Stochastic models - time series.

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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
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Specified if given F(y1,...,ynt1 ,...,tn )
ProbY(t1)?y1,...,Y(tn )?yn that are
symmetric F(?y?t) F(yt), ? a
permutation compatible F(y1 ,...,ym
,?,...,?t1,...,tm,tm1,...,tn
F(y1,...,ymt1,...,tm)
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Finite dimensional distributions
First-order F(yt) ProbY(t) ? t
Second-order F(y1,y2t1,t2)
ProbY(t1) ? y1 and Y(t2) ? y2 and so on
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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 (-?,?
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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
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Moment functions.
Mean function cY(t) EY(t) ? y dF(yt)
? y f(yt) 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
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Second-moments.
autocovariance function cYY(s,t)
covY(s),Y(t) EY(s)Y(t) - EY(s)EY(t)
non-negative definite ?? ?j?kcYY(tj ,
tk ) ? 0 scalars ? crosscovariance
function c12(s,t) covY1(s),Y2(t)
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Stationarity.
Joint distributions, Y(tu1),...,Y(tuk-1),Y
(t), do not depend on t for k1,2,... Often
reasonable in practice - for some time
stretches Replaces "identically distributed"
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mean EY(t) cY for t in
Z autocovariance function covY(tu),Y(t)
cYY(u) t,u in Z u lag
EY(tu)Y(t) if mean
0 autocorrelation function ?(u)
corrY(tu),Y(t), ?(u) ? 1 crosscovariance
function covX(tu),Y(t) cXY(u)
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joint density Probx lt Y(tu) lt xdx and y
lt Y(t) lt y dy f(x,yu) dxdy
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Some useful models Chatfield notation
Purely random / white noise often mean 0
Building block
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Random walk
not stationary
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()
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Moving average, MA(q)

From ()
stationary

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MA(1) ?01 ?1 -.7
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Backward shift operator
Linear process.
Need convergence condition
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autoregressive process, AR(p)
first-order, AR(1) Markov

Linear process For convergence/stationarity
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a.c.f. From ()
p.a.c.f. corrY(t),Y(t-m)Y(t-1),...,Y(t-m1)
linearly 0 for m ? p when Y is
AR(p)
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In general case,
Useful for prediction
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ARMA(p,q)
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ARIMA(p,d,q).
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Some series and acfs
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Yule-Walker equations for AR(p). Correlate,
with Xt-k , each side of
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Cumulants. multilinear functional 0
if some subset of variantes independent of rest
0 of order gt 2 for normal normal is
determined by its moments