- Email: [email protected]

A Unifying Framework for Location-Production Interaction Modelling

DAVID

F. BATTEN,*

Victoria,

Australia

Abstract: Unification

of a wide class of spatial and industrial interaction models is ac~ompiished by identifying three structural similarities which exist between different facets of urban and regional behaviour. The resulting conceptual framework for location-production interaction modelling is based upon satisfactory information content. The research problem is to identify a set of accounting and behavioural constraints which characterize the system of interest to an acceptable level of accuracy. In this form of model-building, standard measures of uncertainty and information play a fundamental role

interaction probabilities (prs), where pry is the probability that a trip is made from region Y to region s. It has been suggested that the most successful planning applications of the entropy-maximizing paradigm have been based on interaction models (for example, see WEBBER, 1977).

Introduction Two main classes of entropy-maximizing models are now firmly established in the literature of regional science. Location models specify the probability that a particular unit of capital stock occupies a location in a given area. Typical examples include the location of households, firms, shops, public facilities and service centres.’ In such models the mean cost of travel time or distance between supply and demand points is usually given in the form of a constraint, and the pattern of trips to the unit in question is regarded as fixed. Typically, a negative exponential decline of probability with distance from the unit is obtained.

Half a century ago these two classes of problem were formally associated when OHLIN (1933) proposed a unified theory of location and trade. Twenty years later Ohlin’s remarks were echoed by ISARD (1954) and MOSES (196(l), who recognized the same fundamental association in their work on interregional models. More recently, it has been generally agreed that the location and interaction subproblems are so closely related that they should not be solved separately. This has led to the establishment of a welcome class of location-interaction models [for example, see the composite models of MACGILL and WILSON (1979) or BERTUGLIA and LEONARD1 (1980)].

The second major class of entropy-maximizing model consists of interaction models, originally formulated by WILSON (1967), in which the elementary event is a trip between some origin and some destination. In this case, it is the pattern of location of people, firms (jobs), shops and other facilities which is usually assumed to be given, together with the mean cost of all trips in the system. The typical output from such a model is a set of *~ommonweaith Scientific and IndLIstrial Organization, PO ESox Sh. Highett, Victoria, 3190.

of entropy model, the prodztctionhas received much less attention in the literature, although WILSON (1970) formulated a rudimentary input-output version over 10 years ago. in such a model the elementary event is the ,novement of commodities from sector i in region r for use in the production of other commod-

A third

kind

interaction model,

Research Australia

231

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232 ities by sector j in region s: ~7; is the probability of such an event. Production-interaction models actually ponds

generalize

earlier

to an interaction

viewed as a location the production

work since Z: p$J corresprobability,

probability

of commodities

i’p;

these subsystems produce employees, which generate people and money.

15 Number 2/‘1984

goods, services and flows of commodities,

may be

and Z p’$ relates to is by sector i in region Y.

This. paper attempts to unify a broad class of location-interaction and production-interaction modefs by identifying various structural similarities which exist between different facets of urban and regional behaviour. The conceptual foundation for location-production-interaction (LPI) modelling, laid by WILLIAMS and WILSON (1980) and BATTEN and ROY (1982), is built upon by considering all three subproblems or phenomena in a simultaneous formulation. In the following section we address these phenomena by means of a suitably indexed probability distribution and then adopt KULLBACK’s (1959) extension of the entropy concept, namely ~?z~~~~~~ju~ gain, to examine the information added to our prior knowledge of each phenomenon by the inclusion of various constraints. Special attention is paid to the determination of u priori probabilities, and the inclusion of constraints which ensure sufficient model accuracy yet satisfy both the theorems of maximum entropy and minimum information gain. In this way a consistent framework for model building based on satisfactory information content is developed. In a later section we examine the advantages of this approach for one particular system of interest, namely interregional input-output systems. The unifying generality of the proposed framework is further demonstrated by a brief description of some additional models which share similar structural characteristics, including multiple facilities (or activities) and multi-purpose trips in an urban system.

