### 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