Development of a High-Accuracy Statistical Model to Identify the Key Parameter for Methane Adsorption in Metal-Organic Frameworks
Abstract
:1. Introduction
1.1. Scheme Depicting the Main Content of This Work
1.2. Motivation and the Big Picture
1.3. Significance of Our Work and the Gap in the Literature
2. Relevant Literature Background, Specific Aims, and Key Contributions of This Work
2.1. Methane Detection Technologies
2.2. Comparison of MOFs vs. Inorganic Adsorbents and Suitability of MOFs in Terms of CH4
2.3. Studies Relevant to Our Hypothesis and Objective
2.4. Evidence of the Importance of Thermodynamic Factors in Storage Performance
2.5. Key Contributions of Important Research Works Involving Computational, Parameter Estimations, Thermodynamic Approaches, and Their Drawbacks
2.6. Key Contributions of This Work
3. Methods and the Significance/Importance of Each Step to Our Objective
3.1. Data Sets and Splitting
3.2. Software Tools Used
3.3. Steps in the Statistical Approach Followed in This Work
3.3.1. Pearson’s Correlation and Its Significance
3.3.2. SLR and MLR Models—Assumptions, Regression Coefficient Estimates, and Standard Errors
3.3.3. Model Performance Evaluations and the Significance of the Overall Model
3.3.4. Statistical Analysis Procedure: Step-by-Step
- (i)
- Changes in signs or a drastic decrease in the regression coefficient estimates from the SLR to the MLR model for an input variable make it unstable in estimating the output;
- (ii)
- Increase in standard errors of the regression coefficient estimate of that variable indicates that it is rendered insignificant, compared to the other input variable;
- (iii)
- If the standardized coefficients of one variable in the MLR model are higher than that of the other variable, it signifies that the variable with a higher standardized coefficient can explain the variance in the output better and will have a higher prediction accuracy, independent of the less significant variable;
- (iv)
- Partial correlations of the inputs with the output indicate the influence of 1 variable on the output, in the presence of the other variables. The input variable with a higher partial correlation will be the more significant variable;
- (v)
- Variance inflation factors (VIFs) are defined by , where is the coefficient of determination when the variable is regressed on , which represents the set of all other explanatory variables in the model. VIF would be 1 for simple linear regression, while a higher value in the MLR model will indicate that multicollinearity is more prevalent in that particular variable, making the other variable more significant. As with Pearson’s correlation coefficient, there is the risk of a false diagnosis of multicollinearity with VIF as well, since there is no consensus on the threshold value [60]. Kutner et al. [61] suggest a minimum value of 10, while Vatcheva et al. [62] demonstrated that even a value of < 5 could be problematic. More than the absolute value, a change in VIF magnitude toward the higher side could provide evidence supporting multicollinearity, which is what is pursued in this study by comparing multiple regression models with the simple regression counterparts, as detailed in further sections in the manuscript. In addition, VIF can also be compared with to establish whether the correlation between the regressors is stronger than the overall regression model [56].
- (vi)
- Eigenvalues (EV) and Condition Index (CI)The sum of the eigenvalues of the correlation matrix (obtained through eigenvalue decomposition) will equal the number of explanatory variables in the system but the distribution of the eigenvalues across the dimensions of the matrix would point toward the presence or absence of linear dependencies [57]. If the variables are linearly independent, all eigenvalues will equal unity; in the case of correlated variables, certain dimensions would show eigenvalues that are close to 0. The latter situation indicates that the regression parameter estimates, when regressed using these input variables, would be very sensitive to changes in the data. The condition index (CI) helps in amplifying the unequal distribution of the eigenvalues and is given in Equation (1) as:According to Midi et al., [58] if the falls below 15, then multicollinearity is not a serious concern. Johnston [63] proclaimed inconsequential collinearity until < 20. Furthermore, the detection process will also be assisted by the variance decomposition proportions for each predictor, i.e., the proportion of variance for the regression coefficient estimates of each input variable that belongs to every dimension. Significantly correlated variables would have higher variance proportions, concentrated on the same eigenvalue dimension. We have considered this aspect in our study as well. Another diagnostic that has been reported in the literature but that has been used on fewer occasions is the determinant of the correlation matrix, where a lower value indicates multicollinearity. However, this diagnostic is beyond the scope of our study.
