
==== Front
Sci Rep
Sci Rep
Scientific Reports
2045-2322
Nature Publishing Group UK London

71161
10.1038/s41598-024-71161-4
Article
Soft sensing modeling of penicillin fermentation process based on local selection ensemble learning
Huang Feixiang
Li Longhao lilonghao@sdut.edu.cn

Du Chuanxiang
Wang Shuang
Liu Xuefeng
https://ror.org/02mr3ar13 grid.412509.b 0000 0004 1808 3414 School of Electrical and Electronic Engineering, Shandong University of Technology, Zibo, 255000 China
2 9 2024
2 9 2024
2024
14 203496 6 2024
26 8 2024
© The Author(s) 2024
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ Open Access This article is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, which permits any non-commercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.
In the process of penicillin fermentation, there is a strong nonlinear relationship between the input eigenvector and multiple output vectors, which makes the prediction accuracy of the existing model difficult to meet the requirements of chemical production. Therefore, a local selective ensemble learning multi-objective soft sensing modeling strategy is proposed in this study. Firstly, a localization method based on transfer entropy and k-means is proposed to reconstruct the sample set. Then, based on the reconstructed local samples, the local soft sensing model is established by the multi-objective support vector regression method, and the selective ensemble of sub-models and the adaptive calculation of prediction weights are realized. At the same time, to reduce the adverse effects caused by improper selection of model parameters, the sparrow search algorithm is used to realize the tuning of the mentioned model parameters. Finally, the proposed modeling strategy is simulated. The results show that, compared with other methods, the proposed local selective ensemble learning multi-objective soft sensing modeling strategy has better prediction performance.

Keywords

Penicillin fermentation
Nonlinear relationship
Multiple output
Transfer entropy
K-means
Soft sensing
Subject terms

Engineering
Chemical engineering
Electrical and electronic engineering
Shandong Provincial Natural Science FoundationZR2022MF344 issue-copyright-statement© Springer Nature Limited 2024
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pmcIntroduction

Many key variables in the penicillin fermentation process, such as substrate concentration, bacterial concentration, dissolved oxygen concentration, and other product quality indicators, cannot be measured online1. Even if the quality index value can be obtained through the laboratory, its serious lag makes it difficult to meet the requirements of industrial production2. Therefore, it is necessary to soft-sensing modeling methods based on process data3. Because the penicillin fermentation mechanism is difficult to master and cannot be used to build accurate mechanical models, the method of sensor modeling based only on the mapping of the input and output data of the object has received extensive attention from researchers4. Compared to other modeling strategies, data-driven modeling methods built into bio-fermentation engineering have high accuracy for predictive models’ high stability and strong generalization capabilities5. With the development of process control technology, penicillin fermenting process data is stored in large quantities, and more and more research proves that data-driven soft-sensor modeling methods are more suitable for modeling penicillin fermentation processes with their stronger online calibration capability and ease of operation.

With the quality control of penicillin products becoming more and more strict, much research has been carried out to improve the prediction accuracy of product quality data in the penicillin fermentation process. Alsadat et al.6 applied the supervised LSTM network to the penicillin fermentation process to reduce the influence of nonlinear and dynamic characteristics on the prediction accuracy of the soft sensor model. Kumar et al.7 used the LSTM network, which has a good fitting effect on nonlinear sequence data, as the preliminary prediction model and uses the training data to establish the LSTM network prediction model and realize the screening of auxiliary variables for modeling. Muhammad et al.8 employed an LSTM neural network for soft-sensing modeling of a fermentation system, utilizing the measured values of relevant process parameters as features to enhance the accuracy of output results. However, at present, the focus of penicillin fermentation process modeling often focuses on how to stratify the sample data set to obtain an independent function model while ignoring the correlation between sample data9. Xinping et al.10 developed an LSTM-GPR soft sensing model for use in complex industrial environments, incorporating temporal data information into the regression analysis to enhance the predictive accuracy of the model. Zhou et al.11 proposed an active learning algorithm based on a covariance matrix to reduce the cost of manual labeling and solve the problem of low model prediction accuracy caused by a lack of labeled samples in industrial processes. Mohammad et al.12 developed a soft perception modeling approach for Bayesian networks, leveraging hidden space to enhance prediction performance and dynamic tracking capabilities of the models. In the age of big data, scientific progress maps fine-grained spatiotemporal distributions for thousands of species, using deep neural networks (DNNs) and ubiquitous citizen science data13. Based on the summary of the above studies, the modeling of the penicillin fermentation process was often limited to a single target output, such as predicting only single output variables such as substrate concentration, penicillin content, and heat generation, and ignoring the impact of simultaneous interaction between multiple input feature vectors and output vectors on the quality of penicillin products. As a result, the accuracy of the prediction model is not high, the generalization ability is reduced, and finally, the prediction model is abandoned. Therefore, it is necessary to the multi-target soft sensor model of multiple product quality indexes in the penicillin fermentation process.

