Abstract
Food preservatives play a crucial role in extending the shelf life of food products. Understanding their physicochemical properties can help in designing more effective and safer preservatives. In this study, we use a Quantitative Structure Property Relationship (QSPR) approach based on topological indices to develop a predictive model for certain physicochemical properties of food preservatives. We compare the performance of linear and curvilinear regression models to understand which provides the best prediction model. Among the tested models, the cubic regression model demonstrated superior predictive performance. Of all the models tested, the cubic regression model had the best predictive capabilities such as 
for vapour density and 
for molecular weight. To validate our findings, we employ the developed model to estimate the properties of an existing food preservative, the propionic acid. Our results offer valuable insights that can aid in the development of new and improved food preservatives.
Keywords: Topological indices, QSPR study, Curvilinear regression models, Food preservatives
Subject terms: Chemical biology, Chemistry, Mathematics and computing
Introduction
Food additives are substances added to food to enhance its shelf life, nutritional value, flavour, or appearance. These substances can be classified as colorants, emulsifiers, flavour enhancers, antioxidants, preservatives, and sweeteners1,2. Natural or synthetic substances added to food to prolong its shelf life are known as food preservatives3,4. These compounds function as antibacterial agents, lessen oxidation, and inhibit the generation of toxins. Up to 40% of all food produced for human consumption globally is wasted due to microbial food deterioration. Food safety and edible quality are ensured by the addition of preservatives5. The demand for less hazardous preservatives is increasing. As a result, the market for safe and effective preservatives is growing. With increasing demand, developing and evaluating new preservatives is imperative6. Analyzing the physicochemical properties of the compound is crucial for its effectiveness. Predictive models use computational and mathematical techniques without the aid of extensive laboratory testing setups to estimate the biological effects. Here we develop models which help to assess the toxicity, allergenicity, and metabolic effects using the structural and physicochemical properties7,8.
QSPR (Quantitative Structure Property Relationship) and QSAR (Quantitative Structure-Activity Relationship) are computational modeling techniques that use a molecule’s chemical structure to predict its physicochemical properties and biological activity, respectively9,10. With the use of certain molecular descriptors and statistical or machine learning models, QSAR establishes a relationship between a compound’s chemical structure and biological activities, such as toxicity or medicinal efficacy. Rather, QSPR focuses on using the structure to predict physical or chemical aspects like melting or boiling points. Both approaches use quantitative analysis of topological properties to generate prediction models that support material science, environmental chemistry, and drug design11–13.
The study of networks of points connected by lines is the subject of the mathematical field known as graph theory14. The application of graph theory can help solve several real-world problems. Over the nineteenth century, graph theory was used in chemistry. The development of chemical graph theory, a new subfield of graph theory, followed. Chemical graphs are used to depict chemical structures in chemical graph theory15. An atom is represented by a vertex, and the relationship between atoms is represented by an edge. The compound’s chemical structure is linked to the majority of its chemical information. A chemical network is represented by topological indices (TI), which are numerical invariants. The TIs are utilized in QSAR and QSPR research to build models that forecast the physical, chemical, or biochemical properties of certain compounds16–19. As a computational method for drug design and various other structural analysis, this concept is widely applied20–26.
Several studies have used different topological indices to predict specific properties of compounds. In 2002, scientists used certain specific TIs to forecast the biological activity of particular alkoxyphenols27. To forecast the physical characteristics of medications for mental disorders, Ejma et al. utilized ensemble learning in conjunction with topological indices28. Using regression models and topological indices, Huili Li et al. examined the structural characteristics of amino acids in order to forecast their chemical and physical characteristics29. The physical characteristics of some antibacterial medications were recently predicted by Abubakar et al. using two regression models and specific neighborhood sum-based indices30.
Both linear regression models and non-linear regression models can be utilized to execute QSAR and QSPR analysis31,13. The link between the molecular structural property and the desired activity or property determines the type of model to use. When the relationship between the dependent and independent variables is non-linear, curvilinear models are thought to be optimal. In this study, the physicochemical properties (P) of the food preservatives are the dependent variables, and their corresponding topological indices (TI) are the independent variables. Here we consider linear, quadratic and cubic regression models.
Linear:
 |
Quadratic:
 |
Cubic:
 |
where ai’s (
to 9) are constants.
Materials and methods
The study takes into account fourteen chemical compounds that are currently used for efficient food preservation. Acetic acid is a commonly used preservative as it breaks down bacterial and fungal cell membranes, making it impossible for them to survive. It is also vinegar’s primary ingredient. It is a widely accepted chemical that occurs naturally32. Ascorbic acid is a naturally occurring antioxidant that is mostly added to fruits, vegetables, and drinks, in contrast to other preservatives33. An ester derivative of ascorbic acid is ascorbyl palmitate. It is added to sunflower oil to extend its shelf life34. One of the earliest chemical preservatives still in use in the food enterprises is benzoic acid. It works well to stop the growth of yeast35. Generally regarded as antibacterials, butylated hydroxyanisole (BHA) and butylated hydroxytoluene (BHT) are phenolic antioxidants36. As an anti-enzymatic preservative, citric acid slows down food from going through enzymatic reactions long after it has been harvested37. A chemical preservative called dimethyl carbonate is used to prolong the shelf life of goods like salsa38. A cationic surfactant called ethyl lauroyl arinate is used in food preservation to enhance the microbiological safety and quality characteristics of a variety of food products39. Often referred to as methylparaben, methyl 4-hydroxybenzoate is a typical food preservative used in baked goods and jams40. Likewise, processed meat and dairy products include propyl 4-hydroxybenzoate, also referred to as propylparaben, as a food preservative41. A common phenolic antioxidant in the food, cosmetics and pharmaceutical sectors is propyl gallate42. Sorbic acids and their salts are the most widely used preservatives in the food industry due to their neutral flavor and physiological inertness43. Sunflower oil is also preserved using tertiary butylhydroquinone (TBHQ)44. Figure 1 shows the molecular structures of the food preservatives mentioned above.
Fig. 1.
Molecular structures of food preservatives.
Figure 2 shows the molecular graph representation of benzoic acid’s chemical structure. Similarly all of the aforementioned compounds’ chemical structures can be depicted as molecular graphs.
Fig. 2.
Molecular graph of Benzoic acid.
The efficacy, stability, and safety of a food preservative compound depend much on several physical and chemical properties, including molecular weight (MW), boiling point (BP), melting point (MP), pKa value, vapour density (VD), log P and LD50 value. A greater LD50 value suggests less toxicity, so the preservative is safer for use. Table 1 depicts the physicochemical properties and LD50 value of the selected food preservatives. All the data are collected from PubChem45.
Table 1.
Physicochemical properties and LD50-value of food preservatives.
| Compound | Molecular weight (g/mol) | Melting 
|
Boiling point at 760 mmHg | pKa | Vapour density | log P | LD50 (oral, rat) (mg/kg) |
|---|---|---|---|---|---|---|---|
| Acetic acid | 60.05 | 16.6 | 117.9 | – | 2.07 | − 0.17 | 3310 |
| Ascorbic acid | 176.12 | 191 | 552.7 | 4.7 | – | − 1.85 | 11900 |
| Ascorbyl Palmitate | 414.5 | 116.5 | – | – | – | – | – |
| Benzoic acid | 122.12 | 122.4 | 249.2 | 4.207 | 4.21 | 1.87 | 1700 |
| BHA | 360.494 | 51 | 267 | – | – | – | – |
| BHT | 220.35 | 70.5 | 264.5 | 12.23 | 7.6 | 5.32 | 2930 |
| Citric acid | 192.12 | 156 | 310 | 2.79 | – | − 1.64 | 5500 |
| Dimethyl Dicarbonate | 134.09 | 17 | 175 | – | – | – | 260 |
| Ethyl Lauroyl Arginate | 384.6 | – | – | – | – | – | – |
| Methyl 4-hydroxybenzoate | 152.98 | 125.2 | 275 | – | – | 1.96 | 5600 |
| Propyl 4-hydroxybenzoate | 180.2 | 97.5 | 294.5 | – | – | 3.04 | – |
| Propyl gallate (PG) | 212.199 | 150 | – | 7.94 | 7.3 | 1.8 | . |
| Sorbic acid | 112.13 | 134.5 | 228 | 4.76 | 3.87 | 1.33 | 7360 |
| TBHQ | 166.217 | 128 | 273 | 10.8 | . | – | – |
The study’s methodology flow chart is presented in Fig. 3.
