Transcript Intro PPT

UNR * STAT 758 * Fall 2014
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/
William Playfair (1759-1823)
http://www.math.yorku.ca/SCS/Gallery/
Prices, wages, and reigns
William Playfair (1759-1823), A Letter on our Agricultural Disaster, 1821.
http://www.math.yorku.ca/SCS/Gallery/
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.
2-D Brownian motion
Y
X
5 10
0
-5
-15
X-coordinate
20
X-coordinate of a chosen particle: 1-D Brownian motion
0
200
400
600
Time
800
1000
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/
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
0
-1
-2
sin(t)
1
2
Pure oscillations
0
10
20
30
Time, t
40
50
60
1
0
-1
-2
sin(t)+e(t)
2
Oscillations with superposed random fluctuations, sd=0.05
0
10
20
30
Time, t
40
50
60
1
0
-1
-2
sin(t)+e(t)
2
Oscillations with superposed random fluctuations, sd=0.1
0
10
20
30
Time, t
40
50
60
1
0
-1
-2
sin(t)+e(t)
2
Oscillations with superposed random fluctuations, sd=0.5
0
10
20
30
Time, t
40
50
60
-1 0 1 2
-3
sin(t)+e(t)
3
Oscillations with superposed random fluctuations, sd=2
0
10
20
30
Time, t
40
50
60
Up to 1925 or thereabouts, there was a common belief that…
1
0
-1
-2
Observed Process
2
Observations
20
=
30
40
50
60
40
50
60
40
50
60
Time
0.0 0.5 1.0
10
-1.0
Oscillatory Process
0
0
10
20
30
+
0.0 0.5 1.0
-1.0
Errors
Time
0
10
20
30
Time
George Udny Yule
(18 Feb 1871, Scotland -- 26 June 1951, England)
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.
100 150 200 250
50
0
1750
1800
1850
1900
1950
Time
0
50
150
250
Monthly Sunspot Data, 1950-1997
Number of sunspots
Number of sunspots
Monthly Sunspot Data, 1749–1997
1950
1960
1970
1980
Time
1990
2000
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.
100 150 200 250
50
0
Number of sunspots
Monthly Sunspot Data, 1749–1997
1750
1800
1850
1900
1950
2000
Time
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
http://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.
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
Andrei Kolmogorov (1903-1987), Russia
Founder of modern theory of
probability (1933)
Norbert Wiener (1894-1964), USA
Founder of cybernetics and
information theory
Process
X (t )
is predicted by
t
Xˆ t (t )
with forward lag t
t+t
Time
Process
S
X (t )
is predicted by
E X (t )  Xˆ t (t )
E X (t )
2
2
Xˆ t (t ) with forward lag t
S  min
General solution: conditional expectation
arg min S  E  X (t ) | X ( s), s   , t  t  
For Markov Gaussian
is a linear function of
X (t )
conditional expectation
X (t - t ) : X̂t (t) = AX (t - t ) + B
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.