# Towards a Faster Symbolic Aggregate Approximation Method.

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TOWARDS A TIGHTER, MORE INTUITIVE SYMBOLIC

AGGREGATE APPROXIMATION METHOD

Keywords: Time Series Information Retrieval, Symbolic Representation of Time Series, Symbolic Aggregate

Approximation, Updated Minimum Distance.

Abstract: The similarity search problem is one of the main problems in time series data mining. Traditionally, this

problem was tackled by sequentially comparing the given query against all the time series in the database,

and returning all the time series that are within a predetermined threshold of that query. But the large size

and the high dimensionality of time series databases that are in use nowadays make that scenario inefficient

to tackle this problem. There are many representation techniques that aim at reducing the dimensionality of

time series so that the search can be handled faster at a lower-dimensional space level. Symbolic

representation is one of the promising techniques, because symbolic representation methods try to benefit

from the wealth of search algorithms used in bioinformatics and text mining communities. The symbolic

aggregate approximation (SAX) is one of the most competitive methods in the literature. SAX utilizes a

similarity measure, which is easy to compute because it is based on pre-computed distances obtained from

lookup tables. In this paper we present a new similarity distance that is as easy to compute as the original

similarity distance, it is also tighter because it uses updated lookup tables. In addition, it is more intuitive.

We conduct several experiments, which show that this new similarity distance gives better results than the

original one.

1 INTRODUCTION

Similarity search is a fundamental problem in

computer science. This problem has many

applications in multimedia

bioinformatics, pattern recognition, text mining,

computer vision, medicine, data mining, machine

learning and so on. With the advert of the internet

and the increasing use of it, this problem has

received more attention from researchers. The

richness of information available on the internet

could not be manageable if search engines were not

present. The usefulness of information depends

highly not only on its quality, but also on the speed

at which it is retrieved, which, in turn, depends upon

the way it is represented and indexed. This all poses

questions on indexing, representation and retrieval

methods. Small databases that contain simple data

objects can be handled easily. But managing large

databases,

databases, like many of the databases in use today,

requires serious effort, especially when they contain

complex data types.

Another substantial change is the kind of queries

launched, which is, somehow, related to the two

latter changes; searching for data objects that are

identical to a given query in unstructured, weakly-

structured, or imprecise data bases, like many

databases in use today, may not be very meaningful.

Besides, in many cases the user may not be sure of

what they are looking for when they launch the

initial query. Hence range query, in which the user

is interested in retrieving all the data objects that are

within a predefined threshold of a given query, or k-

nearest neighbor, in which the user tries to retrieve

the k closest data objects to the query, have become

popular. All these problems make traditional

retrieving techniques inadequate that it is inevitable

to think of new ways to handle these databases.

hal-00690016, version 1 - 21 Apr 2012

Author manuscript, published in "ICSOFT 2010 - Fifth International Conference on Software and Data Technologies, Athens :

Greece (2010)"

Page 2

Time series are data types that appear in many

applications, which vary from medicine through

science and technology to business and economics.

Time series data mining includes many tasks such as

classification, clustering, similarity search, motif

discovery, anomaly detection, and others. One key

to performing these

representation methods that can represent the time

series efficiently and effectively. Another key is

indexing time series in appropriate structures, which

direct the query process towards regions in the

search space, where similar time series to the query

are likely to be found, which makes the retrieving

process faster.

Time series are high dimensional data types, so

even indexing structures can suffer from the so-

called “dimensionality curse”. One of the best

solutions to deal with this problem is to utilize a

dimensionality reduction technique to reduce the

dimensionality of these data objects, then to utilize a

suitable indexing structure on the reduced space.

There have been different suggestions to

represent time series. To mention a few; DFT

(Agrawal et al . 1993) and (Agrawal et al . 1995), DWT

(Chan and Fu 1999), SVD (Korn et al . 1997),

APCA (Keogh et al . 2001), PAA (Keogh et al .

2000) and ( Yi and Faloutsos 2000), PLA( Morinaka

et al . 2001)...etc.

Among dimensionality reduction techniques,

symbolic representation has several advantages,

because it allows researchers to benefit from text-

retrieval algorithms and techniques (Keogh et al .

2001).

Similarity between two data objects can be

computed by means of a similarity distance. There

are quite a large number of similarity distances;

some are applied to a particular data type, while

others can be applied to different data types. Among

the different similarity distances, there are those that

can be used on symbolic data types. At first they

were available for data types whose representation is

naturally symbolic (DNA and proteins sequences,

textual data…etc). But later these symbolic

similarity distances were also applied to other data

types that can be transformed into strings by using

some symbolic representation technique.

Of all the symbolic representation methods in

the times series data mining literature, the symbolic

aggregate approximation method (SAX) (Jessica et

al . 2003) stands out as one of the most powerful

methods. The main advantage of this method is that

the similarity distance that it uses is easy to

compute, because it uses statistical lookup tables. In

this paper we present an improved similarity

tasks successfully is

distance to be used with SAX. It has the same

advantages as the original similarity distance used in

SAX. But our new similarity distance gives better

results in different time series mining tasks

The rest of this paper is organized as follows: in

section 2 we present background on dimensionality

reduction, and on symbolic representation of time

series in general, and SAX in particular. The

proposed similarity distance is presented in section

3. In section 4 we present some of the results of the

different experiments we conducted. The conclusion

is presented in section 5.

