

Here we consider T, the tightness of lower bounds.
This is calculated as:
T = LowerBoundDist(a,b)
/ TrueEuclidianDist(a,b)
We randomly sample a and b (with replacement)
1,000 times for each combination of parameters. We vary the
time series length [480, 960, 1440, 1920] and the number of bytes
available per time series [16, 24, 32, 40]. We assume that each real
valued representation requires 4 bytes per coefficient thus they use
[4, 6, 8, 10] coefficients. For iSAX, we hard code the cardinality
to 256, and we thus have [16, 24, 32, 40] words. For each experiment
we created a bar chart and table. In addition, we include a link to
the dataset if the dataset is not already in the public domain (UCR
Archive data is
here, warning 500meg file, password peggy).
Recall that larger values are better. If the value of T is
zero, then any indexing technique is condemned to retrieving
every object from the disk. If the value of T is one, then
there is no search, we could simply retrieve one file from disk and
guarantee that we we had the true nearest neighbor. Note that the
speedup obtained is generally nonlinear in T, that is to say
if one representation has a lower bound that is twice as large as
another, we can usually expect a much greater than twofold decrease
in disk accesses.

 Pen: Image,
Original Time Series Data.
Data came from this paper: Behzad Malek, Mauricio
Orozco and Abdulmotaleb El Saddik "Novel
ShoulderSurfing Resistant Hapticbased Graphical Password"
Proc. of the EuroHaptics 2006 conference, July 36 Paris, France.
 Koski ECG: Image,
Original Time Series Data.
Data came from this paper: Antti Koski: Primitive
coding of structural ECG features.
Pattern Recognition Letters 17(11): 12151222 (1996)
 Steamgen: Image,
Original Time Series Data. Data came from this paper:
G. Pellegrinetti and J. Benstman, Nonlinear Control Oriented Boiler
Modeling A Benchmark Problem for Controller Design, IEEE Tran.
Control Systems Tech. Vol.4No.1 Jan.1996
 Star Light Curve Image,
Original Time Series Data.
Data came from Harvard University Time Series
Center.
 Mallet: Image, Original
Time Series Data is in UCR Archive (here,
warning 500meg file).
Data came from this paper: M. K. Jeong, J. C. Lu, X.
Huo, B. Vidakovic, and D. Chen (2006), "Waveletbased
Data Reduction Techniques for Process Fault Detection,"
Technometrics, 48(1), 2640
 Power_Data : Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file). Data came
from this paper: van Wijk J. J. and van Selow E. R. Cluster and
calendarbased visualization of time series data. In Proc. IEEE
INFOVIS, pages 49, Oct. 2526, 1999.
 Clorine (cl2fullLarge)
Image: Original Time Series
Data. Data
comes from this
website.
 Kung Fu MoCap:
Image, Original Time
Series Data.
Data comes from this
paper.
 PostureCentroidB
Image, Original Time Series Data.
Data comes from Jessica Lin, Eamonn J. Keogh,
Stefano Lonardi, Jeffrey P. Lankford, Donna M. Nystrom: Visually
mining and monitoring massive time series. KDD 2004: 460469, see
fig 13, 15.
