Synopsis

Residential Location Choice Modelling: Gaussian Distributed Stochastic Utility Functions FRY. T. J. GRIGG TA 1 , U4956 search Report No. CE33 July, 1982 III��\� l\Il I I I I I I I I I�II I���� IllI 3 4067 03255 7828 CIVIL ENGINEERING RESEARCH REPORTS This report is one of a continuing series of Research Reports published by the Department cif Civil Engineering at the University of Queensland. This Department also publishes a continuing series of Bulletins. Lists of recently published titles in both of these series are provided inside the back cover of this report. Requests for copies of any of these documents should be addressed to the Departmental Secretary. The interpretations and opinions expressed herein are solely those of the author(s). Considerable care has been taken to ensure the accuracy of the material presented. Nevertheless, responsibility for the use of this material rests with the user. Department of Civil Engineering, University of Queensland, St Lucia, Q 4067, Australia, (T el:(07) 377-3342, Telex:UNIVQLD AA40315] RESIDEr!TIAL LOCATION CHOICE f10DELLHlG: GAUSSIAN DISTRIBUTED STOCHASTIC UTILITY FUNCTIONS by Trevor J. Grigg, B E, BEcan, PhD, Senior Lecturer in Civil Engineering R E S EARCH REPORT NO. CE 33 Department of Civil Engineering University of Queensland July 1982 Synopsis Modelling urban residential location choice using the theory of constrained utility maximiei�� behaviour by individual locators requires a probabilistic approach. One such approach using random utility theory assumes utilities are stochastic but that decis1:on making is deterministic. This approach can only be made operational by adopting explicit distributional forme for the stochastic utility functions. The extreme value type I (Gumbel) distribution is almost universally adopted, principally for mathematical convenience rather than for any particular conceptual reasons. This paper examinee the derivation of a choice model based on Gaussian distributed stochastic utility functions. Expr•eeeione for• locator's realised utility are also developed. The presentation is confined to the case of mutually independent alternatives. Although the paper specifically addressee urban residential location choice, the results are equally applicable in the context of retail shopping mod�le or trip distribution modele. CONTENTS Page 1. INTRODUCTION 2. THE CHOICE FRAt-1El�ORK 3 3. THE PREFERRED CHOICE IN AN AREA 5 3.1 The Prob ability Districution of the S tochastic Utility Function 5 3.2 The Utility of the Preferred Alternative in an Area 6 4. SELECTION OF THE PREFERRED AND REALISED UTILITY 5. FURTHER CONSIDERATIONS 6. RESIDENTIAL AREA 11 15 5.1 Constancy of a i 15 5.2 A Useful Approximation 15 5.3 C omparison with the Multinomial Log it 1·1odel 16 5.4 Mutually Dependent Choices 18 5.5 Other Applications 13 CONCLUSIONS 19 APPENDIX A - NOMENCLATURE APPENDIX B - REFERENCES UNWERSiflf o�:. C: ..:. ·- · · r�K3 fNrA · • ;?vuPr_C.:.:Z. <ft'{t' . . :. � : ->�iftt ./ --_:...� - l. 1 - INTRODUCTION To model spatial choice and, in particular, urban residential location choice, one must first clearly specify the search and selection process. Focussing on the individual choice maker, that is, adopting a micro-behavioural level of resolution, would appear to be necessary if this is to be achieved. However, one also needs to account for the interplay of choice makers in this process and the mechanics of how individual demands are reconciled with supply (residential location opportunity) constraints at the aggregate level. A useful approach to the individual search and selection process is to use the notion of constrained utility maximising behaviour, wherein it is postulated that the decision taken will be that which maximises the searcher's utility. Such a theory to be operational requires that the utility functions be specified, constraints defined, and that the attributes of opportunities can be measured. The decentralized nature of decision making for individual locators is emphasized in such an approach. However, it is not realistic to apply the theory in a deterministic manner, rather