Modelos dinâmicos e simulação estocástica

r-

o

-!

Ｌｾ＠

i o

º

i 11

!

E

o

,

-\marC

FUNDAÇÃO

" .,.. Getulio Vargas

EPGE

Escola de Pós-Graduação em Economia

Seminários de Pesquisa Econômica II

(Ia

parte)

"MODELOS DINÂMICOS E

SI1VIULAÇÃO ESTOCÁSTICA"

J

DANI GAMERMAN

(Inst. de Matemática e Estatística - UFRJ)

Coordenação: ProC Pedro Cavalcanti Ferreira

Te!: 536-9353 -.

I _; _ •.

i

i .

_ｾ＠

I

I

_l

1

I •

Monte Carlo Markov chains

for dynamic generalized

linear models

Dani Gamerman - DME -IM/UFRJ

No 86 ₁₉₉₅

I

-..

I •

Ｎｾ＠

MONTE CARLO MARKOV CHAINS FOR DYNAMIC GENERALIZED LINEAR MODELS

Dani Gamerman

Instituto de :\latemática. Universidade Federal do Rio de Janeiro Caixa Postal 68530. 21945-970 Rio de .Janeiro. RJ. Brazil

Abstract

This paper presents new methodology for making Bayesian inference about dy-ｾｯＡｳ＠ for exponential famiIy observations. The approach is simulation-based

｟ｾｴ＾＠ use of ｾｶｬ｡ｲｫｯｶ＠ chain Monte Carlo techniques. A yletropolis-Hastings ｩＺｕｾｕｮｌｬｬｬｬｬ＠ 1::; combined with the Gibbs sampler in repeated use of an adjusted

ver-sion of normal dynamic linear models. Different alternative schemes are derived and compared. The approach is fully Bayesian in obtaining posterior samples for state parameters and unknown hyperparameters. Illustrations to real data sets with sparse counts and missing values are presented. Extensions to accommodate for general distributions for observations and disturbances. intervention. non-linear models and rnultivariate time series are outlined.

Ime key word3: :\.djusted Time Series: Bayesian: )'letropolis-Hastings ｡ｬｾｯｲｩｴｨｭｳＺ＠

Reparametrization: Sampling schemes: System disturbances.

1. INTRODUCTION

ｾｯｲｭ｡ｬ＠ dynamic linear mo deIs received a great deal of attention in the Bayesian lit-erature after the work by Harrison and Stevens (1976). They consist on an observation equation in regression form

Vt - N(O, V';)

and a system equation reIating succesive regression coefficients or state parameters via

(1)

and system disturbances Vt and Wt are mutually independent. \Vhen ｬｾ＠ and lVt (or

at least l-Vr/V_{t )} are known. inference can be performed analytically. The so-called

I\.alman filter provides on-line distribution of the state parameters .;/ and smoothing the information back to all state parameters is provided by a recursin> alg;orithm. Thus. the smoothed distributions ( 3t1yll) '"

.v

(Tnr,

Cr)

are obtained with y" = í.lh ... i}n). Full

details. expressions for Tnt and C_t and implications to Bayesian inference in g;eneral are

discussed in \Vest and Harrison (1989). Classical inference with maximum likelihood estimation of the hyperparameters V;, lVt , a and R is studied by Har .... ey (1989).

\Vhen both ｾＮｾ＠ and tt"t are unknown. inference can not he performed analytically and

. Ｇｾ｡ｴｩｮｧ＠ procedures are sought. Recent developments in ).Iarko .... r.hain Monte

.u techniques provide answers explored in a few recent papers. Carlin. Polson

... J Stoffer (1992) suggested the use of Gibbs sampling (Gelfand and Smith. 1990:

Geman and Geman. 1984) in a more general context of mixture of normal disturbances. Inference is performed in blocks consisting of the state parameters and the \"ariances of the disturbances. )'lore ｲ･｣･ｮｴｬｹｾ＠ Fruewirth-Schnatter (1994). Carter and 1\:ohn (1994)

and Shephard (H)94) showed that the procedures can be substantially improved if all state parameters are updated in a single multimove inside a Gibbs step.

Dynamic mo deIs can be generalized to account for different observational distribu-tions. The obsef\'ation equation is replaced by

I

II ) {Yt8t

+

b(8t )}

p( Yt Ut ex: exp

Ot

and the sucessi .... e means J1.t

=

E(!lt 18t ) = b'( 8t ) are related to the state parameters .... ia

t,he link relation g\ ILt)

=

Tlt

=

F: 3_{t •} The functions b and 9 are at least differentiable

twice and the weights <Dt are assumed known. This is a dynamic '.-ersion af the

gen-eralized linear models (McCullagh and Nelder. 1989). The models are called dynamic generalized linear models or exponential family state space models. The analysis can not be performed exactly and \Vest. Harrison and Migon (1985) proposed an approx-imation based on linear Bayes' and conjugate priors. Kitagawa (1987) proposed a non-parametric type of approximation while Fahrmeir (1992) has worked on obtaining the posterior mode of state parameters. Fruewirth-Schnatter (1995) also suggested some analytic approximations to inference in these models.

In this paper. general Bayesian inference (ie posterior distribution for state param-eters and disturbances variances) for dynamic generalized linear models is obtained based on Monte Carlo Markov chain techniques. Some ideas were explored in this

.:.

1 li!

:-ection bv Carlin and Polson

_.

! 1992) and Fahrmeir. Hennevoe:l and Klemme _- (1992) but

In limited setups. In the next section. the Gibhs sampler for dynamic linear models

is revisited to facilitate the introduction of an alternative methodology for generalized mo deIs is section 3. Examples are provided in section -l anel some extensions and ('onduding remarks are made in section 5.

2.

GIBBS SAMPLING FOR DYNAMIC LINEAR MODELS REVISITED

From now on. assume we are studying a dynamic linear model for a univariate time series of size n where Gt

=

G. ｾＮｾ＠

=

F and iVt

=

t.-V. for alI t. These restrictions are

only imposed for clarity of exposition although comments about the general case are provided below when appropriate. Carlin, Polson and Stoffer i 1992) considered a Gibbs sampling approach to parameters divided in blocks 3_{1 , ....} .3n •

t·

and

n

7 and provided expressions for the (normal) fulI conditional distributions for 3t • Fruewirth·Schnatter ( 1994 " Carter and Kohn (1994) and Shephard (1994) showed how one can block alI state parameters and sample directly from the (nonnal) fulI conditional distribution of 3n _{= (.8}

1 , .••• Pn). The computational importance of this multimove lies in removing

separate sampling from the correlated state parameters and hence improving the speed of convergence. Shephard (1994). in particular. stressed the detrimental effect of the prior correlation in the convergence of the sampler. This prior correlation in the model is induced by the system equation relating successive state parameters. It decreases with the increase in magnitude of the system disturbances controlIed hy

n'.

The correlation of state parameters is dealt with hy the above papers by hlocking rhem in Gibbs multimoves. A.nother suggestion for the removal of the correlation is reparametrization. Consideration the system disturbances as functions of the state parameters gives W1

=

31 and Wt =f3t - G Jt - 1 , t

=

2 ... n. The inverse transfonnation rewrittes state parameters as

t

;31

=

W1 and dt

=

2:

Gt-jWj , t

=

2, ... ,n

j=1

(2)

When the transition matrix G depends on time, Gt-j is replaced by G t ••• Gj+1 in (2).

Then. the dynamic linear model can be rewritten as

Yt

=

F:

Li=1 Gt-iwj

+

Vt Vt '" ｾｖＨｏＬ＠ V)

Wt '" N(O, W) , t

=

2 .... , n and W1 '" N( a, R)

The joim density of aH quantities is given hy

11 n

ptyn. w".l-.

n-)

=

IIp(Ytlwt. 1°) IIP(U'rin-)Ptu'l IP( V

n-)

c=1

where xJ denotes (.rl' .... x J)' The fuH conditional density of tc •. t = 2 ... 1/ IS ＷｩＢｾＨ＠ I.vr)

ｾｾｮｾ＠ ｾ＠ .

. ｾ＠

.

where H: j

=

F:GJ-C and ktJ

=

F:

L.;=l.i:#!

Gj-iWj . Similar calculatiolls hold for WI' It is then easy to show that the fuH conditional for Wt is

/-1)

with

"

bt = Bt

L

\-T-I Hrj(Yj - k_{t ))}

J=t

and

B

-

(Hl-

1

+

ｾ＠

,,°-I

H H')

-I

t - L t ) ! )

;=t

for t = 2 ... n and

rznd B

=

(R-I..!..