 Ann_gun_CentroidA
Image, Original Time Series
Data.
Data comes from Ratanamahatana, C. A.
and Keogh. E. (2004). Making Timeseries Classification More
Accurate Using Learned Constraints. SDM '04 see
fig 8
 Converted_ERP Image, Original Time Series
Data.
Data comes from this
paper.
 Tickwise Image,
Original Time Series Data.
Data comes from
here.
 Foetal_ecg Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) L. De Lathauwer, B. De Moor, J.
Vandewalle, Fetal Electrocardiogram Extraction by Source Subspace
Separation. Proc. IEEE SP / ATHOS Workshop on HOS, 1995.
 Buoy Sensor Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) The Center for Coastal Studies'
Data Zoo.
 Winding Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) This is one of the
DaISy
datasets.
 CSTR Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) This is one of the
DaISy
datasets.
 Burst Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) This is one of the
DaISy
datasets.
 Fluid_dynamics
Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) Data comes from, Garabadu,
Thompson, Lindstrom and Klewicki (2003). Fast and Accurate NN
Approach for MultiEvent Annotation of Time Series . UUCS03021.
 Tide Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file). From paper: Analysis of
Subtidal Coastal Sea Level Fluctuations Using Wavelets" by D. B.
Percival and H. O. Mofjeld, JASA September 1997, Vol. 92, No. 439,
868880.
 Motorcurrent Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) From papers of Richard J.
Povinelli.
 Muscle Activation Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file) From Hoos, O., Department of
Sports Medicine, PhilippsUniversity Marburg, Germany Morchen, F.,
Department of Mathematics and Computer Science, PhilippsUniversity
Marburg, Germany
 Physiodata (npo141)
Image,
Original Time Series Data is in UCR Archive (here,
warning 500meg file)
 Respiration Image,
Original Time Series Data is
here,
as used in E. Keogh, J. Lin and A. Fu (2005). HOT
SAX: Efficiently Finding the Most Unusual Time Series Subsequence (ICDM
2005)
 ECG (mitdbx_mitdbx_108)
Image,
Original Time Series Data is
here,
as used in E. Keogh, J. Lin and A. Fu (2005). HOT
SAX: Efficiently Finding the Most Unusual Time Series Subsequence (ICDM
2005)
 ECG (chfdb_chf01_275)
Image,
Original Time Series Data is
here,
as used in E. Keogh, J. Lin and A. Fu (2005). HOT
SAX: Efficiently Finding the Most Unusual Time Series Subsequence (ICDM
2005)
 TOR( Italy power demand 97
Image, Original Time Series
Data is in UCR Archive (here,
warning 500meg file)
 Trajectory3_7_2006.bmp
Image, Original Time
Series Data is in UCR Archive (here,
warning 500meg file) From this paper: Pokrajac,
Lazarevic, Latecki: Incremental Local Outlier Detection for Data
Streams. CIDM 2007: 504515
 Synthetic Lightning EMP
Image, Data and explanation
is here.
Synthetic Lightning EMP Classification Data Sets
 Lightning (forte6class) Image, Data and explanation
is here.
FORTE Time Series Classification Data Sets.
Notes:

There is an apparent paradox in
the some datasets, including Muscle Activation dataset.
The tightest of the lower bounds improves from 480 to 960. How is
this possible? The data consist of noisy steps, like this [1, 2, 0,
1, 2, 0, 101, 100, 101, 103, 100, 0, 1, 2, 1, ...]. If a subsequence
straddles at least one step, then the representation devotes most of
its energy in approximating that step. However, if the subsequence
falls within a single step, the znormalization "expands" the signal
to pure noise, which is hard to approximate. In this case, only the
subsequences of length 480 can fall within a single step.

In general, there is little
difference between DCT, DFT, PAA, iPLA and CHEB. This has been noted
before (see table 1 of this
paper).
APCA is slightly worse that the rest if the data is completely
stationary, but can be significantly better on datasets that are
bursty, see Synthetic Lightning
EMP as an example. Note that this dataset exhibits the apparent
paradox discussed above. The Foetal_ecg
dataset also exhibits this burstyness and corresponding improvement
for APCA.

Here is the code to convert DNA to time
series. Here are some small sample datasets, the mitochondrial DNA of
the Hippo, and the
Polar Bear. The Contig
NT_005334.15 of human chromosome 2 is
here (Local version here). Below in
blue (human) and red (chimp) are the two similar sequences plotted in
the paper.
GTCAATGGCCAGGATATTAGAACAGTACTCTGTGAACCCTATTTATGGTGGCACCCCTTAGACTAAGATAACACAGGGAGCAAGAGGTTGACAGGAAAGCCAGGGGAGCAGGGAAGCCTCCTGTAAAGAGAGAAGTGCTAAGTCTCCTTTCTAAGGCACATGATGGAT
TCAAGGGAAAGCCACATTTGACTAAAGCCCAAGGGATTGTTGCTTCTAATCCGATTTCTTGGCAGAAGATATTACAAACTAAGAGTCAGATTAATATGTGGGTGCCAAAATAAATAAACAAATAATTGAATAATCCCTGGAGGTTTAAGTGAGGAGAAACTCCTCCAC
AGCTTGCTACCGAGGCAGAACCGGTTGAAACTGAAATGCATCCGCCGCCAGAGGATCTGTAAAAGAGAGGTTGTTACGAAACTGGCAACTGCCAACCAAAGTCCACCAATGGACAAGCAAAAAAGAGCACTCATCTCATGCTCCCAAGGATCAACCTTCCCAGAGTTT
TCACTTAAGTGGCCACCAAGCCAGTTGTCAATCCAGGGCTTTGGACTGAAATCTAGGGCTTCATCCGCTACCTCAGAGTGTCTTCTATTTCTTCCAGCCAGTGACAAATACAACAAACATCTGAGATGTTTTAGCTATAAATCCTTTACAATTGTTATTTATGTCTTA
ACTTTTGTTATACCTGGAAAAGTAGGGGAAACAATAAGAACATACTGTCTTGGCCAAGCATCCAAGGTTAAATGAGTTATGGAAATTCATTTGGGAGCCAAGACATTGCACGTGGTTATTTATTAGTCACCCAAGCATGTATTTTGCATGTCCATCAGTTGTTCTTGG
CCAAAAGAGCAGAATCAATGAGCCGCTGCAGATGCAGACATAGCAGCCCCTTGCAGGGACAAGTCTGCAAGATGAGCATTGAAGAGGATGCACAAGCCCGGTAGCCCGGGAAATGGCAGGCACTTACAAGAGCCCAGGTTGTTGCCATGTTTGTTTTTGCAACTTGTC
TATTTAAAGAGATTTG
GGCAATGGCCAGGATATTAGAACAGTACTCTGTGAACCCTATTTATGGTAGCACCCCTTAGACTAAGATAACACAGGGAGCAAGAGGTTGACAGGAAAGCCAGGGGAGCAGGGAAGCCTCCTGTAAAGAGAGAAGTGCTAAGTCTCCTTTCTAAGGCACATGATGGAT
TCAAGGGAAAGTCACATTTGACTAAAGCCCAAGGGATTGTTGCTTCTAATCCGATTCTTGGCAGAAGATATTGCAAACTAAGAGTCAGATTAATATGTGGGTGCCAAAATAAATAAACAAATAATTGAATAATCCCTGGAGGTTTAAGTGAGGAGAAACTCCTCCACA
GCTTGCTACCGAGGCAGAACCGGTTGAAACTGAAATGCACCCGCTGCCAGAGGATCTGTAAAAGGGAGGTTGTTACCGAACTGGCAACTGCCAACCAAAGTCTACCAATGGACAAGCAAAAAAGAGCACTCATCTCATGCTCCCAAGGATCAACCTTCCCAGAATTTT
CACTTAAGTGGCCACCAAGCCAGTTGTCAATCCAGGGCTTTGGACTGAAATCTAGGGCTTCATCCACTACCTCAGAGTGTCTTCCATTTCTTCCAGCCAGTGACAAATACAACAAACATCTGAGATGTTTTAGCTATAAATCCTTTACAATTGTTATTTATGTCTTAA
CTTTTGTTATACCTGGAAAAGTAGGGGAAACAATAAGAACATACTGTCTTGGCCAAGCATCCAAGGTTAAATGAGTTATGGGAATTCATTTGGGAGCCAAGACATTGCGCGTGGTTATTTATTAGTCACCCAAGCATGTATTTTGCATGTCCATCAGTTGTTCTTGGC
CAAAAGAACAGAATCAATGAGCCGCTGCAGATGCAGACATAGCAGCCCCTTGCAGGAACAAGTCTGCAAGATGAGCATTGAAGAGGATGCACAAGCCCGGTAGCCCGGGAAATGGCAGGCACTTACAAGAGCCCAGGTTGTTGCCATGTTTGTTTTTGCAACTTGTCT
ATTTAAACAGATTTGA