In traditional regression problems, sample data was often represented by a feature vector and a continuous output vector, and the sample data had a unique and corresponding relationship with the feature vector and the output vector and was represented by unique semantics in the modeling process14. However, the penicillin fermentation process is a nonlinear, time-varying, and severely coupled system, and the product quality is not only affected by the interaction of multiple feature vectors but also by the mutual interference of multidimensional output vectors. Therefore, the linear or nonlinear relationship from the feature space to the multidimensional target output needs to be dealt with in the modeling process15 Multi-objective regression (MTR) deals with the relationship between the same input space and multi-dimensional output targets16. Compared with the prediction of a single objective output variable, multi-objective regression can better explain the dependency relationship between variables, so it is more convincing to represent the complex relationship between input and output.

Considering the significant impact of multi-objective regression problems on the modeling of complex chemical processes, various prediction models have been proposed. Kihan et al.17 proposed a generation model that combines a supervised variational autoencoder (SVAE) and a Wasserstein generative adversarial network with a gradient penalty to generate virtual label samples. The superiority of the multi-objective optimization modeling strategy for predicting output variables was verified in industrial aureomycin fermentation and simulated penicillin fermentation. Surendra et al.18 proposed a method for multi-objective optimization in the design of feature selection and set member composition. Wenle et al.19 introduced a novel soft sensor designed to simultaneously measure linear and rotational displacements using Soft Pneumatic Sensing Chambers (SPSCs). Jia liang et al.20 proposed a design of soft sensor arrays that can operate with a drastically reduced number of wires without degrading the original performance. Xu et al.21 presented a sensing technology capable of decoding omnidirectional bending, compression, stretch, binary changes in temperature. This multi-modal deformation and temperature sensor harnesses chromaticity and intensity of light as it travels through patterned elastomer doped with functional dyes. However, due to the strong nonlinear relationship between input and output caused by the high-level semantic concept of high-dimensional input represented by multiple targets, the soft sensor model cannot perform well in data prediction.

To mitigate the impact of nonlinear factors on the modeling process in complex industrial processes, the concept of machine learning has been employed for training and updating predictive models. Wang et al.22 proposed a deep learning-based atomic defect detection framework (DL-ADD) to efficiently detect atomic defects in molybdenum disulfide (MoS2) and generalize the model for defect detection in other TMD materials. A large-scale dataset of 3D solar magnetic fields of active regions is built by using the nonlinear force-free magnetic field (NLFFF) extrapolation from vector magnetograms of Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO)23. Deng et al.24 demonstrated the formation of dipolar exciton polaritons in bilayer MoS2 resulting in unprecedented nonlinear interaction strengths. Wang et al.25 proposed a water quality prediction method based on the GAT transformer to capture the complex relationships among various elements in water through GAT. All the studies have mitigated the impact of nonlinear factors on the modeling process. However, for the intricate industrial process exemplified by penicillin fermentation, both input and output variables not only exhibit a high degree of intercorrelation but also a significant level of autocorrelation. These characteristics pose significant challenges to establishing stable and accurate models.

With the advancement of deep learning, researchers have initiated an exploration into the correlation between autocorrelation and cross-correlation among diverse variables in complex industrial processes. This is aimed at constructing more compelling and representative predictive models Zhang et al.26 proposed a hybrid deep learning model to demonstrate the Fowler-Nordheim tunneling effect of monodisperse pointed carbon nanospheres array on polydimethylsiloxane as a strain sensor array. Zhao et al.27 proposed that the development of a Matrix Time Series Threshold Autoregressive Model (MTARX) with exogenous variables could effectively reduce the residual variance. Henrique et al.28 employed a dynamic regression time series model to elucidate the validity of the association between hospital antibiotic consumption and bacterial resistance. While the time series-based deep learning model can effectively predict target data, it often encounters issues such as slow data writing speed and processing residual information as normal feature information during sample data mining. This leads to poor interpretability of experimental results. To enhance the model's generalization ability, further segmentation and screening of the experimental dataset is necessary.

In the process of modeling, the processing method and efficiency of sample data have a significant impact on the prediction performance of the model. To enhance the predictive ability of the model in extracting characteristic information from sample data, researchers conducted a comprehensive on the segmentation of sample data. Jin et al.29 proposed a robust bootstrapping algorithm for constructing prediction intervals and prediction regions in univariate and multivariate autoregressive time series. Li et al.30 presented a comprehensive optimization framework employing the Multi-Objective Multi-Verse Optimization (MOMVO) algorithm for the optimal integration of Distributed Generations (DGs) and Capacitor Banks (CBs) into electrical distribution networks. Wang et al.31 utilized Aspen HYSYS 10 to investigate the optimal partitioning of data sets in the dimethyl carbonate production process. The refinement and screening of sample data enhance the accuracy of prediction and the generalization ability of the prediction model. However, many predictive models encounter issues with local optimization in target output, requiring researchers to optimize model parameters using intelligent algorithms.

