SPATIAL INTERACTION PATTERNS Waldo Tobler July 1975
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SPATIAL INTERACTION PATTERNS Waldo Tobler July 1975
Research Reports are publications reporting on the work of the author. Any views or conclusions are those of the author, and do not necessarily reflect those of IIASA.
Contents
I. 11. 111. IV.
V. VI.
Introduction Background Algebraic Development From Vectors to Potentials Examples Caveats
L i s t of Figures
E x p o r t a t i o n o f S t u d e n t s by S t a t e , 1968.
11
F i e l d of Asymmetric Student Flows,1968.
12
Weighted S t u d e n t Flow F i e l d , 1968.
14
Flow F i e l d Computed f r o m a Random I n t e r a c t i o n T a b l e . 15
I n t e r p o l a t e d S t u d e n t Flow F i e l d , 1968.
17
S c a l a r P o t e n t i a l of S t u d e n t Flows, 1968.
19
V e c t o r P o t e n t i a l of S t u d e n t Flows, 1968.
20
G r a d i e n t F i e l d o f E d u c a t i o n a l E x p e n d i t u r e s , 1968.
2 1
B.A. t o Ph.D. Flow F i e l d , 1968.
22
Ph.D. t o Employment Flow F i e l d , 1968.
23
Commuting F i e l d f o r Munich, 1939.
25
Commuting F i e l d f o r Belgium, 1970.
26
European T o u r i s t e x c h a n g e s , 1973.
27
Telephone Flow F i e l d f o r Z u r i c h , 1970.
28
Business Contact F i e l d f o r Sweden.
29
Flows Between Psychological J o u r n a l s .
30
Spatial Interaction Patterns
Waldo Tobler*
Abstract
An algebraic examination of spatial models leads to the conclusion that a convenient description of the pattern of flows implicit in a geographical interaction table is obtained by displaying a field of vectors computed from the relative net exchanges. The vector field approximates the gradient of a scalar potential, and this may be invoked to explain the flows. The method can be applied to asymmetrical tables of a nongeographical nature.
I. Introduction
Empirical measurements of the interaction between geographical areas are often and conveniently represented by "fromto" tables, usually with asymmetric entries. Many mathematical models offered as descriptors of these geographical interaction patterns do not treat this situation adequately. Specifically, they quite frequently can predict only symmetrical interaction tables, a glaring contrast to the empirical observations. In the present essay an attempt is made to overcome this difficulty through the introduction of a flow field, which one may wish to think of as a "wind." This wind is interpreted as facilitating interaction in particular directions. The algebra allows one to estimate the components of this hypothetical flow field from the empirical interaction tables. Plotting the flow field provides a simple, convenient, and dramatic cartographic representation of the asymmetry of the exchanges, even for extremely large tables of interaction data. A table of countytocounty interactions in the United States, for example,.would yield nearly 10' numbers, an incomprehensible amount. A flow field, on the other hand, showing these data as a set of vectors might be more tractable. Going one step further, it should be possible to infer an estimate of the forcing function, the "pressure," which might be said to have given rise to the interaction asymmetry. Data collected for several time periods may allow one to infer the dynamics of the relation of the forcing function and the flows.
*This work was initiated in 1972 under National Science Foundation Grant GS 34070X, "Geographical Patterns of Interaction", at the Department of Geography, The University of Michigan, Ann Arbor. My appreciation is also extended to the many individuals who provided data matrices and comments. Each of them will be able to recognize their contributions in the pages which follow.
11. Background
The foregoing objectives were motivated by previous papers [27, 281 in which geographical locations were predicted from empirical interaction data by inverting models which contained spatial separations as one of the explanatory variables. The resulting spatial separations were then converted to latitude and longitude positions by a procedure analogous to trilateration, as practiced in geodesy. The empirical data in each instance were indicators of the amount of interaction between the locations in question. These interactions were given in the form of matrices, Mij, in which the rows
are the "from" places and the columns the destination places.