2 BACKGROUND

2.1 Dimensionality Reduction

Handling high-dimensional data is difficult to

achieve. One of the main paradigms to overcome

this problem is to embed the data objects of the

original space into a lower dimensional space. Time

series are highly correlated data, so representation

methods that aim at reducing dimensionality by

projecting the original data onto lower dimensional

spaces and processing the query in those reduced

spaces is a scheme that is widely used in time series

data mining community.

When embedding the original space into a lower

dimensional space and performing the similarity

query in the transformed space, two main side-

effects may be encountered; false alarms, also called

false positivity, and false dismissals. False alarms are

data objects that belong to the response set in the

transformed space, but do not belong to the response

set in the original space. False dismissals are data

objects that the search algorithm excluded in the

transformed space, although they are answers to the

query in the original space. Generally, false alarms

are more tolerated than false dismissals, because a

post-processing scan is usually performed on the

results of the query in the transformed space to filter

out these data objects that are not valid answers to

the query in the original space. However, false

alarms can slow down the search time if they are too

many. False dismissals are a more serious problem

and they need more sophisticated procedures to

avoid them.

False alarms and false dismissals are dependent on

the transformation used in the embedding. If f is a

transformation from the original space

),(

original

S

into another space

originald

hal-00690016, version 1 - 21 Apr 2012

Page 3

),(

no false dismissals this transform should satisfy:

( ),((

1

ufufd

d transforme

Suu

∈∀

2

(1)

The above condition is known as the lower-

bounding lemma. ( Yi and Faloutsos 2000)

If a transformation can make the two above

distances equal for all the data objects in the original

space, then similarity search produces no false

alarms or false dismissals. Unfortunately, such an

ideal transformation is very hard to find. Yet, we try

to make the above distances as close as possible.

The above condition can be written as:

d transformed transforme dS

then in order to guarantee

),())

212

uud

original

≤

,

original

1,

alphabet_size=number(breakpoints)+1 .

Their locations are determined by statistical lookup

tables, so that these breakpoints produce equal-sized

areas under the Gaussian curve. The interval

between two successive breakpoints is assigned to a

symbol of the alphabet, and each segment of the

PAA that lies within that interval is discretized by

that symbol. The last step of SAX is using the

following similarity distance;

1

),(

))( ),

u

((

0

21

21

≤≤

ud

ufufd

original

d transforme

(2)

A tight transformation is one that makes the above

ratio as close as possible to 1.

2.2 Symbolic Representation

One of the dimensionality reduction schemes in time

series data mining is symbolic representation.

Symbolic representation of time series uses an

alphabet A (usually finite) to reduce the

dimensionality of the time series. This can be

defined formally as follows: Given a time series

nttt TS

,...,,21

=

. The symbolic representation

scheme can be considered as a map

[]

Attf

kkji

∈→αα

,:

(3)

Symbolic representation of time series has been a

hot research topic, because by using this scheme we

can not only reduce the dimensionality of time

series, but can also benefit from the numerous

algorithms used in bioinformatics and text data

mining. However, first symbolic representation

methods were ad hoc and did not give satisfactory

results. But later more sophisticated methods

emerged.

The symbolic aggregate approximation method

(SAX) is one of the most powerful methods of

symbolic representation of time series. SAX is based

on the fact that normalized time series have highly

Gaussian distribution (Larsen and Marx 1986), so by

determining the breakpoints that correspond to the

alphabet size, one can obtain equal sized areas under

the Gaussian curve.

SAX is applied in the following steps: in the first

step the time series are normalized. In the second

step, the dimensionality of the time series is reduced.

This is obtained by using the PAA (Piecewise

Aggregate Approximation) ( Keogh, et al . 2000). In

PAA the times series is divided into equal sized

frames and the mean value of the points within the

frame is computed. The lower dimensional vector of

the original time series is the vector whose

components are the means of all successive frames.

In the third step, the PAA representation of the time

series is discretized. This is achieved by determining

the number and the location of the breakpoints. The

number of breakpoints is related to the desired

alphabet size (which is chosen by the user), i.e.

2

1

)),((),(

∧

=

∧

its

∧∧

S

∑

i

≡

i

w

dist

w

n

TMINDIST

(4)

Wheren is the length of the original time series,w is

the length of the strings (the number of the frames),

∧

S and

T are the symbolic representations of the two

time series S andT respectively, and where the

function

)(

dist

is implemented by using the

appropriate lookup table.

