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Geography Vol 3, No. 3, pp. IS%lM. IYYS Copyright @ 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 096h92319.5 $10.00 + O.W

0966-6923(95)00013-5

Spatial interaction modeling using artificial neural networks’ William R Black Department

of Geography,

Indiana University,

Bloomington,

IN 47405, USA

Artificial neural networks using traditional gravity model components are proposed as an alternative to the fully constrained gravity model. Drawn primarily from the cognitive science area of artificial intelligence, these non-linear models are exceptionally good at recognizing patterns in data. At the same time these neural network models are more flexible and more accurate. When used with the traditional components of the gravity model, neural network models require the estimation of far fewer coefficients than fully constrained gravity models. In comparison with those latter models, the application of the gravity artificial neural network (GANN) models to seven sets of commodity flows yields errors that are 30% to 50% smaller. Questions of input significance and network autocorrelation are also addressed. Keywords:

neural

networks,

spatial

interaction,

gravity

Artificial neural networks were initially developed during the 1960s in an attempt to simulate neural network activity of the brain for computer-based artificial intelligence systems. These networks were very successful at learning or recognizing patterns, but they proved to be inflexible, ie they were too slow to recognize changes in input. As a result, neural networks were by and large discarded by artificial intelligence developers for more than a decade and a half. Approximately seven years ago there was a resurgence of interest in neural networks for their predictive capabilities. Neural networks are not unlike statistical methods such as regression analysis 1992). The basic similarity is that both (White, attempt to predict some specified output by minimizing an error term, which measures the difference between input and output. However, in contrast with regression methods, neural networks make no assumptions regarding the form or distributional attributes of the variables used. In this sense neural networks may be viewed as non-parametric methods. In addition, neural networks become better predictors as they are ‘trained’ with more and more data. An area of fundamental importance in transport geography is the general field of spatial interaction modeling. In this area we attempt to replicate flows in order to better understand the factors that affect ‘An earlier version of this paper was presented at the annual meeting of the Association of American Geographers, San Francisco, 1994.

models, commodity flows, migration

these flows, as well as to develop methods that will enable planners and regional analysts to forecast future flows. Gravity and intervening opportunities models, and multiple regression methods are a few of the approaches used for this purpose. The accuracy of these methods is often within acceptable limits, but it is probably fair to say that the methods are better at identifying major patterns in the data under given assumptions than they are at predicting flows. The most successful of the models is the fully constrained gravity model. For this model the sum of the estimated flows from any origin and the sum of the estimated flows to any destination are equal to these values in the original flow data. Obviously total estimated flow is equal to the total original (observed) flow. In addition, the flows generated by a calibrated model have the same average flow length as the original data. The basis for calling such a model fully constrained should be apparent. This model also has data needs, eg the previously noted average flow length for the system under analysis, which require a full originaestination matrix to calculate, and this makes it, if not difficult, at least costly to use in planning and similar contexts. Since the basic purpose of a neural network is to identify patterns in the data submitted to it and to replicate those patterns for new data, evaluating the use of neural networks in replicating flow patterns seems appropriate. This is the purpose of the present paper. Other regional analysts have already used neural networks to examine large interaction problems, and there is no need to replicate their work 159

Neural network jlow modeling:

W R Black

here (Openshaw, 1993; Fischer and Gopal, 1994). Instead this research will address several problems related to the use of basic neural networks for spatial interaction modeling. There should be no doubt that neural networks can be developed that yield better results than conventional models. However, can basic neural network models be developed using the same inputs as existing methods and perform as well as or better than those methods? The emphasis is on basic, because we can always add more layers or neurons to these models to increase their accuracy. This

is the

primary

research

question

examined

here. A second research question seeks to assess how input (variable) significance can be assessed. A third question will address the extent to which errors from a spatial interaction neural network model are randomly distributed over the links of the implicit network that they occupy. In the pages that follow the reader will be introduced to neural networks and their use illustrated on a small sample problem, which uses a neural network for replicating flows. The results of developing neural networks for commodity flows and interregional migration are also presented here and compared with results obtained by other methods. A method of determining the significance of inputs is presented using the migration data. This is followed by a discussion of the network autocorrelation in residuals of the neural network model. A discussion of the general utility of neural networks for replicating flows concludes the paper.

