On the Reverse Problem of Fechnerian Scaling

Transcription

1 On the Reverse Prolem of Fehnerian Saling Ehtiar N. Dzhafarov Astrat Fehnerian Saling imposes metris on two sets of stimuli relate to eah other y a isrimination funtion sujet to Regular Minimality. The two sets of stimuli usually represent the same set of stimulus values presente to an oserver in two istint oservation areas. A isrimination funtion assoiates with every pair of stimuli a nonnegative numer interpretale as the egree or proaility with whih, from the oserver s point of view, the two stimuli iffer from eah other, overall or in a speifie respet. Regular Minimality is a priniple aoring to whih the relation to e the est math for aross the two stimulus areas has ertain uniqueness symmetry properties. Fehnerian istanes are ompute y means of a issimilarity umulation proeure: isrimination values are first onverte into an appropriately efine issimilarity funtion, the sums of the latter s values are ompute for all finite hains of stimuli onneting a given pair of stimuli, the infimum of suh sums (umulative issimilarities) is taken to e the istane from one element of this pair to the other. The Fehnerian istane etween two stimuli in one oservation area is the same as the Fehnerian istane etween the orresponing (est mathing) stimuli in the other oservation area. This hapter eals with the reverse prolem of Fehnerian Saling: uner whih onitions one an ompute the isrimination funtion values given the Fehnerian istanes the isrimination values etween the est mathing stimuli. 1 Bakgroun Some familiarity with the moern theory of (generalize) Fehnerian Saling is esirale ut not neessary for reaing this hapter: the akgroun information neee will e reapitulate. The reaer intereste in the latest pulishe version of the theory is referre to Dzhafarov Colonius (2007) Dzhafarov (2008a, 2008, 2010). For historial etails the origins of the ajetive Fehnerian the reaer an onsult Dzhafarov (2001) Dzhafarov Colonius (2011). A prototypial example of an experiment to whih Fehnerian Saling pertains is this: an oserver is presente various pairs of stimuli (souns, olor pathes, rawings, photographs of faes) aske to iniate for every pair whether the two stimuli are the same or ifferent (possily, in a speifie respet, as in o these photographs epit the same person? ). The assumption is that eah pair (a, ) is assoiate with the proaility ψ (a, ) = Pr [a are juge to e ifferent]. Every stimulus is haraterize y its value (e.g., the shape of a line rawing) its oservation area (or stimulus area), usually a spatiotemporal loation, serving to istinguish the two stimuli to e ompare (e.g., two line rawings an e presente in istint loations, one to the left of the 1

2 other, or suessively, one efore the other). The pairs (a, ) therefore are efine as { } value of a in oservation area 1, (a, ) =, value of in oservation area 2 so that (a, ) (, a). Assuming, as we o throughout this hapter, that the two oservation areas are fixe, a in (a, ) elong to ifferent sets even if their values are iential. We enote these sets A B, so that the isrimination proaility funtion ψ is ψ : A B [0, 1]. This efinition an e immeiately generalize. First, we an replae [0, 1] with the set of all nonnegative reals, ψ : A B R +. This allows us to inlue isrimination funtions whih are not proailities, suh as expete values of numerial estimates of issimilarity given in response to pairs of stimuli. Although the present treatment is onfine to proailisti ψ, its extension to aritrary (oune or unoune) funtions is straightforwar. Seon, we an interpret A B as eing aritrary sets, not neessarily A = V {1}, B = V {2}, with V eing a ommon set of stimulus values. Thus, one an onsier A = V 1 {1}, B = V 2 {2}, where V 1 V 2 are ifferent susets of a set V of possile stimulus values. This may e onvenient if the mathing pairs, as efine elow, involve onstant error, that is, if V 2 is the set of mathes for the elements of V 1 V 2 V 1. We an even onsier the possiility that A B are sets of ifferent nature, with the relation are the same eing efine in speial ways. For instane, A may e a set of examinees B the set of tests, with the relation a are the same meaning that the prolem is neither too iffiult nor too easy for the examinee a (in the opinion of a juge, or as ompute from performane ata). For other non-traitional examples of pairwise omparisons, see Dzhafarov Colonius (2006). Without loss of generality, let us assume that no two istint stimuli in A or in B are equivalent, in the following sense: if a 1 a 2 in A, then for some B; if 1 2 in B, then ψ (a 1, ) ψ (a 2, ) ψ (a, 1 ) ψ (a, 2 ) for some a A. (If this is not the ase, then A B an always e reue to the requisite form.) We say that a A is mathe y B write am if ψ (a, ) = min ψ (a, y). y B 2