Location-Production~ntera~tion Modelling In LPI modelling we are principaIly concerned with the following three phenomena: (1) the location of capital stock; (2) the productive activities occurring within this stock; and (3) the flows generated by these activities, Typical examples of the first subproblem include the location of residential housing. business firms, shopping centres and public facilities or service centres. The activities undertaken within

To date, most urban and regional models currently in use represent the application of one or more standard theoretical techniques to describe one or, at most, two of these phenomena. Popular theorctical approaches range from the entropy-maximizing paradigm (which is an aggregate one), through classical and random utility theories to disaggregate choice models, among which the logit model is perhaps the most familiar (for example, see MCFADDEN, 1974; DOMENCICH and MCFADDEN, 1975). Despite the apparent differences among these approaches, it is not uncommon for them to yield the same model structure. This is hardly surprising if we recall that the logit model is simply a multinomial logistic distribution, and is therefore related directly to probability distributions emanating from the maximum entropy formalism. THEIL (1972) concludes that the logit model measures the sensitivity of entropy to variations in probabilities. This theoretical consistency permits us to work with just one mathematical formulation. We shall adopt the probabilistic or information-theoretical stance advocated by SNICKARS and WEIBULL (1977) and WEBBER (1979), in which the entropymaximizing paradigm is generalized to cope with non-uniform prior probabilities. The present objective is to develop a general methodological framework by addressing the subproblems of location, production and interaction in a simultaneous formulation. Our spatial system of interest is defined by a set of origin and destination regions (or zones) within which various productive activities occur and between which interaction takes place. Within each origin r the activities occurring at facility i are responsible for the production (or reproduction) of factor k. Inside each destination region s, factor k is used (or may just be serviced) by facility j. In such a model the elementary event is the movement of any mobile factor of production k from facility i in region r to facility j in region s; let p,$I be the probability of such an event. This interaction can be regarded as a measure of the covariation between random variables defining each set of regional activities, and as such it is fully described by the probability distribution {p$).

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Particular variables defining (p$} will be introduced in later sections, where a number of illustrative models are presented. In this context the three subscripts (i, j and k) denote non-spatial components of the system, whereas the two superscripts (r and s) determine its spatial resolution. It is assumed that this probability distribution is normalized in the usual manner, namely (I)

where the summations in this paper,

are implicit

over the ranges

marginal probability distributions given by (PZ2, M> and (P”> are of particular interest and are defined as

A major limitation of the maximum entropy principle is the assumption that all microstates satisfying the constraints are equally probable. SNICKARS and WEIBULL (1977) have demonstrated that the classical gravity model may be inferior to models based on a priori trip patterns for describing changes in trip distribution. FISK and BROWN (1975) have also questioned the equiprobability assumption, whereas WEBBER (1979) has stressed that changes in spatial structure over time should be constrained by known information and the a priori state of the region. These and other similar suggestions have prompted a generalization of (5), known as the principle of minimum information gain, to cater for non-uniform prior probabilities.

The

(2)

(3)

If we denote by (4i~~) that a priori probability distribution which reflects the past state of our spatial system, or perhaps describes an anticipated form of Cp$}, the minimum information principle asserts that the most likely posterior distribution is that which minimizes the amount of information yielded by the observed data relative to {qi/i}. To find this we simply minimize the information gain

(4)

(6)

These derived distributions decide, respectively, the probability of facility i locating in region r, the probability of region r producing factor k and the probability of region r interacting with region s. For this reason the model can be aptly described as a location-production-interaction (LPI) model.

between the prior and posterior distributions subject to the available data on (pi,?;} in the form of constraints. Once again the resulting assignment of probabilities corresponds to that prediction which is maximally non-committal with respect to missing data. It has been shown that this minimum information principle reduces to the maximum entropy principle for fundamentally symmetrical systems, that is, systems in which the prior probabilities are uniform and each distribution is defined over an identical interval size (for example, see WEBBER, 1977; EVANS, 1979).

