A general forulation of the cross-nested logit odel Michel Bierlaire, Dpt of Matheatics, EPFL, Lausanne Phone: Fax:

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1 A general forulation of the cross-nested logit odel Michel Bierlaire, EPFL Conference paper STRC 2001 Session: Choices

2 A general forulation of the cross-nested logit odel Michel Bierlaire, Dpt of Matheatics, EPFL, Lausanne Phone: Fax: Eail: Abstract The cross-nested logit (CNL) odel has been entioned for the first tie by Vovsha (1997) in the context of a ode choice survey in Israel. Actually, it is alost identical to the Ordered GEV odel proposed by Sall (1987). This odel, eber of the GEV faily (McFadden, 1978), is appealing for its ability to capture a wide variety of correlation structures. Papola (2000) has shown that a specific CNL odel can be derived for any given hooschedastic variance-covariance atrix. Therefore, the CNL odel, with a closed for forulation derived fro the GEV odel, becoes a serious copetitor for the probit odel. In this paper, we develop on the general forulation of the Cross Nested Logit odel proposed by Ben-Akiva and Bierlaire (1999). We show that the forulations by Sall (1987), Vovsha (1997) and Papola (2000) are particular cases of our general forulation. We also provide soe new insights in the CNL odel based on theoretical analysis and epirical tests. We show that one of the conditions iposed by Sall (1987), Vovsha (1997) and Papola (2000) is not necessary for the CNL odel to be consistent with rando utility theory. This condition iposes that the su of the paraeters capturing the degree of ebership of an alternative to a nest is equal to one. Weprove that the CNL odel is a GEV odel without using that condition. It is actually a noralization condition, iportant for paraeters identification, but not for odel forulation. Finally, we use a siplistic odel to illustrate the role of the noralization condition. We propose a variant of the condition proposed in the literature which is slightly ore general. Keywords: logit odel, cross-nested odel, GEV odel, rando utility, transportation deand Swiss Transport Research Conference STRC 2001 Monte Verita

3 1 Introduction Discrete choice odels play a aor role in any fields involving a huan diension, including transportation deand analysis (the recent Nobel prize to D. McFadden is a perfect illustration of that stateent). Their nice and strong theoretical properties, and their flexibility to capture various situations, provide a vast topic of interest for both researchers and practitioners, that has (by far) not been totally exploited yet. The particular structure of transportation related choice situations are not always fully consistent with the underlying odeling theory (Ben-Akiva and Bierlaire, 1999), requiring to enhance and adapt existing odels. The theory on GEV odels have been introduced by McFadden (1978). It provides a treendous potential, as it defines a whole faily of odels, consistent with rando utility theory. It appears that only a few ebers of this faily have been exploited so far. Aong the, the cross-nested logit (CNL) odel has been used by Vovsha (1997) in the context of a ode choice survey in Israel. Actually, Vovsha's odel is alost identical to the Ordered GEV odel proposed by Sall (1987). This odel is appealing for its ability to capture a wide variety of correlation structures. Papola (2000) has shown that a specific CNL odel can be obtained for any given hooschedastic variance-covariance atrix. Therefore, the CNL odel, with a closed for forulation derived fro the GEV odel, becoes a serious copetitor for the probit odel. It has been shown to be specifically appropriate for route choice applications (Vovsha and Bekhor, 1998), where topological correlations cannot be captured correctly by the ultinoial and the nested logit odels. Unfortunately, no satisfactory procedure has been proposed for the estiation of the odel paraeters. 1

4 The reaining of this section introduces the GEV odel and presents various forulations of the Cross-Nested Logit odel. In Section 2, we analyze the ost general forulation. We prove that it is consistent with the GEV odel faily. 1.1 The GEV odel The Generalized Extree Value (GEV) odel has been derived fro the rando utility odel by McFadden (1978). This general odel consists of a large faily of odels that include the Multinoial Logit and the Nested Logit odels. The probability of choosing alternative i within the choice set C of a given choice aker is P (ic) = i (y 1 ;:::;y J ) (1) G(y 1 ;:::;y J ) where J is the nuber of available alternatives, y i = e V i, V i is the deterinistic part of the utility function associated to alternative i, and G is a non-negative differentiable function defined on R J + with the following properties: 1. G is hoogeneous of degree >0, that is G(ffy) =ff G(y), 2. li yi!+1 G(y 1 ;:::;y i ;:::;y J )=+1, for each i =1;:::;J, 3. the kth partial derivative with respect to k distinct y i is non-negative ifk is odd and non-positive ifk is even that is, for any distinct indices i 1 ;:::;i k 2 f1;:::;jg, we have ( 1) k i1 :::@x ik (x)» 0; 8x 2 R J +: (2) Note that the hoogeneity of G and Euler's theore give P (ic) = e V i+ln G i (:::) P J =1 ev +ln G (:::) ; (3) where G i. It is well known that the Multinoial Logit and the Nested Logit odels are instances of this odel faily. We present now several forulations of the Cross-Nested Logit odel, derived fro the GEV odel. 2