+

1--1

HH,.)-I

I . ｾ＠ I) I)

;=1

\Yhen F and iV depend on tIme. the expressions abovp have 1° alld Ir respectively replaced hy 1:. and IV_r•

The ful! conditional distributions for F and

ir

wil! depend on the form of the prior. In the case these are independent inverse Gamma and inverse \Vishart distributions with kernels V'-{LlV /2)-lexp( -Sv /2V) and

IWI-

Llw/2exp( -tr( t·V-1 Sw )/2) respectively, the full conditionals 7rv( V) and 7rw( W) are in conditional conjugate form with

:Tv( V) ex: ｲｲｾＭｬ＠ p(Ytlwt. V)p( V)

::x:

ｲｲｾｬ＠

V-l/2 exp { -

Ｒｾ＠

(Yt -

F:

ｌＮｾ］Ｑ＠

Gt-jWj

r!}

Y--(Llv/2)-lexp {

-iV

}

::x: 1--((Llv+n)/2)-lexp

ｻＭＲｾ＠

(Sv

+

ｌＮｾＱ＠

(Yt -

F:L.i=1

Gt _j

Wj)2)}

and similarly

r·· _.

-co

I "

Vague priors are obtained by letting the hyperparameters R-I. Sv. Sw. VI". Vw --- O.

Prediction is also given in a sampling-based approach. The predictive distribution

for I Yn+h Iyn) (h

2:

1) is obtained by sampling Yn+h from the observational distribution

of I Yn+h IJn+h' l-)' The sampled "alue of Bn+h is obtained from (6) through

h

.3n+h

=

Ch.'in

+

L

ch-I U"HI

1=1

by sampling Jn, Wn+l, .... Wn+h. The sampled value of 3_n is given by the above

proce-dure while the future disturbances Wn+i are independently sampled from a S(O. IV) ｾ［Ｂｴｲｩ｢ｬｬｴｩｯｮＮ＠ Finally, sampling from the normal density (Yn+hIPn+h, V) is trivial .

. ｾｩｬｬ＠ conditional distributions for Wl, ..•• W_{n ,} F and W are alI in standard form

. _ .... mpling can be easily done thus completing an alternative Gibbs sampling scheme. The state parameters are immediately available through the transformations (2). The relative efficiency of this scheme is expected to lie hetween the efficiency obtained through sampling state parameters separately and jointly (Liu. \Vong and Kong, 1994). Empirical evidence obtained from simulation studies in highly correlated systems or slowly varying parameters suggests that convergence speed is of same order of magni-tude asthat from multimove sampling and both are orders of magnimagni-tude faster than

"'')rat.e ｾ｡ｭｰｬｩｮｧＮ＠ A possible explanation is that most system correlation comes from

.. ,ecification and this is removed by the reparametrization in terms of system u.,turoances. Hence the improvement in speed of convergence. The empirical results below also suggest superiority of sampling procedures based on system disturbances.

For low correlation systems or more erratic parameter trajectories. the reparametriza-tion is less likely to produce substantial gains in efficiency. However, low correlareparametriza-tion systems are less frequent (Shephard. 1994) and is very difficult from the onset to ｡ｳ｣･ｲｾ＠

tain under which regime the process is. It seems unwise not to use the reparametriza-tion as a general safeguard. The price paid by the reparametrizareparametriza-tion is the computa-tional cost of evaluating the moments in (4).

3. INFERENCE PROCEDURES

3.1 Reparametrization of dynamic generalized linear models

A.ny dynamic g;eneralized linear model can also he rewritten in terms of system rlisturbances with

p(YtiBt ) ex:

q(J.lt)

=

{ yr"r+6(l/r)} lvith

éXp Or

77 _t = F' _t ')t Gt-J·_{w ·}

"'-)::1 J

Wt '" _V(O.

n--) .

t

=

2 ... n and W1""'" S( a. R)

where now (}t and J.lt are functions of wt_.

The posterior distribution for mode! parameters isgiven hy

" ....• U'n, IV) x

x

ｉｔｾＱ＠ éXp ｻｶｲ｡､ｷＧＩＺｾｉｉＯｲＡｷｲＩＩｽ＠ ｉｔｾＺＺＲＱｬｖｉＭＱｲＲｆｸｰ＠

{-±IL<n--I'L't}

ｆＮｘｰｻｾＨｗｬ＠ - a)'R-1(Wl -

aI}

Il-VI-

vW

/2exp{ Ｍｾｴｲｻ＠ n--ISlVi} ( 6)

( 7)

for the parametrization in (6), In terms of state pararneters. rhe posterior is the sarne with (}t and Wt written as functions of

:r

using (2). In either case. the form of

the posterior and the resulting fu11 conditional distributions for rnodel pararneters are non-standard. Direct sarnpling procedures required in a Gibbs scherne are difficult to derive arid likely to have lirnited use or not take account of the structure of mo deI. Fahrrneir. Hennevogl and Klernrne (1992) used rejection sampline; which has lirnited use due to. difficulty in finding efficiem envelopes in general. Carlin and Polson (1992) also proposed methodology hased on latem \ã.riables only suitable Íor discrete data. For the hyperparameter IV. the fuH conditional ii"w is an im-erse \Vishart distribution

as given in (5) anel easy to sarnple from.

3.2 Outline of Metropolis-Hasting algorithms

The general approach proposed here is based on taking the structure of the problem imo consideration and to our benefit. The first step is to give up the idea of deriving an efficient Gibbs sampler for mode! parameters. Fortunately, if good but not necessarily enveloping approximations to their full conditional distribution can be found, the :Aetropolis-Hastings algorithm (Metropolis et al .. 1953: Hastings, 1970) can be called in to provide efficient sampling schernes. This can be cornbined with the Gibbs sarnpler for W in a hybrid scherne (Tierney, 1995). When sampling state pararneters one at a

time, an iteration cycle in such scheme moves the chain from state (/3_{1 , ....} Jn , W)(old)

to ({31' ... , (3n, w)(new) by:

-•

a) sampling

3;

from a proposal transition density (lt( :3!(Oldl. ';1) and accepting this \-alue. ie setting ｊｾｮ･ｷＩ＠ =

:3;.

with probability

. { 7"(d 3;l/.qd .;;vid). :1;) }

rnzn 1. ·dl I·)

. J (vI , / :J. J \.:> a \

7It\ t ) qd, I ' ! '

where 7r! is the fuU conditional density of 3_t for t

=

1. .... 12:

b) sampling w(new) from 7rw.

In terms of system disturbances. the iteration cyele is conceptuaUy the same with

IVt repIacing

Pt

throughout step (a). :"'lultimove sampIing of aU state parameters is .. " nossible with a new set of values (di, .... ｊｾＩ＠ jointly sampIed from a joint proposal ;ty and jointly accepted with probability depending on this joint proposal and

. ue joint conditional density of ,3n. The relative merits of each of these schemes is

discussed beIow.

:"larkov chain theory ensures that virtuaUy any proposal tranSltlOn density \ViU lead to the posterior density as the equilibrium distribution of the chain (Roberts and Smith. 1994: Tierney, 1995). Hence. repeatedIy iterating through steps (a) and (b) above \ViU eventually lead to a sample from the posterior distribution.

Empirical work suggests that a good choice of proPosal can lead to substantial effi-("iency gains and should if possible incorporate the structure of the model. The eloser are the proposal and the fuH conditional densities. the higher is the acceptance prob-abili ty. When they are equaL sampling is done directly from the fuU c:ondi tional and the acceptance probability becomes 1. The next section concentrates on specification of a proposal that takes into account the mo deI structure.

3.3 Specification of the proposal transition density

West (1985) showed that repeated use of Bayesian inference for weighted least squares to adjusted observations leads to the posterior mo de of the regression coeffi-cients in generalized linear models. This idea was used in unpublished works by P.

Muller and G. Rosner and by the author to fonn the base of proposal transition densi-ties for a Metropolis-Hastings algorithm in the more general case of generalized linear mixed models. Empirical evidence provided suggests this procedure leads to efficient sampling schemes for inference.

use of a nonual dynamic linear mo deI for an adjusted time series forms a Fisher scoring step that leads to the posterior mo de of the state parameters. Once again. )'Ietropolis-Hastings algorithms incorporating this information are likely to perform just as well as in the static case.

