Figure 1. Jaw RMS target-tracker difference for a.9hz sinusoidal target.

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1 Approaches o Bayesian Smooh Unimodal Regression George Woodworh Dec 3, 999 (Draf - please send commens o george-woodworh@uiowa.edu). Background Speech Ariculaion Daa The daa in Figure were obained by asking 5 subecs ranging in age from 8 o 73 years o rack, using aw movemens alone, a do moving sinusoidally in one dimension a.6 Hz on a video monior. Their aw movemens were capured elecronically by means of srain guages and ranslaed ino he moion of a cursor which was o rack (follow) he moving do. Fideliy of racking was measured in several ways, including TTD, he RMS difference beween arge and racker shown in Figure. Low values of TTD indicae ha he subec has high conrol of he speech ariculaor ((lips, aw, or voice). The invesigaors believed abiliy o conrol speech ariculaors o be unimodal in age, reaching a broad opimum in early o middle adulhood. Hence, hey wished o fi unimodal regressions o such daa. The purpose was o esablish agenorms agains which o compare he performance of neurologically compromised individuals. Targe-Tracker Difference Age Figure. Jaw RMS arge-racker difference for a.9hz sinusoidal arge.

2 Unimodal Regression Unimodal regression is a ype of order resriced inference (ORI). In he mos common formulaion of ORI, y, a px vecor of homoskedasic, independen, normally disribued random variables, has mean vecor µ which is know a priori o lie in a cone C. I.e. if µ and ν are in C hen µa νb is also in C for any non-negaive scalars a and b. The se of unimodal vecors is no a cone unless he posiion of he mode is known and herefore he mos powerful ools of ORI are no available in his case. To work around his difficuly, Frisen (986), suggesed esimaing µ for all possible posiions of he mode and selecing he mode wih he smalles residual sum of squares. He suggesed ha when a smooh esimae is required he iniial esimae could be smoohed wih a unimodal, non-negaive kernel, which preserves unimodaliy under convoluion. Convex funcions are eiher unimodal or monoone and lie in a cone; consequenly, he problem of covex (or concave) regression has received more aenion. However, convexiy (or concaviy) is considerably more resricive han unimodaliy. In his paper, I propose a Bayesian approach o smooh unimodal regression.. Bayesian, Smooh, Unimodal Regression Le y, n, be independen observaions from n subecs. Le x,!, xn be values of a covariae, e.g. age. y is normally disribued wih mean ( x ) µ and precision τ. The mean funcion µ () is assumed a priori o be smooh; i.e., i has a coninuous second derivaive. Le,!, p, p n, be he disinc covariae values. Sufficien saisics are n, y, and ss = ( y y ), and he log likelihood funcion is i i i i x= i p n τ ( ( )) l( µτ, ) = ln( τ ) ni yi µ i ssi (.) i= Bayesian analysis requires he specificaion of a prior disribuion for τ > and a prior condiional disribuion for µ () τ over he space of smooh, unimodal funcions (or some subse of ha space).

3 Ses of Smooh, Unimodal Funcions Le u ( ) be a smooh unimodal funcion over he inerval [,]. If he mode is an endpoin, hen u () is monoone, oherwise here will be an inerior mode u = u( ) a all poins in he modal inerval. If u is sricly unimodal, hen = =. Define Clearly () ( ) m ( ) = m sign( ) u u ( ) (.) m is non-decreasing and is smooh for [, ], and has a leas one coninuous derivaive a and. Thus if m() is any smooh, monoone funcion, hen he funcion v () u ( m m ()) = (.3) is smooh and unimodal; consequenly, funcions of form (.3) are a subse of smooh, unimodal funcions. An example of a smooh, unimodal funcion which canno be expressed his way is v () 3 =. Represenaion (.3) reduces he problem o esimaion of a smooh monoone funcion, which has been exensively invesigaed. Ramsay (998) proved ha any smooh monoone funcion has he represenaion s m( ) c c exp( w( u) du) ds =, (.4) where wu ( ) is any square inegrable funcion. He proposed a penalized maximum likelihood esimaion based on he likelihood funcion, n λ i i i= l( βτ,, w) = τ ( y β βm( )) w ( ) d (.5) The Bayesian inerpreaion of he penaly is ha w=dw, where W is a Wiener process wih precision λ. Combining (.3) and (.4), I propose he prior specificaion, s.5 u ( ) = u u u exp( λ dwu ( )) ds where W () is a sandard Wiener process., (.6)