The aim of the modelling exercise is to estimate the unknown distribution (p$} subject to whatever information is already known. This given information is normally expressed in the form of constraints on certain elements or subsets of elements. The entropy-maximizing formalism asserts that the most likely distribution is that which maximizes the function H =-cc 2 Z cp;;i i j ,k r s

logp$

(3

subject to the available data expressed as constraints. This approach can be interpreted as choosing that probability distribution which is maximally non-committal with regard to missing information; that is, it is the least biased estimate possible using the given information (JAYNES, 1957).

The minimum information principle is justified on logical rather than empirical grounds. An information-minimizing assignment of probabilities is not necessarily more likely to be correct than some other assignment because such a likelihood depends in part on the assumption that the data available are sufficient to characterize {pied}. The minimum information assignment need not be right, but a more accurate assignment requires more data than were available in predicting {p,‘ll}. The principle simply determines what one is entitled to conclude from the given data, and therefore defines a

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234 method of drawing inferences or of organizing thoughts; it tells what the observer is legitimately entitled to conclude from the constraints. What the observer is legitimately entitled to conclude is a probability distribution in which no private bias (information not contained in the constraints) enters the prediction (WEBBER, 1977, 1979). The LPI model builder now faces two kinds of problem. One, which we might call hypothesis selection, is to formulate valid rules whereby the prior distribution (41~~) is chosen. In the literature this assignment of prior probabilities has already fulfilled several different roles. The second, or research, problem is to decide upon the nature and extent of that set of constraints which yields sufficiently accurate estimates of (p$} for the purpose at hand. Furthermore, these two problems may be interrelated, since information which is thought to determine the distribution of system elements across possible states (i, j, k, r, s) can be embodied in the prior distribution {qi~~} or stated in the form of constraints. Constraints impose certain restrictions on the possible values which the probability assignment (p$} may take. The question of deciding upon constraints which are consistent with both the theorems of maximum entropy and minimum information gain is further complicated by the linear constraint theorem, which rules out nonlinear constraints in the maximum entropy formalism (EVANS, 1979). Although higher-order moments of (p$} can be used as constraints on the minimum Table 1. A sample of marginal

Constraints Three-way

Marginal

(PSI

(Pi)

{l-$X,’

;f;

: (,,q; Two-way

information gain, we shall limit our present sion to linear constraints.*

Apart from accounting constraints, more general sets of linear restrictions have also been adopted to influence the distribution of systems elements across possible states (i, j, k, r, s). The well-known cost constraint is a restriction belonging to this class. So is the network or nodal capacity constraint. Such a family of restrictions may aptly be referred to as behavioural constraints, since they are often used to introduce measures of utility, attractiveness or congestion into the accounting formulations. In this way they play a similar role to the terms in an objective model.

function of a mathematical With their inclusion the

distributions

in LPI modelling

Type of probability production attraction location interaction production-attraction location-production location-attraction location-interaction productioninteraction

One-way

discus-

There are two permissible kinds of linear constraints. Firstly, accounting constraints refer in general either to the independent estimates or restrictions which must be satisfied by partial sums of the matrix (that is, marginal constraints), or to logical relationships which exist between different elements of (p$}. Only unit coefficients are permitted in accounting constraints, which are always specific to the problem at hand and independent of any theories of spatial interaction or dispersion (WILLIAMS and WILSON, 1980). Equations (2), (3) and (4) are examples of marginal accounting constraints. In our present LPI framework up to 30 different marginal probability distributions could be specified exogenously using marginal accounting constraints. Those of particular interest are identified in Table 1.

