This document discusses waves and wavelets and their application in analyzing time series data. It provides examples of how wavelets can be used to: 1) Break down time series data into different time periods and analyze the frequency characteristics within each period. 2) Apply various frequency "scales" repetitively over time periods to analyze data at different scales or resolutions. 3) Decompose a signal into different frequency components and analyze how those components change over time.

1. WAVES AND WAVELETS
Shamelessly stolen from R. Greenhalgh, Electrical
Engineering, University of Western Australia.

2. WINDOWING – SHORT TERM FOURIER TRANSFORM

3. Frequency
Time
Cumulative Probability (x10-2)
0.1 1 10 30 50 70 90 99 99.9
Annual
Flood
0
200
400
600
800
1000
1200
k = -0.4
k = 0
k = +0.4
 
 
 
 
 
F x k x k
F x x k
k
( ) exp / .............
( ) exp exp / ................
   





 
    
1 0
0
1
 
 
No Differentiation of Process by occurrence in time or frequency
Domain. Purely stochastic process.

4. Frequency
Time
Simple analysis of Time Series with break down of time
periods but no break down of frequencies within each group.
MENDOZA
Year
1920 1940 1960 1980
Mean
Annual
Discharge
(m
3
s
-1
)
20
40
60
80
100
120

5. Frequency
Time
10 100 1000
10 100 1000
Relative
Spectral
Density
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Simple analysis of frequencies over spectrum periods but
no break down of time and changing frequency characteristics.
Tallahassee Monthly
Precipitation

6. Frequency
Time
Period (Months)
10
Period (Months)
10
Period (Months)
10
Spectral
Density
1e+5
2e+5
3e+5
Breakdown of spectrum of frequencies over discrete time periods.
Jax 1872-1911 Jax 1912-1951 Jax 1951-1990

7. Jacksonville Monthly Precipitation
Frequency Characteristics in 3 Time Periods
Period (Months)
10 100
10 100
Spectral
Density
1e+5
2e+5
3e+5
6
12
1872-1911
1912-1951
1952-1990
Period (Months)
10
10
Spectral
Density
1e+5
2e+5
3e+5
6
12
1872-1911
1912-1951
1952-1990

8. Scale
Time
Various frequencies or “scales”applied repetitively over appropriate
(to “scale”) time periods.

9. Scale
Time
Basic wavelet form applied over various time scales by “extending” and
“compressing” wavelet over different time periods observations. Compression/ extension
Being analogous to various periods of oscillation.

11. 1872 1882 1892 1902 1911
1872 1882 1892 1902 1911

13. 53 58 63 65 73
77
83 87

14. Wenchi, 1950-2001.
Month
1 2 3 4 5 6 7 8 9 10 11 12
Monthly
Precipitation
(mm)
0
100
200
300
400
500

15. ANNUAL AND SEASONAL
PRECIPITATION TOTALS
Wenchi, 1950-2001.
Year
1950 1960 1970 1980 1990 2000
Precipittaion
Total
(mm)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Annual
AMJ
JA
SO
Regression
Lines

16. PERCENTAGE OF ANNUAL
PRECIPITATION FALLING
IN EACH SEASON.
Wenchi, 1950-2001.
Year
1950 1960 1970 1980 1990 2000
Percentage
of
Annual
Precipitation
0
10
20
30
40
50
60
70
AMJ
JA
SO
Regression Lines

17. WENCHI
Month
1 2 3 4 5 6 7 8 9 10 11 12
Monthly
Precipitation
(mm)
0
100
200
300
400
500
1950-1969
1980-2000

21. STANDARDIZED MONTHLY PRECIPITATION TOTALS
Kew Gardens, 1697-1797.

22. STANDARDIZED MONTHLY PRECIPITATION TOTALS
Kew Gardens, 1797-1897.

23. STANDARDIZED MONTHLY PRECIPITATION TOTALS
Kew Gardens, 1897-1980.