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Stochastic models - time series. PowerPoint PPT Presentation


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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}. Specified if given

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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,...,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)


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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


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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) = 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


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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

jkcYY(tj , tk )  0 scalars 

crosscovariance function

c12(s,t) = cov{Y1(s),Y2(t)}


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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"


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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)


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joint density

Prob{x < Y(t+u) < x+dx and y < Y(t) < y+ dy}

= f(x,y|u) dxdy


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Some useful modelsChatfield 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)

0=1 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.

corr{Y(t),Y(t-m)|Y(t-1),...,Y(t-m+1)} 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 acf’s


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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 > 2 for normal

normal is determined by its moments


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