To consider it in more detail. adjusted observations

Ut

and associatt"d \'arianct"s ｶｾ＠ are defined as

They are functions of the current vãlue of the state parameters .Jt or ｾｹｳｴ･ｭ＠ distur-l)ances wt depending on the parametrization adopted through the functional

depen-... llce on Tlt and Il-t. Considering the state parameter fonu. an adjusted normal clynamic linear model can be created with observation equation

- J(o/d)) F' J

yt!,Jt

=

t·}t

+

t't t· . t ,."" -\'{O • T .. "( I t /J J(t.l/d))} t í8)

and system equation (1). The posterior mode algorithm is based on repeated use of smoothing an iteratively adjusted time series as follows:

i) start with Jt

=

ｭｾｏＩＮ＠ t

=

1. .... n and set 1=1:

ii) construct the normal DL::V! with adjusted observations Yt and weights Ｑｾ＠ de-pending on dt = ｭｾＯＭｬＩＮ＠ obsenãtion equation (8) and system equatioIl ( 1 ):

iii) obtain the adjusted smoothed distributions I

"t

1fT)

"'-'

S( 111 ｾＯＩＮ＠

C;

I) ). t = 1. .... 1l:

iv) increase I by 1 and return to (ii).

The proposal densities suggested by the above reasolllllg are based 011 r.he full (onditional distributions obtained at each step given by the adjusted normal dynamic linear mo deI above. These densities are called adjusted full conditional and are denoted beIow by 7i- to distinguish them from the correspondin,e; fuU conditiollal 7l'. Although the full conditional density is difficult to sample from in general. the adjusted full

conditional densities are always normal for the three sampling schemes considered.

3.4 Expressions of the proposal transition density

Wben sampling the state parameters one at a time. the proposal densities qtt ＮＳｾＯＭＱＩＬ＠ :3t) at iteration I are given by the adjusted full conditional densities

ir_{tU3t )} oc p(Ytlpdp(ptlpt-l. W)p(pt+llpt, W)

:'or t

=

2 ... n - 1 with fit

=

fid.3:I-I)) and ｬｾ＠

=

ｬｾＨＳＩｉＭｉＩＩＮ＠ Simple c-alculations

:-.:au

to a _V{{F;l"t-1Ft

+

H7- 1

+

C'lV-IC)-1 ＨｆＯｬｾＭｉｦｩｴ＠

+

H--1C3t_1

+

C'H,"-13t+d.

ＨｾＧｬｾＭｉｆｴ＠

+

H/--1 _{+C'l-V-1C)-I} distribution. The endpoint state parameters also have}

similar expressions for their adjusted fuH conditional distributions. These distributions coincide with those given by Carlin. Polson and Stoffer (1992) with replacement of .lJt

and ｬｾ＠ by fidJ11- 1)) and ｬＭｾＨ［ＮＱＱＱＭＱＩＩＮ＠ ｾｉｩｳｳｩｮｧ＠ values are easily dealt \vith by removal

of terms ｆＺｖｾＭｉ＠ F t and ｆＺｾＭｉ＠ fit from moments above. In this ca<;e. the adjusted full

conditional coincides with the full conditional and the corresponding state parameter is Gibbs sampled.

\Vhen sampling from the system disturbances. the proposal densities qd ｷｾｉＭｉＩＮ＠ Wt)

"m:> !!;iven by adjusted fuH conditionals 1rt{

wtl,

(t = 1. .... n) in (<1) with replacement of Yt

, _ＧＮｾ＠ fidJ11- 1)) and ｖｾＨｰＩｉＭｉＩＩ＠ and 31 /-1) depend on W)'-I) "ia (2). A missing value

,lue t interferes with the adjusted full conditional for UJ3 • • ｾ＠

=

1. .... t hy removal of

terms ｬｾＭｉ＠ ｈｾｴｻｙｴ＠ - ｫｾ､＠ and ｬｾＭｉ＠ H 3t H;t from the expression of b$ and B$'

When multimove sampling, the joint proposal q(( 31' .... Jn )(1-1). (;.11, .... .in)) is given by the adjusted fuH conditional 1ra(j3_1, ...• ;.1n)

=

7r(;31, .... ,3_n l.iin. l-V) obtained by

Fruewirth-Schnatter (1994) and Carter and Kohn (1994). These papers showed that

n

:T <3(;31, ...• Jn ) = íT{ dn lfin• l-V)

rr

íT( .3_{t 1.3t+}1

d/

t. IV) (9)

/=1

The densities to the right of (9) are obtained hy repeatedly updating the state

param-t"fS to give the on-line distributions íT(

dtl!?

IV) and then incorporating the system

)fi 7r( 3 t+₁

i

.3_{t . l-V) .} . \ssuming the normal on-line distributions of (.3_{t lfit,} ｾｖＩ＠ with

lllUlllents _ãt and Rt. it is easy to show that the distributions of (3t Idt+l' fi!. n:) are

nor-mal with means _{à t}

+

RtC'( CRtC'

+

J,V)-I _(8t+1 - Cà t ) and variances Rt - RtC'( CRtC'

+

IV)-ICRt, t

=

1. ... , n - 1. Hence sampling from the proposal involves sampling ｪＳｾ＠

from 7r(;3n l.iin, W) and then

a;

from 7r(

.BtI.B;+

l'

i?

W) for t

=

n - 1, .... 1. A missing \'alue at time t is taken care of by setting àt

=

Cãt-1 and Rt

=

CR t-1C'

+

W.

The adjusted full conditional densities are obtained in each case with results for normal dynamic linear models. They depend on the previous value of the chain through the adjusted observation and weights. These are iteratively evaluated at each step of the Markov chain.

predictive distribution (Yn+h iY") (h

2

1) is carried out hy ｳ｡ｭｰｬｩｮｾ＠ ｦｊｾＫｨ＠ usine;

I-I -I I h. h-I

{ (

!1 ) }

fJ 71 + h = b 9 F G 3"

+

ｾ＠ G W,,+i

anel then ｳ｡ｭｰｬｩｮｾ＠ Y71+h from ( Yn+h 18;'+h) which is in the t'xponential family and easy

ro sample from. •

4. APPLICATIONS

The schemes derived aboye where applied to three datasets. 80th the separate ... ., .. ?meter émd system disturbance sampling schemes led to very ｨｩｾｨ＠ acceptance teral. typically accepting on average more than 90

%

of the proposed . mg Lue adequacy of the proposals used. A. somewhat surprisine; result found in the applications concerns the multimove scheme. Despite its anticipated ";llperiority. it seems to produce an inefficient ｾＱ･ｴｲｯｰｯｬｩｳＭｈ｡ｳｴｩｮ･［ｳ＠ ale;orithm due to

\'ery small acceptance rates. The use of a very high-clilllensional parallleter space at each multimove leads to densities that tend to be concentrated in a small region of the parameter space. So. it becomes extremely difficult for any multimove to get accepted and as a result the chain virtua11y does not move. It may he worth t:'xploring alternatives for the multimoye such as thicker tail proposals hut this was not pursued here,

As for the other schemes. despite similar acceptances rates. results were as antic-ipated. The high correlation between state parameters slows C'onsiderably their con-\'ergence just as in the normal ca'5e with the disturhances salllpline; scheme showine; a :'iuperior performance. In what fo11ows. only results for these schellles are reported.

Example 1. The first example is derived from the dataset of coal-minine: clisastf.'rs given in Jarrett (1979). The dataset consists of times of occurrence of sf.'rious disasters in

Ｈｾｯ｡ｬ＠ mines in England and 'Vales. Dynamic moclels were used by Gamerman (1992)

in a point process analysis. Here. the data is discretized by year similarly to what was done by Carlin. Gelfand and Smith (1992). The observatioll model is Poisson and the system equation is a random walk for the log means. The average trajectories over ·}OO chains is depicted along with the data in Figure 1 where the slowness of the state pararneter sampling scheme is evident albeit IlOt excessive.

:-igure 1 about here >

E.J:lLmple 2. This data set comes from a study on advertising awareness by :"Jigon anel Harrison (1985), Samples of size nt

=

66 are weekly polled to inform on their awareness of an advertising campaign and the relevam covariate here is a cumulative measure of advertising expenditure. both depicted on Figure 2. Features of interest are a change of campaign before week 41 and a few missing data points. A dynarnic logistic regression is used wi th

Again. convergence was empirically monitored by simple averages over 200 parallel rhains as shown in Figures 2 and 3. The slow ｣ｯｮｶ･ｲＬｾ･ｮ｣･＠ of the state parameter :;cheme is more evidem here than in the previous example specially for expenditure (·oetficiem. Figure -l: shows the estimated trajectory of this parameter along with llncertaimy limits. The campaign change is captured in the model hy a chane;e of leveI and greater uncertainty over the final weeks of the study can also be noted.

