Contributions to the general theory of movement

Regional Science and Urban Economics 11 (1981) 157-173. North-Holland

CONTRIBUTIONS TO THE GENERAL THEORY O F M O V E M E N T Oscar FISCH The Ohio State University, Columbus, OH 43210, USA Received August 1980 One of the purposes of this paper is to contribute to the ongoing efforts to formulate a mathematical framework for a general theory of movement. It shows that in a particular model, and from a different perspective and a slight reinterpretation, AIonso's seminal articulation holds.

1. Introduction

William Alonso (1976) has concluded, in a highly seminal paper, that the accumulated mass of the various models of movement tha~ have been developed in the social sciences, during the preceding decades, have supported his attempt to articulate a common, logical, and mathematical framework for a general theory of movement. In the interim, a lively discussion about the merits of s qch an articulation has taken place. This discussion has revolved around whether the mathematical framework was general enough to encompass all the accumulated mass of models. In other words, it was postulated that Alonso's mathematical framework was a particular model and was thus, not supporting a general theory of movement. This discussion continues unabated. One of the purposes of this paper is to contribute to Alonso's efforts to formulate a mathematical framework by showing that, in a particular model and from a different perspective (i.e., with a very slight reinterpretation), Alonso's articulation holds. In the reinterpretation process, the paper answers and raises some conceptual and empirical questions that, hopefully, may enrich the ongoing discussion. In addition, the paper analyzes the performance of the general theory under the demands of predicting movement in a particular system. Section 2 of the paper presents a particular modal and its derivation source. This is the theoretical portion of the paper which illustrates the theoretical compatibility of the model with the general theory. Some causality questions are raised in the current discussion about the general 0166-0462/81/0000-4X)00/$02.50 ¢~ North-Holland Publishing Company

158

O. Fisch, Contributions to the general theory2 of movement

theory and are answered. In addition, the theoretical reinterpretations are formulated. The third section of the paper presents the empirical behavior of the model. This empirical portion was recently made feasible by the release of a very useful set of interprovincial migration flows in Canada for three consecutive five-year periods: 1961-1966; 1966-1971; and 1971-1976. These data sets will support an empirical testing of the general theory as framed by the theoretical aspects of the second section.

2. The system of movers

2.1. Total accounting of movers The system is characterized by r/nodes that are simultaneously origins and destinations of movers in a given time interval t. These movers are people in models of urban traffic, migration, social mobility, and mobility within specific organizations; in the Leontief system, the movers are goods and services in a common currency value. The nodes are geographic jurisdictions in models of urban traffic and migration, class/status in social and organizational mobility and economic sectors in the case of goods and services. The basic element of the social accounting of movers is a matrix M(to) =[Miflto)]. Miflto) being the number of movers from node i to node j at interval to. This basic element of information is rendered by an exhaustive {total) survey. The social accounting is completed with M~.(to), total outmovers from node i at to; M.~(to), total inmovers to node j at to and M..(to), total movers in the system at time to. Mathematically, these are defined as

M,.(to)= ~ Mi~(to), all i,

(1)

J

M.flto)= ~ Mii(to), all j,

(2)

i

M..tto)= Y Mi.ito )= i

M.jito)

(3)

j

in terms of shares, we can define a matrix m(to)--[mi~(to)], being mij(to) the share of total movers in the system moving from node i to node j at to; m,.ito) as the share of total movers in the system moving out from node i at t; and m.j(to) as the share of total movers in the system moving into node j at to. By definition we have that

miflto)-, Mo(to)/M..(to),

all

i,j,

(4)

O. Fisch, Contributio~ to the general theory of movement

159

mi.(to)=Mi.(to)/M..(to), all i,

(5)

m.j(to)- M°j(to)/M..(to),

(6)

all./.

And it follows that m..(to)=~imi.(to)=~.m.~(to)= 1. Furthermore, in terms of proportions, we can define p~'(t), as the proportion of the share of total movers moving out from node i that have as destination node j; and we can define pfl(t), as the proportion of the share of total movers moving into node j that have as origin node i. Mathematically, these are defined as

p~'(tD)=mq(to)/mi.(to),

all

i,j,

(7)

pfl(to)-mu(to)/m.j(to),

all

j,i.

