UNR * STAT 758 * Fall 2014

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/

Balance of Trade

William Playfair (1759-1823)

http://www.math.yorku.ca/SCS/Gallery/

William Playfair (1759-1823), A Letter on our Agricultural Disaster, 1821.

http://www.math.yorku.ca/SCS/Gallery/

Prices, wages, and reigns

Jan Ingenhousz (December 8, 1730 – September 7, 1799),a British physiologist, botanist and physicist. First observed BM, 1765.

Robert Brown (December 21, 1773 – June 10, 1858), a British botanist. Observed and studied BM, 1828.

Albert Einstein (March 14, 1879 – April 18, 1955),a German-born theoretical physicist. Mathematical and physical description of BM, 1900.

Louis Jean-Baptiste Alphonse Bachelier (March 11, 1870 - April 28, 1946),a French mathematician. Mathematical description of BM, 1900.

Thorvald Nicolai Thiele (December 24, 1838 – September 26, 1910),a Danish astronomer, actuary, and mathematician. First mathematical description of BM, 1880.

Fat in milk

0 200 400 600 800 1000

-15

-50

510

20

Time

X-c

oord

inat

e

X

Y

X-coordinate of a chosen particle: 1-D Brownian motion

2-D Brownian motion

http://www.math.yorku.ca/SCS/Gallery/

Simple pendulum

Oscillatory motion is widely observed in

Astronomy

Physics

Mechanics

Electricity

Biology

Human behavior

Socio-economics

Climate and geophysics

Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830),

a French mathematician and physicist

Fourier decomposition of a periodic identity function

http://en.wikipedia.org/wiki/Fourier_series

0 10 20 30 40 50 60

-2-1

01

2

Time, t

sin(

t)

Pure oscillations

0 10 20 30 40 50 60

-2-1

01

2

Time, t

sin(

t)+

e(t)

Oscillations with superposed random fluctuations, sd=0.05

0 10 20 30 40 50 60

-2-1

01

2

Time, t

sin(

t)+

e(t)

Oscillations with superposed random fluctuations, sd=0.1

0 10 20 30 40 50 60

-2-1

01

2

Time, t

sin(

t)+

e(t)

Oscillations with superposed random fluctuations, sd=0.5

0 10 20 30 40 50 60

-3-1

01

23

Time, t

sin(

t)+

e(t)

Oscillations with superposed random fluctuations, sd=2

0 10 20 30 40 50 60

-2-1

01

2

Time

Obse

rved P

roce

ss

0 10 20 30 40 50 60

-1.0

0.0

0.5

1.0

Time

Osc

illato

ry P

roce

ss

0 10 20 30 40 50 60

-1.0

0.0

0.5

1.0

Time

Err

ors

Observations

=

+

Up to 1925 or thereabouts, there was a common belief that…

George Udny Yule

(18 Feb 1871, Scotland -- 26 June 1951, England)

Monthly Sunspot Data, 1749–1997

Time

Num

ber

of s

unsp

ots

1750 1800 1850 1900 1950 2000

050

100

150

200

250

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.

Monthly Sunspot Data, 1950-1997

Time

Num

ber

of s

unsp

ots

1950 1960 1970 1980 1990

050

150

250

Monthly Sunspot Data, 1749–1997

Time

Num

ber

of s

unsp

ots

1750 1800 1850 1900 1950 2000

050

100

150

200

250

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

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

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.

http

://vi

sual

iseu

r.bn

f.fr/

Vis

ualis

eur?

Des

tinat

ion=

Gal

lica&

O=

NU

MM

-560

31

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

Norbert Wiener (1894-1964), USAFounder of cybernetics and

information theory

Andrei Kolmogorov (1903-1987), RussiaFounder of modern theory of

probability (1933)

Timet t+

( )X tProcess is predicted by ˆ ( )X t with forward lag

( )X tProcess is predicted by ˆ ( )X t with forward lag 2

2

ˆ( ) ( )

( )

E X t X tS

E X t

minS

General solution: conditional expectation

argmin ( ) | ( ), ,S E X t X s s t

For Markov Gaussian ( )X t conditional expectation

is a linear function of

Andrei Andreyevich Markov (June 14, 1856 – July 20, 1922),

a Russian mathematician.

CandyLand Game: an example of Markov process

Markov process: all information aboutthe history of the process that will affect its

future is included in the process current state.