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UNR * STAT 758 * Fall 2014

Time Series Analysis. fragmentary and subjective historical remarks. UNR * STAT 758 * Fall 2014. SP500 dynamics for 2007-2008 (finance.yahoo.com). First Time Series. Balance of Trade.

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UNR * STAT 758 * Fall 2014

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  1. Time Series Analysis fragmentary and subjective historical remarks UNR * STAT 758 * Fall 2014

  2. SP500 dynamics for 2007-2008 (finance.yahoo.com)

  3. First Time Series

  4. Balance of Trade William Playfair (1759-1823), a Scottish engineer and political economist, is generally viewed as the inventor of line plots, bar chart, and pie chart. His The Commercial and Political Atlas, published in 1786, contained a number of interesting time-series charts. Playfair: "On inspecting any one of these Charts attentively, a sufficiently distinct impression will be made, to remain unimpaired for a considerable time, and the idea which does remain will be simple and complete, at once including the duration and the amount." http://www.math.yorku.ca/SCS/Gallery/

  5. William Playfair (1759-1823) http://www.math.yorku.ca/SCS/Gallery/

  6. Prices, wages, and reigns William Playfair (1759-1823), A Letter on our Agricultural Disaster, 1821. http://www.math.yorku.ca/SCS/Gallery/

  7. Brownian Motion First & Very Useful Stochastic Process

  8. Brownian Motion Jan Ingenhousz (December 8, 1730 – September 7, 1799), a British physiologist, botanist and physicist. First observed BM, 1765. Fat in milk Robert Brown (December 21, 1773 – June 10, 1858), a British botanist. Observed and studied BM, 1828. Thorvald Nicolai Thiele (December 24, 1838 – September 26, 1910), a Danish astronomer, actuary, and mathematician. First mathematical description of BM, 1880. Louis Jean-Baptiste Alphonse Bachelier (March 11, 1870 - April 28, 1946), a French mathematician. Mathematical description of BM, 1900. Albert Einstein (March 14, 1879 – April 18, 1955), a German-born theoretical physicist. Mathematical and physical description of BM, 1900.

  9. 2-D Brownian motion Y X X-coordinate of a chosen particle: 1-D Brownian motion

  10. Oscillatory Behavior

  11. Oscillations Oscillatory motion is widely observed in Astronomy Physics Mechanics Electricity Biology Human behavior Socio-economics Climate and geophysics Simple pendulum http://www.math.yorku.ca/SCS/Gallery/

  12. Oscillations Fourier decomposition of a periodic identity function http://en.wikipedia.org/wiki/Fourier_series Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830), a French mathematician and physicist

  13. Pure oscillations

  14. Oscillations with superposed random fluctuations, sd=0.05

  15. Oscillations with superposed random fluctuations, sd=0.1

  16. Oscillations with superposed random fluctuations, sd=0.5

  17. Oscillations with superposed random fluctuations, sd=2

  18. Up to 1925 or thereabouts, there was a common belief that… Observations = +

  19. Time Series (Proper) Begin...

  20. George Udny Yule (18 Feb 1871, Scotland -- 26 June 1951, England)

  21. G. Udny Yule, "On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers", Philosophical Transactions of the Royal Society of London, Ser. A, Vol. 226 (1927), pp. 267–298.

  22. G. Udny Yule, "On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers", Philosophical Transactions of the Royal Society of London, Ser. A, Vol. 226 (1927), pp. 267–298. Observations can not be explained as oscillatory behavior plus noise; they are affected not by superposed fluctuations, but by true disturbances, which are incorporated into the future behavior of the system

  23. http://www.junglephotos.com/galapagos/satimages/othersatimages/elnino.shtmlhttp://www.junglephotos.com/galapagos/satimages/othersatimages/elnino.shtml

  24. http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56031http://visualiseur.bnf.fr/Visualiseur?Destination=Gallica&O=NUMM-56031 G. Udny Yule, "On a Method of Investigating Periodicities in Disturbed Series, with Special Reference to Wolfer's Sunspot Numbers", Philosophical Transactions of the Royal Society of London, Ser. A, Vol. 226 (1927), pp. 267–298.

  25. Objectives of TS Describe behavior of the series (construct a mathematical model) Explain behavior of the system that produced the series (construct a physical model) Forecast (predict) the system behavior using its model Control the system using its forecast behavior

  26. Prediction problem Andrei Kolmogorov (1903-1987), Russia Founder of modern theory of probability (1933) Norbert Wiener (1894-1964), USA Founder of cybernetics and information theory

  27. Prediction problem Process is predicted by with forward lag t t t+t Time

  28. Prediction problem with forward lag t Process is predicted by General solution: conditional expectation For Markov Gaussian conditional expectation is a linear function of

  29. Markov Processes Markov process: all information about the history of the process that will affect its future is included in the process current state. CandyLand Game: an example of Markov process Andrei Andreyevich Markov (June 14, 1856 – July 20, 1922), a Russian mathematician.

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