Notation and naming

Time, time interval, time scale

A time is noted with a t, a time interval with dt,δt, Δt and as τ in process parameters. The distinction should be clear between a time (or an epoch), like in ”today at 1 o’clock”, and a time interval, like in ”1 hour ago”. A time interval is a fairly abstract concept, measured essentially by the beats of a pendulum. The ratio of 2 time intervals gives a real number. For example, for a time interval dt of 1 week, the ratio dt ∕ 1h = 168 (a real number), where 1h is a time interval of 1 hour. Such ratios appear in exponential moving average (EMA), or in process definitions, in the form dt ∕ τ, with τ a characteristic time interval of the process. In finance, it is usual to ’annualize’ volatility. This is done by multiplying with the factor ∘ ------
  1y∕dt, where 1y denotes a 1 year time interval.

Time and time intervals are measured on a given time scale. Usually, this is the physical time scale, as assumed in the discussion in the previous paragraph. In finance, other time scales can be more appropriate, like a business time scale, or a transaction time scale. The deseasonalization of the empirical data is done using a continuous version of the business time scale (which contains 5 days per week of daily data). We use the words ’time scale’ for a scale to measure times and time intervals, and the words ’time interval’ or ’time horizon’ for time differences.

Notation for time series

Historical, realized and centered