For example, if
is the amount of migration from place i
to place j, then the social gravity model predicts that
Mi j
=
kP.P.f(d.. )
1 3 13
,
where the P's denote the populations, d stands for distance, and k is a scale factor. Then the inversion is
From the adjustment procedures used in surveying one learns how to calculate the latitude and longitude coordinates of positions when their separations have been measured [31]. A similar procedure has recently been used in psychology [221. The social gravity model of course is symmetrical in the sense that if
then Mij must equal Mji, and the converse. In prac
ij = d j i
tice, however, interaction matrices are asymmetrical, and Mij # Mji. This would imply, if the model is inverted as was
done above, that dij # dji, with the consequence that the
trilateration solution can result in more than one geometrical configuration [12], or that the standard errors of the position determination are increased. In order to overcome this difficulty it is natural to introduce a wind, or current of some type, which facilitates interaction in particular directions. This vector field is to be estimated from the empirical data, and of course reflects their influence. At the moment the wind need not be given any interpretation other than that of a mathematical artifact which allows the problem to be solved. Later we can look for independent evidence which might confirm (or deny) its existence.
111. A l a e b r a i c Development
ai A s a s i m p l e example, p o s t u l a t e t h a t a t r a v e l e f f o r t
( t i m e , c o s t , e t c . ) ti i s a i d e d by a flow
in the
d i r e c t i o n of movement from p l a c e i t o p l a c e j . Then we can
write
that
tij = d . . / ( r
+
b
cij),
where
r
is a
rate
of
travel,
13
independent of position and of direction, and is i n t h e
b
same u n i t s a s c. An i n t e r p r e t a t i o n might be t h a t t i j i s
t r a v e l t i m e f o r someone rowing on a l a k e , r i s t h e rowing
speed
in meters
per
second,
and
b
c
is
a
current
in
the water;
o r t h a t j is u p h i l l from i, and t h a t t h i s r e s u l t s i n a
difference i n travel speed; o r t h a t there e x i s t s a grain, a s when s k i n s a r e p l a c e d u n d e r n e a t h s k i s , and movement i s
a r e n d e r e d e a s i e r i n one d i r e c t i o n . Whatever t h e i n t e r p r e t a t i o n ,
solving f o r one obtains
Here u s e h a s been made o f t h e r e l a t i o n cb i j =  cb j i which must
h o l d f o r c u r r e n t s , and of t j i = d j i / ( r + cb j i ) .
The same argument can b e a p p l i e d t o t h e g r a v i t y model. S u b s t i t u t e t f o r d i n t h a t model, w i t h f ( t ) = ti1j f o r s i m 
plicity,
P .P
P.P.
=kA=kk(r+:
,
Mj i
tij
dij
i j
and
S o l v i n g f o r cb i j , r e c a l l i n g t h a t d i j  d j i , o n e f i n d s
 c o n v e n i e n t c h o i c e o f u n i t s w i l l make r E 1 and t h e n
It is encouraging that this quantity has already been found useful in studies of migration [21], albeit without the present derivation. The original objective, inversion of the model, follows immediately:
Reversing i and j does not change this quantity. Algebraically
which is the same result as would have been obtained if
had been assumed initially. A solution to the original problem has thus been achieved, in the sense that any asymmetric interaction table can be made to yield a unique distance estimate to be used in further computations involving locations.
Two difficulties remain. First, only one interaction
zij model has been examined. Secondly, can a reasonable inter
pretation be provided for the
when the interaction con
sists of, say, telephone calls between exchanges?
In the first instance, a more general gravity model might be written using
Mi. + M.i
P.P
1= k A ,
2
a
dij
the development of which is straightforward. vein, an exponential model
bd../(r+g I
Mij = k(Pi + P.) e 1I
i j
3
In a similar
yields
P
'ij =
In M S i In Mi.