5 1.2 Forulations of the Cross-Nested Logit odel The liitations of the Nested Logit odel has been observed by several authors (Willias, 1977, Forinash and Koppelan, 1993). The requireent ofunabiguous assignent of alternatives to nests does not allow to capture ixed interactions across alternatives. It sees that the first Cross-Nested Logit odel has been proposed by Sall (1987) in the context of departure tie choice. Sall's odel, called the Ordered GEV odel, isbased on the following function:! J+M ρr G(y 1 ;:::;y )= w r y 1=ρr ; (4) r=1 ψ Br where M is a positive integer, ρ r and w are constants satisfying 0 < ρ r» 1, w 0and M =0 The B r are overlapping subsets of alternatives: w =1: (5) B r = f 2f1;:::;Jgr M»» rg: (6) Vovsha (1997) applies the Cross-Nested Logit odel to a ode choice application, where the park&ride" alternative is allowed to belong to the coposite auto" and the coposite transit" nests. Vovsha derives the Cross-Nested Logit fro the GEV odel with the generating function: G(y 1 ;:::;y J )= ψ 2C ff y! (7) where is the nest index, and ff are odel paraeters such that 0» ff» 1 8; ; (8) and Vovsha (1997) iposes also that ff > 0 8i: (9) ff i =1 8i: (10) 3

6 Ben-Akiva and Bierlaire (1999) ention the CNL as an instance of a GEV odel, and based on the following generating function: G(y 1 ;:::;y J )= ψ 2C ff y! : (11) A siilar forulation is used by Papola (2000), based on the following generating function: G(y 1 ;:::;y J )= k ψ 2Ck! k 0 ff 0= k ik e V i= k ; (12) with 0» k» 0. Papola iposes also that 2 Theoretical analysis k ff ik =1 8i: (13) Aong these forulations, (11) is the ost general. Indeed, Vovsha's and Sall's forulations are specific cases of (11). We obtain Sall's forulation (4) with = 1 and = 1=ρ. Vovsha's forulation (7) is obtained fro (11) with =1for all. Papola's odel (12) is equivalent to (11), with = 1= 0, = 1= and ff = ff 0=. However, Papola's constraint (13) is not required by our forulation. Note that Sall (5) and Vovsha (10) ipose the sae constraint. The following theore shows that (11) is indeed a GEV generating function. Theore 1 The following conditions are sufficient for (11) to define a GEV generating function: 1. ff 0, 8;, 2. P ff > 0, 8, 3. >0, 4. > 0, 8, 5.», 8. 4

7 Proof. We show that, under these assuptions, (11) verifies the four properties of GEV generating functions. 1. G is obviously non negative, if y 2 R n G is hoogeneous of degree. Indeed, G(fiy) = = = ψ 2C ψ = fi = fi G(y): ff fi y fi 2C ff y ψ fi ff y 2C ψ ff y 2C!!!! 3. The liit properties hold fro assuption 2, that guarantees that there is at least one non zero coefficient ff for each alternative. li yi!1 G(y 1 ;:::;y J ) = li yi!1 = P = 1 P li yi!1 P 2C ff y P 2C ff y (14) 4. The condition for the sign of the derivatives is obtained fro (21) in Lea 2 (see Section 6). We distinguish three cases, considering only y 0. (a) If k =1,we = ff y 1 A 0: (15) (b) If k>1 and =, we k G(y)=@y i1 :::@y ik =0: (16) 5

8 Indeed, k 1 Y n=0 contains a zero factor when n =1. n (17) (c) If k > 1 and <, the sign of (21) is entirely deterined by the sign of (17). For n > 0, we have n < 0 (assuption 5). Therefore, there are k 1 negative and one positive factors in the product. We obtain that k 1 Y n n=0 ( 0 if k is odd» 0 if k is even (18) Therefore, in any case, we have k G(y)=@y i1 :::@y ik 0 if k is odd» 0 if k is even (19) Λ This theore otivates the absence in our forulation of a constraint siilar to (5), (10) and (13). Indeed, such a condition is not required to obtain a valid GEV odel and, consequently, to be consistent with discrete choice theory. Instead, they are used to enable paraeter estiability. Indeed, it is ipossible to estiate all paraeters of the GEV odel, exactly as it is ipossible to estiate all Alternative Specific constants in a MNL or NL odel (see Bierlaire, Lotan and Toint, 1995). 3 Estiation procedure The estiation procedures proposed by Sall (1987) and Vovsha (1997) are based on heuristics. Sall reduces the nuber of free paraeters by iposing arbitrary restrictions on the paraeters: w = 1 M+1, 8, andρ r = ρ, 8r. Vovsha proposes a coplicated heuristic, where each observation is artificially substituted with n observations (Vovsha proposes n =100). We prefer to use a pure axiu likelihood procedure. However, the proble of estiability reains open. A detailed theoretical analysis of odel overspecification, siilar to what Bierlaire et al. (1995) has done for the Alternative Specific 6