4 .5 The smooh monoone funcion m( ) exp( dw ( u)) ds has he nonlinear sae space represenaion, where ( ) i i λ s = evaluaed a he poins,!, p m = m m e i i i m = m e i i u, v, i p, are independen random vecors defined by, vi i i i ui, (.7) i.5 ui = ln exp( λ W( s)) ds i. (.8) v = W( ) W( ) Alhough his approach is promising, is implemenaion requires a good approximaion o he oin disribuion of ( u v ),. For ha reason, I propose a second approach using b-splines. i i 3. B-spline represenaion of a smooh unimodal funcion. The normalized cubic b-spline basis funcions are smooh b (),!, b () wih knos a T,!, Tk have he propery ha he funcion K = K v () = vb(), (3.) has no more srong sign changes han does he finie weigh sequence v,!, vk (Schumaker98, Theorem 4.76). Since he basis funcions sum o one for all, his implies ha if he finie series is v,!, v K is unimodal, hen he smooh funcion v () is eiher unimodal or monoone. Thus funcions of he form (3.) wih unimodal weigh vecors v,!, vk are a subse of smooh monoone funcions. For a Bayesian analysis, he daa precision, τ, can be given a conugae (gamma) prior h( τ ) The problem is o specify a prior disribuion g( v τ ) for he weighs v,!, vk over he se of unimodal vecors. Once he priors are specified, he log poserior disribuion, up o an addiive funcion of he daa, is, n ( βτ,, ) = ( i β β ( i)) ( ) ( τ) i= l v τ y u g v h, (3.)

5 where K u () = vb() (3.3) = To specify he prior disribuion of v,!, vk, noe ha he vecor v,!, vk is unimodal if and only if i can be expressed in he form = * ( * ), where v v m m m m " m K is non-decreasing. There are several possibiliies for specifying a prior disribuion over he space of monoone vecors; order saisics of independen and idenically disribued random variables, cumulaive sums of independen, non-negaive random variables, ec. Informal invesigaion suggess ha if he prior is fairly diffuse, he exac specificaion is no criical; however, his poin requires furher work. Figure shows he poserior mean unimodal b-spline curve wih knos a 6,4,3,...,8. Daa precision was given a diffuse gamma prior wih mean 5 and shape parameer.5, which means ha he residual sandard deviaion had prior median. and wih prior probabiliy.95 was beween.6 and 4.5. The prior disribuion of he unimodal weighs was specified by, v = v ( v m ), m = z " z, (3.4).5 * * where z,!, z K, are independen, normally disribued random variables wih zero mean and precision., and v * has a Normal disribuion wih mean and precision. runcaed a zero. The poserior mean is shown in Figure ; he poserior disribuion of he residual sandard deviaion had mean.6 and 95% poserior credible inerval.48 o.8. The poserior disribuion was no sensiive o he hyperparameers (he mean and shape of he gamma disribuion of τ, he precision of z,!, z K, and he mean and precision of v * ). Compuaions were carried ou via Markov chain Mone-Carlo (Gelman, e al., 996; Gilks, e al. 996) using WinBUGS. (Spiegelhaler, e al., 999).

6 Targe-Tracker Difference Age Figure. Unimodal b-spline fi. References Frisen, M. "Unimodal Regression," The Saisician, 35, , 986. Ramsey, J.O. "Esimaing Smooh Monoone Funcions," J.R.Saisi.Soc. B, 6, Par, , 998 Schumaker, L.L. Spline Funcions: Basic Theory, New York, John Wiley & Sons, 98. Gelman A. and Rubin DB. Markov chain Mone Carlo mehods in biosaisics. Saisical Mehods in Medical Research 996; 5: Gilks, W.R., Richardson, S., and Spiegelhaler, D.J. Markov Chain Mone Carlo in Pracice, London, Chapman and Hall, 996. Spiegelhaler DJ, Thomas A, Bes NG. WinBUGS Version. User Manual. MRC Biosaisics Uni, Cambridge, UK, 999.