Fig. 3.
Methodological framework of the study.
Let G be a simple connected graph with vertex set V(G) and edge set E(G). Let u be any arbitrary vertex in the set V(G). Then 
, the degree of the vertex u, is the number of vertices adjacent to u. To examine the molecular structure of the above mentioned compounds, we employ sixteen degree based indices. Table 2 shows the selected topological indices used for the study.
Table 2.
Topological indices.
| Topological index | Notation | Formula | References |
|---|---|---|---|
| First Zagreb Index |  |
 |
Gutman and Trinjstic46 |
| Second Zagreb Index |  |
 |
Gutman and Trinjstic46 |
| Reduced Zagreb Index |  |
 |
Furtula et al.47 |
| Hyper Zagreb Index | HM(G) |  |
Shirdel et al.48 |
| Augmented Zagreb Index | AZ(G) |  |
Ali et al.49 |
| Randić Zagreb Index | R(G) |  |
Randić50 |
| Reciprocal Randić Zagreb Index | RR(G) |  |
Gutman et al.51 |
| Reduced Reciprocal Randić Index | RRR(G) |  |
Gutman et al.51 |
| Harmonic Index | H(G) |  |
Fajtlowicz52 |
| Sum Connectivity Index | SC(G) |  |
Trinjastic53 |
| Geometric Arithmetic Index | GA(G) |  |
Vukicevic et al.54 |
| Inverse Sum Index | IS(G) |  |
Vukicevic et al.55 |
| Forgotten Index | F(G) |  |
Gutman and Trinjstic46 |
| Symmetric Division Index | SD(G) |  |
Vukicevic et al.55 |
| Atom Bond Connectivity Index | ABC(G) |  |
Estrada et al.56 |
| Sombor Index | So(G) |  |
Gutman57 |
For calculating the topological indices, we take the aid of the edge partitioning method. This is a useful technique for computing topological indices by categorizing edges based on the degree of their end vertices. Let 
represent the set of edges joining the vertices u and v. Here, r and s represent the degree of vertices u and v, respectively. For every chemical graph, the maximum degree of a vertex is 4. Here in this study, the molecular graphs of the selected compounds have edges of types 
. 
represents the cardinality of the set 
. Table 3 gives the edge partitions of the dataset of compounds.
Table 3.
Edge partitions of the compounds.
| Compound |  |
 |
 |
 |
 |
 |
 |
 |
|---|---|---|---|---|---|---|---|---|
| Acetic acid | – | 3 | – | – | – | – | – | – |
| Ascorbic acid | 1 | 4 | – | – | 3 | – | 4 | – |
| Ascorbyl Palmitate | 1 | 5 | – | 14 | 5 | – | 4 | – |
| Benzoic acid | – | 2 | – | 4 | 2 | – | 1 | – |
| BHA | 1 | 1 | 3 | 1 | 5 | – | 1 | 1 |
| BHT | – | 2 | 6 | – | 4 | – | 2 | 2 |
| Citric acid | – | 6 | 1 | – | 2 | 2 | – | 1 |
| Dimethyl Dicarbonate | 2 | 2 | – | – | 4 | – | – | – |
| Ethyl Lauroyl Arginate | 2 | 4 | – | 13 | 6 | – | 1 | – |
| Methyl 4-hydroxybenzoate | 1 | 2 | – | 2 | 5 | – | 1 | – |
| Propyl 4-hydroxybenzoate | 1 | 2 | – | 4 | 5 | – | 1 | – |
| Propyl gallate (PG) | 1 | 4 | – | 2 | 5 | – | 3 | – |
| Sorbic acid | 1 | 2 | – | 3 | 1 | – | – | – |
| TBHQ | – | 2 | 3 | 1 | 3 | – | 1 | 1 |
Results and discussion
In this section, we computed the topological indices of all selected compounds. As a representative example, we consider the case of benzoic acid, and the results are summarized in the following theorem.
Theorem 3.1
Let
represent the molecular graph of benzoic acid. Then, (1) 
, (2) 
, (3) 
, (4) 
, (5) 
, (6) 
, (7) 
, (8) 
, (9) 
, (10) 
, (11) 
, (12) 
, (13) 
, (14) 
, (15) 
, (16) 
.
Proof
Let 
denote the molecular graph of benzoic acid. Then evaluating each indices from Table 2, using the edge partition from Table 3, we get,
We determine the degree-based topological indices of each of the chosen compounds using MATLAB and a methodology similar to that in the previously discussed Theorem. The compounds’ computed indices are shown in the Tables 4 and 5.
Table 4.
Calculated topological indices of the compounds (Part I).
| Compound |  |
 |
 |
HM | AZ | R | RR | RRR |
|---|---|---|---|---|---|---|---|---|
| Acetic acid | 12 | 9 | 0 | 48 | 10.125 | 1.7321 | 5.1962 | 0 |
| Ascorbic acid | 58 | 68 | 22 | 292 | 91.0625 | 5.5746 | 27.6909 | 12.2426 |
| Ascorbyl Palmitate | 128 | 139 | 40 | 582 | 222.4375 | 13.9684 | 62.3219 | 29.0711 |
| Benzoic acid | 40 | 43 | 12 | 182 | 66.1406 | 4.3045 | 19.3631 | 8.8284 |
| BHA | 64 | 72 | 21 | 326 | 91.7007 | 5.9477 | 29.8576 | 12.5206 |
| BHT | 84 | 96 | 28 | 452 | 103.4015 | 7.0317 | 38.1903 | 14.5558 |
| Citric acid | 58 | 62 | 16 | 292 | 68.4444 | 5.7764 | 26.4122 | 8.7420 |
| Dimethyl Dicarbonate | 34 | 34 | 8 | 150 | 54.75 | 4.2019 | 16.0905 | 5.6569 |
| Ethyl Lauroyl Arginate | 110 | 113 | 29 | 476 | 192.8906 | 13.0064 | 53.4536 | 28.4853 |
| Methyl 4-hydroxybenzoate | 50 | 55 | 16 | 234 | 82.1406 | 5.2364 | 24.1258 | 11.0711 |
| Propyl 4-hydroxybenzoate | 58 | 63 | 18 | 266 | 98.1406 | 6.2364 | 28.1258 | 13.0711 |
| Propyl gallate (PG) | 70 | 79 | 24 | 338 | 111.6719 | 7.0577 | 33.5899 | 15.0711 |
| Sorbic acid | 28 | 26 | 5 | 114 | 46.7500 | 3.7701 | 13.3278 | 4.4142 |
| TBHQ | 55 | 61 | 17 | 283 | 71.0757 | 5.0015 | 25.2767 | 9.6921 |
Table 5.