3 Analogously, if ψ (a, ) = min ψ (x, ), x A we say that B is mathe y a A write Ma. The spae (A, B, ψ), or the isrimination funtion ψ itself, is sai to satisfy Regular Minimality (Dzhafarov, 2002; Dzhafarov & Colonius, 2006; Kujala & Dzhafarov, 2008) if (RM1) for every a A there is one only one B suh that am; (RM2) for every B there is one only one a A suh that Ma; (RM3) am if only if Ma, for all (a, ) A B. If this is the ase, we an relael the stimuli in A B so that any two mathing stimuli reeive the same lael. Formally, if Regular Minimality hols, then one an fin (non-uniquely) ijetive funtions h : A S suh that g : B S am h (a) = g (). Any suh mapping (h, g) is alle a anonial transformation of the spae (A, B, ψ), it reates a anonial isrimination spae (S, ψ ), with the funtion efine y ψ : S S R + ψ (a, ) = ψ ( h 1 (a), g 1 () ). Clearly, the funtion ψ satisfies Regular Minimality in the simplest (anonial) form: for any istint a, S, ψ (a, a) < min {ψ (a, ), ψ (, a)}. Although the anonial isrimination spae (S, ψ ) is not uniquely etermine y (A, B, ψ), it an e viewe as eing essentially unique, in the following sense. Any two anonial spaes, (S 1, ψ 1) (S 2, ψ 2), are relate to eah other y a ijetive transformation t : S 1 S 2 suh that ψ 1 (a, ) = ψ 2 (t (a), t ()). In other wors, the anonial transformation is unique up to trivial renaming of the stimuli. Heneforth we will eal with anonial isrimination spaes (S, ψ ) only. We rop the asterisk for the anonial isrimination funtion write ψ in plae of ψ. We also use the notational onventions aopte in most of the author s previous puliations on generalize Fehnerian Saling: 1. for any inary funtion f : S S R, we write fa instea of f (a, ) (in partiular, the isrimination funtion is written as ψa); 3

4 2. if a inary funtion f is followe y a string (or hain) X = x 1..., x n of more than one point, then n 1 fx = fx 1... x n = fx i x i+1 ; 3. for a hain X = x 1..., x n with n = 1 or n = 0 the expression fx is set equal to zero; 4. any two hains of points X Y an e onatenate into a hain XY (e.g., if X = x 1..., x n, then ax is the hain ax 1..., x n ). A issimilarity funtion D : S S R is efine y the following properties: for any a, S, any sequenes a n, a n, n, n in S, any sequene of hains X n with elements in S (n = 1, 2,...), (D1) Da 0; (D2) Da = 0 if only if a = ; (D3) if max {Da n a n, D n n} 0 then Da n n Da n n 0; (D4) if Da n X n n 0 then Da n n 0. Given any hain X = x 1..., x n, the quantity i=1 n 1 DX = Dx i x i+1 i=1 is alle the umulative issimilarity for this hain. In Fehnerian Saling the role of issimilarity funtions is playe y the psyhometri inrements of the first seon kin, efine as, respetively, Ψ (1) a = ψa ψaa, Ψ (2) a = ψa ψaa. (It is learly unneessary to onsier separately the versions ψa ψ = Ψ (1) a ψa ψ = Ψ (2) a.) In other wors, a anonial isrimination spae (S, ψ) inues a oule-issimilarity spae ( S, Ψ (1), Ψ (2)). We use the notation a n n (as n ) to esignate any of the pairwise equivalent onvergenes Ψ (1) a n n 0, Ψ (1) n a n 0, Ψ (2) a n n 0, Ψ (2) n a n 0. The isrimination funtion ψ is uniformly ontinuous with respet to the uniformity inue y this onvergene: } a n a n = ψa n n n ψa n n 0. n 4

5 In partiular, a n n implies ψ n n ψa n a n 0. A metri (or istane funtion) G : S S R an e efine as a issimilarity funtion satisfying the triangle inequality: for all a,, S, Ga + G Ga. This efinition iffers from the lassial Frehét s efinition in that it oes not require gloal symmetry, Ga = Ga. However, it is more speifi than the notion of a quasimetri (efine y ropping from the lassial efinition of a metri the gloal symmetry requirement). Namely, sine G is a issimilarity it has the following symmetry-in-the-small property: for all sequenes a n, n in S, Ga n n 0 G n a n 0. In Fehnerian Saling, the Fehnerian metri G 1 inue y the issimilarity funtion Ψ (1) is efine as G 1 a = inf X Ψ(1) ax. Analogously, G 2 a = inf X Ψ(2) ax is the Fehnerian metri inue y the issimilarity funtion Ψ (2). Both G 1 G 2 are well-efine metris, G 1 a + G 1 a = Ga. G 2 a + G 2 a This sum, Ga, is a onventional (symmetri) metri. It is referre to as the overall Fehnerian metri. The asymmetri ( oriente ) metris G 1 G 2 are also relate to eah other y the ientities G 1 a G 2 a G 2 a G 1 a = ψ ψaa. This follows from the proeure of omputing G 1 G 2 from Ψ (1) Ψ (2) from the immeiately verifiale ientities Ψ (1) a Ψ (2) a = ψ ψaa. Ψ (2) a Ψ (1) a For any sequenes a n n in S, we have, as n, a n n G 1 a n n 0 G 2 a n n 0. It follows that G 1 G 2 are uniformly ontinuous with respet to the uniformity inue y. 5

6 .1 a Figure 1: Stimulus spae of Example 1. For eah stimulus pair (x, y), the numer attahe to the arrow from x to y is the value of ψxy. a a Figure 2: Psyhometri inrements of the first seon kin ompute for the stimulus spae shown in Figure 1. For eah stimulus pair (x, y), the numer attahe to the arrow from x to y is the value of Ψ (1) xy (on the left) or Ψ (2) xy (on the right). 6

