1. Geology 5600/6600 Signal Analysis 11 Sep 2015 © A.R. Lowry 2015 Last time: The Central Limit theorem : The sum of a sequence of random variables tends to a normal distribution as N , regardless of the pdf’s of individual rv’s. A stochastic process is a family of functions of random variables (i.e., having multiple realizations ). An ensemble is a collective group of functions & measurements (e.g. ); the ensemble average is the average over n. If average over n (large) equals average over t, the process is ergodic. Time series analysis extends multivariate statistics to ensembles that are sampled in time, and has its own special terminology (e.g., Autocorrelation ) reflecting structure imposed by time-dependence…

2. E.g., the mean and standard deviation of Arctic sea ice is much different for September 11 than for an unspecified day… And the mean and standard deviation for possible extents is much smaller than that for a specific date if we know what the value was on the day previous! But that won’t be true of all time series…

3. Stationary Processes are processes for which the statistical properties are time-invariant. Strict-sense definition: ALL statistical properties are time-invariant, i.e., for all and N, pdf’s: (However this has limited practical utility…) Wide-sense definition: Only the first- and second-order statistical properties need be time-invariant: i.e., Constant Mean and autocorrelation depends only on the lag  = t 1 – t 2 : In this class we will generally assume wide-sense stationary processes…

4. Properties of Autocorrelation: (1) (2) R(  ) is maximum at the origin: (3) R(  ) is an even function: R(–  ) = R(  ) R()R() 

5. Autocorrelation is even (proof): Substitute = t –  (thus, t = +  ): Then Autocorrelation is maximum at the origin (proof): Anticipate that: Then By stationarity: So: And:

6. Exercise: find R(  ) for

7. General white noise has (constant) zero-mean, time-variable variance, and the property: or equivalently, Here  denotes the Dirac’s delta function :

8. Wide-sense stationary white noise has both mean and variance that are time-invariant: (Note that white-noise by definition also has zero-mean!) A Quasi-Stationary Process has mean and autocorrelation that are slowly-varying functions of time: R 

9. A random process is Nearly Wide-Sense Stationary if the mean varies as a function of time,, but the autocovariance is time-invariant: (Note that we’ve introduced a new notation here: C xx (  ) is the autocovariance of the random process.) In this case we can form a new random variable,, with. Since the mean is constant for z, it's a stationary process with R zz (  ) = C zz (  ). ~

10. The Cross-Correlation for two wide-sense stationary processes is given by: Properties: (1) Even: R xy (  ) = R yx (–  ) (Show as exercise!) (2) Bounded: R xy 2 (  ) ≤ R xx (0)R yy (0) (*Note that cross-correlation is among the most widely-used tools in seismology! Including seismic interferometry [e.g., “ambient noise” methods], NMO correction, automated picking of travel times, & much more)

11. Example: What is the autocorrelation of a linear combination of two random variables? Consider e.g.: The autocorrelation: (Exercise…) If the random variables are uncorrelated and zero-mean (i.e., R xy (  ) = 0 !), then These relations hold for discrete as well as continuous random variables…