Dynamic correction of soft measurement model for evaporation process parameters based on ARMA

1.   Introduction

The alumina evaporation process is a complex industrial procedure where key parameters (such as the concentration of sodium aluminate solution) fluctuate due to production conditions (like acid washing, water washing, equipment failure) and other factors. As a result, the input-output relationship remains unstable, with process variable data dynamically changing around process values ^[1]. When the production process is in a steady state, it can be considered roughly static. However, in actual production, it is challenging to maintain a steady operational state, and dynamic changes are evident ^[2,3]. Consequently, the predictive performance of static soft measurement models established by traditional LSSVM (least squares support vector machine) methods is often unsatisfactory and, in severe cases, can even lead to model failure ^[4,5,6]. Additionally, the time-varying nature of industrial production processes causes the performance of soft measurement models to gradually degrade over time, reducing their lifespan ^[7]. Therefore, how to improve and maintain the predictive accuracy of soft measurement models has become a pressing issue. Hence, establishing predictive models that reflect the dynamic changes in the production process is essential, especially for processes with significant dynamic characteristics. Employing dynamic predictive models can effectively enhance the estimation accuracy and robustness of the model ^[8].

Currently, many scholars have conducted research on nonlinear dynamic predictive modeling, leading to the development of various dynamic prediction methods ^[9,10,11]. Suykens et al. ^[12] used historical input data combined with time variables to form a new input data vector, aiming to integrate dynamic information into this vector, utilizing nonlinear static models to represent the dynamic characteristics between input and output variables. ^[13] introduced the concept of a moving window, using a finite length of historical input to form a new input vector for a feedforward neural network. This model is similar to a nonlinear finite impulse response model, where the moving window length of historical input must be sufficiently long to fully reflect the object's dynamic characteristics. This method was applied to predictive modeling in a polymerization reactor, accurately estimating the concentration of the polymer. Li et al. ^[14] proposed a prediction model based on local model networks, using an online subtractive clustering method to recursively update the local model structure and parameters, accordingly updating the cluster centers and local models, generating new clusters and local models and establishing a predictive model using a nonlinear moving average method. This was applied to the predictive modeling of a para-xylene distillation column, effectively estimating the purity of para-xylene. ^[15] proposed using a feedback neural network to establish a predictive model. Unlike traditional networks, this model included output feedback as an additional input, reflecting the object's dynamic characteristics; and it has been successfully applied to industrial boiler emissions, the concentration of mycelium in erythromycin fermentation and the viscosity and lipid content of epoxy resin and graphite fiber composites during the curing process. ^[16], based on a particle swarm optimization process neuron network, proposed a new learning algorithm. The new algorithm, after expanding the network input function and connection weight function with orthogonal basis functions, integrated the structural parameters and other parameters of the network into a particle and then globally optimized using the particle swarm optimization algorithm. This new algorithm does not rely on function gradient information and requires manual adjustment of the network structure, better utilizing the approximation performance of the process neuron network. Studies ^[17,18,19] utilized the MA model, widely applied in the field of engineering technology, capable of describing the correlation between a set of time-related random variables, predicting the direction and continuity of the target development and predicting future values based on past and current values in time series. Bayesian analysis was used to study the moving average model under time series, enriching the theoretical research on time series models. ^[20] used an exponential weighted moving average (EWMA) model structure to solve the problem of traffic congestion monitoring.