<

Place Figure 2 about here

>

<

Place Figure 3 about here

>

< Place Figure 4 about here

>

;/', "" .V(O. tV) for .it = logit( iit l, Figure;) shows the estimated trajectory of iit which

;..: \"E'ry similar to the pstimates obtained by Kitagawa (1987) anel by the \\'ork of

L. Fahrmeir and S, \Vagenpfeil. Cnlike these approaches, the presem analysis also provides complete Baypsian inference for the hyperparameter

n-

depicted in Fi2;ure 6, Tht' mean and median for Hl are 0,034 and 0,032 which compare \\,pll \\"ith the point t>stimate of 0.032 quoted by L. Fahrmeir and S. \Vagenpfeil.

<

Place Figure 5 about here

>

<

Place Fil?;ure 6 about here >

5. EXTENSIONS

5.1 Intervention and outlier detection

The approach can be extended to accommodate more general distri butions for the ｾｹｳｴ･ｭ＠ disturbances. A. simple extension is to allow for mixture of normais. Carlin. Polson and Stoffer (1992) detailed the use of scale mixture of normaIs by introduc-tion of supplementary hyperparameters Àt such that Wt '" "V(O. l,V/ ÀI l, For instance.

independent Gamma priors for ,\, with small scale index lead to the fattPr-tailed t dis-tributions for system disturbances that can be more appropriate for periods of sudden rather than gradual change in leve!.

.-\lternatively. discrete mixture of normals \vould aliow for specific departures from the prescribed model. For mixtures of the form tL't "'-' 2:,/\;X(O.

a-, ')

one !),'ets the multiprocess mo deIs class I whereas dynamic mixtures I,Ct '" 2:"\;1'\"(0. BC,) lead to Illultiprocess models dass

n.

These can incorporate specific departure from standard model as exemplified by \Vest and Harrison (1989. ch. 121.

:"Iore direct intervention procedures can also be used. \Vhen the times for

illterven-tion are known in advance. appropriate changes such as system variance infiaillterven-tion can !:.

be set. When it is not known. independent indicator parameters tit '" be1'noulli(p) are

imroduced in the system equation as dt = G{3t-l

+

btZt

+

Wt and Zt is the magnitude

of the change. A related approach is given in McCulloch and Tsay (1994).

5.2 Estimation of other unknown hyperparameters

.-'uü:nown hyperparameter such as the weights (Dt in (6) or the intervention probabili ty

:J

:liJlJ\-e. For such unstructured univariate parameters. adaptive rejection sampling (Gilks and Wild. 1992) or adaptive rejection :Vletropolis sampling (Gilks. Best and Tan. 1995 1 provide efficient sampling schemes that can be combined with the procedures in this paper.

5.3 Multivariate observations

The approach is directly applicable to multivariate observations in the exponential family. A. vector of means J.1.t is related to a vector of canonical parameters IJ_t via

J.1.t

=

8b(Btl/8B

t. These are related to a vector of state parameters

d

t through g(J.1.d =

'li = Ft 3t where the link relation now is a vector valued function and Ft is a matrix of

ﾷｾｲｩ｡ｴ･ｳＮ＠ An adjusted multivariate time series

Yt

and associated \'ariance matrices üre formed by

and

Results follow .from proposaI based on normal dynamic linear model analysis of the above observations with their associated weights. Similar comments also apply to the /'ase where a hierarchical structure is superimposed on the dynamic models ( Gamerman

ＧｲｩｾｯｮＮ＠ 1993 l.

5.4 Observations outside the exponential family

ACKNOWLEDGMENTS

This research was supported by Conselho .\'acional cle Desenyolyimento Científico

e' Tecnológico. Brazil. The initial part of the work was done while the amhor was

\·isiting Imperial CoUege London with a grant from Coordenação de .-\.perfeiçoamento de Pessoal de Ensino Superior. Brazil. The author is grateful to Adrian Smith .. Jon

'°1.kefield and ｾ･ｩｬ＠ Shephard for useful comments.

REFERENCES

Carlin. B. P .. Gelfand. o-\.. E. and Smith. A .. F. :"1. (19921. Hierarchical Bayesian

analysis of changepoint problems. Appl. Statist. 42. 389-405.

Carlin. B. P. and Polson. _ｾＮ＠ G. (1992). 1Ionte Carlo ｂ｡ｾＧ･ｳｩ｡ｮ＠ methoels for discrete regression models and categorical time series. In Bayesian Statisties

4.

Eds. .J. ),1. Bernardo. J. O. Berger. .-\.. P. Dawid and A. F. ).,1. Smith. pp. 577-86. Oxford: Oxford Cniversity Press.

Carlin. B. P .. Polson. _ｾＮ＠ G. and Stoffer. D. (1992), .-\. 1Ionte Carlo approach to nonnormal and' non-linear state-space modeling. J. Am. Statist. Assoe. 87. 4930500.

Carter. R. and Kohn. R. \ 1994). On Gibbs ｳ｡ｭｰｬｩｮｾ＠ for state-space models.

Biometrika 81. 541-53.

Fahrmeir. L. (1992). Posterior mode estimation by extended Kalman filtering for multiyariate dynamic generalized linear models. J. Am. Statist. Assoe. 87. 501-9.

Fahrmeir. L.. Hennevoe:l. \V. and Klemme. K. (1992\. Smoothine; in dynamic generalized linear mo deIs by Gibbs sampling. In Ad"ances in GLnI anel Statistical :\Iodelling, Ed. L. Fahrmeir. B. Francis. R. Gilchrist and G. Tutz. Leeture Notes in Statisties 78. 85-90. Springer.

Fruewirth-Schnatter. S. (1994). Data augmentation and dynamic linear models. J.

Time Ser. Anal. 15. 183-202.

Fruewirth-Schnatter. S. ( 1995). Applied state space modeling of non-Gaussian time series using integration-based Kalman filtering. To appear in Statistics and Compv.ting.

Gamerman. D. (1992). .-\. dynamic approach to the statistical analysis of point processes. Biometrika 79. 39-50.

Gamerman, D. and Migon. H. S. (1993). Dynamic Hierarchical Models. J. R.

Selfand. A. E. and Smith. A. F. :\1. (1990). Sampling hased approaches to

calcu-.atillg marginal densities . .J. A m. StatiJt. AJJoc. bf 85. 398-409.

Geman. S. and Geman. D. (1984). Stochastic relaxation. Gibbs distributions and rhe Bayesian restoration of

ｩｭ｡ｾ･ｳＮ＠

IEEE TranJ. Pattern AnaL. Machine Intel.

PAMI-6. 721-4l.

Gilks. vV. R .. Best. "X. G. and Tan. K. K. C. (19951. A.dapti'·e rejection :\Ietropolis sampling for Gibbs sampling. To appear in AppL. StatiJt.

Gilks. W. R. and \Vild. P. (1992). Adaptive rejection sampline; for Gibbs sampling. AppL. Statüt. 41. 337-48.

Harrison. P. J. and Stevens. C.

F.

(1976). Bayesian

ｦｯｲ･｣｡ｳｴｩｮｾ＠

(with discussion).

J. R. Statüt. Soc. B 34. 1-4l.

Harvey, A. C. (1989). ForecaJting, StructuraL Time Series lvfodeLs and the Kalman Filter. Cambridge: ｃ｡ｭ｢ｲｩ､ｾ･＠ Cniversity Press.

Hastings, Vv·. K. (1970). :\Iome Carlo sampling methods

ｵｳｩｮＮｾ＠

:"Iarkov chains and their applications. Biometrika 57. 97-109 .

.larrett. R. G. (1979). A note on the interval between coal-mining clisasters. Biometrika 66. 191-3.

Kitagawa. G. (1987). "Xon-Gaussian state-space modeling of nonstationary time .;;pnes . ./. Am. StatiJt. ASJoc. 82. 1032-63.

- -:. \Vong, \V. H. and Kong, A. (1994). Covariance structure of the Gibbs

.• th applications to estimators and ｡ｵｾｭ･ｮｴ｡ｴｩｯｮ＠ schemes. Biometrika 81. \IcCullagh. P. and

ｾ･ｬ､･ｲＮＮｬＮ＠

A. (1989). GeneraLized linear ModelJ (2nd. edition). London: Chapman and Hall.

:\IcCulloch. R. E. and Tsay, R. S. (1994). Bayesian analysis of autoregressive time series via the Gibbs sampler. J. Time Ser. Anal. 15. 235-50.

