# Input Selection Methods for Soft Sensor Design: A Survey
## 6 Summary and Conclusions
\begin{table}
\begin{tabular}{p{56.9 pt} p{56.9 pt} p{56.9 pt} p{56.9 pt}} \hline \hline
**Class** & **Type** & **Pros** & **Cons** \\ \hline
**FE** & \multirow{2}{*}{**Linear**} & **Nonlinear** & \multirow{2}{*}{Reduced computational demand.} & \multirow{2}{*}{Unsupervised. Final projections do not have any physical meaning and all measurement sensors are still needed.} \\  & PCA, SPCA, PPCA. & NLPCA, KPCA, ICA, MDS, PCoA, Isomap, LLE, E SOM. & & \\ \hline
**FS** & \multirow{2}{*}{**Linear**} & **NonLinear** & \multirow{2}{*}{Simplest, fastest, model-independent.} & \multirow{2}{*}{Inputs are considered individually. Dependencies and interactions are disregarded.} \\ _Filter_ & CC analysis. (Pearson, Spearman) & & & \\ \multirow{2}{*}{_Filter_} & **Linear** & **NonLinear** & \multirow{2}{*}{\
\begin{tabular}{} \end{tabular} } \\  & CC analysis. (Pearson, NMIFS, MIFS-U, MIFS, ITSS), Lipschitz coeff. & & & \\ \multirow{2}{*}{_Wrapper_} & **Deterministic** & **Random** & \multirow{2}{*}{
\begin{tabular}{} \end{tabular} } \\  & FS, SFFS, HE, SBFS. & Heuristic search (GA, ACO, SA). & & \\ \multirow{2}{*}{_Embeddad_} & **RFE, RFEST, EnRFE, Sensitivity analysis, Evolutionary ANNs, LASSO, LASSO-MLP, Elastic net. & & & \\ \multirow{2}{*}{_Hybrid_} & **Every possible combination of methods from different classes.** & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Classification of the methods, considering pros and cons. Given \(n\) number of samples and \(p\) number of inputs. tests must be performed by the designer in an effort to find the most suitable subset for the application, making the input selection step time and computationally consuming. For this reason, the problem of variable selection is highly demanded, making it a topic that still needs to be researched. \begin{table}
\begin{tabular}{l l l} \hline \hline \multicolumn{2}{l}{**Methods References Table**} \\ \hline  & PCA & [105; 106; 107; 108; 109; 110; 117] \\  & MDS & [118; 119] \\  & PCoA & [120] \\
**FE** & Isomap & [121; 122; 123] \\  & LLE & [124; 125] \\  & LE & [126] \\  & SOM & [127] \\ \hline  & CC & [136; 138; 139; 140; 141] \\  & Univariate MI & [129] \\  & Multivariate MI & [149; 150; 152; 153; 154; 155; 156; 159] \\  & ITSS & [160] \\  & Lipschitz quot. & [161] \\ \hline
**FS** & FS, BE & [130] \\  & SFFS, SFBS & [131] \\  & Random & [173; 174; 175] \\  & ACO based & [132; 133; 134] \\ \hline  & RFE & [129; 180; 181] \\  & Sensitivity analysis & [182; 183; 184] \\  & EANN & [187] \\  & LASSO & [188; 198; 192] \\ \hline
**Semi-supervised** & & [46] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Classification of references of theory and procedure of the main input selection methods. \begin{table}
\begin{tabular}{l l l} \hline \hline \multicolumn{2}{l}{**Real Case Applications References Table**} \\ \hline \multicolumn{2}{l}{**Plant experts’ knowledge**} & [7; 8; 9; 10; 22; 26; 64] \\ \hline \multirow{5}{*}{**FE**} & PCA & [11; 19; 28; 34; 36; 37; 56; 58; 89; 111; 112] \\  & Distributed PCA & [31] \\  & Kernel PCA & [113; 114; 115] \\  & PLS & [32; 84] \\  & Discriminant anal. & [16] \\ \hline \multirow{5}{*}{**FS**} & CC & [6; 12; 23; 30; 49; 137] \\  & Univariate MI & [143] \\  & Multivariate MI & [75; 77; 90; 91; 144; 145; 146; 147; 148; 149; 150; 151; 152; 153; 154; 155; 156; 157; 158] \\  & ITSS & [13; 49] \\  & Lipschitz quot. & [49; 162] \\ \hline \multirow{5}{*}{**FS**} & FS, BE & [14; 99; 165] \\  & SFFS, SFBS & [164] \\ \cline{1-1}  & Random & [166; 167; 168; 170; 176; 177] \\ \hline \multirow{2}{*}{_Embedded_} & RFE & [178; 179] \\  & Sensitivity analysis & [185] \\ \cline{1-1}  & LASSO & [162; 49; 163] \\ \hline \hline \multicolumn{2}{l}{**Hybrid**} & [29; 49; 83; 193; 194; 195; 196; 197] \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Classification of references of application of the methods on a real case study. **Author Contributions**: Conceptualization, F.C., G.F., M.G.X. ; Writing-review & editing F.C., G.F., M.G.X. All authors have read and agreed to the published version of the manuscript.