(8)

These proportions can be interpreted as node specific probabilities of destination of any mover departing from ,~,de i and node specific probabilities of origin of any mover arriving to node j, respectively. These proportions define the following probabilities vector functions:

p,.(to)=(P~'(to), P~'(to),...,p~'(to)),

all i

(7')

p.i(to)=(p~(to), p~(to),...,p~J(to),

all ].

(8')

2.2. Margb~al accounting of movers Assume ,+hat at a later interval t t, instead of an exhaustive survey, a minimum suivey renders only Mi.(tt) and M.fltl) for all nodes in the system. The empirical question is- is it possible to estimate ~/i~(t,) from Mu(to) in such a way that tl

Mq(tl)=Mi.(tl),

all i,

(9)

~. ~4ij(tl)=M.j(tl),

all j.

(10)

j--I

n

i=l

In terms of shares, we have that

M..(tl) = ~ Mi.(tl)=~ M.j(t), i

(II)

j

mi.(t, )= Mi.(tl )/M..(tl ),

all i,

(12)

m.j(tl)=M,~(tl)/M..(tl),

all j.

(13)

160

O. Fisch~ Contributions to the general theory of movement

Now. the empirical question is: is it possible to estimate ~jj(tt)from m,j(to~ in such a way that

~. ~i~(tt )" m,.(tt ), all i,

(9')

J

~.j(tt)=m.j(tt),

all j.

(10')

i

The problem with (9') and (10') is that we have rlZmu(t! ) unknowns in a 2r/ system of equations, Faced with a similar problem of updating an InputOutput table, a Cambridge group of economists [Stone (1962), Bacharach (1965)] reduced the problem to one of 2r/unknowns in a 2~/system of nonlinear equations, solving the following system:

ri ~ s~mq(to) = m,.(tz),

all i,

(14)

sj ~ rimo(t o) = rn.j(tt ), all j.

(15)

J

i

The method of solution, by successive approximations, took tt:e generic name of RAS. Given the solution of the system (14), (15), then the estimation is given by

~fiis(t~)= r, mo(to)S s, all i,j.

(16)

In terms of input-output coefficients, r i explains the changes in the degree to which commodity i has been uniformly substituted by other intermediate inputs and sj explains the degree of change of intermediate inputs in the fabrication of commodity j.

2.3. General theory Alonso defines two sets of forces acting in each node. One set is the node's specific push (I,~(to)) and pull (Wj(to)) factors. The other set of factors are system's specific: D,(to), the system's pull acting at node i; and Cj(to), the system's push acting at node j. The first set defines the state (level) of the fa~.,ors acting at interval to. That state of the factors acting at one interval later t, can be defined as

Vl(tl )= 1.+ v,)Vt(to), Wj(t I )= (1.+ wj)~(to),

all i,

(17)

all j,

(18)

O. Fisch, Contributions to the general theory of mo~ment

161

where v~ is the interperiod change rate of the push factors acting at node i and w~ is the interperiod change rate of push factors acting at node j. In terms of the solution of system (14), (15), then

all i,

(19)

wj = s~- 1, all j.

(20)

vi = ri -

1,

If we define further, v~ as the interperiod change rate of the share of total movers departing from node i, and toj, as the interperiod change rate of the share of total movers arriving at node j, then the system (14) and (15) and its solution, can be rewritte~l as ( 1 . + v i ) ~ (l.+w~)mij(to)=(1.+vi)mi.(to),

all i,

(14')

J

(1. + wi) ~ (1. + v~)mo(to)= (1. +coj)m.j(to),all ).

(15')

i

Let us define the vectors v=(vi), w=(w~), v=(vi) and co=(wj), then from the solution of the system of equations (14') and (15') we can state

1.,emma I. If every node maintains the same share of outmovers and, simultaneously, every node maintains the same share of inmovers, then, independently of the total number of movers in the system, every node shows the same (unchanged) state (level) of push and pull. Briefly: l]" v=(O) and co=(O)

-,

v=(O) and w=(O).