7 Example 1. Let S e the set of four stimuli, {a,,, }, 1 with the values of ψ shown in Figure 1. Fehnerian omputations are illustrate in Figures 2-5. Thus, the numer 0 attahe to the arrow from to in Figure 2, left, is Ψ (1) = ψ ψ = The numer 0.7 attahe to the orresponing arrow on the right is Ψ (2) = ψ ψ = 1 0. For any finite stimulus set, the psyhometri inrements Ψ (1) Ψ (2) are issimilarity funtions (i.e., satisfy the properties D1-D4 aove) if only if ψ satisfies Regular Minimality. The Fehnerian istanes are shown in Figure 3. For instane, the numer 0 attahe to the arrow from a to on the left is ompute y forming all possile hains leaing from a to, alulating their umulative issimilarities hoosing the smallest. Omitting hains ontaining loops, as we oviously o not nee them in searhing for the minimum, we get the list hain umulative Ψ (1) a 0 a 0+0 a 0+0 a a in whih the iret link a is learly a geoesi (a shortest path). We onlue therefore that G 1 a = 0. Note that geoesis generally are not unique if they exist (they have to exist in finite stimulus sets ut not generally). The analogous alulations for, say, the stimuli on the right yiel hain umulative Ψ (2) 0.7 a a a Here, the geoesi is a we onlue that G 2 = 0.6. This geoesi is shown in Figure 4, right, together with the two other iniret geoesi paths, those onsisting of more than two stimuli. Figure 5 presents the values of the overall Fehnerian istane, otaine as G 1 xy + G 1 yx = G 2 xy + G 2 yx 1 As a rule we use symols a (interhangealy with x y, respetively) to generially refer to stimuli in, respetively, the first seon oservation areas. Thus, in an expression like Ψ (1) a > 0, a are variales, aritrary memers of S. However, in some of our examples a are use to esignate speifi stimuli, together with other speifi stimuli (here, ). The two uses of a shoul e easily istinguishale y the ontext. 7

8 a a Figure 3: Fehnerian istanes of the first seon kin ompute from the psyhometri inrements in Figure 2. The numer attahe to the arrow from x to y is the value of G 1 xy (on the left) or G 2 xy (on the right). The frame numers iniate Fehnerian istanes that are smaller than the orresponing psyhometri inrements: the geoesis for them are shown in Figure 4. for every stimulus pair (x, y). Thus, for (x, y) = (, ), we have G 1 + G 1 = G = G = G 2 + G 2 = Note that G an e viewe as the umulative issimilarity Ψ (1) a = Ψ (2) a for the geoesi loop a otaine y onatenating the geoesi paths from to ak; the loop shoul e rea in the opposite iretions for Ψ (1) Ψ (2). 2 Prolem We have seen that a anonial isrimination spae (S, ψ) inues a oule-metri spae (S, G 1, G 2 ) from whih one an form the spae (S, G) with a onventional, symmetri metri G. It is easy to see that these omputations annot generally e reverse. The following example shows that (S, G) oes not allow one to reonstrut (S, ψ) uniquely. Example 2. Let (S, ψ) inue (S, G 1, G 2 ) (S, G), let ψaa e some nononstant funtion of a. Denote y ω a an aritrary funtion suh that ω a ψaa 8

9 a a Figure 4: Geoesi paths of the first seon kin orresponing to the frame values of the Fehnerian istanes in Figure 3. Eah geoesi onsists of three stimuli onnete y two onseutive arrows. a Figure 5: The overall (symmetri) Fehnerian istanes ompute from the Fehnerian istanes shown in Figure 3. 9

10 (that is, ω a ψaa are not iential), 0 ω a min { 1 sup Ψ (1) a }, ω ω a Ψ (1) a. All three inequalities an always e ahieve, for instane, y putting ω a 0 (a nononstant ψaa has to e positive at some a, then 1 sup Ψ (1) a > 0 ). The funtion ψa = Ψ (1) a + ω a is learly oune y 1 from aove. It satisfies Regular Minimality eause ψaa = ω a ψa ω a ψa = Ψ (1) a + ω. The funtion ψ inues preisely the same metri spae (S, G) as the original funtion ψ. Inee, from the efinition of ψ it follows that Ψ (1) a = Ψ (1) a, where Ψ (1) is the psyhometri inrement of the first kin ompute from ψ. This means that G 1 ompute from Ψ (1) oinies with G 1. But then also G 1 a + G 1 a = G 1 a + G 1 a. Figures 6 7 illustrate the proeure just esrie on the stimulus spae of Example 1. The left panel of Figure 7 oinies with that of Figure 3 eause the psyhometri inrements of the first kin remain the same. By aing the numers attahe to opposite arrows one an verify that although the Fehnerian istanes of the seon kin o hange, they yiel the same overall Fehnerian istanes. Is it possile then that ψ (or at least the psyhometri inrements Ψ (1) Ψ (2) ) an e reonstrute if one knows oth G 1 G 2? The next example shows that this is not the ase. Example 3. Let S e a ountale set of stimuli enumerate s 1, s 2,..., let 0 if i = j 1/3 if i j = 1 ψs i s j = 2/3 + γ i if i j = 2 2/3 + γ i if j i = 2 1 if i j 3, where 0 γ i, γ i < 1 /3. This is an example of a uniformly isrete stimulus spae, onsiere in Setion 4. The psyhometri inrements Ψ (1) Ψ (2) ψ 10

11 .1 a Figure 6: Stimulus spae of Example 1 moifie in aorane with Example 2. a a.1.6 Figure 7: Fehnerian istanes of the first seon kin ompute from the psyhometri inrements in Figure 6. 11

