The AQ Algorithm For The Solution Of A Sparse System

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.