Input Variable Selection Method Research Articles

Monthly Streamflow Prediction of the Source Region of the Yellow River Based on Long Short-Term Memory Considering Different Lagged Months

Accurate and reliable monthly streamflow prediction plays a crucial role in the scientific allocation and efficient utilization of water resources. In this paper, we proposed a prediction framework that integrates the input variable selection method and Long Short-Term Memory (LSTM). The input selection methods, including autocorrelation function (ACF), partial autocorrelation function (PACF), and time lag cross-correlation (TLCC), were used to analyze the lagged time between variables. Then, the performance of the LSTM model was compared with three other traditional methods. The framework was used to predict monthly streamflow at the Jimai, Maqu, and Tangnaihai stations in the source area of the Yellow River. The results indicated that grid search and cross-validation can improve the efficiency of determining model parameters. The models incorporating ACF, PACF, and TLCC with lagged time are evidently superior to the models using the current variable as the model inputs. Furthermore, the LSTM model, which considers the lagged time, demonstrated better performance in predicting monthly streamflow. The coefficient of determination (R2) improved by an average of 17.46%, 33.94%, and 15.29% for each station, respectively. The integrated framework shows promise in enhancing the accuracy of monthly streamflow prediction, thereby aiding in strategic decision-making for water resources management.

Incorporating novel input variable selection method for in the different water basins of Thailand

Selecting appropriate input variables for developing a rainfall prediction model is significantly difficult. The present study proposed an innovative framework for input variable selection (IVS) model bootstrapped long short-term recurrent neural network (BTSP-LSTM-RNN) to identify relevant variables for monthly rainfall forecasting. Monthly meteorological and large-scale climatic variables (LCVs) from 1993 to 2022 at two selected river basins in the northern region of Thailand were used for model development. The proposed BTSP-LSTM-RNN model results were compared with the support vector regression with recursive feature elimination (SVR-RFE) and Gradient boosting (GB) by statistical metrics such as coefficient of determination (R2), mean absolute error (MAE), relative root mean squared error (RRMSE), Pearson’s correlation coefficient (r) and mean absolute percentage error (MAPE). BTSP-LSTM-RNN demonstrated exceptional performance, boasting a higher R2 (0.84), MAE (92.28), RRMSE (10.36) in the Wang basin, and R2 (0.83), MAE (242.60), RRMSE (9.93) in the Nan basin. BTSP-LSTM-RNN also achieved the lowest MAPE of 29.82%. Based on this IVS model results, two input variable combinations (IVCs) were designed. IVC-1 is based on BTSP-LSTM-RNN selection, and IVC-2 is an original set of variables. LSTM-RNN, multi-layer perceptron artificial neural network (MLP-ANN), and ensemble model with bootstrapping on random forest (RF) were employed for monthly prediction. When BTSP-LSTM-RNN's selected input variables from IVC-1 are utilized, the BSTP-RF model demonstrates robust performance. It achieves a high R2 (0.82), a low RRMSE (10.13%) suggests accurate predictions, and the r of 0.91 further supports the model's strong linear relationship with observed rainfall data. Based on prediction model results, the BTSP-LSTM-RNN (IVS) model plays a pivotal role in the selection of input variables for rainfall forecasting and its impact on the performance of prediction models (BSTP-RF, MLP-ANN, and LSTM-RNN). These results consistently underscored the pivotal role of the BTSP-LSTM-RNN IVS model in enhancing the precision and reliability of rainfall predictions.

Prediction, modelling, and forecasting of PM and AQI using hybrid machine learning

This paper proposes a combination of hybrid models like Input Variable Selection (IVS), Machine Learning (ML), and regression method to predict, model, and forecast the daily concentrations of particulate matter (PM1, PM2.5, PM10) and Air Quality Index (AQI). A sensor placed in the centre of Craiova, Romania, provides a two-year dataset for training, testing, and validation phases. The analysis identifies the most important predictor variables for PM prediction and forecasting. The coefficient of determination (R2) values in this stage exceeded 0.95 (95%), indicating a strong correlation between PM concentrations. The performance of the proposed models is evaluated by objective measures, including root mean squared error (RMSE) and standard deviation (σ). RMSE ranged between 0.65 and 1 μg/m3, while σ has values between 2.75 and 4.1 μg/m3, reflecting a high level of precision and a successful performance of the proposed models. Furthermore, 13 multivariable-based PM models are developed in this study and adjusted using a hybrid Least Square - Decision Tree approach. The R2 values for these adjusted models range from 0.66 to 0.75, while the RMSE and σ vary between 8 and 9.1 μg/m3. Finally, a handled application for multistep-ahead time series forecasting is elaborated by combining the Nonlinear System Identification (NARMAX) approach with Decision Tree machine learning. This application allows for forecasting PM concentrations and AQI for the next periods. The R2 values obtained in this stage surpass 0.93, indicating almost a high level of accuracy. The RMSE ranged between 4.43 and 6.25 μg/m3, while the σ ranged between 4.44 and 6.26 μg/m3, further validating the precision of our forecasting model.