12 in this ase are issimilarity funtions eause ψ satisfies Regular Minimality. ontaining s i s i+2 as suessive elements, For every hain the umulative issimilarity Xs i s i+2 Y or Xs i+2 s i Y, Ψ (ι) Xs i s i+2 Y or Ψ (ι) Xs i+2 s i Y where ι sts for 1 or 2, annot inrease if one replaes this hain with respetively: Xs i s i+1 s i+2 Y or Xs i+2 s i+1 s i Y, Ψ (ι) Xs i s i+2 Y Ψ (ι) Xs i s i+1 s i+2 Y = γ i Ψ (ι) Xs i+2 s i Y Ψ (ι) Xs i+2 s i+1 s i Y = γ i. Hene in omputing G ι as the infimum (here, minimum) of umulative issimilarities aross a set of hains, one an onfine one s onsieration to hains in whih for any two suessive elements s i s j, either i j = 1 or i j 3. This means that G ι annot epen on the funtions γ i γ i. In fat, as one an easily verify, G 1 s i s j = G 2 s i s j = { j i /3 if i j < 3 1 if i j 3, irrespetive of the funtions γ i γ i. The values of ψs is i+2 ψs i+2 s i an e aritrarily hosen etween 2 /3 1. The next example emonstrates the same point, that (S, G 1, G 2 ) etermines neither ψ nor the psyhometri inrements uniquely, for a ontinuous stimulus spae. Example 4. Let S e the interval [0, 1], let ψa = 2 a 4 + a p 2 if a, 3a a p 4 if a >, where a are the numerial values of stimuli a, respetively, p > 1. We have here a Ψ (1) 2 + a p 2 if a, a = a 2 + a p 4 if a >, Ψ (2) a = 3( a) 4 + a p 4 if a <, a 4 + a p 2 if a. We omit a emonstration that these psyhometri inrements are issimilarity funtions: it an e one using the methos presente in Dzhafarov (2010). For any a < m < in [0, 1], Ψ (1) a > Ψ (1) am 12

13 Ψ (2) a > Ψ (2) am. This is an example of a spae with intermeiate points, onsiere in Dzhafarov (2008a). The inverse triangle inequalities imply that, for any a, the umulative issimilarities Ψ (1) ax Ψ (2) ax erease as one progressively refines the hain X = x 1... x k whose elements values partition the interval etween a (i.e., for ( a <, a < x 1 < )... < x k <, analogously for a > ). As n the maximal gap in a, x (n) 1,..., x(n) k n, for hains X n = x n 1... x n k n tens to zero, Ψ (1) a ax n G 1 a = 2 3( a) Ψ (2) 4 if a <, ax n G 2 a = a 4 if a. As these values o not epen on p, the funtion ψ annot e reonstrute even if one knows all values of G 1 a G 2 a. The question arises: an one impose on the funtion ψ ertain onstraints uner whih ψ an e uniquely restore from (S, G 1, G 2 ) the set of self-isrimination proailities 2 {ω a = ψaa : a S}? On this level of generality the prolem is too iffiult, however. Its formulation oes not exlue the possiility that ψa for a given pair of stimuli (a, ) is etermine y the values of G 1, G 2 ω on some suset of pairs in S S, if not the entire Cartesian prout. The prolem we pose in this paper is more restrite: uner what onitions an one ompute ψa from the quantities G 1 a, G 2 a, G 1 a, G 2 a, ω a, ω? We refer to this as the reverse prolem of Fehnerian Saling (in the restrite sense). The soalle Fehner s prolem (as formulate in Lue & Galanter, 1963) is losely relate to the reverse prolem in the restrite sense ut is left outsie the sope of this hapter. The reaer intereste in the issue is referre to Falmagne (1985) Dzhafarov (2002a). 3 General Consierations The formulation of the reverse prolem immeiately suggests the following representation for ψa. The reverse prolem has a solution if only if ψ an e presente in either of the two equivalent forms, ψa = ω a + G 1 a + R (G 1 a, G 2 a, G 1 a, G 2 a, ω a, ω ) or ψa = ω + G 2 a + R (G 1 a, G 2 a, G 1 a, G 2 a, ω a, ω ), where R is uniformly ontinuous, nonnegative, vanishing at a =. 2 One shoul keep in min that ue to a anonial transformation of the spae, the first the seon a in ψaa may e stimuli physially ifferent in value, even if not, they always have ifferent oservation areas. Therefore the self in self-isrimination is a onvenient ut potentially misleaing prefix. 13

14 Proof. It is ovious that any funtion of G 1 a, G 2 a, G 1 a, G 2 a, ω a, ω an e presente in either of the two forms. That they are equivalent follows from whih is a onsequene of Sine ω a + G 1 a = ω + G 2 a, G 1 a G 2 a G 2 a G 1 a = ω ω a. (ψa ω a ) G 1 a = Ψ (1) a G 1 a is the ifferene of uniformly ontinuous funtions, R is uniformly ontinuous. R is nonnegative eause ψa ω a = Ψ (1) a G 1 a R vanishe at a = eause ψa ω = Ψ (2) a G 2 a. ψa = ω a = ω G 1 a = G 2 a = 0. This proves the only if part of the theorem. The if part is ovious. The examples in the previous setion show that the representation given in this theorem oes not have to exist. Moreover, espite its formulation in the form of a neessary suffiient onition, it is not ovious whether a funtion ψ satisfying this onition an in fat e onstrute: the onition in question relates ψ to G 1 G 2, whih are themselves ompute from ψ. The situation is remeie y the following examples. Example 5. Let S e the interval [0, 1], ψa = a 6 + a (a )2 (a + ). The funtion is easily heke to e etween 0 1 satisfy Regular Minimality. The psyhometri inrements of the first kin are Ψ (1) a = 1 3 a ( a 2 2) (a ), they an e shown to form a issimilarity funtion (we skip this emonstration). Sine, for any a, [0, 1] m etween a, Ψ (1) am < Ψ (1) a, 14