Studies ^[21,22,23] achieved effective data prediction by constructing an ARX-RBF model. ^[24] used the ARX model structure to solve the problem that different time ranges require different data sample sets for prediction. Baptista et al. ^[25] and Shen et al. ^[26] employed the ARMA model respectively for predicting aircraft component failures and polar motion. Additionally, the ARMA model has been used for forecasting photovoltaic power and particulate matter motion ^[27]. However, the difficulty in determining the length of historical input data and the problem of too many input nodes leading to model ill-conditioning limited the application of this multi-point input soft measurement modeling method. Therefore, some scholars have approached dynamic weighted input soft measurement modeling by using dynamic weights on multi-dimensional historical input data to form new input variables, integrating process dynamic information, as demonstrated in ^[28]. This work used the fuzzy curve method to weight historical inputs at each moment based on their impact on the output variable at the current moment, optimizing the soft measurement model. Wang et al. ^[29] proposed a multi-objective model correction method based on dynamic weighting coefficients, dynamically adjusting the weighting coefficients according to the gap between each local optimum solution and the sub-goal expectation, thereby optimizing the weighting coefficients during the evolution process. Jiang et al. ^[30] achieved real-time and effective prediction of blast furnace charge level information by dynamically weighting the constructed piecewise linear regression model. Similarly, Di et al. ^[31] accomplished real-time monitoring of complex power system component data by developing a time series prediction method based on dynamic weighted probability support vector regression. Peesapati ^[32] implemented dynamic weighting of a fuzzy system based on the S-PSO (stochastic particle swarm optimization) method in a dynamic weighted input model. Compared to other methods, the dynamic weighted input modeling approach does not increase the input vector of the static model, thereby reducing the model's complexity and making it more straightforward. However, overall, determining the length of historical data and the fusion weight values of multidimensional historical data remains a challenging aspect of dynamic weighted input soft measurement modeling methods.

Through this comprehensive review and our proposed methodology, this paper proposes a dynamic correction method for soft measurement models of evaporation process parameters based on the ARMA. Initially, the Powell-DE algorithm is employed to identify the autoregressive order and moving average order of the ARMA model. Subsequently, the prediction error of a mechanism-reduced robust least squares support vector machine ensemble model is used as input. An error time series prediction model, which compensates for the errors in the autoregressive moving average model, is then applied for dynamic estimation of the prediction error. Finally, through an entropy method integration strategy, the static soft measurement model based on the mechanism-reduced robust least squares support vector machine is integrated with the dynamic correction soft measurement model, which is based on the error time series compensation of the ARMA model. The models are analyzed and validated using continuous production data from the evaporation process over two months. Experimental results show that the new model offers excellent predictive performance and high accuracy, and it is fully capable of online soft measurement for evaporation process parameters, specifically the concentration of sodium aluminate solution.

2.   Process flow

This paper studies the process flow of the four-effect three-flash falling film evaporation process in an alumina plant's evaporation workshop, as depicted in Figure 1.

The four-effect, three-flash evaporation process mainly consists of four evaporators, three preheaters, three flash evaporators and several condensate water pumps. According to the process flow shown in Figure 1, the evaporation feed first enters the third- and fourth-effect evaporators. The feed liquid flows from the fourth-effect evaporator into the third-effect preheater, where the temperature of the solution is increased. When the temperature of the feed liquid approaches the boiling point temperature of the third-effect evaporator, the liquid is transferred from the preheater to the third-effect evaporator. In a similar manner, the feed liquid sequentially passes through the third-, second-, and first-effect evaporators, continually increasing in temperature and concentration. After exiting the first-effect evaporator, the evaporated mother liquor flows through the three flash evaporators. The concentrated mother liquor is then pumped from the evaporation process to a blending tank.

The primary heat source for the evaporation process is fresh steam from the thermal power plant. The fresh steam first enters the first-effect evaporator, where it indirectly heats the feed liquid on the outer wall of the heating pipes, producing primary steam condensate. Subsequently, the secondary steam generated by the first-effect evaporator enters the second-effect evaporator, and the process continues with the secondary steam from the second-effect entering the third-effect evaporator, and so on, until the secondary steam from the third-effect evaporator enters the fourth-effect evaporator. Finally, the secondary steam generated by the fourth-effect evaporator is fed into a water cooler to ensure a smooth discharge of steam from the evaporation system.

3.   Identification of ARMA model parameters based on the Powell-DE algorithm

The ARMA (autoregressive moving-average model) model is a classic method used for system identification and prediction. In the ARMA(p, q) model, p and q represent the autoregressive order and the moving average order, respectively. Once these two parameters are determined, the estimation model becomes relatively straightforward. Common algorithms for estimating p and q in the ARMA(p, q) model include the least squares method, the Marple algorithm, Levinson's method and the U-C method. However, determining the order of the model, i.e., selecting p and q, is a challenging aspect of the ARMA model, especially when it comes to error compensation and enhancing the accuracy of error compensation.

