Tag moving average

[Time Series] Correlation and Stationarity

Archit Vora Leave a comment

Co-variance vs Correlation

  • Correlation is co-variance divided by standard deviation of both variables
  • Hence it is independent of units and is always between -1 and 1, which makes comparison easier
  • Formula on the right is time series specific
    • It is auto correlation coefficient at lag k
    • It is define as ration of auto-correlation at lag k divide by auto-correlation at lag 0
    • This values are plotted on correlogram  (See one for MA(2) process below)

acf

 

Stationary Time Series

  • No systematic change in mean (No trend)
  • No systematic change in Variance
  • No periodic variation (Seasonality)

If time series is not stationary we apply several transformation to make it stationary.

For example applying difference operator to random walk makes it stationary.

 

 

Random Walk

  • Previous value of noise
  • If first value is zero then current value is summation of all the noises so far
  • X(t) = X(t-1) + Z(t)
  • Z(t) = Normal (mu, sigma2)
  • if X(0) = 0 then X(t) = sum(Z(k)) k form 0 to t
  • Expectation[X(t)] = t*mu   – –  Changes with time
  • Variance[X(t)] = t*sigma2   – – Changes with time
  • Not a stationary process
  • let Y(t) = X(t) – X(t-1) = Z(t)  – – Y(t) is a stationary process

 

Example of Stationary Process

Moving average and Auto regressive processes described here can be stationary under conditions described here.

 

 

References

 

Further reading