15 we use the same argument as in Example 4 to arrive at One an hek now that G 1 a = a. 3 a 6 + a (a )2 (a + ) = a 6 + a 3 + ( 2 3 a + ( a 6 (( 6)) 6 ) a a ) ( a 6 + ) 6 = ω a + G 1 a + (2G 1 a + (ω a ω )) ((ω ω a ) + 2G 1 a) (ω a + ω ). That is, ψ an e presente in a form require in Theorem 3, with R = ω a + G 1 a + (2G 1 a + (ω a ω )) ((ω ω a ) + 2G 1 a) (ω a + ω ). This an e rewritten to involve G 1 G 2 symmetrially: using the ientities we get 2G 1 a + (ω a ω ) = G 1 a + G 2 a (ω ω a ) + 2G 1 a = G 1 a + G 2 a, ω a + G 1 a + (G 1 a + G 2 a) (G 2 a + G 1 a) (ω a + ω ) ψa = ω + G 2 a + (G 2 a + G 1 a) (G 1 a + G 2 a) (ω + ω a ). This representation is of interest in view of the symmetry onsierations to e invoke elow. The next example provies another emonstration of the same nature, argualy the simplest possile eause it orrespons to R 0. Example 6. Consier the spae shown in Figure 8. One an hek that, for any pair of stimuli x, y {a,,, } any hain X with elements in {a,,, }, Ψ (1) xy Ψ (1) xxy Ψ (2) xy Ψ (2) xxy. This means that the psyhometri inrements shown in Figure 9 oinie with the orresponing Fehnerian istanes, we have ω a + G 1 a ψa = ω + G 2 a, whih are speial ases of the representations in Theorem 3. 15

16 .1 a Figure 8: Stimulus spae of Example 6. a a Figure 9: Psyhometri inrements (in this ase oiniing with Fehnerian istanes) of the first seon kin ompute for the stimulus spae shown in Figure 8. 16

17 Theorem 3 enompasses a wealth of possile speial ases. One an restrit this lass y onsiering stimulus spaes with speial properties (as we o in the next setion) or y imposing ertain symmetry onstraints on the funtion R iretly. The latter an e one as follows. First, we an eliminate from the expression R (G 1 a, G 2 a, G 1 a, G 2 a, ω a, ω ) quantities eterminale from other quantities. Knowing G 1 a, G 1 a, ω ω a one an ompute G 2 a, G 2 a as G 2 a = G 1 a (ω ω a ) G 2 a = G 1 a + (ω ω a ) (these ientities were use in Example 5). This leas to R 1 (G 1 a, G 1 a, ω a, ω ) R (G 1 a, G 2 a, G 1 a, G 2 a, ω a, ω ) = R 2 (G 2 a, G 2 a, ω, ω a ). Now, it may sometimes e natural to posit (e.g., in psyhophysial appliations, when two oservation areas ontain the same set of stimulus values) that the two oservation areas are interhangeale, so are the stimuli a. More preisely, one an assume that R 1 R 2 remain invariant (1) if one exhanges a in all their arguments; (2) if one replaes G 1 with G 2 or vie versa. Sine the arguments of R 2 an e otaine from those of R 1 y suessively applying these two rules, we have R 1 R 2 R. The assumption in question an now e formulate y saying that any asymmetry etween the two oservation areas is only in the first two summs of the four equivalent representations for ψ: ω a + G 1 a + R (G 1 a, G 1 a, ω a, ω ) ω ψa = a + G 1 a + R (G 1 a, G 1 a, ω, ω a ) ω + G 2 a + R (G 2 a, G 2 a, ω, ω a ) ω + G 2 a + R (G 2 a, G 2 a, ω a, ω ). Let us all suh a funtion R symmetri. If a reverse prolem has a solution with a symmetri funtion R, then G 1 a G 2 a = ψa ψa, G 1 a G 1 a = Ψ (1) a Ψ (1) a, G 2 a G 2 a = Ψ (2) a Ψ (2) a. 17

