Etropy 2003, 5, 205-23 Etropy ISSN 099-4300 2003 by MDPI www.mdpi.org/etropy The Iformatioal Patters of Laughter José A. Bea ad Pedro C. Marijuá 2 Mechaical Egieerig Departmet, CPS-Uiversity of Zaragoza; María de Lua, 3. E 5008, Zaragoza, Spai. E-mail: jabea@posta.uizar.es 2 Fudació CIRCE, CPS-Uiversity of Zaragoza, Zaragoza 5008, Spai. E-mail: marijua@posta.uizar.es Received: 0 Jauary 2003 / Accepted: 2 February 2003 / Published: 30 Jue 2003 Abstract: Laughter is oe of the most characteristic ad eigmatic commuicatioal traits of huma idividuals. Its aalysis has to take ito accout a variety of emotioal, social, cogitive, ad commuicatioal factors desely itercoected. I this article we study laughter just as a auditive sigal (as a eutral iformatio carrier), ad we compare its structure with the regular traits of liguistic sigals. I the experimetal records of huma laughter that we have performed, the most oticeable trait is the disorder cotet of frequecies. I compariso with the soograms of vowels, the iformatio cotet of which appears as a characteristic, regular fuctio of the first vibratio modes of the dyamic system formed, for each vowel, by the vocal cords ad the accompayig resoace of the vocalizatio apparatus, the soograms of laughter are highly irregular. I the episodes of laughter, a highly radom cotet i frequecies appears, reaso why it caot be cosidered as a geuie codificatio of pattered iformatio like i liguistic sigals. I order to umerically gauge the disorder cotet of laughter frequecies, we have performed several "etropy" measures of the spectra tryig to uambiguously idetify spotaeous laughter from "faked", articulated laughter. Iterestigly, Shao s etropy (the most atural cadidate) performs rather poorly. Keywords: Iformatio, Laughter, Soograms, Plosives, Etropy of Laughter, Power Spectral Desity.
Etropy 2003, 5 206 A Brief Itroductio to Laughter Followig several classic authors, we may describe huma laughter as the vocal output of a "cofluece" produced i a variety of social, emotioal, cogitive, ad commuicatioal processes that somehow cotai a abrupt trasitio or a oppositio of evaluatios, ad are fially resolved ito a cacellatio of effective actio (e.g., for Kat, laughter derives from a sudde trasformatio of a straied expectatio ito othig. ) []. I a slightly differet iterpretatio, we will cosider here that laughter works as a automatic mechaism for the ad hoc miimizatio of a high level of euroal excitatio that circumstatially suddely becomes irrelevat from the behavioral poit of view as a focus of attetio [2-4]. Thus, a potetial problem, ad so a focus of dedicated problem solvig processes, is suddely discarded by the laughig idividual, ad as a cosequece the accompayig euroal processes have to be cacelled or discharged prematurely. This lie of explaatio is remiiscet of Zajoc s approach to emotioal processes e.g., his explaatio of the regulatio of facial blood flow i emotioal expressios such as blushig ad pallor, that are cotributig to the appropriate level of brai irrigatio; whe we blush, for istace, there is a bypass of a uecessary surge of brai blood related to a cacelled actio of escape which is redirected towards the face [5,6]. I the case of laughter, its geuie meaig as a solved problem is accompaied by the regular reward or biological pleasure that esues the culmiatio of problem solvig activities at ay behavioral or cogitive level. So, laughter is pleasurable, ad is actively looked upo by huma subjects, although it cotais that curious ier tesio of opposed processes, seemigly beyod volutary productio. Huma laughter is performed socially, i a sort of mutual admiistratio of biological reward. It is a highly characteristic trait of huma groups. Neurodyamically, laughter is also closely related to its opposite: cryig [3,6]. They both appear evolutioarily as additioal automatic mechaisms cooperatig i the social problem solvig of huma groups (which approximately cout three of four times more idividuals tha other athropoid groups). There is good evidece (Bachorowsky, persoal commuicatio) that huma laughter directly derives from the prelaughter traits foud i juveile chimpazees ad gorillas, but applied to wider social cotexts []. So, the evolutioary reutilizatio of the acestral pathway used by primates for emotioal commuicatio i playful cotexts, becomes trasformed i a automatic miimizatio mechaism of euroal excitatio withi huma groups, parallelig the emergece of laguage [6]. We have to take ito accout that symbolic commuicatio by meas of laguage is a powerful problem-solvig tool, but as the same time it is eve more powerful as a trouble-makig istrumet. So the social eed of additioal problem solvig tools, ivolutary oes, leadig to the resolutio of social tesios ad favorig the creatio of iter-idividual bods. I sum, laughter becomes a privileged chael to promote social bodig by reutilizatio of the pseudo solutio processes related to discardig commuicative irrelevaces, edowed with pleasurable reward. Laughter ad cryig cotribute to the creatio of powerful exus i huma groups, ad are edowed with a itriguig backgroud of associated molecular processes at the syaptic level. Laughter actively cotributes to the fabricatio of the most importat memories ad bods of
