analisis bayesiano de series temporales
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Analisis Bayesiano de Series Temporales
Analisis Bayesiano de Series Temporales
Raquel Prado
Universidad de California, Santa Cruz
Julio, 2013
Analisis Bayesiano de Series Temporales
Definitions
BASIC DEFINITIONS AND INFERENCE
Analisis Bayesiano de Series Temporales
Definitions
Applications and Objectives
Univariate time series analysisModeling and inference: Describing the latent structure of a single series
time
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Analisis Bayesiano de Series Temporales
Definitions
Applications and Objectives
Multivariate time series analysis
What if we have multiple time series or a time series vector,yt= (y1,t, . . . , yk,t)
,at each time t?For instance, the electroencephalogram (EEG) time series just
displayed is one of several EEGs recorded at different locations
over the scalp of a patient. We are interested in discovering
common features accross the multiple EEG signals.
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Analisis Bayesiano de Series Temporales
Definitions
Applications and Objectives
Univariate and multivariate time series analysisForecasting. Example: Annual per capita gross domestic product (GDP).
1950 1960 1970 1980
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Austria
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USA
Analisis Bayesiano de Series Temporales
Definitions
Applications and Objectives
Online monitoring and control
Monitoring a time series to detect possible changes in real time.
Example: TAR(1)
yt= (1)yt1 +
(1)t , +ytd>0 (M1)
(2)
yt1 +
(2)
t , +ytd 0 (M2),where
(1)t N(0, v1)and (2)t N(0, v2).
Analisis Bayesiano de Series Temporales
Definitions
Applications and Objectives
y1:T and1:T
0 500 1000 1500
4
0
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6
time(a)
0 500 1000 1500
1.0
1.4
1.8
time(b)
Here(1) =0.9, (2) = 0.3,d=1, = 1.5,and v1= v2= 1.Also,t=1 ifyt M1 andt=2 ifyt M2.
Analisis Bayesiano de Series Temporales
Definitions
Applications and Objectives
Goals of time series analysis in the exampleOnline monitoring and control
If transitions between modes occur in response to acontrollers action1:Tis known and so, we can:
infer the parameters of the stochastic models that controlthe settings, i.e., infer(i) andvi,and
learn about transition rule, i.e., infer and d;
If transitions do not occur in response to a controllers
action we need to make inferences on1:Tas well.
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Analisis Bayesiano de Series Temporales
Definitions
Applications and Objectives
Other goals
Clustering
Time series models as components of models with
additional structure: e.g., regression models,
spatio-temporal models, etc.
Tracking and online learning.
Analisis Bayesiano de Series Temporales
Definitions
Stationarity
Stationarity
Definition{yt, t T } iscompletely or strongly stationaryif, for anysequence of timest1, . . . , tnand any lag h,the distribution of(yt1 , . . . , ytn)
is identical to the distribution of (yt1+h, . . . , ytn+h).
Definition{yt, t T } isweakly or second order stationaryif for anysequencet1, . . . , tn,and anyh,the first and second jointmoments of(yt1 , . . . , ytn)
and those of(yt1+h, . . . , ytn+h) exist
and are identical.
Complete stationarity implies second order stationarity but the
converse is not necessarily true.
Analisis Bayesiano de Series Temporales
Definitions
Stationarity
StationaritySecond order stationarity
If {yt} is weakly stationary E(yt) = ,V(yt) =v,andCov(yt, ys) =(s t).
Gaussian time series processes: strong and weak
stationarity are equivalent.
Analisis Bayesiano de Series Temporales
Definitions
Stationarity
Stationarity
time
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Analisis Bayesiano de Series Temporales
Definitions
The ACF
The autocorrelation function (ACF)
DefinitionThe autocovarianceof {yt} is defined as
(s, t) = Cov(yt, ys) =E{(yt t)(ys s)}.
If {yt} is stationary we can write (h) =Cov(yt, yth).Definition
Theautocorrelation function(ACF) is given by
(s, t) = (s, t)(s, s)(t, t)
.
For stationary processes we can write (h) =(h)/(0).
Analisis Bayesiano de Series Temporales
Definitions
The ACF
The sample autocorrelation function
DefinitionThesample autocovarianceis given by
(h) = 1
T
Tht=1
(yt y)(yt+h y),
wherey= Tt=1 yt/T is the sample mean.Definition
Thesample autocorrelationis given by (h) = (h)(0) .