4. Results and Discussion
4.1. Bivariate Pearson’s Correlations
4.1.1. Input-Output Correlations
4.1.2. Input-Input Correlations
4.2. Ascertaining the Prediction Accuracy and Best Predictor for GD from SLR Models
4.2.1. SLR Models Applied to the Calibration Set
4.2.2. SLR Models Applied to the Validation Set
4.3. Assessing the Interdependencies between the Input Variables Using the MLR Models
5. Future Extensions of Our Work
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Authors | Description/Agenda | Parameters Considered | Drawbacks/Motivation for Our Work |
---|---|---|---|
Wu et al. [33] | Recommended minimization of larger pores and an increase in the percentage of small cages and open metal sites | Pore volume, accessible surface area, density | Purely experimental study and takes longer. Correlates with our results that density can be a crucial parameter |
Wu et al. [34] Sezginel et al. [36] | Molecular simulations to visualize the site structures | Assumptions on pore size, volume, density to get adsorption energies | Does not account for dynamic changes in structure during experiments |
Hechinger et al. [37] | DFT simulations provided 3-D structural descriptors that were taken as inputs to QSPR models | Pore size, pore volume, surface area | Assumption of most stable conformation can lead to erroneous energies |
Peng et al. [38] | Methane uptakes of 6 promising MOFs were determined to meet the DOE 2012 target and also seen to vary linearly with their pore surface area and pore volume but inversely proportional to density | Pore surface area, pore volume, packing density | The analysis was not comprehensive enough to find out the best descriptor of the MOF that can predict the uptakes with the highest accuracy |
Li et al. [39] (2016) | The computational screening of MOFs -> revealed physical limitation to storage | Structural descriptors | Thermodynamic descriptors not considered Molecular simulations consume a great deal of time |
Becker et al. [48] (2017) | Compared polarizable force fields vs. orbital interaction energies DFT -> adsorption isotherms -> adsorption energies Calculated for CO2 and CH4 on MOFs with 10 different metal ions | , assumptions of structural descriptors such as pore size, volume, density | Computational tools involving quantum mechanics consume lots of time and computer power and memory |
Vandenbrande et al. [44] (2017) | Computed force fields for Zr-based MOFs adsorbing CH4 | General stable form of MOF assumed | Significant quantitative differences in methane uptakes, predictions are highly sensitive to the computed PES |
Wiersum et al. [43] (2013) | Developed API considering weighting factors for performance sensitivity towards temperature and pressure | Adsorption energies, working capacities as inputs | Other material and pore properties of the MOF not considered whereas our model considers 8 input parameters |
Tahmooresi and Sabzi [47] (2014) | Methane sorption in five different MOFs was modeled from experimental data PHSC) equation of state used—thermodynamic approach | Molecular dimensions and interaction energies used as inputs | Only 5 MOFs were considered, whereas our model considers 83 MOFs Other material and pore properties of the MOF were not considered Molecular parameters were estimated using a group contribution method—may not represent a real system |