Calculated topological indices of the compounds (Part II).
| Compound | H | SC | GA | IS | F | SDI | ABC | So |
|---|---|---|---|---|---|---|---|---|
| Acetic acid | 1.5 | 1.5 | 2.5981 | 2.25 | 30 | 10 | 2.4495 | 9.4868 |
| Ascorbic acid | 5.2 | 5.5520 | 11.3463 | 13.2667 | 156 | 30.3333 | 8.7611 | 42.6724 |
| Ascorbyl Palmitate | 13.5 | 13.9464 | 28.1719 | 30.4167 | 304 | 66 | 20.8913 | 92.6438 |
| Benzoic acid | 4.1333 | 4.3027 | 8.6916 | 9.4 | 96 | 21 | 6.5423 | 29.0926 |
| BHA | 5.4857 | 5.9413 | 12.0976 | 14.031 | 182 | 35.5 | 9.6765 | 47.8665 |
| BHT | 6.2381 | 7.0446 | 14.4307 | 17.5286 | 260 | 49 | 12.2819 | 63.9707 |
| Citric acid | 5.1524 | 5.5361 | 10.8311 | 12.081 | 168 | 35.6667 | 9.2389 | 44.2521 |
| Dimethyl Dicarbonate | 3.9333 | 3.9436 | 7.5369 | 7.6333 | 82 | 20.3333 | 5.8756 | 25.2189 |
| Ethyl Lauroyl Arginate | 12.5667 | 12.7462 | 25.2285 | 26.0333 | 250 | 519.3333 | 18.7819 | 79.7667 |
| Methyl 4-hydroxybenzoate | 5 | 5.2217 | 10.5738 | 11.6667 | 124 | 26 | 7.9565 | 36.4879 |
| Propyl 4-hydroxybenzoate | 6 | 6.2217 | 12.5738 | 13.6667 | 140 | 30 | 9.3707 | 42.1447 |
| Propyl gallate (PG) | 6.6667 | 7.0382 | 14.3059 | 16.1667 | 180 | 36.6667 | 10.9228 | 51.2977 |
| Sorbic acid | 3.5667 | 3.5246 | 6.6547 | 6.3667 | 62 | 17.333 | 5.1685 | 20.6515 |
| TBHQ | 4.519 | 4.9695 | 10.0612 | 11.7143 | 161 | 3 | 8.3717 | 41.5816 |
Tables 6, 7, 8, 9, 10 and 11 show the linear, quadratic, and cubic regression models that use different degree-based topological indices as predictors to predict the physicochemical properties of the food preservtaives (MW, MP, BP, pKa, VD, Log P, and LD50). All the regression models are developed using MATLAB progamming. Here we consider the following metrics for evaluation:
R (regression coefficient): Measures the strength of the relationship between the actual value and value predicted by the model. The range of values for R is
to 1. When the R-value is higher than 0.9, it indicates a strong positive correlation and a strong influence of the predictor variable on the dependent parameter.
(coefficient of determination): Indicates the proportion of the variance in the dependent variable the model can account for. A higher 
value denotes a better fit and implies that the majority of the variation is explained by the model.RMSE (Root Mean Square Error): Demonstrates average error in the predictions. A smaller RMSE indicates a better model.
F-statistic: It evaluates the overall significance of the model. A low p value and a higher F-statistic indicate that the independent factors largely account for the variance in the dependent variable.
p value: It evaluates the model’s statistical significance. It is suggested that the predictor is statistically significant if the p value is less than 0.05.
In linear regression analysis, most of the computed indices show a positive correlation with the physicochemical properties. The Reduced Reciprocal Randić index shows the highest correlation with LD50 value and melting point. Vapour density and molecular weight have the strongest correlations with the Atom bond connectivity Index. Randic index, Hyper Zagreb index and Harmonic index shows a high association with boiling point, pKa value and Log P value respectively. Although they do not predominate, descriptors such as RM2, AZ, and GA exhibit reasonable relationships across many attributes. The regression coefficient between each property and index is displayed in Table 12. The linear regression plots of the physicochemical property and index pairs with the highest coefficient of determination are displayed in Fig. 4. These plots illustrate the strong predictive relationship between specific topological indices and selected physicochemical properties.
Table 6.
Linear regression analysis.
| Equation | R |  |
RMSE | F-statistic | p value |
|---|---|---|---|---|---|
 |
0.9019 | 0.8135 | 44.3986 | 52.3308 | 0 |
 |
0.0266 | 0.0007 | 54.0850 | 0.0085 | 0.9280 |
 |
0.5211 | 0.2715 | 87.8094 | 3.3543 | 0.1003 |
 |
0.6406 | 0.4103 | 2.4482 | 4.1753 | 0.087 |
 |
0.9876 | 0.9754 | 0.3332 | 118.7533 | 0.0017 |
 |
0.4110 | 0.1689 | 1.9440 | 1.4228 | 0.2718 |
 |
0.1829 | 0.0334 | 3370.5546 | 0.2076 | 0.6647 |
 |
0.8896 | 0.7914 | 46.9468 | 45.5369 | 0.0000 |
 |
0.0773 | 0.0060 | 53.9424 | 0.0721 | 0.7929 |
 |
0.5540 | 0.3069 | 85.6467 | 3.9861 | 0.0770 |
 |
0.6262 | 0.3921 | 2.6053 | 3.2246 | 0.1325 |
 |
0.9847 | 0.9696 | 0.3704 | 95.5742 | 0.0023 |
 |
0.3938 | 0.1551 | 1.9601 | 1.2850 | 0.2943 |
 |
0.2128 | 0.0453 | 3349.8183 | 0.2846 | 0.6129 |
 |
0.7932 | 0.6292 | 62.5971 | 20.3630 | 0.0007 |
 |
0.2454 | 0.0602 | 52.4491 | 0.7693 | 0.3977 |
 |
0.5843 | 0.3415 | 83.4874 | 4.6665 | 0.0590 |
 |
0.2900 | 0.0841 | 3.1979 | 0.4589 | 0.5282 |
 |
0.8649 | 0.7480 | 1.0657 | 8.9061 | 0.0584 |
 |
0.2766 | 0.0765 | 2.0493 | 0.5799 | 0.4712 |
 |
0.1350 | 0.0182 | 3396.9600 | 0.1114 | 0.7499 |
 |
0.8758 | 0.7670 | 53.5982 | 39.4997 | 0.0000 |
 |
0.2202 | 0.0485 | 54.2124 | 0.5603 | 0.4698 |
 |
0.4810 | 0.2313 | 99.7177 | 2.7086 | 0.1342 |
 |
0.7680 | 0.5898 | 2.5321 | 7.1900 | 0.0437 |
 |
0.7582 | 0.5748 | 1.7942 | 5.4080 | 0.0806 |
 |
0.5536 | 0.3065 | 2.2337 | 1.7676 | 0.2544 |
 |
0.0041 | 0.0000 | 3972.8655 | 0.0001 | 0.9916 |
 |
0.8901 | 0.7923 | 47.5157 | 50.0762 | 0 |
 |
0.4063 | 0.1651 | 54.1512 | 2.3727 | 0.1479 |
 |
0.5875 | 0.3452 | 95.1462 | 3.7175 | 0.0805 |
 |
0.5394 | 0.2909 | 2.5915 | 2.8629 | 0.1235 |
 |
0.9872 | 0.9745 | 0.3411 | 152.0418 | 0.0003 |
 |
0.3287 | 0.1080 | 1.9427 | 0.9726 | 0.3487 |
 |
0.3365 | 0.1132 | 3095.1422 | 1.0211 | 0.3418 |
 |
0.8764 | 0.7681 | 50.0842 | 46.2338 | 0 |
 |
0.4373 | 0.1912 | 53.2048 | 2.8042 | 0.1266 |
 |
0.6982 | 0.4875 | 87.1746 | 9.5063 | 0.0115 |
 |
0.5874 | 0.3450 | 2.5132 | 3.7121 | 0.0807 |
 |
0.9845 | 0.9693 | 0.3676 | 125.2836 | 0.0004 |
 |
0.3461 | 0.1198 | 1.9024 | 1.0843 | 0.3238 |
 |
0.3744 | 0.1402 | 3050.1256 | 1.1392 | 0.3125 |
 |
0.8921 | 0.7969 | 46.6978 | 50.2742 | 0 |
 |
0.4312 | 0.1859 | 53.3772 | 2.7131 | 0.1317 |
 |
0.7108 | 0.5052 | 85.1373 | 10.3018 | 0.0101 |
 |
0.5932 | 0.3519 | 2.4874 | 3.8476 | 0.0781 |
 |
0.9886 | 0.9773 | 0.3216 | 169.4287 | 0.0002 |
 |
0.3567 | 0.1272 | 1.8953 | 1.1433 | 0.3106 |
 |
0.3842 | 0.1476 | 3031.2456 | 1.1986 | 0.3001 |
 |
0.9024 | 0.8144 | 44.7321 | 55.1387 | 0 |
 |
0.4378 | 0.1917 | 53.1954 | 2.7934 | 0.1268 |
 |
0.7172 | 0.5144 | 83.8572 | 10.7691 | 0.0092 |
 |
0.5771 | 0.3331 | 2.5107 | 3.7461 | 0.0821 |
 |
0.9898 | 0.9797 | 0.3079 | 180.6244 | 0.0002 |
 |
0.3723 | 0.1386 | 1.8731 | 1.2298 | 0.2943 |
 |
0.4086 | 0.1670 | 2995.3287 | 1.3315 | 0.2745 |
Table 7.
