Long Tailed functions

Transcription

1 Log Taled fuctos Log tal fuctos are desrable for fttg may physologcal data sets A geeral example s fttg the respose of a system to a mpulse put Most passve systems have u modal rght skewed respose fuctos I multple dcator dluto studes, the put s a bref solute jecto pulse to the arteral flow to the orga ad the output s the cocetrato curve vs tme the outflowg blood The complexty of traorga hadlg of the solute gves rse to a large varety of outflow curve forms Those for utrasformed solutes ted to be u modal but the tals are prologed, ofte beg multexpoetal or eve fractal wth power law scalg The partcular log tal fuctos modeled here are composed of two parts: a tal u modal probablty desty fucto (PDF), F(t), ad a tal fucto, T(t), ether the sum of decayg expoetals or the sum of decayg power law fuctos The tal fuctos are joed to the dowslope sde of the PDFs The fuctos F(t) ad T(t) have matchg values ad slopes where they are joed so there s o apparet dscotuty A graphcal user terface (GUI) for LTFs allows chagg may parameters (Fgure ) The selectos made wth the GUI are: () the type of PDF ( PDF ), () the type of tal fucto, expoetal or power law, ( exporpow, ad () the place o the PDF where the tal s attached ( torfr ), specfed as ether a specfc tme, tjo, or as a fracto of the peak heght, frjo Fgure : Graphcal User Iterface for the log taled fuctos showg default parameters PDF parameters: Clck o PDF to choose the leadg part of a log taled fucto The choces are () Lagged Normal Desty, () Gaussa, () Posso, (4) Radom Walk ad (5) Gamma Varate PDFs are descrbed detal at where lmtatos o the

2 parameters are gve The default choce for PDF s the Lagged Normal desty fucto, a Gaussa dstrbuto lagged by a sgle expoetal The parameters used for the varous PDFs ad ther default values are summarzed Table The area s the tegrated area for the leadg part of the curve before jog the log tal to t Evetually area wll used to ormalze the etre curve cludg the tal The mea trast tme of the curve, tmea s usually slghtly later tha where the peak occurs, except whe the Gaussa PDF s chose RD, the relatve dsperso, s the square root of the varace of the curve, ormalzed by the mea trast tme The skewess of the curve s gve by skew The fractoal of the peak heght, frpeak, s the cutoff for calculatg the PDF The upslope parameter s oly used for the Lagged Normal Desty curve to replace the begg of the curve wth a lear upslope for ths partcular PDF Table : Default parameters for the PDFs PDF parameters Lag Normal Desty Gaussa Posso Radom Walk Gamma Varate area tmea RD skew Not used Not used frpeak E 6 E 6 E 6 E 6 E 6 upslope Regular Not used Not used Not used Not used Area uder the LTF: The user defes the area that s used to ormalze the etre curve The LTF s brought to a ed whe T(t) s less tha frpeak, the fracto of the peak heght of F(t), the PDF The ormalzato accouts exactly for the completeess of T(t) gog to zero, ad the area s exactly what the user has chose The parameter frpeak s usually set to less tha 00 Cotuty at the jog pot: A key codto for smoothess at the jog pot s that the fuctos match ad ther dervatves match F t=tjo=t t=tjo ad df t=tjo =dt t=tjo/dt dt To eforce the cotuty codtos, the weghts ( w ' s ) ad the decay rates ( k ' s ) for the sum of expoetals are scaled For the power fuctos, the weghts ( wpow ' s ) are scaled ad the argumets to the power fucto are tme shfted as descrbed below The user has the opto of

3 specfyg at what tme (tjo) or at what fracto of the peak heght (frjo) the log taled fucto wll jo the chose PDF Ths choce s labeled torfr The default choce s frjo whch s set at 5% of the peak heght Multexpoetal ad power law tal fuctos: The user ca choose ether expoetal or power law fuctos for the exteded tal Ths choce s labeled exporpow ad the default settg s expoetal The expoetal choce allows sums of up to four expoetals The power law choce allows sums of up to four power law curves of the form w t tjoshft Sgle or multexpoetal fuctos for T(t): The user ca specfy from to 4 expoetal fuctos all of whch are postve fuctos decayg wth tme The expoetal fuctos have ampltudes (w, = to Exp) ad decay rates (k, = to Exp) The actual fucto joed to the PDF at ether tjo or frjo s gve by T t=a w exp b k t tjo = where tjo s ether the specfed tme, or the tme where frjo occurs ad Exp =, s the umber of expoetals wated Desgatg F tjo=f ad df t/dt=s at t=tjo, the costats a ad b are chose so that at the jog pot, the value of the PDF ad ts dervatve are matched from whch we derve T t=tjo=a = w =F ad dt t=tjo/dt=a b w k =S = a= F = w ad b= S w = F w k = If the weghts are chose so that w =, the a=f ad b= S F = w k For smplcty, the k ' s should be ordered descedg magtude It s mportat to remember the the weghts ad the decay rates are relatve to each other, ot absolute All the chose rate costats are modfed by b, ad b does ot chage the weghtg scheme, the w ' s If adherg to specfc rate

4 costats s mportat, the b ca be set equal to by adjustg the weghts so that = w k = S /F ad w = For =, ths requres that k S/ Fk order that both w ad w are postve Sgle or multple power law fuctos for T(t): If the exporpow choce s set to PowerLaw, a sum of power law fuctos are used for the log tal exteso The parameter Pow = s the umber of power law fuctos used ad ca rage from to 4 We wll use w for coveece here to represet wpow set by the user the GUI (Fgure ) We use the prevous deftos of F, the value of F(tJo), ad S, the dervatve df(t)/dt at t=tjo ad derve T t=a w t tjots, ts = F F a=, ad = S w = The coeffcet a ca be expaded as whch becomes a= F w ts w ts w ts F a= w ts w ts w Whe t=tjo, the sum becomes whch s the equal to T tjo= = T tjo= a w = F w w ts w ts ts ts w wth each ts beg caceled by ts the umerator of the 'th term The sum of the terms the umerator whe dvded by the deomator equals F

5 The dervatve at t=tjo s gve by dt tjo dt N = = a w ts Choosg ts = F, the part of the term S = = S ts F F ad the summato becomes dt tjo = dt = S a w S F = S F a w =S = Examples of the LTFs usg default parameters from Fgure are dsplayed Fgure ExpORpow has bee swtched from Expoetal to PowerLaw to produce the curves Fgure : LTFs usg default parameters The umber of expoetals, EXp, ad the umber of power law fuctos, Pow are vared from to 4 ad dcated by the umbers adjacet to the curves

6 Caveats about optmzato: Do ot attempt to optmze wth frjo The route for fdg the fracto of a peak returs the dex of the pot, ot the tme whe ths occurs Hece small perturbatos aroud the value of frjo wll retur the same pot ad there wll be o model sestvty to perturbg frjo Optmze wth tjo stead If usg expoetal fuctos wth Exp=, t s potless to attempt to optmze w ad k because they get ormalzed out of the equato T t, Exp==F exp S t F If optmzg wth two expoetal fuctos, t s best to optmze oly oe weght ad oe decay rate As the expoetals have hgh covarace wth each other ad the resultat cofdece lmts wll be uacceptably large Smlarly, f usg oe power law fucto, t s potless to optmze wpow because t s ormalzed out of the equato Normalzg o beta s useful T t, Pow==ts t tjots Optmzg excessve umbers of expoetals or power fuctos mafests tself overly large cofdece lmts wth the covarace matrx returg Ifs ad NaNs Do't use more tha you eed