Analisis Bayesiano de Series Temporales
Definitions
The ACF
The ACF: Examples
White Noise. Letyt N(0, v)for allt,withCov(yt, ys) =0 ift=s.Then,(0) =v,(h) =0 for allh
=0, and so,(0) =1
and(h) =0 for allh =0.
AR(1). Letyt=yt1+ t, t N(0, v).Then,
(0) = v
(1 2) ,
(h) = |h|(0).
Analisis Bayesiano de Series Temporales
Definitions
The ACF
ACF of AR(1)
0 10 20 30 40 50
1.0
0.
5
0.0
0
.5
1.
0
h
= 0.9
= 0.9
= 0.3
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Analisis Bayesiano de Series Temporales
Definitions
The ACF
Sample ACF of AR(1)
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
(a)Lag
ACF
= 0.9
0 5 10 15 201.0
0.5
0.0
0.5
1.0
(b)Lag
ACF
= 0.9
0 5 10 15 20
0.0
0.2
0.4
0.6
0.8
1.0
(c)Lag
ACF
= 0.3
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayes theorem: Independent Observations
p(|y1:T) =
likelihood p(y1:T|)
priorp()
p(y1:T) predictive
,
with
p(y1:T) = p(y1:T|)p()d.Alternatively, we can write:
p(|y1:T) p(yT|y1:(T1),) p(|y1:(T1)) p(yT|)
likelihood
p(|y1:(T1)) predictive
.
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayes theorem: Dependence on(t 1)
p(|y1:T) p()prior
p(y1|)T
t=2
p(yt|yt1,) likelihood
.
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayes theorem: Dependence on(t 1)Example
AR(1): yt=yt1+ t, t N(0, v),and so = (, v). Conditional likelihood: p(yt
|yt1,) =N(yt
|yt1, v);
p(y1|) =N(0, v/(1 2));Then,
p(|y1:T) p()
(1 2)(2v)T/2
exp
Q
()
2v
,
with
Q() = y21 (1 2) +T
t=2
(yt yt1)2 Q()
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Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayes theorem: Dependence on(t 1)
Example
AR(1)(cont.): We can also use theconditional likelihoodp(y2:T|, y1)as an approximation to the full likelihood andobtain the posterior
p(|y1:T) p()v(T1)/2expQ()2v
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Estimation
Maximum likelihood estimation (MLE):Find = MLE thatmaximizes p(y1:T|).
Maximum a posteriori estimation (MAP):Find= MAPthat maximizesp(|y1:T).
Least squares estimation (LSE):Write the model as
y= F+ , N(0, vI).
with dim(y) = nand dim() =pso that
p(y|F,, v) = (2v)n/2exp(Q(y,)/2v) ,
and find that minimizesQ(y,).
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayesian Estimation
Consider again the modely = F+ ,with N(0, vI).Theposterior density is given by
p(, v|y) p(, v) p(y|, v) p(, v) (2v)n/2exp(Q(y,)/2v)
where
Q(, y) = (y F)(y F) = ( )(FF)( ) +R,
with = (FF)1Fyand R= (y F)(y F). The MLE of is ; The MLE ofv isR/n,however,s2 =R/(n p)is used
instead.
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayesian Estimation
Reference prior:p(, v) 1/v
p(|y, F)is Student-t withn pdegrees of freedom, modeand density
p(|y, F) |FF|1/2
1 + ( )FF( )/(n p)s2n/2
Whenn p(|y, F) N(|, s2(FF)1). p(v|y) =IG
(np)2 ,
(np)s2
2 .
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Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayesian EstimationConjugate Prior:
p(, v) = p(|v)p(v) = N(|m0, vC0) IG(v|n0/2, d0/2)
p(, v|F, y) v{(p+n+n0)/2+1}
e(m0)C
1
0 (m0)+(yF
)(yF
)+d02v
(y|F, v) N(Fm0, v(FC0F + In)); (|F, v) N(m, vC),with
m = m0+ C0F[FC0F + In]
1(y Fm0)C = C0 C0F[FC0F + In]1FC0,
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayesian Estimation (Conjugate prior)
(v|F, y) IG(n/2, d/2)withn =n+n0 andd =eQ1e +d0,with
e= (y Fm0), and Q= (FC0F + I).
(|y1:n, F) Tn [m, dC/n].
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Estimation
Example
ML, MAP, and LS estimators for the AR(1) model.
yt=yt1+ t, witht
N(0, 1). In this case = .