Linear regression analysis.
| Equation | R |  |
RMSE | F-statistic | p value |
|---|---|---|---|---|---|
 |
0.8840 | 0.7814 | 51.9136 | 42.8963 | 0.0000 |
 |
0.2294 | 0.0526 | 51.8268 | 0.6667 | 0.4301 |
 |
0.0677 | 0.0046 | 103.8197 | 0.0552 | 0.8182 |
 |
0.4519 | 0.2042 | 3.5270 | 1.2828 | 0.3088 |
 |
0.9836 | 0.9674 | 0.4950 | 88.9777 | 0.0025 |
 |
0.4853 | 0.2355 | 1.9948 | 1.8484 | 0.2228 |
 |
0.2288 | 0.0523 | 3853.7237 | 0.3314 | 0.5858 |
 |
0.8940 | 0.799 | 49.761 | 47.749 | 0.00001 |
 |
0.2608 | 0.068 | 53.653 | 0.803 | 0.389 |
 |
0.5536 | 0.306 | 94.717 | 3.978 | 0.077 |
 |
0.5238 | 0.274 | 3.368 | 1.891 | 0.228 |
 |
0.9936 | 0.987 | 0.310 | 230.76 | 0.0006 |
 |
0.4011 | 0.161 | 2.215 | 1.342 | 0.285 |
 |
0.1165 | 0.0136 | 3462.3901 | 0.0825 | 0.7836 |
 |
0.8935 | 0.7984 | 46.1575 | 47.5214 | 0.0000 |
 |
0.2757 | 0.0760 | 49.1407 | 0.9052 | 0.3618 |
 |
0.5532 | 0.3060 | 85.7052 | 3.9684 | 0.0775 |
 |
0.6499 | 0.4223 | 2.5397 | 3.6553 | 0.1141 |
 |
0.9914 | 0.9828 | 0.2782 | 171.6443 | 0.0010 |
 |
0.4071 | 0.1657 | 1.9477 | 1.3906 | 0.2768 |
 |
0.1129 | 0.0127 | 3463.8341 | 0.0774 | 0.7901 |
 |
0.8904 | 0.7927 | 46.8009 | 45.8961 | 0.0000 |
 |
0.3746 | 0.1403 | 47.4005 | 1.7953 | 0.2073 |
 |
0.6058 | 0.3670 | 81.8504 | 5.2186 | 0.0482 |
 |
0.1019 | 0.0104 | 3.3240 | 0.0525 | 0.8279 |
 |
0.9714 | 0.9437 | 0.5038 | 50.2757 | 0.0058 |
 |
0.0640 | 0.0041 | 2.1281 | 0.0288 | 0.8701 |
 |
0.2232 | 0.0498 | 3398.2034 | 0.3145 | 0.5953 |
 |
0.8619 | 0.7429 | 52.1226 | 34.6773 | 0.0001 |
 |
0.2463 | 0.0606 | 49.5482 | 0.7101 | 0.4173 |
 |
0.4575 | 0.2093 | 91.4826 | 2.3821 | 0.1571 |
 |
0.6654 | 0.4428 | 2.4942 | 3.9734 | 0.1028 |
 |
0.9559 | 0.9137 | 0.6238 | 31.7523 | 0.0111 |
 |
0.3997 | 0.1598 | 1.9547 | 1.3311 | 0.2865 |
 |
0.1203 | 0.0145 | 3460.7911 | 0.0881 | 0.7766 |
 |
0.8951 | 0.8012 | 45.8347 | 48.3627 | 0.0000 |
 |
0.2301 | 0.0529 | 49.7510 | 0.6149 | 0.4495 |
 |
0.4307 | 0.1855 | 92.8499 | 2.0493 | 0.1861 |
 |
0.6364 | 0.4050 | 2.5774 | 3.4037 | 0.1244 |
 |
0.9674 | 0.9359 | 0.5373 | 43.8302 | 0.0070 |
 |
0.3969 | 0.1575 | 1.9573 | 1.3091 | 0.2902 |
 |
0.0962 | 0.0093 | 3469.9437 | 0.0561 | 0.8207 |
 |
0.9028 | 0.8151 | 44.2025 | 52.9029 | 0.0000 |
 |
0.2584 | 0.0668 | 49.3861 | 0.7871 | 0.3940 |
 |
0.5152 | 0.2655 | 88.1719 | 3.2529 | 0.1048 |
 |
0.5940 | 0.3528 | 2.6881 | 2.7259 | 0.1596 |
 |
0.9949 | 0.9898 | 0.2146 | 290.5792 | 0.0004 |
 |
0.4103 | 0.1684 | 1.9447 | 1.4171 | 0.2727 |
 |
0.1116 | 0.0125 | 3464.3409 | 0.0757 | 0.7925 |
 |
0.8998 | 0.8096 | 44.8602 | 51.0137 | 0.0000 |
 |
0.2621 | 0.0687 | 49.3354 | 0.8114 | 0.3870 |
 |
0.5007 | 0.2507 | 89.0574 | 3.0104 | 0.1168 |
 |
0.6369 | 0.4056 | 2.5761 | 3.4119 | 0.1240 |
 |
0.9827 | 0.9657 | 0.3931 | 84.4840 | 0.0027 |
 |
0.4086 | 0.1669 | 1.9464 | 1.4026 | 0.2749 |
 |
0.1222 | 0.0149 | 3459.9941 | 0.0909 | 0.7732 |
Table 8.
Quadratic regression analysis.
