Stochastic models - time series. Random process. an infinite collection of consistent distributions probabilities exist Random function. family of random variables, {Y(t;), t  Z, } Z = {0,±1,±2,...},  a sample space

Specified if given F(y1,...,yn;t1 ,...,tn ) = Prob{Y(t1)y1,...,Y(tn )yn } n = 1,2,... E.g. t in Z or R F's are symmetric in the sense F(y;t) = F(y;t),  a permutation F's are compatible F(y1 ,...,ym ,,...,;t1,...,tm,tm+1,...,tn} = F(y1,...,ym;t1,...,tm) m+1  n = 2,3,...

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

There may be densities joint density Prob{x < Y(t+u) < x+dx and y < Y(t) < y+ dy} = f(x,y,u) dxdy

Extensions. The Y(t) may be discrete, angles, proportions, vectors, ... Kolmogorov extension theorem. To specify a stochastic process give the distribution of any finite subset {Y(1),...,Y(n)} in a consistent way,  in some A

The algebra. (*) Extend to case of > 2 variables

Normal process/series. Finite dimension distributions multivariate normal Multivatiate normal. Entries linear combinations of i.i.d standard normals Y =  +  Z : s by 1 : s by r Y: s by 1 Z: Nr(0,I) I: r by r identity E(Y) =  var(Y) = ' s by s Y = [Y1τ Y2τ]τ Conditional marginals linear in Y2 when condition on it

Other methods i) Y(t;), : random variable measurability ii) urn model iii) probability on function space iv) analytic formula Y(t) =  cos(t + ) , : fixed : uniform on (-,]

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) Signal plus noise model: Y(t) = S(t) + Z(t) e.g. S(.) fixed/random E{Y(t)} = S(t) fixed case

Y S^ N^

Shumway and Stoffer (2011)

DRB (1975)

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  var { jY(ti)} crosscovariance function c12(s,t) = cov{Y1(s),Y2(t)}

Shumway and Stoffer (2011)

Stationarity. Joint distributions, {Y(t+u1),...,Y(t+uk-1),Y(t)}, do not depend on t for k=2,3,... Often reasonable in practice, particularly for some time stretches Replaces "identically distributed" (i.i.d.)

Cryer and Chan (2008)

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)

DRB

DRB (1975)

acf’s and ccf