1. Time Series Indexing II

2. Time Series Data
A time series is a collection of observations made sequentially in time. .. 50
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value
axis
time axis

3. TS Databases
A Time Series Database stores a large number of time series
Similarity queries
Exact match or sub-sequence match
Range or nearest neighbor
But first we should define the similarity model
E.g. D(X,Y) for X = x1, x2, …, xn , and Y = y1, y2, …, yn

4. Similarity Models Euclidean and Lp based Edit Distance and LCS based
Probabilistic (using Markov Models)
Landmarks
The appropriate similarity model depends on the application

5. Euclidean model Query Q Database Distance 0.98 0.07 0.21 0.43 Rank 4 1
n datapoints
Database
n datapoints
Distance
0.98
0.07
0.21
0.43
Rank 4 1 2 3 S Q
Euclidean Distance between
two time series Q = {q1, q2, …, qn}
and S = {s1, s2, …, sn}

6. Similarity Retrieval Range Query Nearest Neighbor query
Find all time series S where
Nearest Neighbor query
Find all the k most similar time series to Q
A method to answer the above queries: Linear scan … very slow
A better approach GEMINI

7. GEMINI Solution: Quick-and-dirty' filter:
extract m features (numbers, eg., avg., etc.)
map into a point in m-d feature space
organize points with off-the-shelf spatial access method (‘SAM’)
discard false alarms

8. GEMINI Range Queries
Build an index for the database in a feature space using an R-tree
Algorithm RangeQuery(Q, e)
Project the query Q into a point in the feature space
Find all candidate objects in the index within e
Retrieve from disk the actual sequences
Compute the actual distances and discard false alarms

9. GEMINI NN Query Algorithm K_NNQuery(Q, K)
Project the query Q in the same feature space
Find the candidate K nearest neighbors in the index
Retrieve from disk the actual sequences pointed to by the candidates
Compute the actual distances and record the maximum
Issue a RangeQuery(Q, emax)
Compute the actual distances, keep K

10. GEMINI GEMINI works when:
Dfeature(F(x), F(y)) <= D(x, y)
Note that the closer the feature distance to the actual one the better.

11. Problem How to extract the features? How to define the feature space?
Fourier transform
Wavelets transform
Averages of segments (Histograms or APCA)

12. Fourier transform DFT (Discrete Fourier Transform)
Transform the data from the time domain to the frequency domain
highlights the periodicities
SO?

13. DFT A: several real sequences are periodic Q: Such as? A:
sales patterns follow seasons;
economy follows 50-year cycle
temperature follows daily and yearly cycles
Many real signals follow (multiple) cycles

14. How does it work? value x ={x0, x1, ... xn-1} time n-1 1
Decomposes signal to a sum of sine (and cosine) waves. Q:How to assess ‘similarity’ of x with a wave?
value
x ={x0, x1, ... xn-1}
time 1
n-1

15. How does it work? value value freq. f=1 (sin(t * 2 p/n) ) freq. f=0
A: consider the waves with frequency 0, 1, ...; use the inner-product (~cosine similarity) 1
n-1
time
value
freq. f=1 (sin(t * 2 p/n) ) 1
n-1
time
value
freq. f=0

16. How does it work? value freq. f=2 time 1 n-1
A: consider the waves with frequency 0, 1, ...; use the inner-product (~cosine similarity) 1
n-1
time
value
freq. f=2

17. How does it work? 1 n-1 cosine, f=1 sine, freq =1 1 n-1 1 n-11
n-1
‘basis’ functions 1
n-1
cosine, f=1
sine, freq =1 1
n-1
cosine, f=2
sine, freq = 2 1
n-1 1
n-1

18. How does it work?
Basis functions are actually n-dim vectors, orthogonal to each other
‘similarity’ of x with each of them: inner product
DFT: ~ all the similarities of x with the basis functions

19. How does it work?
Since ejf = cos(f) + j sin(f) (j=sqrt(-1)), we finally have:

20. DFT: definition
Discrete Fourier Transform (n-point):
inverse DFT

21. DFT: definition
Good news: Available in all symbolic math packages, eg., in ‘mathematica’
x = [1,2,1,2];
X = Fourier[x];
Plot[ Abs[X] ];

22. DFT: properties
Observation - SYMMETRY property: Xf = (Xn-f )* ( “*”: complex conjugate: (a + b j)* = a - b j ) Thus we use only the first half numbers

23. DFT: AMPLITUDE SPECTRUM
Intuition: strength of frequency ‘f’
count Af
freq: 12
time
freq. f

24. DFT: Amplitude spectrum
excellent approximation, with only 2 frequencies!
so what?

25. DFT: Amplitude spectrum
excellent approximation, with only 2 frequencies!
so what?
A1: compression
A2: pattern discovery
A3: forecasting

26. DFT: Parseval’s theorem
sum( xt 2 ) = sum ( | X f | 2 ) Ie., DFT preserves the ‘energy’ or, alternatively: it does an axis rotation: x1
x = {x0, x1} x0

27. Lower Bounding lemma
Using Parseval’s theorem we can prove the lower bounding property!
So, apply DFT to each time series, keep first coefficients as a vector and use an R-tree to index the vectors
R-tree works with euclidean distance, OK.