The conditional MLE is found by maximizing
exp{ Q()/2} (or by minimizingQ()). Therefore,= ML=
nt=2 ytyt1/
nt=2 y
2t1.
MLE of unconditional likelihood is obtained by maximizing
p(y1:n|)or by minimizing
0.5[log(1 2) Q()].
We need methods such as Newton-Raphson or scoring to
findML.
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
AR(1)conditional and unconditional likelihoods; simulateddata with = 0.9;MLEs = 0.9069and = 0.8979.
0.6 0.7 0.8 0.9 1.0
140
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60
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Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
AR(1)conditional and unconditional posteriors with priors
N(0, c),c= 1 and c=0.01
0.6 0.7 0.8 0.9 1.0
140
120
100
80
60
40
0.6 0.7 0.8 0.9 1.0
140
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40
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayesian Estimation (Conjugate Analysis)
Reference analysis in the AR(1) model. For the conditional likelihoodML=
nt=2 yt1yt/
n1t=1 y
2t .
Under the reference priorMAP= ML.
Also,
R=n
t=2
y2t(n
t=2 ytyt1)2
n1t=1 y
2t
,
and sos2
=R/(n 2)estimatesv. Marginal posterior for : Student-t withn 2 degrees of
freedom, centered atMLwith scales2(FF)1.
Marginal posterior forv :Inv 2(v|n 2, s2)or,equivalently,(n 2)s2/2IG(v|(n 2)/2, (n 2)s2/2).
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
AR(1)reference analysis; 500 simulated observations with= 0.9and v=100.
Frequency
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0
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1000
(a)v
Frequency
90 100 120 140
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1400
(b)
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
Bayesian Estimation: Non-Conjugate Analysis
AR(1)with full likelihood: The prior p(, v) 1/vdoes notlead to a closed form posterior distribution when the full
likelihood is used. We obtain
p(, v|y1:n) v(n/2+1)(1 2)1/2expQ()
2v
.
How can we summarize posterior inference in this case?
Via simulation-based methods such as Markov chain
Monte Carlo...
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Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
MCMC: The Metropolis Hastings Algorithm
Creates a sequence of random draws,
(1)
,
(2)
, . . . ,whosedistributions converge to the target distribution, p(|y1:n).1. Draw(0) withp((0)|y1:n)> 0 fromp0().2. Form= 1, 2, . . . ,(until convergence):
(a) Sample J(|(m1))(b) Compute the importance ratio
r= p(|y1:n)/J(|(m1))p((m
1)|y1:n)/J((m1)|).
(c) Set
(m) =
with probability= min(r, 1)
(m1) otherwise.
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
MCMC: AR(1)case
MCMC for AR(1)with full likelihood.
Samplev(m) from(v|, y1:n) IG(n/2, Q()/2)(Gibbsstep, every draw will be accepted)
Sample
N(m1), c .
Analisis Bayesiano de Series Temporales
ML and Bayesian Inference
MCMC: AR(1)example
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0.0
0.2
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0.6
0.8
1.0
iteration
(a)
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0.0
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1
.0
1.5
2.0
iteration
v
(b)
Frequency
0.86 0.90 0.94
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v
Frequency
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Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR(p)Models
An autoregression of order p,or AR(p),has the form
yt=
p
j=1
jytj+ t,
wheretis a sequence of uncorrelated error terms typicallyassumed Gaussian, i.e.,t N(0, v).Under Gaussianity, ify = (yT, yT1, . . . , yp+1)
,we have
p(y
|y1:p) =
T
t=p+1 p(yt|y(tp):(t1)) =T
t=p+1 N(yt|ft, v) = N(y
|F, vIn)
with = (1, . . . , p),ft= (yt1, . . . , ytp)
,F = [fT, . . . , fp+1].
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Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Causality and Stationarity
DefinitionAn AR(p)processyt iscausalif it can be written as
yt= (B)t=
j=0
jtj,
withBthe backshift operator Bt=t1, 0= 1 and
j=0 |j| < .DefinitionTheAR characteristic polynomialis defined as:
(u) =1 p
j=1
juj.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Causality and Stationarity
ytis causal only when(u)has all its roots outside the unitcircle (or the reciprocal roots inside the unit circle). In other
words,yt is causal if(u) =0 only when |u| >1. Causality
Stationarity.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: State-space representation
yt AR(p)can be written as
yt = Fxt
xt = Gxt1+ t,
withxt= (yt, yt1, . . . , ytp+1), t= (t, 0, . . . , 0)
,F= (1, 0, . . . , 0) and
G=
1 2 3 p1 p1 0 0 0 00 1 0 0 0... . . . 0
...
