Multiple Linear Regression (MLR) Model: A Tool for Water Quality Interpretation

The lack of standard water analysis equipment as well as inadequate trained personnel especially in the developing countries has discouraged many researchers in such countries to execute water quality researches. Hence, this paper presents developed mathematical relationship among some physicochemical parameters in order to aid the determination of the concentrations of certain parameters with the use of minimal equipment. This was achieved by weekly analyzing 7 physicochemical parameters of two sources of potable water (tap water and borehole water) stored in different containers for a period of 6 weeks using standard methods. The storage containers used were black plastic tank, blue plastic tank, green plastic tank, coated steel metal tank, uncoated steel metal tank and clay pot. The parameters examined were turbidity, electrical conductivity (EC), pH, alkalinity, chloride ion (Cl), dissolved oxygen (DO) and total hardness. Results showed that the relationship between electrical conductivity (EC), alkalinity (Alk), total hardness (TH) and chloride ion (Cl) is given as; EC = -224.8066493 + 6.244028022(Alk) + 0.28204735(TH) + 0.000518108(Cl). A programing language was written on the models using Visual Basic.Net (VB.Net) version 2018.


INTRODUCTION
Due to the presence of impurities in the global water resources, only a very few proportion of the world's available water is potable or fit for human consumption (Andrew, 2014;Nala and Jagals, 2013;and Ochekpe, 2011). Knowing the concentrations of the physicochemical and bacteriological parameters of a given sample of water is very important because, comparing the known concentrations with the permissible limits set by regulatory bodies will help in deciding whether such water is potable or not. However, the exact concentration of water quality parameters can only be known by thorough analysis of the water with standard equipment as well as reagents, which could be expensive, time consuming and risky. Hence, it is necessary to develop a means of reducing the time usually spent in analyzing water quality as well as the cost incurred. This could be achieved by building an equation (mathematical model) out of previous studies that will act as a representation of reality without interfering directly with the quality of the result. A mathematical model will actually reduce the risk, cost and also the time spent in analyzing the quality of water samples. A well written programing language of the model will further reduce the time spent when installed in a computer.
Many researchers have used MLR model successfully to determine factors influencing groundwater flow (e.g. Yan et al., 2018;Zomlot et al., 2015;Sahoo and Jha, 2013) and for checking water quality (Mustafa and Fuaad, 2019;Salam et al., 2018;Magda and Marco, 2018;Chen and Liu, 2015;Ahamad et al., 2015;Narayanan et al., 2015). However, the efficacy of this model for potable water quality is not yet studied. Hence, this paper presents the results of the investigation carried out to check the possibility of using MLR model in understanding the relationship between the quality parameters of potable water.
Where is the ith observation of the dependent variable and , is the ith observation of the jth independent variable.
The concept of sum of squares of errors was applied on equation (2) to yield equation (3).
Thereafter, the associated error e was minimized by partial differentiation of equation (3) with respect to 0 , 1 , 2 ,…, , and equating the values to zero. The resulted equations were transformed into matrix form as shown in equation (4) Equation (4)

RESULTS AND DISCUSSION
The laboratory results of the 7 physicochemical parameters monitored during the six (6) weeks retention period for both sources of water are shown in table 1.

Regression Analysis
Curves were plotted between the parameters of the results shown in table 1 in a linear graph, and it was observed that the electrical conductivity (EC) was a function of alkalinity (Alk), total hardness (TH) and chloride ion (Cl -) present in the water samples as represented in equation (5).
In order to fit a relationship between the dependent variable (EC) and independent variables (Alk, TH and Cl -), equation (5) was transformed into equation (6) as; Where, 0 , 1 , 2 and 3 are constants.
Assuming the line of best fit in equation (6) is associated with an error e, then by applying the concept of Sum of Squares of Errors (SSE) to equation (6), it resulted to equation (7) as follows.

Trend in Water Quality During Retention Period
The turbidity of both water sources (tap water and borehole water) in all the storage containers during the retention period were within the WHO permissible limit (5.00 NTU) except for water stored in uncoated steel metal tanks (USMt and USMb). Likewise, the pH of all the water samples were within the WHO limit (6.5 -8.5) in the first four weeks of storage thereafter, little deviations from the permissible limit occurred especially for the water samples stored in uncoated steel metal tanks. The electrical conductivity, chloride ion and alkalinity of the water samples throughout the retention period were within the WHO standards (750 µS/cm, 200 mg/l and 250 mg/l respectively) irrespective of the type of water and storage container. Dissolved oxygen (DO) is an important water quality parameter as it is a respiratory gas. The DO content in all the water samples improved (increased) during the first week of storage and afterward, it fluctuated between values higher and lower than the permissible limit (1.5 mg/l). Notwithstanding, the minimum concentration of DO (1.0 mg/l) was recorded on the second week of retention in tap water stored in uncoated steel metal tank (USMt) while the maximum concentration (2.7 mg/l) was noted on the first week in tap water stored in clay pot (CLPt). The variations in hardness (total) level of the water samples during the retention period were not much. Nevertheless, water samples drawn from storage containers filled with borehole water recorded higher concentrations (308.77 mg/l to 661 mg/l) than samples drawn from storage containers filled with tap water (77.44 mg/l to 127.60 mg/l). Water hardness above 200 mg/l may cause scale deposition in treatment works and pipe distribution system as well as excessive soap consumption (Ogbozige et al., 2018b).

CONCLUSION
Based on the results obtained in this research, it could be concluded that the relationship between electrical conductivity (EC), alkalinity (Alk), total hardness (TH) and chloride ion (Cl -) is given as; EC = -224.8066493 + 6.244028022(Alk) + 0.28204735(TH) + 0.000518108(Cl -). Hence, prospective researchers on water quality of the selected sampling points could make use of the model while researchers on water samples drawn from other sources should calibrate the model before making use of it.

ACKNOLEDGEMENTS
The role played by Prof. O.J. Mudiare in writing this paper is well appreciated.

CONFLICT OF INTERESTS
There is not conflict of Interests.