Projection

The functionality is primarily used if origin or destination total values of a zone are to be multiplied by a particular value, or a particular expected value is to be attained, which can be necessary in some circumstances after origin-destination studies. Matrices collected are often just random samples and must be projected to census values.

Matrix values can be projected per row (singly-constrained projection regarding the production), per column (singly-constrained projection regarding the attraction) or by row and column (doubly-constrained projection) (User Manual: Projecting matrix values).

Singly-constrained projection means that each row or column is multiplied by a fixed value. This value can be a procedure parameter or – for zone and main zone matrices – an attribute of the zone or main zone. The complexity of doubly-constrained projection is illustrated in the example below.

Objective: projection of origin and destination demand as follows:

  • zone 1 by 10 %
  • zone 2 by 20 %

Table 73: Basic matrix

Zone

1

2

Origin demand

1

20

30

50

2

40

50

90

Destination demand

60

80

140

Line by line multiplication, therefore for purely singly-constrained projection of the demand regarding production originating from zone 1 by 10% and zone 2 by 20%, produces the following matrix.

Zone

1

2

Origin demand

1

22

33

55

2

48

60

108

Destination demand

70

93

163

While the origin traffic has been increased correctly, the destination traffic has not.

For the doubly-constrained projection, the Matrix editor uses an iterative process, also called a Multi-procedure. During this iterative procedure, a solution to how the target values are best reached is generated stepwise (The multi-procedure according to Lohse (Schnabel 1980)).

The Matrix Editor thus provides the following solution which correctly projects the origin and destination traffic.

Table 74: Output

Zone

1

2

Origin demand

1

21

34

55

2

45

62

107

Destination demand

66

96

162

The multi-procedure according to Lohse (Schnabel 1980)

With the multi-procedure new traffic flows are calculated in each iteration step Fij (Schnabel 1980).

The iteration formula applied is as follows

Fij(n+1) = Fij(n) • qi(n) • zj(n) • f(n)

with

Qip

Desired origin traffic zone i

Zjp

Desired destination traffic zone j

Gp

Desired total traffic

Fij(n)

Traffic flow from zone i to zone j in iteration n

Qi(n)

Origin traffic zone i, iteration n

Zj(n)

Destination traffic zone j, iteration n

G(n)

Total traffic, iteration n

This iterative calculation is done repeatedly until the following conditions are met for all boundary values (origin and destination expected values).

for all zones i

for all zones j

The threshold ε suggested by Lohse was used. It states that

or

QF: quality factor