| Equation | R |  |
RMSE | F-statistic | p value |
|---|---|---|---|---|---|
 |
0.9022 | 0.8139 | 44.3444 | 24.0571 | 0.0001 |
 |
0.4622 | 0.2136 | 45.3338 | 1.3584 | 0.3007 |
 |
0.6446 | 0.4155 | 78.6519 | 2.8438 | 0.1167 |
 |
0.7061 | 0.4985 | 2.3662 | 1.9883 | 0.2515 |
 |
0.9899 | 0.9799 | 0.3010 | 48.7421 | 0.0201 |
 |
0.5513 | 0.3039 | 1.7792 | 1.3097 | 0.3373 |
 |
0.2221 | 0.0493 | 3399.0454 | 0.1297 | 0.8812 |
 |
0.8919 | 0.7954 | 46.4962 | 21.3847 | 0.0002 |
 |
0.4787 | 0.2291 | 44.8847 | 1.4863 | 0.2722 |
 |
0.6551 | 0.4292 | 77.7285 | 3.0074 | 0.1062 |
 |
0.7104 | 0.5047 | 2.3516 | 2.0379 | 0.2453 |
 |
0.9870 | 0.9742 | 0.3409 | 37.7717 | 0.0258 |
 |
0.5323 | 0.2834 | 1.8052 | 1.1863 | 0.3680 |
 |
0.2122 | 0.0450 | 3406.7461 | 0.1178 | 0.8912 |
 |
0.8607 | 0.7409 | 52.3281 | 15.7261 | 0.0006 |
 |
0.4913 | 0.2413 | 44.5283 | 1.5906 | 0.2513 |
 |
0.8630 | 0.4396 | 77.0152 | 3.1378 | 0.0986 |
 |
0.6676 | 0.4457 | 2.4876 | 1.6085 | 0.3072 |
 |
0.9840 | 0.9683 | 0.3782 | 30.5181 | 0.0317 |
 |
0.4546 | 0.2067 | 1.8994 | 0.7815 | 0.4993 |
 |
0.2221 | 0.0493 | 3399.0239 | 0.1298 | 0.8812 |
 |
0.8817 | 0.7774 | 48.4989 | 19.2101 | 0.0003 |
 |
0.4883 | 0.2384 | 44.6139 | 1.5653 | 0.2562 |
 |
0.6616 | 0.4377 | 77.1479 | 3.1132 | 0.1000 |
 |
0.7223 | 0.5218 | 2.3107 | 2.1821 | 0.2287 |
 |
0.9858 | 0.9717 | 0.3569 | 34.3884 | 0.0283 |
 |
0.5480 | 0.3003 | 1.7838 | 1.2875 | 0.3426 |
 |
0.2479 | 0.0615 | 3377.2959 | 0.1637 | 0.8534 |
 |
0.8899 | 0.7920 | 46.8887 | 20.9364 | 0.0002 |
 |
0.4693 | 0.2202 | 45.1444 | 1.4119 | 0.2883 |
 |
0.6223 | 0.3872 | 80.5331 | 2.5278 | 0.1410 |
 |
0.5134 | 0.2636 | 2.8674 | 0.7160 | 0.5423 |
 |
0.9847 | 0.9696 | 0.3703 | 31.8779 | 0.0304 |
 |
0.4609 | 0.2124 | 1.8925 | 0.8091 | 0.4885 |
 |
0.1600 | 0.0256 | 3441.1868 | 0.0657 | 0.9372 |
 |
0.8972 | 0.8049 | 45.4067 | 22.6903 | 0.0001 |
 |
0.4580 | 0.2098 | 45.4456 | 1.3272 | 0.3082 |
 |
0.5949 | 0.3539 | 82.6973 | 2.1906 | 0.1743 |
 |
0.5418 | 0.2936 | 2.8084 | 0.8312 | 0.4990 |
 |
0.9984 | 0.9968 | 0.1199 | 312.2868 | 0.0032 |
 |
0.4660 | 0.2172 | 1.8868 | 0.8322 | 0.4797 |
 |
0.1922 | 0.0370 | 3421.0900 | 0.0959 | 0.9102 |
 |
0.9024 | 0.8143 | 44.3045 | 24.1103 | 0.0001 |
 |
0.4653 | 0.2165 | 45.2515 | 1.3816 | 0.2953 |
 |
0.6387 | 0.4080 | 79.1602 | 2.7562 | 0.1229 |
 |
0.6918 | 0.4786 | 2.4127 | 1.8361 | 0.2718 |
 |
0.9918 | 0.9837 | 0.2715 | 60.1651 | 0.0163 |
 |
0.5447 | 0.2967 | 1.7883 | 1.2659 | 0.3478 |
 |
0.2028 | 0.0411 | 3413.6492 | 0.1073 | 0.9003 |
 |
0.8910 | 0.7939 | 46.6689 | 21.1861 | 0.0002 |
 |
0.4775 | 0.2280 | 44.9183 | 1.4766 | 0.2742 |
 |
0.6277 | 0.3940 | 80.0849 | 2.6011 | 0.1348 |
 |
0.5752 | 0.3309 | 2.7332 | 0.9891 | 0.4477 |
 |
0.9816 | 0.9636 | 0.4051 | 26.4614 | 0.0364 |
 |
0.5030 | 0.2531 | 1.8430 | 1.0163 | 0.4167 |
 |
0.1545 | 0.0239 | 3444.2302 | 0.0612 | 0.9414 |
Table 9.
Quadratic regression analysis.
| Equation | R-value |  |
RMSE | F-statistic | p value |
|---|---|---|---|---|---|
 |
0.8907 | 0.7934 | 46.7232 | 21.1240 | 0.0002 |
 |
0.4683 | 0.2193 | 45.1713 | 1.4043 | 0.2901 |
 |
0.5878 | 0.3455 | 83.2326 | 2.1112 | 0.1835 |
 |
0.4698 | 0.2207 | 2.9497 | 0.5664 | 0.6073 |
 |
0.9893 | 0.9786 | 0.3102 | 45.8277 | 0.0214 |
 |
0.4487 | 0.2014 | 1.9057 | 0.7565 | 0.5094 |
 |
0.1805 | 0.0326 | 3428.8522 | 0.0842 | 0.9205 |
 |
0.8980 | 0.8064 | 45.2267 | 22.9152 | 0.0001 |
 |
0.4635 | 0.2149 | 45.2989 | 1.3683 | 0.2984 |
 |
0.6028 | 0.3633 | 82.0902 | 2.2825 | 0.1643 |
 |
0.5733 | 0.3287 | 2.7377 | 0.9794 | 0.4506 |
 |
0.9974 | 0.9948 | 0.1524 | 193.0910 | 0.0052 |
 |
0.4979 | 0.2479 | 1.8493 | 0.9889 | 0.4254 |
 |
0.1907 | 0.0364 | 3422.1640 | 0.0943 | 0.9116 |
 |
0.8950 | 0.8010 | 45.8624 | 22.1329 | 0.0001 |
 |
0.4677 | 0.2188 | 45.1862 | 1.4001 | 0.2910 |
 |
0.6057 | 0.3668 | 81.8638 | 2.3173 | 0.1607 |
 |
0.7258 | 0.5268 | 2.2985 | 2.2266 | 0.2239 |
 |
0.9945 | 0.9889 | 0.2232 | 89.4665 | 0.0111 |
 |
0.5159 | 0.2661 | 1.8268 | 1.0879 | 0.3952 |
 |
0.1733 | 0.0300 | 3433.3551 | 0.0774 | 0.9266 |
 |
0.8904 | 0.7928 | 46.7905 | 21.0475 | 0.0002 |
 |
0.5403 | 0.2920 | 43.0172 | 2.0617 | 0.1779 |
 |
0.6340 | 0.4019 | 79.5616 | 2.6882 | 0.1279 |
 |
0.5632 | 0.3172 | 2.7610 | 0.9292 | 0.4662 |
 |
0.9719 | 0.9446 | 0.4996 | 17.0555 | 0.0554 |
 |
0.4339 | 0.1883 | 1.9212 | 0.6959 | 0.5348 |
 |
0.2338 | 0.0546 | 3389.5317 | 0.1445 | 0.8689 |
 |
0.8635 | 0.7457 | 51.8433 | 16.1249 | 0.0005 |
 |
0.5069 | 0.2569 | 44.0682 | 1.7289 | 0.2265 |
 |
0.6625 | 0.4390 | 77.0595 | 3.1296 | 0.0991 |
 |
0.7207 | 0.5194 | 2.3165 | 2.1612 | 0.2310 |
 |
0.9867 | 0.9736 | 0.3449 | 36.8963 | 0.0264 |
 |
0.5528 | 0.3056 | 1.7770 | 1.3203 | 0.3348 |
 |
0.2778 | 0.0772 | 3348.8623 | 0.2091 | 0.8180 |
 |
0.8952 | 0.8014 | 45.8124 | 22.1932 | 0.0001 |
 |
0.4445 | 0.1976 | 45.7952 | 1.2310 | 0.3327 |
 |
0.6440 | 0.4148 | 78.7032 | 2.8349 | 0.1173 |
 |
0.6839 | 0.4677 | 2.4379 | 1.7573 | 0.2833 |
 |
0.9937 | 0.9875 | 0.2378 | 78.6844 | 0.0125 |
 |
0.5340 | 0.2852 | 1.8029 | 1.1970 | 0.3652 |
 |
0.2819 | 0.0795 | 3344.7401 | 0.2158 | 0.8130 |
 |
0.9038 | 0.8170 | 43.9735 | 24.5578 | 0.0001 |
 |
0.4533 | 0.2055 | 45.5676 | 1.2934 | 0.3165 |
 |
0.6266 | 0.3927 | 80.1759 | 2.5861 | 0.1361 |
 |
0.6678 | 0.4459 | 2.4872 | 1.6097 | 0.3070 |
 |
0.9949 | 0.9899 | 0.2136 | 97.8067 | 0.0101 |
 |
0.5382 | 0.2897 | 1.7973 | 1.2234 | 0.3584 |
 |
0.2223 | 0.0494 | 3398.8996 | 0.1299 | 0.8810 |
 |
0.9003 | 0.8106 | 44.7418 | 23.5344 | 0.0001 |
 |
0.4618 | 0.2132 | 45.3456 | 1.3551 | 0.3015 |
 |
0.6483 | 0.4203 | 78.3304 | 2.9001 | 0.1129 |
 |
0.7084 | 0.5019 | 2.3583 | 2.0152 | 0.2481 |
 |
0.9892 | 0.9786 | 0.3109 | 45.6368 | 0.0214 |
 |
0.5504 | 0.3030 | 1.7803 | 1.3040 | 0.3386 |
 |
0.2364 | 0.0559 | 3387.3384 | 0.1479 | 0.8661 |
Table 10.