Development of an efficient input selection method for NN based streamflow model

In this paper, using a neural network-based streamflow simulation model (NNSSM), we simulate the runoff of the Ajichai River. The selection of suitable inputs is an essential step toward developing NNSSM. For this aim, we investigate a novel application of the Genetic Classification Algorithm (GCA) as an input variable selection (IVS) method in comparison with the Self-Organizing Map (SOM) and Binary Fully Informed Particle Swarm Optimization (BFIPSO). In another innovative application, we establish Social Choice (SC) for the final ranking of selected data using SOM. Next, the model was improved by adding seasonality indexes. The results indicate the superiority of GCA. The average (maximum) Nash-Sutcliffe index for GCA was found to be 0.63(0.84), while it was 0.55(0.71) and 0.58(0.77) for SOM-SC and BFIPSO, respectively. Moreover, GCA took less than 30 min for each run, while for SOM-SC and BFIPSO at least 3 and 48 h were needed under the same circumstances.

A Hybrid Approach to Improve Flood Forecasting by Combining a Hydrodynamic Flow Model and Artificial Neural Networks

Climate change is driving worsening flood events worldwide. In this study, a hybrid approach based on a combination of the optimization of a hydrodynamic model and an error correction modeling that exploit different aspects of the physical system is proposed to improve the forecasting accuracy of flood water levels. In the parameter optimization procedure for the hydrodynamic model, Manning’s roughness coefficients were estimated by considering their spatial distribution and temporal variation in unsteady flow conditions. In the following error correction procedure, the systematic errors of the optimized hydrodynamic model were captured by combining the input variable selection method using partial mutual information (PMI) and artificial neural networks (ANNs), and therefore, complementary information provided by the data was achieved. The developed ANNs were used to analyze the potential non-linear relationships between the considered state variables and simulation errors to predict systematic errors. To assess the hybrid forecasting approach (hydrodynamic model with an ANN-based error correction model), performances of the hydrodynamic model, two ANN-based water-level forecasting models (ANN1 and ANN2), and the hybrid model were compared. Regarding input candidates, ANN1 considers the historical observations only, and ANN2 considers not only the historical observations that used in ANN1 but also the prescribed boundary conditions required for the hydrodynamic forecast model. As a result, the hybrid model significantly improved the forecasting accuracy of flood water levels compared to individual models, which indicates that the hybrid model is able to take advantage of complementary strengths of both the hydrodynamic model and the ANN model. The optimization of the hydrodynamic model allowing spatially and temporally variable parameters estimated water levels with acceptable accuracy. Furthermore, the use of PMI-based input variable selection and optimized ANNs as error correction models for different sites were proven to successfully predict simulation errors in the hydrodynamic model. Hence, the parameter optimization of the hydrodynamic model coupled with error correction modeling for water level forecasting can be used to provide accurate information for flood management.

An improved input variable selection method of the data-driven model for building heating load prediction

