Transcript Lecture 1

Time series: discrete vs. continuous
Prime rate TS
is discrete
Temperature TS
is continuous
0.5
0.0
Sine TS
is deterministic
-1.0 -0.5
5
10
15
0
-1
iid rvs TS
is stochastic
1
2
Time
-2
0
Measurement
Measurement
1.0
Time series: deterministic vs. stochastic
0
20
40
60
Time
80
100
Time series: deterministic + stochastic !
Time series: deterministic + stochastic !
GPS measurement (location in N. America)
Time series: deterministic + stochastic !
GPS measurement (location in N. America)
250
150
50
Price Index
350
Beveridge Wheat Price Index, 1500-1869
1500
1550
1600
1650
1700
Time, years
1750
1800
1850
300
200
100
1810
1820
1830
1840
1850
Time, years
100
150
200
250
300
350
400
Beveridge Wheat Price Index, 1800-1860
Price Index
Price Index
400
Beveridge Wheat Price Index, 1800-1860
1810
1820
1830
1840
Time, years
1850
1860
1860
250
150
50
1550
1600
1650
1700
1750
1800
100
150
200
250
300
350
Time,
years
Beveridge Wheat
Price
Index, 1500-1869
50
1500
Price Index
Price Index
350
Beveridge Wheat Price Index, 1500-1869
1500
1550
1600
1650
1700
Time, years
1750
1800
1850
1850
Strong linear trend
100
80
60
40
20
0
0.0
0.2
0.6
0.4
Time
0.8
1.0
Weak linear trend
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4
3
X
2
1
0
-1
-2
-3
0.0
0.2
0.4
0.6
Time
0.8
1.0
Non-linear trend
5.0
X
2.5
0.0
-2.5
-5.0
0.0
0.2
0.4
0.6
Time
0.8
1.0
Trend analysis
5
4
3
2
X
1
0
-1
-2
-3
-4
0.0
0.2
0.4
0.6
Time
0.8
1.0
Annual cycle: 1 year
El-Nino cycle: 4-7 years
Business cycle: A cartoon
Business cycle: Real data (6-7 years)
Sun activity cycle: 11 years
100 150 200 250
50
0
Number of sunspots
Monthly Sunspot Data, 1749–1997
1750
1800
1850
1900
Time
1950
2000
Climate cycle: 150 Kyears
Changing variance
3
2
X
1
0
-1
-2
-3
0.0
0.2
0.4
0.6
Time
0.8
1.0
Time-dependent variance
3000
2500
1500
1000
500
0
0.0
0.2
0.4
0.6
0.8
1.0
Time
Log-transformed data
8
6
4
Log(X)
X
2000
2
0
-2
0.0
0.2
0.4
0.6
Time
0.8
1.0
30
10
Signal
50
70
Signal = rand(t)*(1.1 + sin(p*t/3))
0
200
400
600
800
Time
Variation depends on position within the cycle
1000
0.5
0.0
Log of Signal
1.0
Signal = Log(rand(t)) + Log (1.1 + sin(p*t/3))
0
200
400
600
800
Time
Variation is independent of position within the cycle
1000