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Abstract: The Surface Free Energy SFE of a material is defined as the energy needed to create a new surface unit under vacuum conditions. This property is directly related to the resistance to fracture and recovery of material and the ability to create strong adhesion with other materials. This value can be used as a complementary parameter for the selection and optimal combination of materials for asphalt mixtures, as well as in the micromechanical modeling of fracture and recovery processes of said mixtures.
This document describes the results of the implementation of the use of machine learning and Random Forest prediction techniques for the estimation of surface free energy based correllation data from previous studies. Keywords: asphalt cement, surface free energy, asphalt mixtures, machine learning, random forest, how to find correlation between two variables python highway research plan.
Resumen: La energía libre de superficie de un material se define como la energía necesaria para crear una nueva unidad de superficie en condiciones de vacío. Asphalt mixtures used in pavement variabbles, are porous materials that result from the combination of multiple aggregates among many of these you can have crushed rock in various sizes and with a density of different proportions and asphalt cement.
The strength and durability of an asphalt mix depend to a large extent on the quality of the adhesion between the cement and the aggregates. In this way, the adhesion between these two or more materials usually turns out to be a function how to find correlation between two variables python their mineralogical and chemical composition, the morphology of the aggregates shape and texturesand in addition to the conditions in which they are mixed is prepared.
Often how to find correlation between two variables python happens is that the asphalt mixtures are deficient and how to find correlation between two variables python their performance in the works for which they were made is also deficient. This could be due to inadequate conditions in the preparation of the mixtures by not ensuring compatibility of the compounds. In addition, very high or very low temperatures, or outside the preparation behween, could also affect the integrity of the mixtures define an example of a casual relationship well as their performance on the work.
From a physical point of view, adhesion in a mixture of liquid and solid materials such as asphalt is defined in terms of the physical surface properties of the materials that how to find correlation between two variables python the liquid to wet or coat the solid component. This how to find correlation between two variables python is known as wettability [ 1 ] [ 2 ] [ 3 ] [ 4 ] it is defined as the resistance of a liquid droplet to stay in balance when in contact with a solid body.
The ability of liquids to coat solid bodies, and solids to be coated by liquids, is directly related to the surface tension or Surface Free Energy SFE of the materials i. SFE is a fundamental property of materials, and its quantification is done through the application of advanced characterization techniques, such as the Wilhelmy Plate Method WPM [ 5 ] [ 6 ], the Sessile Drop Method [ 7 ] [ 8 ] [ 9 ] [ 10 ] [ 11 ], the Universal Adsorption Method UAM [ 12 variablex, among others.
The main motivation to dorrelation the adhesion in asphalt mixtures is the growing need for better material selection techniques i. It has been shown that by studying adhesion in aggregate-asphalt cement systems, is possible to identify combinations of materials that produce high adhesion systems and high resistance to moisture damage. This type of damage in asphalt mixtures is defined as the decrease in adhesion between the asphalt cement and the aggregate or the reduction of the cohesion within the asphalt cement [ 13 ].
By determining the SFE of the materials and applying the basic theory of surface physics, it is possible to identify combinations of aggregates and asphalt cement with high adhesion in the dry state and with low susceptibility to moisture damage. From the thermodynamic point of view, the SFE of a material is defined as the work required to create a new unit of area in said material, under vacuum conditions [ 12 ].
Gray correlation analyses were carried out to determine which chemical components and chemical elements of asphalt binders are most related to surface free energy SFE measurements of asphalt binders. The measurement of the contact angle was carried out using a Drop Shape Analysis betweej, manufactured by Kruss How to find correlation between two variables python.
The Owens Wendt theory was applied to determine the surface free energy. The experimental procedure, as well as the determination of the surface free energies for these twenty-three asphalt samples, can be consulted in the manuscript [ 14 ]. The asphalt identification codes, the contents of four fractions, the wax content, and the elemental analysis of these asphalt samples are given in Fijd 2. Furthermore, from the analyses performed in [ 15 ], a simple and multiple regression analysis was carried out to correlate and obtain parametric mathematical relationships between the free energy of the surface and the chemical compositions of the asphalt binders, including group type analysis saturated, naphthenic, polar aromatics and asphalteneswax content and elemental content, based on published data on chemical composition [ 14 ].
The present manuscript explores a different line varlables action for betweeb relationships between SFE and the chemical characteristics of asphalt samples. In this case, the main objective of our study is to implement a non-parametric methodology with the use of machine learning tools for the estimation and prediction of SFE in terms of the dependent variables measured in previous studies. The use of machine learning for these purposes is aimed at improving the quality of predictions and estimates.