0 0 1 0
.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: State-space representation
The eigenvalues of the matrix G, 1, . . . , pare thereciprocal roots of the AR characteristic polynomial theAR characteristic polynomial can written as:
pj=1
(1 ju).
The expected behavior of the process in the future is given
by
ft(h) =E(yt+h|y1:t) = FGhxt=p
j=1ct,j
hj,
withct,j=djet,j,and dj,et,jelements ofd = EF,and
et=E1xt,whereE is an eigenmatrix of G.
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Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Forecast function
Ifyt is such that |j|
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Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: PACF
Let(h, h)be thepartial autocorrelation coefficient at lag h,given by
(h, h) =
(y1, y0) =(1) h= 1
(yh yh1h , y0 yh10 ) h> 1,
withyh1h the minimum mean square linear predictor ofyhgiven
yh1, . . . , y1,and yh10 the minimum mean square linearpredictor ofy0 giveny1, . . . , yh1.
Result:Ifyt AR(p), (h, h) =0 for allh> p.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Computing the PACF
n
n=
n,with
nann
nmatrix with elements
{(hj)}nj=1, n= ((1), . . . , (n)),andn= ((n, 1), . . . , (n, n))
.
Durbin-Levinson recursion. Forn= 0 (0, 0) = 0 and forn 1
(n, n) =(n) n1h=1 (n 1, h)(n h)
1
n1h=1 (n
1, h)(h)
,
with
(n, h) =(n 1, h) (n, n)(n 1, n h),
forn 2 andh= 1 : (n 1).Sample PACF can also be computed using these algorithms.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Yule-Walker Estimation
p= p, v= (0) p1p p.
It can be shown that
T( ) N(0, v1p ),
and thatv is close tov whenT is large.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: MLE and Bayesian estimation
MLE.Find that maximizes
p(y|, v, y1:p) =T
t=p+1p(yt|, v, y(tp):(t1))
=T
t=p+1
N(yt|ft, v) = N(y|F, vIn).
Bayesian. Combinep(y|, v, y1:p)with priorp(, v). Reference priorp(, v) 1/v.
Conjugate priorp(|v) =N(|m0, vC0)andp(v) =IG(n0/2, d0/2). Non-conjugate.
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Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: EEG data analysis
time
voltage(mcv)
0 100 200 300 400
-300
-200
-100
0
100
200
Posterior mean from AR(8) reference
analysis (n= 392):
= (0.27, 0.07, 0.13, 0.15,0.11, 0.15, 0.23, 0.14)
ands= 61.52.These estimates leadto the following estimates of thereciprocal characteristic roots:
(0.97, 12.73); (0.81, 5.10);(0.72, 2.99); (0.66, 2.23).
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: EEG data analysisForecast function
time
voltage(mcv)
0 100 200 300 400 500 600
-300
-200
-100
0
100
200
Future sample
time
voltage(mcv)
0 100 200 300 400 500 600
-300
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0
100
200
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Model Order Assessment
Choose a valuep and for allp p compute Akaikes Information Criterion (AIC):
2p+nlog(s2
p).
Bayesian Information Criterion (BIC):
log(n)p+nlog(s2p).
Marginal:
p(y(p
+1):T|y1:p , p) = p(y(p+1):T|p, v, y1:p)p(p, v)dpdv.Heren= T p.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
Order Assessment in EEG Example: Takep =25 andn= 400 p.
mm
m
m
m
m
m m m m mm
mm
m
m mm
mm m
mm
mm
a a
a
a
a
a
aa a
a a a a a a a a a a a a a a
a a
b
b
b
b
b
b
b bb b
bb
b b
bb
b
bb
bb
bb
bb
AR order p
log-
likelihood
5 10 15 20 25
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Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Initial ObservationsFull likelihood:
p(y1:T|, v) = p(y(p+1):T|, v, y1:p)p(y1:p|, v)= p(y|, v, xp)p(xp|, v).
What aboutp(xp|, v)? N(xp|0, A)withA known. N(xp|0, vA())withA()depending on through the
autocorrelation function and
p(y1:T|, v) vT/2|A()|1/2exp(Q(y1:T,)/2v),where
Q(y1:T,) =T
t=p+1
(yt ft)2 + xpA()1xp.