a probabilistic approach will be required if the inevitable dispersion in behaviour which is observed is to be accommodated. There are two well known versions of probabilistic choice theory - the constant utility approach and the random utility approach. In the constant utility approach (Luce, 1959) it is assumed that the utility function is deterministic but that the decision making is stochastic (that is, not all individuals make the utility maximising choice). The lack of a rational choice behaviour -2assumption, in addition to the assumed constancy of perceived utility for all individuals, does not make this approach conceptually very attractive. In the random utility approach each individual is assumed to make his decisions rationally, in a manner consistent with his own preferences (utility) (Williams, 1977). That is, utilities are stochastic but the decision making is deterministic. not assumed to be identical, Individuals are but at the cost of assumed rationality. The dispersion in observed behaviour is obtained from the aggregation of individual conceptually. decisions. This approach is far more attractive To make the random utility theory operational it is necessary to choose explicit parametric distributions for the stochastic component of utility functions. The random utility approach will be adopted here. This approach, which has become the standard method of travel-demand analysis (Domencich and McFadden, 1975; W illiams, 1977), has also been applied to residential location choice (McFadden, without exception the extreme value type 1978). Almost I (Gumbel) distribution is chosen for the stochastic utility function. (A number of papers in the literature have been refering to this distribution as a Weibull distribution. type The Weibull distribution is actually the extreme value Ill distribution - a three parameter distribution as opposed to the two parameter type I distribution.) T his choice has been guided more by considerations of mathematical convenience and tractability than by any conceptual considerations. The implications of adopting this distribution form to the exclusion of others have not been addressed. There may be ramifications for model application and parameter interpretation. The aim of this paper is to derive the choice model which results when Gaussian distributed stochastic utility functions are assumed. -3THE CHOICE FRAMEWORK 2. Suppose an idividual at i face s a residential location decision, with a perceived choice of residential areas indexed j k l,2, . ,N and Hj locational opportunities indexed . . residential area j. . . . ,Hj in For each individual in i being considered, each alternative opportunity attributes x jk" i = 1,2, k in j will have a vector of observed These observed attributed could include dwelling and/or site characteristics, accessibility of the opportunity to zone i, purchase price and residential area (neighbourhood) quality. utility of this opportunity k The in j for the individual will be a function of these observed attributes x jk of the opportunity, a i vector s of the individual's characteristics (such as socioeconomic status and income), the unobserved attributes of the alternative, and other unobserved factors determining tastes. The vector n jk of i unobserved attributes and factors is assumed to have been drawn from Let the individual's utility some random probability distribution. u jk be defined by the utility function, i (1) Then U jk is stochastic, depending not just on the non-stochastic i vectors x jk and s, but also on the random vector n j k" i i If the individual behaves rationally, then the location opportunity which maximises his utility will be chosen. would prefer opportunity in j if U k ij > U ij m k That is, the individual at i in j in preference to any other opportunity for m f. k , m = 1,2,... ...,H . j lf the utility derived from the preferred opportunity in each area q, q = l,N is given by U = max (U kl for iq iq k = 1,2, . • . ,H , then an opportunity in j q will be chosen for residential location if U j i q=l,2, . • . ,N. > U iq for q I j, - 4 - Since the utility