Time series of the day

Stat 153 - 11 Sept 2008 D. R. Brillinger

Simple descriptive techniques

Trend

Xt = + t + t

Filtering

yt = r=-qs ar xt-r

Simple moving average

s = q , ar = 1/(2q+1)

Filters may be in series

Differencing

yt = xt - xt-1 = xt

"removes" linear trend

Seasonal variation model

Xt = mt + St + t

St St-s

12 xt = xt - xt-12 , t in months

Stationary case, autocorrelation estimate at lag k, rk

t=1N-k (xt- )(xt+k - )

over

t=1N (xt - )2

autocovariance estimate at lag k, ck

t=1N-k (xt - )(xt+k - ) / N

x

x

x

x

x

Departures from assumptions

Nonstationarity

Trend - OLS

Seasonality - trig functions

Missing values

Outliers