Commonly used intelligent algorithms for machine learning include the particle swarm optimization32, the genetic algorithm33, the Gray Wolf algorithm, and the Archimede annealing algorithm34. Guo et al.35 proposed a method combining density clustering with particle swarm optimization (DCNPSO) to improve UWB positioning in the presence of noise. Kabiri et al.36 conducted multi-objective optimization using Non-dominated Sorting Genetic Algorithm II (NSGA-II) based on jEPlus and EA. Xu et al.37 utilized a combination of Grey Wolf Optimization (GWO) and the multi-scale cascade forest algorithm to enhance the accuracy of mechanical fault diagnosis in high-voltage circuit breakers. However, most traditional optimization algorithms require manual parameter setting, which not only complicates the operation process but also often leads to slow convergence of the model. This results in a limited response speed for the model. To address this issue, this study proposes a sparrow search algorithm that enables adaptive updating of the target. The algorithm not only overcomes the drawbacks of manual parameter setting but also significantly reduces computational requirements during prediction model establishment and allows for quick response to local optima predicted by the model.

In conclusion, due to the time-varying, nonlinear, and high coupling of the penicillin fermentation process, the following three difficulties need to be overcome to establish a soft sensor prediction model with stable operation and high precision.The quality of penicillin products is the result of the cooperation of multiple product indicators, considering not only the effects of multiple input characteristic vectors on the product but also the interaction between multidimensional output vectors; however, the current modeling methods for penicillin product fermentation processes are mostly targeted at a single target, resulting in poor accuracy of existing predictive models and a lack of stability and generalization.

The penicillin fermentation process data set not only presents continuity in spatial sequences but also a multi-time correlation. Currently, many methods of research are merely simple excavations of the relationship between spatial sequence factors, resulting in poor explanations of the characteristics obtained.

Due to the complexity of the penicillin fermentation process and the randomness of the data, the sub-models produced during the modeling process are highly susceptible to the interference of the fermenting environment, resulting in the predictive results being interfered with by varying degrees of rough information.

Methods

A LOC-SEL-MLSSVR modeling strategy during the study

To address these problems and improve the stability and predictive accuracy of soft sensor models in the penicillin fermentation process, the study proposes a local selective ensemble learning multi-objective soft sensing modeling strategy (LOC-SEL-MLSSVR). First, in the processing phase of sample data, the sample set of the penicillin fermentation process is divided into multiple local sample regions according to time and space guidelines. The multi-target regression modeling strategy is used to establish soft sensor sub-models in the various regional sample areas, and the prediction weights are adapted from adaptive calculations and updated according to the predictable accuracies of the sub-models. The overall methodological framework is shown in Fig. 1.Fig. 1 The methods and frameworks.

In the process of penicillin fermentation, there is a strong nonlinear relationship between multiple input feature vectors and output vectors38, and the fermentation process of penicillin is particularly susceptible to the influence of external conditions such as equipment aging, production environment changes, and catalyst activity, making it difficult to maintain a steady state of operation. The process of penicillin fermentation has strong, time-varying characteristics. To reduce the influence of these factors on the established prediction model, the penicillin fermentation process was divided into several local regions, soft sensor models were established in different regions, and then ensemble learning was carried out on the sample data set. Ensemble learning usually includes two steps: the generation of ensemble members and the combination of predictive data set39. In the generation stage of localized sample sets, the output of penicillin fermentation sample data sets has strong time continuity40. Therefore, when processing samples, this study divides the data set according to the time continuity and spatial correlation of the sample set, to improve the clustering accuracy of the localized data in the penicillin fermentation process. The operation flow chart is shown in Fig. 2.Fig. 2 Data set processing based on k-means and transfer entropy.

To maintain the stability of the established model's forecasting performance during the predictive dataset fusion phase, it is necessary to establish selective mechanisms for local ensemble learning to integrate sub-models selectively and adapt adaptively based on the sub-models' predicted weight. The procedure is shown in Fig. 3.Fig. 3 Adaptive updating of sub-models’ weight and establishment of ensemble learning model.

The innovative aspects of the study, including its contributions to the domain, can be summarized as follows:Due to the instability and low prediction accuracy of the existing penicillin fermentation process prediction model, this study employs multi-target MLSSVR to establish a sub-model and utilizes the sparrow search algorithm to optimize the model parameters. Compared with a single target forecasting model, this not only enhances the stability of the soft sensor model but also improves the prediction accuracy and generalization ability of the model.

Based on the time continuity and spatial correlation of the sample set, a time–space norm and a localization method based on the transmission of the equal values of magnesium and K have been proposed to localize the samples of the space–time sequence set compared to samples obtained only from the space sequence, making the output of the predictive model more convincing and interpretative.

To improve the anti-interference capacity of the sub-models, a multi-target method of predicting and updating the calculation of the weight of the predicted sub-model has been proposed.