Cubic regression analysis.
| Equation | R |  |
RMSE | F-statistic | p value |
|---|---|---|---|---|---|
 |
0.9022 | 0.8139 | 44.3423 | 14.5817 | 0.0006 |
 |
0.5681 | 0.3227 | 42.0728 | 1.4294 | 0.2973 |
 |
0.6759 | 0.4569 | 75.8189 | 1.9628 | 0.2082 |
 |
0.7142 | 0.5100 | 2.3390 | 1.0409 | 0.4873 |
 |
0.9918 | 0.9837 | 0.2711 | 20.1015 | 0.1622 |
 |
0.7385 | 0.5454 | 1.4378 | 1.9995 | 0.2327 |
 |
0.4046 | 0.1637 | 3188.0429 | 0.2610 | 0.8507 |
 |
0.8920 | 0.7957 | 46.4641 | 12.9829 | 0.0009 |
 |
0.5573 | 0.3106 | 42.4484 | 1.3513 | 0.3183 |
 |
0.6949 | 0.4829 | 73.9811 | 2.1789 | 0.1785 |
 |
0.7160 | 0.5127 | 2.3325 | 1.0521 | 0.4838 |
 |
0.9881 | 0.9763 | 0.3270 | 13.7198 | 0.1953 |
 |
0.7380 | 0.5446 | 1.4390 | 1.9933 | 0.2336 |
 |
0.4538 | 0.2059 | 3106.5286 | 0.3457 | 0.7953 |
 |
0.8611 | 0.7414 | 52.2726 | 9.5583 | 0.0028 |
 |
0.5286 | 0.2794 | 43.3958 | 1.1634 | 0.3762 |
 |
0.9071 | 0.8999 | 72.7515 | 2.3327 | 0.0005 |
 |
0.6896 | 0.4756 | 2.4197 | 0.9069 | 0.5311 |
 |
0.9842 | 0.9687 | 0.3758 | 10.3027 | 0.2242 |
 |
0.6887 | 0.4743 | 1.5462 | 1.5035 | 0.3214 |
 |
0.4656 | 0.2167 | 3085.2781 | 0.3690 | 0.7805 |
 |
0.8818 | 0.7775 | 48.4913 | 11.6472 | 0.0013 |
 |
0.5658 | 0.3201 | 42.1537 | 1.4124 | 0.3018 |
 |
0.6797 | 0.4620 | 75.4574 | 2.0041 | 0.2021 |
 |
0.7224 | 0.5218 | 2.3107 | 1.0912 | 0.4723 |
 |
0.9888 | 0.9778 | 0.3164 | 14.6756 | 0.1890 |
 |
0.7519 | 0.5654 | 1.4059 | 2.1680 | 0.2102 |
 |
0.4951 | 0.2451 | 3028.8685 | 0.4330 | 0.7412 |
 |
0.8903 | 0.7925 | 46.8217 | 12.7346 | 0.0009 |
 |
0.5189 | 0.2693 | 43.7000 | 1.1057 | 0.3963 |
 |
0.6790 | 0.4611 | 75.5263 | 1.9962 | 0.2033 |
 |
0.5223 | 0.2728 | 2.8495 | 0.3751 | 0.7790 |
 |
0.9848 | 0.9699 | 0.3685 | 10.7289 | 0.2199 |
 |
0.4892 | 0.2393 | 1.8598 | 0.5244 | 0.6843 |
 |
0.3162 | 0.1000 | 3307.2395 | 0.1481 | 0.9257 |
 |
0.8972 | 0.8050 | 45.3987 | 13.7577 | 0.0007 |
 |
0.5271 | 0.2779 | 43.4434 | 1.1543 | 0.3793 |
 |
0.6779 | 0.4596 | 75.6282 | 1.9845 | 0.2050 |
 |
0.7051 | 0.4972 | 2.3694 | 0.9888 | 0.5036 |
 |
0.9986 | 0.9973 | 0.1113 | 120.9979 | 0.0667 |
 |
0.6634 | 0.4401 | 1.5957 | 1.3099 | 0.3685 |
 |
0.4017 | 0.1613 | 3192.5405 | 0.2565 | 0.8537 |
 |
0.9026 | 0.8146 | 44.2580 | 14.6500 | 0.0005 |
 |
0.5608 | 0.3145 | 42.3260 | 1.3765 | 0.3114 |
 |
0.6827 | 0.4660 | 75.1783 | 2.0363 | 0.1975 |
 |
0.7131 | 0.5084 | 2.3427 | 1.0344 | 0.4892 |
 |
0.9920 | 0.9840 | 0.2686 | 20.4962 | 0.1606 |
 |
0.7361 | 0.5419 | 1.4433 | 1.9716 | 0.2367 |
 |
0.3845 | 0.1478 | 3218.1667 | 0.2313 | 0.8705 |
Table 11.
Cubic regression analysis.
