Signal 1Mscale1(7,‘db1’)
0 200 400 600 800 10001
2
3
4
5
6
7Wavelet Tree
0 2 4 6 8100
150
200
250
300Approximation
0 2 4 6 8-10
0
10
20
30
40Detail
0 500 1000 15005
10
15
20
25
30Segementation
Signal 1 Division Plots
0 50 100 150 20010
20
30
0 100 200 30015
20
25
30
0 50 100 15015.5
16
16.5
17
0 20 40 6010
12
14
16
0 20 40 60 80 1006
8
10
12
0 100 200 300 4005
10
15
20
Signal 2 Mscale1(7,‘db1’)
0 200 400 600 800 10001
2
3
4
5
6
7Wavelet Tree
0 2 4 6 8120
140
160
180
200
220
240Approximation
0 2 4 6 8-60
-40
-20
0
20
40Detail
0 500 1000 15005
10
15
20
25
30Segementation
Signal 2 Division Plots
0 100 200 300 40010
15
20
25
0 20 40 6020
22
24
26
0 100 200 30015
20
25
0 20 40 60 8012
13
14
15
16
0 10 20 30 4010
11
12
13
0 50 100 150 200 25010
12
14
16
Test Pattern
0 5 10 15 20 25 30 35 40-1
-0.5
0
0.5
1
1.5
2
Results (Mscale1(2,‘cs1’)) - Different Templates Discovered?
5 10 15 20 25 30 350
0.5
1
1.5
2Wavelet Tree
0 5 10 15 20-1
-0.5
0
0.5
1Approximation
0 5 10 15 20-1.5
-1
-0.5
0
0.5
1
1.5Detail
0 10 20 30 40-2
-1
0
1
2Segmentation
Other Patterns Mscale1(6,‘cs1’)
0 200 400 6001
2
3
4
5
6Wavelet Tree
0 2 4 6 8 1030
35
40
45
50
55Approximation
0 2 4 6 8 10-30
-20
-10
0
10Detail
0 200 400 600 80030
40
50
60Segmentation
Other patterns contd. Mscale1(9,‘cs1’)
Timing - Two plots of Mscale time with increasing values of scale (m)
1 2 3 4 5 6 70.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5Plot of Time (in seconds) as a function of scale (m)
Scale (m)
Time (s)
1 2 3 4 5 6 7 80
0.2
0.4
0.6
0.8
1
1.2
1.4
Different Wavelets - Mscale2(7,cs2)
0 200 400 600 800 10001
2
3
4
5
6
7Wavelet Tree
0 200 400 600 800 100012
14
16
18
20Approximation
0 200 400 600 800 1000-0.5
0
0.5
1
1.5Detail
0 500 1000 15005
10
15
20
25
30Segmentation
Different Wavelets - Mscale2(7,D-2)
0 200 400 600 800 10001
2
3
4
5
6
7Wavelet Tree
0 200 400 600 800 100010
12
14
16
18
20
22Approximation
0 200 400 600 800 1000-1.5
-1
-0.5
0
0.5
1
1.5Detail
0 500 1000 15005
10
15
20
25
30Segmentation
Different Wavelets - Mscale2(7,D-5)
0 200 400 600 800 10001
2
3
4
5
6
7Wavelet Tree
0 200 400 600 800 100010
15
20
25Approximation
0 200 400 600 800 1000-4
-2
0
2Detail
0 500 1000 15005
10
15
20
25
30Segmentation
Different Wavelets - Mscale2(7,D-8)
0 200 400 600 800 10001
2
3
4
5
6Wavelet Tree
0 200 400 600 800 100010
15
20
25Approximation
0 200 400 600 800 1000-2
-1
0
1
2Detail
0 500 1000 15005
10
15
20
25
30Segmentation
Different Wavelets - Mscale2(7,BO1)
0 200 400 600 800 10000
2
4
6
8Wavelet Tree
0 200 400 600 800 100010
15
20
25Approximation
0 200 400 600 800 1000-0.5
0
0.5
1
1.5
2Detail
0 500 1000 15005
10
15
20
25
30Segmentation
Different Wavelets - Mscale2(7,BO3)
0 200 400 600 800 10000
2
4
6
8Wavelet Tree
0 200 400 600 800 100010
15
20
25Approximation
0 200 400 600 800 1000-1
0
1
2Detail
0 500 1000 15005
10
15
20
25
30Segmentation
Different Wavelets - Mscale2(6,cs2)
0 200 400 600 8001
2
3
4
5
6Wavelet Tree
0 200 400 600 80040
45
50
55Approximation
0 200 400 600 800-6
-4
-2
0
2Detail
0 200 400 600 80030
40
50
60Segmentation
Different Wavelets - Mscale2(6,D-2)
0 200 400 600 8001
2
3
4
5
6Wavelet Tree
0 200 400 600 80040
45
50
55Approximation
0 200 400 600 800-2
-1
0
1
2
3Detail
0 200 400 600 80030
40
50
60Segmentation
Different Wavelets - Mscale2(6,D-5)
0 200 400 600 8001
2
3
4
5
6Wavelet Tree
0 200 400 600 80035