9 Constants in the Nested Logit odel, is currently ongoing. Meanwhile, we have developed an optiization process which is able to handle overspecified odels. A new odel estiation package called Biogee (BIerlaire's Optiization routines for GEv Model Estiation) has been developed. It is designed to estiate a wide variety of discrete choice odels. Actually, any odel out of the GEV odel faily can be estiated. Moreover, non linear utility functions can be handled. In particular, a specific scale paraeter can be associated with different groups in the saple. The optiization algorith is based on a quasi-newton BFGS ethod in a trust-region fraework (Conn, Gould and Toint, 2000). The trust-region subproble is solved with a conugate gradient ethod, which does not require the solution of the Newton equations. These equations do not have a solution when the approxiation of the hessian atrix is singular (as it is the case with overspecified odels). Instead, the conugate gradient involves only atrix-vector coputations. The Newton ethod is known to be very slow when the obective is not strictly convex at the solution. Also, it stops when a local axiu of the log-likelihood function has been reached. The search for a global axiu is out of the scope of the ethod. Despite these shortcoings, this optiization algorith is sufficiently robust to epirically analyze the structure of the CNL odel. An efficient estiation procedure will be derived later on, when the structural analysis will be coplete. 4 Preliinary epirical analysis We have perfored a preliinary epirical analysis of the odel estiability. Our ain obective is to assess the relevance of conditions (5), (10) and (13) entioned in the literature. We analyze a trivial odel, with 3 alternatives. The utility functions are ust coposed of the ASCs. The saple contains 3 observations, each one corresponding to a different alternative. It is well know that the axiu likelihood estiator for a MNL odel is obtained with all ASC being equal (say to zero). The associated log-likelihood is 3ln3 ß 3: Clearly, using a CNL odel for such a trivial odel cannot iprove the log-likelihood. But we see that the sae log-likelihood can be obtained with various cobinations of paraeters values. 7

10 asc asc asc ff ff ff ff ff ff Table 1: Optial paraeters for the trivial odel We conecture that if ff = i = K; 8i; K > 0; (20) then the interpretation of the ASC is consistent with rando utility theory. Note that condition (5) by Sall, condition (10) by Vovsha and condition (13) by Papola are equivalent to (20), with K =1. If (20) is not verified, we still have a valid odel with a perfect predictive capability. However, the usual interpretation of the ASC paraeters is not relevant anyore. We illustrate that conecture in Table 1, which contains sets of axiu likelihood estiators of the paraeters. The paraeters contained in each colun provide the exact sae log-likelihood ( 3 ln 3). Condition (20) is verified for coluns 1 to 3. We observe that the optial value of the ASC are the sae as the MNL. In colun 4, where the condition is not verified, the values of the ASC are copletely different. We note also that the ff paraeter ay take values larger than 1, and that the value of K in(20) is not necessarily 1 (K=1 in colun 1, and K = 2 in coluns 2 and 3). We have observed siilar behavior with a non trivial odel on a cobined RP/SP data set for ode choice in Switzerland (Bierlaire, Axhausen and Abbay, 2001). In that case, it sees that the regular" paraeters (for cost or tie) are independent fro the validity of 20, while the 8

11 ASCs are clearly affected. We have shown that (20) is not necessary for the CNL to be part of the GEV odel faily. Therefore, it ay be adequate to ignore it. We suggest that condition (20) is violated for estiation purposes, as it provides ore degrees of freedo to the algoriths and avoid to handle linear constraints. But once an estiated odel is obtained, an equivalent odel verifying (20) ust be derived, in order to provide values of the paraeters that can be analyzed in a usual way. We also note that the constant K in (20) does not necessarily need to be one. Foral proofs of these stateents ust of course be provided. Such proofs are straightforward for the trivial odel used for epirical analysis, but not interesting. The proof for the general case is uch ore difficult. This research is currently ongoing. 5 Conclusion and perspectives In this paper, we have provided and analyzed a general forulation of the crossnested logit odel. It generalizes existing forulations in the literature. We have proved that, under ild conditions, the forulation is consistent with GEV odels. Also, wehave perfored preliinary analysis of a condition on the paraeters iposed on forulations published in the literature. We have proved that this condition is not necessary for the theoretical properties of the odel. But it is iportant for practical properties, related to paraeter estiability, and odel coparison. We have identified an extension of this condition that is iportant in order to correctly interpret the estiated paraeters. The condition is therefore desirable for practical purposes, but not necessary to have a odel with prediction capabilities. The CNL odel is appealing to capture coplex situations where correlations cannot be handled by the Nested Logit odel. Even with few alternative and nests, the use of a CNL instead of a NL odel ay significantly iprove the estiated odel (Bierlaire et al., 2001). The price to pay is the difficulty to estiate the odel paraeters. We have adopted a robust ethod with does not require the odel to be fully estiable in order to find an optial value of the paraeters. However, because the second derivative atrix of the log-likelihood function is singular at the solution, we cannot copute a reliable estiation of the 9