probability

distributions

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location-interaction location-production-interaction location-production-interaction

programming information-

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15 Number 211984

minimizing model can be viewed as a device for adding dispersion from some optimal base to obtain the most probable prediction. The errors associated with our prediction {p$} are also of two kinds. On the one hand errors may arise because the data (or assumptions) are incorrect. This source of error appears in all model building and must be controlled in the usuaI manner. Because accounting constraints are generally introduced to guarantee consistency, they are unlikely to contain such mistakes. Incorrect information is more likely to appear in the behavioural constraints. Alternatively, the predictions may go astray because insufficient constraints are used to determine choice probabilities accurately. It is in handling this latter source of error that the i~fo~ation-minimizing (or ~ntr~py~maximizing) paradigm can offer some methodological assistance. Assuming that no errors of measurement occur, the modeller usually begins with a small collection of data about which he is highiy uncertain. As additional data are added to the constraint set, the uncertainty contained in the predicted probability distribution falls, and the prediction becomes more reasonable. As information accumulates it is therefore possible to implement a decision rule which determines the point at which uncertainty has been reduced suffieiently~ or enough new information gained, to regard the resulting constraint set as one which characterizes (p$) to an acceptable degree of accuracy. The research problem is then to identify that set of Iconstraints which achieves this desired level of accuracy. Because accounting constraints rarely suffer from errors of measurement or assumption, it seems appropriate to ensure that all the pertinent constraints of this type have been included before the search for behavioural constraint begins. Nevertheless, it is evident that the desired leve’l of model accuracy may not be reached using accounting constraints alone.’ A second (and more difficult) step is then to identify those behavioural constraints which can further improve our prediction by increasing G (or reducing H) inI comparison to earlier models. The search for informative constraints has an interesting parallel in the related field of disaggregate choice modelling. The behavioural modellers frequently test a number of different model specifications involving various unknown variables via maximum likelihood estimation. Their inclusion of

235

additional variables to improve the explanatory capability of disaggregate models is analogous to our inclusion of additional constraints in aggregate information-minimizing models. But the accuracy of our posterior prediction {p$) will also depend on the extent to which any prior dist~bution (@) reflects the past state of affairs or describes an anticipated form of @ij{) . Prior distributions have been compiled in a number of different ways. In forecasting problems, (4;;) most usefully describes the present state of affairs: SNICKARS and WEIBULL (1977) used historical data to derive prior distributions for their models of metropolitan commuters and intermediate shipments. In theoretical problems {q$} may be used to test a variety of interesting hypotheses. For example, prior distributions which reflect disaggregate demand functions have been confronted with aggregate supply constraints to produce a ~o~~~~~~~~ the most probable pattern of consumption. Atternatively, SNICKARS (1979) and BATTEN (1982a. b) have adopted log-linear models based on the contingency table approach to provide a priori estimates of the interrcgional shipments between economic sectors. More generally, BATTY (1978) has examined the hypothesis that the relative population distribution of a region is directly proportional to its area. In all of the above examples we can identify two distinct stages in the appii~ation of the minimum information principle. We can now formalize these within our LPI framework. Firstly, a complete prior distribution (q$ is determined. This primary stage will be labelled hypothesis seiection, although it may simply correspond to the collection of historical data. The second stage consists of confronting this u priori distribution with more limited information about the posterior distribution {pi;;} in the form of constraints. The desired solution is achieved by an efficient addition of information to complete the u posteriori distribution. A fundamental need is for consistency between these two stages of the modelling process. The adoption of some arbitrary assumption like the gravity hypothesis for the estimation of a priori probabilities is inconsistent with our desire to minimize bias in the second stage, In the absence of historical probabilities, we have previously advocated use of the maximum-likelihood assumption (e.g. maximum entropy or contingency table solution) for hypothesis selection in the first stage (for example, see BATTEN, 1982a, b). By so

