Continuous Allocation

problem formulation

the following text comes from Koomen & Borsboom-van Beurden (2011)

The original version of the Land Use Scanner model had a 500 m resolution with heterogeneous cells, each describing the relative proportion of all current land-use types. In this form, it is referred to as a continuous model since it uses a continuous description of the amount of land covered by each type of use in a cell. In the past, this approach has also been described as probabilistic to reflect that the outcomes essentially describe the probability that a certain land use will be allocated to a specific location. This is different from most land-use models, which only describe one dominant type of land use per cell.

The original, continuous model employs a logit-type approach derived from discrete choice theory. Nobel prize winner McFadden made important contributions to this approach of modelling the choices made by actors between mutually exclusive alternatives (McFadden, 1978). In this theory, the probability that an individual selects a certain alternative is dependent on the utility of that specific alternative in relation to the total utility of all alternatives. Given its definition, this probability is expressed as a value between 0 and 1, although it will never reach either of these extremes. When translated into land use, this approach explains the probability of a certain type of land use at a certain location based on the utility of that location for that specific type of use in relation to the total utility of all possible uses.

The utility of a location can be interpreted as its suitability for a certain use. This suitability is a combination of positive and negative factors that approximate benefits and costs. The higher the utility or suitability for a land-use type, the higher the probability that the cell will be used. Potential users assess suitability and can also be interpreted as a bid price. After all, the user deriving the highest benefit from a location will offer the highest price. Furthermore, the model is constrained by two conditions: the overall demand for each land-use function and the amount of available land. By imposing these conditions, a doubly constrained logit model is established in which the expected amount of land in cell $i$ that will be used for land-use type $j$ is essentially described by the formula:

• $M_{ij} = a_j \cdot b_i \cdot e^{\beta \cdot S_{ij}}$

in which:

• $M_{ij}$ is the amount of land in cell $i$ expected to be used for land-use type $j$;
• $a_j$ is the demand balancing factor (condition 1) that ensures that the total amount of allocated land for land-use type $j$ equals the sector-specific claim;
• $b_i$ is the supply balancing factor (condition 2) that ensures that the total amount of allocated land in cell $i$ does not exceed the amount of land that is available for that particular cell;
• $\beta$ is the monetary scaling factor of suitability. Also known as the inverse of the temperature of the system.
• $S_{ij}$ is the suitability of cell $i$ for land-use type $j$ based on its physical properties, operative policies and neighbourhood relations. The importance of the suitability value can be set by adjusting the scaling factor $\beta$.

The appropriate $a_j$ values that meet the demand of all land-use types are found in an iterative process, as is also discussed by (Dekkers & Koomen, 2007). This iterative approach simulates, in fact, a bidding process between competing land users (or, more precisely, land-use classes). Each use will try to satisfy its total demand but may be outbid by another category that derives higher benefits from the land. Thus, it can be said that the model, in a simplified way, mimics the land market. Governmental spatial planning policies restricting the free functioning of the Dutch land market can be included in this process when they are interpreted as either taxes or subsidies that cause an increase or decrease of the local suitability values, respectively. In fact, the simulation process sort of produces shadow prices of land in the cells. This is discussed in more detail in the literature (Koomen & Buurman, 2002).

In reality, the process of allocating use is more complex than this basic description suggests. In brief, the most important extensions to the model are:

