Question:
You are hired by the city planning department to calibrate a multiple regression model for trip productions. The department has collected base-year data for the following variables:
P = trip productions
X1= zonal population
X2 = median income
X3 = median age
X4 = car registrations
X5 = number of dwelling units
A preliminary analysis of the data resulted in the following correlation matrix:
P X1 X2 X3 X4 X5
P 1.00 0.95 0.83 0.41 0.82 0.85
X1 1.00 -0.21 0.22 -0.29 0.91
X2 1.00 0.82 0.89 -0.43
X3 1.00 -0.19 -0.15
X4 1.00 -0.22
X5 1.00
Specify at least three possible equations that may be tried and give the specific reasons for their selection.
Answer:
Given that-
P = trip productions
X1= zonal population
X2 = median income
X3 = median age
X4 = car registrations
X5 = number of dwelling units
A preliminary analysis of the data resulted in the following correlation matrix:
P X1 X2 X3 X4 X5
P1.000.950.830.410.82 0.85
X1 1.00-0.210.22-0.290.91
X2 1.000.820.89-0.43
X3 1.00-0.19-0.15
X4 1.00-0.22
X5 1.00
Based on these data following equations can be formed-
X1 + x2 + x3 + x4 + x5 = 1 ………………………………………….i
-0.21x1 + 0.82x2 – 0.19x3 – 0.22x4 = 0.95 …………………..ii
0.22x1 + 0.89x2 -0.15x3 = 0.83 ………………………………….iii
-0.29x1 – 0.43x2 = 0.41 ………………………………………………iv
0.91x1 = 0.82 …………………………………………………………….v
From equation v-
i.e. x1 = 0.90
Substituting the value in equation iv-
-0.29*0.90 – 0.43*x2 = 0.41
i.e. x2 = -1.56
Substituting the values in equation iii gives-
0.22*0.90 + 0.89(-1.56) -0.15x3 = 0.83
i.e. x3 = -13.47
Substituting the values in equation ii gives-
-0.21(0.90) + 0.82(-1.56) – 0.19(-13.47) – 0.22x4 = 0.95
i.e. x4 = 0.644
Substituting all the values in the equation i gives-
x5 = 1 – [ x1 + x2 + x3 + x4 ]
= 1 – [ 0.90 – 1.56 -13.47 + 0.644 ]
= - 12.48
Therefore possibly equation i, ii, iii can be used as for these equation the trip production rate is higher as compared to the equation iv and v.
References: -
BORLAND, R. E. The AQ algorithm for the solution of a sparse system of linear equations In-text: (Borland, 1979) Bibliography: Borland, R. (1979). The AQ algorithm for the solution of a sparse system of linear equations. Teddington: National Physical Laboratory.
CODENOTTI, B. AND LEONCINI, M. Parallel complexity of linear system solution In-text: (Codenotti and Leoncini, 1991) Bibliography: Codenotti, B. and Leoncini, M. (1991). Parallel complexity of linear system solution. Singapore: World Scientific.