:\tletropolis, ｾＮＬ＠ Rosenbluth. A. W .. Rosenbluth. :\1. )I., Teller, A. H. and Teller, E.

( 1953). Equations of state calculations by fast computing machines. J. Chem. Phy3. 21. 1087-92.

Migon, H. S. and Harrison. P. J. (1985). An application of non-linear Bayesian forecasting to television advertising. In Baye3ian Stati3tic3 2. Eds. .l. :\1. Bernardo,

:vI. H. DeGroot. D. V. Lindley and A. F. M. Smith. pp. 681-696. Amsterdam: North

Holland.

-he Gibbs sampler and ｾｬ･ｴｲｯｰｯｬｩｳＭｈ｡ｳｴｩｮｧｳ＠ algorithm. ｓｴｯ｣ｨ｡Ｌｾｴｩ｣＠ Proce,Q,Qe3 and .ar Application3 49. 207-16.

Shephard. :\. (1994\. Partial non-Gaussian state space. Biometrika 81. 115-3l. Tierney, L. I 1995\. ｾＮｦ｡ｲｫｯ｜Ｇ＠ chains for exploring posterior clisrributiollS. To appear

in A nn. Stati3t.

\Vest. ).tI. (1985), Generalized linear models: olltlier accommodation. :'icale param-eters and prior distriblltions. In Baye3ian Stati3tic3 2. Eds. .l.

:-1.

Bernardo.

:-1.

H. DeGroot. D. Lindley and A. F. :V!. Smith. pp 531-58. Amsterdam: ::'\orth Holland.

\Vest. :'1. and Harrison. P . .l. (1989). Baye3ian Foreca3ting and Dynamic Model3. )jew York: Springer.

\Vest. :'1.. Harrison. P . .J. and YIigon. H. S. (1985), Dynamic Generalized Linear :-lodels and Bayesian Forecasting (with discussion). J. A m. Stati3t . .. Ls,qoc. 80. 73-96.

16

Caption for the figures

Figure 1. :Ylean leveI trajectory for the sampling schemes with number of iterations with data (dots).

Figure 2. :Ylean leveI trajectory for the sampling schemes with ntll11ber of iterations \vith data (dots) and covariate (vertical bars).

Figure 3. :YIean trajectory of regression coeflicient for the sampling schemes with number of iterations.

Figure 4. Estimated trajectory of regression coeflicient obtained after 200 iterations

".p!" 200 rhains with the disturbance sampling scheme.

ｾｳｴｩｭ｡ｴ･､＠ trajectory of rain probability obtained in 200 samples with the

\.u;:, l ",I" Dance sam pling scheme.

Figure 6. Posterior histogram of system variance l-V obtained in 200 samples with the disturbance sampling scheme.

I I

..

I I I

ｾ＠ "

(a) sampling trom system

､ｩｾ＠

CJ) ｾ＠ •

-

.

•• • •

ri

'"'

te

.

ｾ＠ ｾ＠

c _(!1

ｾ＠ • • ; ; ｾＮ＠ 4,_

.

_C!:

3

Wi\

!! " " ,

&

' "

.:

.'; j ｾ＠ I

I

ii ' , '

_..

.'-.. •

I

<

.

.: . .

!!.

N

o

-

...

..

---1860 1880 1900 1920

t-IGURE 1

.;es

1 10 50 100 m C/)

-ｾｦ＠

11)

-

(1) Q. CD < (1)

-

--- ____ e

1940 1960

mj.

ｾ＠

N o

..

.

, ",' 1860

ｾ＠

,]

..

I

Ipling

trom

state parameters

1 10 50

-

100 500

..

...

..

•

•

-(/)

•

=

ｾ＠

￪ﾧｾｾｾ＠ •

_o

Q) • _•

--

Q) _{• •} • • - ex>

E

-!

ro

•

---ｾ＠

ro

c..

•

o

• _• co

Q) _•

-

ro

• -._--,--,-

:,-:;;;::::--

• .li::

r,;; (/) CU

• CU

E

•

_--

_{O ｾ＠}

O _•

• ｾ＠

ｾ＠

•

-

_C>

_•

_•

₌

-•

c: _{• •}

c..

•

_-

-

_O

--

N •

---• ... • .:l

-

---• O

30

20

10

O

N

UJ

Estimated levei

"'V'

....

_'.

--LL

(/)

•

•

-Q)

•

=

u ₌

c: 0 0 8 li! 8

•

o

ro

- oi)

--

.... •

-

ex>

ｾ＠ •

ｾ＠

--:=I

-

(/)

---1:J _•

_o

E

co

Q) _•

-

(/) .li:: _cu

>- _{• • cu}

(/)

o

ｾ＠

o:-J

E

_ｾ＠

o

₌

-.

-

ｾ＠

•

=

!õ' C>

•

-c:

_o

c..

N

E

•

-

-

-_

.•.. -.

---ro

(/) .-.":..:. ｾ＠ •

-

_ro

_•

---o

C"

!l

ｾ＠

o o

la

o

o

I 9 o o I\) I

§

i

I

9

o

ｾ＠

o

!!.I ｾＨＮ＠ l!

FIGURE

J

(a) sampling from system disturbances

,

I

,

20

10

50

100

200

: ... ' , ..

---...

40 60 80

C"

(I)

ｾ＠

o o o I\) o o I o o o I\) I o o o

ｾ＠

I o o o m

o

"

(b) samfJl.

_Im

_{state parameters}

20

500

1000

1500

2000

40 I;

1 \

I ,

,

, 1 ,

"

1 ,\

\",

,··1 i

60

,

_,

. "

80

ｾ＠

"

o

o o m

o

o

o

ｾ＠

o

o

I

o

o

o

ｾ＠

o

FIGURE 4

-"'"-- ....

_-

. . ; -....

----20 ₄₀

_0_.

60

90% limits 80% limits median

---"

....

_

r--"O

ｾ＠

o

o-11)

Q:

-<

.;

ｾ＠

o

o

00

o

ｾ＠

o

Í\J

o

O

o

FIGURE 5

100 ₂₀₀

90% limits median

300

-,

_1

ｲｾ＠

w

O

J

O

ｾ＠

(]'I

ｾ＠

O

(]'I

o

FIGURE 6

...

.

INDICE

DE

RELATÓRIOS

TÉCNICOS

1. ESTCDOS DE ｣ｾｲ＠ CASO DE DISCRüII:\"AÇAo QCADR . ..\ TICA ｣ｯｾｲ＠

ESTüIADORES ROBCSTOS.

David Dorigo e ｾｲ｡ｲｬｯｳ＠ :-\.G. Viana - nrjlTRJ - 1980.

"2. ｃｏｾｉｂｉＺ｜Ｂａￇａｯ＠ DE ｉＺ｜Ｂｆｏｒｾｲａ￧ￕｅｓ＠ SCBJETIVAS E \IÉTODOS

ｑｃａｾｔｉｔａｔｉｖｏｓＮ＠ PARA PREVISÕES DE TAXAS :\"0 OPE:\"-\L-\RKET.

Basílio de B. Pereira - ｉｾｉ＠ e COPPE/CFRJ. Rocardo Cesar Otero e Antônio Horácio Vicente Perrota - DETEC. ｏｐｅＺ｜ＢＭｾｉａｒｋｅｔＮ＠ ｂａｾｉｅｒｉＺ｜ＢｄｃｓＮ＠ - 1981.

.3. REPRODCçAo DE LAGOSTA - A ａｾ￁ｌｉｓｅ＠ DE ｃｾｉａ＠ SÉRIE ｔｅｾｉｐｏｒａｌＮ＠

Basílio de B.Pereira - üI e COPPE/CFRJ. Ricardo C.F.Bezerra e J.B.F. Gomes :\"eto - ｄｅｾｬａＯｃｆｃ＠ - 1982 .

.1. üIPROVED· LIKELIHOOD RATIO STATISTICS FOR GE:\"ERALIZED

ｌｉｾｅａｒ＠ ｾｉｏｄｅｌｓＮ＠

Gauss ｾｉＮ＠ Cordeiro - Imperial College e nl/CFRJ - 1982.

C). ｉｄｅｾｔｉｆｉｃａￇａｯ＠ ECOCARDIOGRÁFICA DE PADRÕES EVOLCTIVOS

DE HIPERTROFIA CARDÍACA ｅｾｬ＠ PACIE:\TES ｃｏｾｬ＠ ｈｉｐｅｒｔｅｾｳａｯ＠

ARTERIAL.