• The location of a selected number of land-use types (e.g. infrastructure, water) is considered static and cannot be changed during simulations. Anticipated developments in these land-use types (e.g. the construction of a new railway line) are supplied exogenously to the simulations; that is, they are directly included as simulation results and are not derived from the iterative simulation process;
• The land-use claims are specified per region, and this regional division may differ per land-use type, thus creating a more complex set of demand constraints;
• Minimum and maximum claims are introduced to ensure that the model can find a feasible solution. It is possible to allocate more land for land-use types with a minimum claim. With a maximum claim, it is possible to allocate less land. Maximum claims are essential if the total of all land-use claims exceeds the available amount of land;
• To reflect the fact that urban functions will, in general, outbid other functions at locations that are equally well suited for either type of land use, a monetary scaling of the suitability maps has recently been introduced (Borsboom-van Beurden et al., 2005; Groen, Koomen, Ritsema & Piek, 2004). In this approach, the maximum suitability value per land-use type is related to a realistic land price, ranging from, for example, 2.5 euros per square metre for nature areas to 35 euros per square metre for residential areas. The merits of this approach are currently being studied by others (Dekkers, 2005 and Koomen & Borsboom-van Beurden (2011)).

A more extensive mathematical description of the basic model and its extensions can be found in the literature (Hilferink & Rietveld, 1999).

The continuous model directly translates the probability that a cell will be used for a certain type of land use into an amount of land. A probability of 0.4 will thus, in the case of a 500 m × 500 m grid, translate into 10 ha. This straightforward approach is easy to implement and interpret but has the disadvantage of potentially providing very small surface areas for many different land-use types in a cell. This will occur especially if the suitability maps have little geographical variation in their values, a problem that can be solved by making the suitability maps more distinctive and pronounced. Another possible solution for this issue is the inclusion of a threshold value in the translation of probabilities into surface areas. Allocation can then be limited to land-use types that, for example, have a probability of 0.2 or higher. Including such a threshold value calls for an adjustment of the allocation algorithm to ensure that all land-use claims are met. This is feasible, however, and has been applied in the Natuur Plan Generator model that aims to find an optimal spatial allocation of different types of nature within an area (Van Eupen & Nieuwenhuizen, 2002), which is in many ways similar to Land Use Scanner. Experience with this threshold value shows that insignificant quantities of land use are indeed set to zero, but if the threshold value is increased, the model will have difficulties finding an optimal solution. This is due to the possibility that all probabilities are below the threshold value. The application of a threshold value in land-use simulation with Land Use Scanner remains to be tested and is a topic for further research.

For the visualisation of results, the simulation outcomes are normally aggregated and simplified in such a way that each cell portrays the single dominant category among a number of major categories. This simplification has, however, a substantial influence on the apparent results and may lead to a serious over-representation of some categories and an under-representation of others. To prevent the above-mentioned issues related to the translation and visualisation of the probability-related outcomes, an allocation algorithm was introduced that deals with homogenous cells – see the description of the discrete model in the following paragraphs.

specification

Continuous allocation in the context of the GeoDMS is solving the following equation given the suitability $S_{ij}$ for each land use type $j$ and land unit $i$:

• $X_{ij} := a_i * b_j * e^{S_{ij}}$

subject to:

• $L_{jr} \le \sum\limits_{i}{X_{ij}} \le H_{jr}$ for each $j$ and $r$ for which claims are specified;
  • with $b_j < 1$ only if $H_{jr}$ is binding
  • and $b_j > 1$ only if $L_{jr}$ is binding
• and for each land unit $i$: $\sum\limits_{j}{X_{ij}} = L_{i}$

in which:

• $X_{ij}$ is the amount of land allocated to cell $i$ to be used for land-use type $j$;
• $S_{ij}$ is the suitability of cell $i$ for land-use type $j$;
• $L_{jr}$ is the minimum claim for land-use type $j$ in region $r$;
• $H_{jr}$ is the maximum claim for land-use type $j$ in region $r$; and
• $L_i$ is ?? for cell $i$

Compare this with discrete allocation.

