Biological modelling / Biomodé lisation Stochastic monotony signature and biomedical applications Signature de monotonie ale´atoire et applications biome´dicales

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  Biological modelling / Biomodé lisation Stochastic monotony signature and biomedical applications Signature de monotonie ale´atoire et applications biome´dicales
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  Biologicalmodelling/Biomode´lisation Stochastic   monotony   signatureandbiomedicalapplications Signature   de   monotonie   ale´ atoire   et    applications   biome´ dicales  JacquesDemongeot a, *,GiulianaGalliCarminati a ,FedericoCarminati b ,MustaphaRachdi a,c a Laboratoire    AGIM,   FRE    CNRS-UJF     3405,   Faculte´  de   me´ decine,   Universite´  Joseph-Fourier,    38700   LaTronche,   France b CERN,   1211   Gene` ve    23,   Switzerland c Universite´  Grenoble    Alpes,   UFR   SHS,   BP47,    38040   Grenoble   cedex   09,   France 1.   Introduction In   many   biomedical   applications,   semi-quantitative   orqualitative   variables   are   observed,   which   are   valued   on   asmall   number   of    levels   (typically   the   integers   from   0to   10),but   are   comparable   through   astrategy   of    monotonytesting.   If    only   the   succession   of    intervals   of    monotonyofthe   function,   between   the   times   of    observation   (eitherdiscrete   or   continuous),   isimportant   for   comparing   thesevariables,   we   only   consider   the   sequence   of    signs   of    theseintervals   (‘‘+1’’,   if    the   function   is   increasing   or   constant,   and‘‘  1’’if    the   function   is   decreasing)   called   the   stochasticmonotony   signature,   and   then   we   calculate   the   probabilitythat   asequence   of    signs   is   similar   ornot   to   another C.   R.   Biologies   xxx   (2015)   xxx–xxx A   R    T   I   C   L    E   I   N   F   O  Articlehistory: Received   18   December   2014Accepted   after   revision   8   September   2015 Availableonlinexxx Keywords: Stochastic   monotony   signatureMonotony   statistical   testComparison   of    functions Mots   cle´ s   : Signature   de   monotonie   ale´atoireTest   statistique   de   monotonieComparaison   de   fonctions A   B   STR    A   C   T Weintroduceanewconcept,thestochasticmonotonysignatureof    a   function,made   ofthesequenceofthe   signsthatindicateif    thefunctionis   increasingor   constant(sign+),ordecreasing(sign  ).Ifthefunctionresultsfromtheaveragingofsuccessiveobservationswitherrors,themonotonysignisarandombinaryvariable,whosedensityis   studiedundertwohypothesesforthedistributionoferrors:uniformandGaussian.Then,we   describeasimplestatisticaltestallowingthecomparisonbetweenthemonotonysignaturesoftwofunctions(e.g.,oneobservedandtheotheras   reference)andwe   applythetestto   fourbiomedicalexamples,coming   fromgenetics,psychology,gerontology,andmorphogenesis.  2015   Acade´miedessciences.   PublishedbyElsevierMassonSAS.   Allrightsreserved. R    E´ SU   M   E´ Nousintroduisonsunnouveauconcept,lasignature   demonotonieale´atoired’unefonction,constitue´edelase´quencedessignesindiquantsiunefonctionestcroissanteouconstante(signe+),oubien   de´croissante(signe    ).Silafonctionre´sultedelamoyennisationd’observationssuccessivesentache´esd’erreurs,lesignede   monotonieestunevariable   ale´atoirebinaire,dontnouse´tudionslaloideprobabilite´ sousdeuxhypothe`sesde   distributiondes   erreurs:   uniformeetgaussienne.Nousde´crivonsensuiteunteststatistiquesimple   permettantdecomparerlessignatures   de   monotonie   dedeuxfonctions(parexemple,l’uneobserve´eetl’autreservantdere´fe´rence)etnous   l’appliquonsa` quatreexemplesde   fonctions,issuesdelage´ne´tique,de   lapsychologie,de   lage´rontologieetde   lamorphogene`se.  2015Acade´miedessciences.Publie´ parElsevierMassonSAS.Tousdroitsre´serve´s. * Corresponding   author. E-mail   address:    Jacques.Demongeot@agim.eu   (J.   