In building energy conservation, HVAC systems have great potential, and load prediction plays an important role in optimizing system control and improving system energy efficiency. Therefore, improving the accuracy of load prediction is of great significance. This paper aims to improve the prediction accuracy of the data-driven model of load prediction through optimizing the input variable selection. Factors such as outdoor temperature, wind speed, solar radiation, indoor temperature and user behavior have different mechanisms of action to influence the heating load, and there exists difference among the influence degree of each factor on the heating load at each moment. Therefore, the optimal input variable used by the heating load prediction models at different moments will be different. Based on this, this paper proposes to establish an input variable selection method based on hourly correlation analysis. Taking an office building as the case building, this paper uses the back propagation neural network (BPNN) model and the support vector regression (SVR) model to compare the selection methods of hourly correlation analysis and non-hourly correlation analysis. The results show that the influence degree of solar radiation intensity and the historical load at the moment of 24 h ahead on the load at each moment have great difference. Compared with non-hourly correlation analysis, hourly correlation analysis can reduce the root mean square error (RMSE) of the prediction results of the BPNN model and the SVR model by an average of 10.4% and 12.9%, respectively, and increase the coefficient of determination (R2) by an average of 11.7% and 7.9%, respectively, while all the calculation time is about 0.4 s. This shows that the heating load prediction method based on hourly correlation analysis can improve the prediction accuracy of model while keeping the prediction rate unchanged.

Examination and comparison of binary metaheuristic wrapper-based input variable selection for local and global climate information-driven one-step monthly streamflow forecasting

The use of data-driven models to forecast streamflow has received substantial attention from scholars in recent years. However, systematic studies have not been performed to examine binary metaheuristic wrapper-based input variable selection (BMWIVS) in real-world streamflow forecasting. In this study, we explored binary metaheuristic-based shallow machine learning wrappers for one-step monthly streamflow forecasting using local weather information and global climate indices from three catchments with different hydroclimatic conditions. First, the maximal information coefficient (MIC) was employed to investigate the correlations among the forecasting target, streamflow and candidate input variables, which included both local and global climate information. Then, the BMWIVS models obtained by combining eight binary metaheuristic algorithms, five commonly used shallow machine learning algorithms, two combined filter-based input variable selection (FIVS) methods, and two forecasting methods were examined. Finally, the performance of each model was compared with the performance of typical benchmark models, including the univariate seasonal autoregressive integrated moving average model, five machine learning algorithms with no input variable selection, and five machine learning algorithms that use five different FIVS methods. The experimental results emphasized three significant findings. First, an appropriate input variable selection method should be selected in practice because several examined wrappers were inferior to the benchmark models. Second, the BMWIVS model that combined the regularized extreme learning machine method, binary gray wolf optimizer, FIVS results-based initialization method, and forecasted values averaged over multiple runs yielded the best performance in the three cases studied. Third, the correlations in terms of the MIC between the global climate indices and streamflow were lower than those between local weather information and streamflow, and the best wrapper and FIVS would select more local weather information variables than global climate index variables, which suggests that global climate information can be complementary to local weather information for one-step monthly streamflow forecasting. These findings have remarkable practical applications for forecasting monthly streamflow.

Predicting water quality during urbanization based on a causality-based input variable selection method modified back-propagation neural network.

Rapid urbanization has been recognized as the primary cause of deteriorating water quality. Thus, it is crucial to take into account urbanization in water quality forecasting. The present study aims at finding the causal relationship between urbanization and water quality, and then predicting water quality based on this causality. For this purpose, nine urbanization indicators and 12 water quality parameters from 2006 to 2018 in Nanjing were collected as urbanization and water quality indices. Correlation and path analyses were firstly used to identify causal relationships between urbanization and water quality indices. Based on these causal relationships, comprehensive water quality indicators and their correlated urbanization parameters were input into a back-propagation neural network (BPNN) to predict water quality. In the improved BPNN, the R2 of the training sets were all greater than 0.99, and those of the test sets were all greater than 0.76, demonstrating that the optimized model is able to predict the water quality with reasonable accuracy. It also showed that the overall water quality in Nanjing will remain good from 2019 to 2028, which means that, when undergoing future urbanization process, water quality is not necessarily negatively affected. The transfer of industrial structure can have a positive influence on water quality. After 2028, the biological water environment index remained in a good state but the volatile phenol index continued to increase, making it a potential threat to future water quality. Industrial wastewater and fertilizer usage, as the primary sources of volatile phenols, should be prioritized for continued governmental control and monitoring into the future. This study provides new insight into the relationship between urbanization and water quality, and the presented models can assist in future-proofing water management strategies.