Lemma 1'. I f every node maintains the same. state of push and, simultaneously, every node maintains the same state of pull, then, independently of the total number of movers in the system, every node will show the same (unchanged) share of outmovers and inmovers. Briefly: If

v=(l)andw=(O)

~

v=(O) andco=(O).

Combining both lemmas, we have a reciprocal relationship that states briefly, that independently of the number of movers in the system: v = (0) and o~= (0)

~

v = (0) and w = (0).

The above reciprocal relationship can settle (we hope) some causality arguments in the ongoing discussion about interpreting the general theory [Anselin and Isard (1978), Ledent (1980)].

162

O. Fisch, Contributions to the general theory of movement

Recalling (7) and (8), the solution of the system of eqs. (14') and (15') can be expressed as all i,

(14")

all j.

(15-)

(l.+vi)~,(l.+wj)p~'(to)=l.+vi, J

(l.+wj)~(l.+vi)pt~(to)=l.+%, i

Recalling (7')and (8') and defining

E,.[w]=E[wlp,.(to)],

all i,

(21)

E.j[v]=E[vlp.i(to)], all j,

(22)

where E is the expected value operator, then the solution of the system of eqs. (14") and (15") can be expressed as

I 1. + ri )( 1. + Ei . [ w ] ) = 1. + v i,

all i,

(14'")

(l.+wj)(l.+E,j[v])=l.+coj,

all j.

(15'")

Ei.[w] is the expected value of the rate of change of the pull acted by the system at node i, as perceived from there with prior information given by the vector of destination-probabilities of outmovers pi.(to); Ei.[v] is the expected value of the rate of the push acted by the system at node j, as perceived from there with prior information given by the vector of origin-probabilities of inmovers p.i(to). In Alonso's terms, we define these rates of change as

Di{t,)=(l+di)Di(to), all i,

(23)

Cflt, l=(1 +c~)Cflto), all j,

(24)

di=E,,[w], all i,

(25)

c~=Ei.[v], all j,

(26)

where

and the solution of the system of eqs. (14'") and (15'") is reduced to ( l . + v j ) ( l . + d ~ ) = l . + v i,

aiii,

(27)

(1.+%)(1.+ci)=I.+%,

allj.

(28)

o. Fisch, Contributions to the generaltheory of movement

163

In terms of change rates, the system is reduced to vj+d~+vid~=vl,

all i,

(27')

wj+cj+w~cj=co~,

all j.

(28')

From (27)and (28) we can state"

Lemma 2. The change in the share of total movers departing from any node is equal to the product of the change of the node's push times the change of the system's pull acting at that node. Lemma 2'. The change in the share of total movers arriving to any node is equal to the product of the change of the node's pull times the change of the system's push acting at that node.

3. Empirical behavior of the model 3.1. On nodal and systemic changes The data used to test the behavior of the model corresponds to the Canadian national system of interprovincial migration flows, for three consecutive five-year periods, with the provinces as nodes. Time indexing these peri.~ds, we have to for the period 1961-1966; t~ for 1966= 1971 and t2 for 1971-1976. In addressing the nodal and systemic changes, the following three systems ~,f equations was solved:

[1.+v~(l)]Y'El.+wj(l)]mo(to)=ml.(tl), J

all j,

(29)

[l.+wj(1)]~El.+vl(l)]mo(to)=m,i(tl),

all j,

(29')

[ 1. + v~(2)] ~ [ 1. + wi(2)] too(to )= mi.(t2 ), J

all i,

(30)

[1.+w~(2)]~[1.+vt(2)]mil(to)=m.~(t2) ,

all j,

(30')

all i.

(31)

allj.