values are stochastic, the choice of a location in area j by this individual at i will occur with some probability, given by (2) With complete generality it is always possible to write the stochastic utility function in the form, u = U(x,s,n) (j) = V(x,sl + E(x,s) (4) where Vis the so-called 'representative utility', a constant for an alternative and a choice maker, and E is the random disturbance term reflecting unobserved factors affecting the choice (with a mean value independent of the alternative under consideration). That is, (5) To be able to specify the probability in Equation (2) in terms of the observed attributes and characteristics of alternatives it is necessary to specify both the explicit functional form and the probability distribution for the stochastic utility function U(x,s,n)·. This paper confines its attention to the probability distribution of the stochastic utility function. -53. THE PREFERRED CHOICE IN AN AREA 3.1 The Probability Distribution of the Stochastic Utility Function The most general assumption possible concerning the distribution of the random utility component E ij ' in the absence of any evidence to the contrary, is to assume it is a Gaussian ]. (The extreme value type I (Gumbel) ij distribution is usually chosen, principally for reasons of distribution E . . - lJ N[E . . lJ , o E mathematical convenience.) This distribution applies to all alternatives in j for a 1 ocator at i, it being assumed the. variance o[ . in the random term and its mean value t ' are independent of the ij lJ particular alternative under consideration. This random term E may ij . be mutually dependent among the alternatives in j. For opportunity k in j the stochastic utility function will be given by u " - N[(V .. +E ), o lJ k E lJ iJ k . . ij ]. (6) While the residential areas j ,j = 1,2,.,N may have been chosen with the aim of minimising the observed heterogeneity in locational opportunity attributes, there will inevitably still be variability in their 'representative' utilities V " ij k As for the random utility term, it will be assumed that these 'representative' utilities within ' ov. _]. Hence, the lJ stochastic utility function for an opportunity in j selected at �ndom an area j are Gaussian distributed as V ij k - N[V ij will be (7) {8) - 6 - and (9) Label the probability distribution N[U ' o ] as g(U ) and let ij ij ij G(U ij 3.2 ) be its cumulative form. The Utility of the Preferred Alternative in an Area One plausible hypothesis for the search and choice process is that it is sequential. First, the individual perceives a number of areas, each of essentially homogeneous character, and chooses one of them in which to initiate a more detailed search. (That is, the areas are viewed as essentially independent alternatives.) The second stage of the process is the selection of the preferred, utility maximising opportunity in the chosen area. If the modeller's interest is only with the area of choice then the second stage is of little consequence. However, this does not mean that the characteristics of the opportunities within the area can be ignored. In choosing an area, the individual will have 'summarised' the attributes of each of the alternative areas in terms of a few broad measures, based on an imperfect and approximate perception of opportunity and neighbourhood quality. For derivation purposes it is convenient to consider initially the selection of the preferred location opportunity in each of the areas j,j = 1,2, ... ,N. However, interest centres not so much on the probability that a particular opportunity is preferred but rather on the probability distribution of the utility to be derived from what is perceived to be the utility maximising opportunity in j. That is, we -7seek the probability distribution f(U ) given by f(max {U ijl ij u U ijk ' U , .. ij2 . }). The form of this distribution will depend on whether the j In this paper only the 's are independent or correlated. lJ H . . independent case· will be considered. Kendall and Stuart (lgS8) have shown that the characteristic limiting form (that is, the form for H + j oo, or in practical terms, the form for larg e H ) of the required distribution f(U ) in cumulative ij j form F(U ) is ij (10) where U .. lJO value of U ij expected to be exceeded just once in a sample of size H , given j implicitly by H.