A method of localization based on time–space norms for the transfer entropy and k-means

The penicillin fermentation process has a serious time delay characteristic, which often results in poor rationality in locating samples after clustering, resulting in low predictive performance of the soft-sensing model41. This study proposes a spatiotemporal criterion localization method combining transfer entropy42 and k-means based on the traditional k-means positioning method to eliminate such influence and better quantify the transfer relationship between information. Since the probability that penicillin fermentation sample data is arranged at the time point n+1 according to the time series and is only related to the first k time points, it meets the requirements of the k-order Markov process43, which is expressed as follows:1 pxi1xik=pxin+1xin,xin-1,…,xin-k+1

For the variables X and Y satisfying the Markov process, Xi and Yi are used to represent the time sequence of the moment I; xi satisfies the k-step Markov, Yi satisfies the step Markov, and the transmission between the X and Y is as follows:2 tXY=∑PXi+1,Xik,YillogPXi+1Xik,YilPXi+1Xik

In general, to avoid the introduction of complex high-dimensional probability density in the process of calculation, K = 1 is expressed as:3 PXi+1Xi,Yi=PXi+1,Xi,YiPXi+1,Xi

Therefore, the expression of Eq. (2) can be written as:4 tXY=∑Xi+1,Xi,YiPXi+1,Xi,YilogPXi+1,Xi,YiPXi+1XiPXi,Yi

where PXi+1,Xi,Yi represents the joint probability of Xi+1,Xi,Yi; PXi+1Xi is the conditional probability, which represents the conditional probability of work Xi+1 when i time Xi is known; PXi,Yi represents the joint probability of Xi and Yi.

Given the directionality of the transfer entropy, the information flow transfer relationship tX→Y from variable X to variable Y is expressed as:5 tX→Y=tYX-tXY

If t>0, the information is transferred from variable X to Y; If t<0, the information is passed from variable Y to X.

To sum up, the transfer entropy of variable X and variable Y can be obtained, and the time correlation of time series data sets is measured by the transfer entropy method.

In the next stage, the traditional k-means clustering method will be introduced.

Let X=x1,x2,…,xn be a known dataset, x1,x2,…,xn in X be n data objects, and each data object is N-dimensional data, as xi=vi,1,vi,2,…,vi,N. The algorithm needs to find a set containing k clustering centers:6 C=c1,c2,…,ck=v1,1,v1,2,…,v1,N,v2,1,v2,2,…,v2,N,…,vk,1,vk,2,…,vk,N

Traditional target functions are:7 JC,X=min∑i=1k∑j=1ndci,xj

The calculating equation where dci,xj is the oscillation distance between the data object and the cluster center.8 dci,xj=∑m=1Nvi,m,xj,m2

The goal function of the conventional k-means technique is enhanced in tandem with the mercury transmission to better represent the influence of the relationship between time sequence data, as follows:9 J′C,X= min∑i=1k∑j=1ntcixj⊗dci,xj

The time difference between two clusters of clustering data is tcixj.

Finally, a method of localization based on the transmitting spacetime, and k-means guidelines were used to divide the dataset into classes k, making the target function minimal.

Approaches for multi-target LSSVR modeling

The mapping relationship between multiple input variables and multiple target variables based on single target LSSVR44 is known as multi-target LSSVR, and it may be explained as follows:10 Y^1t+n=∑i=1Nw0+v1,i·Kz,zi+b1,i

11 Y^2t+n=∑i=1Nw0+v2,i·Kz,zi+b2,i

12 Y^mt+n=∑i=1Nw0+vm,i·Kz,zi+bm,i

13 Y^Mt+n=∑i=1Nw0+vM,i·Kz,zi+bM,i

14 w0=λM∑m=1Mvm

The predicted value of t+n in the time range is m=1,2…M, where M is the total number of multi-targetable and vm=[vm,1,vm,2,…,vm,N] is the vector of the weight parameter of the outputs of the Mth variable. Among them, Y^1(t+n),Y^2(t+n), Y^m(t+n) and Y^M(t+n) are the output variables of 1,2,…M. If multiple objects are very similar to each other,vm,i is a very small number that is close to 0. The correlation is represented by w0, which is the multi-target variable's average weight in the M-LSSVR model. The average weight normalization parameter is denoted by λ. To minimize the goal function under constraints, the best search power vector and deviation vector parameters should be used, specifically: 15 minGw0,V,ε=12w0Tw0+12λMVTV+γ12εTε

16 Y^1t+n=∑i=1Nw0+v1,i·Kz,zi+b1,i+ξ1

17 Y^2t+n=∑i=1Nw0+v2,i·Kz,zi+b2,i+ξ2

18 Y^mt+n=∑i=1Nw0+vm,i·Kz,zi+bm,i+ξm

19 Y^Mt+n=∑i=1Nw0+vM,i·Kz,zi+bM,i+ξM

The multi-target prediction problem's target function is G(w0,V,ε).ε=(ξ1,ξ2,...,ξM) is the loose variable's vector. Reassembling the Eqs. (15–19) yield the following linear matrix equation group:20 S00HBH-1PB+w0=PTH-1YY

21 S=PTH-1P

22 H=ZTZ,M,M+1γI+MλZTZ

where H is the transform matrix, I is the unit of the M line, P is the unit matrix of column M, and S is the fixed matrix. Z=(Z1,Z2...ZN).Y=Y^1(t+n),Y^2(t+n),...,Y^M(t+n) is the multi-target variable matrix. It is the matrix of the input variable.