18 Proof. The first of these ientities is otaine y sutrating ψa = ω a + G 2 a + R (G 2 a, G 2 a, ω a, ω ) from ψa = ω a + G 1 a + R (G 1 a, G 1 a, ω a, ω ). The representations for the ifferenes of the psyhometri inrements follow then from the ientities G 2 a = G 1 a (ω ω a ) G 2 a = G 1 a + (ω ω a ). Example 7. The following funtions satisfy the symmetry requirement for R : ω a + G 1 a + (G 1 a + G 2 a) (G 2 a + G 1 a) (ω a + ω ) ψa = ω + G 2 a + (G 2 a + G 1 a) (G 1 a + G 2 a) (ω + ω a ) ω a + G 1 a + f (Ga,S (ω a, ω )) ψa = ω + G 2 a + f (Ga,S (ω a, ω )), where S is some ommutative funtion (whih may, as a speial ase, e ientially onstant, say, zero). 4 Speial Stimulus Spaes 4.1 Diretly linke spaes Let us say that a point a in a stimulus spae (S, ψ) is iretly 1-linke to point (or iretly linke to it in the first oservation area) if Ψ (1) a = G 1 a. Analogously, a point a is iretly 2-linke to point (or iretly linke to it in the seon oservation area) if Ψ (2) a = G 2 a. A point a is iretly 1-linke to a point if only if is iretly 2-linke to a. Proof. The equality means that, for every hain X = x 1... x k, Ψ (1) a = G 1 a k 1 Ψ (1) ax =Ψ (1) ax 1 + Ψ (1) x i x i+1 + Ψ (1) x k Ψ (1) a. i=1 18

19 This an e written as k 1 (ψax 1 ω a ) + (ψx i x i+1 ω xi ) + (ψx k ω xk ) (ψa ω a ). i=1 Replaing ω a with ω on oth sies rearranging the ω-terms as we get ω ω x1 ω xi ω xk 1 ω xk ω x1 ω x2 ω xi+1 ω xk ω, k 1 ( ) (ψax 1 ω x1 ) + ψxi x i+1 ω xi+1 + (ψxk ω ) (ψa ω ). i=1 In other wors, for every hain Y = x k... x 1, But this means proving the theorem. k 1 Ψ (2) Ya =Ψ (2) x k + Ψ (2) x i+1 x i + Ψ (2) x 1 a Ψ (1) a. i=1 Ψ (2) a = G 2 a, Points a are iretly 1-linke to eah other if only if they are iretly 2-linke to eah other. In a spae with any two points iretly 1-linke any two points are iretly 2-linke ( vie versa). The spae referre to in this orollary is alle a iretly linke spae. The reverse prolem for suh a spae has its simplest possile solution: G 1 a + ω a ψa = G 2 a + ω. Example 8. Consier a spae (S, ψ) suh that ψa = Ma + r 1 (a) + r 2 (), where M is a symmetri metri r 1, r 2 some nonnegative funtions. Then To ensure Regular Minimality, we posit Ψ (1) a = Ma + r 2 () r 2 (a) Ψ (2) a = Ma + r 1 (a) r 1 (). r 1 (a) r 1 () < Ma r 2 (a) r 2 () < Ma. 19

20 We verify that for any a,, m in S, Ψ (1) am = Mam + r 2 (m) r 2 (a) + Ψ (1) m = Mm + r 2 () r 2 (m) Ma + r 2 () r 2 (a) = Ψ (1) a, proving therey Analogously we prove Sine we onlue that G 1 a + G 1 a G 2 a + G 2 a Ψ (1) a = G 1 a. Ψ (2) a = G 2 a. = Ψ(1) a + Ψ(1) a Ψ (1) a + Ψ (1) a Ma = 1 2 Ga, = 2Ma, so that the efinition of ψ an e given as ψa = 1 2 Ga + r 1 (a) + r 2 (). This is one possile form of presenting the quarilateral issimilarity moel (Dzhafarov & Colonius, 2006). The statement of the following theorem is ovious given without proof. We enote (referring to Theorem 3) R (G 1 a,g 2 a,g 1 a,g 2 a,ω a, ω ) = R a. If the reverse prolem has a solution, then point a is iretly 1-linke to point ( is iretly 2-linke to point a) if only if R a = 0. This oservation agrees with Theorem 4.1: oth iret linkages are equivalent to R a = 0. Example 6 shows that a iretly linke spae an e easily onstrute. 4 Spaes with metri-in-the-small issimilarities A issimilarity funtion D is sai to e metri-in-the-small if, whenever a n n with a n n, The onvergene is from the right eause a n n & a n n Da n n Ga n n 1. Da n n Ga n n. Applying this efinition to psyhometri inrements, = Ψ(1) a n n G 1 a n n 1 20

21 Clearly, these onvergenes imply a n n & a n n = Ψ(2) a n n G 2 a n n 1. Ψ (ι) a n n + Ψ (ι) n a n Ga n n 1. Reall that a n n means any of the equivalent onvergenes Ψ (ι) a n n 0, Ψ (ι) n a n 0, where ι sts for 1 or 2. If the reverse prolem has a solution, then the issimilarities Ψ (1) Ψ (2) are metri-in-the-small if only if R (x, y, u, v, a, ) is of a higher egree of infinitesimality than either of the arguments x v. Proof. Rewrite the expressions as We know that, as a n n, oth ψa n n = ω an + G 1 a n n + R an n = ω n + G 2 n a n + R an n Ψ (1) a n n G 1 a n n = 1 + R (G 1a n n,g 2 a n n,g 1 n a n,g 2 n a n,ω an,ω n ) G 1 a n n Ψ (2) n a n G 2 n a n = 1 + R (G 1a n n,g 2 a n n,g 1 n a n,g 2 n a n,ω an,ω n ) G 2 n a n. G 1 a n n 0 G 2 n a n 0. The left-h sies ten to 1 if only if the ratios on the right ten to zero, proving the theorem. If we make use of the symmetry onstraint of Setion 3 present the funtion R in the aove theorem as R (G 1 a,g 1 a,ω a, ω ) = R (G 2 a,g 2 a, ω, ω a ), then the onition of the higher-orer infinitesimality aquires a simpler form. If the reverse prolem has a solution with a symmetri R, then Ψ (1) is metri-in-the-small if only if so is Ψ (2) if only if R (x, y, a, ) is of a higher egree of infinitesimality than oth x y. 21