236 doing, we ensure that both stages remain “maximally non-committal with regard to missing information” (see JAYNES, 1957, p. 620). Also of concern is the nature of any behavioural constraints which are introduced in the second stage, efficient infarrnation adding. The inclusion of cost or capacity constraints (with weighted coefficients) to restrict some marginal probabilities to certain expected vaIues is not as easily justified as the inclusion of similar historical information in the first stage. The influence of earfier costs or capacities will be embodied (to some extent) in the a priori probabilities {qi;i). If the relative magnitudes of these costs or capacities do not alter greatly over time, we would not expect the addition of such constraints to be particularly informative. This inability to improve our prediction substantially would be reflected by littte change in the value of G compared with the solution based on accounting constraints alone. The various steps in our general

LPI modelfing

Figure 1.

procedure can now be summarized (see Figure 1). The first stage of hypothesis selection not only distinguishes the model’s purpose (forecasting or theoretical hypothesis testing), but also quantifies our a priori knowledge concerning the model’s parameters {q$}. At this stage any missing entries are completed using the entropy-maximizing assumption of equiprobability, since this is the least biased statistical stance we can take. The second stage of efficient information adding begins by identifying all the accounting constraints which pertain to the problem. This usually consists of examining which marginal probabilities are either known or can be estimated exogenously, and which logical relationships exist between different elements of Cpi~~},Next, we minimize the information gain (G) subject to these accounting constraints and record the chosen value of G. If the desired level of model accuracy (Gmin) has not been achieved, behavioural constraints are then sought. The addition of these shoutd improve the model further; their effect can be monitored by changes in G.

General LPf modelling procedure.

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Should the model fail to achieve the desired level of accuracy with its full quota of accounting and behavioural constraints, then it is deemed unacceptable statistically. In such circumstances the level of spatial aggregation or activity representation in the data sets (constraints) requires revision or the amount of data available is inadequate.

be related directly to the aggregate structure of the economy and movements generated by imbalances between production and demand inside the different regions. The basic relationships may be viewed as a collection of accounts in which the additional notations given in the appendix apply.

In the following pages we demonstrate the overall approach by reference to two classes of LPI model. The next section discusses constraint information in the context of interregional input-output analysis. A later section deals with the case of multiple facilities (or activities) and multi-purpose trips in an urban system. Further examples of LPI modelling may also be considered.

To simplify the present analysis we shall restrict our attention to a single commodity by dropping the subscript k. The primary aim is then to estimate the interregional pattern of intermediate flows {

An Example: Interregional Systems

The basic set of accounting constraints ation can be summarized as follows:

Input-Output

A particularly unrealistic feature of input-output analysis in its traditional Leontief form is that it assumes a square matrix of technical coefficients, thereby implying that each industry or sector produces just one homogenous product. Although no commodity is strictly homogeneous, it may be regarded as such for most practical purposes. However, the one-to-one sector-product assumption is clearly unacceptable and greatly confounds the aggregation problem. We must abandon this assumption if realistic commodity flow models are to be developed. Fortunately, the development of rectangular inputoutput tables has necessitated a distinction between sectors and products or industries and commodities. LPI modelling permits a similar distinction by drawing together the pertinent information embodied in rectangular input-output theory. Subsets of elements in (p$} can be related directly to the elements of the absorption (attraction) and make (production) matrices by appropriate summations. Our spatial system of interest is defined by a set of origin regions r and a set of destination regions s. Within each origin region r the activities undertaken by industry i are responsible for the production of commodity k. Inside each destination region s, commodity k is absorbed or consumed by industry j. In such a model the elementary event is the movement of commodity k from industry i in region r to industry j in region s; (p$J} is the probability of such an event. The full set of interregional shipments can

C Xx7 is

+

Cy:” s

for this situ-

= XT* + rn: -

er

(7)

(8)

(9)

CCY:” rs

=yi.