To enhance the MLSSVR45 model's stability and generalization capabilities, the radius-to-base function is assigned to the core function in the following manner:23 Kxi,xj=exp-xi-xj22σ2

Establishment and refinement of the sparrow search algorithm

The configuration of model parameters significantly impacts the predictive performance of the model. Researchers employ optimization algorithms to fine-tune the model's parameters. However, traditional optimization algorithms are plagued by issues such as complex manual operation processes, extensive computational requirements, and slow convergence speeds. Therefore, this study introduces a sparrow search algorithm that can automatically adapt to dynamic changes in the model. This algorithm not only requires minimal computational resources but also facilitates adaptive updates based on changes in the predicted target variable. Additionally, it enables rapid responses to model outputs. The objective function expression is as follows46:24 minFx=f1x,f2x,…,fmx,…,fMxTFx=1M∑l=1Mfl2xflx=1H∑k=1Hy^tk-ytk2,1≤l≤Ms.t.gix≤0i=1,2,…,phjx=0j=1,2,…,qx=x1,x2,…,xDT

where x is the d-dimensional decision vector, F(x) is the target vector, and M is the optimized target total;gi(x)≤0 is the first i inequality constraint;hj(x)=0 is the j inequality constraint; and fm(x), the first m target function; gi(x)≤0 and hj(x)=0 are the feasible areas for determining the solution.

This study employs the sparrow search algorithm to optimize the model, thereby preventing the predictive model from falling into local optimum due to improper parameter settings. In the sparrow search algorithm (SSA), the detective is depicted as follows during each iteration:25 xi,jt+1=xi,jt·exp-iα·itermaxifr2<STxi,jt+Q·Lifr2≥ST

where xi,jt represents the position of the ith sparrow in dimension j of tth iteration, xi,jt+1 represents the position of the ith sparrow in dimension j of (t+1)th iteration.r2∈0,1, ST∈0.5,1 represent early warning value and safety value respectively. α is the uniform random number in (0,1], itermax is the maximum number of iterations. Q is the random number (0,1] obeys the normal distribution, and L is the matrix with elements 1. When the early warning value r2 is less than the safety value ST, the searcher makes a large-scale jump search. When the warning value exceeds the safety value ST, the searcher moves to other locations for search.

The location update of followers is:26 xi,jt+1=Q·expxworset-xi,jti2ifi>nn22xPt+1+xi,jt-xPt+1·A+·Lotherwise

where xworset represents the global worst position found by the discoverer. xi,jt represents the position of the ith sparrow in the dimension j of iteration t, and xi,jt+1 represents the position of the ith sparrow in the dimension j of iteration t+1. A represents a matrix of 1×D, where each element is randomly assigned 1 or -1, and A+=ATAAT-1. The sparrows with i>nn22 have low fitness value and do not get food. They need to jump to the minimum value, and other followers’ sparrows move to the optimal position found by the explorer. The watchers are some sparrows randomly selected from the explorers and followers of the sparrow population and avoid falling into local optimization through an anti-predation strategy.

The location update equation of warning persons is:27 xi,jt+1=xbestt+βxi,jt-xbesttiffi>fgxi,jt+Kxi,jt-xworsetfi-fw+εiffi=fg

where xbestt is the global best location; β is the iteration step; fi is the fitness of the current sparrow; fg and fw are the global best and worst fitness values and K is the random number between [− 1,1]; ε is the smallest constant against the division of zero.

A round of position updates can be completed according to Eq. (27), and each update will make the population change in the direction of better fitness value. After several iterations, the best fitness value and the corresponding parameters can be obtained.

Establishment of the LOC-SEL-MLSSVR sub-models

A combination of generated LOC-MLSSVR model estimates was used to determine the multi-target estimate value of y(m,l),y^m for the query sample xt in the integrated investigation of the penicillin fermentation sample. The LOC-SEL-MLSSVR model provided by xt is as follows, according to the ensemble learning framework:28 y^1,l=∑l=1LαlflMLSSVRxty^2,l=∑l=1LαlflMLSSVRxty^m,l=∑l=1LαlflMLSSVRxty^M,l=∑l=1LαlflMLSSVRxt

where flMLSSVRxt is the prediction of the ith LOC- MLSSVR model given xt, are the combined weights of sub-models.29 RMSE=1N∑i=1Nyt^-yt2

30 MAE=1N∑i=1Nyt^-yt

where N is the total number of samples.