:\"elson de Souza e Silva - Faculdade de ｾｉ･､ｩ｣ｩｮ｡Ｏ＠ CFRJ e \Iarlos A.G. Viana -I:'l/CFRJ-1982.

6. A :\"OTE ｏｾ＠ THE I:\"TRACLASS CORRELATION COEFICE:\"T FOR STAN-DARD ｓｙｾｉｍｅｔｒｉｃ＠ ［Ｂ［ｏｒｾｬａｌ＠ ｄｉｓｔｒｉｂｃｔｉｏｾｓＮ＠

ｾｉ｡ｲｬｯｳ＠ A.G. Viana - nI/CFRJ - 1982.

7. CONCORDÂNCIA ｅｾｔｒｅ＠ JCÍZES.

ｾｉ｡ｲｬｯｳ＠ A.G. Viana - CFRJ - 1982.

8. A XECESSARY ｃｏｾｄｉｔｉｏｎ＠ FOR THE EXISTE:\"CE OF ｾｬａｉＺ｜Ｂ＠ EFFECT PLCS O;";E PLA;";S FOR 2 FACTORIALS.

B.C. Gupta e S.S. Ramirez Carvajal - I:'ljCFRJ - 1982.

9. A FAST ANALYTICAL :-'lETHOD FOR THE ADDITION OF VARIABLES.

RANDON

V. de Senna e Ruy Luiz Milidiú - IM/UFRJ. P.V. Fleming, ｾｉＮｒＮ＠ Salles e L.F.S Oliveira - COPPE/UFRJ - 1982.

10. PLANOS BAYESIANOS DE INSPEÇÃO POR AMOSTRAGEM. Paulo Carneiro Bravo - IM e COPPE/UFRJ - 1982.

11. XEARLY COXST.-\.\"T I.\"SPECTIO.\" I.\"TERVALS FOR THE DETECTIO.\" OF FAILCRE.

V. de Senna - CFRJ e A.h:. Shahani - C. Southamptom - 1983.

12. ｾｉｏｄｅｌｌｉＮ｜Ｂｃ＠ THE TOTAL SCSPEXDED PARTICCLATES .-\XD RELATED ｾｉｅｔｅｏｒｏｌｏｃｉｃ＠ AL VARIABLES.

David Dorigo - CFRJ e José Edilson Almeida - CFPb - 1983.

13. SECCROS E Í.\"DICES DE PREÇOS .

.-\nnibal Parracho SantO .-\nna - ｉｾｉｪｌＭｆｒｊ＠ - 1983.

14 . .-\LCC.\"S TRABALHOS ｅｾｉ＠ SÉRIES ｔｅｾｉｐｏｒａｉｓ＠ X A CFRJ. Basílio de B. Pereira - nIjCFRJ - 1983.

1·). ａｐｅｒｆｅｉￇｏａｾｉｅＮ｜Ｂｔｏ＠ DE TESTES ESTATÍSTICOS PARA :\IODELOS LI:-';EARES CE.\"ERALIZADOS.

Causs ｾｉｯｵｲｴｩｮｨｯ＠ Cordeiro - nl e COPPEjCFRJ - 1984.

16. A SD,IPLE I:-';SPECTIOX POLICY FOR THE DETECTIO.\" OF FAILCRE. V. de Senna - ｉｾｉｪｃｆｒｊ＠ - 1984.

17. r:\SPECTIO.\"S POLICIES FOR THE DETECTIO.\" OF FAILCRE. V. de Senna - nljCFRJ - 1984.

18. ESTCDO ｃｒｏｾｉｏｓｓￔｾｈｃｏ＠ DA ｌｅｃｃｅｾｉｉａ＠ E ｐｒ￉ＭｌｅｃｃｅｾｉｉａＮ＠

David Dorigo e ｾｉ￴ｮｩ｣｡＠ ｾｉ｡ｲｩ｡＠ F. ｾｉ｡ｧｮ｡ｮｩｮｩ＠ - nljCFRJ - 1984.

19. ESTCDOS DOS PADRÕES DE SEXSIBILIDADE A ａＮ｜ＢｔｄＮｈｃｒｏｂｉａｾｏｓ＠ ｅｾ＠

AMOSTRAS DE BACTERIÓIDES FRAGILIS ISOLADA DA ｾｉｉｃｒｏｂｉｏｔａ＠ ｉｾﾭ

TESTI.\" AL ｈｃｾｉａＮ｜Ｂ＠ A.

David Dorigo e Christina C. Tiziano - nljCFRJ - 1985.

:20. ESTCDOS DA FLORA BACTERIA.\"A TRA.\"SITÓRIA DAS ｾｉￃｏｓＮ＠

David Dorigo - nljCFRJ e Celso Luiz Cardoso - ｉＮｾｉｩ｣ｲｯ｢ｩｯｬｯｧｩ｡ＯｃｆｒｊＮ＠

21. PREDICTIVE ECHOCARDIOGRAPHIC PATTER.\"S.

ｾｉ｡ｲｬｯｳ＠ A.C. Viana - ｲｾｉＯｃｆｒｊ＠ . .\"elson Souza e Silva HCjCFRJ e Frank Soler -Associated Research Consultants - 1985.

22. VARIÁVEIS QCE ｃｏＺＭＧ［ｔｒｉｂｃｅｾｬ＠ PARA A QCALIDADE DO PÃO: ｃｾｉ＠ ES-TUDO DE SELEÇÃO DE VARIÁVEIS.

David Dorigo - ｉｾｬｪｕｆｒｊ＠ - 1985.

23. DIABETES ｾｉｅｌｉｔｖｓ＠ EM POPULAÇÃO RESIDENTE ｾａ＠ X X!! REGIÃO

AD-ｾｈｎｉｓｔｒａｔｉｖａ＠ DO :VIUNICÍPIO DO RIO DE JANEIRO: ESTCDO CLÍNICO.

David Dorigo. Ana Nlaria V. Botelho e Flávia Landim - IMjUFRJ - 1986.

24. ｾｉｏｄｅｌｏ＠ ESTATÍSTICO DE AUDITORIA.

David Dorigo, Annibal Parracho Sant' Anna IMjtIFRJ e Valter de Senna -IMjUFRJ e DATAPREV - 1986.

2

.

..

r

ＮｾＮ＠

2.3. :\IODELAGE:\1 ｾｓｔｒｃｔｃｒａｌ＠ E PREVISÃO BAYESIA.\'A: C:\IA APLICAÇÀO E CO:\IPARAÇAO CO:\1 :\IODELOS ARI:\IA.

Gutemberg H. Brasil/ DEE-peC/RJ e Hélio S. :\Iigon/CFRJ/SERPRO - 1987.

26. PESQCISA ELEITORAL - C:\IA A.\ÁLISE BAYESIA.\A.

Izabel G. S. Furtado de :\Iendonça - I:\I/CFRJ E' Hélio S. :\Iigon - I:\I1CFRJ /SERPRO

- 1987.

n.

CSO DE I.\VERSA GE.\'ERALIZADA E:\I ESTnIAç'ÃO RECCRSIVA DE PARÀ:\IETRt DO :\IODELO LI.\EAR.

F.A.:\Ioura - IBGE e A.P. Sant'Anna - I:\I/CTRJ - 1987.

18. ESTCDO DESCRITO DO .\ÚCLEO DE CALCA.\EO.

Gastão Coelho Gomes - I:\I/CFRJ e Cacilda Behmer - HC/CFRJ - 1987.

19. PESQCISA DE Ope'HÃO SOBRE TE:\IAS DE I.\TERESSE DA ASSE:\IBLÉIA .\ ACIO:'\ AL CO:\'STITCI:'\TE.

Annibal P. Sant'Anna - nI/C'FRJ. Gastão Coelho Gomes - I:\I/CFRJ e Joel G. da Silva Júnior - nl/C'FRJ - 1987.

:30. ALERGIA AO LEITE DE VACA - A.\ÁLISE CLI:'\ICA E LABORATORIAL. Gastão Coelho Gomes I:\l/C'FRJ e Rafael Del Castillo Villalba HC /CFRJ -1987.

:n.

CC'RSO DE ESTATÍSTICA PARA CIÉ:'\CIAS DA SAl-DE. Basílio de Bragança Pereira - nI/COPPE/CFRJ -1987.

32. PERFIL DOS ESTABELECI:\IENTOS DE ASSISTÉ:'\CIA :\IÉDICA. Annibal P. Sant'Anna - I:\I/C'FRJ e David Dorigo - I:\I/C'FRJ - 1987.