Demongeot). GModel CRASS3-3401;   No.   of    Pages   7 Please   cite   this   article   in   press   as:    J.Demongeot,   et   al.,   Stochastic   monotony   signature   and   biomedical   applications,   C.   R.Biologies   (2015),   http://dx.doi.org/10.1016/j.crvi.2015.09.002 Contents   lists   available   at   ScienceDirect ComptesRendusBiologies www.sciencedirec   t.com http://dx.doi.org/10.1016/j.crvi.2015.09.0021631-0691/    2015   Acade´mie   des   sciences.   Published   by   Elsevier   Masson   SAS.   All   rights   reserved.  reference   sequence.   We   willrestrict   the   present   study   tothecase   where   one   cannot   observe   the   same   biologicalobject   at   different   times   or   locations   on   aone-dimensionalscale,   because   itis   experimentally   destroyed,   censored   orindividually   hidden   by   adouble   blind   procedure.   Hence,   itis   observed   a   series   of    empirical   distributions   and   themonotony   intervals,   bounded   by   observation   times.   Thesupport   of    the   empirical   distributions   isan   interval   in   theuniform   case   and   an   estimate   of    the   95%-confidenceinterval   in   the   Gaussian   case.Atest   will   be   built   under   the   assumption   ofindepen-denceof    the   components   and   increments   of    the   observedvariables   at   the   observation   times   which   constitute   thefrontiersof    the   monotony   intervals   and   we   will   presenttypical   examples   infour   application   domains:   one   will   focuson   the   comparison   of    intervals   of    monotony   of    histogramscorresponding   to   the   observation   of    physiological   events(crossing-overs),   observed   for   men   and   womenandcomparedon   a   single   human   chromosome.   The   secondexample   concerns   the   answers   of    agroup   of    individualsduring   double   blinded   exercises   of    choice   of    an   imageamong   a   pair   of    images,   along   asuccession   of    image   pairspresented   successively.   Thethird   example   isrelated   to   theevolution   during   the   nychthemeron   (day/night   24   h   inter-val)   of    thenumber   of    entrances   ina   given   room,   observed   fordifferentrooms   and   successive   25   days.   The   final   exampleconcerns   microscopic   data   about   segregation   and   transportofcolloidal   particles   during   microtubule   morphogenesis,phenomena   compared   with   and   without   gravity.The   former   biomedical   methodologies   concerningmonotony   comparison   are   essentially   coming   from   theLD50   toxicological   bioassays,   in   which   there   is   noindividual   horizontal   sampling   because   the   animals   testedare   not   reused   (because   death   or   pathologic—even   minor—reaction)   after   each   dose   administration.   These   bioassaysproduce   data   susceptible   to   benefit   from   amonotonysignature   testing,   notably   in   their   sequential   version,   inwhich   the   experimental   procedure   consists   in   choosingincreasing   toxic   doses   d 1 , .   .   . , d n ,giving   lethal   effects  X  1 , ... ,  X  n  (measured   by   the   percentage   of    death)   based   onformer   experiments   done   on   aknown   reference   drug   of    thesame   chemical   family   giving   lethal   effects   Y  1 , .   .. , Y  n ,procedure   called   prediction   and   based   on   referencechemicals   testing   [1].X 1 , .   .. ,  X  n  and   Y  1 , ... , Y  n  are   consideredasrandom   variables   observed   at   the   same   times.The   first   attempt   to   compare   the   monotony   intervalshas   been   to   use   the   rank   statistics   correlation   test[2,3].More   precisely,   if  r  (  X  )   denotes   the   decreasing   rankstatistics   of     X  ,   we   have   for   the   sign   of    the   i th   monotonyinterval   of     X  ,   denoted   sgn  X  i : 8   I  ¼   1 ; ..   . ; n  1 ; sgn  X  i  ¼   1I r     X  ðÞ i  < r     X  ð   Þ i þ 1 fg 1I rX  ðÞ i  > r     X  ðÞ i þ 1 f   g : Hence,   we   have:sgn  X    ¼   s  f   g   ¼   \ i ¼ 1 ; ... ; n  1  X  i    s  i  1 ðÞ = 2  <  X  i þ s  i þ 1 ð   Þ = 2     and P    sgn  X    ¼   s  f   gð   Þ¼ P    \ i ¼ 1 ; ... ; n  1  X  i  s  i  1 ðÞ = 2  <    X  i þ s  i þ 1 ð   Þ = 2     ¼ Z   ... Z  A s  ðÞ  f  j  ðÞ d j  (1)where    f    isthe    joint   distribution   function   of     X  and    A   s  ðÞ   ¼\ i ¼ 1 ; ... n  1  j  ; j  i  s  i  1 ðÞ = 2  < j  i þ s  i þ 1 ðÞ = 2 no Let   us   suppose   that  X    is   a   random   vectorwithindependentcomponentsand   increments;   then   (1)   becomes   (2): P  sgn  X    ¼   s  f   gð   Þ¼ Y i ¼ 1 ; ... ; n  1 PX  i  s  i  1 ð   Þ = 2  <    X  i þ s  i þ 1 ðÞ = 2 ðÞ  ¼ Z   ... Z  A s  ð   Þ Y i ¼ 1 ; ... ; n  1  f  i þ   s  i þ 1 ðÞ = 2  j  i ðÞ F  i  s  i  1 ð   Þ = 2  j  i ð   Þ¼ Y i ¼ 1 ; ... ; n  1 Z  Sf  i ð   Þ  f  i  j  i ðÞ 1   þ   sgn  X  i ðÞ = 2  sgn  X  i F i þ 1  j  i ð   Þ½ d j  i (2)where    f  i  (resp.   F  i )is   the   distribution   (resp.   Cumulativedistribution)   function   of     X  i ,and   S  (  f  i )denotes   its   support.Forexample,   ifsgn  X    =   (  1, ... ,  1),then   we   have: P  ({sgn  X  =   (–1, .   .. ,–1)})   =   P  ( \ i   =1, ..   . , n –1 {  X  i +   1 –  X  i <   0})   =   P i =1, ... , n –1  P  ({  X  i   +   1 <    X  i })   = P i   =   1, ... , n –1  P  ( [ j  i 2 S(f  i ) ({  X  i = j  i } \ {  X  i   +   1 <   j  i }))   = P i   =1, ... , n –1  R S  (  fi )  f  i ( j  i ) F  i +1 ( j  i )d j  i .Theformulas   above   show   that   the   knowledge   about   therank   statistics   r  (  X  )   gives   the   monotony   signature   sgn  X  ,   butthat   the   converse   isfalse.   Then,   an   identity   test   betweensgn  X  and   sgn Y    ispossible   when   the   monotony   intervals   areobserved,   even   if    the   rank   statistics   remain   unknown.   Wewill   build   such   a   test   in   the   following. 2.   Stochastic   monotony    signature:   definition   and   study of    the   distribution  2.1.   Definitions Let   usconsider   the   graphs   of    real   functionsof    time,  X  ( t  )and   Y  ( t  ),   recorded   at   the   same   observationinstantsbelonging   to   the   discrete   time   set   { t  1 , ... t  n }    giveninFig.   1.   Wesuppose   that   the   studied   phenomenon   involves   in Fig.   1.   (Color   online.)   Temporal   profiles   of    an   observed   signal    X    (red)   andareference   signal   Y    (blue)   over   time   t  ,   indicating   monotony   segmentsbetween   the   successive   averages   of     X  ,   each   average   being   the   centre   of    theempirical   95%   confidence   interval   of    adistribution   oferrors   on    X  supposed   tobe   symmetrical.   