| Equation | R |  |
RMSE | F-statistic | p value |
|---|---|---|---|---|---|
 |
0.8908 | 0.7935 | 46.7183 | 12.8058 | 0.0009 |
 |
0.5202 | 0.2706 | 43.6604 | 1.1131 | 0.3937 |
 |
0.6951 | 0.4831 | 73.9624 | 2.1812 | 0.1782 |
 |
0.4709 | 0.2218 | 2.9477 | 0.2850 | 0.8350 |
 |
0.9940 | 0.9881 | 0.2315 | 27.6890 | 0.1386 |
 |
0.5289 | 0.2797 | 1.8098 | 0.6473 | 0.6176 |
 |
0.4675 | 0.2185 | 3081.7808 | 0.3728 | 0.7781 |
 |
0.8981 | 0.8066 | 45.2057 | 13.9040 | 0.0007 |
 |
0.5367 | 0.2880 | 43.1374 | 1.2135 | 0.3597 |
 |
0.6826 | 0.4659 | 75.1846 | 2.0356 | 0.1976 |
 |
0.6711 | 0.4504 | 2.4771 | 0.8195 | 0.5630 |
 |
0.9991 | 0.9982 | 0.0900 | 185.2779 | 0.0539 |
 |
0.6750 | 0.4557 | 1.5733 | 1.3953 | 0.3466 |
 |
0.3920 | 0.1537 | 3207.1254 | 0.2421 | 0.8633 |
 |
0.8951 | 0.8012 | 45.8322 | 13.4360 | 0.0008 |
 |
0.5351 | 0.2863 | 43.1877 | 1.2037 | 0.3629 |
 |
0.6626 | 0.4391 | 77.0527 | 1.8263 | 0.2302 |
 |
0.9419 | 0.8854 | 2.2406 | 1.2240 | 0.0060 |
 |
0.9998 | 0.9996 | 0.0437 | 786.3431 | 0.0262 |
 |
0.6803 | 0.4628 | 1.5630 | 1.4356 | 0.3369 |
 |
0.3521 | 0.1240 | 3262.8268 | 0.1887 | 0.8990 |
 |
0.8904 | 0.7929 | 46.7865 | 12.7588 | 0.0009 |
 |
0.5427 | 0.2945 | 42.9390 | 1.2525 | 0.3474 |
 |
0.6531 | 0.4266 | 77.9033 | 1.7360 | 0.2464 |
 |
0.7171 | 0.5143 | 2.3288 | 1.0587 | 0.4819 |
 |
0.9759 | 0.9523 | 0.4634 | 6.6616 | 0.2757 |
 |
0.4970 | 0.2470 | 1.8504 | 0.5468 | 0.6716 |
 |
0.2548 | 0.0649 | 3371.0144 | 0.0926 | 0.9602 |
 |
0.8635 | 0.7457 | 51.8395 | 9.7746 | 0.0026 |
 |
0.5669 | 0.3214 | 42.1138 | 1.4208 | 0.2996 |
 |
0.6659 | 0.4435 | 76.7495 | 1.8592 | 0.2247 |
 |
0.7217 | 0.5209 | 2.3129 | 1.0871 | 0.4734 |
 |
0.9897 | 0.9795 | 0.3038 | 15.9476 | 0.1816 |
 |
0.7677 | 0.5894 | 1.3665 | 2.3923 | 0.1847 |
 |
0.5101 | 0.2603 | 2998.3607 | 0.4691 | 0.7199 |
 |
0.8987 | 0.8076 | 45.0872 | 13.9947 | 0.0007 |
 |
0.5744 | 0.3300 | 41.8469 | 1.4773 | 0.2852 |
 |
0.6455 | 0.4166 | 78.5783 | 1.6664 | 0.2598 |
 |
0.6987 | 0.4882 | 2.3904 | 0.9539 | 0.5150 |
 |
0.9976 | 0.9951 | 0.1479 | 68.3555 | 0.0886 |
 |
0.7426 | 0.5514 | 1.4283 | 2.0486 | 0.2258 |
 |
0.4125 | 0.1702 | 3175.6327 | 0.2735 | 0.8424 |
 |
0.9039 | 0.8170 | 43.9733 | 14.8837 | 0.0005 |
 |
0.5634 | 0.3175 | 42.2353 | 1.3954 | 0.3063 |
 |
0.6649 | 0.4421 | 76.8411 | 1.8493 | 0.2263 |
 |
0.7154 | 0.5118 | 2.3346 | 1.0485 | 0.4849 |
 |
0.9958 | 0.9917 | 0.1933 | 39.8914 | 0.1157 |
 |
0.7418 | 0.5502 | 1.4302 | 2.0388 | 0.2272 |
 |
0.3680 | 0.1354 | 3241.5237 | 0.2088 | 0.8856 |
 |
0.9004 | 0.8107 | 44.7272 | 14.2747 | 0.0006 |
 |
0.5711 | 0.3261 | 41.9659 | 1.4520 | 0.2915 |
 |
0.6698 | 0.4487 | 76.3899 | 1.8988 | 0.2182 |
 |
0.7125 | 0.5077 | 2.3446 | 1.0311 | 0.4902 |
 |
0.9923 | 0.9846 | 0.2630 | 21.3801 | 0.1574 |
 |
0.7393 | 0.5466 | 1.4359 | 2.0091 | 0.2313 |
 |
0.4187 | 0.1753 | 3165.7712 | 0.2835 | 0.8358 |
Table 12.
Linear regression coefficients of properties with molecular descriptors.
M
|
M
|
RM
|
HM | AZ | R | RR | RRR | H | SC | GA | IS | F | SDI | ABC | So | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| MW | 0.9019 | 0.8896 | 0.7932 | 0.8758 | 0.8901 | 0.8764 | 0.8921 | 0.9024 | 0.8840 | 0.8940 | 0.8935 | 0.8904 | 0.8619 | 0.8951 | 0.9028 | 0.8998 |
| MP | 0.0266 | 0.0773 | 0.2454 | 0.2202 | 0.4063 | 0.4373 | 0.4312 | 0.4378 | 0.2294 | 0.2608 | 0.2757 | 0.3746 | 0.2463 | 0.2301 | 0.2584 | 0.2621 |
| BP | 0.5211 | 0.5540 | 0.5843 | 0.4810 | 0.5875 | 0.6982 | 0.7108 | 0.7172 | 0.0677 | 0.5536 | 0.5532 | 0.6058 | 0.4575 | 0.4307 | 0.5152 | 0.5007 |
| pKa | 0.6406 | 0.6262 | 0.2900 | 0.7680 | 0.5394 | 0.5874 | 0.5932 | 0.5771 | 0.4519 | 0.5238 | 0.6499 | 0.1019 | 0.6654 | 0.6364 | 0.5940 | 0.6369 |
| VD | 0.9876 | 0.9847 | 0.8649 | 0.7582 | 0.9872 | 0.9845 | 0.9886 | 0.9898 | 0.9836 | 0.9936 | 0.9914 | 0.9714 | 0.9559 | 0.9674 | 0.9949 | 0.9827 |
| log P | 0.4110 | 0.3938 | 0.2766 | 0.5536 | 0.3287 | 0.3461 | 0.3567 | 0.3723 | 0.4853 | 0.4011 | 0.4071 | 0.0640 | 0.3997 | 0.3964 | 0.4103 | 0.4086 |
| LD50 | 0.1829 | 0.2128 | 0.1350 | 0.0041 | 0.3365 | 0.3744 | 0.3842 | 0.4086 | 0.2288 | 0.1165 | 0.1129 | 0.2232 | 0.1203 | 0.0964 | 0.1116 | 0.1222 |
Bold values indicate the highest correlation.
Fig. 4.
Linear regression plots.
In quadratic regression analysis, the Atom bond connectivity index has the highest predictive ability (highest regression coefficient) for molecular weight. Geometric arithmetic index and inverse sum connectivity index have the highest predictive ability for pKa value and melting point, respectively. Symmetric division index shows the highest predictive ability for LD50 value. Unlike linear regression, RM2 shows high predictive ability for boiling point. Forgotten index and sum connectivity index have the highest predictive ability for log P value and vapour density respectively. The regression coefficient between each property and index is displayed in Table 13. The quadratic regression plots of the physicochemical property and index pairs with the highest coefficient of determination are displayed in Fig. 5.
Table 13.
Quadratic regression coefficients of properties with molecular descriptors.
M
|
M
|
RM
|
HM | AZ | R | RR | RRR | H | SC | GA | IS | F | SDI | ABC | So | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| MW | 0.9022 | 0.8919 | 0.8607 | 0.8817 | 0.8899 | 0.8972 | 0.9024 | 0.8910 | 0.8907 | .8980 | 0.8950 | 0.8904 | 0.8635 | 0.8952 | 0.9038 | 0.9003 |
| MP | 0.4622 | 0.4787 | 0.4913 | 0.4883 | 0.4693 | 0.4580 | 0.4653 | 0.4775 | 0.4683 | 0.4635 | 0.4677 | 0.5403 | 0.5069 | 0.4445 | 0.4533 | 0.4618 |
| BP | 0.6446 | 0.6551 | 0.8630 | 0.6616 | 0.6223 | 0.5949 | 0.6387 | 0.6277 | 0.5878 | 0.6028 | 0.6057 | 0.6340 | 0.6625 | 0.6440 | 0.6266 | 0.6483 |
| pKa | 0.7061 | 0.7104 | 0.6676 | 0.7223 | 0.5134 | 0.5418 | 0.6918 | 0.5752 | 0.4698 | 0.5733 | 0.7258 | 0.5632 | 0.7207 | 0.6839 | 0.6678 | 0.7084 |
| VD | 0.9899 | 0.9870 | 0.93840 | 0.9858 | 0.9847 | 0.9984 | 0.9918 | 0.9816 | 0.9893 | 0.9974 | 0.9945 | 0.9719 | 0.9867 | 0.9932 | 9949 | 0.9892 |
| log P | 0.5513 | 0.5323 | 0.4546 | 0.5480 | 0.4609 | 0.4660 | 0.5447 | 0.5030 | 0.4487 | 0.4979 | 0.5159 | 0.4339 | 0.5528 | 0.5340 | 0.5382 | 0.5504 |
| LD50 | 0.2221 | 0.1222 | 0.2221 | 0.479 | 0.1600 | 0.192 | 0.2028 | 0.1545 | 0.1805 | 0.1907 | 0.1733 | 0.2338 | 0.2778 | 0.2817 | 0.2223 | 0.2364 |
Bold values indicate the highest correlation.