Neural networks: an introduction There are numerous artificial neural network models in the literature. The three major neural network models are backpropagation (McClelland and Rumelhart, 1988)) counter-propagation (Hecht-Nielson, 1987), and the bidirectional associative memories systems (Kosko, 1988,1992). The following discussion will focus on the first of these. The backpropagation or backprop model has three (and sometimes four) layers of neurons, which

[

1

[l+gxli

Figure 1 Neural network

160

connectivity

are the basic processors in neural networks. The layers are an input layer, one or two hidden layers and an output layer. These layers are connected by two or three layers of synaptic weights. Neural network modelers may also specify a learning rate, alpha, a momentum term, eta, and a tolerance or error level. The learning rate indicates how much the weights change on each pass. The momentum term indicates the influence of previous weight changes on current weight changes. Both alpha and eta usually lie in the range from 0 to 1. There may also be bias terms on the hidden and output layers, although these are not always used. Diagrammatically, the backpropagation neural network is given in Figure 2 for the case of a two-neuron input layer [Z, and Z,], a one-neuron hidden layer [Z!Zl,and a one-neuron output layer [O], and biases. This diagram differs from typical neural network connection systems; it represents the connection process and the modification of the weights that occurs. These terms are probably all new to transport geographers. They stem primarily from the original research in this area of artificial intelligence, which sought to understand the manner in which the human brain functions. We can easily view input neurons as variables in statistical analysis. The weights and biases are not so amenable to interpretation, eg the weights are randomly generated and they are adjusted to better replicate target values. While we sometimes interpret coefficients in regression analysis, we rarely acknowledge that such interpretations only have meaning if the variables are linearly related. So we will not attach transport interpretations to the terms since this would be somewhere between misleading and wrong. The net input to the hidden layer for such a system, X,, where k = 1, is given by n-1

x, =

(1)

2 ZjiwIjXk + BH j=o

where Zjiis the value of the input for the jth neuron, is the weight attached to the jth input as it WfjXk

j-+0

1

r

]

Ll+CfYli

for initial problem

1

Oli

Neural network flow modeling: W R Black

moves to the hidden layer, X,, and BH is a bias attached to the hidden layer. A threshold function is used with the backpropagation method. The most common of the transfer functions used with backprop neural networks is the sigmoid or squashing function represented by the logistic

A

~ 1 + eeX”

B

C

Total

15

a4

01

20

ffp$Jq

,;,, 1 Hkl=

A

(2)

where Hki is the value of the kth hidden neuron for observation i. The net input from the hidden layer to the output layer is represented as

Figure

Flows and marginal totals for the 3

2

X 3

sample

problem

A

B

C

4

5

2

where Oki is the output for the kth neuron for the ith observation and C

n-l

Yk =

x HkiWH, k=O

V, +

BO

Figure

is as defined above, w,k, is the weight attached to the value from the hidden layer as it moves to the output layer, Yki, and B. is a bias term attached to the output layer. The output values for the kth (k = 1) output neuron and observation i, Oki, i = 0 . . . n-l, is then compared with the actual or target values, and the error term from this comparison is used to adjust the weights of equation (1) and (4) in a manner described in detail elsewhere (see inter afia, Blum, 1992; Eberhart and Dobbins, 1990; Lawrence, 1993; Masters, 1993; McClelland and Rumelhart, 1988; Rumelhart et al, 1986). Hki

A three-region

flow problem

Consider the flows between three regions as given in the matrix of Figure 2. The distances between these regions are given in Figure 3. Setting this problem up as productions, attractions and distances being used to estimate flows, we have the values of Table I. Normalizing these data so that they are less than unity comes next. There appear to be no rules about how this normalization is done. I used total flow to normalize productions, attractions and flows. Since the total flow is 100 this simply meant that a decimal point was placed before each of the flow values above. Distances were normalized by taking the longest distance, five units, and dividing all values by this. As is apparent from the above, the neural network pattern set up for this problem included three input channels (flow produced, flow attracted and flow length), one hidden layer with three neurons and one output layer - the observed flows. The results of this exercise are displayed in Figure 4. The root mean squared error (RMSE) resulting

