The Importance of the Chosen Technique to Estimate Diffuse Solar Radiation by Means of Regression

The Ordinary Least Squares (OLS) is one of the widely used methods which is used for estimating the diffuse solar radiation. However, in order to use the OLS method in the estimation, the dataset must provide certain assumptions. In this study, alternative robust methods have been described and they were compared with the OLS method, which is used for estimating diffuse radiation frequently in an application. At the end of the analysis, the R² value obtained by the OLS method is less than the values obtained by M regression models. In other words, the explanation of the dependent value is weak when the OLS method is used. Finally, it can be said that the most appropriate method is Andrews for estimating the diffuse solar radiation.     

PARAMETRIC ESTIMATION OF COVARIANCES OF REGIONALIZED VARIABLES1

ABSTRACT: The usefulness of stochastic models in describing the spatial variability of hydrogeologic quantities, such as permeability, storativity, piezometric head, seepage velocity, and solute concentrations is now widely recognized. In practice, these quantities are represented as the sum of a well-structured component, or drift, and a more erratic fluctuation component which is described statistically through its covariance function. This paper reviews some of the most recent and most promising methods for the estimation of parameters of these covariances from existing data. They are maximum likelihood, restricted maximum likelihood, minimum-variance unbiased quadratic estimation, and minimum-norm (weighted least squares) estimation. The applicability of such methods to conditional and unconditional probability problems is discussed.     

AN ILLUSTRATION OF MODEL STRUCTURE IDENTIFICATION1

ABSTRACT: The purpose of this article is to discuss the importance of uncertainty analysis in water quality modeling, with an emphasis on the identification of the correct model specification. A wetland phosphorus retention model is used as an example to illustrate the procedure of using a filtering technique for model structure identification. Model structure identification is typically done through model parameter estimation. However, due to many sources of error in both model parameterization and observed variables and data, error-in-variable is often a problem. Therefore, it is not appropriate to use the least squares method for parameter estimation. Two alternative methods for parameter estimation are presented. The first method is the maximum likelihood estimator, which assumes independence of the observed response variable values. In anticipating the possible violation of the independence assumption, a second method, which coupled a maximum likelihood estimator and Kalman filter model, was presented. Furthermore, a Monte Carlo simulation algorithm is presented as a preliminary method for judging whether the model structure is appropriate or not.     

Revisiting a statistical shortcoming when fitting the Langmuir model to sorption data

The Langmuir model is commonly used for describing the sorption behavior of reactive solutes to surfaces and is often fit to sorption data using nonlinear least squares regression. An important assumption of least squares regression is that the predictor variable is error free. In the case of sorption data, this assumption is not valid, and therefore the potential for parameter bias exists. Although alternative regression methods exist that either explicitly account for error in the predictor variable (Model II regression) or minimize the error in the predictor variable, these methods are not commonly used. Therefore, this paper more fully explores the differences in fitted parameters and model fits between these different data fitting methods by fitting P sorption data collected on 26 different soil samples using three different regression methods. For a majority of soils tested in this study, the differences in model fits between the three regression methods were not statistically significant. Statistical differences were observed in over a third of the soils, however, suggesting that errors in the predictor variable may be large enough to produce biased parameter estimates. These results suggest that multiple regression methods should be used when fitting the Langmuir model to sorption data to better assess the potential impact of error on model fits.     

PREDICTING FECAL COLIFORM BACTERIA LEVELS IN THE CHARLES RIVER,MASSACHUSETTS, USA1

In Massachusetts, the Charles River Watershed Association conducts a regular water quality monitoring and public notification program in the Charles River Basin during the recreational season to inform users of the river's health. This program has relied on laboratory analyses of river samples for fecal coliform bacteria levels, however, results are not available until at least 24 hours after sampling. To avoid the need for laboratory analyses, ordinary least squares (OLS) and logistic regression models were developed to predict fecal coliform bacteria concentrations and the probabilities of exceeding the Massachusetts secondary contact recreation standard for bacteria based on meteorological conditions and streamflow. The OLS models resulted in adjusted R²s ranging from 50 to 60 percent. An uncertainty analysis reveals that of the total variability of fecal coliform bacteria concentrations, 45 percent is explained by the OLS regression model, 15 percent is explained by both measurement and space sampling error, and 40 percent is explained by time sampling error. Higher accuracy in future bacteria forecasting models would likely result from reductions in laboratory measurement errors and improved sampling designs.     

