stochastic models - time series. random process. an infinite collection of consistent distributions...
DESCRIPTION
Finite dimensional distributions First-order F(y;t) = Prob{Y(t) t} Second-order F(y 1,y 2 ;t 1,t 2 ) = Prob{Y(t 1 ) y 1 and Y(t 2 ) y 2 } and so onTRANSCRIPT
Stochastic models - time series.
Random process.
an infinite collection of consistent distributions
probabilities exist
Random function.
a family of random variables, e.g. {Y(t), t in Z}
Specified if given
F(y1,...,yn;t1 ,...,tn ) = Prob{Y(t1)y1,...,Y(tn )yn }
that are symmetric
F(y;t) = F(y;t), a permutation
compatible
F(y1 ,...,ym ,,...,;t1,...,tm,tm+1,...,tn} = F(y1,...,ym;t1,...,tm)
Finite dimensional distributions
First-order
F(y;t) = Prob{Y(t) t}
Second-order
F(y1,y2;t1,t2) = Prob{Y(t1) y1 and Y(t2) y2}
and so on
Other methods
i) Y(t;), : random variable
ii) urn model
iii) probability on function space
iv) analytic formula
Y(t) = cos(t + )
: fixed : uniform on (-,]
There may be densities
The Y(t) may be discrete, angles, proportions, ...
Kolmogorov extension theorem. To specify a stochastic process give the distribution of any finite subset {Y(1),...,Y(n)} in a consistent way, in A
Moment functions.
Mean function
cY(t) = E{Y(t)} = y dF(y;t)
= y f(y;t) dy if continuous
= yjf(yj; t) if discrete
E{1Y1(t) + 2Y2(t)} =1c1(t) +2c2(t)
vector-valued case
mean level - signal plus noise: S(t) + (t) S(.): fixed
Second-moments.
autocovariance function
cYY(s,t) = cov{Y(s),Y(t)} = E{Y(s)Y(t)} - E{Y(s)}E{Y(t)}
non-negative definite
jkcYY(tj , tk ) 0 scalars
crosscovariance function
c12(s,t) = cov{Y1(s),Y2(t)}
Stationarity.
Joint distributions,
{Y(t+u1),...,Y(t+uk-1),Y(t)},
do not depend on t for k=1,2,...
Often reasonable in practice
- for some time stretches
Replaces "identically distributed"
mean
E{Y(t)} = cY for t in Z
autocovariance function
cov{Y(t+u),Y(t)} = cYY(u) t,u in Z u: lag
= E{Y(t+u)Y(t)} if mean 0
autocorrelation function (u) = corr{Y(t+u),Y(t)}, |(u)| 1
crosscovariance function
cov{X(t+u),Y(t)} = cXY(u)
joint density
Prob{x < Y(t+u) < x+dx and y < Y(t) < y+ dy}
= f(x,y|u) dxdy
Some useful models Chatfield notation
Purely random / white noise
often mean 0
Building block
0 ,0 0 ,1)(
kkk
,...2/,1/ ,0 0 ,),()( 2
kkZZCovk Zktt
Random walk
not stationary
0, 01 XZXX ttt
tXE t )(
2)( Zt tXVar
randompurelyZXXX tttt ,1
t
i it ZX1
(*)
)()( )( YbEXaEbYaXE
)(),(2)( )( 22 YVarbYXabCovXVarabYaXVar
),(),(),(),( ),(
VYbdCovUYbcCovVXadCovUXacCovdVcUbYaXCov
Moving average, MA(q)
qtqttt ZZZX ...110
otherwisek
kkMA
,0 1/ ),1/(
0 ,1)( ).1(2
11
From (*)
0)( ,0)( tt XEZEIf
stationary
)(
,...,1,0 ,
,0)(
0
2
k
qk
qkk
kq
i kiiZ
MA(1)
0=1 1 = -.7
Backward shift operator
Linear process. )(MA
jttj XXB
0i itit ZX
Need convergence condition
tq
q
tt
BBB
ZBBZBX
qMA
...)(
)...( )(
)(
10
10
autoregressive process, AR(p)
first-order, AR(1) Markov
Linear process
For convergence/stationarity
1||
tt
tptptt
ZXB
ZXXX
)(
...11
ttt ZXX 1
... )(
22
1
21
ttt
tttt
ZZZXZZX
*
a.c.f. From (*)
||
22||
)(,...2/,1/,0 ),1/()(
k
Zk
kkk
p.a.c.f.
corr{Y(t),Y(t-m)|Y(t-1),...,Y(t-m+1)} linearly
= 0 for m p when Y is AR(p)
In general case,
Useful for prediction
tptptt ZXXX ...11
tystationarifor 1||in 0(z) of roots need)(
zZXB tt
ARMA(p,q)
)()( tt ZXB
qtqtttptptt ZZZXXX ...... 11011
ARIMA(p,d,q).
0)ARIMA(0,1,
)1( walkRandom
1
tt
ttt
ZXXBXX
q)ARMA(p, stationary a is td X
Some series and acf’s
Yule-Walker equations for AR(p).
Correlate, with Xt-k , each side of
tptptt ZXXX ...11
0 ),(...)1()( 1 kpkkk p
Cumulants.
multilinear functional
0 if some subset of variantes independent of rest
0 of order > 2 for normal
normal is determined by its moments