Identification of ARMA model parameters fundamentally involves testing the model's applicability. Theoretically, the most basic condition for the validity of an ARMA model is that its residual series e(t) should be white noise. Therefore, verifying whether the model is effective essentially means checking if e(t) is white noise. Theoretically, the auto-correlation function of white noise can be used to calculate e(t), but practically, this is quite difficult to achieve as there is usually a significant deviation between estimated and actual values. Consequently, experts and scholars have researched various criteria and methods for testing the applicability of the model, such as the FPE criterion, BIC criterion, AIC criterion, F-criterion and Box-Jenkins method. However, these criteria are not universally applicable and have their limitations to varying degrees. Recently, researchers have continued to explore the structure of the ARMA model and have proposed some new criteria and methods that represent significant improvements over previous approaches. Nonetheless, a universal method for determining the order of the ARMA model, i.e., the values of p and q, has not yet been found.

This section proposes a new algorithm to address the problem of identifying ARMA model parameters, namely, the use of the Powell-DE algorithm for parameter identification in the model. Utilizing the equivalence in output between the ARMA model and actual models, a variable directly reflecting the difference between model predictions and actual time-series measurements is used as the fitness function for the Powell-DE algorithm. The fast convergence rate and strong global optimization capability of the Powell-DE algorithm are leveraged to evolve the best ARMA model with optimal prediction effectiveness through successive generations.

3.1.   Introduction to ARMA(p, q) model

The basic principle of the ARMA model involves establishing a mathematical model through time series to achieve system state identification and future prediction. Here, a time series refers to a stable random signal generated during the operation of a system, which is processed to form a sequentially arranged series over time. This series includes information about the system's operational state.

Typically, an ARMA(p, q) model is represented as follows: For a stable time series xt, a linear difference equation can be established, as shown in Eq (3.1).

In this equation, $x_t$ represents the time series, where t = 1, 2, 3, ..., N. $\alpha_i$ are the autoregressive parameters, for i = 1, 2, 3, ..., p. $\beta_j$ are the moving average parameters, for j = 1, 2, 3, ..., q. $a_t, a_{t-1}, \ldots a_{t-q}$ are pairwise independent white noise inputs. By determining the parameters p, q, $\alpha_{i({\mathrm{i}} = 1, 2, 3 \cdots \cdots, {\mathrm{p}})}$ (for i = 1, 2, 3, ..., p) and $\beta_{i({\mathrm{i}} = 1, 2, 3 \cdots \cdots, {\mathrm{p}})}$ (for j = 1, 2, 3, ..., q), the ARMA model is defined. The autoregressive model AR(p) and the moving average model MA(q) are special cases within the ARMA(p, q) model.

3.2.   ARMA(p, q) parameter identification and method analysis

In the ARMA(p, q) model, the determination of orders p and q is crucial. Many scholars, both domestic and international, have dedicated significant time and energy to researching this, proposing several effective criteria and methods, such as the FPE criterion, AIC criterion, BIC criterion and F criterion. Box and Jenkins suggested that the structure of the model could be determined using the sample's autocorrelation function (SACF) and partial autocorrelation function (SPACF). In 1984, Tsay and Tiao proposed that the model order could be identified using the extended sample autocorrelation function (ESACF) method. The FPE criterion and Box-Jenkins method are suitable for AR models, while the AIC and BIC criteria are applicable to both AR and MA models. However, the upper limit of the model order must be manually set, and when comparing AIC and BIC criterion values, local minima (i.e., pitfalls) occur. Determining the best model requires repeated comparisons. The F criterion, which uses a statistical approach, requires a manually set confidence level during the F test.