Fig. 5.
Quadratic regression plots.
In cubic regression analysis, the Atom bond connectivity index exhibits the greatest predictive capacity, indicated by the highest regression coefficient, for molecular weight. The best predictor of pKa value and vapour density is the geometric arithmetic index. The best predictor of LD50 value and melting point is the symmetric division index. High boiling point prediction ability is demonstrated by RM2. The best predictor of the log P value is the forgotten index. The regression coefficient between each property and index is displayed in Table 14. The cubic regression plots of the physicochemical property and index pairs with the highest coefficient of determination are displayed in Fig. 6.
Table 14.
Cubic regression coefficients of properties with molecular descriptors.
M
|
M
|
RM
|
HM | AZ | R | RR | RRR | H | SC | GA | IS | F | SDI | ABC | So | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| MW | 0.9022 | 0.8920 | 0.8611 | 0.8818 | 0.8908 | 0.8972 | 0.9026 | 0.8911 | 0.8908 | 0.8981 | 0.8951 | 0.8904 | 0.8635 | 0.8987 | 0.9039 | 0.9004 |
| MP | 0.5681 | 0.5573 | 0.5286 | 0.5658 | 0.5189 | 0.5272 | 0.5608 | 0.5371 | 0.5202 | 0.5367 | 0.5351 | 0.5427 | 0.5669 | 0.5744 | 0.5634 | 0.5711 |
| BP | 0.6759 | 0.6949 | 0.9071 | 0.6797 | 0.6790 | 0.6779 | 0.6827 | 0.6645 | 0.6951 | 0.6826 | 0.6626 | 0.6531 | 0.6659 | 0.6455 | 0.6649 | 0.6698 |
| pKa | 0.7142 | 0.7160 | 0.6896 | 0.7224 | 0.5223 | 0.7051 | 0.7131 | 0.5765 | 0.4709 | 0.6711 | 0.9419 | 0.7171 | 0.7217 | 0.6987 | 0.7154 | 0.7125 |
| VD | 0.9918 | 0.9881 | 0.9842 | 0.9888 | 0.9848 | 0.9986 | 0.9920 | 0.9919 | 0.9940 | 0.9991 | 0.9998 | 0.9759 | 0.9897 | 0.9976 | 9958 | 0.9923 |
| log P | 0.7385 | 0.7380 | 0.6887 | 0.7519 | 0.4892 | 0.6634 | 0.7361 | 0.5676 | 0.5289 | 0.6750 | 0.6803 | 0.4970 | 0.7677 | 0.7426 | 0.7418 | 7393 |
| LD50 | 0.4046 | 0.4538 | 0.4656 | 0.4950 | 0.3162 | 0.4017 | 0.3845 | 0.2781 | 0.4675 | 0.3950 | 0.3521 | 0.2548 | 0.5101 | 0.4125 | 0.3680 | 0.4187 |
Bold values indicate the highest correlation.
Fig. 6.
Cubic regression plots.
For molecular weight, the cubic model has the highest 
value, 0.9039, suggesting that it explains most variance, with ABC as the best predictor. For melting point, cubic model has the highest 
-value (0.5744) but has a high p value of 0.2852, making the model statistically weak. For the boiling point in the cubic model, 
and p value < 0.05, making it the best model, with the predictor index RM2. For the pKa value, 
for cubic model, showing the best predictive ability. For vapor density, the cubic model is the best (
) with the lowest RMSE. All the models are highly significant. For log P values and LD50 values, the models are statistically insignificant due to high p values ( > 0.05).
To develop the most predictive model, cubic regression offers the maximum accuracy for the four of molecular parameters, molecular weight (MW), boiling point (BP), pKa, and vapour density (VD), as indicated by the highest 
values. The models for BP, MW, pKa, and VD have p values less than 0.05 in terms of statistical significance, indicating that they are dependable in forecasting these properties. However, because of their weaker correlations and statistical insignificance, the models for Melting Point (MP), LogP, and LD50 show lower 
values and higher p values, making them less accurate predictors. Table 15 shows the best predictive model.
Table 15.
Best predictive model.
| Equation | R |  |
p value |
|---|---|---|---|
 |
0.9039 | 0.8170 | 0.0005 |
 |
0.9071 | 0.8999 | 0.0005 |
 |
0.9419 | 0.8854 | 0.0060 |
 |
0.9998 | 0.9996 | 0.0262 |
We take into consideration propionic acid, a different food preservative, in order to validate the predictive model. Propionic acid is a fungicide used to stop bacteria and fungi from growing in grain that has been stored for use by animals and poultry58. Propionic acid’s chemical structure is depicted in the Fig. 7. We compute the degree-based topological indices of propionic acid, as in Theorem 3.1, and the results are shown in Table 16. By substituting the index values in the predictive model, the physicochemical characteristics of propionic acid are estimated. The physiochemical characteristics’ experimental values are gathered from PubChem45. The experimental and predicted values for propionic acid’s physicochemical characteristics are compared in Table 17, which indicates that they are approximately equal. These results show how well the model predicts the characteristics of novel food preservative candidates. This technique can be used to screen a large number of chemicals and identify those that have a high potential for usage as food preservatives.
Fig. 7.
Propionic acid.
Table 16.
Topological indices of Propionic acid.
M
|
M
|
RM
|
HM | AZ | R | RR | RRR |
|---|---|---|---|---|---|---|---|
| 16 | 14 | 2 | 66 | 22.75 | 2.2701 | 7.3278 | 1.4142 |
| H | SC | GA | IS | F | SDI | ABC | So |
| 2.0667 | 2.0246 | 3.6547 | 3.3667 | 38 | 11.3333 | 3.0472 | 12.1662 |
Table 17.
Comparison between experimental and predicted values.
| Property | Experimental value | Predicted value |
|---|---|---|
| Molecular Weight(g/mol) | 74.08 | 78.2329 |
| Boiling point (°C) at 760 mmHg | 141 | 142.3753 |
| pKa | 4.88 | 4.82 |
| Vapour Density | 2.56 | 2.7514 |
Conclusion
In this study, topological indices are effectively utilized to create prediction models for evaluating the physicochemical characteristics of food preservatives. We determined the most reliable method for predicting these attributes by comparing the linear and curvilinear regression models. From the findings, it can be concluded that cubic regression models are better than linear and quadratic models as they yield 
values such as 0.9998 for vapor density and 0.9039 for molecular weight, which indicates high predictive capability. The best predictive model is also identified. Furthermore, the applicability and dependability of the model are confirmed by its validation using propionic acid, an existing food preservative. In further studies, this predictive model can be utilized in the screening of food preservative compounds. This study demonstrates how computational approaches can be used to screen and design new food preservatives, providing an economical and effective substitute for experimental testing.
Author contributions
GKB: Conceptualization; writing-original draft; validation;software RS: Supervision; validation; conceptualization.
Funding
Open access funding provided by Vellore Institute of Technology.
Data availability
All data generated or analysed during this study are included in this published article.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Data Availability Statement
All data generated or analysed during this study are included in this published article.