Math 2.0

Statistics Definition

Time series

Builds on Stochastic processSequence

Definition

Let TT be a set of time points, usually T=ℤT = \Z, T={1,…,n}T = \set{1, \dots, n} or an interval of ℝ\R. A time series is a family of observations (xt)t∈T(x_t)_{t \in T} indexed by time, together with the order of TT. As a statistical model, a time series is a realization of a stochastic process (Xt)t∈T(X_t)_{t \in T}. The same name is often used for the process itself. The series is in discrete time when TT is a subset of ℤ\Z, and in continuous time when TT is an interval of ℝ\R.

Remarks

Example: the yearly maximum temperatures x1990,…,x2020x_{1990}, \dots, x_{2020} at one weather station form a discrete-time series with T={1990,…,2020}T = \set{1990, \dots, 2020}. Non-example: the heights of 3030 pupils, listed in alphabetical order of their names, are not a time series. The index is not time, and its order carries no meaning. A discrete-time series is a finite or infinite sequence of real numbers or vectors; when its values are vectors it is called multivariate. Many methods assume that the underlying process is stationary, possibly after removing a trend or a seasonal component.

Reading path

To understand this page from the ground up, read these 41 pages in order. The foundations come first.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Ordered field definition
  5. Proposition concept
  6. Random experiment concept
  7. Rational numbers object
  8. Real numbers object
  9. Sequence concept
  10. Set definition
  11. Ring of integers statement
  12. Disjunction definition
  13. Implication definition
  14. Predicate definition
  15. Ordered pair definition
  16. Union definition
  17. Sample space definition
  18. The rationals form a field statement
  19. Existential quantifier definition
  20. Universal quantifier definition
  21. Cartesian product definition
  22. Subset definition
  23. The rationals form an ordered field statement
  24. Uniqueness quantifier definition
  25. Binary relation definition
  26. Universal set definition
  27. Order on the reals definition
  28. Power set definition
  29. Complement definition
  30. Function definition
  31. Interval definition
  32. Preimage of a set definition
  33. Sigma-algebra definition
  34. Event definition
  35. Borel sigma-algebra definition
  36. Probability axioms concept
  37. Probability measure definition
  38. Probability space definition
  39. Random variable definition
  40. Stochastic process concept
  41. Time series definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Time_series
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TagDBDVPNMD46
URIm2://DBDVPNMD46
Kinddefinition
WritingW0 stub
MSC62M10, 91B84
DefinesTime series
UsesStochastic process, Sequence

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