WATER QUALITY MODEL CALIBRATION: A COMPARISON OF INPUT AND OUTPUT ERROR CRITERIA1

ABSTRACT: A common framework for the analysis of water resources systems is the input-parameter-output representation. The system, described by its parameters, is driven by inputs and responds with outputs. To calibrate (estimate the parameters) models of these systems requires data on both inputs and outputs, both of which are subject to random errors. When one is uncertain as to whether the predominant source of error is associated with inputs or outputs, uncertainty also exists as to the correct specification of a calibration criterion. This paper develops and analyzes two alternative least squares criteria for calibrating a numerical water quality model. The first criterion assumes that errors are associated with inputs while the second assumes output errors. Statistical properties of the resulting estimators are examined under conditions of pure input or output error and mixed error conditions from a theoretical perspective and then using simulated results from a series of Monte Carlo experiments.     

SIMULTANEOUS EQUATION SYSTEMS: A CONSISTENT ESTIMATOR FOR UNKNOWN PARAMETERS IN CONFINED AQUIFERS1

ABSTRACT: This paper presents criteria for establishing the identification status of the inverse problem for confined aquifer flow. Three linear estimation methods (ordinary least squares, two-stage least squares, and three-stage least squares) and one nonlinear method (maximum likelihood) are used to estimate the matrices of parameters embedded in the partial differential equation characterizing confined flow. Computational experience indicates several advantages of maximum likelihood over the linear methods.     

GENERALIZED LOW-FLOW FREQUENCY RELATIONSHIPS FOR UNGAGED SITES IN MASSACHUSETTS1

ABSTRACT: Regional hydrologic procedures such as generalized least squares regression and streamflow record augmentation have been advocated for obtaining estimates of both flood-flow and low-flow statistics at ungaged sites. While such procedures are extremely useful in regional flood-flow studies, no evaluation of their merit in regional low-flow estimation has been made using actual streamflow data. This study develops generalized regional regression equations for estimating the d-day, T-year low-flow discharge, Q_{d, t}, at ungaged sites in Massachusetts where d = 3, 7, 14, and 30 days. A two-parameter lognormal distribution is fit to sequences of annual minimum d-day low-flows and the estimated parameters of the lognormal distribution are then related to two drainage basin characteristics: drainage area and relief. The resulting models are general, simple to use, and about as precise as most previous models that only provide estimates of a single statistic such as Q_7,10. Comparisons are provided of the impact of using ordinary least squares regression, generalized least squares regression, and streamflow record augmentation procedures to fit regional low-flow frequency models in Massachusetts.     

DERIVATION OF THE GAMMA DISTRIBUTION BY USING THE PRINCIPLE OF MAXIMUM ENTROPY (POME)1

The principle of maximum entropy (POME) was used to derive the two-parameter gamma distribution used frequently in synthesis of instantaneous or finite-period unit hydrographs. The POME yielded the minimally prejudiced gamma distribution by maximizing the entropy subject to two appropriate constraints which were the mean of real values and the mean of the logarithms of real values of the variable. It provided a unique method for parameter estimation. Experimental data were used to compare this method with the methods of moments, cumulants, maximum likelihood estimation, and least squares.     

ESTIMATING PARAMETERS OF EV1 DISTRIBUTION FOR FLOOD FREQUENCY ANALYSIS1

ABSTRACT: The parameters of the extreme value type 1 distribution were estimated for 55 annual flood data sets by seven methods. These are the methods of (1) moments, (2) probability weighted moments, (3) mixed moments, (4) maximum likelihood estimation, (5) incomplete means, (6) principle of maximum entropy, and (7) least squares. The method of maximum likelihood estimation was found to be the best and the method of incomplete means the worst. The differences between the methods of principle of maximum entropy, probability weighted moments, moments, and least squares were only minor. The difference between these methods and the method of maximum likelihood was not pronounced.     