3

Distances between regions for the sample

problem

Table 1 Data for the three-region

problem

Productions

Attractions

Distance

Flow

Dyad

20 20 20 40 40 40 40 40 40

50 30 20 50 30 20 50 30 20

2 3 4 3 2 5 4 5 2

15 04 01 18 21 01 17 05 18

1-I 1-2 l-3 2-1 2-2 2-3 3-1 3-2 3-3

from the comparison of the (non-normalized) observed flows and the estimated flows was 1.0. A value of zero would represent a perfect replication. For comparative purposes this same set of data was fitted with an unconstrained and a fully constrained gravity model. The RMSE in these cases was 2.91 and 0.00, respectively. While the latter error level is impressive, one should not lose sight of the fact that the matrices are only 3 X 3, or nine flow values. The fully constrained gravity model as noted above places constraints on flow origins, flow destinations and flow length. In effect, six marginal totals and average flow length or distance represent seven constrained values which are used to estimate nine flow values. This is not the case with the unconstrained gravity models or the neural network model and since the latter outperforms the unconstrained gravity model, further tests on larger data sets were merited. 161

Neural network flow modeling:

W R Black

A

B

C

Total

A

15

4

0

19

B

20

20

2

42

C

16

5

18

39

Total

51

29

20

Figure 4 Estimated flows for the 3 X 3 sample problem

Tests with commodity

flows

Data exist for commodity flows between the nine census regions of the United States from the 1977 US Census of Transportation and seven sets of these were used as input to an unconstrained gravity model, a fully constrained gravity model, and what we will call a gravity artificial neural network (GANN) model. Input to the first two models is well known and consists of flow production, flow attraction and distances between all flow regions. For the fully constrained model the average shipping distance for

the item being examined is also necessary. Input to the GANN includes regional flow production, regional flow attraction, and the inter-regional distance between the origin and destination region. In addition, the amount shipped (in thousands of tons) between any two regions (or within a region) is used in the fitting process. The GANN had three input neurons (corresponding to the three variables noted above), a hidden layer of three neurons and a single output neuron representing the flow that took place (see Figure 5). Table 2 displays the results of this experiment. The table includes total flow, the average shipping distance, and the non-standardized root mean squared errors obtained from each of the three models for each set of commodity flows. As can be seen, the accuracy of the modeling improves as we move from the unconstrained gravity model to the fully constrained gravity model to the GANN model. This was obviously the desired result, but the improvement of the GANN model over the fully constrained gravity model was actually much better than expected. The poorest improvement

n

BH

Flow

Attraction

Figure 5 Gravity

Table 2 Commodity

artificial

neural

network

connectivity

flow experiments

Commodity

Av. dist.

Volume (tons)

UGM - RMSE

CGM - RMSE

GANN - RMSE

Chemicals Textiles Transport equipme :nt Stone products Machinery Meat products Lumber products

938 632 698 41.5 765 477 913

97 622 14 483 6 162 134 316 15 821 38 675 61 846

817 190 87 1740 190 477 885

571 72 35 864 96 398 479

337 44 27 437 66 199 270

162

Neural network flow modeling:

is for transport equipment flows; the error was 23% lower in that case. At the same time the error for the flow of meat products decreased by 50%. We do not have a generally acceptable index of the accuracy of neural network models. For that matter we do not have a goodness-of-fit index for fully constrained gravity models (Black, 1991). Since the gravity models used here are generally accepted as reasonably accurate, we have measured accuracy by comparing the neural network models with those models. It is difficult to make comparisons across different commodities since their tonnages differ. In order to eliminate this problem the total flow of all commodity groups was standardized to 1000 units and the fully constrained gravity mode1 and GANN mode1 were calibrated to these data. The results of this procedure appear as Table3. The table displays the improvement from one mode1 to the next and the decrease in the error from this activity. With regard to commodities, we can see that the GANN mode1 works best for textiles and worst for meat products. This is not particularly meaningful, but the uniformly high accuracy level is notable. There seems to be no relationship between model accuracy and the average length of shipments or the total weight of shipments. There has been no discussion here of the number of iterations necessary to fit the GANN models. This varied from about 5000 iterations to 150 000 iterations. Although this is of interest, it is somewhat meaningless to use in comparison of different models since the process of weight (or coefficient) identification starts with a set of randomly generated numbers. As a result, some models may require fewer iterations to calibrate since the random numbers were closer to the desired weights. Input significance:

