Time series concepts and terminology

• A time series is a sequence of observations x_t
• The times t usually range over a discrete index set, often equally spaced
• Examples:
  • Weather data
  • Stock market data
  • Light curves for stars
  • Sound level samples
  • DNA or protein sequences
• Time series can be modeled as a sequence of random variables X_t
• Examples:
  • Random noise, such as an iid sequence of Bernoulli(p) random variables
  • Random walk, such as partial sums of the above
  • X_t = m_t + Y_t, where m is non-random and Y has zero mean
  • X_t = s_t + Y_t, where s is periodic (seasonal) and Y has zero mean
• Mean: m_X(t) = E[ X_t ]
• Covariance: Cov( X_r, X_s) = E[ (X_r- m_X(r))(X_s- m_X(s)) ]
• Weakly stationary if m_X(t) and Cov( X_t+h, X_t) don't depend on t