Based on the values of RMSE and MAE of ith sub-models for M prediction objects, the prediction performance value PPVl is defined as:31 PPVl=1M∑i=1MRMi2x+1M∑i=1MMAi2x

where RMi and MAi are the RMSE and MAE values of the ith sub-model for M prediction objects, respectively.

Since the PPVl can reflect the prediction performance of the ith sub-model for M prediction objects, the likelihood of the ith sub-model is selected based on LHi which is defined as:32 LHl=1-PPVl∑j=1LPPVj

where L is the number of sub-models.

In Eq. (32) larger LHll=1,2,…,N represents higher estimation accuracy, and the normalization term in the denominator will not influence the combination results. Still, it can scale LHi into the range of (0,1], which can facilitate the implementation of selective ensemble learning. Up to now, selective ensemble learning which combines part of sub-models can achieve better generation performance, as it can reach a good equilibrium between the prediction bias and variance.

Adaptive weight updating of sub-models

To perform the selective ensemble learning, which models should be selected at each prediction round must be determined. Intuitively, only those ensemble members whose estimation accuracies are higher than a certain satisfaction degree are fused, and the rest will not be involved. Thereby the sub-models denoted as LHthr is determined as:33 LHthr=LH12+LH22+⋯+LHL2L

However, multiple product quality indexes of complex chemical processes have a common impact on product quality, and each product quality index has strong importance. Therefore, each prediction target in multi-objective prediction needs to maintain high prediction accuracy. This requires that the selected sub-model should not only meet the requirements of LHi, but also meet the prediction requirements of monomer prediction objectives LHRMl,m and LHMAl,m.

The values of RMSE and MAE of ith sub-models for mth prediction target is respectively shown as:34 RMl,m=1N∑i=1Ny^l,m-yl,m2

35 MAl,m=1N∑i=1Ny^l,m-yl,m

Furtherly, the LHRMl,m and LHMAl,m are shown as:36 LHRMl,m=1-RMl,m∑j=1LRMj,m

37 LHMAl,m=1-MAl,m∑j=1LMAj,m

Then sub-models denoted as LHRMthr and LHMAthr are determined as:38 LHRMthr=LHRM1,m2+LHRM2,m2+⋯+LHRML,m2L

39 LHMAthr=LHMA1,m2+LHMA2,m2+⋯+LHMAL,m2L

Thereby the sub-models that can be selected are determined as:40 Js=jLHl>LHthr&LHRMl>LHRMthr&LHMAl>LHMAthr

where S is the number of selected sub-models. Moreover, due to the dynamic and time-varying characteristics of complex chemical processes, Js are usually not fixed but adaptively determined depending on the values of LHi ,LHRMl,m and LHMAl,m at each prediction round.

Finally, the equation of computing yt^ with those sub-models as:41 y^1,s=∑s=1SαsfsMLSSVRxty^2,s=∑s=1SαsfsMLSSVRxty^m,s=∑s=1SαsfsMLSSVRxty^M,s=∑s=1SαsfsMLSSVRxt,s=0,1,2,⋯,S

where S is the number of sub-models.

The calculation of sub-models’ prediction value fusion weight such as αs in Eq. (41) are the second stage of ensemble learning, which also affects the prediction performance of target variables, and inappropriate fusion weight value is easy to reduce the prediction performance of the model. The values of RMSE and MAE of ith sub-models for the Mth prediction target is shown by Eqs. (34) and (35). Furthermore, the prediction performance of the sub-models for the Mth prediction target can be obtained by:42 Pl,m=RMl,m2+MAl,m2

For the multi-objective regression problem, although a nonlinear regression model is used to predict the data of multiple objectives, the prediction weight of the sub-model needs to be calculated and adjusted adaptively according to the prediction performance of different prediction objectives. The prediction weight indicates the importance of each sub-model to the final prediction value. Therefore, for the Mth prediction target, the sub-models weight calculation equation is defined as follows:43 αs,m=Ps,mRM1,m2+RM2,m2+…+RMs,m2S+MA1,m2+MA2,m2+…+MAs,m2S-1

where S is the number of sub-models.

Ultimately, the sub-models weight calculation can be obtained by Eq. (43). And further, realize the data prediction for multiple objects.

Results and discussion

In this research, we propose a local selection ensemble Learning multi-objective soft sensor modeling approach to predict key output variables in penicillin fermentation. The experimental data is obtained from the Pensim simulation platform. To validate the effectiveness of our proposed modeling method, we utilized BP neural network (BPNN) modeling combined with the K-means method to establish a local multiple local objective (EL-MLSSVR) model. Furthermore, the local multi-dimensional target (LOC-EL-MLSSVR) method is developed by applying entropy and K-means methods to model several fundamental and accurate soft sensing techniques. RMSE and MAE indicators were employed for comparing the performance of the LOC-SEL-MLSSVR model on a test dataset to quantify its prediction accuracy. Figure 4 presents an illustration of the penicillin fermentation process, depicting some of the key variables, aimed at enhancing readers' comprehension of this intricate chemical process.Fig. 4 An illustration of the penicillin fermentation process.