(31')

i

i

[ 1. + vi(3)] ~ [ 1. + wl(3)]mo(t I )= mi.(t z ), J

[1.+wj(3)]~.,[1.+vi(3)]mo(tl)=m,j(t~) i

,

O. Fisch, Contributions to the general theory of movement

164

The solution of the first system (29), explicitly gives the vectors v(1) = [rill )) and w(l )= (wj(l)), and implicitly, the vectors d ( l ) = (di(1)) and c(1) =(cj(l)), in terms of the vectors v(1)=(v~(1)) and co(l)=(coj{1)), using the prior information given by the matrix of flows re(to)= [too(to)], to predict the flows at the second period t l (see table 1). The solution of the second system (30), gives a similar output for the third period (t2) using the same re(to) (see table 2). The solution of the third system t31), gives a similar output for t.he third period (t~), using m(tt) (see table 3). Overall, the sign of the nodal push and systemic pull coincides 15 times. In the remaining 15 cases, the sign of the nodal push prevails in 13 cases and in 2 cases, the sign of systemic pull prevails. The sign of nodal pull and systemic push coincides 12 times; in the remaining 18 cases, the sign of the nodal pull prevails in 15 cases and in 3 cases, the sign of the systemic push prevails. When there is no coincidence of sign, the effect of the nodal push or pull is reduced by the systemic puli or push, in relation to change in nodal share of outmovers and inmovers respectively. Table 1 System solution for the period 1966-1971, based on prior information (1961-1966). Province

Node r(1)

d(l)

v(l)

w(l)

c(l)

co(l)

Newfoundland Prince Edward Is. Nova Scotia

I 2 3 4 5 6 7 8 9 10

- 0.01944 -0.01240 0.00590 - 0.03947 0.03392 - 0.07272 0.02860 0.04389 0.07059 -0.01719

0.15283 -0.13257 -0.14093 - 0.12019 0.15992 ---0.05008 - 0.01543 0.05696 - 0.05574 0.11249

0.16549 -0.07105 -0.06018 - 0.06945 - 0.23345 0.04444 - 0.09964 - 0.11739 0.02869 0.22729

- 0.01175 -0.03625 0.02177 0.00302 - 0.00046 0.01250 0.00977 - 0.02633 0.04678 -0.04696

0.15180 -0.10452 -0.03972 -0.06664 -0.23380 0.05750 -0.09084 -0.14063 0.07681 0.16966

New Brunswick Quebec

Ontario Manitoba .qaskatchewan Alberta

British Columbia

0.17568 - 0.12168 -0.14597 - 0.08404 0.12187 0.02441 - 0.04281 0.01252 - 0. ! 1800 0.13195

Table 2 System solution for the period 1971--1976, based on prior information (1961-1966). Province

Node r(2)

d(2)

v(2)

w(2)

c(2)

to(2)

Newfoundland PrmceEdward Is. Nova Scotia

! 2 3 4 5 6 7 8 9 10

-0.06834 -0.01721 -0.00219 -0.08036 -0.00672 -0.15994 0.02830 0.10804 0.04618 0.04045

0.16873 -0.21259 -0.26563 -0.26924 -0.04268 0.11404 -0.08829 -0.13648 0.04345 0.29123

0.66197 0.14724 -0.03416 0.03833 -0.33729 -0.06886 -0.11543 -0.11044 0.23954 0.22220

-0.01815 -0.04875 0.05266 -0.00093 0.08393 -0.06272 0.03187 0.03571 0.02944 -0.01994

0.63181 0.09131 0.01670 0.03736 -0.28167 -0.12726 -0.08724 -0.07867 0.27603 0.19783

New Brunswick Quebec Ontario

Manitoba Saskatche~'an Alberta British Columbia

0.25446 -0.19880 -0.26400 -0.21626 -0.03620 0.18507 -0.11338 -0.22068 -0.00261 0.24103

O. Fisch, Contributions to the general theory of movement

165

After this preliminary analysis of the results, some general discussion of the meaning of the nodal and systemic changes are in order. First, if we are given the vectors v-(vi) and to={toi), in which specific situation of movement, are the estimation of the flows mo still relevant? In terms of population forecasts for each node, it seems that the estimation of flows per se is not critical and the given vectors v and to are the output we generally seek. In terms of travel demand forecasting, the estimation of the flows are at the core of the transportation planning process, with the given vectors v and ca representing the changes in trip production and in trip attractions respectively, for eaca traffic zone. Table 3 System solution for the period 1971-1976, based on prior information (1966-1971). Province