22 Example 9. In partiular, if R an e presente as in the seon funtion of Example 7, R = f (Ga,S (ω a, ω )), the onition of the higher-orer infinitesimality reues to R (x, a) x 0 as x 0 (x 0). Suh funtions as ω a + G 1 a + ωa ω ω a+ω (Ga) 2 ψa = ω + G 2 a + ωa ω ω a+ω (Ga) 2 provie examples. ω a + G 1 a + k 2 (Ga) 2 + k 3 (Ga) k r (Ga) r ψa = ω + G 2 a + k 2 (Ga) 2 + k 3 (Ga) k r (Ga) r 4 Uniformly isrete spaes A spae (S, ψ) is uniformly isrete if This onition is equivalent to eause, as we know, inf x y Ψ(1) xy > 0. inf x y Ψ(2) xy > 0 Ψ (1) x n y n 0 Ψ (2) x n y n 0. Any finite spae is uniformly isrete. In the following we will taitly assume, with no loss of generality, that all hains X = x 1...x k onsiere are non-wasteful, in the following sense: for no i = 1,..., k 1, x i = x i+1. A hain X = x 1...x k is alle 1-asi if, for any 1 i < k, x i is iretly 1-linke to x i+1. A 2-asi hain is efine analogously. The lass of all 1-asi (2-asi) hains onneting a to ontaining k elements, not ounting a, is enote y Ck 1a (respetively, C2 ka). Clearly, k = 0, 1,.... For any a, in a uniformly isrete spae one an fin k 1 k 2 suh that G 1 a = inf Ψ (1) ax ax Ck 1 a 1 G 2 a = inf Ψ (2) ax. ax Ck 2 a 2 22

23 Proof. Let ι st for 1 or 2. Consier all hains X (k) ontaining k = 0, 1,..., elements, efine Denoting we have hene also Therefore, for some K > 0, we have to have G (k) ι a = inf Ψ (ι) ax (k). X (k) s ι = inf x y Ψ(ι) xy > 0, Ψ (ι) ax (k) (k + 1) s ι, G (k) ι a (k + 1) s ι. G ι a < G (K) ι a. Consier a numer k with the following property: in some sequene of hains ax n suh that Ψ (ι) ax n G ι a the hains with k elements our infinitely often. Denote y k 0 the largest numer with this property (whih exists eause k < K). By onstrution, there is a sequene of hains ax (k0) n suh that ut there is no sequene of hains ax (>k0) n that Ψ (ι) ax (k0) n G ι a, in whih eah X (>k0) n Ψ (ι) ax n (>k0) G ι a. has more than k 0 elements suh We will show that all ut a finite numer of these hains ax n (k0) are ι-asi. Suppose this is not the ase. Then one an hoose a sequene of hains ax (k0) n all of whih are not ι-asi, with Ψ (ι) ax (k0) n G ι a. Let x in,nx in+1,n e a link in eah of these hains with x in,n not iretly ι-linke to x in+1,n (where i n may e 0 or k 0, with x 0,n = a x k0+1,n = ). Then one an fin nonempty hains Y n suh that Ψ (ι) ax 1,n...x in,ny n x in+1,n...x k0,n < Ψ (ι) ax n (k0), sine the onvergene of Ψ (ι) ax (k0) n to G ι a is from the right, Clearly, Y n must ontain an element whene the hains Ψ (ι) ax 1,n...x in,ny n x in+1,n...x k0,n G ι a. m n / {x in,n, x in+1,n}, x 1,n...x in Y n x in+1...x k0,n ontain more than k 0 elements. But this ontraits the efinition of k 0. 23

24 With ι sting for 1 or 2, in a uniformly isrete spae any point an e iretly ι-linke to some other point, any two points an e onnete y an ι-asi hain. To formulate another immeiate onsequene of Theorem 4, we nee a new onept. A ase for G 1 is a suset suh that if for all (a, ) D, then D S S G 1 a = G 1a G 1 a = G 1a for all (a, ) S S. The efinition of a ase for G 2 is analogous. With ι sting for 1 or 2, in a uniformly isrete spae the set of all iretly ι-linke orere pairs of points forms a ase for G ι. We onlue this hapter y onsiering the reverse prolem for a speial ase of uniformly isrete spaes, those with geoesis. All finite spaes fall within this ategory. A uniformly isrete spae is sai to e with geoesis if, for any points a in it, G 1 a = min X Ψ(1) ax G 2 a = min X Ψ(2) ax. In other wors, the requirement is that for any a one e ale to fin hains X 1 X 2 suh that G 1 a = Ψ (1) ax 1 G 2 a = Ψ (2) ax 2. The hains ax 1 ax 2 in this efinition are referre to as geoesis of the first seon kin, respetively. That geoesis nee not exist in all uniformly isrete spaes is shown y the following example. Example 10. Let S onsist of a,, 1, 2,.... Let Ψ (1) e a symmetri funtion with the following values: for i, j {1, 2,...}, The spae is uniformly isrete eause For the sequene of hains a n, as n, Ψ (1) a = 2, Ψ (1) a i = /i, Ψ (1) i = 1 /2, Ψ (1) i j = i j. inf x y Ψ(1) xy = 1 2. Ψ (1) a n = n 3 2, 24