Heuristic Input Variable Selection in Multi-Objective Reservoir Operation

Deriving operating rules for multi-objective cascade reservoir systems is an important challenge in water resources management. To address, this study combines a radial basis function network with an evolutionary algorithm to propose a heuristic input variable selection (HIS) method that extracts reservoir operating rules based on feature selection. For a case study of the Hanjiang cascade reservoirs in China, we initially describe the operating rules with radial basis functions and subsequently refine them based on the HIS method. We select the most suitable input variables for each reservoir conditioned on water supply and power generation targets to derive and optimize the rules with a Pareto-archived dynamically dimensioned search algorithm. From this we can analyze input variable selection and the corresponding impact on multi-objective cascade reservoir operations. The results demonstrate that the HIS method selects the input variables accurately and the reservoir operating rules refined by the method could increase water supply by up to 6.6% and power generation by up to 1.2%. The most suitable input variables for reservoir operation vary depending on reservoir objective, however the HIS method appears effective at selecting the appropriate input variables for individual reservoirs in a cascade system.

Online parameter estimation considering input errors for synchronous generation systems

Parameter estimation of synchronous generation systems helps system operators develop control schemes to enhance system stability and reliability. However, the input signals used in the online estimation are measured with errors and their direct usage may produce biased parameters. To handle this problem, an online parameter estimation method considering input errors is proposed. A Lagrange multiplier-based input variable selection method is further proposed to reduce the scales of estimation problem and improve the computational efficiency. In order to avoid ill-conditioned problem resulted from input error over-fitting, a Tikhonov regularisation approach is applied and the optimal regularisation parameter is determined via cross-validation. In the case study, the numerical experiments and real-world results validate the effectiveness and advantages of the proposed method.

Use of data-driven model to analyse the occurrence patterns of an indicator fish species in river: A case study for Alburnoides eichwaldii (De Filippi, 1863) in Shafaroud River, north of Iran

The present study aims to integrate multinomial logistic regression with an input variable selection method, genetic algorithm, GA, to select the most important explanatory variables for evaluating the occurrence patterns of the bleak (Alburnoides eichwaldii) in river. Seven different sampling sites (from the source to the mouth of the Shafaroud River, north of Iran) were considered to analyse the probability of occurrence of the fish during one year sampling campaign. The abundance of bleak (based on 42 fish presence and 42 fish absence data, as outputs of model) together with a set of physical-chemical water characteristics and river morphology (84 instances as inputs of model) were monthly and repeatedly recorded at each sampling site. Two-third of instances (56) was used for training and the remaining of instances (28) as test set. The results of paired Student’s t-test showed that the predictive performances of model (% correctly classified instance and Kappa statistics) were improved after variable selection method. GA selected 9 of 18 input variables including dissolved oxygen, pH, water temperature, river depth, electric conductivity, total hardness, nitrite, orthophosphate and sulphate. The curves of binary logistic regression confirmed that increasing three of the selected variables (dissolved oxygen, water temperature and pH) might increase the probability of bleak presence while increasing concentration of other selected variables might decrease the probability of fish occurrence in the river basin (p < 0.05 for all). Information on the occurrence patterns of indicator fish species in relation to physico-chemical water quality and habitat variables is important to develop conservation and management programs for the study river, where water resources are facing with high risk of pollution due to increasing of agricultural and industrial development activities.

Seasonal streamflow forecasts using mixture-kernel GPR and advanced methods of input variable selection

Abstract Gaussian Process Regression (GPR) is a new machine-learning method based on Bayesian theory and statistical learning theory. It provides a flexible framework for probabilistic regression and uncertainty estimation. The main effort in GPR modelling is determining the structure of the kernel function. As streamflow is composed of trend, period and random components. In this study, we constructed a mixture-kernel composed of squared exponential kernel, periodic kernel and a rational quadratic term to reflect different properties of streamflow time series to make streamflow forecasts. A relevant feature-selection wrapper algorithm was used, with a top-down search for relevant features by Random Forest, to offer a systematic factors analysis that can potentially affect basin streamflow predictability. Streamflow prediction is evaluated by putting emphasis on the degree of coincidence, the deviation on low flows, high flows and the error level. The objective of this study is to construct a seasonal streamflow forecasts model using mixture-kernel GPR and the advanced input variable selection method. Results show that the mixture-kernel GPR has good forecasting quality, and top importance predictors are streamflow at 12, 6, 5, 1, 11, 7, 8, 4 months ahead, Nino 1 + 2 at 11, 5, 12, 10 months ahead.