 In Mij + In Mji 2 i n k (Pi + P.) 3
and this is a much more complicated result. One could continue further by, for example, considering the entropy model elaborated by Wilson, or the migration model published by Lowry [30]. These models are in fact already more general in that they do yield asymmetrical interaction tables, but they also require supplementary information before one can solve for the distances. The Lowry model is
u. W. P.P
=k2llj,
Mij u. wi 7
where U is related to unemployment and W to wages. This can be rewritten as
and implies, if dij = dji, that
and also (solve for a and substitute) that j
Thus the second half of the interaction table carries no
information. Furthermore, if the distances are known, one
can infer the wage and unemployment ratios from the empiri
cal migration data. Such a result has recently also been
 achieved for another model by
sent instance there are n(n
CordeyHayes [ 4 1 . 1 ) / 2 equations of
In the prethe form
and n unknowns, the ails The system is overdetermined unless some of the equations can be shown to be dependent.
In a comparable manner, given only an empirical inter
n
action table, then the row sums 0 = 1 Mij and column sums
n
i=l
D
=
1
a
Mij can all be computed. The simplest sort of model
j i=1
is then that Mij = kO .D.f (d.. ) , and Mji = kO .D.f (d..) where
1 3 13
3 1 11
the origin and destination s&s now take the place of the
dpiojpuladtjiionsi.t
In is
order to obtain a consistent value for necessary that M../Mji = O.D./ojDi, and
this
13
13
is a hypothesis which can be tested.
Another interesting model has been proposed by Somermeijer [23]. This is
is the difference in attractiveness between areas
Here Qij i and j.
Qij
=
Aj

Ai.
Clearly
Qij
=

Qji;
solving
for
this quantity using
and adding, then subtracting, the equations for Mij and Mji, one finds
=.( ~i~M m i Mi.
Qij b
+ Mji
This is a very interesting relation because, although there
is much speculation in the literature, no one really knows
by how much areas differ in attractivity. The model allows
an estimate to be made of this quantity. One notices that
a Qii = 0, although usually Mii # 0, and a desirable property
would be that Qi = Qik +
for all i, j, k. In this case

the attractivity of area j, call it A;, would simply be
>
for some base level Ai. If this relation does Aj = Ai + Qij not hold for all i and j then an approximate estimator must
be devised, which does not appear difficult. One may then
wish to draw contour maps of the scalar field A(x,y), on
the assumption that attractivity is a continuous variable.
Solving for distances in this model leads to
Research Reports are publications reporting on the work of the author. Any views or conclusions are those of the author, and do not necessarily reflect those of IIASA.
Contents
I. 11. 111. IV.
V. VI.
Introduction Background Algebraic Development From Vectors to Potentials Examples Caveats
L i s t of Figures
E x p o r t a t i o n o f S t u d e n t s by S t a t e , 1968.
11
F i e l d of Asymmetric Student Flows,1968.
12
Weighted S t u d e n t Flow F i e l d , 1968.
14
Flow F i e l d Computed f r o m a Random I n t e r a c t i o n T a b l e . 15
I n t e r p o l a t e d S t u d e n t Flow F i e l d , 1968.
17
S c a l a r P o t e n t i a l of S t u d e n t Flows, 1968.
19
V e c t o r P o t e n t i a l of S t u d e n t Flows, 1968.
20
G r a d i e n t F i e l d o f E d u c a t i o n a l E x p e n d i t u r e s , 1968.
2 1
B.A. t o Ph.D. Flow F i e l d , 1968.
22
Ph.D. t o Employment Flow F i e l d , 1968.
23
Commuting F i e l d f o r Munich, 1939.
25
Commuting F i e l d f o r Belgium, 1970.
26
European T o u r i s t e x c h a n g e s , 1973.
27
Telephone Flow F i e l d f o r Z u r i c h , 1970.
28
Business Contact F i e l d f o r Sweden.
29
Flows Between Psychological J o u r n a l s .
30
Spatial Interaction Patterns
Waldo Tobler*
Abstract
An algebraic examination of spatial models leads to the conclusion that a convenient description of the pattern of flows implicit in a geographical interaction table is obtained by displaying a field of vectors computed from the relative net exchanges. The vector field approximates the gradient of a scalar potential, and this may be invoked to explain the flows. The method can be applied to asymmetrical tables of a nongeographical nature.