28. Wavelets - DWT value time
DFT is great - but, how about compressing opera? (baritone, silence, soprano?)
value
time

29. Wavelets - DWT Solution#1: Short window Fourier transform
But: how short should be the window?

30. Wavelets - DWT
Answer: multiple window sizes! -> DWT

31. Haar Wavelets subtract sum of left half from right half
repeat recursively for quarters, eightths ...

32. Wavelets - construction
x0 x1 x2 x3 x4 x5 x6 x7

33. Wavelets - construction
level 1
d1,0
d1,1
s1,1 + -
x0 x1 x2 x3 x4 x5 x6 x7

34. Wavelets - construction
level 2
d2,0
s1,0
d1,0
d1,1
s1,1 + -
x0 x1 x2 x3 x4 x5 x6 x7

35. Wavelets - construction
etc ...
s2,0
d2,0
s1,0
d1,0
d1,1
s1,1 + -
x0 x1 x2 x3 x4 x5 x6 x7

36. Wavelets - construction
Q: map each coefficient
on the time-freq. plane f
s2,0
d2,0 t
s1,0
d1,0
d1,1
s1,1 + -
x0 x1 x2 x3 x4 x5 x6 x7

37. Wavelets - construction
Q: map each coefficient
on the time-freq. plane f
s2,0
d2,0 t
s1,0
d1,0
d1,1
s1,1 + -
x0 x1 x2 x3 x4 x5 x6 x7

38. Wavelets - Drill:
Q: baritone/silence/soprano - DWT? f t
time
value

39. Wavelets - Drill:
Q: baritone/soprano - DWT? f t
time
value

40. Wavelets - construction
Observation1:
‘+’ can be some weighted addition
‘-’ is the corresponding weighted difference (‘Quadrature mirror filters’)
Observation2: unlike DFT/DCT,
there are *many* wavelet bases: Haar, Daubechies-4, Daubechies-6, ...

41. Advantages of Wavelets
Better compression (better RMSE with same number of coefficients)
closely related to the processing of the mammalian eye and ear
Good for progressive transmission
handle spikes well
usually, fast to compute (O(n)!)

42. Feature space Keep the d most “important” wavelets coefficients
Normalize and keep the largest
Lower bounding lemma: the same as DFT

43. PAA and APCA
Another approach: segment the time series into equal parts, store the average value for each part.
Use an index to store the averages and the segment end points

44. Feature Spaces X X' DFT X X' DWT X X' SVD20 40 60 80
100
120
140 X X'
DFT X X'
DWT 20 40 60 80
100
120
140 X X'
SVD 20 40 60 80
100
120
140
eigenwave 0
eigenwave 1
eigenwave 2
eigenwave 3
eigenwave 4
eigenwave 5
eigenwave 6
eigenwave 7
Haar 0
Haar 1
Haar 2
Haar 3
Haar 4
Haar 5
Haar 6
Haar 7 1 2 3 4 5 6 7
Agrawal, Faloutsos, Swami 1993
Chan & Fu 1999
Korn, Jagadish, Faloutsos 1997

45. Piecewise Aggregate Approximation (PAA)
value
axis
time axis
Original time series
(n-dimensional vector)
S={s1, s2, …, sn}
sv1
sv2
sv3
sv4
sv5
sv6
sv7
sv8
n’-segment PAA representation
(n’-d vector)
S = {sv1 , sv2, …, svn’ }
PAA representation satisfies the lower bounding lemma
(Keogh, Chakrabarti, Mehrotra and Pazzani, 2000; Yi and Faloutsos 2000)

46. Can we improve upon PAA? n’-segment PAA representation (n’-d vector)
sv1
sv2
sv3
sv4
sv5
sv6
sv7
sv8
n’-segment PAA representation
(n’-d vector)
S = {sv1 , sv2, …, svN }
Adaptive Piecewise Constant
Approximation (APCA)
sv1
sv2
sv3
sv4
sr1
sr2
sr3
sr4
n’/2-segment APCA representation
(n’-d vector)
S= { sv1, sr1, sv2, sr2, …, svM , srM }
(M is the number of segments = n’/2)

47. APCA approximates original signal better than PAA
Reconstruction error PAA Reconstruction error APCA
Improvement factor =
1.69
3.77
1.21
1.03
3.02
1.75

48. APCA Representation can be computed efficiently
Near-optimal representation can be computed in O(nlog(n)) time
Optimal representation can be computed in O(n2M) (Koudas et al.)