If    it   is   supposed   tobe   uniform,   the   weightedmonotony   signature   of    X   is   equal   to   (0.4,1,0,0,1,1,0,0,1).   Monotonyintervals   circled   in   orange   correspond   to   the   difference   between    X  and   Y  monotonies.  J.Demongeot    et    al.    /    C.R.   Biologies    xxx   (2015)    xxx–xxx 2 GModel CRASS3-3401;   No.   of    Pages   7 Please   cite   this   article   in   press   as:    J.Demongeot,   et   al.,   Stochastic   monotony   signature   and   biomedical   applications,   C.   R.Biologies   (2015),   http://dx.doi.org/10.1016/j.crvi.2015.09.002  general   the   censoring   of    the   observed   system(individual,cell,   chromosome .   .. )   by   loss,   death,   destruction ... Hence,weonly   getfrom   the   experiments   theempirical   distribution   of thedata   recorded   onasample   of    individuals   differentateachtime(like   in   theL  D 50   experiments).   We   call   monotonysignatureof   X    (resp.   Y  )the   sequenceof    the   values   ‘‘+1’’   and‘‘  1’’corresponding   to   their   successive   monotony   intervals:sgn  X  i =   +1   (resp.    1)corresponds   to   the   increase   orconstancy   (resp.decrease)   ofthe   function    X  onits   i thmonotonyinterval,   and   the   monotony   signature   of   X  (resp.   Y  )fornine   successive   intervals   ofmonotony   inFig.   1   equals:{sgn  X  i } i =   1,9 =(+1,  1,+1,+1,  1,  1,+1,+1,  1)(resp.{sgn Y  i } i   =1,9 =(  1,  1,+1,+1,+1,  1,+1,+1,  1).Wecan   decide   thatthe   monotony   signature   of    theobserved    X    profile   is   significantly   different   from   the   Y  reference   profile   (Fig.   1),   after   testing   the   similarity   of    thesesignatures   due   to   acommon   causality   between    X    and   Y  (hypothesis   H1)   againsta   random   choice   ofthe   values   of    thesuccessivesgn  X  i ’s(hypothesis   H0),   by   using,   whensgn i Y    =  1,   the   probability   P  _ i decreasing   from   avalue   takenon   thesupport   in   (denoted   above S  (  f  i ))   ofthe   distribution  f  i  of     X  i  to   a   value   of     X  i +1  on   S  (  f  i   +1 )(cf.   Fig.   2   for   i =   1): P   i  ¼   P     X  i þ 1  <    X  i f   gðÞ   ¼ P    [ j    2 S  f  i ð   Þ  X  i  ¼   j  f   g\    X  i þ 1  < j  f   gð   Þ     ¼ Z   s  f  i ðÞ  f  i  j  ð   Þ F  i þ 1  j  ðÞ d j  (3)where   F  i +1  is   the   cumulative   distribution   function   of   X  i +1 .The   sequence   ofthe   probabilities   { P  _ i } i   =   1, ..   . , n –1 of    decayof     X    on   its   ith   monotony   interval   iscalled   the   weightedmonotony   signature   of   X  .  2.2.   Gaussian   errors Proposition   1   In   the   Gaussian   case,   where   thedistribution   of     X  i  is   N  (  x i , s i ),   we   have: P   i  ¼ Z   10 exp       g  i  z  ð   Þ b ðÞ 2 þ   a 2  g     z  ðÞ 2  = 2 a 2 h   i = a   d  z    (4)where   a = s  i  /  s  i+ 1 , b   =   (  x i –  x i +1 )/ s  i   +   1  and    g  i = F  i  1 . Proof    1 If    the   density   distribution   of    errors   is   Gaussian,the   formula   (3)   becomes   (by   neglecting   the   index   i ): P  _   =   R IR   f  ( j  ) F  ( a j  +   b )   d j  ,where    f  (resp.   F  )   is   the   distribution   (resp.   cumulativedistribution)   function   of    the   standard   Gaussian   law   N  (0,1), a   =   s  1 / s  2  and   b =   ( m 1 –   m 2 )/ s  2 .By   changing   the   integrationvariable   j  in    z  =   F  ( ax +   b ),   we   have:d  z    =   af  ( a j  + b )   d j  and   j  =(  g  (  z  )   –   b )/ a ,where    g  =   F   1 .Then   we   can   write   P  _   under   the   followingformula: P  ¼ Z   10  f     gz  ð   Þ b ð   Þ = a ðÞ = af     g     z  ðÞðÞ½ d  z  ¼ Z   10 exp       g     z  ð   Þ b ð   Þ 2 þ   a 2  g     z  ðÞ 2     = 2 a 2 hi = a   d  z  ¼ Z   10 exp   b 2  gz  ð   Þ b ð   Þð   Þ = 2 ðÞ d  z  if  a ¼   1Because   the   cumulative   distribution   functions   oftheuniform   law   on   [–2 s  ,   2 s  ]and   ofthe   Gaussian   law N  (0, s  )arevery   close   (cf.   