N c d e v(3)

d(3)

v(3)

w(3)

c(3)

~o(3)

Newfoundland Prince Edward Is. N o v a Scotia New Brunswick Quebec Ontario Manitoba Saskatchewan Alberta British Columbia

1 z 3 4 5 6 7 8 9 10

- 0.06920 -0.01827 -0.02040 - 0.04949 -0.05571 0.02110 0.00092 0.06217 - 0.02510 0.04444

0.01379 -0.09224 -0.14514 - 0.18077 -0.17467 0.17276 - 0.07400 - 0.18302 0.10504 0.16066

0.37787 0.22068 0.01755 0.09962 -0.12892 -0.12160 -0.00659 0.00493 0.20615 - 0.00240

0.02821 -0.00164 0.04951 0.01074 0.07627 -0.06048 0.01062 0.06683 0.01752 0.02654

0.41674 0.21868 0.05875 0.11143 - 0.06248 - 0.17472 0.00396 0.07209 18502 0.02,;08

0.08915 -0.07535 -0.12734 - 0.13820 -0.12598 0.14852 - 0.07485 - 0.23084 0.13348 0.11127

Second, if we are given the vectors v=(vi) and o9=(ogj), in which specific situation ~r movement are the explicit computation of the vectors v=(vi) and w=(wj) ai, d the imphcit computation of the vectors d=(di) and c = (c~) still relevant? In terms of population studies, the most relevant contribution is in the area of social indicators of nodal and systemic changes of migrational pull and push characteristics at each node. In transportation systems, the above vectors are the components of the zonal and systemic changes affecting trip production and attraction at each traffic zone, indicating the changing characteristk:s of households, employment and in modal-split choices. Third, if we are not given the vectors v=(vi) and o9 =(o9~), then, given the reciprocal relationship established in the theoretical section, we have to have explicitly given the nodal share change vectors v = (v~) and w= (w~) with the implicit result of the system share change vectors d = (d~) and c = (cj), based on a prior information of flows in the system. In this case, what kind of social, economic and demographic variables (or combination thereof) can be considered as explanatory of nodal push and pull changes? Below, in the rest of this subsection, we will address this question.

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O. Fisclg Contributions to the general theory of movement

In relation to the third argument, the solution of the system of equations, is in terms of shares of total movers moving in or moving out of a particular node, but the total number of movers in the system in a predicted period is unknown. The main empirical question that remains is: which set of changes in socioeconomic variables at the node, can we associate with nodal push and pull changes? The usual set are changes in population size (1. + pp), changes in per capita income ( l . + y ) or in wages (l.+ww), and changes in unemployment rate (1. +ur). Furthermore, which subset of these variables can we associate with the nodal push change (l.+v~), associating the complementary subset with the nodal pull change (1. + w~)? From our cross-sectional analysis of the Canadian data (see appendix A.5), a week pattern emerges: changes in population size and changes in unemployment rate are the variables that show a significant positive association with the computed nodal push change; and changes in per capita income shows a significant positive association with our computed nodal pull change. in terms of explanatory power, the results are mixed, with the best pair of results given by

(I. + ri(3)) = - 2.701 +0.794(1. + y~(3)) + 2.094(1. + pp~(3)) (2.67) (6.43) +0.154(1.+ ur(3)),

R 2 =0.893,

(2.49~ (1. + w~(l ))= -2.739 +0.799(1. + y~(l )) + 2.468(1. + pp~(l )),

(5.13)

(5.21) R 2 =0.857.

In reduced form, this pair is explained by change rates as follows: Vi~3)= -0.659 + 0.794y~(3) + 2.094ppi (3) + 0.154uri(3), wj(l )= -0.472 + 0.799y~(1 )+ 2.468ppj(l ). The unsatisfactory aspect of this specification is 1the fact that both income and population change are explanatory variables at the same time of both nodal push and pull changes.