This is possible thanks to the advantages of having deep and automatic learning methods with algorithms that try to learn from the data, and the more data available to learn and richer and more complete the algorithm will work better. Data used in this study for the asphalt identification codes, four fractions' contents, wax content, elemental analysis of these asphalt samples are given in Table 1 [ 14 ].
Source: [ 14 ] [ 15 ]. Generally, when trying to statistically study the behavior of a variable alone, a process of analysis of the distribution of this variable is required. This analysis provides information from the systematic exploration of the properties of each variable under study. Probability density diagrams can visually as a first what is the purpose of action research methodology to study the general behavior of the variable under study.
One way to obtain this empirical estimate of density which is certainly a nonparametric methodology is by using histograms of individual counts or relative frequencies. Often, this preliminary step can reveal what type of distribution the variable follows and thus characterize the central properties ot the entire possible range of variable values. This will determine if the distribution is completely symmetric and if the central measures represent a good estimator, which is particularly useful because many times, some known probability density functions are applicable to be modeled by the data set.
In this case, we present scatter plots for each input variable with the output variable. Since the data studied here are non-Gaussian, the Spearman rank correlation coefficient can be used to obtain a statistical metric concerning the strength of association of each input variable with the output. The Spearman rank correlation coefficient can characterize general monotonic relationships and is in the range of -1 to 1, where the negative sign indicates that it is inversely proportional and the positive sign indicates a proportional relationship, while the magnitude denotes what is very strong in this relationship.
In addition, we evaluate if this relationship is statistically significant with the p-values and verify the importance at the 0. When the how to find correlation between two variables python studied bstween properly interrelated or intends to be related with another set of variables, which we call predictors, the multivariate factorial analysis is convenient to establish and expose the underlying structure in a data matrix that precisely measures this degree correlaton relationship.
The first is to determine all the relationships between each pair of these variables without making a priori distinction of which is the dependent or independent variable, or in other words, which is the predictor variable, and which is predicting. Therefore, it is a data reduction technique, the information contained in the data matrix can be expressed, without much distortion, in a smaller number of dimensions represented by said FACTORS. To evaluate the significant differences between the sites for all the water quality variables, the data were analyzed through the analysis of variance.
The multivariate analysis of the water quality data sets was done through hierarchical group analysis HCA and principal components analysis PCA [ 16 ]. The objective of clustering is to divide the objects into homogeneous groups so that the similarities within the group are large compared to the similarities between groups. The Principal Components, on the other hand, are extracted to represent the patterns that encode the highest variance in the data set and not to maximize the separation between groups of samples directly.
The statistical package used in this case is R version 3. In many practical applications, the inputs may show a complicated functional relationship to determine the output. The classification and regression tree method CART, for its acronym in English of its Classification and Regression Tree is a method conceptually simple, although powerful nonlinear, which often provides reasonable results [ 20 ] how to find correlation between two variables python 21 ].
CART works by successively dividing the space of the input entity into smaller and smaller subregions. This procedure can be visualized as a tree that is divided into successively smaller branches, each of which represents a subregion finf the ranges of the input variables. The tree grows until it is not possible to divide it further or a certain criterion has been fulfilled.
A natural extension of CART is the methodology of its random forests RFwhich is simply a why does my verizon hotspot say connected but no internet of many trees [ 22 ]. The training procedure is the same as in CART with the difference that a subset of candidate variables chosen at random can be used to select the optimal variable for each division; the practice has shown that the RF algorithm works extremely well in many different applications [ 20 ] [ 21 ] [ 22 ].
In addition, RF has the desirable ability to promote the most important input variables to predict variavles output variable as part of its inherent learning strategy [ 21 ]. We emphasize that the importance of the variable is not evaluated independently for each variable; instead, it is evaluated jointly for the subset of characteristics used in the RF, making use of the concepts of relevance strength of association of variable and responseredundancy strength of association between variables and complementarity force of joint association of variables with the answer.
The results of the Principal Component Analysis revealed that the first two independent variables that result from how to find correlation between two variables python decomposition study add up to The analysis of the correlation between the four fractions contents, wax content, elemental finf of these asphalt samples, and their contributions to these what does it mean if a gene is recessive Figure 1 shows that Wax and Aromatics content is positively correlated with dimensions 2 and negatively correlated with dimensions 1; Carbon and Hydrogen contents are positively correlated with dimensions 1 and 2.
Figure 1 Projection of asphalt binder samples and independent variables on factorial variables determined for principal component analysis. On the other hand, Resin, Oxygen, and Nitrogen are positively correlated with dimensions 1 and negatively correlated with dimensions 2; and finally, Sulfurs, Aspens, and Vanadium content are correlaion correlated with dimensions 1 and 2. The analysis of the projection of the different groups of samples Figure 2 on the dimensions of independent variables shows that there are three very well-identified groups.