33. AVALIAÇÃO DA REDE DE ATE:'\DI:\IE:'\TO DO I:'\PS.

David Dorigo - I:\I/C'FRJ e Luiz Otávio Langlóis - 1:\1./C'FRJ - 1987 .

. 14. RECCRSIVE LEAST SQCARES \VITH SI:'\GC'LAR DESIG:'\ :\IATRICES. Annibal P. Sant'Anna - I:\IjCOPPE/CFRJ - 1987.

:3.3. :\IODELO DE RISCO :'\EONATAL PRECOCE.

David Dorigo - nIjCFRJ e E\'elise Poshmann Sih'a - :\Iaternidade Escola/CFRJ - 1988.

36. CTILIZAÇÃO DO STATGRAPHICS.

Gastào Coelho Gomes - IM/UFRJ. Ana Beatriz Soares Monteiro - IM/UFRJ e André Cláudio V.C. de Mendonça - E\1/UFRJ - 1988,

37. SWITCHING :\10DELS IN SURVIVAL ANALYSIS. Dani Gamermam - IM/UFRJ - 1988.

38. SOBRE O CALCl"LO DOS ｂｅｾｅｆￍｃｉｏｓ＠ DA ｐｒｅｖｉｄￊｾｃｉａ＠ OFICIAL: :-'IÉ-TODOLOGIA E PROGRA:\IA E:-'I LI::\Gl"AGE:-'I PASC.\L.

J.J. sa Serra Costa - I:..I/L"FRJ. Heitor C. Borges Riqueira - I:..I/L"FRJ. Sônia :-'Iaria de Resende - I:-'IE/l"ERJ e Joel G. da Silva Júnior - I:..I1L"FRJ - 1988.

:39. :-'IÍXI:\IOS QL"ADRADOS.

Basílio de Bragança Pereira - I:..I e COPPEíL"FRJ - 1988 .

. !D. ｇｅｾｅｒａｌｉｚｅｄ＠ ｉｾｖｅｒｓｅｓ＠ ｉｾ＠ RECl-RSIVE LEAST SQL"ARES ｅｓｔ｛｜ｉａｔｉｏｾ＠ \VITH :-'IL"LTIPLE ａｄｄｉｔｉｏｾａｌ＠ ｏｂｓｅｒｖａｔｉｏｾｓＮ＠

.\nnibal Parracho Sant' Anna - I:.. I /L"FRJ - 1988.

-lI. GENERALIZED ｅｘｐｏｾｅＺ｜ｔｉａｌ＠ GRO\VTH :-'IODELS - A ｂａｙｅｓｉａｾ＠

APPRPACH.

Hélio dos Santos :'Iigon I:..l/l"FRJ/SERPRO e Dani Gamerman I:.. I L"FRJ -1988.

-l2. :-'IÉTODOS AL"TO:-'IATICOS DE PREVISÃO.

Basílio de Bragança Pereira - I:..I/COPPEjL"FRJ - 1988.

-l:3. IXTRODl"ÇÃO A ａｾａｌｉｓｅ＠ DE ｖａｒｉￃｾｃｉａＮ＠

Basílio de Bragança Pereira - I:"l/COPPEjl"FRJ - 1988.

-l4. ｇｅｾｅｒａｌｉｚｅｄ＠ ｉｾｖｅｒｓｅ＠ ｅｓｔｮｉａｔｉｏｾ＠ ｉｾ＠ THE ｌｉｾｅａｒ＠ :-'IODEL \\-ITH

ｓｉｾｇｬＢｌａｒ＠ ｄｅｓｉｇｾ＠ :-'IATRIX.

Annibal P. Sant'Anna - I:..1/COPPE/l"FRJ -1988.

-l.). ｅｘｐｅｒｉＺ｜ｬｅｾｔｏｓ＠ CO:-'IPARATIVOS: ｔ￉ｃｾｉｃａｓ＠ PARA REDl"ZIR ｔｅｾｄｅｎﾭ

CIOSIDADE.

David Dorigo - I:"I/l"FRJ. Santiago S. Ramirez Can'ajal et alii - 1988.

46. EFEITO DE ｄｉｆｅｒｅｾｔｅｓ＠ TIPOS DE ａｌｮｉｅｾｔａ￧ￃｏ＠ ｾｏ＠ ｃｒｅｓｃｮｉｅｾｔｏ＠

DEC A:"IARÕES.

Santiago S. Ramirez Carvajal - nljl"FRJ. José Vieira .\'unes - I:..I1l"FRJ . :"Iaria Alice :-'Iedeiros - I:..l/l"FRJ e :-'Iaria Fernanda P. Pádua - IB/l"FRJ - 1988.

·H. O DEBATE _{ｒｅｃｅｾｔｅ＠} DA _{ｉｾｆｌａￇￃｏ＠} BRASILEIRA: C\IA _{ｃｏＮ｜Ｇｔｒｉｂｬｾｉￇￃｏ＠}

SOB O E:\FOQCE DE :"lODELOS BAYESIA.\'OS DE PREVISÃO.

Annibal P. Sant'Anna - nl/CFRJ. Gutemberg Hespanha Brasil- Pl"C jRJ. :"laurício

Koki Matsutani - DATAPREV-:\IPAS -1989.

48. ａｌｇｏｒｉｔｾｉｏ＠ DE ＺＢｉￍｾｉｍｏｓ＠ QUADRADOS REGCRSIVOS PARA O ｾｉｏｄｅﾭ

LO LINEAR GERAL.

Annibal Parracho Sant'Anna - IM/UFRJ. Victor Pêgo Hottum - I:Yl/UFRJ - 1989.

49. ESTUDO DA HIPERTENSÃO ARTERIAL ｾｏ＠ QUADRO FUNCIONAL DO

HOSPITAL ｕｾｉｖｅｒｓｉｔ￁ｒｉｏ＠ - UFRJ.

Dani Gamerman - IM/CFRJ. Rosângela Aparecida G. :"Iartins - IM/UFRJ - 1989.

4

, '

·")0. VERSÃO DI2'\Ã:\IICA DO :\IODELO DE ÍXDICE DO SHARPE E SELEÇAo

DE CARTEIRAS ÓTI:\IAS.

E. Su:vama - ICExjCF:\IG e Hélio dos Santos :\Iigon - I:\IjlTRJ - 1990 .

. ,)l. ｐｌａｾｅｊａＺ｜ｉｅｘｔｏｓ＠ DE BCSCA DE RESOLCçAo .'3.1 CO:\l BCSC.-\

RESTRI-TA A ｉｾｔｅｒａￇ￠ｅｓ＠ DE I!! e 1!! ORDEXS. Santiago S. Ramirez Can'ajal - I\I/CFRJ - 1990.

·')2. DIFEREXCIAIS DE REXDA EXTRE PROFISSIOXAIS DE ｾｩｖｅｌ＠ SCPERIOR

ｾｏ＠ RIO DE JAXEIRO .

. -\nnibal Parracho Sant' Anna - I:\I/COPPE/CFRJ e .Jacira Guiro Can'alho da Rocha - DEjCFPe - 1990.

·'):3. ESTCDO SOBRE DESEXVOLVI:\IEXTO DE IXTERFACES ｔ￉ｃｾｉｃａｓ＠ DE

CLASSIFICAÇÃO E DE AVALIAÇAo BASEADAS

ｾａ＠

SATISFAÇ.'\O DO CSCÁRIO.

Flávia ;,\/!! P. F. Landim - I:\ljCFRJ e Rita de Cássia O. Este"am - COPPEjCFRJ - 1990 .

. ').±. A STATISTICAL AXALYSIS OF THE ｉｔａｌｉａｾ＠ AIDS DATA.

Dani Gamerman - I:\ljCFRJ - 1990.

n. ｏｾ＠ THE ｊｏｉｾｔ＠ ｅｓｔｉＺ｜ｉａｔｉｏｾ＠ OF ｉｘｆｅｃｔｉｏｾ＠ RATE ａｾｄ＠ IXCCBATION

DISTRIBCTIO:\' OF IXFECTIOCS DISEASE. Dani Gamerman - I:\I/CFRJ - 1990 .

.56. GEOESTATÍSTICA APLICADA.

Luis Paulo Vieira Braga - I:\-l/CFRJ - 1990 .

. ')7. DYN A:\uC HIERARCHIC AL :\10DELS.

Dani G amerman e Hélio dos Santos :\1igon - I:\I/CFRJ - 1991.

·")8. ESCOLHA DE :\10DELOS PARA PREDIÇAo DE AIDS.

