In a Multiple linear regression model, there are instances where one has to update the regression parameters. In such models as new data become available, by adding one row to the design matrix, the least-squares estimates for the parameters must be updated to reflect the impact of the new data. We will modify two existing methods of calculating regression coefficients in Multiple linear regression to make the computations more efficient. By resorting to an initial solution, we first employ the Sherman-Morrison formula to update the inverse of the transpose of the design matrix multiplied by the design matrix. We then modify the calculation of the product of the transpose of design matrix and the design matrix by the Cholesky decomposition method to solve the system. Finally, we compare these two modifications by several appropriate examples.

In this paper, we consider the issue of data classification in which the response (dependent) variable is two (or multi) valued and the predictor (independent) variables are ordinary variables. The errors could be nonprecise and random. In this case, the response variable is also a fuzzy random variable. Based on this and logistic regression, we formulate a model and find the estimation of the coefficients using the least squares method. We will describe the results with an example of one independent random variable. Finally, we provide recurrence relations for the estimation of parameters. This relation can be used in machine learning and big data classification.

Evaluating the cutting rate (CR) of stones is important in the cost estimation and the planning of the stone processing plants. This research used regression models to estimate the stones’ CR based on their physico-mechanical characteristics. Stone processing factories in Mahallat City (Markazi province, Iran) were visited, and the CR of diamond circular saws was recorded on six different travertine stones. Next, the stone block samples were collected from the quarries for laboratory tests. Stones’ porosity (n), uniaxial compressive strength (UCS), and Schmidt hammer hardness (SH) were determined in the laboratory as their physico-mechanical characteristics. Correlation relationships of CR with physico-mechanical characteristics were evaluated using simple and Multiple regression analyses, and estimator models were developed. Results showed that Multiple regression models are more reliable than simple regression for estimating the stones’ CR. The validity of the developed Multiple regression models was verified with the published data of one researcher. The findings indicated that these models are accurate enough for estimating the CR of stones. Consequently, the Multiple regression models provide practical advantages for estimating the CR and save time and cost during the planning and design of the stone processing factories.

In this study, two pairs of OEM brake disc-pads have been used. One of these discs belongs to a passenger car, and the other one belongs to a light commercial vehicle. The disc-pad pair of the passenger car (PC) has been subjected to global brake effectiveness test by full scale inertia dynamometer according to SAE J2522 test standard; and the other one has been subjected to the tests by full-scale inertia dynamometer according to FIAT 7-H4020 and 7-H2000 standards. During these tests, 13 variables for passenger car disc-pad pair and 11 variables for light commercial vehicle disc-pad pair have been measured and recorded. Interrelation of the parameters has been analyzed with Multiple regression method and importance levels have been determined. In this study, dependent variables in Multiple regression method are selected as braking time, friction coefficient, disc final temperature, brake speed and brake pressure for each braking pair. In Multiple regression analysis for PC, for each unit increase in deceleration and friction coefficient, braking time decreases with 7. 3 and 60. 9 units, respectively. Also, for each unit increase in brake pressure and friction coefficient for LCV, braking time increases with 1. 267 and 91. 887 units, respectively.

A novel approach to the problem of regression modeling for fuzzy input-output data is introduced. In order to estimate the parameters of the model, a distance on the space of interval-valued quantities is employed. By minimizing the sum of squared errors, a class of regression models is derived based on the interval-valued data obtained from the a-level sets of fuzzy input-output data. Then, by integrating the obtained parameters of the interval-valued regression models, the optimal values of parameters for the main fuzzy regression model are estimated. Numerical examples and comparison studies are given to clarify the proposed procedure, and to show the performance of the proposed procedure with respect to some common methods.

Identification of regions having potential for landslide occurrence is one of the basic measures in natural resources management. Different landslide hazard zonation models are proposed based on the environmental condition and goals. In this research landslide hazard zonation map using Multiple regression model was provided and applicability of this model is investigated in Bagh Dasht watershed. Dependent variable is landslide inventory map and independent variables consist of information layers as Geology, slope, aspect, distance from river, distance from road, distance from fault and land use. For doing this, existing landslides have been identified and an inventory map made. The landslide hazard zonation map is based on the Multiple regression provided. The level of similarity potential hazard classes and figures of this model were compared with the landslide inventory map in the SPSS environments. Results of research showed that there is a significant correlation between the potential hazard classes and figures with the area of landslides.The Multiple regression model is suitable for application in the Bagh Dasht Watershed.

Aims Karun River, which is the largest river in Iran, represents a unique ecosystem. However, increased anthropogenic activities result in the formation of this river is seriously affected by a large range of pollutants especially the heavy metal pollutants which may be toxic to human and aquatic fauna. Therefore, there is a need for continuous monitoring of pollution levels in the river. Materials & Methods In this study, water, sediment, and algae samples were collected from six different stations along the course of the river in September 2015 to investigate the quality of Karun’ s River in terms of heavy metals (Pb, Zn, Cr, and Cd) at the basin of drinking water treatment in Ahwaz and Mollasani cities. After drying and digestion of samples, heavy metal concentrations were determined using an atomic absorption spectrophotometry (Perkin Elmer-Analyst 300). Findings The highest concentration of trace metals was found in sediment samples with Zn having the highest mean concentration values in all stations. The heavy metal concentrations in the downstream indicated an increase in the pollution load due to the flow of water from upstream to downstream of the river resulted in the movement and accumulation of all contaminants to the river in the downstream; hence, there was the highest concentration of metals in basin of the Kut Abdollah treatment (downstream) and the lowest in Mollasani (upstream). Conclusion Comparison of the concentration of metals in the sediments with some universal standards including EPA3050 and the criterion of sediments quality standard from NOAA and Canadian Environment Agency showed that the concentration of chromium and cadmium in stations was higher than the allowable limit of EPA3050 standards and some environmental standards of Canada among all metals. Since algae samples have been able to accumulate a significant amount of heavy metals, therefore, these are suitable bio-indicators to determine the concentration of heavy metals in this aquatic ecosystems.