a migration

example

A point of some concern in the development of artificial neural network models in general is that it is often difficult to interpret the contribution of the different variables (input neurons) that go into a model. There are weights attached to the various neurons, but these may sometimes act to cancel out Table 3 Root mean squared errors for fully constrained gravity models and gravity artificial neural network models for standardized commodity flows

Commodity

CGM RMSE

GANN RMSE

Error reduction W)

Chemicals Textiles Transport equipment Stone products Machinery Meat products Lumber products

5.86 4.91 5.63 6.38 6.12 10.28 7.75

3.45 3.03 4.34 3.25 4.17 5.14 4.36

41 38 23 49 32 50 44

W R Black

each other’s influence on the output neuron, and therefore their magnitude has little meaning in a significance sense. One approach to the question of variable or input neuron contribution or importance is provided by Eberhart and Dobbins (1990). They suggest that the contribution of different variables may be obtained by calibrating the model for all possible combinations of input variables. Although this approach will often be computationally intensive, the GANN flow application has only three variables - production, attraction and length of movement - so we can pursue it here. This involves developing seven models: three single-input neuron models, three models using two input neurons, and one model using all three input neurons. For the flows we will make use of migration data for the United States from 1965 to 1970. These migration data have seen much use in the spatial interaction modeling literature by Tobler (1983), Dorigo and Tobler (1983), Williams and Fotheringham (1984), Ledent (1985), Harker (1986), Baxter (1987), and Black (1992). The flows are for migration between the nine major census regions in the United States between 1965 and 1970 as reported by the United States Census for 1970. Intraregional migration is not included so that II is equal to 72 in this case. The data are reproduced in normalized form in the Appendix. Once again we will work with exactly the same GANN model consisting of three input neurons, a hidden layer with three neurons and a single output neuron. The input neurons consist of productions, attractions and distance. The flows were standardized to 1000 units to allow comparisons with the previous analysis. The neural network software yielded a root mean squared error of 5.879. In effect, this migration mode1 was the worst of the neural network models developed. Table 4 gives the root mean squared errors for each combination of the three input neurons for the standardized migration data examined here. The error level for productions and attractions is nearly equal. This is as it should be since these are taken from marginal values (column and row sums) for the flow matrix of interest. Put another way, the

Table 4 Migration example: contribution neuron combinations

of different neurons and

Neurons

GANN RMSE

OLS R2

Production Attraction Distance Production/Attraction Production/Distance Attraction/Distance Production/Attraction/ Distance

10.752 10.788 9.414 7.997 9.300 6.998

0.0048 0.1454 0.0813 0.1594 0.1016 0.2968

5.879

0.3532

163

Neural network flow modeling:

W R Black

production and attraction input neuron can have only m different values each, where m is the number of regions. The contribution of distance is stronger than either of these input neurons. On the other hand, the joint contribution of productions and attractions (Production/Attraction in Table 4) is greater than that of the distance neuron. While the joint importance of production and distance is less than the joint contribution of production and attraction, the latter’s joint contribution is less than the joint contribution of attraction and distance. In effect, the most important of these three variables is distance. This is followed by attraction and then production. Using ordinary least-squares (OLS) regression on these data yields the R* values of Table 4. The four multivariate models yield the same ranking in terms of accuracy, but the importance of individual variables (neurons) is quite different. The relative order of importance of these variables in the neural network case, ie distance, production and attraction, is replaced by attraction, distance and production in the OLS case. In effect, although the use of regression methods may give some idea of the importance of different variables (neurons), it is not a substitute for the all-possible-combinations approach of neural networks. Since the contribution of neurons and neuron combinations is a function of the weights, and the weights are a function of the number of neurons in the hidden layer and the number of iterations, it is probably unreasonable to assume that OLS regression will yield an index of the importance of neurons and neuron combinations in most cases. Network autocorrelation