O. Fisch, Contributions to the general theory of movement

167

3.2. Predicting movements in the system The solution of the system,,; (29), (30), and (31), allows the estimation of the flow matrices in the system; One estimation matrix r~(tt)=[r~o(t~)], for the second period t l based on prior information of share flows from point to

rho(t 1) = [ 1. + vl(1 )]toO(to )[ 1. + w~(l )1

all i, j,

and t w o estimation matrices for the third period t2, one matrix =[~o(t,)], based on prior information of share flows from period to

r~o(t2)=[l.+vi(2)]mo(to)rl.+w~(2)],

all i,j,

(32)

r~(t,) (33)

and one matrix vh(t2)=[rho(t2) ] based on prior information of share flows from period tl ~ho(t2)= [ 1. + vi(3 )]mij(tl )[ 1. -I- wj(3 )],

all i, j.

(34)

It can be seen, from the equations stated above [i.e., (32), (33) and (34)1, that the current estimated share flow is the prior actual share flow, times the change of nodal push at the origin i, times the change of nodal pull at the destination j. The systemic changes of pull working at the node i and the systemic changes of push working at node j are not explicitly included in our estimating of share flows procedure. We are making the conjecture that they are imp!icitly included in the transformation from a priori probabilities to a priori share flow, used in our estimation. The usual way of measuring the efficiency of the estimation procedure, is to produce the deviation matrices for each period. In our case~ it is simply given by (see appendix)

e.ij(l)=mij(tl)-rho(tl),

all i,j.

(35)

e,o(2)=mo(t2)-thij(tz) ,

all i,j,

(36)

~ij(3)=mo(t2)-rho(t2),

all i,j.

(37)

But the ennumeration of deviations of the estimations, is a rather descriptive procedure. The availability of three sequential periods, allows comparing j'lst 1~:o(2)i~1eo(3)i and we can ask: which a priori information, to or tl, is a superior base for estimation at t2? Our ex-ante hypothesis was that, given that time deteriorates the quality of information, we expected that tl will be superior to Zo as a base for the estimation of every movement in the system. In our empirical case, tl was superior only on 63?/o of the cases, with to

168

O. Fisch, Contributions to

the general theory of movement

superior in the remaining 37% of the cases. One 'possible explanation for these results, is that there is an oscillating inter-period behavior that our intra-period model does not capture (see table 4). Table 4

Superiority of prior information base for estimatin 8 movements in period t2. to

to tt

tt

--

tt

tt

to

tt

to

to

to

to

to

6

3

tt tt to tt

-tt to to

tt -to t~

tt to -tt

to to tt ~

tt tt tt tt

tI tI tt to

tt tt tt to

tt tt t, t~

to tt tt tt

2 2 3 3

7 7 6 6

tt

to

to

to

tt

--

tt

tt

to

tt

4

5

to

tt

tl

tt

to

tt

~

to

tt

to

4

5

to tt to

to tt tt

tt tt tt

tt tt t!

to tt t!

to to tt

to tt tt

--tt to

tI ~ tt

to tI ~

6

3

| 2

8 7

4 5

4 5

2 7

3 6

4 5

3 6

3 6

4 5

2 7

4 5

33 57

Table 5 System solution for the period 1971-1976, based on prior estimated information (1966-1971). Province

Node v(4)

d(4)

v(4)

w(4)

c(4)

oJ(4)

Newfoundland Prince Edward is. Nova Scotia N e w Brunswick Quebec Ontario Manitoba Saskatchewan Alberta British Columbia

I 2 3 4 5 6 7 g 9 10

- 0.05374 -0~,891 -0.U1208 - 0.04645 -0.04321 0.00999 -0.00435 0.05714 -0.02677 0.05434

0.01379 -0.09224 - 0.14514 -0.18077 -0.17467 0.17276 - 0.07400 - 0.18302 0.10504 0.16066

0.42018 0.22996 0.02350 0.11128 -0.13910 - 0.11209 -- 0.02154 0.00378 0.20007 - 0.00819

-0.00243 - 0.0091~ 0.03444 0.00013 0.08888 - 0.07054 0.02606 0.06805 - 0.01254 0,03254