Ten variables and related data are chosen, as indicated in Table 1, based on the penicillin fermentation process to create an input variable sample data set. Concurrently, the output variable sample dataset is formed by selecting pertinent data and four variables, as shown in Table 2. In addition, the platform utilized to simulate the penicillin fermentation process generates 2000 data sets, of which the first 1000 are used as training data sets and the last 1000 as test data sets, with a simulation length of 2000 h and a sampling time of one hour. The RMSE and MAE values for the target y1, y2, y3, and y4 are anticipated based on the various local sampling sets acquired, and PPV values are created in conjunction with Eq. (31), as indicated in Table 3.Table 1 Input variables.

No.	Variable description	No.	Variable description	
u1	Aeration rate	u6	Culture volume	
u2	Agitator power	u7	CO2 concentration	
u3	Substrate feed rate	u8	pH	
u4	Substrate feed temperature	u9	Reactor temperature	
u5	Dissolved oxygen concentration	u10	Coiling water flow rate	

Table 2 Output variables.

No.	Variable description	
y1	Substrate concentration	
y2	Biomass concentration	
y3	Penicillin concentration	
y4	Generated heat	

Table 3 The values of PPV for different sub-models.

Sub-models	J1	J2	J3	J4	J5	J6	
The Values of PPV	0.81943	1.0118	0.12719	0.53703	0.70729	2.778	

Then, as shown in Table 4, we apply Eq. (32) to get the LHi of the Nth sub-model. Combining the values of LHi in Table 4 yields an LHthr value of 0.2604. Table 5 lists the six sub-models generated for the forecast objectives y1, y2, y3, and y4 RM(l.m) and MA(l.m) values. Equations (34) and (35) are used to derive these models, together with the RMSE and MAE values from Table 6. Additionally, Eqs. 36 and 37 yield the values of LHRM(l.m) and LHMA(l.m), which are 0.099829 and 0.99744, respectively.Table 4 The values of the LHi for different sub-models.

Sub-models	J1	J2	J3	J4	J5	J6	
The Values of LHi	0.2328	0.27566	0.21667	0.24209	0.4135	0.003807	

Table 5 The values of RMSE and MAE with different sub-models for prediction targets.

Sub-models	y1	y2	y3	y4	
RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	
J1	0.009	0.0010	0.128	0.0133	0.111	0.012	0.093	0.0095	
J2	0.067	0.0067	0.13	0.013	0.071	0.0072	0.055	0.006	
J3	0.213	0.022	0.017	0.004	0.129	0.013	0.043	0.004	
J4	0.022	0.0022	0.055	0.005	0.062	0.0078	0.128	0.013	
J5	0.08	0.0087	0.076	0.0122	0.045	0.005	0.054	0.006	
J6	0.07	0.007	0.002	0.0015	0.137	0.014	0.11	0.011	

Table 6 The expected performance values for various local sample sets.

Local sample sets	y1	y2	y3	y4	
RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	
1	0.0134	0.0018	0.0251	0.0025	0.0135	0.0013	0.0491	0.0051	
2	0.0009	0.00046	0.0011	0.0009	0.0298	0.003	0.0206	0.00211	
3	0.0166	0.0017	0.002	0.0002	0.018	0.0022	0.0651	0.00671	
4	0.0021	0.0004	0.1018	0.0102	0.0040	0.0005	0.0425	0.00421	
5	0.0318	0.0032	0.0306	0.0031	0.0274	0.003	0.0159	0.00151	
6	0.01703	0.002	0.0271	0.0029	0.0393	0.0041	0.0036	0.00071	

Consequently, the sub-model J2 of the model that can be chosen is derived from the analysis and combined with the selection criteria suggested in Eqs. (40), (41). The next step is to apply Eq. (42), as shown in Table 5, together with the predictive properties of the sub-models for the prediction targets.

According to Tables 3, 4 and 5, the sub-models utilize Eq. (43) in combination with the predictive performance P(l.m) values shown in Table 5 to determine the fusion weights for the different predicting targets, y1, y2, y3, and y4. Additionally, it is evident from Fig. 5a–d that the sub-models established in different sample set regions have yielded improved prediction results, leading to a 21.7% increase in accuracy of the RMSE for the four output targets. In comparison with the error of weight fusion of the sub-models presented in Table 5, there has been an almost 27.6% enhancement in prediction accuracy. Furthermore, as depicted in Fig. 5a–d, the changing trend of the prediction curve aligns with actual value changes, thus indicating that this chapter's model exhibits strong prediction accuracy.Fig. 5 Prediction curves based on LOC-SEL-MLSSVR for prediction objects.