40
45
50
55Approximation
0 200 400 600 800-3
-2
-1
0
1
2Detail
0 200 400 600 80030
40
50
60Segmentation
Different Wavelets - Mscale2(6,D-8)
0 200 400 600 8001
2
3
4
5
6Wavelet Tree
0 200 400 600 80040
45
50
55Approximation
0 200 400 600 800-6
-4
-2
0
2
4Detail
0 200 400 600 80030
40
50
60Segmentation
Different Wavelets - Mscale2(6,BO1)
0 200 400 600 8001
2
3
4
5
6Wavelet Tree
0 200 400 600 80040
45
50
55Approximation
0 200 400 600 800-6
-4
-2
0
2
4Detail
0 200 400 600 80030
40
50
60Segmentation
Different Wavelets - Mscale2(6,BO3)
0 200 400 600 8001
2
3
4
5
6Wavelet Tree
0 200 400 600 80040
45
50
55Approximation
0 200 400 600 800-6
-4
-2
0
2Detail
0 200 400 600 80030
40
50
60Segmentation
Wavelet Comparison
• Performance depended very much on original signal
• For example Debauchies was best for tag1s but not so good for others
• Best overall wavelet for patterns on tag1s, tag3 and tag5 = Cubic Spline 2.
The Primitives
1 2 3
4 5 6
7
Primitives discovered using sum of mean sq error and Mscale2(s2,7,’cs2’)
0 500 10000
2
4
6
8Wavelet Tree
0 500 100012
14
16
18
20Approximation
0 500 1000-0.5
0
0.5
1
1.5Detail
0 500 1000 15005
10
15
20
25
30Segmentation
0 10 20 30 4013
13.5
14
0 50 100 15013
13.5
14
0 100 200 30010
20
30
0 10 20 3020
22
24
0 100 200 300 40010
20
30
0 10 20 30 4010
11
12
0 100 200 30010
15
20
1 3 3 6 44 1
Primitives discovered MScale2(s1,8,’ cs2’)
5 7 6 1 12
0 500 1000 1500 20000
2
4
6
8Wavelet Tree
0 500 1000 1500 200012
14
16
18
20Approximation
0 500 1000 1500 2000-1
-0.5
0
0.5Detail
0 1000 2000 30005
10
15
20
25
30Segmentation
0 50 100 15014.5
15
15.5
0 200 400 60010
20
30
0 50 100 150 20014
16
18
0 50 100 1500
10
20
0 200 400 6000
10
20
0 200 400 60014
15
16
Problems to still address: 1) Improve Tree Path Heuristic
10 20 30 401
1.2
1.4
1.6
1.8
2Wavelet Tree
0 5 10 15 20-1
-0.5
0
0.5Approximation
0 5 10 15 20-0.5
0
0.5
1Detail
0 10 20 30 40-2
-1
0
1
2Segmentation
Tree Heuristic
• Crossover should not be allowed
• Some improvement to take into account the magnitude (as well as position) of extrema on the detail signal.This should help determine the corresponding point on the next level.
Problems to still address:2) Determining further refinement (e.g. segmenting at extrema)
0 20 40 60 80 1007
8
9
10
11
0 200 400 600 800 1000 12005
10
15
20
25
30
0 5 10 15 20 257
8
9
10
11
0 20 40 60 807
7.5
8
8.5
9
9.5
10
Signal 1 Division(599:684)
Further segment refinement
• Should detect if pattern within segment is an extrema or not
• If it is then split the segment again at the extrema
Problems to still be addressed:3) The distortion of the approximation and detail signals at lower levels
related to tree path heuristic
0 200 400 600 800 1000 1200-1
-0.5
0
0.5
1Detail Signal
0 200 400 600 800 1000 120010
15
20
25
30Scaled Signal
Problems to still be addressed:4) Confusion between primitives
• Primitives 1 & 3 & 5 are confused
• Primitives 2 & 4 & 6 are confused
• An association amongst these could be made in determining the complete pattern
Work since Return
• Coded up a representation of a Dynamic Bayesian Network
• Updated the GA to work with a Bayesian Network metric rather than Pearson’s Correlation Coefficient
• Now looking at different discretizations to learn the best structure from the data
Typical model learnt from the data
A
D
C
B
0-1-2-3-16