Measuring Plant Cover in Sagebrush Steppe Rangelands: A Comparison of Methods

Methods that are more cost-effective and objective are needed to detect important vegetation change within acceptable error rates. The objective of this research was to compare visual estimation to three new methods for determining vegetation cover in the sagebrush steppe. Fourteen management units at the US Sheep Experiment Station were identified for study. In each unit, 20 data collection points were selected for measuring plant cover using visual estimation, laser-point frame (LPF), 2 m above-ground-level (AGL) digital imagery, and 100-m AGL digital imagery. In 11 of 14 management units, determinations of vegetation cover differed (P < 0.05). However, when combined, overall determinations of vegetation cover did not differ. Standard deviation, corrected sums of squares, coefficient of variation, and standard error for the 100 m AGL method were half as large as for the LPF and less than the 2-m AGL and visual estimate. For the purpose of measuring plant cover, all three new methods are as good as or better than visual estimation for speed, standard deviation, and cost. The acquisition of a permanent image of a location is an important advantage of the 2 and 100 m AGL methods because vegetation can be reanalyzed using improved software or to answer different questions, and changes in vegetation over time can be more accurately determined. The reduction in cost per sample, the increased speed of sampling, and the smaller standard deviation associated with the 100-m AGL digital imagery are compelling arguments for adopting this vegetation sampling method.     

OPTIMAL IDENTIFICATION OF MUSKINGUM ROUTING COEFFICIENTS1

Traditionally, identification of the Muskingum routing coefficients has been based on observations of the linearity of a loop formed by graphically plotting a forward and a reverse path. This graphical procedure is time-consuming and may not minimize the error of estimation. A procedure was developed to improve the drawbacks of the graphical method. This procedure calls for (a) the use of least square regression on the forward and reverse paths to determine their respective slopes, and (b) the use of statistical t-test to evaluate the hypothesis that these two slopes are equal. The computational procedure is repeated, using incremental values of the flow weighting coefficient, x. A graph of the computed t-value versus x can be constructed. The optimal value of x, as read from the graph, occurs at the minimum computed t-value. The procedure has been demonstrated superior to the graphical method for three illustrative examples, resulting in a reduction of the error squares by factors ranging from 5 to 6.     

A study of total beta specification through symmetric regression: the case of the Johannesburg Stock Exchange

A topic of recent interest is risk management in equity investments from emerging markets. One traditional measure for systematic risk of an asset is beta, which is constructed through ordinary least squares (OLS) regression between historical returns on an individual asset and an index representing the overall market. OLS regression assumes all the error lies within the asset returns. Tofallis (Eur J Oper Res 187(3):1358–1367, 2008) made the case for constructing a systematic risk measure through symmetric regression, where error is assumed to be present in the returns of both the asset and the index. In this paper, we construct a systematic risk measure using symmetric regression for the case of the Johannesburg Stock Exchange (JSE). This paper makes the case that the so-called ‘total beta’ parameter provides a more realistic and stable estimator for market-related risk and return. The total beta estimate, explicitly allowing for error in both variables, is less likely to underestimate the magnitude of the beta parameter.     

Evaluation of various global solar radiation models for Nigeria

This study assessed the performance of six solar radiation models. The objective was to determine the most accurate model for estimating global solar radiation on a horizontal surface in Nigeria. Twenty-two years meteorological data sets collected from the Nigerian Meteorological agency and the National Aeronautics and Space Administration for the three regions, covering the entire climatic zones in Nigeria were utilized for calibrating and validating the selected models for Nigeria. The accuracy and applicability of various models were determined for three locations (Abuja, Benin City, and Sokoto), which spread across Nigeria using seven viable statistical indices. This study found that the estimation results of considered models are statistically significant at the 95% confidence level, but their accuracy varies from one location to another. However, the multivariable regression relationship deduced in terms of sunshine ratio, air temperature ratio, maximum air temperature, and cloudiness performs better than other relationships. The multivariable relationship has the least root mean square error and mean absolute bias error, not exceeding 1.0854 and 0.8160 MJ m^?2 day^?1, respectively, and monthly relative percentage error in the range of ± 12% for the study areas.     

DYNAMIC MODELING OF STREAM QUALITY BY INVARIANT IMBEDDING1

ABSTRACT. The estimator equations obtained using invariant imbedding is used to estimate the parameters in river or stream pollution. By using these equations, the parameters can be estimated directly from differential equations representing the pollution model and from measured noisy data such as BOD and DO. Another advantage of this approach is that a sequential estimation scheme is obtained. By using this sequential scheme, only current data are needed to estimate current or future values of the unknown parameters. Consequently, a large amount of computer time and computer memory can be saved. Furthermore, not only the parameters but also the concentrations of pollutants can be estimated. Thus, it also forms an effective forecasting technique. The classical least squares criterion is used in the estimation. Several examples are solved to illustrate the technique. (KEY WORDS: dynamic modeling; water pollution; invariant imbedding; forecasting; least squares criterion; estimation)     