in the residuals

When the migration data were previously analyzed by Black (1992), it was noted that the residuals displayed significant dependence in terms of network autocorrelation. Using the three inputs (production, attraction and distance) utilized here resulted in standard z scores of 5.19 using the equations for regression residuals and 4.60 for simple network autocorrelation. Both standard scores are very significant at the 0.01 alpha level. It is reasonable to ask whether the GANN model eliminates or increases the level of network autocorrelation. Subjecting the residuals from the three-input neural network model developed here to a simple network autocorrelation analysis results in a standard normal deviate of 4.11. Once again the value is very significant and in the same direction. The use of the GANN model does not appear to alter the level of network autocorrelation. Openshaw (1993) has suggested that if spatial autocorrelation is present in the data used, then neural network models will learn to represent it. This is unlikely in the spatial or network autocorrelation case since there is no explicit recognition of the spatial contiguity or network connectivity 164

in the GANN model developed. We could add such terms to the neural network models and eliminate autocorrelation from the analysis, but the same is true of regression methods. Before concluding, one point of clarification is necessary. The GANN model used here was essentially used in what the neural network literature refers to as a training context. The networks were trained, but they were not tested. In effect, the GANN models were used as we tend to use regression methods, ie to evaluate how well they are able to fit the data. Testing a neural network model is much like cross-validating a regression model. It is the application of the model to data that were not used in its development. This will be done with additional data sets in the future. It was not done here because of the desire to compare the goodnessof-fit of the different models. Another point worth noting is the fact that large neural network models and problems have a reputation for taking considerable amounts of computer time. For those concerned with computer time, this is surely only a temporary condition that will cease to be a problem in the near future. Programs are developing daily that claim 90% reductions in training time. In addition, once a model has been developed it is possible to design unique computer chips that significantly decrease computation time.

Discussion

and extensions

The GANN model discussed in this paper has the potential of revolutionizing flow modeling. It has been demonstrated here that the errors of this model are as much as 50% less than the next best general model that is available. Although the problems examined here were small, neural network models seem to be capable of high levels of accuracy based on their use in other fields. It has been noted that one does not need to know the average length of movements to calibrate GANN models, as you must in fully constrained gravity models. The latter models also have normalizing factors on productions and attractions that are equal to twice the number of flow regions or zones. The GANN model, using the same three input neurons, a three-neuron hidden layer and a one-neuron output layer, has only 16 weights that must be estimated and this only increases if the GANN model becomes more complex; it is independent of the number of regions. In a planning context the model can be used to estimate future flows for new regions that were not included in the initial model development. Also, there are often situations where the flows are unknown between certain regions owing to errors or survey non-response. In this case we can simply ignore these flow dyads and fit the model to those dyads and marginal totals for which the data are of a higher quality. These are just a few

Neural

of the reasons for using this model in flow analysis and forecasting situations. There is a concern emerging in this area regarding overfitting of neural network models. Actually this problem is found throughout the modeling literature (Leahy, 1994). All models can be subject to overfitting. Generally, simple models are less susceptible to overfitting. In this case the gravity model has always included values for two masses and one spatial separation factor. These are all that have been included here. In addition, the neural network model specified is about the least complex available. So we have less concern for overfitting in the present applications. With regard to future research a few comments are in order. There is every reason to believe that a practical (as opposed to theoretical) general model may be within our grasp. It seems unreasonable to assume that all flows will be estimatable by the same model. This does not mean that all flows are different. It does mean that there are most likely classes of flows that need to be treated in a slightly different manner. Openshaw (1993) has made a similar suggestion noting ‘that there are only so many different patterns of spatial interaction that can exist’. Black (1973) also noted the existence of a set number of interaction patterns and that these were a function of origin and destination regions in the interaction patterns under analysis. If flows are influenced by value, by attributes of their supply and demand, or by the concentration of production and consumption, artificial neural networks offer the possibility that these phenomena could be operationalized and used as the basis for yet another input neuron that could activate or deactivate the appropriate weights to handle any type of flow system.

