Lesson 3:
Spatial Interaction Models

Dr. Kam Tin Seong
Assoc. Professor of Information Systems(Practice)

School of Computing and Information Systems,
Singapore Management University

30 Nov 2023

Content

• Characteristics of Spatial Interaction Data
• Spatial Interaction Models
  • Unconstrained
  • Origin constrined
  • Destination constrained
  • Doubly constrained

What Spatial Interaction are?

Spatial interaction describe quatitatively the flow of people, material, or information between locations in geographical space.

Conditions for Spatial Flows

• Three interdependent conditions are necessary for a spatial interaction to occur:

• Complementarity. There must be a supply and a demand between the interacting locations. A residential zone is complementary to an employment zone because the first is supplying workers while the second is supplying jobs. The same can be said concerning the complementarity between a store and its customers and between an industry and its suppliers (movements of freight). An economic system is based on a large array of complementary activities.
• Intervening opportunity (lack of). Refers to a location that may offer a better alternative as a point of origin or as a point of destination. For instance, in order to have an interaction of a customer to a store, there must not be a closer store that offers a similar array of goods. Otherwise, the customer will likely patronize the closer store and the initial interaction will not take place.
• Transferability. Mobility must be supported by transport infrastructures, implying that the origin and the destination must be linked. Costs to overcome distance must not be higher than the benefits of the related interaction, even if there are complementarity and no alternative opportunity.

Representation of a Movement as a Spatial Interaction

Representing mobility as a spatial interaction involves several considerations:

• Locations: A movement is occurring between a location of origin and a location of destination. i generally denotes an origin while j is a destination.
• Centroid: An abstraction of the attributes of a zone at a point.
• Flows: Flows are generally expressed by a valued vector Tij representing an interaction between locations i and j.
• Vectors: A vector Tij links two centroids and has a value assigned to it (50) which can represents movements.

• Locations. A movement is occurring between a location of origin and a location of destination. i generally denotes an origin while j is a destination. The representation of origins and destinations commonly involves centroids.
• Centroid. An abstraction of the attributes of a zone at a point. This is of particular relevance when the attributes generating mobility are zonal (e.g. ZIP codes, cities, states, etc.) while the graphic representation requires specific origins and destinations. For instance, showing flows between ZIP codes would implicitly require the generation of one centroid for each ZIP code.
• Flows. Flows are generally expressed by a valued vector Tij representing an interaction between locations i and j.
• Vectors. On the above figure, two areas, zone i and zone j, are represented as two centroids, i and j. A vector Tij links two centroids and has a value assigned to it (50) which can represents movements such as tons of freight, numbers of passengers per day, or number of phone calls.

Constructing an O/D Matrix

• The construction of an origin / destination matrix requires directional flow information between a series of locations.
• Figure below represents movements (O/D pairs) between five locations (A, B, C, D and E). From this graph, an O/D matrix can be built where each O/D pair becomes a cell. A value of 0 is assigned for each O/D pair that does not have an observed flow.

Three Basic Types of Interaction Models

Gravity Models

The general formula (also known as unconstrained):

• 𝛵_𝑖𝑗 is the transition/trip or flow, 𝑇, between origin 𝑖 (always the rows in a matrix) and destination 𝑗 (always the columns in a matrix). If you are not overly familiar with matrix notation, the 𝑖 and 𝑗 are just generic indexes to allow us to refer to any cell in the matrix.
• 𝑉 is a vector (a 1 dimensional matrix – or, if you like, a single line of numbers) of origin attributes which relate to the emissivity of all origins in the dataset, 𝑖 – this could be any of the origin-related variables.

• 𝑊 is a vector of destination attributes relating to the attractiveness of all destinations in the dataset, 𝑗 – similarly, this could be any of the destination related variables.
• 𝑑 is a matrix of costs (frequently distances – hence, d) relating to the flows between 𝑖 and 𝑗.
• 𝑘, 𝜆, 𝛼 and 𝛽 are all model parameters to be estimated. 𝛽 is assumed to be negative, as with an increase in cost/distance we would expect interaction to decrease.

Note

Spatial interaction models seek to explain existing spatial flows. As such it is possible to measure flows and predict the consequences of changes in the conditions generating them. When such attributes are known, it is possible to better allocate transport resources such as conveyances, infrastructure, and terminals.

Effects of beta, alpha and lambda on Spatial Interactions

Variations of the beta, alpha, and lambda exponents have different impacts on the level of spatial interactions.

A Family of Gravity Models

• Unconstrained
• Origin constrined
• Destination constrained
• Doubly constrained

The Unconstrained (Totally constrained) Model

The formula:

where

In the Unconstrined Model,

• 𝑣_𝑖 represents the origin propulsiveness variable(s).
• 𝓌_𝑗 represents the destination attractiness variable(s).
• 𝒹_𝑖𝑗 represents distance.
• 𝑘, the scale parameter.
• 𝜆, 𝛼, and 𝛽 represent exponents or parameters to be estimated for the origin propulsiveness variable(s), the destination attractiveness variable(s) and the distance respectively.

Unconstrained (Totally constrained) case

The O-D Matrix

and distance matrix:

The estimated O-D matrix:

and the calculation T11

The Origin (Production) Constrained Model

The formula:

where

and

In the Origin Constrained Model,

• 𝑂𝑖 does not have a parameter as it is a known constraint.
• 𝐴𝑖 is known as a balancing factor and is a vector of values which relate to each origin 𝑖 which do the equivalent job as 𝑘 in the unconstrained/total constrained model but ensure that flow estimates from each origin sum to the know totals 𝑂𝑖 rather than just the overall total.

Origin (Production) constrained case

The O-D Matrix

and distance matrix:

The estimated O-D matrix:

A1 is calculated as shown below:

Hence, T11 is calculated as shown below:

The Destination (Attraction) Constrained Model

The formula:

where

and

In the Destination Constrained Model,

• Dj does not have a parameter as it is a known constraint.
• Bj is known as a balancing factor and is a vector of values which relate to each destination j which do the equivalent job as 𝑘 in the unconstrained/total constrained model but ensure that flow estimates from each origin sum to the know totals Dj rather than just the overall total.

Destination (Attraction) constrained case

The O-D Matrix

and distance matrix:

The estimated O-D matrix:

B1 is calculated as shown below:

Hence, T11 is calculated as shown below:

The Doubly Constrained Model

where

and

Note

Note that the calculation of 𝐴𝑖 relies on knowing 𝐵𝑗 and the calculation of 𝐵𝑗 relies on knowing 𝐴𝑖 – something of a conundrum to which the solution is elegantly described by Senior (1979), who sketches out a very useful algorithm for iteratively arriving at values for 𝐴𝑖 and 𝐵𝑗 by setting each to equal 1 initially and then continuing to calculate each in turn until the difference between successive iterations of the 𝐴𝑖 and 𝐵𝑗 values is small enough not to matter.

Doubly constrained case

The O-D Matrix

and distance matrix:

The estimated O-D matrix:

Hence, T11 is calculated as shown below:

Notice that A1 and B1 are computed by using computer.