Semi-seasonal groundwater forecast using multiple data-driven models in an irrigated cropland

Abstract In agricultural areas where groundwater is the main water supply for irrigation, forecasts of the water table are key to optimal water management. However, water management can be constrained by semiseasonal to seasonal forecast. The objective is to create an ensemble of water table one- to five-month lead-time forecasts based on multiple data-driven models (DDMs). We hypothesize that data-driven modeling capabilities (e.g., random forests, support vector machines, artificial neural networks (ANNs), extreme learning machines, and genetic programming) will improve naïve and autoregressive simulations of groundwater tables. An input variable selection method was used to carry the maximum information in the input (precipitation, crop water demand, changes in groundwater table, snowmelt, and evapotranspiration) and output relationship for the DDMs. Five DDMs were implemented to forecast groundwater tables in an unconfined aquifer in the Northern High Plains (Nebraska, USA). Root mean squared error (RMSE) and Nash–Sutcliffe efficiency index were used to evaluate the skill of the model and three hydrologic regimes were determined statistically as high, mid, and low groundwater table levels. Additionally, varying storage regimes were used to construct rising and falling limbs in the tested well. Results show that all models outperform the baseline for all the lead times, ANNs being the best of all.

Application of PCA and Clustering Methods in Input Selection of Hybrid Runoff Models

This study has proposed and investigated a novel input variable selection method for nonlinear modelling based on principle component analysis (PCA) and cluster analysis. The proposed approach was applied to daily rainfall-runoff modelling of the Brue catchment of the United Kingdom using wavelet based hybrid forms of two nonlinear models, Artificial Neural Networks (ANNs) and Local Linear Regression (LLR), to identify meaningful wavelet decomposed sub-series. The homogenous group formation capability of cluster analysis and redundancy assessment capability of PCA were applied effectively in this study to solve input selection uncertainties associated with wavelet based hybrid models. Though this concept has been represented in the selection of effective wavelet decomposed subseries in runoff modelling, the application has gotten wider implications in time series modelling with highly redundant and large input space. The study revealed the weakness of conventional forms of cross-correlation analysis and also suggested that input selection could be improved by making sufficient natural clusters (equal to the desired number of input data series) of input space and restricting the search within each cluster according to silhouette or correlation value. The study also highlighted the higher modelling capability of ANN over traditional LLR models in rainfall-runoff modelling of the Brue catchment.

Using Random Forests to Select Optimal Input Variables for Short-Term Wind Speed Forecasting Models

Achieving relatively high-accuracy short-term wind speed forecasting estimates is a precondition for the construction and grid-connected operation of wind power forecasting systems for wind farms. Currently, most research is focused on the structure of forecasting models and does not consider the selection of input variables, which can have significant impacts on forecasting performance. This paper presents an input variable selection method for wind speed forecasting models. The candidate input variables for various leading periods are selected and random forests (RF) is employed to evaluate the importance of all variable as features. The feature subset with the best evaluation performance is selected as the optimal feature set. Then, kernel-based extreme learning machine is constructed to evaluate the performance of input variables selection based on RF. The results of the case study show that by removing the uncorrelated and redundant features, RF effectively extracts the most strongly correlated set of features from the candidate input variables. By finding the optimal feature combination to represent the original information, RF simplifies the structure of the wind speed forecasting model, shortens the training time required, and substantially improves the model’s accuracy and generalization ability, demonstrating that the input variables selected by RF are effective.

Multiobjective reservoir operating rules based on cascade reservoir input variable selection method

AbstractThe input variable selection in multiobjective cascade reservoir operation is an important and difficult task. To address this problem, this study proposes the cascade reservoir input variable selection (CIS) method that searches for the most valuable input variables for decision making in multiple‐objectivity cascade reservoir operations. From a case study of Hanjiang cascade reservoirs in China, we derive reservoir operating rules based on the combination of CIS and Gaussian radial basis functions (RBFs) methods and optimize the rules through Pareto‐archived dynamically dimensioned search (PA‐DDS) with two objectives: to maximize both power generation and water supply. We select the most effective input variables and evaluate their impacts on cascade reservoir operations. From the simulated trajectories of reservoir water level, power generation, and water supply, we analyze the multiobjective operating rules with several input variables. The results demonstrate that the CIS method performs well in the selection of input variables for the cascade reservoir operation, and the RBFs method can fully express the nonlinear operating rules for cascade reservoirs. We conclude that the CIS method is an effective and stable approach to identifying the most valuable information from a large number of candidate input variables. While the reservoir storage state is the most valuable information for the Hanjiang cascade reservoir multiobjective operation, the reservoir inflow is the most effective input variable for the single‐objective operation of Danjiangkou.