References Baxter, M (1987) ‘Testing for misspecification in models of spatial flows’ Environment and Planning A 19 1153-1160 Black, W (1973) ‘Toward a factorial ecology of flows’ Economic Geography 48 59-67

Black,

network

flow modeling:

W (1991) ‘A note on the use of correlation

assessing goodness-of-fit

in spatial interaction

W R Black coefficients for models’ Trans-

portation 17 25-31 Black, W (1992) ‘Network autocorrelation in transport network and flow systems’ Geographical Analysis 24 207-222 Blum, A (1992) Neural Networks in C+ + Wiley, New York Dorigo, G and Tobler, W (1983) ‘Push-pull migration laws’ Annals of the Association of American Geographers 73 1-K Eberhart, R and Dobbins, R (eds) (1990) Neural Network PC Tools, A Practical Guide Academic Press, San Diego, CA

Fischer, M and Gopal, S (1994) ‘Artificial neural networks: a new approach to modeling interregional telecommunication flows’ Journal of Regional Science 34 503-527 Harker, P (1986) ‘The use of expert judgements in predicting interregional migration patterns: an analytic hierarchy approach’ Geographical Analysis 18 62-80 Hecht-Nielson, R (1987) ‘Counterpropagation networks’ Proceedings of the IEEE Networks

First International Conference

on Neural

B (1988) ‘Bidirectional associative memories’ IEEE Transactions on Systems, Man, and Cybernetics 18( 1) 49-60 Kosko, B (1992) Neural Networks and Fuzzy Systems PrenticeKosko,

Hall, Englewood Cliffs, NJ Lawrence, J (1993) Introduction to Neural Networks: Design, Theory, and Applications California Scientific Software, Nevada City, CA Leahy, K (1994) ‘The overfitting problem in perspective’ A/ Expert 9( 10) 35-36

Ledent, J (1985) ‘The doubly constrained interaction: a more general formulation’

model of spatial Environment

and

Planning A 17 253-262

D (1988) Explorations in Parallel Distributed Processing: A Handbook of Models, Programs, and Exercises MIT Press, Cambridge, MA Masters, T (1993) Practical Neural Network Recipes in C+ + McClelland, J and Rumelhart,

Academic Press, San Diego, CA Openshaw, S (1993) ‘Modeling spatial interaction using a neural net’ in Fischer, M and Nijkamp, P (eds) Geographic Information Systems, Spatial Modeling and Policy Evaluation SpringerVerlag, New York Rumelhart, D and McClelland, J and the PDP Research Group (1986) Parallel Distributed Processing: Explorations in the Microstructure of Cognition Vol 1: Foundations MIT Press, Cambridge,

MA

Tobler, W (1983) ‘An alternative formulation for spatial interaction modeling’ Environment and Pfanning A 15 693-703 White, H (1992) ‘Neural network learning and statistics’ Artificial Neural Networks Blackwell, Cambridge, MA 90-131 Williams, P and Fotheringham, A S (1984) ‘The calibration of spatial interaction models by maximum likelihood estimation with program SIMODEL’ Geographic Monograph Series 7, Department of Geography, Indiana University, Bloomington, IN, USA

Appendix Table A: Standardized migration data for the USA 1965-70 Origin+iestination’

Production*

Attraction’

Distance“

Flows

l-2 1-3 IL4 l-5 l-6 1-7 1-8

0.053770 0.053770 0.053770 0.053770 0.053770 0.053770 0.053770

0.093996 0.014550 0.076622 0.201886 0.068251 0.101387 0.086493

0.068998 0.317895 0.477000 0.306868 0.399495 0.565532 0.762444

0.014642 0.006442 0.002186 0.016114 0.001463 0.002892 0.002482 165

Neural network flow modeling: l-9 2-1 2-3 24 2-5 2-6 2-7 2-8 2-9 3-1 3-2 H 3-5 3-6 3-7 3-8 3-9 4-l 4-2 4-3 4-5