Etropy 2003, 5 207 huma life: i betwee parets ad childre, i betwee sexual parters, ad amog the members of stable, close-kit groups [,4, 7-9]. The Soograms of Laughter I relatio with the possible orchestratio of laughter out from the "old" primate emotioexpressig path [, 0,], we have already foud i our iitial aalysis of laughter soograms that there is a strog discrepacy i betwee the mathematical forms of laughter ad laguage acoustic elemets. Surprisigly, there seems to be few formal studies of laughter soograms []. As ca be easily appreciated, the occurrig frequecies, the pauses, ad the duratio of the laughter bouts (plosives) are completely differet betwee laughter ad laguage, ad this differece is a firm argumet for attributig their activity to completely differet euroal systems. Eve i the same idividual, we cofrot two very differet kids of soograms (see below, Figures, 2, ad 3). The values correspodig to laughter are far more disperse ad etropic tha those of laguage. I a "volutary" or o-spotaeous laughter the soograms easily detect the presece of laguage ordered elemets, ad the relative absece of etropic or "chaotic" compoets. Ituitively this differece is very easy to detect by ay subject, just listeig to the respective souds. But, formally, thigs are more complicated. I this study we attempt precisely the measuremet of the disorder i the soud frequecies that appear i the soogram. The first method we have attempted is the atural cadidate: Shao s etropy. Shao s Etropy A classical measure of disorder ad ucertaity is Shao s etropy. Its goal is to obtai the disorder i the distributio of a sigal. Thus, our iitial proposal is to use Shao s etropy as a measure of the disorder of laughter. Let X be a discrete radom variable takig a fiite umber of possible values x, x, 2, with associated probabilities p, p, 2, K p respectively. The p 0 i =,2, K, i p i i= = K Let h be a fuctio defied o the iterval (0,) ad h ( p) be iterpreted as the ucertaity associated with the evet X =, i =,2, K, or the iformatio coveyed by revealig that X has x i take o the value x i i a give performace of the experimet. For each, let be defie a fuctio H over the variables p, p2, K, p. The fuctio H ( p, p 2, K, p ) is goig to be iterpreted as X =, i =,2,K,. The the average ucertaity associated with the evet { } H x i ( p, p2, K, p ) = pih( pi ) i= (2.) x
Etropy 2003, 5 208 To arrive to the exact expressio of H ( p, p2,, ) ecessary: H should be cotiuous i the. p i p K some axiomatic characterizatio is If all the pi are equal, pi =, the H should be a mootoic icreasig fuctio of. If a choice is broke dow ito two successive choices, the origial should be weighted sum of the idividual values of H. Shao showed [Shao, 948] tha the oly form: H H ( p, p2, K, p ) = k p log where k is a positive costat, a >, ad 0 log 0 = 0 a i= H satisfyig the three above assumptios is of the i a p i (2.2). The costat k plays a role of the electio of a uit of measuremet of the etropy. If k = the the expressio (2.2) is the Shao s etropy. The Etropy Cotet of Huma Voice Versus Laughter Let us focus first o the iformatioal cotet of huma voice. I Fig. there is a sample of a soogram of the five (Spaish) vowels: a, e, i, o, u from a adult male subject. The soogram shows the perceived frequecies associated with the vibratioal modes of the vocal chords. The characteristic vibratio modes are subject to active amplificatio ad resoaces alog the vocal tract so that the resultig souds become highly distiguishable, as it is clear from the forms of the respective spectra. They are very differet, ad so it is very easy to distigue each vowel from the others. I the iformatioal cotext described by Shao, it seems that the vibratio modes of the vocal cords are used to covey iformatio, that is, huma idividuals use the form of the sigal, i frequecy domai, to sed iformatio. Fig. 2 shows a soogram of a complete laughter episode. It is observed that there is ot a predomiace either of ay fudametal frequecy or of its multiples, except for very short periods of time. To measure the iformatioal cotet of this laughter ad its relatioship with the above pieces of laguage, we are applyig Shao s etropy. It is clear that there exists a discrete radom variable X associated with this process. Obviously this radom variable is discrete because the sigal is discrete i time domai, so is discrete i frequecy domai. The possible values x, x2,, are the amplitude of each harmoic, with K associated probabilities p, p2, K, p respectively. The spectral desity fuctios play for a radom process q() t a role aalogous to the Fourier aalysis of a determiistic fuctio, ad describe the frequecy cotet of the process. With referece to statioary processes, the power spectral desity (PSD) is a real o-egative fuctio defied as the Fourier trasform of the auto-correlatio fuctio. To fid the probabilities p, p2,, associated with the radom process q t its ecessary to K ormalize the PSD of the sigal, to fulfill the coditio x p () i= p =. The commads used to implemet i
Etropy 2003, 5 209 this algorithm were take from MatLab. This procedure was applied to the etire laughter episode showed i Fig. 2, ad to the a vowel, showed i Fig.. The results are show i Table. SIGNAL: Spaish a vowel 0.404 Laughter 0.427 SHANNON S ENTROPY: Table l. Shao s etropy. Surprisigly for the very differet visual impressio they covey, Shao s etropy for both sigals is quite similar. Rather tha the Shao s etropy, the Power Spectral Desity (PSD) of both sigals becomes far more sigificat. We show i Figures 3 ad 4 the average PSD correspodig to a plosive of laughter i Fig. 2 ad to the Spaish vowel a i Fig.. While the vowel a shows several of its characteristic plateau, laughter appears very close to a power law distributio. Figure. Soogram of the five Spaish vowels, a-e-i-o-u; i Eglish they correspod to the souds: [ah], [eh], [ee], [oh], [oo].