25 it is easy to see that G 1 a = 3 2. There is, however, no hain X suh that Ψ (1) ax = 3 2. Therefore, this uniformly isrete spae is not with geoesis. In a uniformly isrete spae with geoesis, we say that a is strongly 1-linke (strongly 2-linke) to if a is the only geoesi of the first (respetively, seon) kin onneting a to. Clearly, strong ι-linkage implies iret ι-linkage (ι = 1, 2). A hain X = x 1...x k is alle strongly ι-asi if, for any 1 i < k, x i is strongly ι-linke to x i+1 (ι = 1, 2). With ι sting for 1 or 2, in a uniformly isrete spae with geoesis, (i) any two points an e onnete y a strongly ι-asi geoesi hain; (ii) any point an e strongly ι-linke to some other point; (iii) the set of all strongly ι-linke orere pairs of points forms a ase for G ι. Proof. Assume (i) is not true. Then, for some a, all geoesis ax are not strongly ι-asi. Choose a geoesi ax 1... x k with the largest numer of elements k. It shoul exist y the same argument as in Theorem 4: Ψ (ι) ax 1... x k (k + 1) inf x y Ψ(ι) xy > 0. Let x i x i+1 e a link in ax with x i not strongly ι-linke to x i+1. Then there exists a geoesi hain x i Yx i+1 with a nonempty Y, whene ax 1... x i Yx i+1... x k is a geoesi hain from a to with more than k elements. This ontraition proves (i). The statements (ii) (iii) are immeiate orollaries. With ι sting for 1 or 2, in a uniformly isrete spae with geoesis, a point a is strongly ι-linke to a point if only if, for any point m istint from a, G ι am > G i a. Proof. We prove the equivalene of the negations of the two statements: (1) a is not strongly ι-linke to means a is onnete to y a geoesi other than a; (2) the negation of the inequality in the formulation is (sine G ι is a metri) the equality G ι am = G i a. If a an e onnete to y a geoesi ax with a nonempty X, then hoosing an element m of this hain, we get the equality aove. Conversely, if this equality is satisfie for some point m, then the onatenation of the geoesis from a to m from m to is a geoesi other than a. 25

26 We emonstrate the use of these results y the following example. Example 11. Consier Figure 3 again. Inspeting, say, the link from a to in the left panel we see that { G1 a = G 1 a = 0 < G 1 a = Sine this exhausts all trias of the form am in this spae, we onlue that a is strongly 1-linke to, Ψ (1) a = G 1 a = 0. We ome to the same onlusion restore the values of Ψ (1) Ψ (2) for all links in Figure 3 with unframe values of istane. For the links with frame values, say, from to in the right panel, we have G 2 = 0.6 = G 1 a = , whene we onlue that a is not strongly 2-linke to. The value of Ψ (2) therefore annot e reonstrute uniquely: it an e any value 0.6. Aknowlegments This researh has een supporte y AFOSR grant FA to Purue University. Referenes Dzhafarov, E. N. (2001). Fehnerian psyhophysis. In N. J. Smelser P. B. Baltes (Es.), International Enylopeia of the Soial Behavioral Sienes (vol. 8, pp ). NY: Pergamon Press. Dzhafarov, E. N. (2002a). Multiimensional Fehnerian saling: Proaility-istane hypothesis. Journal of Mathematial Psyhology, 46, Dzhafarov, E. N. (2002). Multiimensional Fehnerian saling: Pairwise omparisons, regular minimality, nononstant self-similarity. Journal of Mathematial Psyhology, 46, Dzhafarov, E. N. (2008a). Dissimilarity umulation theory in ar-onnete spaes. Journal of Mathematial Psyhology, 52, [see Dzhafarov, E. N. (2009). Corrigenum to Dissimilarity umulation theory in ar-onnete spaes. Journal of Mathematial Psyhology, 53, 300.] Dzhafarov, E. N. (2008). Dissimilarity umulation theory in smoothly onnete spaes. Journal of Mathematial Psyhology, 52, Dzhafarov, E. N. (2010). Dissimilarity, quasiistane, istane. Journal of Mathematial Psyhology, 54, Dzhafarov, E. N., & Colonius, H. (2006). Regular Minimality: A funamental law of isrimination. In H. Colonius E. N. Dzhafarov (Es.), Measurement Representation of Sensations (pp. 1 46). Mahwah, NJ: Erlaum. Dzhafarov, E. N., & Colonius, H. (2007). Dissimilarity Cumulation theory sujetive metris. Journal of Mathematial Psyhology, 51,

27 Dzhafarov, E. N., & Colonius, H. (2011). The Fehnerian iea. Amerian Journal of Psyhology, 124, Falmagne, J. C. (1985). Elements of Psyhophysial Theory. Oxfor: Oxfor University Press. Kujala, J. V., & Dzhafarov, E. N. (2008). On minima of isrimination funtions. Journal of Mathematial Psyhology, 52, Lue, R. D., & Galanter, E. (1963). Disrimination. In R. D. Lue, R. R. Bush, E. Galanter (Es.), Hook of Mathematial Psyhology, vol. 1 (pp ). NY: John Wiley & Sons. 27