4-6 4-l 4-8 4-9 5-l 5-2 5-3 5-t 5-6

g-7 8-9 9-1 9-2 9-3 9-4 9-5 9-fj 9-7 9-8

W R Black 0.053770 0.152429 0.152429 0.152429 0.152429 0.152429 0.152429 0.152429 0.152429 0.173570 0.173570 0.173570 0.173570 0.173570 0.173570 0.173570 0.173570 0.098648 0.098648 0.098648 0.098648 0.098648 0.098648 0.098648 0.098648 0.143592 0.143592 0.143592 0.143592 0.143592 0.143592 0.143592 0.143592 0.080191 0.080191 0.080191 0.080191 0.080191 0.080191 0.080191 0.080191 0.093239 0.093239 0.093239 0.093239 0.093239 0.093239 0.093239 0.093239 0.080295 0.080295 0.080295 0.080295 0.080295 0.080295 0.080295 0.080295 0.124267 0.124267 0.124267 0.124267 0.124267 0.124267 0.124267 0.124267

0.170708 0.055001 0.014550 0.076622 0.201886 0.068251 0.101387 0.086493 0.170708 0.055001 0.093996 0.076622 0.201886 0.068251 0.101387 0.086493 0.170708 0.055001 0.093996 0.014550 0.201886 0.068251 0.101387 0.086497 0.170708 0.055001 0.093996 0.014550 0.076622 0.068251 0.101387 0.086493 0.170708 0.055001 0.093996 0.014550 0.076622 0.201886 0.101387 0.086493 0.170708 0.055001 0.093993 0.014550 0.076622 0.201886 0.068251 0.086493 0.170708 0.055001 0.093996 0.014550 0.076622 0.201886 0.068251 0.101387 0.170708 0.055001 0.093996 0.014550 0.076622 0.201886 0.068251 0.101387 0.086493

l.OOOOOO 0.068998 0.261814 0.420920 0.237870 0.330497 0.496534 0.706364 0.943919 0.317895 0.261814 0.159105 0.321046 0.208569 0.293950 0.457151 0.694707 0.477000 0.420920 0.159105 0.431632 0.279773 0.206049 0.298046 0.535601 0.306868 0.237870 0.321046 0.431632 0.151858 0.360428 0.717706 0.901701 0.399495 0.330497 0.208569 0.279773 0.151858 0.208569 0.565532 0.749842 0.565532 0.496534 0.293950 0.206049 0.360428 0.208569 0.405482 0.560491 0.762444 0.706364 0.457151 0.298046 0.717706 0.565532 0.405482 0.237555 1 .OoOOOO 0.943919 0.694707 0.535601 0.901701 0.749842 0.560491 0.237555

0.009010 0.023019 0.024425 0.005471 0.058446 0.004480 0.007598 0.007155 0.021832 0.007097 0.019292 0.022916 0.044849 0.018768 0.014517 0.014045 0.032081 0.002429 0.004934 0.023306 0.011699 0.004057 0.015095 0.014790 0.022334 0.010639 0.031112 0.028171 0.007507 0.020509 0.015632 0.007269 0.022749 0.001743 0.004373 0.023368 0.004052 0.025751 0.011522 0.002229 0.007151 0.002463 0.005257 0.013145 0.011790 0.016221 0.009870 0.010916 0.023574 0.001744 0.003557 0.007954 0.009245 0.007303 0.002079 0.012849 0.035559 0.005864 0.010826 0.018685 0.013451 0.021657 0.005393 0.020497 0.027890

‘1 = New England; 2 = Middle Atlantic; 3 = East North Central; 4 = West North Central; 5 = South Atlantic; 6 = East South Central; 7 = West South Central; 8 = Mountain; 9 = Pacific. *Standardized flow productions/lOOO. %tandardized flow attractions/lOOO. 4Distance/maximum distance in matrix. ’ Standardized flow/lOOO.

166

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