Section 3 - Spatial interaction models - What they are and where they're used - Calibration and "what-if" - Trade area analysis and market penetration:

• The Huff model and variations.

• Site modeling for retail applications - regression, analog, spatial interaction.

• Modeling the impact of changes in a retail system.

• Calibrating spatial interaction models in a GIS environment.

• a model used to explain, understand, predict the level of interaction between different geographic locations

• examples of interactions:
• • migration (number of migrants between pairs of states)
  • phone traffic (number of calls between pairs of cities)
  • commuting (number of vehicles from home to workplace)
  • shopping (number of trips from home to store)
  • recreation (number of campers from home to campsite)
  • trade (amount of goods between pairs of countries)

• interaction is always expressed as a number or quantity per unit of time

• interaction occurs between defined origin and destination
• • these may be the same or different classes of objects
  • e.g. the same class in the case of migration between states
  • e.g. different classes in the case of journeys to shop or work
  • the matrix of interactions can be square or rectangular
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• some measure of the origin (its propensity to generate interaction)

• some measure of the destination (its propensity to attract interaction)

• some measure of the trip (its propensity to deter interaction)

• these measures are assumed to multiply
•  

I_ij = a E_i A_j C_ij
 

• that is, interaction can be predicted from the product of a constant, emissivity, attraction and deterrence

• such ideas of social physics have long since gone out of fashion, but the name is still sometimes used

• even in the form above, the model bears some relationship to Newton's Law of Gravitation
•  

• e.g. the value of a might be unknown in a given application

• its value would be calibrated by finding the value that gives the best fit between the observed interactions and the interactions predicted by the model

• the conventional measure of fit is the total squared difference between observation and prediction, that is, the summation over i and j of (I_ij - I*_ij)²

• this is known as least squares calibration

• other unknowns might be the method of calculating deterrence (C_ij) from distance, or the attraction value to give to certain retail stores

C_ij

• deterrence is often strongly related to distance
• • the further the distance, the less interaction and thus the lower C_ij

• a common choice is a decreasing function of distance:
• C_ij = d_ij^-b            (C_ij = 1 / d_ij^b)
  or C_ij = exp (-bd_ij)            (exp denotes 2.71828 to the power)

• generally the fit of the model is not sufficiently good to distinguish between these two, that is, to identify which gives the better fit

• the negative exponential has a minor technical advantage in not creating problems when d_ij = 0 (origin and destination are the same place)

• the b parameter is unknown and must be calibrated
• • its value depends on the type of interaction, and also probably on the region
  • b has units in the negative exponential case (1/distance) but none in the negative power case

• other measures of deterrence include:
• • some function of transport cost
  • some function of actual travel time
  • in either case the function used is likely to be the negative power or negative exponential above
  • there are examples where distance has a positive effect on interaction

• how to measure the propensity of each origin to emit interaction?

• gross population P_i

• some more appropriate measure weighting each cohort, e.g. age and sex cohorts
• • some cohorts are more likely to interact than others

• gross income

• E_i could be treated as unknown and calibrated
•  

• the propensity of each destination to attract interaction

• could be unknown and calibrated

• for shopping models, gross floor area of retail space is often used

• some forms of interaction are symmetrical
• • flow from origin to destination equals reverse flow
  • e.g. phone calls
  • requires E_i and A_j to be the same, e.g. population

• what happens when a new destination is added?
• • interactions with existing destinations are unaffected
  • assumes outflow from origins can increase without limit
  • in practice, in many applications flow from origin to existing destinations will be diverted
  • we need some form of "production constraint"

•  
• summing interaction to all destinations from a given origin:
• I_ij = E_i

• the Huff model is consistent with the axiom of Independence of Irrelevant Alternatives (IIA)
• • the ratio of flows to two destinations from a given origin is independent of the existence and locations of other destinations

• it is often desirable to predict how much business a new store will draw from existing ones
• • e.g. how much will a new mall draw business away from downtown?

• population of a neighborhood increases by x%

• ethnic mix of a neighborhood changes

• a new bridge is constructed

• an earthquake takes a freeway out of operation

• a mall adds new space

• an anchor store moves out

• a store changes its signage
•  

• three major areas:
• • use of the spatial interaction model
  • analog techniques
  • regression models
  •  

• the business done by a new store or an old store operating under changed circumstances is best estimated by finding the closest analog in the chain

• requires a search

• criteria include:
• • physical characteristics of each store
  • intangibles such as management, signage
  • local market area
  • a GIS can help compare market areas (local densities, street layouts, traffic patterns)
  • a multi-media GIS can help with the intangibles
  • • bring up images of site, layout, signage...

• identify all of the factors affecting sales, and construct a model to predict based on these factors

• an enormous range of factors can affect sales

• some factors are exogenous
• • determined by external, physical, measurable variables
  • some of these travel with the store if it moves (site factors), others are attributes of place (situation factors)

• other factors are endogenous
• • determined by crowding, types of customers, trends, advertizing
  • unpredictable, determined by the state of the system
  •  

• site layout - on a corner? parking spaces, etc.

• inside layout

• trade area - number of households in primary, etc

• characteristics of neighborhood

Sales per 2-week period for convenience store:
 

• use of surrogate variables

• problems in use of model for prediction in planning
•  

• many different circumstances

• major issues involved in calibration

• specific tools are available
• • SIMODEL

• possible to use standard tools in e.g. SAS, GLIM

• calibration possible using aggregate flows or individual choices
•  

• transformations to make the right hand side of the equation a linear combination of unknowns, the left hand side known

• suppose the E_i are known, the A_j unknown
• • the constant a can be absorbed into the A_j (i.e. find aA_j)

• suppose we use the negative power deterrence function
•  I_ij = E_i A_j / d_ij^b

• move the E_i to the left:
• I_ij/E_i = A_j / d_ij^b

• take the logs of both sides:
•  log (I_ij/E_i) = log A_j - b log d_ij

•   now a trick - introduce a set of dummy variables u_ijk, set to 1 if j=k, otherwise zero:

• now the left hand side is all knowns, the right hand side is a linear combination of unknowns (the logs of the As and b)

• the model can now be calibrated (the unknowns can be determined) using ordinary multiple regression in a package like SAS

• it may be easier to avoid linearizing altogether by using the nonlinear regression facilities in many packages

• normally, we would try to maximize the fit of the observed and predicted interactions

• linearization changes this
• • e.g. we minimize the squared differences between observed and predicted values of log (I_ij/E_i) if ordinary regression is used on the linearized form above
  • this is easy in practice, but makes no sense

• intuitively, an error of 30 in a prediction of 1000 trips is much more acceptable than an error of 30 in a prediction of 10 trips

• these ideas are formalized in the technique of Poisson regression, which assumes that I_ij is a count of events, and sets up the objective function accordingly
• • the function minimized to get a good fit is roughly the difference between observed and predicted, squared, divided by the predicted flow