[1 J G(U - . . l JO )] = 1, (11) (12) and ( 13) c If 1 im c u ..,. oo ij = 0, then 1 im F(U) ( 14) Ui j ..,.,., Distributions for which lim c distributions. g(U ij = 0 as U +oo are termed "exponential type" ij The Gaussian distribution is one such distribution. l is Gaussian distributed, where -8exp[ -�[(U g(U1.J.) lJ . . -u lJ . . ) ;(o . l J . )/ J/o lJ . . . (2IT);,J (15) By using Equation (15) in conjunction with Equations (11), (12) and (l�) it can be shown that the limiting form of F(U j) is an extreme i value type I (Gumbel) distribution. The actual mathematical form of F(U j) is, i u with mode u . . lJO = U j i + [(2 ln Hj) - \(ln.ln Hj + J l O . . )} ] (16) J, \ ln 4IT)/(2 ln Hj) ].o j i (17) (l!l) mean (19) coefficient of skewness = 1.14 (20) J, (21) "' (2 ln Hj) Jo j. i with The expression for F(U j) and its statistical parameters were derived i \ omitting terms of magnitude less than (2 ln Hj )- The distribution has the same general 'bell' shape as the Gaussian distribution but is positively skewed. and (21) indicate that both the mean Di j Equations (18), (19) and the varianceo2( j) are i functions of the number of opportunities Hj in j, the former increasing and the latter decreasing with increasing Hj. In Figure the mean and variance of F(U j) are plotted in standardised form as a i function of Hj; ,_ �" 1,0 ,F 5 b ,.. C:J > w 0 0,6� <( 0 z <( 1- 0 w Vl 0,4 1./) 02 0::: <( 0 z a"'(u) w Vl �1 Vl 02 � , Ti pp e tt (1925) ---- 0 0 10 Figure 1 100 1000 NUMBER OF ALTER NATIVES I 10000 H Expected utility and utility standard deviation as a function of number of alternatives <( 0 z <( 0 t/i I <.0 I - 10 - (22) Tippett (lY25) has calculated the exact distribution of F(U ) for ij varying sample sizes. His results are also plotted on Figure 1. Only for sample sizes less than 10 do the approximations inherent in the derivation of Equation (16) exceed five percent. For sample sizes greater than 1000 the error is less than one percent. It is concluded that the terms omitted in deriving Equation (16) have not adversely affected the accuracy of the estimates of the distribution parameters. -114. SELECTION OF THE PREFERRED RESIDENTIAL AREA AND REALISED UTILITY It is hypothesized that the rational individual at i will choose as his preferred residential area, that area j which .contains the preferred opportunity of maximum utility. Hence, the expected probability of that individual locating in area j, p ' will be given ij by, =f = :::f = Pr(U ij (24) U, U iq < U, V q£N).dU (25) U, (�6) 0 00 Pr(U ij U, U iq since the pr obability of < V q£N).dU the utility of the preferred alternative being negative is very small. If each of the residential areas are viewed as independent, unique localities by choice makers then we can write p ij . lJ lJ oo f -co where f as = -Joo [PPrr(U(U .. <U)= U)J· [klN= ' Pr(Uik <U)l U rr co p ij .d [fij (U)(U ] [ F.k Fij j k=l,N --- . (27) - n (28) 1 (U) is the p.d.f. of F; j (U ) . ij (Writing, for c onvenience, -12- Solution of Equation (28) for F ij (U), as defined by Equation (lb), is not mathematically tractable except for some special cases; for example, unless it is assumed that a residential areas jEN. is constant across all _ From Equation (21), it can be seen that for a ij ij to be constant for all j then either (a) H and o have to be constant for all j; j ij or (b) the ratio of (2 ln H )�/o has to be constant j ij for all j. Whilst it is perhaps within the control of the modeller to achieve approximate constancy of H through the choice of zoning system, it j would be pure coincidence if potential locators perceived the available areas of choice in the same way. for all j Constancy of o ij is again unlikely because the degree of homogeneity (similarity) of the available opportunities H cry .. will vary by zone. lJ on o ") ij satisfied. j in each j will no doubt vary. That is, (Equation (9) specifies the