When $L_i = 1$ for each land unit $i$, we call this a probabilistic allocation problem. When $L_{jr} = H_{jr}$, we call this an iterative proportional fitting problem.

corollaries

note that

• from substituting $X_{ij}$ in the land unit restriction it follows that $a_i = L_i / \sum\limits_{j}{b_j e^{\beta S_{ij}}}$
• and similarly,
  • if $H_{jr}$ is binding then $b_j = H_{jr} / \sum\limits_{i}{a_i e^{\beta S_{ij}}}$
  • if $L_{jr}$ is binding then $b_j = L_{jr} / \sum\limits_{i}{a_i e^{\beta S_{ij}}}$
• $ln(X_{ij}) = ln(a_i) + ln(b_j) + βS_{ij}$
• $βS_{ij} = ln(X_{ij}) − ln(a_i) − ln(b_j)$

shadow price interpretation

• $β^{−1} ln(a_i)$ can be interpreted as the shadow price of land unit $i$, thus very suitable land units have a high price.
• $β^{−1} ln(b_j)$ can be interpreted as the shadow price of Claim $j$: taxation or subsidy of the claimed land may be required to get the allocation not to exceed the maximum claim, nor remain below the minimum claim respectively.

entropy maximization

The solution $X_{ij}$ maximizes the following entropic quantity:

$E := \sum\limits_{ij}{X_{ij} (1 + \beta S_{ij})} - X_{ij} \ln(X_{ij})$ subject to the same restrictions.

This is shown by the fact that ${∂E \over ∂X_{ij}} = βS_{ij} − ln(X_{ij})$ and the KKT condition ${∂E \over ∂X_{ij}} + ln(a_i) + ln(b_j) = 0$

implies that $βS_{ij} = ln(X_{ij}) − ln(a_i) − ln(b_j)$

usage

Continuous allocation (also probabilistic allocation) is used to find the allocation of land use to land units that fit best to the suitability maps when endogenous interactions are disregarded, but some form of beta-dependent entropy is allowed.

Given suitabilities $S_{ij}$ for land unit $i$ and land use type $j$, the share of land of unit $i$ allocated to type $j$ is usually defined by the logit transformation ${\exp(\beta S_{ij})}\over{\sum\limits{k}\exp(\beta S_{ik})}$.

This solution can be considered as the expected amount of land use for each type when the actual suitabilities are assumed to have a Weibull distributed error term, and the land users with the highest suitability always get to use the land unit.

In the GeoDms, continuous allocation can be implemented using the loop or for_each operator. An earlier version of the Land Use Scanner applied only continuous allocation. Later versions also included discrete allocation.

bibliography

• Borsboom-van Beurden, J. A. M., Boersma, W. T., Bouwman, A. A., Crommentuijn, L. E. M., Dekkers, J. E. C., & Koomen, E. (2005). Ruimtelijke Beelden. Visualisatie van een veranderd Nederland in 2030. RIVM rapport 550016003.
• Dekkers, J. E. C. (2005). Grondprijzen, geschiktheidkaarten en parameterinstelling in de Ruimtescanner. Technisch achtergrondrapport bij Ruimtelijke Beelden. MNP report, 550016005.
• Dekkers, J., & Koomen, E. (2007). Land-use simulation for water management: Application of the Land Use Scanner in two large-scale scenario studies. Modelling land-use change: Progress and applications, 355-374.
• Groen, J., Koomen, E., Ritsema van Eck, J., & Piek, M. (2004). Scenario's in kaart; Model-en ontwerpbenaderingen voor toekomstig ruimtegebruik.
• Hilferink, M., & Rietveld, P. (1999). Land Use Scanner: An integrated GIS based model for long term projections of land use in urban and rural areas. Journal of Geographical Systems, 1, 155-177.
• Koomen, E., & Borsboom-van Beurden, J. (2011). Land-use modelling in planning practice (p. 214). Springer Nature.
• Koomen, E., & Buurman, J. (2002, April). Economic theory and land prices in land use modeling. In 5th AGILE Conference on Geographic Information Science, Palma (Balearic Islands Spain) April 25th-27th (Vol. 7).
• McFadden, D. (1978). Modelling the Choice of Residential Location, Spatial Interact ion Theory and Planning Models (A. Karlqvist et al., eds.).