Fig.   3,right),   the   results   concerning   thecalculations   of    P  _   are   similar.   Then,   in   any   case   of    errors,   wehavechosen   uniformly   100   couples   of    values(  x i ,  x i +1 ) i   =   1, .   .. ,100  in   [0,10] 2 .Then,   we   simulated   100   sam-ples   of    100   couples   of    values   ( j  ik , j  ( i   +   1) k ) k   =1, .   .   . ,100  by   using100couples   of    Gaussian   distributions   { N  (  x i ,1), N  (  x i   +   1 ,1)} i   =1, ..   . ,100 ,and   we   calculated   the   difference   D i between   the   probability   P  _ i  calculated   from   the   integralformula   (1)   and   the   empirical   frequency   P  _ i *obtained   fromtheobservation   of    the   events   { j  ik >   j   ( i +1) k } k =   1, ... ,100 .Theresult   is   given   in   Fig.   3   (left),   showing   as   expected   that   theempirical   distribution    f  D *   of    the   random   variable   D i = P  _ i – P  _ i *is   asymptotically   (in   sample   size)   Gaussian   N  (0,1).  2.3.   Uniform   errors Proposition   2 In   the   uniform   case,   let   denote   by   [0, d ](resp.   [ D 1 , D 2 ])   and    x 1 (resp.    x 2 )the   interval   and   the   mean   of theuniform   lawof    X 1  (resp.   X 2 ).   Then,   there   are   sixdifferent   configurations   (cf.   Fig.   4):1)   D 1 < 0  d    D 2 ,then   P_   =[( d –   D 1 ) 2 –   D 12 ]/  2 d D=   d  /  2 D–   D 1  /D   I2)   D 1 < 0    D 2 <   d ,then   P_   = 1   –   D 22  /  2 d D   II3 Þ   D 2     d  D 1     0 ;   then   P  ¼   ð d - D 1 Þ 2 = 2 d D III   (5)4)   0  D 1    D 2 <   d ,   then   P_   =   1- (D 22 - D 12 )/  2 d D=( 2 d - (D 2 +D 1 ))/  2 d IV5)   D 1 > d ,then   P_   = 0   V6)   D 2 < 0,   then   P_   = 1VI Proof    2   In   the   uniform   case,   the   formula   (3)   becomes: P  ¼ Z   inf    d ; D 2 ðÞ sup0 ; D 1 ð   Þ  f    j  ðÞ F    j  ðÞ d j  ; Hence,   we   have:inf( d , D 2)If  D 2    d ; P    ¼ Z   inf  d ; D 2 ðÞ sup0 ; D 1 ð   Þ j   D 1 ðÞ d j  = d D inf( d , D 2)If  D 2  <   d ; P    ¼   1  Z   inf  d ; D 2 ðÞ sup0 ; D 1 ð   Þ d j  = d D Fig.   2.   (Color   online.)   Calculation   of    the   probability   P  _   of    negativemonotony,   where   [0, d ]   (resp.   [ D 1 , D 2 ])   and    x 1 (resp.   X 2 )   denote   the   intervaland   the   mean   of    the   uniform   law   of     X  1  (resp.    X  2 ).  J.   Demongeot    et    al.    /    C.R.   Biologies    xxx   (2015)    xxx–xxx   3   GModel CRASS3-3401;   No.   of    Pages   7 Please   cite   this   article   in   press   as:    J.Demongeot,   et   al.,   Stochastic   monotony   signature   and   biomedical   applications,   C.   R.Biologies   (2015),   http://dx.doi.org/10.1016/j.crvi.2015.09.002  Then,   by   considering   all   the   possibilities   of    values   of    theextrema   inf( d , D 2 )   and   su  p (0, D 1 ),   we   get   the   six   differentformulas   (5)   (cf.   Fig.   4) & 3.A   statistical   test   ofmonotony  Wewill   supposein   thefollowing   thatthe   distributionfunctionof     X  i  is   uniform   on   [ a i 1 , a i 2 ]andthat  X    is   astochasticprocess   with   independent   componentsandincrements.   