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Initial ObservationsIt can be shown (e.g., see Box, Jenkins, and Reinsel, 2008) that
Q(y1:T,) =a 2b + C,witha, b,and C obtained from
D=
a bb C
,
andD a (p+1)
(p+1)withDij= T+1jir=0 yi+ryj+r. If |A()|1/2 is ignored when computing p(y1:T|, v),the
likelihood function is that of a standard linear model form
and so, ifp(, v) 1/vwe have a normal/inverse-gammaposterior with
=C1b.
Jeffreys prior is approximatelyp(, v) |A()|1/2v1/2.Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Structured non-conjugate priors
Ifyt AR(p),ytis causal and stationary if all the AR reciprocalroots have moduli less than one.
Huerta and West (1999) proposed priors on the reciprocalcharacteristic roots as follows.
LetCbe the maximum number of pairs of complex roots
andRthe maximum number of real roots with p= 2C+ R.
Denote the complex roots as (rj, j),forj=1 : Cand thereal roots asrj,forj= (C+1) : (R+C).
Then...
Analisis Bayesiano de Series Temporales
Time Domain Models
Autoregressions
AR Models: Structured non-conjugate priors Prior on the real reciprocal roots.
rj r,1I(1)(rj) + c,0I0(rj) + r,1I1(rj) ++(1
r,0
r,1
r,1)gr(rj),
withgr()a continuous distribution on(1, 1), e.g.,gr() =U(| 1, 1).
Prior on the complex reciprocal roots.
rj c,0I0(rj) + c,1I1(rj) + (1 c,1 c,0)gc(rj),j h(j),
withgc(rj)and h(j)continuous distributions on 0< rj
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Analisis Bayesiano de Series Temporales
Time Domain Models
ARMA Models
ARMA Models
ytfollows an autoregressive moving average model,ARMA(p, q),if
yt =
pi=1
iyti+
qj=1
jtj+ t,
We can also write
(1 1B . . . pBp) (B)
yt = (1 + 1B+ . . . + qBq) (B)
t
We typically assumet N(0, v).If q= 0 yt AR(p)and ifp= 0 yt MA(q).
Analisis Bayesiano de Series Temporales
Time Domain Models
ARMA Models
ARMA Models
DefinitionA MA(q)process is identifiable or invertibleif the roots of theMA characteristic polynomial(u)lie outside the unit circle. Inthis case is possible to write the process as an infinite order AR.
Example
Letyt MA(1)with MA coefficient . The process is stationaryfor all and
(h) =
1 h= 0
(1+2) h= 1
0 otherwise.
Note that a MA process with coefficient 1/has the same ACF the identifiability condition is 1/ >1.
Analisis Bayesiano de Series Temporales
Time Domain Models
ARMA Models
An ARMA(p, q)process iscausalif the roots of (u)lie outsidethe unit circle. In this case:
yt= 1(B)(B)t= (B)t=
j=0
jtj,
with(B)(B) = (B).The js can be found by solving thehomogeneous difference equations
jp
h=1
hjh= 0, j max(p, q+1),
with initial conditions
jj
h=1
hjh=j, 0 j
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Analisis Bayesiano de Series Temporales
Time Domain Models
ARMA Models
MA coefficient index
coefficient
0 2 4 6 8
-0.5
0.0
0.5
1.0
1.5
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Note:the optimal ARMA(p, q)model for these data, among allthe models withp, q 8,is an ARMA(2, 2).The MLEs for theMA coefficients are 1= 1.37 and 2= 0.51.
Analisis Bayesiano de Series Temporales
Time Domain Models
ARMA Models
ARMA Models: Inference
MLE and least squares estimation. See, e.g., Shumway
and Stoffer, 2006.
Inference via state-space representation. E.g., Kohn and
Ansley (1985), Harvey (1981, 1991).
Bayesian estimation: Monahan (1983); Marriott & Smith
(1992); Chib and Greenberg (1994); Box, Jenkins, and
Reinsel (2008); Zellner (1996); Marriott, Ravishanker,Gelfand, and Pai (1996); Barnett, Kohn and Sheather
(1997) among others...
Analisis Bayesiano de Series Temporales
Time Domain Models
ARMA Models
Other Related Models
ARIMAyt ARIMA(p, d, q)
(1 1B . . . pBp
)(1 B)d
yt= (1 + 1B+ . . . + qBq
)t.
SARMA
(1 1Bs . . . pBps)yt= (1 + 1Bs + . . . + qBqs)t. Multiplicative Seasonal ARMA
p(B)P
(Bs)(1
B)dyt= q(B)Q
(Bs)t.
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