# Multivariate Regression Modeling for Coastal Urban Air Quality Estimates
## 3 Results
### Definition of Variables in the Multivariate Regression Model
\begin{array}{l}Y_{1}={\rm a}_{11}X_{1}+{\rm a}_{12}X_{2}+{\rm a}_{13}X_{3}+{ \rm a}_{14}X_{4}+{\rm a}_{15}X_{5}+{\rm a}_{16}X_{6}+{\rm a}_{17}X_{7}+{\rm a}_{18 }X_{8}+{\rm a}_{19}X_{9}\\ +{\rm a}_{110}X_{10}+{\rm a}_{111}X_{11}+{\rm a}_{112}X_{12}+{\rm a}_{113}X_{13}+ {\rm a}_{114}X_{14}+{\rm a}_{115}X_{15}+{\rm b}_{1}\\ Y_{2}={\rm a}_{21}X_{1}+{\rm a}_{22}X_{2}+{\rm a}_{23}X_{3}+{\rm a}_{24}X_{4}+{ \rm a}_{25}X_{5}+{\rm a}_{26}X_{6}+{\rm a}_{27}X_{7}+{\rm a}_{28}X_{8}+{\rm a}_{2 9}X_{9}\\ +{\rm a}_{210}X_{10}+{\rm a}_{211}X_{11}+{\rm a}_{212}X_{12}+{\rm a}_{213}X_{13}+ {\rm a}_{214}X_{14}+{\rm a}_{215}X_{15}+{\rm b}_{2}\\ Y_{3}={\rm a}_{31}X_{1}+{\rm a}_{32}X_{2}+{\rm a}_{33}X_{3}+{\rm a}_{34}X_{4}+{ \rm a}_{35}X_{5}+{\rm a}_{36}X_{6}+{\rm a}_{37}X_{7}+{\rm a}_{38}X_{8}+{\rm a}_{3 9}X_{9}\\ +{\rm a}_{310}X_{10}+{\rm a}_{311}X_{11}+{\rm a}_{312}X_{12}+{\rm a}_{313}X_{13} +{\rm a}_{34}X_{14}+{\rm a}_{315}X_{15}+{\rm b}_{3}\\ Y_{4}={\rm a}_{4}X_{1}+{\rm a}_{12}X_{2}+{\rm a}_{13}X_{3}+{\rm a}_{14}X_{4}+{ \rm a}_{15}X_{5}+{\rm a}_{16}X_{7}+{\rm a}_{18}X_{8}+{\rm a}_{19}X_{9}\\ +{\rm a}_{100}X_{10}+{\rm a}_{111}X_{11}+{\rm a}_{112}X_{12}+{\rm a}_{113}X_{1 3}+{\rm a}_{114}X_{14}+{\rm a}_{115}X_{15}+{\rm b}_{1}\\ Y_{5}={\rm a}_{51}X_{1}+{\rm a}_{22}X_{2}+{\rm a}_{23}X_{3}+{\rm a}_{24}X_{4}+ {\rm a}_{25}X_{5}+{\rm a}_{26}X_{6}+{\rm a}_{27}X_{7}+{\rm a}_{28}X_{8}+{\rm a }_{29}X_{9}\\ +{\rm a}_{210}X_{10}+{\rm a}_{211}X_{11}+{\rm a}_{212}X_{12}+{\rm a}_{213}X_{1 3}+{\rm a}_{214}X_{14}+{\rm a}_{215}X_{15}+{\rm b}_{2}\\ Y_{6}={\rm a}_{61}X_{1}+{\rm a}_{32}X_{2}+{\rm a}_{33}X_{3}+{\rm a}_{34}X_{4}+ {\rm a}_{35}X_{5}+{\rm a}_{36}X_{6}+{\rm a}_{37}X_{7}+{\rm a}_{38}X_{8}+{\rm a }_{39}X_{9}\\ +{\rm a}_{310}X_{10}+{\rm a}_{311}X_{11}+{\rm a}_{312}X_{12}+{\rm a}_{313}X_{1 3}+{\rm a}_{314}X_{14}+{\rm a}_{315}X_{15}+{\rm b}_{3}\\ \end{array}  \tag{1}\]  
Equation (1) can be rewritten in a matrix form as below. \[ \begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\\ X_{4}\\ X_{5}\\ X_{6}\\ X_{7}\\ X_{8}\\ X_{9}\\ X_{10}\\ X_{11}\\ X_{12}\\ X_{13}\\ X_{14}\\ X_{15}\end{bmatrix} = \begin{bmatrix}{\rm a}_{11}&{\rm a}_{12}&\ldots&\ldots& \ldots&\ldots&{\rm a}_{115}\\ {\rm a}_{21}&{\rm a}_{22}&\ldots&\ldots&\ldots&{\rm a}_{215}\\ {\rm a}_{31}&{\rm a}_{32}&\ldots&\ldots&{\rm a}_{315}\\ {\rm a}_{41}&{\rm a}_{42}&\ldots&\ldots&\ldots&{\rm a}_{415}\\ {\rm a}_{51}&{\rm a}_{52}&\ldots&\ldots&{\rm a}_{515}\\ {\rm a}_{61}&{\rm a}_{62}&\ldots&\ldots&\ldots&{\rm a}_{615}\\ \end{bmatrix} + \begin{bmatrix}{\rm b}_{1}\\ {\rm b}_{2}\\ {\rm b}_{3}\\ {\rm b}_{4}\\ {\rm b}_{5}\\ {\rm b}_{6}\end{bmatrix}  \tag{2}\]  
where \({\rm a}_{ij}\) (i = 1 to 6; \(j\) = 1 to 15) indicates the coefficient of the matrix and the measured values of \(X_{\rm j}\) (j; represents the number of independent variables; j = 1 to 15) such as PM\({}_{10}\)_K_, PM\({}_{2.5}\)_K_, TEMP_K (air temperature), WIND_K (wind speed), RH_K (relative humidity), SO\({}_{2}\)_K_, CO_K, O\({}_{3}\)_K_, NO\({}_{2}\)_K_, PM\({}_{10}\)_C_, PM\({}_{2.5}\)_C_, SO\({}_{2}\)_C_, CO_C, O\({}_{3}\)_C and NO\({}_{2}\)_C_. The real-time predictive values \(Y_{\rm i}\) (i = 1 to 6; PM\({}_{10}\)_K_(N), PM\({}_{2.5}\)_K_(N), SO\({}_{2}\)_K_(N), CO\({}_{K}\)(N), O\({}_{3}\)_K_(N) and NO\({}_{2}\)_K_(N)) are dependent variables and b\({}_{\rm i}\) (i = 1 to 6) is an error term that intercepts in the equations of \(Y_{\rm i}\). In this study, our prediction models comprise multiple predictive regression equations on each PM\({}_{10}\)_K_, PM\({}_{2.5}\)_K_, SO\({}_{2}\)_K_, CO\({}_{K}\), O\({}_{3}\)_K_ and NO\({}_{2}\)_K value that are influenced by not only meteorological and gas parameters of the city but also PM\({}_{10}\)_C_, PM\({}_{2.5}\)_C_, SO\({}_{2}\)_C_, CO\({}_{C}\), O\({}_{3}\)_C and NO\({}_{2}\)_C concentrations. Tables 2-4 shows multiple correlation coefficients and predictive regression equations on each PM and gas concentration at Kangnung before, during and after the yellow sand periods in three classifications through multiple regression analyses using SPSS statistics software v27.