Another outlook into energy-growth nexus in Mexico for sustainable development: Accounting for the combined impact of urbanization and trade openness

This study corroborates the importance of United Nations Sustainable Development Goal 7 (SDG-7), intended to ensure access to affordable, reliable, sustainable energy for all, and SDG-8, designed to promote decent work and sustainable economic growth. This article is motivated by the highlighted SDGs and empirically explores the long-run and causality relationship between energy consumption, urbanization, trade openness, and economic growth for annual frequency data from 1965 to 2021 for the case of Mexico. To this end, we leverage the use of fully modified ordinary least squares, dynamic ordinary least squares, and canonical regression estimation methods, while for the direction of causality, the gradual shift and wavelet coherence methods are used. According to the Autoregressive distributed lag (ARDL), the bounds test traces a long-run relationship between the outlined variables over the sampled period. Empirical evidence validates the energy-induced growth hypothesis. This result resonates with the causality analysis, where energy consumption drives economic growth one way in Mexico. This suggests that Mexico cannot embark on energy-conservative policies, as such actions will hurt economic progress. In addition, unidirectional causality is seen between urbanization, trade openness, and economic growth. These findings have far-reaching implications for economic growth and macroeconomic indicators in Mexico. More insights are highlighted in the concluding section.     

Predicted uranium and radon concentrations in New Hampshire (USA) groundwater—Using Multi Order Hydrologic Position as predictors

Two radioactive elements, uranium (U) and radon (Rn), which are of potential concern in New Hampshire (NH) groundwater, are investigated. Exceedance probability maps are tools to highlight locations where the concentrations of undesirable substances in the groundwater may be elevated. Two forms of statistical analysis are used to create exceedance probability maps for U and Rn in NH groundwater. The first, Boosted Regression Tree (BRT), was selected for estimating U exceedance values. It computes exceedance values directly using the Bernoulli distribution function. The second method of statistical analysis used for Rn to determine exceedance probabilities is ordinary least squares (OLS) regression. In the process of determining exceedance probabilities for U and Rn, the utility of a new dataset is investigated. That new predictor dataset is the Multi-Order Hydrologic Position (MOHP) dataset. MOHP raster datasets have been produced nationally for the conterminous United States at a 30-m resolution. The concept behind MOHP is that, for any given point on the earth's surface, there is the potential for a longer groundwater flow path as one goes deeper beneath the land surface. MOHP predictors were tested in both models. Three MOHP predictors were found useful in the BRT model and two in the OLS model. MOHP data were found useful as predictors along with other site characteristics in predicting U and Rn exceedance probabilities in New Hampshire groundwater.     

A measurement of community disaster resilience in Korea

Building a community that is resilient to disasters has become one of the main goals of disaster management. Communities that are more disaster resilient often experience less impact from the disaster and reduced recovery periods afterwards. This study develops a methodology for constructing a set of indicators measuring Community Disaster Resilience Index (CDRI) in terms of human, social, economic, environmental, and institutional factors. In this study, the degree of community resilience to natural disasters was measured for 229 local municipalities in Korea, followed by an examination of the relationship between the aggregated CDRI and disaster losses, using an ordinary least squares (OLS) regression method and a geographically weighted regression (GWR) method. Identifying the extent of community resilience to natural disasters would provide emergency managers and decision-makers with strategic directions for improving local communities' resilience to natural disasters while reducing the negative impacts of disasters.     

UNIT HYDROGRAPHS – A COMPARATIVE STUDY1

ABSTRACT. Unit hydrographs derived by using two methods, linear programming and least squares, are compared. Test data comprise rainfall and runoff information from four storms over the North Branch Potomac River near Cumberland, Maryland. The mathematical bases of these methods for unit-hydrograph derivation are explained. The linear programming method minimizes the sum of absolute deviations, and the least squares method minimizes the sum of the squares of deviations. Computer subroutines are readily available for application of these methods. The unit hydrographs derived with the two methods are practically the same for storms 2 and 3, but differ somewhat for storms 1 and 4. However, the reconstituted direct surface runoff hydrographs are similar to those observed with the exception of the hydrograph for storm 4 which had a relatively more non-uniform rainfall excess of a considerably larger duration.