I. Introduction
Empirical measurements of the interaction between geographical areas are often and conveniently represented by "fromto" tables, usually with asymmetric entries. Many mathematical models offered as descriptors of these geographical interaction patterns do not treat this situation adequately. Specifically, they quite frequently can predict only symmetrical interaction tables, a glaring contrast to the empirical observations. In the present essay an attempt is made to overcome this difficulty through the introduction of a flow field, which one may wish to think of as a "wind." This wind is interpreted as facilitating interaction in particular directions. The algebra allows one to estimate the components of this hypothetical flow field from the empirical interaction tables. Plotting the flow field provides a simple, convenient, and dramatic cartographic representation of the asymmetry of the exchanges, even for extremely large tables of interaction data. A table of countytocounty interactions in the United States, for example,.would yield nearly 10' numbers, an incomprehensible amount. A flow field, on the other hand, showing these data as a set of vectors might be more tractable. Going one step further, it should be possible to infer an estimate of the forcing function, the "pressure," which might be said to have given rise to the interaction asymmetry. Data collected for several time periods may allow one to infer the dynamics of the relation of the forcing function and the flows.
*This work was initiated in 1972 under National Science Foundation Grant GS 34070X, "Geographical Patterns of Interaction", at the Department of Geography, The University of Michigan, Ann Arbor. My appreciation is also extended to the many individuals who provided data matrices and comments. Each of them will be able to recognize their contributions in the pages which follow.
11. Background
The foregoing objectives were motivated by previous papers [27, 281 in which geographical locations were predicted from empirical interaction data by inverting models which contained spatial separations as one of the explanatory variables. The resulting spatial separations were then converted to latitude and longitude positions by a procedure analogous to trilateration, as practiced in geodesy. The empirical data in each instance were indicators of the amount of interaction between the locations in question. These interactions were given in the form of matrices, Mij, in which the rows
are the "from" places and the columns the destination places.
For example, if
is the amount of migration from place i
to place j, then the social gravity model predicts that
Mi j
=
kP.P.f(d.. )
1 3 13
,
where the P's denote the populations, d stands for distance, and k is a scale factor. Then the inversion is
From the adjustment procedures used in surveying one learns how to calculate the latitude and longitude coordinates of positions when their separations have been measured [31]. A similar procedure has recently been used in psychology [221. The social gravity model of course is symmetrical in the sense that if
then Mij must equal Mji, and the converse. In prac
ij = d j i
tice, however, interaction matrices are asymmetrical, and Mij # Mji. This would imply, if the model is inverted as was
done above, that dij # dji, with the consequence that the
trilateration solution can result in more than one geometrical configuration [12], or that the standard errors of the position determination are increased. In order to overcome this difficulty it is natural to introduce a wind, or current of some type, which facilitates interaction in particular directions. This vector field is to be estimated from the empirical data, and of course reflects their influence. At the moment the wind need not be given any interpretation other than that of a mathematical artifact which allows the problem to be solved. Later we can look for independent evidence which might confirm (or deny) its existence.
111. A l a e b r a i c Development
ai A s a s i m p l e example, p o s t u l a t e t h a t a t r a v e l e f f o r t
( t i m e , c o s t , e t c . ) ti i s a i d e d by a flow
in the
d i r e c t i o n of movement from p l a c e i t o p l a c e j . Then we can
write
that
tij = d . . / ( r
+
b
cij),
where
r
is a
rate
of
travel,
13
independent of position and of direction, and is i n t h e
b
same u n i t s a s c. An i n t e r p r e t a t i o n might be t h a t t i j i s
t r a v e l t i m e f o r someone rowing on a l a k e , r i s t h e rowing
speed
in meters
per
second,
and
b
c
is
a
current
in
the water;
o r t h a t j is u p h i l l from i, and t h a t t h i s r e s u l t s i n a
difference i n travel speed; o r t h a t there e x i s t s a grain, a s when s k i n s a r e p l a c e d u n d e r n e a t h s k i s , and movement i s
a r e n d e r e d e a s i e r i n one d i r e c t i o n . Whatever t h e i n t e r p r e t a t i o n ,
solving f o r one obtains
Here u s e h a s been made o f t h e r e l a t i o n cb i j =  cb j i which must
h o l d f o r c u r r e n t s , and of t j i = d j i / ( r + cb j i ) .
