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732A34 Time series analysis Fall semester 2012

732A34 Time series analysis Fall semester 2012. 6 ECTS-credits Course tutor and examiner: Anders Nordgaard Course web: www.ida.liu.se/~732A34 Course literature: Cryer J.D., Chan K-S.: Time Series Analysis – With Applications in R. 2nd ed. ISBN 978-0-387-75958-6.

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732A34 Time series analysis Fall semester 2012

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  1. 732A34 Time series analysis Fall semester 2012 • 6 ECTS-credits • Course tutor and examiner: Anders Nordgaard • Course web: www.ida.liu.se/~732A34 • Course literature: • Cryer J.D., Chan K-S.: Time Series Analysis – With Applications in R. 2nd ed. ISBN 978-0-387-75958-6. • Complementary handouts

  2. Organization of this course: • Weekly “meetings”: Mixture between lectures, (computer) exercises and seminars • A great portion of self-studying • Assignments (every second week) • Individual written exam • Access to a computer is necessary. • Optimal: Bring your own laptop to the meetings

  3. Examination The course is examined by 1.Assignments (3 in total) 2.Final written exam Assignments will be marked Passed or Failed. If Failed, corrections must be done for the mark Pass. Written exam marks are given according to ECTS grades. The final grade will be the same grade as for the written exam.

  4. Communication Contact with course tutor is best through e-mail: Anders.Nordgaard@liu.se. Office in Building B, Entrance 27, 2nd floor, corridor E (the small one close to Building E), room 3E:485. Normal working hours: When teaching E-mail response almost all weekdays and occassionally in weekends All necessary information will be communicated through the course web. Always use the English version. The first page contains the most recent information (messages) Solutions to assignments should be e-mailed. Note! Course tutor is away from Linköping on 3-4 September 12-14 September 25-29 September

  5. Assignments A number of exercises will be given as assignments to be individually carried out. There will not be any supervision for these assignments since they are part of the examination, but they can be carried out in the computer rooms or at home. No other statistical software than R will be needed. The solutions to the assignments should be submitted in forms of written reports. The core text of these reports may contain graphs and tables, but the latter should be constructed from scratch (i.e. no copying and pasting from R or other software). Besides such components the text should be completely your own and easy to read. Direct outputs from the software (except graphs) can only be included in form of attachments. In the marking of these reports, emphasis will be put on the English language. It will not be sufficient to simply give short answers to the detailed questions of the exercises.

  6. Time series • What kind of patterns can visually be detected? • Is the development stable or non-stable?

  7. What kind of patterns can visually be detected? • Is the development stable or non-stable?

  8. Characteristics: • Non-independent observations (correlations structure) • Systematic variation within a year (seasonal effects) • Long-term increasing or decreasing level (trend) • Irregular variation of small magnitude (noise)

  9. Economic indicators: Sales figures, employment statistics, stock market indices, … Meteorological data: precipitation, temperature,… Environmental monitoring: concentrations of nutrients and pollutants in air masses, rivers, marine basins,… Sports statistics? Electromagnetic och thermal fields Where can time series be found? Time series analysis • Estimate/Investigate different parts of a time series in order to • understand the historical pattern • judge upon the current status • make forecasts of the future development • judge upon the quality of data

  10. Methodologies

  11. yt Decomposition yt A time series can be thought of as built-up by a number of components

  12. What kind of components can we think of? Long-term? Short-term? Deterministic? Purely random?

  13. Decomposition – Analyse the observed time series in its different components: • Trend part (TR) • Seasonal part (SN) • Cyclical part (CL) • Irregular part (IR) • Cyclical part: State-of-market in economic time series • In environmental series, usually together with TR • Multiplicative model: • yt=TRt·SNt ·CLt ·IRt • Suitable for economic indicators • Level is present in TRtor in TCt=(TR∙CL)t • SNt , IRt(and CLt) works as indices •  Seasonal variation increases with level of yt

  14. Additive model: • yt=TRt+SNt+CLt +IRt • More suitable for environmental data • Requires constant seasonal variation • SNt , IRt(and CLt) vary around 0 Additive or multiplicative model?

  15. sales figures jan-98-dec-01 observed (blue), deseasonalised (magenta) 50 50 40 40 30 30 20 20 10 10 0 0 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 observed (blue), estimated trend (green) observed TR SN fitted IR 50 50 40 40 30 30 20 20 10 10 0 0 -10 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 Example 1: Sales figures, additive decomposition

  16. A more theoretical description A time-series is a special case of a stochastic process. A stochastic process is a family of random variables coupled with a deterministic index variable t: t can be continuous or discrete. Y can be continuous- or discrete-valued.

  17. Examples • Yt = Number of events (e.g. number of telephone calls arrived) up to time t.  Point process (usually modelled as a Poisson process) • Yt =Number of customers in a queue at time-point t (Birth-and-death process) • Yn = The number of offspring in generation n of a population starting with an initial population Y0.  Markov chain (Yn depends only on Yn – 1 ) • Assume you score +1 if you toss a coin and get “heads” and –1 if you get “tails”. Let Yn = The sum of scores after n tosses.  Random walk • Yt = The temperature outdoors at time point t (infinitesimal resolution)

  18. When t is discrete a stochastic process is called a sequence and constitute a model for an observed time series. (Sometimes the sequence itself is referred to as the time series) Mean (value) function: (Auto)covariance function: (Auto)correlation function:

  19. Example Random walk

  20. Note that t,sand t,s are symmetric functions, i.e.

  21. Stationarity A stochastic process is said to be strictly stationary if the joint probability distribution of is the same as the joint probability distribution of for any set of time points (t1, … , tn )no matter of the value of k

  22. A stochastic process is said to be weakly stationary (or second-order stationary) if stationary non-stationary Roughly: Constant mean and constant variance

  23. White noise A stochastic process that is a sequence of independent and identically distributed (i.i.d.) random variables e1, e2, … is called a white noise process. By definition a white noise process is strictly stationary Independent random variables Of interest in the construction of models for general processes

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