Identifying the structure of ARMA models is complex due to the need to estimate model parameters for all possible model structures. This can lead to substantial computational effort when using the above criteria. Based on the ESACF method, a pattern vector can be obtained, and its discrimination function can be determined using the perceptron algorithm, combined with the AIC and BIC criteria, to identify the structure of the ARMA model. First, the ESACF is calculated, then the parameters of multiple model structures (for models with larger discrimination function values) are estimated, and finally, the best model is determined in combination with the AIC and BIC criteria. In ^[33], the structure of the ARMA model is identified through a neural network algorithm. The widely used modeling approach currently is the one proposed by Professor Wu (University of Wisconsin, USA). The specific process starts with selecting the model form as ARMA(p, p-1) and then progressively increasing the order from low to high to find a suitable model. If the model to be fitted is complex, an ARMA(2p, 2p-1) model can be used, a two-step approach, first jumping two orders at a time. Once a practical model is obtained, the process switches to jumping one order at a time to find the lowest applicable model.

The identification process for the ARMA(p, q) model is tedious, akin to a trial-and-error or exhaustive search method, and its stability is relatively poor. Therefore, there is a need to find effective and feasible criteria and methods for determining the order of ARMA models.

3.3.   Powell-DE algorithm

The Powell algorithm consists of three directions: basic search, adjustment search and accelerated search. The basic search is a crucial step where one-dimensional searches are conducted from a base point along n straight search directions to determine a new base point. This methodical approach ensures a thorough exploration of the solution space, enhancing the likelihood of identifying optimal solutions. The accelerated search is another critical component, involving one-dimensional searching along the direction of the line connecting two base points. This method is particularly effective in rapidly decreasing the function value, thereby speeding up the convergence process. Finally, one of the known n search directions is replaced with the line direction of the base points, forming a new group of search directions for the next round of iterative search, exhibiting strong local search capabilities. This replacement of search directions is a unique feature of the Powell algorithm, allowing for an adaptive and dynamic search strategy that significantly enhances the local search capabilities. Integrating this into the DE (differential evolution) algorithm can overcome the drawbacks of the differential evolution algorithm, which has a relatively slow search speed and weak local search capabilities.

The fundamental idea of the Powell-DE algorithm is a two-phase optimization process. The first phase uses the differential evolution algorithm for coarse optimization to obtain an optimal solution. This phase is crucial for exploring the broader solution space efficiently. The second phase involves fine optimization using the Powell algorithm, which refines the solution obtained in the first phase to achieve the global optimum. This two-phase approach synergizes the strengths of both algorithms, ensuring both a wide search coverage and precise local optimization. The steps of the algorithm are as follows:

1) Initialization: This includes initializing the population $N_G$, scaling factor $F$, crossover factor $CR$, maximum number of iterations $G_{\text {max}}$ and setting a precision threshold $\varepsilon > 0$.

2) Initial population assessment: For the initial population, use formula (3.2) as the fitness function to calculate fitness values. Record the best individual $X_{\text {best}}$, its corresponding optimal value $F_{\text {best}}$ and its index value index $\left(X_{\text {best }}\right)$.

Here, $y_{o j}$ represents the training output, $y_{i j}$ represents the actual value, and $N$ is the total number of samples.

3) Mutation, crossover and selection: Perform mutation, crossover and selection operations according to the formulas in ^[34] to eventually obtain a new individual $X_i^{G+1}$.

4) Iteration and repetition: Set $G = G + 1$; repeat step 3 until the algorithm reaches the maximum number of iterations, concluding the coarse optimization phase.

5) Fine optimization with Powell search: Use the extremum point obtained from the differential evolution optimization as the starting point for the Powell search. (ⅰ) Let $y^1 = x^0$ (the extremum point from differential evolution), and set $n$ linearly independent directions: $d^{(1, 1)}, d^{(1, 2)}, \cdots, d^{(1, n)}$. Allow for an error $\varepsilon > 0$, and set $k = 1$. (ⅱ) Set $x^{(k, 0)} = x^{k-1}$ and start from $x^{(k, 0)}$, sequentially searching along the specified directions $d^{(k, 1)}, d^{(k, 2)}, \cdots, d^{(k, n)}$. Let

to obtain the points $x^{(k, 1)}, x^{(k, 2)}, \cdots, x^{(k, n)}$. Then, set $d^{(k, n+1)} = x^{(k, n)}-x^{(k, 0)}$ and conduct a one-dimensional search starting from $x^{(k, n)}$ along $d^{(k, n+1)}$ to find the point $x^k$. (ⅲ) If $\left\|x^k-x^{k-1}\right\| < \varepsilon$, stop and take the point $x^k$ as the final result. Otherwise, let $d^{(k+1, j)} = d^{(k, j+1)}, j = 1, 2 \cdots n$. Let $k: k+1$, and repeat from step (ⅱ) to find the global optimal solution.

