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UNR * STAT 758 * Fall 2006

Time Series Analysis. historical remarks. UNR * STAT 758 * Fall 2006. S&P500 composite index for Monday, August 28, 2006. UNR * STAT 758 * Fall 2006. First Time Series. UNR * STAT 758 * Fall 2006. William Playfair (1759-1823). UNR * STAT 758 * Fall 2006.

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UNR * STAT 758 * Fall 2006

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  1. Time Series Analysis historical remarks UNR * STAT 758 * Fall 2006

  2. S&P500 composite index for Monday, August 28, 2006 UNR * STAT 758 * Fall 2006

  3. First Time Series UNR * STAT 758 * Fall 2006

  4. William Playfair (1759-1823) UNR * STAT 758 * Fall 2006 http://www.math.yorku.ca/SCS/Gallery/

  5. Prices, wages, and reigns William Playfair (1759-1823), A Letter on our Agricultural Disaster, 1821. UNR * STAT 758 * Fall 2006 http://www.math.yorku.ca/SCS/Gallery/

  6. 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." UNR * STAT 758 * Fall 2006 http://www.math.yorku.ca/SCS/Gallery/

  7. Probability Gerolamo Cardano (September 24, 1501 – September 21, 1576) Author of the first book on probability “De Ludo Aleae” ~ “On the dice game” written in 1560s, published in 1663 UNR * STAT 758 * Fall 2006

  8. Statistical Mechanics Ludwid Eduard Boltzmann Inventor of statistical mechanics (February 20, 1844 – September 5, 1906) Boltzmann’s tomb in Vienna UNR * STAT 758 * Fall 2006

  9. Brownian Motion Simple, First, Very Useful Stochastic Process UNR * STAT 758 * Fall 2006

  10. Brownian motion of plastic spheres (913 nm in diameter) in water http://physics.ius.edu/~kyle/K/Brownian/Brownian.html UNR * STAT 758 * Fall 2006

  11. 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. UNR * STAT 758 * Fall 2006

  12. 2-D Brownian motion Y X X-coordinate of a chosen particle: 1-D Brownian motion UNR * STAT 758 * Fall 2006

  13. Oscillatory Behavior UNR * STAT 758 * Fall 2006

  14. Oscillations Oscillatory motion is widely observed in Astronomy Physics Mechanics Electricity Biology Human behavior Socio-economics Climate and geophysics Simple pendulum UNR * STAT 758 * Fall 2006 http://www.math.yorku.ca/SCS/Gallery/

  15. 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 UNR * STAT 758 * Fall 2006

  16. Pure oscillations UNR * STAT 758 * Fall 2006

  17. Oscillations with superposed random fluctuations, sd=0.05 UNR * STAT 758 * Fall 2006

  18. Oscillations with superposed random fluctuations, sd=0.1 UNR * STAT 758 * Fall 2006

  19. Oscillations with superposed random fluctuations, sd=0.5 UNR * STAT 758 * Fall 2006

  20. Oscillations with superposed random fluctuations, sd=2 UNR * STAT 758 * Fall 2006

  21. Up to 1925 or thereabouts, there was a common belief that… Observations = + UNR * STAT 758 * Fall 2006

  22. Time Series (Proper) Begin... UNR * STAT 758 * Fall 2006

  23. George Udny Yule (18 Feb 1871, Scotland -- 26 June 1951, England) UNR * STAT 758 * Fall 2006

  24. 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. UNR * STAT 758 * Fall 2006

  25. 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 UNR * STAT 758 * Fall 2006

  26. 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. UNR * STAT 758 * Fall 2006

  27. 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 UNR * STAT 758 * Fall 2006

  28. Prediction problem Andrei Kolmogorov (1903-1987), Russia Founder of modern theory of probabilities (1933) Norbert Wiener (1894-1964), USA Founder of cybernetics and information theory UNR * STAT 758 * Fall 2006

  29. Prediction problem Process is predicted by with forward lag t t t+t Time UNR * STAT 758 * Fall 2006

  30. Prediction problem with forward lag t Process is predicted by General solution: conditional expectation For Markov Gaussian conditional expectation is a linear function of UNR * STAT 758 * Fall 2006

  31. 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. UNR * STAT 758 * Fall 2006

  32. Extreme Events UNR * STAT 758 * Fall 2006

  33. 86% of total 97% of total UNR * STAT 758 * Fall 2006 http://neic.usgs.gov/neis/eq_depot

  34. Extreme events Sometimes, we need to predict not smooth dynamics of a process, but some rare extreme events largest EQ which cause most casualties/damage (not seismic rate) largest droughts (not temperature/humidity) El-Nino event (not temperature/pressure) market crashes (not price dynamics ) largest insurance payments that lead to default (not regular payment flow) etc. UNR * STAT 758 * Fall 2006

  35. Predictor X(t) Extreme Event Extreme events Process X(t) Time Very good (in classical sense) predictor is useless in predicting level crossing. Thus, in predicting rare events (extreme events) we need to introduce alternative approach to testing quality. UNR * STAT 758 * Fall 2006

  36. 2: Determine their interrelationships 3: Choose appropriate mathematical apparatus 5: Add new primary factors 4: Determine outcome given primary factors Deterministic Approach (how classical Sciences worked) 1: Single out a set of primary factors UNR * STAT 758 * Fall 2006

  37. Turbulence Da Vinci (1452-1519) UNR * STAT 758 * Fall 2006

  38. Hurricane Floyd, 1999 UNR * STAT 758 * Fall 2006

  39. Turbulence Phenomenon Probabilistic approach Da Vinci (1452-1519) Classical (deterministic) approach George Gabriel Stokes Andrei Kolmogorov (1903-1987): A founder of modern theory of probabilities (1933) Claude Louise Mary Henry Navier (1821) Navier-Stokes EQs UNR * STAT 758 * Fall 2006

  40. Theory of Probabilities & Statistics Probability (a fair coin will show about 50% of tails) Model Observations Statistics (a coin that shows 90 tails out of 100 throws is probably not fair) UNR * STAT 758 * Fall 2006

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