influence of o v .. Consequently, condition (a) above is unlikely to be Condition (b) would be satisfied if o ij lJ was directly � proportional to (2 ln H l . with the constant of proportionality being j a constant fraction of 1/a " ij If it can be argued that the diversity of available opportunities in an area will increase as the number of opportunities increase then condition (b) may well capture (albeit very approximately) such a relationship. reasonable line of argument. This appears to be a more Alternatively, one could adopt a purely mathematical approach and argue that since (2 ln H )"'2 will vary 1 ittle j for moderate variation of H over all j and provided the degree of j homogeneity also does not vary widely over all j, then to sufficient accuracy, condition (b) will be satisfied. It is assumed in the 13. remainder of this derivation that u That is , Substituting ij is constant across all jEN. a j = a i , for a11 j . i (29) Equations (16) and (2�) into Equation (28), we obtain after integration, p This ij = exp (a O l/ I exp (a u kl i i i ij k=l,N expression for p ij (30) has the form of the familiar multinomial legit model (Domencich and McFadden, 1975). Note however, that U ' the ij expected maximum utility attainable from the preferred opportunity in j, and not O ' the expected utility of a randomly selected ij opportunity in j, is the appropriate utility to enter into the multinomial legit in this case. The cumulative probability distribution �(U ) of the maximum utility ij to be derived from the preferred locational opportunity in j, conditional on area j being chosen is uij J Pr(U ij = U, U k i < U, IJ ksN).dU]/p ij (31) which, using Equations (16) and (30), yields �(U .. ) = exp [-{I exp( a. u .,, )} exp(-a. U . .)] 1J 1 ho 1 1J k=l,N (32) This is another Gumbel distribution with a mode u .. a mean u 1JO ij = = ) (1/a ).ln[ I exp(a u i i .k ] 1 0 k=l,N u .. 1J0 + 0.577/a. 1 (jJ) (34) 1�. (1/a;l.lnLI exp(a ; k=l,N (J5 ) Uik)] {36) and a skewness of 1.14. This result, which comes from the stability of the Gumbel distribution under maximisation, shows that the statistical properties of the utility enjoyed, conditional on an opportunity in a particular area j being chosen, are independent of the area in question. That is, U ij = Note also that em U for all j i a(U lJ ) . . a(u .) 1 for all j The constancy of U for all j is a not unexpected result. i (J8) Simply, it means that from a probabilistic point of view choices are made in such a way that choice equilibrium is attained. 15. S. FURTHER CON$!UERATIONS 5.1 Constancy of a i The choice model derived in Equation (JO) indicates that ai may need to be determined for each search origin i. However, provided that only the same 'type' of individual choice maker is searching from all i, then there is no need to differentiate a search origin. a i i by That is, (39) a for all A Useful Approximation 5.2 By substituting for U ij as defined by Equation (lH), and by using Equation (J9), the area choice probability given by Equation (JO) can be re-expressed as, P. lJ . = exp(aO . . lJ + 2 ln H.-� J ln.ln H.)/ J (40) (41) There is no reason why Equation (41) could not be applied in the form chosen. However, a useful approximation can be made. The terms (H /(ln H )�) for all j are dominated by the variations in the j numerator. For example (ln H l� only increases by eleven percent if H j j j 16. increases from 1000 to 5000. (rl } j 2 on· the other hand increases by �400 percent. Consequently, to reasonably accuracy, Equation (41} can be written as p � - .. }/( \L Hk exp(a Ui H.� exp(aU lJ k}} J k=1,N (42} . iJ. where � takes a value in the range 1.�2 10 000. - 1.�b for H in the range 10 - The actual variation of � as a function of H, the number of alternatives in an area, is plotted in Figure 2. It can be seen that � varies little over quite wide ranges of H. Since in most applications H will not vary greatly from one area to another, a single average value of �applying to all areas should suffice. 