3.4.   ARMA model parameter identification based on Powell-DE algorithm

The application of the Powell-DE algorithm in the identification of parameters for the ARMA model is based on the following principle: By selecting an appropriate fitness function, the algorithm conducts global and local searches within the solution space of all $p$ and $q$ values to identify the optimal $p$ and $q$, thus optimizing the ARMA model. This method avoids the complexities and intricacies of traditional approaches and enhances the model's accuracy. The steps for identifying the parameters of the ARMA(p, q) model using the Powell-DE algorithm are outlined in Figure 2.

1) Initialization of basic parameters: Set the population size N = 70, the mutation operator F, the crossover factor CR and the maximum number of iterations Gmax for the differential evolution algorithm. Initialize step size, acceleration factor, reduction rate and precision.

2) Determining the fitness function: The fitness function primarily reflects the individual's adaptability to the environment. Individuals with high fitness values are retained during evolution, while those with low fitness are eliminated or updated. In the context of the ARMA model, the appropriateness of the choice of p and q is theoretically verified by checking if the residual series is white noise, which is a complex calculation. Practically, to test the validity of the ARMA model, one can compare the actual and predicted values of the model. If the difference is trivial, the model is considered correct; otherwise, the model is incorrect.

In practical applications of the ARMA model, especially when observing the predicted time series $\left\{{\mathrm{x}}_1, {\mathrm{x}}_2, \ldots\right\}$, part of the data is used for modeling, and the rest is used for validation. If the predicted values closely match the actual values, then the model is considered correct. The chosen fitness function is as follows:

Here, ${\mathrm{V}}_{{\mathrm{AR}}}$ is referred to as the average relative change value. $N$ is the number of data points used for comparison, $x(i)$ represents the original data, $\hat{x}(i)$ represents the predicted values, and $\bar{x}(i)$ is the average of the original data. Clearly, the smaller ${\mathrm{V}}_{{\mathrm{AR}}}$ is, the better the prediction effect. The determination of the fitness function is closely related to ${\mathrm{V}}_{{\mathrm{AR}}}$. A higher value of the fitness function indicates better prediction accuracy, aligning with the computational principles of the Powell-DE algorithm.

3.5.   Experimental validation

In order to validate the effectiveness of ARMA parameter identification based on the Powell-DE algorithm, a set of 120 groups of data representing the concentration of mother liquor at the outlet of an evaporation process was selected. This data was used for validation and compared with methods from ^[33,35]. The first 60 sets of data were used for ARMA modeling, and the remaining 60 sets were used for model validation.

The parameters for the Powell-DE algorithm were set as follows: population size N = 70, scaling factor and crossover rate at 0.54 and 0.9, respectively, and the number of evolutionary generations at 100. During the simulation process, it was observed that the fitness function value reached its maximum after 60 generations, at which point the optimal autoregressive order $p$ and moving average order $q$ were determined. In Table 1, the top 10 values optimized for the fitness value after 100 generations of evolution are presented. The optimal fitness value was found to be 2.5786 with an average relative change value (VAR) of 0.3878, the autoregressive order at 7 and moving average order at 5.

For comparison, the ARMA(6, 4) and ARMA(2, 0) models from ^[35] and ^[33] were selected. The fitness function values for these models were calculated to be 0.6306 (the determination of the fitness function is closely related to ${\mathrm{V}}_{{\mathrm{AR}}}$ = 1.5859) and 0.4844 (the determination of the fitness function is closely related to ${\mathrm{V}}_{{\mathrm{AR}}}$ = 2.0645), respectively. Figures 3–5 respectively show the comparisons between the actual measurements and predicted values of the mother liquor concentration using ARMA(7, 5), ARMA(6, 4) and ARMA(2, 0) models. Evidently, from the figures, the ARMA(7, 5) model's prediction performance is superior to both the ARMA(6, 4) and ARMA(2, 0) models. This also validates the accuracy and effectiveness of the ARMA model parameter identification method based on the Powell-DE algorithm.