Gomez-Gualdron et al. [50] | Identified high-performance MOFs with Zr as central metal and OH and CO2 as ligands from a computational screening of 204 MOFs | Requires exact structures of MOFs as inputs—very difficult and time-consuming in the real system The MOFs used in the screening were hypothetical |
Input or Output | GD 8 | GU 9 | VD 10 | VU 11 |
---|---|---|---|---|
ASA 1 | 0.96 (<0.001) | 0.947 (<0.001) | 0.399 (0.059) | −0.592 (<0.001) |
Density | −0.941 (<0.001) | −0.908 (<0.001) | −0.406 (0.003) | 0.597 (<0.001) |
BET SA 2 | 0.924 (<0.001) | 0.889 (<0.001) | 0.298 (0.049) | −0.491 (<0.001) |
VP 3 | 0.925 (<0.001) | 0.883 (<0.001) | 0.249 (0.075) | −0.558 (<0.001) |
AV 4 | 0.918 (<0.001) | 0.885 (<0.001) | 0.206 (0.346) | −0.613 (<0.001) |
LCD 5 | 0.819 (<0.001) | 0.778 (<0.001) | 0.090 (0.684) | −0.501 (0.009) |
PLD 6 | −0.687 (<0.001) | −0.642 (<0.001) | −0.731 (<0.001) | 0.246 (0.226) |
Qst 7 | −0.68 (<0.001) | −0.374 (0.014) | −0.185 (0.24) | 0.727 (<0.001) |
Input Variables | ASA | Density | BET SA | VP | AV | LCD | PLD | Qst |
---|---|---|---|---|---|---|---|---|
ASA | 1 | −0.99 (<0.001) | 0.944 (<0.001) | 0.952 (<0.001) | 0.956 (<0.001) | 0.804 (<0.001) | −0.566 (0.003) | −0.881 (<0.001) |
Density | −0.99 (<0.001) | 1 | −0.89 (<0.001) | −0.898 (<0.001) | −0.917 (<0.001) | −0.745 (<0.001) | 0.653 (<0.001) | 0.572 (<0.001) |
BET SA | 0.944 (<0.001) | −0.89 (<0.001) | 1 | 0.979 (<0.001) | 0.982 (<0.001) | 0.871 (<0.001) | −0.252 (0.284) | −0.49 (0.002) |
VP | 0.952 (<0.001) | −0.898 (<0.001) | 0.979 (<0.001) | 1 | 0.989 (<0.001) | 0.863 (<0.001) | −0.345 (0.084) | −0.484 (<0.001) |
AV | 0.956 (<0.001) | −0.917 (<0.001) | 0.982 (<0.001) | 0.989 (<0.001) | 1 | 0.902 (<0.001) | −0.346 (0.084) | −0.781 (<0.001) |
LCD | 0.804 (<0.001) | −0.745 (<0.001) | 0.871 (<0.001) | 0.863 (<0.001) | 0.902 (<0.001) | 1 | 0.653 (<0.001) | −0.563 (0.01) |
PLD | −0.566 (0.003) | 0.653 (<0.001) | −0.252 (0.284) | −0.345 (0.084) | −0.346 (0.084) | −0.237 (0.244) | 1 | 0.644 (0.002) |
Qst | −0.881 (<0.001) | 0.572 (<0.001) | −0.49 (0.002) | −0.484 (<0.001) | −0.781 (<0.001) | −0.563 (0.01) | 0.644 (0.002) | 1 |
Output | Input | Top 2 Best Performing Fits | F-Statistic | p-Value for bo * | p-Value for b1 ** | Best Fit according to the Calibration Set | Sign of the Correlation | |
---|---|---|---|---|---|---|---|---|
GD | ASA | Linear | 0.92 | 243 | 0.047 | <0.001 | Linear | Positive |
Quadratic | 0.88 | 151 | <0.001 | <0.001 | ||||
GD | Density | Exponential | 0.91 | 490 | <0.001 | <0.001 | Exponential | Negative |
Linear | 0.89 | 381 | <0.001 | <0.001 | ||||
GD | BET SA | Linear | 0.85 | 240.5 | <0.001 | <0.001 | Linear | Positive |
Quadratic | 0.74 | 116 | <0.001 | <0.001 | ||||
GD | AV | Linear | 0.84 | 113 | 0.011 | <0.001 | Linear | Positive |
Exponential | 0.74 | 60 | <0.001 | <0.001 | ||||
GD | LCD | Linear | 0.67 | 42 | 0.184 | <0.001 | Quadratic | Positive |
Quadratic | 0.66 | 41 | <0.001 | <0.001 | ||||
GD | Qst | Cubic | 0.54 | 46 | <0.001 | <0.001 | Cubic | Negative |
Exponential | 0.52 | 42 | <0.001 | <0.001 |
Best-Fit of the Input | RMSEP | RMSEC | abs (RMSEP—RMSEC) | ||
---|---|---|---|---|---|
Density (Exponential) | 0.88 | 0.015 | 0.91 | 0.011 | 0.004 |
BET (Linear) | 0.85 | 0.02 | 0.85 | 0.015 | 0.012 |
ASA (Linear) | 0.76 | 0.021 | 0.92 | 0.013 | 0.008 |
AV (Linear) | 0.75 | 0.031 | 0.84 | 0.018 | 0.013 |
LCD (Quadratic) | 0.36 | 0.053 | 0.66 | 0.027 | 0.026 |
Qst (Cubic) | 0.26 | 0.046 | 0.54 | 0.027 | 0.019 |
(a) | ||||||
---|---|---|---|---|---|---|
Input Variable | Coefficient Estimates for SLR Models | Coefficient Estimates for MLR Models | ||||
M1 | M2 | M3 | M4 | M5 | ||