0.41674 0.21868 0.05875 0.11143 -0.06248 - 0.17472 0.00396 0.07209 0.18502 0.02408

0.07136 -0.08408 -0.13469 - 0.14086 -0.13740 0.16156 - 0.06995 - 0.22718 0.13544 0.10084

The availability of these three sequential data sets, allows us also to ask how prior information based on estimated flows compares with actual prior information. Solving the system with estimated share flows at tl (see table 5)

[l.+vi(4)]~[l.+wj(4)]~hij(tl)=mi.(t2),

all i,

(38)

J

[l.+w~(4)]~[l.+v,(4)]ph,j(tl)=m.j(t2), J

all

j.

(38')

O. Fisch, Contributions

~o

I! II ~

the general theor~ of movemem

,,,.=(~,~ ,,.?,~ I

(,,7 II o

II II II II °~

II II

o

o,~

IN

II

II

It II

II U II II

o

II II o

II

H

0 II H

II

II II II II

II

169

O. Fisch, Contributions to the general theory of movement

! 70

We can estimate the matrix ~(t~)=[mi~(te)] of share flows at te

(39)

~o(tz)=[l. + v~(4)]~fio(tl )[l. + w~(4)], all i, j, and the corresponding matrix of deviations (see appendix)

f4o)

~,O(4)---mii{t2)--610(t~), all i,j.

The first comparison is [ei~(2)[~[ei~(4)[ for the period t2, and w,,.~ can ask which a priori information on movements, the actual at to or the estimated at t~ is a superior base for estimation at t~. In our case (see table 6): in 33% of the cases both were equivalent; in 36 % of the cases to was a SUl~:,'rior base; and in the remaining 31 0.o, t l was a superior base.

Table 7 Superiority of prior information base for estimating m o v e m e n t s in period t2.

t_! t~

t~ t~ t~

--t~ t~

t~ .... ~

t~ t~

t, t: t~

t~ tl t~

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58

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32

The second comparison is le, 3)lXle, t4)l for the period t 2, and we can ask which a prior information on movements, the actual at tl or the estimated at /'1 is a superior base for estimation at t2. Again, we expected that the actual will be superior to the estimated base throughout, but this was the ca~e in only 64° 0 of the cells, with the remaining 36% the estimated flow are the superior base (see table 7)u In general, time partially deteriorates the quality of prior information of movement in the system, and the estimated flows at t l work as an equivalent information base with to.

O. Fisch, Contributions to the general theory of movement

171

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O. Fisch, Contributions to the general theory of movement

173

A.5. Pearson correlation coefficients.

1 +v(l) 1 + w(l )

l+y(l) -0,3761 0.5490

1 +PP(I) 0.602 0.564

l+ww(l) 0.161 0.209

1 +ur(l) N/A N/A

1 + v(2) 1 + w(2)

1 + y(2) 0.0591 0.7272

1 + PP(2) 0.814 0.275

1 + ww(2) 0.366 0.338

1 + ur(2) N/A N/A

1 +v(3) 1 +w(3)

1 + y(3) 0.2156 0.4222

1 + PP(3) 0.868 -0.035

1 + ww(3) 0.376 0.153

1 + ur(3) 0.093 -0.386

"['he transformed data was computed from Ledent's (1980) original data.

References Alonso, William, 1976, A theory of movements, IURD-WP-266 (University of California, Berkeley, CA). Anselin, L. and W. lsard, 1978, On Aionso's general theory of movement, Man, Environment, Space and Time 1, no. 1, forthcoming. Bacharach, Michael, 1965, Estimating nonr,egative matrices from marginal data, In',ernationai Economic Revi~:w,293-310. Fisch, O. and S. Gordon, 1979, RAS vs. MAN in minimum survey methods of input-output tables, Modeling and Siraulation 10, 1405-1412. Ledent, J., 1980, Calibrating Alonso's general theory of movement: The case of interprovincial mig~'ation flows in Canada, WP-80-41 (IIASA, Vienna). Stone, R. et al., 1962, A programme for growth, Vols. I and 2, Department of Applied Economi~s, Cambridge (Chapman and Hall, London).