4.   Dynamic error compensation of soft measurement model for evaporation process based on ARMA

The soft measurement of alumina evaporation process outlet mother liquor concentration based on an integrated model is considered from a static perspective. However, the production conditions of the alumina evaporation process are often affected by factors such as acid washing and equipment failures, causing fluctuations that significantly reduce the prediction accuracy of static soft measurement models and, in severe cases, lead to model mismatch. Therefore, it is necessary to establish a soft measurement model that can reflect the dynamic changes in the production process. In particular, for industrial production processes with distinct dynamic characteristics, the use of dynamic soft measurement models can enhance the model's estimation accuracy and robustness ^[36].

Unlike previous models that predominantly focus on refining static aspects, our approach integrates the ARMA model's capability to adaptively adjust to time-varying process changes, offering a more realistic and responsive representation of the dynamic industrial environment. This adaptability is rooted in the ARMA model's unique ability to amalgamate both autoregressive and moving average components, providing a more nuanced and comprehensive understanding of process dynamics compared to conventional static models.

Several scholars have studied nonlinear dynamic soft measurement modeling ^[37,38]. ^[39] integrates dynamic information into a nonlinear static model to achieve the dynamic characteristics of the static model. ^[40] introduces the concept of moving windows into soft measurement modeling for polymer reactors, improving the accuracy of polymer concentration estimation. ^[41] uses a nonlinear moving average method to construct a soft measurement model for a xylene distillation column, and the experimental results prove the method's ability to accurately estimate the purity of xylene. ^[42] proposes the implementation of a feedback neural network to construct a dynamic soft measurement model of the mycelium concentration in the erythromycin fermentation process.

These soft measurement models are based on improvements to static models. The process neural networks (PNN) ^[43], proposed by He Xingui, can effectively express the cumulative effect of input information over time and are a relatively practical dynamic soft measurement model. Based on this, our model leverages the ARMA process mechanism, which uniquely addresses the continuity and cumulativity of process parameters, a feature not adequately captured by previous models. The inclusion of error compensation based on this mechanism, combined with a reduced robust LSSVM, provides a theoretical innovation in dynamic modeling, offering practical advantages in terms of improved prediction accuracy and adaptability to process fluctuations.

4.1.   Principle of error time series prediction based on ARMA

The ARMA model ^{[44,45,46,47]} predicts future series values based on the current time series changes, i.e., a time series fitting model. The response $X_t$ at time $t$ is not only related to its own value at the previous moment but also to the disturbance entering the system at the previous moment. The ${\rm{ARMA}}(p, q)$ model can be described as

Introducing the backward operator $B$, its operation includes $B X_t = X_{t-1}, \quad B^2 X_t = X_{t-2}$ and so on. Thus, Eq (4.1) can be transformed into

In this formula, $a_t$ represents a white noise series, and $\varphi(B)$ and $\theta(B)$ are the $p$th and $p$th-order polynomials of the backward operator $B$, respectively.

4.2.   Dynamic compensation modeling of soft measurement for outlet concentration of evaporation based on ARMA

The dynamic compensation model for soft measurement of outlet concentration of evaporation based on ARMA uses the error time series of the reduced robust LSSVM model ^[48] as the model input. The relationship between the reduced robust LSSVM and the ARMA model is shown in Figure 6. Based on the static soft measurement model of reduced robust LSSVM outlet mother liquor concentration, first, the difference between the output value of the reduced robust LSSVM model $y_n^{\prime}(t)$ and the actual outlet mother liquor concentration measurement value $y_n(t)$ is used to obtain the time series value of the output error $\Delta y_n$. Then, the ARMA model is used to predict the error time series. The difference between the output value of the reduced robust least squares support vector machine integrated model and the predicted error value of the ARMA model gives the prediction result of the outlet mother liquor concentration.

