l for experimental and predicted (BM) data were obtained against 2002 and 1991 mg/l for experimental and predicted (LSM) data. This suggested method could be generalized in several application fields in biological sciences.

No MeSH data available.