5.3 Comparison with the Multinomial Logit Model By way of comparison, the expression obtained for p when the ij stochastic utility functions are assumed to be extreme value type I (Gumbel} distributed and the location alternatives are assumed to be independent is (Luce, 1959} (4::S} Equation (4::S) is noticeably different form Equation (42) as far as the exponent on the area alternatives is concerned. This difference arises directly from the choice of the assumed distributional form of the stochastic utility functions. Equation (43) is not equal to a in Equation (42). al in Grigg (l9�2b} compares the two distributional assumptions in much greater detail. Ci >--' " 0 10 100 1000 1000 0 NUMBER OF ALTERNATIVES Figure 2 a as a function of H , H 100000 18. 5.4 Mutually Uependent Choices McFadden (1�7b) has addressed the case of residential location choice modelling when groups of alternatives are perceived similarly. He applies the multinomial logit model to groups of similar alternatives. The derivation for independent alternatives described in this paper can be extended to cover this case of choice alternative mutual dependence (Grigg, 1Y82a). In the special case where all alternatives within any set are perceived as identical by an individual choice maker, then � takes a value of zero and Equation (4j) becomes, (44) since the locator is then merely choosing between areas. is not equal to 5.5 (Note that a. l Other Applications While the discussion in this paper has centred on residential location choice models, the results are equally applicable to trip destination choice (trip distribution) and retail shopping choice models. The exponent on the retail shopping models. Hj term is of some significance for Many such models take the form of Equation (42), with an exponent on the so-ca11 ed attraction term. The relationship of the size of the exponent to intra-area utility correlation may provide new insight into interpreting such models (Grigg, 1Y!J2c). ac 19. 6. CONCLUSIONS The extreme value type I (Gumbel) distribution has been almost the only parametric distribution assumed for stochastic utility functions in random utility choice modelling. This paper has shown that the assumption of Gaussian distributed stochastic utility leads to a choice model of similar form, though different in detail, to that of the multinomial logit model. While only the case of independent alternatives was examined in detail, brief reference to the implications of mutually dependent alternatives was made. may also provide new insight into the shopping models. 'attraction' The results term of retail 20. APPENDIX A - NOMENCLATURE �1eani ng f,F p.d.f. and c.d.f. of utility of randomly selected alternative g,G p.d.f. and c.d.f. of utility of area preferred alternative i ,j ,k ,m area labels p area choice probability q area label vector of choice maker characteristics X vector of alternative characteristics utility random disturbance term, its expected value and standard deviation H number of alternatives N number of search areas u,O,o total utility, its expected value and standard deviation expected utility and standard deviation for area preferred alternative U,o(U) expected utility and standard deviation for preferred alternative expected maximum utility from a sample of alternatives 'representative' utility, its expected value and standard deviation parameters parameters c.d.f. of utility of preferred alternative 2 1. APPENDIX B - REFERENCES 1. DOnENCICH, T. A. and McFADDEN, D. "Urban Travel Demand: a Behavioural Analysis", North Holland, Amsterdam, 1975. 2. GRIGG, T.J. atives", "Spatial choice modelling with mutually independent altern­ Research Report No. CE35, Department of Civil Engineering, University of Queensland, Brisbane, Australia, 3. GRIGG,. T.J. functions: 1982a. "The choice of parametric distribution for stochastic utility effect on the structure and parameter interpretation of aggregate choice models", Research Report No. CE 39, Department of Civil Engineering, 4. GRIGG, T.J. University of Queensland, Brisbane, Australia, 1982b. "Parameters of the retail trade model: a utility based interpretation", Research Report No. CE37, Department of Civil Engineering, University of Queensland, Brisbane, Australia, 1982c. 5. KENDALL, f·1.G. and STUART, A. Hafner, New York, 6. LUCE, R.D. "Individual Choice Behaviour: a Theoretical Analysis", Hi 1 ey, New York, 7. lkFADDEN, D. 1959. "1·1odelling the choice of residential location", Transportation Research Record, Vol. 8. "The Advanced Theory of Statistics: Volume 1", 1958. TIPPETT, L.H.C. 673, 1978, pp 72-77. "On the extreme individuals and range of samples taken from a normal population", Biometrika, Vol. 9. 