It is evident from test results a through d that the suggested sub-model has successfully verified the output feature prediction from y1 to y4. The change curve for the anticipated value is essentially consistent with the actual value curve when 1000 test samples are compared between their true and predictive values. The error range between the expected and actual values is minimal and controlled within a stable range, as seen by comparing the Table’s MAE and RMSE.Additionally, Table 7 quantifies the predictive performance of soft sensor models based on comparative modeling methodologies using RMSE and MAE to better depict the benefits and efficacy of the proposed LOC-SEL-MLSSVR strategy.Table 7 The prediction performances of different targets with the LOC-SEL-MLSSVR method and compared methods.

Compared modeling methods	The prediction performances of y1	The prediction performances of y2	The prediction performances of y3	The prediction performances of y4	
RMSE	MAE	RMSE	MAE	RMSE	MAE	RMSE	MAE	
EL-MLSSVR	0.172	0.017	0.188	0.019	0.135	0.014	0.12	0.12	
LOC-EL-MLSSVR	0.177	0.011	0.108	0. 065	0.197	0.02	0. 13	0.015	
BPNN	1.22	0.074	1.2012	0. 07	1.063	0.065	1.4	0.082	
LOC-SEL-MLSSVR	0.016	0.0017	0.019	0.002	0.075	0.008	0.037	0.004	

In conjunction with Table 7, Fig. 6 illustrates a comparison of error curves between the LOC-SEL-MLSSVR prediction model proposed in this study and three other comparative groups.Fig. 6 Error curves of prediction results of different prediction models for output targets.

Figure 6 presents the error prediction curve of each model. In combination with the RMSE and MAE values computed in Table 7, the subsequent conclusions can be derived. In the process of penicillin fermentation modeling, this project adopts LOC-EL-MLSSVR modeling, EL-MLSSVR modeling, and BPNN modeling to compare the predictive performance of the LOC-SEL-MLSSVR model in the penicillin fermentation process. By comparing the prediction curves of different predicted targets y1, y2, y3, and y4 in Fig. 6, it can be concluded that although all kinds of soft sensor models have achieved data prediction, by further combining Table 7 and Fig. 6, it can be concluded that the LOC-SEL-MLSSVR proposed by this project has a smaller error fluctuation. Combined with the error calculation in Table 7, RMSE and MAE increased by 37.3%, while compared with EL-MLSSVR and LOC-EL-MLSSVR, RMSE and MAE increased by 28.5% and 21.7%, respectively. The fundamental reason is that the BPNN modeling strategy does not cluster the sample data in the implementation process, and the model is seriously disturbed by the gross error information of the sample set. Further combined with Fig. 6a–d, the errors generated by the LOC-SEL-MLSSVR model fluctuate around 0 errors. At the same time, the other three control methods, under the influence of different interference factors, can improve the model's performance. All the models were subjected to strong error information interference, which further demonstrated that the LOC-SEL-MLSSVR model has good stability and high prediction accuracy, and its performance was far superior to other modeling strategies in the control group. Meanwhile, the model also has a certain reference value for modeling other biochemical reaction processes.

Conclusion

In this study, a soft sensing modeling strategy for the penicillin fermentation process based on local selection ensemble learning was proposed to mitigate the influence of randomness, nonlinearity, and aperiodicity on the modeling procedure. The model is founded on k-means, which conveys transfer entropy, the sparrow search algorithm, and an intelligent optimization algorithm for the adaptive update of sub-model weights. Firstly, a feature dataset encompassing multiple input variables of the fermentation process is established. After selecting ensemble learning samples, a sample localization approach based on k-means and transfer entropy is employed to handle the sample data, enhancing the clustering performance of the sample set compared to traditional ensemble learning methods. Secondly, to enhance the stability and generalization ability of the ensemble learning model, this study establishes adaptive weight-updating sub-models in multiple sample data regions. This approach can better reflect the characteristics of the target outputs in different fermentation stages. Thirdly, to improve the global search capability of the model algorithm and prevent it from converging to the local optimum, this study combines the sparrow search algorithm with the model parameter setting to enhance the model's adaptive ability and generalization performance. The experimental results indicate that compared with LOC-EL-MLSSVR, EL-MLSSVR, BPNN, and other prediction models, the improvement rates of RMSE and MAE of this model are higher than 20%. Simultaneously, the comparative analysis of different models reveals that the proposed LOC-SEL-MLSSVR modeling strategy can effectively reduce the impact of the randomness and nonlinearity of the fermentation process on model performance. Additionally, the proposed model holds significant potential in practical applications, particularly in complex industrial processes. The application of the model to the prediction of key data in complex industrial processes can not only reduce the production cost of enterprises and enhance product quality but also further domesticate the performance of the model to better serve the production of enterprises.

Acknowledgements

This work was supported by Shandong Provincial Natural Science Foundation (Grant No. ZR2022MF344).

Author contributions

Feixiang Huang: providing scientific research ideas and technical routes; Longhao Li: data collection and model modeling; Chuanxiang Du: drawing and table arrangement; Shuang Wang: Experimental verification of the proposed scientific research method; Xuefeng Liu; Writing documents and organizing solutions.

Data availability

The datasets used and analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.

Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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