Stochastic models - time series.

1. 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}

2. 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)

3. 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

4. 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 (-,]

5. 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

6. 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

7. 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)}

8. 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"

9. 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)

10. joint density Prob{x < Y(t+u) < x+dx and y < Y(t) < y+ dy} = f(x,y|u) dxdy

11. Some useful modelsChatfield notation Purely random / white noise often mean 0 Building block

12. Random walk not stationary

13. (*)

14. Moving average, MA(q) From (*) stationary

15. MA(1) 0=1 1 = -.7

16. Backward shift operator Linear process. Need convergence condition

17. autoregressive process, AR(p) first-order, AR(1) Markov * Linear process For convergence/stationarity

18. 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)

19. In general case, Useful for prediction

20. ARMA(p,q)

21. ARIMA(p,d,q).

22. Some series and acf’s

23. Yule-Walker equations for AR(p). Correlate, with Xt-k, each side of

24. Cumulants. multilinear functional 0 if some subset of variantes independent of rest 0 of order > 2 for normal normal is determined by its moments