InFig.1,theprobability P  _of    decayof     X    isequalto   0.4forthefirstintervalof    monotonyand1   forthefifth(both   circled   in   orange). P  _   equalsalso   1for   thesecond,sixth   andninthintervals,and0   forthethird,fourth,seventhandeighthones.   Letus   denote   by P  ( h )theprobability   ofhaving   h   differences   between   thesignsof    monotonyof    observed    X  ( t  )   andreference   Y( t  )signals.Wecall   H0   the   hypothesissayingthat   monotonysignaturesof     X    and Y  aresimilar   bychancewithindependencybetween  X    and Y  ,theprobabilisticstructurebeingdefinedbytheempiricalestimates   of theirdistributionsand   theindependency   of    the   compo-nents   andincrementsof   X  .Then,if  h =2,   the   probability   P  (2)   under   the   hypothesisH0equals:  P  ð 2 Þ¼   S i ;  j  ¼  1 ; 9 P  6¼ i P  6¼  j P  ðf   8   k 2   f i ;  j g ;   sgn  X  k 6¼   sgn Y  k ;   8   k   = 2   f i ;  j g ;   sgn  X  k ¼   sgn Y  k gÞ (6)where   P  6¼ i  is   the   probability   that   the   monotonies   of     X    and   Y  are   different   on   the   ith   interval:  in   the   case   where   sgn Y    is   deterministic,   P  6¼ i = P  _ i  if sgn Y  i =   +1   and   P  6¼ i =   1– P  _ i  if    sgn Y  i =   –1;  in   the   case   where   sgn Y    israndom   and   independent   of sgn  X  : P  6¼ i =   P  _ i (X)(1– P  _ i ( Y  ))   +   (1– P  _ i (  X  )) P  _ i ( Y  ),where P  _ i (  X  )   denotes   the   probability   that    X    decreases   on   its i thmonotony   interval.In   the   case   of    Fig.   1,we   have,   by   supposing   thesuccessive   monotony   signs   of    Y  known   with   a   certainty   of 3/4:  P  ð 2 Þ   ¼   ð 0 : 4  0 : 75   þ   0 : 6    0 : 25 Þð 1      0 : 75 þ 0 Þð 0 : 75 Þ 7    0 : 045(7)We   can   therefore   consider   that   the   probability   of rejecting   falsely   the   hypothesis   that   monotony   similarity,except   for   h =   2intervals,   is   due   to   chance   with   indepen-dency   between    X    and   Y    is   less   than   5%.   Thistest   is   not   aspowerful   asacorrelation   test,   but   it   isinteresting   in   thecase   of    alownumber   of    longitudinal   observations   in   whichsignal   amplitude   isnot   pertinent   compared   to   monotony,when   the   variance   of    the   empirical   correlation   with   areference   signal   isimportant.   Easy   calculations   aboverequire   that   reference   Y    and   observed   signal    X  are   known   atthesame   instants   of    observation   and   random   process   X( t  )has   independent   components   and   increments.   The   cause   of rejecting   the   hypothesis   of    similarity   with   independencybetween    X    and   Y  isthe   presence   of    a   link   betweensuccessive   values   of    the   observed   and   reference   signals. 4.   Biomedical   applications We   will   give   in   the   following   simple   illustrativeexamples   where   monotony   signature   is   pertinent. 4.1.Genetic    events   localization Fig.   5below   gives   the   localization    X    (resp.   Y  )   ofthephysiologic   crossing-overs   along   chromosome   3for   humanfemales,   with   the   blue   curve   in   Fig.   5   (resp.   males,   with   redbars)   [4].By   comparing   the   two   monotony   signatures   asgiven,   we   reject   the   hypothesis   that   female   and   male Fig.   3.   (Color   online.)   Left:empirical   distribution    f  D *   of    the   random   variable   D = P  _– P  _*,asymptotically   (in   sample   size)   Gaussian   N  (0,1)   (red   curve).   Right:theoretical   cumulative   function   F    of    the   uniform   (blue)   and   Gaussian   (red)   distributions   of    errors. Fig.   4.   (Color   online.)   The   six   different   configurations   of     X  1  and  X  2 supports   in   the   case   of    the   uniform   distributions   of    errors.  J.   Demongeot    et    al.    /    C.R.   Biologies    xxx   (2015)    xxx–xxx 4 GModel CRASS3-3401;   No.   of    Pages   7 Please   cite   this   article   in   press   as:    J.Demongeot,   et   al.,   Stochastic   monotony   signature   and   biomedical   applications,   C.   R.Biologies   (2015),   http://dx.doi.org/10.1016/j.crvi.2015.09.002  crossing-overs   have   similar   localization   by   chance   withindependency   between    X    and   Y  ( P    <   10  7 ),   which   is   infavor   of    the   existence   of    the   same   frailty   domains   along   themale   and   female   chromosome,   explaining   the   frequency   of crossing-over   co-occurrences. 4.2.Individual   choices   and   collective   consciousness During   successive   sessions   of    choice,   participantsbelonging   to   a   defined   group   are   choosing   one   picture   froman   ‘‘absurd   questionnaire’’   consisting   of50   pairs   of    pictures,ineach   of    which   one   picture   has   to   be   chosen   [5].The   resultsgivenin   Fig.   6   areused   to   search   for   evidence   in   favor   oftheinfluence   of    group   dynamics   on   individual   choices   of    thepictures   proposed   inthe   questionnaire.   The   swaps   betweenthetwo   pictures   of    apair,   measured   for   each   pair   of    picturesbetween   the   initial   choice   (called   Achoice)   across   sessionscould   be   interpreted   asamanifestation   of    group   dynamics:for   instance,the   ‘‘honeymoon’’   (dependence   on   the   leader)and   the   successive   ‘‘fight-flight’’   (reaction   against   thedependence   on   the   leader)   attitudes   could   berepresentedby   agreater   number   of    group   dynamics   swaps   (further   awayfrom   pure   randomness).   It   seems   intuitivethat   an   increaseof    simultaneous   swaps   could   be   linked   to   an   increased   groupactivity   or   group   dynamics   event.   Statistics   carried   out   onthetotality   of    swaps   A ! B   or   B ! A   evidencedasignificantincrease   in   the   swap   numbers   between   S01/S02   and   S02/S03session   transitions,   and   between   S03/S04   and   S04/S05session   transitions,   incompatible   witharandomfluctuation.In   other   words,   the   number   ofchanges   inchoices   increased Fig.   5.   (Color   online.)   Crossing-over   numbers   N    (male   red   bars   andfemale   blue   curve)   along   the   human   chromosome   3   (12/63   monotonic   discrepanciesencircled   by   orange   ellipses). Fig.   6. (Color   online.)   Group   dynamics-driven   swaps   from   picture   A   to   Band   from   picture   Bto   A,occurring   over   time   between   11   consecutive   sessions   foreachof    50   pairs   of    pictures,   for   the   two   types   ofswaps   (Ato   Band   Bto   A).   Wesee   inside   orange   ellipses   the   localization   of    discrepancies   between   themonotony   of    the   curves   relative   to   A ! B   and   B ! A   swap   numbers.  J.   Demongeot    et    al.    /    C.R.   Biologies    xxx   (2015)    xxx–xxx   5 GModel CRASS3-3401;   No.   of    Pages   7 Please   cite   this   article   in   press   as:    J.Demongeot,   et   al.,   Stochastic   monotony   signature   and   biomedical   applications,   C.   R.Biologies   (2015),   http://dx.doi.org/10.1016/j.crvi.2015.09.002
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