The specific steps for dynamic compensation in soft measurement modeling are as follows:

1) Select the input-output variable set $\tilde{X} = \left(x_{i j}\right)_{M \times N}$ with $i = 1, 2, \cdots, M, j = 1, 2, \cdots, N$, where $M$ is the dimension of the sample set, and $N$ is the dimension of the auxiliary variable set. First, standardize the samples as follows:

In this formula, ${\rm{var}}\left(X_j\right)$ is the sample variance, and $E\left(X_j\right)$ is the mean of the $j$th column of samples in $\tilde{X}$, resulting in a standardized data set $\tilde{X}^*$ (mean 0, variance 1).

2) Choose the Gaussian radial basis function as the kernel function for the reduced robust least squares support vector machine, initialize model parameters and train the reduced robust LSSVM using the training sample set.

3) Validate the reduced robust LSSVM model through the test sample set, and then return to step 2 until the test ends.

4) Based on the established soft measurement intelligent integration model for the evaporation process, obtain the predicted value of the outlet mother liquor concentration. Use the mean square error as given in Eq (4.5) as the evaluation index, and select the model with the smallest generalized mean square error as the prediction model for the reduced robust least squares support vector machine.

In this formula, $y_n$ represents the actual measurement value on-site, and $y_n^{\prime}(t)$ is the model output value.

5) Obtain a set of model output values $y_n^{\prime}$ from the reduced robust LSSVM model and the actual on-site measurement values $y_n$. Subtract them to get the time series value of the error $\Delta y_n$, and use ARMA to model the error series. After modeling, determine the order of the ${\rm{ARMA}}(p, q)$ model and seek the optimal $p$ and $q$ values.

6) Apply the two models to the test samples, perform the same steps, and obtain the final output value of the soft measurement model as follows: $y^{\prime}(t) = y_n^{\prime}(t)-\Delta y_n^{\prime}(t)$.

4.3.   Industrial production data simulation and analysis of ARMA model

In order to test the effectiveness of the ARMA model, 300 groups of dynamic data samples from the evaporation process were selected as the training set, with an additional 60 groups of dynamic data as the test set. The input to the reduced robust LSSVM model was $x(t) = \left(a_1, a_2, a_3, a_4, a_5, a_6\right)$, and the output was $y(t) = b_1$. Here, $a_1, a_2, a_3, a_4, a_5, a_6$ represent six influential factor variables derived from data preprocessing, grey relational analysis and principal component analysis in the evaporation process, while $b_1$ represents the outlet mother liquor concentration. The model was trained and tested with these input and output values.

The difference between the training output values of the reduced robust LSSVM model and the actual values of outlet mother liquor concentration formed a sequence. This sequence was then used as the input for the ARMA model. After training, the ARMA model predicted the error and compensated for the reduced robust LSSVM model. The parameters of the ARMA model were determined to be ARMA(5, 4) after comparison. The soft measurement results for the outlet mother liquor concentrations of caustic alkali, aluminum oxygen and total alkali are illustrated in Figures 7–9.

5.   Intelligent integration of ARMA model and mechanism-reduced robust LSSVM model

5.1.   Intelligent integrated soft measurement model based on entropy method

Recent years have seen a trend among scholars, both domestically and internationally, to combine several models through a weighted approach, a practice that has been effectively implemented in various fields. ^[49] has introduced methods for weighted harmonic and geometric averaging combinations. These methods, based on the minimization of the sum of squared errors, represent an approach that reflects the internal nonlinear characteristics of systems. Furthermore, ^[50] has applied these methods in practical scenarios, finding that constructing effective nonlinear combination functions is challenging, thereby affecting prediction accuracy. These methods are suitable for both linear and nonlinear integrated forecasting.

This subsection introduces an advanced integration strategy using the entropy method. The entropy method stands out for its ability to effectively balance and integrate multiple models by assigning optimal weights based on the relative performance of each model. This integration aims to closely align the predicted and actual values of the mother liquor concentration at the outlet of the evaporation process over a long-term production period. Drawing from the concept of integrated soft measurement modeling ^[51], the aforementioned two models are intelligently integrated to construct a comprehensive model for predicting the concentration of the mother liquor at the outlet of the evaporation process. The optimal weighting coefficients are determined using the entropy method ^[52,53], and the weighted outputs of the two sub-models are combined to form the final prediction of the mother liquor concentration at the outlet. $\hat{y}_{1 t}$ and $\hat{y}_{2 t}$ represent the predicted values of the mother liquor concentration at the outlet at time $t$ for the mechanism-reduced robust LSSVM model and the dynamic compensation model based on ARMA, respectively. $\hat{y}_{t}$ is the predicted value of the integrated soft measurement model at time $t$.

