UNR * STAT 758 * Fall 2006

S&P500 composite index for Monday, August 28, 2006

UNR * STAT 758 * Fall 2006

William Playfair ( )

UNR * STAT 758 * Fall 2006 William Playfair ( ), A Letter on our Agricultural Disaster, Prices, wages, and reigns

UNR * STAT 758 * Fall 2006 William Playfair ( ), 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." Balance of Trade

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

UNR * STAT 758 * Fall 2006

Brownian motion of plastic spheres (913 nm in diameter) in water

UNR * STAT 758 * Fall 2006 Jan Ingenhousz (December 8, 1730 – September 7, 1799), a British physiologist, botanist and physicist. First observed BM, Robert Brown (December 21, 1773 – June 10, 1858), a British botanist. Observed and studied BM, Albert Einstein (March 14, 1879 – April 18, 1955), a German-born theoretical physicist. Mathematical and physical description of BM, Louis Jean-Baptiste Alphonse Bachelier (March 11, April 28, 1946), a French mathematician. Mathematical description of BM, Thorvald Nicolai Thiele (December 24, 1838 – September 26, 1910), a Danish astronomer, actuary, and mathematician. First mathematical description of BM, Fat in milk

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

UNR * STAT 758 * Fall 2006

Simple pendulum Oscillatory motion is widely observed in Astronomy Physics Mechanics Electricity Biology Human behavior Socio-economics Climate and geophysics

UNR * STAT 758 * Fall 2006 Jean Baptiste Joseph Fourier (March 21, May 16, 1830), a French mathematician and physicist Fourier decomposition of a periodic identity function

UNR * STAT 758 * Fall 2006 Pure oscillations

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

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

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

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

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

UNR * STAT 758 * Fall 2006

George Udny Yule (18 Feb 1871, Scotland June 1951, England)

UNR * STAT 758 * Fall 2006 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 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 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 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 Norbert Wiener ( ), USA Founder of cybernetics and information theory Andrei Kolmogorov ( ), Russia Founder of modern theory of probabilities (1933)

UNR * STAT 758 * Fall 2006 Timet t+  Processis predicted by with forward lag 

UNR * STAT 758 * Fall 2006 Processis predicted by with forward lag  General solution: conditional expectation For Markov Gaussianconditional expectation is a linear function of

UNR * STAT 758 * Fall 2006 Andrei Andreyevich Markov (June 14, 1856 – July 20, 1922), a Russian mathematician. CandyLand Game: an example of Markov process Markov process: all information about the history of the process that will affect its future is included in the process current state.

UNR * STAT 758 * Fall 2006

97% of total 86% of total

UNR * STAT 758 * Fall 2006 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 Time Extreme Event 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. Process X(t) Predictor X(t)

UNR * STAT 758 * Fall : Single out a set of primary factors 2: Determine their interrelationships 3: Choose appropriate mathematical apparatus 4: Determine outcome given primary factors 5: Add new primary factors

UNR * STAT 758 * Fall 2006 Da Vinci ( )

UNR * STAT 758 * Fall 2006 Hurricane Floyd, 1999

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

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