12 variance covariance atrix associated with the estiates. Also, the estiation algorith, based on a quasi-newton approach, can be very slow. An foral analysis of the sources of overspecification, siilar to what Bierlaire et al. (1995) have ade for the Nested Logit Model, is required in order to obtain an efficient estiation process, and a good estiates of the variance-covariance atrix of the estiated paraeters. 6 Annex: Lea The following Lea has been proven by Nicolas Antille, who is gratefully acknowledged. Lea 2 Let i 1,...,i k be k different indices (k > 0) arbitrarily chosen within f1;:::;jg. If G is defined by (11), k G(y)=@y i1 :::@y ik = ψ Yi k k 1 Y! k ffn yn 1 k n A (21) n=i 1 n=0 where A = 2C ff y : (22) Proof. The proofisby induction. We k G(y) = A ff i1 y 1 i1 1 = ff i1 y 1 i 1 A proving the result for k =1. Assuing now that the result is verified for k+1 G(y)=@y i1 :::@y k i1 :::@y ik+1 P P Q = ik k = P k+1 k Q ik n=i1 (ff n y 1 n ) Q k 1 n=0 Q n=i1 (ff n yn 1 k 1 ) Q ik+1 n=i 1 (ff n yn 1 That concludes the proof. Λ n=0 ) Q k n=0 n A n k n A k (k+1) k A : ff ik+1 y 1 i k+1 10

13 7 Annex: Derivatives We provide here the derivatives of the log-likelihood function for GEV odels in general, and for the Cross-Nested Logit odel in particular. Given a saple of observations, the log-likelihood of the saple is L = n2saple ln P (i n C n ); (23) where i n is the alternative actually chosen by individual n, C n is the choice set, and ln P (i n C) = n V in + ln G in (e nv 1 ;:::;e nv J ) (24) ln P e nv G (e nv 1 ;:::;e nv J ) where G i i and n is a scale paraeter associated to individual n. This paraeter allows to estiate odels with heterogeneous saples, without using coplicated nested structures. If fi k is a paraeter appearing in the utility functions V 1 ;:::;V k ln P i P + 1 J G i e k P k G + P n k (25) where = e V G (26) Note that we do not assue here that the V are linear-in-paraeters, so =@fi k is not necessarily a constant. The derivatives with respect to odel paraeters ff k` are k` ln P = 1 G k` 1 k` (27) The derivatives with respect to odel paraeters k are k ln P = 1 G k 1 The derivative with respect to odel paraeter is ln P = 1 G k (28) (29)

14 The derivative with respect to the scale paraeter is ln P = V i + i G 1 (30) = V e (31) V e V G + (32) Finally, we provide the first and second derivatives of (11) with respect to every paraeter. The first derivative with respect to a variable x i is given by! 1 (33) G i = ff i x 1 i ψ ff x The first derivative with respect to the = 1 y ln(y ) (34) where y = ff x (35) 2C The first derivative with respect to the nest paraeter = y ψ and with respect to the ff paraeter is where 2 2 i 2C ff x ln(x = i 1! y 2 ln(y ) (36) x k i (37) y = ff x (38) 2C We now provide the second derivative with respect to x i and x. If i =, we i = y 2 ff i x 2 i (( 1)ff i x i +y ( 1)) (39) 12

15 and if i 6=, we 2 = ( 1)ff i ff y 2 x 1 i x 1 (40) where y = ff x (41) 2C The second derivative withrespect to x i and 2 = y 1 ff i x 1 i (1 + ln(y )) (42) where y = ff x (43) 2C The second derivative withrespect to x i and 2 = y 1 + y 1 ff i x 1 i ff i x 1 i 2 + y y The second derivative withrespect to x i and ff ik 2 G = x k 1 i ik 1 k k ln(y )ff i x 1 i ff i x 1 i ln(x i ): 1+ff ik ( 1)y 1 k x k i k (44) (45) and with respect to x i and ff k (i 6= ) 2 G = ff ik x k 1 i ( k k 2 k k x k (46) where y = ff x (47) 2C References Ben-Akiva, M. and Bierlaire, M. (1999). Discrete choice ethods and their applications to short-ter travel decisions, in R. Hall (ed.), Handbook of Transportation Science, Kluwer, pp

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