17, 1925, p 364. l'i!LL!Af.1S, H. C .H. L. "On the formation of trave 1 evaluation measures of user benefits", 1977, pp 285-344. d2mand mode1 s and economic Enviroment and Planning A, Vol. 9, CIVIL ENGINEERING RESEARCH REPORTS CE No. 20 21 ��uthor( s) Title of Axi-symmetric Rodles Non Axi-syt"metric Loading Consolidation CARTf.R, J.P. Subjected �llOKER, J.R. to KUNJAHBOO, Truck Suspension Houels O'CONNOR, 22 23 Date K.K. CARTER, J.P. Circular Tunnel !lOOKER, J.R. Study of illuckage WEST, 1981 & Harch, & 1981 G.S. April, Effects on Some Bluff Profiles 24 1981 DUX, Inelastic: Beam Buckling Experim0nts P.F. & Hay, KI'fli'ORNCHAI, 25 Critical Assessment of th� February, 1981 C. Elastic Consolidation Around a IJ(•c'p An Experimental J anuary , & International Estimates for Relaxation Losses ·in KO}{J:TSKY, S. A.V. PIUTCHARD, R.W. & 1981 June, 1981 Prestressing S trands 26 Some Predic:a tions of the Non-homogC>nous CARTER, July, .J.P. 1981 Behaviour of Clay in the Triaxial Test 27 The Finite Integral Hethod Analys i s 28 : in Dynamic SWANNELL, August, P. 19 8 1 A Reappraisal Effects of Laminar B ound ary Layer on a l SA/ICS, September, L. T. 1981 Mod el Broad-Crested Weir 29 Blocka g e and Aspec t WEST, Flow Past a APELT, C.J. Time D ep en den t Deformation in P re st r es se d SOK/l.L, Y • .J. Concrete Girder: TYRER, P. � 10 30 31 < R < 10 Ratio Effects on Circu lar Cylinder for 5 Measurement and PreU.ic.tion Non-uniform Alongshore Currents and Sediment Transport - G.S. & October, 1981 November, & 1981 GOURLAY, M.R. January, 1982 a One IHmcnsiona 1 Approach 32 33 A Theor eti cal Study of Pore \-la t e r Prcs<,ure� ISAACS, L.T. Developed in llydraul.ic Fi 11 in MlnL' CARTEH, J.P. Stop"s Residential Location Cltoicl' �odt>llin:.:: GRIGG, T.J. Gaussian Distribution Stochastic Ut i.l i ty & February, 1982 Julv ,1982. Fun c tions 34 The Dynamic Charactt�r is tics of Pressure T r ans d uc e r s Sl>llll' Lo\V WeST, C.S. August, 1982 CURRENT CIVIL ENGINEERING BULLETINS 4 Brittle Fracture of Steel - Perform­ 11 Analysis by Computer - Axisy­ metric solution of elasto-plastic pro­ blems by finite element methods: J. L. Meek and G. Carey (1969) 12 Ground Water Hydrology: J. R. Watkins (1969) 13 Land use prediction in transportation planning: S. Golding and K. B. David­ son (1969) 14 Finite Element Methods - Two dimensional seepage with a free sur­ face: L. T. Isaacs (1971) 15 Transportation Gravity Models: A. T. C. Philbrick (1971) 16 Wave Climate at Moffat Beach: M.R. Gourlay ·(1973) ance of NO 1B and SAA A1 structural steels: C. O'Connor (1964) 5 Buckling in Steel Structures- 1. The use of a characteristic imperfect shape and its application to the buckling of an isolated column: C. O'Connor (1965) 6 Buckling in Steel Structures - 2. The use of a characteristic imperfect shape in the design of determinate plane trusses against buckling in their plane: C. O'Connor (1965) 7 Wave Generated Currents - Some observations made in fixed bed hy­ draulic models: M. R. Gourlay (1965) 8 Brittle Fracture of Steel -2. Theoret­ ical stress distributions in a partially yielded, non-uniform, polycrystalline material: C. O'Connor (1966) 9 Planning and Evaluation of a High Speed Brisbane-Gold Coast Rail Link: K.B. Davidson, et al. (1974) 19 Brisbane Airport Development Flood­ way Studies: C. J. Ape/t (1977) 20 Numbers of Engineering Graduates in Queensland: C. O'Connor (1977) Force Analysis of Fixed Support Rigid Frames: (1968) J. L. Meek and R. Owen of Traffic Lucas and 18 Analysis by Computer -Programmes for frame and grid structures: J. L. Meek (1967) 10 17. Quantitative Evaluation Assignment Methods: C. K. B. Davidson (1974)