non-linear regression

Recently, pseudo-continuous transfer functions (PC-PTFs) have been introduced for estimating soil water retention curve (SWRC). The M5 tree method is similar to regression trees, where linear functions are located in its leaves, and it has a high capability in creating transfer functions. These functions are highly sensitive to the power of machine learning algorithms. However, so far, the powerful M5 tree method has not been used to develop PC-PTFs for a wide range of soil textures. Additionally, the effectiveness of some soil structural variables in improving PC-PTFs has not been investigated, so far. Furthermore, the dependency of the error distribution of PC-PTFs on soil textural triangles to various factors has not been deeply examined. Therefore, the objectives of this study were to develop PC-PTFs using the M5 method, investigate the effect of soil structural variables on the performance of these functions, and examine the error dependence of these functions on different factors. A total of 120 soil samples were collected from depths of 10 to 60 centimeters, with agricultural, orchard, and pastureland uses of Tehran and Hamedan provinces, and soil texture, bulk density (BD), SWRC, saturated hydraulic conductivity (Ks), organic matter (OM), mean weight diameter (MWD) of soil aggregates, and penetration resistance at 300 hectopascals (PR300) were measured. Thirteen PC-PTFs, in three groups of inputs, were developed to estimate SWRC, using M5 tree and non-linear regression methods. The error distribution of all PC-PTFs was plotted on the soil texture triangle, according to root mean square error (RMSE). In the first function, soil suction was used as the only estimator. A non-linear regression model produced an acceptable model for the first function with a R2 of 0.718. In PC-PTFs 3 to 6, components of soil texture, BD, and FC (at 300 hPa matric suction) and PWP (at 15000 hPa matric suction) moisture contents were used to estimate SWRC. The R2for these functions ranged from 0.719 to 0.990, indicating an improvement in the performance of SWRC estimation. In the M5 method, the use of FC significantly improved the model performance and created an optimal model, resulting in RMSE of 0.015 and 0.020 cm³cm⁻³, and R² of 0.987 and 0.973 in the training and validation stages, respectively. In the M5 method, any function using Ks and MWD as estimators showed significant improvement compared to PC-PTF4, which used soil texture components and BD as estimators. The AIC values in both training and validation stages in the M5 method were 37% to 283% and 111% to 157% lower compared to non-linear regression, respectively. The error distribution on the soil texture triangle showed no dependence on soil texture but was related to the method of creating PC-PTFs and relevant input variables. A powerful artificial intelligence methods can be employed to create a comprehensive model for SWRC. This would eliminate the need for users to rely on various SWRC models such as van Genuchten for different soils. Incorporating a set of soil texture and structure variables increases the accuracy of SWRC estimation. However, among structural variables, those indicating pore size distribution were more suitable for SWRC estimation. The greater impact of FC compared to PWP demonstrated the higher efficiency of moisture in intermediate matric suctions for SWRC estimation. The robust algorithm of the M5 tree method identified some patterns of relationships between input and output variables that were not detectable by non-linear regression. Considering the dependence of error distribution on the soil texture triangle to the method of creating PC-PTFs and input variables, categorizing error distribution maps should be done based on the mentioned factors.

Regardingthe climate change and recent droughts, the proper management of irrigation systems and water resources is very important. However, achieving the optimum crop yields due to lack of water requires, the accuracy estimating the amount of water by the plants to prevent water loss and or water stress in plants is needed. In this study, the reference evapotranspiration using lysimeter was measured during three cropping seasons. Applying Ref-ET software, the reference evapotranspiration with 16 empirical equations using meteorological data were estimated, the significant difference between measured and estimated was determined using SPSS software. The empirical equations Hargreaves, Priestley-Taylor and Penman (FAO) were determined as the most suitable methods for the studied area. For the obtained data from two growing seasons and by applying nonlinear regression in SPSS software, optimized coefficients for the two Priestley-Taylor and Hargreaves equations were estimated, and then by using obtained data in the third year, thevalidation was done and RMSE as well as nRMSE values were calculated to assess the equations that for optimized Hargreaves were 0.20 mm/day and 4.60% and for optimized Priestley-Taylor equation were 0.22 mm/day and 4.5%, respectively.

Investigating the process of pollutant dispersion in natural rivers is important in pollution control and its distribution and management of aquatic environments. The process of pollution distribution is a function of dispersion coefficient that has been used in Advection-Diffusion equation. Since the one-dimensional models are applied for river water quality modeling, the first step in modeling the quality of river water is determining the Longitudinal Dispersion Coefficient (LDC). Many methods such as empirical, analytical, statistical, field measurements, and more sophisticated approaches, i.e., artificial intelligence techniques have been suggested for LDC estimation. By now, Laboratory methods are associated with many limitations. These methods are along with high costs and detrimental effects on the aquatic environment due to the application of some specific and harmful tracers. Artificial Intelligence Models have had a good prediction for LDC in recent years, but uncertainty in these models, as an annoying factor, always limited their results for practical purposes. These techniques act exactly like a black box model. Also, although different studies have been carried out for LDC estimation in natural rivers, the absence of a comprehensive study to investigate the effects of different patterns of dimensionless hydraulic and geometric parameters on LDC is still felt. Another challenge is the inaccuracy and less user-friendliness of the provided models to predict LDC. Thus, the main objective of this study is to investigate different types of LDC prediction by application of regression analysis.
The data used in this study for estimation of LDC contains 61 datasets of hydraulic parameters such as velocity U (m/s), shear velocity u*(m/s) and flow rate (Q), and also the geometric specifications measuring in cross-section of rivers like water depth H(m), stream width (W) and curvature of the river σ. This dataset was taken from different sections of 31 rivers in the United States of America. This information was used as input parameters (about 70 percent, i.e 43 data for calibration and 30 percent, i.e. 18 data for testing) for the non-linear regression model.
Many of researchers have used equations based on establishing a relationship between LDC and hydraulic-geometric specifications of rivers. Some researchers have used other parameters such as Q and σ to improve LDC estimation. So, in this research, all combination of input parameters for application of non-linear equations have been investigated. Non-linear equations could be changed to linear equations by a logarithmic method. Thus, all of the unknown coefficients of equations could be known by a linear Least Squares Regression.
Calibration and test steps were carried out using various patterns of hydraulic and geometric data of the rivers in the United States of America. The obtained equations have been compared by the statistics of regression method such as coefficient of determination (R2), t-value, p-value and Variance Inflation Factor (VIF). Higher values of t-value led to the lower value of p-value that indicated the importance of parameter in equation. VIF shows the existence of multi-collinearity in equation. Results showed that the pattern including the river curvature parameter had the best performance for LDC prediction model, while discharge parameter had the least effects on the LDC prediction model. Also, if the constant number is eliminated from the equation, the model’s performance is increased. The R2 of calibration and test stages of the best tuned model were 0.993 and 0.938, respectively. Selected equations in this research (with the best performance) compared with the models suggested by other researchers based on Root Mean Square Error (RMSE), Mean Related Error (MRE), Developed Discrepancy Ratio (DDR), and Threshold Statistics (TS) indices. The comparison of the developed model with the past studies revealed that it had a better performance in LDC estimation. Although the model proposed by Zeng and Huai (2014) had the second best performance, its statistical indices (RMSE and MRE)were about 2 and 83 times greater than that those obtained for the developed model. The best model based on the DDR graphic statistic has the largest crest and lowest width. So the comparison of models based on DDR indicated that the developed model in this research had the best performance. Also, the graph of the TS showed that the model developed in this research had the lowest threshold of errors in 100 percent of datasets. Thus, this research provides an LDC estimator model that outperforms the other existence equations.