The same argument can b e a p p l i e d t o t h e g r a v i t y model. S u b s t i t u t e t f o r d i n t h a t model, w i t h f ( t ) = ti1j f o r s i m 
plicity,
P .P
P.P.
=kA=kk(r+:
,
Mj i
tij
dij
i j
and
S o l v i n g f o r cb i j , r e c a l l i n g t h a t d i j  d j i , o n e f i n d s
 c o n v e n i e n t c h o i c e o f u n i t s w i l l make r E 1 and t h e n
It is encouraging that this quantity has already been found useful in studies of migration [21], albeit without the present derivation. The original objective, inversion of the model, follows immediately:
Reversing i and j does not change this quantity. Algebraically
which is the same result as would have been obtained if
had been assumed initially. A solution to the original problem has thus been achieved, in the sense that any asymmetric interaction table can be made to yield a unique distance estimate to be used in further computations involving locations.
Two difficulties remain. First, only one interaction
zij model has been examined. Secondly, can a reasonable inter
pretation be provided for the
when the interaction con
sists of, say, telephone calls between exchanges?
In the first instance, a more general gravity model might be written using
Mi. + M.i
P.P
1= k A ,
2
a
dij
the development of which is straightforward. vein, an exponential model
bd../(r+g I
Mij = k(Pi + P.) e 1I
i j
3
In a similar
yields
P
'ij =
In M S i In Mi.
 In Mij + In Mji 2 i n k (Pi + P.) 3
and this is a much more complicated result. One could continue further by, for example, considering the entropy model elaborated by Wilson, or the migration model published by Lowry [30]. These models are in fact already more general in that they do yield asymmetrical interaction tables, but they also require supplementary information before one can solve for the distances. The Lowry model is
u. W. P.P
=k2llj,
Mij u. wi 7
where U is related to unemployment and W to wages. This can be rewritten as
and implies, if dij = dji, that
and also (solve for a and substitute) that j
Thus the second half of the interaction table carries no
information. Furthermore, if the distances are known, one
can infer the wage and unemployment ratios from the empiri
cal migration data. Such a result has recently also been
 achieved for another model by
sent instance there are n(n
CordeyHayes [ 4 1 . 1 ) / 2 equations of
In the prethe form
and n unknowns, the ails The system is overdetermined unless some of the equations can be shown to be dependent.
In a comparable manner, given only an empirical inter
n
action table, then the row sums 0 = 1 Mij and column sums
n
i=l
D
=
1
a
Mij can all be computed. The simplest sort of model
j i=1
is then that Mij = kO .D.f (d.. ) , and Mji = kO .D.f (d..) where
1 3 13
3 1 11
the origin and destination s&s now take the place of the
dpiojpuladtjiionsi.t
In is
order to obtain a consistent value for necessary that M../Mji = O.D./ojDi, and
this
13
13
is a hypothesis which can be tested.
Another interesting model has been proposed by Somermeijer [23]. This is
is the difference in attractiveness between areas
Here Qij i and j.
Qij
=
Aj

Ai.
Clearly
Qij
=

Qji;
solving
for
this quantity using
and adding, then subtracting, the equations for Mij and Mji, one finds
=.( ~i~M m i Mi.
Qij b
+ Mji
This is a very interesting relation because, although there
is much speculation in the literature, no one really knows
by how much areas differ in attractivity. The model allows
an estimate to be made of this quantity. One notices that
a Qii = 0, although usually Mii # 0, and a desirable property
would be that Qi = Qik +
for all i, j, k. In this case

the attractivity of area j, call it A;, would simply be
>
for some base level Ai. If this relation does Aj = Ai + Qij not hold for all i and j then an approximate estimator must
be devised, which does not appear difficult. One may then
wish to draw contour maps of the scalar field A(x,y), on
the assumption that attractivity is a continuous variable.
Solving for distances in this model leads to
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