Etropy 2003, 5 20 Figure 2. Soogram of a laughter episode (a very itese ad joyful oe). Figure 3. Power Spectral Desity of a complete laughter episode, obtaied from the sigal showed i Fig. 2.
Etropy 2003, 5 2 Figure 4. Power Spectral Desity of a Spaish vowel ( a ), obtaied from the first part of the sigal showed i Fig.. Closig Commets Our aalysis has barely begu. After Shao s etropy ad our brief PSD aalysis, differet wavelets ad trasforms related to the Fourier decompositio of sigals will be explored by the authors. Our guidig hypothesis is that the mathematical form of laughter cotais relevat cues i order to uderstad laughter s eigmatic behavioral ad evolutioary roles. We also thik that the dyamic process iheret of laughter may be very close to the dyamic processes i other sesory varieties (e.g., visio) ad this opes the possibility of itroducig powerful symmetry tools i the aalysis, as has bee doe for istace by Michael Leyto i his treatmet of forms [2]. Besides, the uiversality of laughter i huma idividuals (eve i blid or deaf people) ad its emergece i the most varied behavioral, cogitive ad emotioal cotexts make this pheomeo oe of the most iterestig places to discuss cadidate eurodyamic priciples orietig huma iformatio processig, ad the ature of emotios [4, 3, 4]. A brief commet about the surprisig poor performace of Shao s etropy may be i order. The iformatioal cotext developed by Shao was very differet from the oe cosidered here. He was focusig the attetio o the umber of bits ecessaries to represet the iformatio trasferred. That
Etropy 2003, 5 22 is, he studied the umber of bits ecessaries i the code used (i.e. the Morse code), because this is the first step to study the etire iformatio trasfer betwee a source ad a receptor, takig ito accout the oise itroduced by the chael used. It is clear (ad well kow) from soogram aalysis, that huma beigs use the waveform i frequecy domai to sed ad recogize phoemes. This process eeds to obtai some particular vibratioal modes of the vowel chords, with further amplificatio ad resoace to establish some clear, highly characteristic plateaus. I the case of laughter it seems that there is simply a eergy trasfer from the expelled air to the vowel chords, o matter the resoace produced (ad, apparetly, the lack of active cotrol leads to the power law sigature we have metioed). Perhaps this absece of ay pattered sigal is the best sigature for a uique process: a miimizatio de gratis of looped euroal excitatios that had arise i a cotext of relative surprise ad have become behaviorally irrelevat (Marijuá, 200b). The oisy public display of problem solved by socializig idividuals has bee precisely pattered upo the cospicuous absece of patters (plosive s duratio is highly irregular too). I the cotext of uambiguously distiguishig, by formal tools, laughter from laguage, it seems ecessary a ad hoc redefiitio of the iformatio-etropy coceptualizatio (which parts of the sigal are really cosidered as sigal?). The authors are ivestigatig several ways to do this, some of them related to Javorszky s, Villarroel s ad Marijuá s approach to biological commuicatio [6]. Cocerig the formal tools right ow available to the authors, the ext way might stem from the study of the frequecy dispersio showed i the PSD: i fact, it is possible to defie several measures of that dispersio. But the use of the wavelets seems highly adequate i order to characterize the disorder i frequecy domai of laughter; actually wavelets are the best formal way to artificially geerate ad to recogize voice. I ay case, they have to be supplemeted by a itesity filter so as to get free of most of the irrelevat harmoics that accompay the vocal cord vibratio (ad which actually add cosiderable oise but very few iformatio to the soud patter). It is our hope that, i future works, we will be able to precisely measure i how much spotaeous laughter distiguishes from the volutary articulatios of voice; by gaiig further uderstadig o the form of laughter we will cotribute to illumiate its really itriguig eurodyamic ature ad social role. Ackowledgmets To Ae Bachorowsky for geeral iformatio o social cotexts of laughter, ad to Agustí Sáchez Vidal for providig laughter records. Refereces. Provie, R.R. Laughter: A scietific ivestigatio. Faber & Faber, Lodo, 2000. 2. Collis, K.P. O the Automatio of Kowledge withi Cetral Nervous Systems. (Upublished mauscript). 99. 3. Collis, K.P. & Marijuá, P.C. El Cerebro Dual. Editorial Hacer, Barceloa, 997.
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