Density_exp 1 | −1.49 | −0.32 | −0.31 | −0.44 | −0.34 | −0.38 |
BET SA_linear 2 | ||||||
ASA_linear 3 | ||||||
AV_linear 4 | 0.092 | −0.02 | ||||
LCD_quad 5 | ||||||
Qst_cubic 6 | ||||||
(b) | ||||||
Input Variable | SEs for SLR Models | Standard Errors (SEs) of Regression Coefficient for MLR Models | ||||
M1 | M2 | M3 | M4 | M5 | ||
Density_exp 1 | 0.016 | 0.058 | 0.018 | 0.085 | 0.038 | 0.027 |
BET SA_linear 2 | ||||||
ASA_linear 3 | ||||||
AV_linear 4 | 0.009 | 0.023 | ||||
LCD_quad 5 | ||||||
Qst_cubic 6 | ||||||
(c) | ||||||
Input Variable | Standardized Coefficients for SLR Models | Standardized Coefficients for MLR Models | ||||
M1 | M2 | M3 | M4 | M5 | ||
Density_exp 1 | 0.94 | 0.84 | 0.82 | 1.19 | 0.92 | 1.03 |
BET SA_linear 2 | 0.92 | 0.12 | ||||
ASA_linear 3 | 0.96 | 0.15 | ||||
AV_linear 4 | 0.92 | 0.23 | ||||
LCD_quad 5 | 0.81 | 0.06 | ||||
Qst_cubic 6 | 0.73 | 0.08 |
(a) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Input Variable | t-Statistics for SLR Models | t-Statistic Values for MLR Models | |||||||||||
M1 | M2 | M3 | M4 | M5 | |||||||||
Density_exp 1 | −23 | −5.6 | −1.7 | −5.2 | −8.9 | −14.3 | |||||||
BET SA_linear 2 | 15.5 | 0.8 | |||||||||||
ASA_linear 3 | 15.6 | 0.3 | |||||||||||
AV_linear 4 | 10.6 | −1 | |||||||||||
LCD_quad 5 | 6.4 | 0.5 | |||||||||||
Qst_cubic 6 | −6.6 | 1.1 | |||||||||||
(b) | |||||||||||||
Input Variable | Confidence Intervals for SLR Models | Confidence Intervals for MLR Models | |||||||||||
Lower Bound (LB) | Upper Bound (UB) | M1 | M2 | M3 | M4 | M5 | |||||||
LB | UB | LB | UB | LB | UB | LB | UB | LB | UB | ||||
Density_exp 1 | −3 | −0.39 | −2.1 | −0.44 | −3.1 | −0.7 | −2.6 | −0.6 | −2.1 | −0.5 | −3.3 | −0.4 | |
BET SA_linear 2 | |||||||||||||
ASA_linear 3 | |||||||||||||
AV_linear 4 | 0.074 | 0.11 | −0.07 | 0.02 | |||||||||
LCD_quad 5 | |||||||||||||
Qst_cubic 6 | |||||||||||||
(c) | |||||||||||||
Input Variable | Partial Correlations for SLR Models | Partial Correlations for MLR Models | |||||||||||
M1 | M2 | M3 | M4 | M5 | |||||||||
Density_exp 1 | −0.94 | −0.66 | −0.45 | −0.76 | −0.9 | −0.92 | |||||||
BET SA_linear 2 | 0.92 | 0.13 | |||||||||||
ASA_linear 3 | 0.96 | 0.07 | |||||||||||
AV_linear 4 | 0.92 | −0.22 | |||||||||||
LCD_quad 5 | 0.81 | 0.13 | |||||||||||
Qst_cubic 6 | −0.73 | 0.18 | |||||||||||
(d) | |||||||||||||
Input Variable | F-Statistics for SLR Models | F-Statistic Values for MLR Models | |||||||||||
M1 | M2 | M3 | M4 | M5 | |||||||||
Density_exp 1 | 490 | −0.66 | −0.45 | −0.76 | −0.9 | −0.92 | |||||||
BET SA_linear 2 | 241 | 0.13 | |||||||||||
ASA_linear 3 | 243 | 0.07 | |||||||||||
AV_linear 4 | 0.92 | −0.22 | |||||||||||
LCD_quad 5 | 0.81 | 0.13 | |||||||||||
Qst_cubic 6 | −0.73 | 0.18 |
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Sivaramakrishnan, K.; Mahmoud, E. Development of a High-Accuracy Statistical Model to Identify the Key Parameter for Methane Adsorption in Metal-Organic Frameworks. Analytica 2022, 3, 335-370. https://doi.org/10.3390/analytica3030024
Sivaramakrishnan K, Mahmoud E. Development of a High-Accuracy Statistical Model to Identify the Key Parameter for Methane Adsorption in Metal-Organic Frameworks. Analytica. 2022; 3(3):335-370. https://doi.org/10.3390/analytica3030024
Chicago/Turabian StyleSivaramakrishnan, Kaushik, and Eyas Mahmoud. 2022. "Development of a High-Accuracy Statistical Model to Identify the Key Parameter for Methane Adsorption in Metal-Organic Frameworks" Analytica 3, no. 3: 335-370. https://doi.org/10.3390/analytica3030024