The relative prediction error $e_{i t}$ for each soft measurement model at time $t$ is calculated as follows:

where \(y_t \) represents the actual measured concentration of the mother liquor at the outlet of the evaporation process.

The steps for determining the weighting coefficients using the entropy method are as follows:

Step 1. Calculate the relative error proportion \(p_{it} \) for the \(i^{th} \) soft measurement model at time \(t \).

Step 2. Calculate the entropy value \(E_{it} \) for the relative error at time \(t \) for the \(i^{th} \) model.

Step 3. Determine the variation coefficient \(d_{it} \) for the relative error sequence of the \(i^{th} \) model.

Step 4. Compute the weighting coefficient \(w_{it} \) for each soft measurement model.

5.2.   Simulation analysis of intelligent integrated soft measurement results

The weighting coefficients determined by the entropy method resulted in the values $w_1 = 0.468$ and $w_2 = 0.532$. These coefficients were used to combine the predictions according to the formula $\hat{y}_t = w_1 \hat{y}_{1 t}+w_2 \hat{y}_{2 t}$. This formula was utilized to predict the outlet mother liquor concentration values from the evaporation process. To validate the robustness of the model, a set of 100 dynamic data points from industrial operations was selected for testing. The results of the integrated soft measurement model for the total alkali and aluminum oxide concentrations at the outlet of the evaporation process are displayed in Figures 10 and 11, respectively. An error analysis and comparison are presented in Table 2.

As observed from Table 2, the intelligent integrated soft measurement model demonstrates higher predictive accuracy compared to the static soft measurement combination model based on mechanisms and data, as well as the dynamic correction soft measurement model based on the ARMA error time series. This is evidenced by improved metrics such as MSE (mean squared error), RMSE (root mean squared error), and RRMSE (relative root mean squared error) in the intelligent integrated soft measurement model. This indicates that the model effectively combines static and dynamic characteristics, exhibiting superior modeling performance. It is well-suited for predicting the concentration of the mother liquor at the outlet of the evaporation process, and its predictive accuracy meets the error requirements for the alumina evaporation process.

6.   Conclusions

This paper addresses the issues arising from static soft measurement modeling methods, which fail to reflect the dynamic information of industrial processes, leading to reduced prediction accuracy and poor robustness of the soft measurement models. A dynamic correction soft measurement modeling method based on the ARMA error time series is proposed to address these issues. Our proposed method stands out in its ability to dynamically and accurately adapt to process changes, a crucial aspect often overlooked in conventional static methods.

By incorporating the ARMA model's strengths in error time series analysis, the proposed method offers a theoretically innovative approach that captures the evolving nature of industrial processes. This approach not only enhances prediction accuracy but also ensures robustness against the inherently dynamic and unpredictable elements of industrial settings. An ARMA model for the error time series of the soft measurement model of the mother liquor concentration at the outlet of the evaporation process was established, focusing on the mechanism-reduced robust LSSVM. This model dynamically estimates the prediction errors of the soft measurement model. Finally, an intelligent integration method is proposed. By intelligently integrating the static and dynamic soft measurement models using the entropy weighting method, soft measurement values of evaporation process parameters are obtained. Simulation validation with actual field data demonstrates that the intelligent integrated soft measurement model has high prediction accuracy and robustness, and the prediction results meet the industrial production requirements.

This intelligent integration of static and dynamic modeling, achieved through the novel application of the entropy weighting method, represents a significant theoretical and practical advancement in the field of soft measurement. The validation of our model with actual field data confirms its superior predictive capabilities and robustness, aligning closely with the requirements of industrial production.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under grant No. 61963036 and in part by the Jiangxi Provincial Department of Education Science and Technology Project under grant No. GJJ2201719.

Conflict of interest

The authors declare there is no conflict of interest.