Content
- Characteristics of Spatial Interaction Data
- Spatial Interaction Models
- Unconstrained
- Origin constrined
- Destination constrained
- Doubly constrained
Lesson 3:
Spatial Interaction Models
Dr. Kam Tin Seong
Assoc. Professor of Information Systems(Practice)
Assoc. Professor of Information Systems(Practice)
School of Computing and Information Systems,
Singapore Management University
30 Nov 2023
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:
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.
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):
- 𝛵𝑖𝑗 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,
- 𝑣𝑖 represents the origin propulsiveness variable(s).
- 𝓌𝑗 represents the destination attractiness variable(s).
- 𝒹𝑖𝑗 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.
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