(10)

Equations (7) correspond to an interregional extension of the traditional Leontief balance, whereas equation (8) defines the inflow factors needed to support the prescribed intraregional production levels. Equations (9) and (10) simply ensure that levels of interregional trade are consistent with aggregated interindustry statistics. We shall investigate four different cases of additional information relating to intraregional demands. In so doing, particular attention will be paid to the uncertainty (H) and information gain (G) associated with each separate formulation. The complete set of five formulations may be summarized as follows: Case A: accounting constraints (7) through (10) only; Case B: accounting constraints (7) through (9) plus the following information about final demands (y*S for each industry and region:

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transformations: (16)

Case C: accounting constraints (7) through (10) plus an interregional version of the LeontiefStrout relationship to define total demands (..Y*:) for each industry’s products in each region: -y+&..

+

x

y'?

$;\; (12)

r

ir

Case D: accounting constraints (7) through (9) plus information about both intermediate (x:“) and final (y*:) demands for each industry’s products in each region: c xx!/

= $2

(13)

zz yr”

(14)

jr c yT r Case E: accounting plus the following ing transportation trade:

constraints (7) through (10) behavioural constraints defincosts incurred in interregional

(15)

To analyse each formulation and determine the most probable pattern of deliveries to both intermediate ($) and final (‘yt) demands, constraints (7) through (1.5) have been normalized using the

where

X

= Z,zZCxY i j r s

Final demand

Sector 2

Sector 3

0.3213E-4 0.7390E - 3 0.6426E-4

0.2635E-2 0.1124 0.29986 - 1

0.3213E-4 0.2047E - 1 0.5751E-2

0.25066-2 0.1543 0.92216-l

1

0.25706-3

04402E-2 0.4498E-3

0.1446E-2 0.6143E- 1 O.l639E-1

0.0000

2 ‘I 3

0.5526B--2 0.1542B-2

0.25066-2 0.1543 0.9221E- 1

3

0.2570E-3 O.l606E- 3 0.3213E-4

O.l571E-1 0.2599E-1 0.7936E-2

0.2570E-3 0.47236-2 O.lSlOE-2

O.l510E- 1 0.3566E- 1 0.2442E - 1

3

O.l510E-2 O.l028E-2 O.l285E-3

0.8578E-2 O.l420E-1 0.43376-2

0.64266-4 O.l285E-2 0.4176E - 3

O.l510E-1 0.3566E- 1 0.2442E - 1

1

lJ 3 Ptz

(17)

Normalized flow estimates (py} for case A are given in Table 2. These distributions are typical of the output from the INTEREG package. Each of our five LPI models is defined by four characteristics (WEBBER, 1979): the number of states over which these probability distributions are defined (48 in this case), the number of constraints, the form of these constraints and the prior probabilities (47). Table 3 summarizes these characteristics for each case, when the prior assumption is that each flow is equally probable (a state of maximum uncertainty

Sector 1 2

i r s

The INTEREG package (BATTEN, 1982a, b) facilitates computer estimation of the interregional flows. It has been applied to a simple system of two regions representing the state of Victoria in Australia. Recent non-survey data derived using the RAS and GRIT procedures provide an initial database, and are drawn from CRUICKSHANK (1979) and JENSEN et al. (1979). The flow tables originating from these two sources have been reconciled. and then aggregated into two regions (Melbourne and the rest of Victoria) and three sectors (primary, secondary and tertiary industries). The resulting data set contains the Victorian input-output table and interindustry estimates for Melbourne and the rest of Victoria.

Table 2. Normalized flow estimates for case A

pu

+ ZZcy’;.