Figure 2 Projection of groups on factorial variables determined for principal component analysis. The correlation coefficient see Table 2 between the dependent variable SFE and the independent variables shows that the most important variables are X3, X4, X6, X9, and X10 with correlation coefficients 0. In the reference [ 15 ], the tto developed a single regression and multiple regression analysis were applied to correlate how to find correlation between two variables python relationships between chemical composition and surface free energy of asphalt binders.
Several regressions were constructed by the authors to examine the behavior by groups of separate variables, however, one fnid the most important estimates that could be made is precisely that which provides the relationship between the SFE and all the measured variables. Corre,ation found relationship takes the mathematical form:.
Our approach is different, and in this case, we find a model built through machine learning tools using Random Forest estimators. Why do i have such a hard time reading out loud this strategy, several decision trees have been used to find the best model that fits the SFE data in terms of the twelve variables vairables. For the evaluation of the pythob and the good approximation of the solution have been found several parameters and metrics that allow measuring the efficiency of the model see Table 2 and 3.
Table 3 Multiple metrics used to evaluate the accuracy and good performance of the Random Forest model in comparison with linear regression Source: [ 15 ]. The calculations are made in the computer from a code designed for the implementation of Random Forest in how to find correlation between two variables python situation. After examining several alternatives, the model is found and saved on the hard disk so that it can be used later in future applications for data of the same species.
A summary of these parameters can be found in Table cprrelation. As can be seen from Table 2the variables with the highest correlation assignment with SFE generally retain how to find correlation between two variables python greater importance in the model developed by Random Forest. This is precisely true for the variables X 3X 4and Correlatioj 10however, for the variables X 6 and X 9there is a loss of importance for these last variables, which is compensated with a gain of importance in the variables X 5X 11and X 12which acquire relevance in the machine learning model despite the little initial correlation they had with the SFE measure.
This implies that the Wax, Nickel, and Vanadium content are variables that should not be neglected, and their weight is very useful to estimate the dependent variable more adequately. This can be confirmed in the principal component analysis observing that most of the samples report high contents of these variables see Figures 1 and 2. On the other hand, the analysis in Table 3 shows that there is considerable improvement in all the metric parameters to evaluate the performance of the model.
All these factors and parameters determine a better performance of the machine learning tools and, the estimation using Random Forest, in the approximation in this case not parametric for the calculation of the surface free energy for asphalt samples. Finally, a visual representation of the behavior of the model that reported in previous studies can be seen in Figure 3.
Figure 3 a A comparison of the relative error f - ywith fi is the model estimation and y is experimental data between the original data and the two constructed models multiple linear regression and random forest. We presented methods for performing Random Forest optimization for hyperparameter selection of general machine learning algorithms for the estimation of Surface Free Energy for twenty-three asphalts binders' experimental samples used in an SHRP.
We introduced full Random Forest treatment algorithms to make a comparison with previous results for this dataset getting good effectiveness of our approaches. Considering the metrics used in this study we can say that the model determined by multiple linear regression estimates the SFE variable vriables an error of 0. Variablfs model developed by Random Forest also rescue importance to variables that had a lower weight and correlation in the approach with multiple linear regression, this is a great improvement of the use how to calculate expectation and variance of a random variable based on machine learning tools.
While it is true that the size of the data and the sample is small for the selection of machine learning techniques for the resolution of this problem and the analysis of the study variable in terms of correaltion predictor variables, the same argument also applies to their study using multivariate analysis, so that for equal conditions of data, the best method used for the analysis will always be the one that provides the least errors in the estimation what is a unicorn when dating the study variable.
However, the purpose of this article is variablees establish criteria that allow us to affirm that machine learning can, and indeed improves, a better estimation of surface energy for the study of asphalt aggregates normally used in construction. With this first scenario, a future work which is carried out at this time for author and collaborationis to increase the database to include other types of asphalt binders and aggregates with other predictor contents, and perhaps more study variables.
Based what are the 3 kinds of relation in math the results of this research, it can be affirmed getween the technique and methodology used will be able to establish very accurate and adequate models for the study of aggregates and asphalt binders used in highway construction.
No funds have been received for the development of this project from any institution. Mittal, Advances in contact angle, wettability and adhesion. De Gennes, "Wetting: statics and dynamics," Reviews of modern physicsvol. Voinov, "Dynamics of a viscous liquid wetting a solid via van der waals forces," Journal of Applied Mechanics and Technical Physicsvol. Btween, "The interpretation of dynamic contact angles measured by the wilhelmy plate method," Journal of colloid and interface what does fundamental.meanvol.
Lander, L.