Basílio de Bragança Pereira e Hélio dos Santos :\ligon - I:\I/CFRJ - 1991.

")9. PREVISÕES DO _{ｾＨＧＺ｜ｉｅｒｏ＠} DE C ASOS _{ｄｉａｇｾｏｓｔｉｃ＠} ADOS DE .-\IDS - DA-DOS ａｾｾｔａｉｓＮ＠

Hélio dos Santos .\1igon - I:\I/CFRJ - 1991.

60. I:\'TRODCçAo A PROBABILIDADE E ESTATÍSTICA. Hélio dos Santos :\-1igon - I:\I/CFRJ - 1991.

61. CONTROLE DE QUALIDADE.

Paulo Carneiro Bravo I11/eFRJ - 1991.

62. PLANEJAMENTO DE EXPERIMENTOS. David Martins Dorigo - IM/UFRJ - 1991.

63. ANÁLISE DE REGRESSÃO.

64. ｉｾｔｒｏｄｃ￧ａｯ＠ AO STATGRAPHICS.

:\Ianuel :\Iartins Filho - SERPRO/I:\I/CFRJ - 1991.

6.3. ｃｾｉ＠ ｾｉｏｄｅｌｏ＠ I:\TEGRADO PARA :\IELHORA:\IE:\TO DA QCALIDADE DAS C:\IVERSIDADES PCBLIC AS .

. \.nnibal P. Sant'Anna I:\I/COPPE/CFRJ, Dani Gamerman e Hélio :\Iigon -I:\I/CFRJ - 1992.

66. EVALCATI:\G ACADE:\IIC PERFOR:\IA:\CE \\'ITH HIERARCHICAL :\IODELS.

Dani Gamerman - I:\I/LTRJ - 1992.

ti7. :\ICLTIVARIATE TRE:\D :\IODELS. Dani Gamerman - I:\I/CFRJ - 1992.

68. PESQCISA DE OPI:\IÃO JC:\TO AOS Ft-:\CIO:\ÁRIOS DO ｓｃｐｅｒｾｉｅｒﾭ

CADO ZO:.'-JA SCL.

João Ismael D. Pinheiro. Gastão C. Gomes e Hélio dos Santos :\Iigon - I:\I/L"FRJ - 1992.

ô9. PESQl"ISA DE OPI:\IÃO - PROJETO: L'OREAL - LI:\'"HA DE PRODCTOS. Gastão C. Gomes e Flá\'ia Landim - ｉｾｉＯｃｆｒｊ＠ - 1992.

70. ESTnIAç ÃO DO PERCE:\TC AL DE TALO E:\I :\IISTCRA DE ｆｬＧｾｉｏ＠ ATRAVÉS DA A:\ÁLISE ESTATÍSTICA :\ICLTIVARIADA DO I:\FRAVER:\IELHO .

. João Ismael D. Pinheiro e Hedibert Freitas Lopes - I\l/CFRJ - 1992.

71. ELEIÇÕES :\·IL":\ICIPAIS ｾｏ＠ RIO DE JA:\EIRO - 1992. PESQCISA DE OPI:"nÃo Hélio do S. :\Iigon, João Ismael D. Pinheiro. Paulo César R. Carrano e Rosãngela Aparecida :\1. :\'"oé - I:\l/CFRJ - 1992.

72. PESQCISA DE I:\TE:\Ç ÃO DE VOTO PARA PREFEITO DA CIDADE DO RIO DE JA:-..'EIRO - ＺＲｾ＠ TCR:\O - 1992.

Hélio dos S. :\ligon et elii - I:\I/CFRJ - 1992.

6

Ｌｾ＠

r

c

..

")

-73. A:\"ÁLISE ESTATÍSTICA DOS DADOS DO ESTl-DO CLÍ:\"ICO DE EFICÁCIA E TOLERÂ:\"CIA AO _{ａｾａＺ｜Ｂｄｒｏｾ＠} (RC .51908l.

João Ismael D. Pinheiro. Flávia Landim. 2\Iônica 2\Iagnanini. Otá\'io H.S. Figueiredo - nI/CFRJ -1992.

74. PROJETO: DESTILADOS 2\IÉDIOS

, .).

76.

João Ismael D. Pinheiro. Gastão C. Gomes. :\na Beatriz S. 2\Ionteiro. Alexandra :-'I. Schmidt e André Flá\'io A. A. de Azevedo - I2\I/lTRJ - 199.1.

:-'IODELOS BAYESI.-\,\:OS ｃｾｉｖａｒｉＮＭ｜ｄｏｓ＠ APLICADOS A PREVISÃO DE SÉRIES ECO:\"02\IICAS.

Hélio dos S. 2\Iigon - 12\I/CFRJ . Ana Beatriz S. :-'Ionteiro - CFF E' .-\..ia.x R. B.

:-'Ioreira - IPEA-RJ - 1993.

ｃｏｎｆｏｃｾｄｉｘｇ＠ VARIABLES IX ｾｃｔｒｉｔｉｏｾ＠ STCDIES. David 2\Iartins Dorigo - 12\I/CFRJ - 1993.

I I. SELECTI:\G EXPLA,\:ATORY VARlABLES I,\: ｂｉｾａｒｙ＠ DATA: ｐｌｉｃａｔｉｏｾ＠ TO CARDIAC STl-DY

David 2\Iartins Dorigo - I2\I/CFRJ - 1993.

78. :-'IEDIDAS E ERROS.

David 2\lartins Dorigo - nljCFRJ e Jorge Flamarion Vasconcelos - ｉｐｒｪｄｾｅｒ＠ -1993.

79. ｋｒｉｇｅａｇｅｾｉ＠ ｃｾｉｖｅｒＮｓａｌＮ＠

Luis Paulo V. Braga - I\IjCFRJ e ｾｉ｡ｲ｣ｯ＠ Antonio Rosa Ferreira - nljCFRJ e Otávio Henrique dos S. Figueiredo - ｉｾＢＱｪｃｆｒｊ＠ -1993.

80. I:\"TRODCçAo AO _{ｾｉａｔｈｃａｄ＠} 2.0.

:-'Iônica Barros - nIjCFRJ - 1993.

81. 2\IÉTODOS .-\PROXnIADOS E2\I :-'IODELOS BAYESIA:\"OS DE RESPOSTA ALEATÓRIZADA.

Hélio dos S. ｾｬｩｧｯｮ＠ nIjCOPPEjCFRJ e \\ilda 2\I. Tachibana FCT/C:\"ESP -1994.

82. _{ｔｅｾｄｅｾｃｉａｓ＠} ESTOCÁSTICAS DO PRODCTO: EFEITO DE FLCTC.-\ÇÕES ｾａ＠ PRODCTIVIDADE E DA TAXA DE JCROS REAL.

Hedibert Freitas Lopes - CFRJ jCFF. Elcyon Caiado Rocha Lima - IPEA-RJ jCSU. Ajax R. B. ｾｉｯｮｲ･ｩｲ｡＠ - IPEA-RJ e Pedro L. VaUs Pereira l7SP /C:\"IC Ai-.lP -1994.

83. ｾＱＰｄｅｌｏｓ＠ DINÃ:\lICOS: Ci-.lA APLICAÇÃO A _{ｾｉｏｄｅｌａｇｅｍ＠} CHUVA-VAZÃO

Hélio dos S. ｾＭｬｩｧｯｮ＠ - CFRJ e Ana Beatriz S. ｾｉｯｮｴ･ｩｲｯ＠ - CFF - 1994.

84. RACS E O MODELO BOOLEANO EM ａｾ￁ｌｉｓｅ＠ DE ｉｍａｇｅｾｓ＠

Maurício Nardone - PES - COPPE/UFRJ

Luis Paulo Vieira Braga - DME - IM/UFRJ - 1994

85. ａｾａｌｙｓｉｓ＠ AND CLASSIFIC ａｔｉｏｾ＠ OF SOIL PROPERTIES BY

GEOESTA-TISTICAL ａｾｄ＠ FCZZY :\IETHODS Luis Paulo Vieira Braga - O ｾｉｅ＠ - I:\ljCFRJ Suzana Druck Fuks - E:\IBRAPA - RJ - 1995

.86. MONTE CARLO IvIARKOV CHAINS FOR DYNAMIC GENERALIZED LINEAR

MODELS

Dani Gamerman - DME - IM/UFRJ - 1995.

I

I

, I

', ... -.--- - . -. - ---_. - ---

-8

... --" ＧＭｾＭＭ

...

,o

)

! 1- _'('

ｾ＠

1.'.1

ｻｾ＠