239

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Table 3. Characteristics

Case

No. of variables

A B c D E

48 48 48 48 48

No. of constraints 24 27 30 33 27

of each model

Form of constraints (7)-(10) P-(9), 01) (7)~(101, (12) (7)_(g)> (131, (141 (7)-(10), (15)

in which H,,,=3.8712). The measures of uncertainty (H) and information gain (G) corresponding to each of the five solutions are also included in Table 3. At least two important conclusions can be drawn from these results. Firstly, when a new constraint is added to an LPI model the maximum level of uncertainty cannot rise; nor can the minimum level of information fall. Comparison of cases A and C or A and E verifies that the information contained in each prediction necessarily rises (or at least remains constant) as additional data are acquired about the flows. Secondly, it is clear that the addition of certain constraints does not alter the prediction greatly. Probabilities, information and uncertainty are largely unaffected by accounting constraints like (12), since the inf(~rmation contained in cases B and C is almost identical. Furthermore, the introduction of behavioural data such as that embodied in constraint (15) is less informative than the same number of additional accounting constraints in case B. Valuable data are those which alter the predictions substantially, so long as the data are accurate. The results in Table 3 highlight the need to consider carefully what useful information can be gleaned from various types of constraints. The general research process now becomes clear. Once the LPI system has been identified, a basic set of accounting constraints is included in the first model of that system. If the results are sufficiently accurate, additional constraints may not be needed. If not, the process of adding constraints must continue, but only those constraints are added which affect the predictions (HABERMAN, 1973). However. a fundamental requirement is that any constraint, be it accounting or behavioural, must itself be accurate for the prediction to be reliable. This may preclude the inclusion of various types cf behavioural assumptions.

Uncertainty H 2.7856 2.5807 2.5800 2.5509 2.6267

Information gain G 1.0856 1.2905 1.2912 1.3203 1.2445

A Second Example It is generally agreed that the most desirable form of location planning models is one with the location of multiple facilities or multiple activities in an urban system. The reason for multi-service systems becomes clear if we drop the assumption of homebased return trips and allow for multi-purpose trips which involve a number of destinations. Multipurpose trips link a number of different facilities and activities together, implying decisions about their optimal location which take all the pertinent links into account. The LPI framework is a convenient and appropriate one for this type of modelling. In this case our spatial system of interest is defined by a set of origin zones r and a set of destination zones s. Within each zone of origin r, the activities in facility i are responsible for the production of a tripmaker k. Inside each destination zone s, tripmaker k is either supplied, serviced or employed by facility j. The elementary event in such a model is a trip by tripmaker k from facility i in zone r to facility j in zone s; p$ is the probability of such an event. The inclusion of the tripmaker subscript k permits the simultaneous consideration of a variety of trip purposes, e.g. work trips, shopping trips, leisure trips, etc. A number of important advantages arise from this approach to the modelling of multi-purpose trips between multiple facilities in an urban system. Firstly, LPI modelling does not rely on any sequential arrangement of submodels, which is certainly a major weakness of the Lowry family of models. Secondly, it is capable of determining the most informative set of constraints to describe the trip patterns to an acceptable level of accuracy. Thirdly, it allows the log-linear approach adopted in multidimensional contingency table analysis to be applied to the prior {q$} trip estimates. The latter

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240 method enables interaction patterns between different facilities and zones to be detected without demanding that these interactions exhibit certain assumed functional forms (like the gravity assumption). It is interesting to note that a number of authors have recently identified certain structural similarities and equivalences between multi-activity location models (such as the Lowry model) and spatial input-output models. There are clearly a number of recognizable circumstances under which seemingly different spatial phenomena may be analysed using an identical methodology. To this extent, the two examples cited herein by no means exhaust the potential for LPI modelling. A wide range of phenomena could be analysed using this powerful unifying framework for spatial modelling.