Hybrid Input Data Selection Technique for Neural Network Based Load Forecast Model

Selection of best model parameters is a key to building an effective neural network based prediction model. This paper focuses on short term load forecasting (STLF) framework emphasizing on a new hybrid data selection preprocessing and selection technique to enhance forecast accuracy. A Multilayer perceptron neural network (MLPNN) is used to predict the load demand of one week ahead. Because of the mapping and memorizing ability of the artificial neural network (ANN) towards the non-linear relations between input and output variables, they are widely used in predictive modeling, such as, electrical load demand prediction models. The non-linear behaviour of the load can only be narrated effectively by ANN, if a group of appropriate inputs is identified prior to the training process. A new input variable selection method based on statistical methods and evolutionary computation technique genetic algorithm (GA) is proposed in this research. A case study is developed to verify the efficacy of the optimized training dataset and input variable selection method. In this approach, we trained the ANN with conventional back propagation (BP) training algorithm with and without using the optimized and reduced dataset. Results show that this dataset selection method is effective and forecasts electrical load demand with reduced mean absolute percentage error (MAPE) and reduced training time.

A hybrid input variable selection method for building soft sensor from correlated process variables

Selection of the most relevant input variables for an inferential predictor is important for good prediction ability. A hybrid variable selection method is proposed for selecting input variables for support vector regression (SVR) model. The proposed method combines Taguchi's experimental design method with backward elimination method to select the most relevant variables from a large set of process variables. Taguchi's design of experiment (DoE) method was used to screen variables, as process variables are highly correlated this poses difficulty to fill in the design matrix of Taguchi's DoE method. The proposed method makes several modifications to Taguchi's method to deal with this problem. Subsequently backward elimination method was used to select the final set of input variables. The efficacy of the proposed methodology is demonstrated on an industrial case study.

Bootstrap rank‐ordered conditional mutual information (broCMI): A nonlinear input variable selection method for water resources modeling

AbstractThe input variable selection problem has recently garnered much interest in the time series modeling community, especially within water resources applications, demonstrating that information theoretic (nonlinear)‐based input variable selection algorithms such as partial mutual information (PMI) selection (PMIS) provide an improved representation of the modeled process when compared to linear alternatives such as partial correlation input selection (PCIS). PMIS is a popular algorithm for water resources modeling problems considering nonlinear input variable selection; however, this method requires the specification of two nonlinear regression models, each with parametric settings that greatly influence the selected input variables. Other attempts to develop input variable selection methods using conditional mutual information (CMI) (an analog to PMI) have been formulated under different parametric pretenses such as k nearest‐neighbor (KNN) statistics or kernel density estimates (KDE). In this paper, we introduce a new input variable selection method based on CMI that uses a nonparametric multivariate continuous probability estimator based on Edgeworth approximations (EA). We improve the EA method by considering the uncertainty in the input variable selection procedure by introducing a bootstrap resampling procedure that uses rank statistics to order the selected input sets; we name our proposed method bootstrap rank‐ordered CMI (broCMI). We demonstrate the superior performance of broCMI when compared to CMI‐based alternatives (EA, KDE, and KNN), PMIS, and PCIS input variable selection algorithms on a set of seven synthetic test problems and a real‐world urban water demand (UWD) forecasting experiment in Ottawa, Canada.

Efficient input variable selection for soft-senor design based on nearest correlation spectral clustering and group Lasso

Appropriate input variables have to be selected for building highly accurate soft sensor. A novel input variable selection method based on nearest correlation spectral clustering (NCSC) has been proposed, and it is referred to as NCSC-based variable selection (NCSC-VS). Although NCSC-VS can select appropriate input variables, a lot of parameters have to be tuned carefully for selecting proper variables. The present work proposes a new methodology for efficient input variable selection by integrating NCSC and group Lasso. The proposed NCSC-based group Lasso (NCSC-GL) can not only reduce the number of tuning parameters but also achieve almost the same performance as NCSC-VS. The usefulness of the proposed NCSC-GL is demonstrated through applications to soft sensor design for a pharmaceutical process and a chemical process.