49. Distance Measure Exact (Euclidean) distance D(Q,S)
Lower bounding distance DLB(Q,S) S Q’ Q
DLB(Q’,S)
DLB(Q’,S)

50. Index on 2M-dimensional APCA spaceR1 R3 R2 R4 R2 R3 R4 R1 S3 S4 S5 S6 S7 S8 S9 S2 S1
2M-dimensional APCA space S6 S5 S1 S2 S3 S4 S8 S7 S9
The k-nearest neighbor search traverse the nodes of the multidimensional index structure in the order of the distance from the query.
We define the node distance as MINDIST which is the minimum distance from the query to any point in the node boundary.
In case of Query Point Movement, MinDist is computed as
the distance between the centroid of relevant points and
the point P which is defined as in this equation.
Any feature-based index structure can used (e.g., R-tree, X-tree, Hybrid Tree)

51. k-nearest neighbor Algorithm
MINDIST(Q,R2)
MINDIST(Q,R3) R1 S7 R3 R2 R4 S1 S2 S3 S5 S4 S6 S8 S9 Q
MINDIST(Q,R4)
The k-nearest neighbor search traverse the nodes of the multidimensional index structure in the order of the distance from the query.
We define the node distance as MINDIST which is the minimum distance from the query to any point in the node boundary.
In case of Query Point Movement, MinDist is computed as
the distance between the centroid of relevant points and
the point P which is defined as in this equation.
For any node U of the index structure with MBR R, MINDIST(Q,R) £ D(Q,S) for any data item S under U

52. Index Modification for MINDIST Computation
APCA point S= { sv1, sr1, sv2, sr2, …, svM, srM }
smax3
smin3 R1
sv3 S2 S5 R3 S3
smax1
smin1
smax2
smin2 S1 S6 S4
sv1 R2
smax4
smin4 R4 S8
sv2 S9
sv4 S7
sr1
sr2
sr3
sr4
APCA rectangle S= (L,H) where
L= { smin1, sr1, smin2, sr2, …, sminM, srM } and
H = { smax1, sr1, smax2, sr2, …, smaxM, srM }

53. MBR Representation in time-value space
We can view the MBR R=(L,H) of any node U as two APCA representations
L= { l1, l2, …, l(N-1), lN } and H= { h1, h2, …, h(N-1), hN }
REGION 2
H= { h1, h2, h3, h4 , h5, h6 } h1 h2 h3 h4 h5 h6
value
axis
time axis l3 l4 l6 l5
REGION 1 l1 l2
REGION 3
L= { l1, l2, l3, l4 , l5, l6 }

54. Regions l(2i-1) h(2i-1) h2i l(2i-2)+1 h3 h5 h2 h4 h6 l3 l1 l2 l4 l6 l5
REGION i
l(2i-1)
h(2i-1)
h2i
l(2i-2)+1
M regions associated with each MBR; boundaries of ith region: h3 h1 h5 h2 h4 h6
value
axis
time axis l3 l1 l2 l4 l6 l5
REGION 1
REGION 3
REGION 2

55. Regions t1 t2 h3 l3 h5 l1 l5 l2 l4 h2 h4 h6 l6
ith region is active at time instant t if it spans across t
The value st of any time series S under node U at time instant t must lie in one of the regions active at t (Lemma 2)
REGION 2 t1 t2 h3
value
axis h1 l3
REGION 3 h5 l1 l5
REGION 1 l2 l4 h2 h4 h6 l6
time axis

56. MINDIST Computation t1 minregion G active at t MINDIST(Q,G,t)
For time instant t, MINDIST(Q, R, t) =
minregion G active at t MINDIST(Q,G,t) t1
MINDIST(Q,R,t1)
=min(MINDIST(Q, Region1, t1),
MINDIST(Q, Region2, t1))
=min((qt1 - h1)2 , (qt1 - h3)2 )
=(qt1 - h1)2
REGION 2 h3 l3 h1
REGION 3 h5 l1 l5
REGION 1 l2 l4 h2 h4 h6
MINDIST(Q,R) = l6
Lemma3: MINDIST(Q,R) £ D(Q,C) for any time series C under node U

57. Images - color
what is an image?
A: 2-d array

58. Images - color
Color histograms,
and distance function

59. Mathematically, the distance function is:
Images - color
Mathematically, the distance function is:

60. Images - color Problem: ‘cross-talk’:
Features are not orthogonal ->
SAMs will not work properly
Q: what to do?
A: feature-extraction question

61. Images - color possible answers: avg red, avg green, avg blue
it turns out that this lower-bounds the histogram distance ->
no cross-talk
SAMs are applicable

62. Images - color
time
performance:
seq scan
w/ avg RGB
selectivity

63. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: how to normalize them?

64. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: how to normalize them?
A: divide by standard deviation)

65. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: other ‘features’ / distance functions?

66. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
(Q: other ‘features’ / distance functions?
A1: turning angle
A2: dilations/erosions
A3: ... )

67. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
Q: how to do dim. reduction?

68. Images - shapes
distance function: Euclidean, on the area, perimeter, and 20 ‘moments’
Q: how to do dim. reduction?
A: Karhunen-Loeve (= centered PCA/SVD)

69. Images - shapes log(# of I/Os) all kept # of features kept
Performance: ~10x faster
log(# of I/Os)
all kept
# of features kept