Concluding

Remarks

Most urban and regional models currently in use represent the application of one or more standard theoretical techniques to describe one or, at most, two types of standard social phenomena. This paper has attempted to unify a broad class of such twophenomena models by identifying certain structural similarities which exist between different facets of spatial behaviour. The result is a general framework known as location-production-interaction (LPI) modeiling, which identifies three groups of phenomena: (1) the location of capital stock; (2) the productive activities occurring within this stock; and (3) the flows generated by these activities. By formulating a suitably indexed probability distribution which caters for all three subproblems simultaneously, we can build a wide variety of information-minimizing LPI models. The research problem is then to identify a set of accounting and behavioural constraints which characterise the system of interest to an acceptable level of accuracy. A two-stage process of hypothesis selection followed by efficient information adding is proposed. In this form of model building standard measures of uncertainty and information play an important role, since they help to identify those constraints which clearly affect the predictions. The result is a consistent framework for model building based upon a satisfactory level of information content. The

two

examples

which

have

been

chosen

to

15 Number 20984

illustrate the advantages of the LPI framework are by no means exhaustive. Many different models share similar structural characteristics, underlining the generality of the proposed framework. In attempting to offer guidelines rather than rules for the unification of such models, it is desirable to argue in terms of examples rather than formal theorems. Notes Residential location models have improved steadily since the early work of WILSON (1960) and BUSSIERE and SNICKARS (1970). Models devised for the location of firms have concentrated mostly on the classical plant location problem (found in the operations research literature). Examples include EFROYMSON and RAY (1966), REVELLE and ROJESKI (1970) and HANSEN and KAUFMAN (1976). Shopping models have attracted considerable attention following the pioneering work of HARRIS (1964), LAKSHMANAN and HANSEN (1965) and HUFF (1966). More recent activity is reported in COELHO and WILSON (1976) and ROY and JOHANNSON (1981). Public facility location modelling has a much shorter history, with significant recent contributions from LEONARD1 (1978, 1981), LEA (1979) and CLARKE and WILLIAMS (1982). Algorithms for the location of multi-level service centres have been proposed by BERTUGLIA and LEONARD1 (1980). The information-minimizing formalistn can produce a variety of well-known distributions, depending upon the mathematical nature of the formulated set of constraints. For example, quadratic terms can lead to normal distributions instead of the standard exponential distribution. Inclusion of logarithmic terms can result in a solution closely resembling the gamma distribution. No attempt will be made here to quantify an ‘acceptable’ level of model accuracy; it represents an important research problem. Appropriate statistics for measuring changes in spatial variation due to spatial aggregation in data sets and in model predictions of spatial interaction are now being developed via information theory; for example, see BATTY and SIKDAR (1982). Until such statistics have been tested empirically, measures of model accuracy remain largely subjective. However, comparable measures of goodness of fit from the parallel field of maximum likelihood estimation (using the chi-square distribution) may provide useful guidelines. For our five-dimensional case, up to 30 terms could be specified in a log-linear mode1 to be applied to contingency tables. Five one-way terms, ten two-way terms. ten three-way terms and five four-way terms must be considered. The contingency table approach parallels our exogenous specification of up to 30 different marginal probability distributions using marginal accounting constraints. For further details of contingency table analysis see HABERMAN (1973) or FIENBERG (1977).

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~eoforum~Volume

242 WILSON,

A. G. (196’) A statistical theory of spatial distribntian models, ~r~~~ort~t~o~ Researciz, 1, 253269. WILSON, A, G. (1969) Developments of some elementary residential location models, lo~rnal of Regional

Science,

9, 377-385.

WILSON, A. G. (1970) Interregional comma&y flows: entropy maximizing approaches, Geographical Analysis, 2, 255-282.

Appendix: Additianaf Rtotation Cri = unit cost of a delivery from industry i in region I to region s; Ci = total cost of all deliveries from industry i; r e, = exports (to abroad) from industry i in region r;

15 Number Z/I984

imports (from abroad) to industry i in region r; probability of a delivery from industry i in region r to industry j in region s; value added ta industry i in region r; gross production of industry i in region r; gross deliveries from industry i to region s; intermediate deliveries from industry i to industry j; intermediate deliveries from industry i to regions; intermediate deliveries from industry i in region r to industry j in region s; final demand (excluding exports) for industry i; final demand deliveries from industry i to region s; final demand deliveries from industry i in region r to region s.