We also need to mention that the similarity distance

used in PAA is:

∑=

N

Wheren is the length of the time series, N is the

number of frames. It is proven in ( Keogh, et al . 2000)

and (Yi and Faloutsos 2000) that the above

similarity distance is a lower bounding of the

Euclidean distance applied in the original space of

time series . This results in the fact that MINDIST

∧

−=

N

i

yx

n

YXd

1

2)(),(

(5)

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Page 4

(1)The original time series

0 50100150200 250 300

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(2) Converting the time series to PAA

0 50100 150200 250300

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(3) Choosing the breakpoints

0 50 100150200250300

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

(4) Discretization the PAA

050100150200250300

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

a

a

c

d

d

d

b

b

aacdddbb

Figure 1: The different steps of SAX

is also a lower bounding of the Euclidean distance,

because it is a lower bounding of the similarity

distance used in PAA. This guarantees no false

dismissals. Figure.1 illustrates the different steps of

SAX.

3 THE UPDATED MINIMUM

DISTANCE (UMD)

The main advantage of SAX, which makes it fast to

apply, is that the similarity distance it uses is easy to

compute, because it is based on pre-computed

distances obtained from corresponding lookup

tables. However, MINDIST has a main drawback;

in order to be lower bounding this similarity distance

ignores all the distances between any successive

symbols of the alphabet and considers them to be

zero. For instance, the lookup table of the MINDIST

for an alphabet size of 3 is the one shown in Table 1.

Table 1: The lookup table of MINDIST for alphabet size

=3. All values between any successive symbols are equal

to zero. The breakpoints in this case (obtained from

statistical tables) are: -0.43 and 0.43. The distance

between them is 0.86

a b c

a

0 0 0.86

b

0 0 0

c

0.86 0 0

This has two consequences: the first is that

MINDIST is not tight enough, which produces many

false alarms. The second consequence can be shown

by the following example. Let the symbolic

representing of the five time series

3

TS ,

4

TS

,

5

TS using SAX with alphabet size =4

be :

aabddTS =

1

,

TS =

2

abbcdTS =

3

,

bacddTS =

4

The MINDIST between any two of these five times

series is zero, which is not only unintuitive, since no

two time series of these five are identical, but this

also produces many false alarms.

In this section we present a modified minimum

distance, which remedies the above problems. The

new minimum distance has all the advantages of the

original distance, in that it is also a lower bounding

of the Euclidean distance, and it is also fast to

compute, as it uses pre-computed distances. But the

new minimum distance is tighter. It is also intuitive,

in that it does not ignore the distances between

1

TS ,

2

TS

,

bacdc

,

,

bbbdcTS =

5

.

hal-00690016, version 1 - 21 Apr 2012

Page 5

c c

c

0.5 value3

value1 b

b b

b value0

b

value2 b b

-0.5 value4 a

a

Figure 2: The PAA representation of two time series: TS1

=cbcbab (bold black) and TS2=bcbbab (bold grey). The

solid arrows show the ignored distances and the dashed

arrow shows the only distance considered by MINDIST :

dist (a,c)=value0 (0.86)

successive symbols.

The principal of our new minimum distance,

which we call the updated minimum distance

(UMD) is to recover the distances between any

successive symbols, which were ignored in

MINDIST. Figure 2 shows an example of the

ignored distances in the case of alphabet size =3, and

which are recovered in UMD.

Figure 2 shows that in the case of alphabet size =3

the breakpoints are -0.43 and 0.43. In this case the

only non-zero distance according to MINDIST is

dist (a,c) which is equal to 0.86 (the distance

indicated by the dashed arrow). The distances

represented by the solid arrows are the distances

between the minima or the maxima of all the

symbols of the alphabet and the corresponding

breakpoint. These distances are ignored in

MINDIST, but as we can see they are not equal to

zero. So dist(a,b), which was zero in MINDIST can

be updated to become value2+value4 , and dist (b,c)

which was also zero in MINDIST can be updated to

become value1+value3, and even dist (a,c) is

updated to become value4+value 3+value0 (value 0

is the original value). Lookup tables of different

alphabet sizes are updated in a like manner.

Obviously, this update of lookup tables results in a

tighter similarity distance. For instance, the lookup

table shown at the beginning of this section can be

updated to become the one shown in Table2.

We can easily notice that this new distance is a

lower bounding of the PAA distance presented in (5)

in section 2.2, since we take the closest distance

between two successive symbols among all the

distances of all the PAA segments of these two

symbols. As a result, our new distance is also a

lower bounding of the Euclidean distance (c.f

section 2.2). This is the same property that

MINDIST has.

Table 2: The updated lookup table for alphabet size =3.

We can see that the distances between successive symbols

are no longer equal to zero. And the distance dist (a,c) is

tighter

a b c

a

0

value2+

value4

0.86+

value4+

value3

b

value2+

value4

0

Value1+

Value3

c

0.86+

value4+

value3

Value1+

Value3

0

The other consequence of this update is that

UMD, which is based on the updated lookup tables,

is intuitive, because it gives non-zero values to

successive symbols, so the UMD of any two of the

five time series presented at the beginning of this

section is not zero, which is what we expect from

any similarity distance applied to these time series

because they are not identical.

In order to obtain the minimum and the maximum

values of each symbol, the SAX algorithm is

modified so that at the step where the different

segments of the PAA are compared against the

breakpoints to decide what symbol is used to

discretize that segment, at that step we modify SAX

so that it keeps a record of the minimum and

maximum values of each segment of that time series.

This is performed offline, so it does not include any

extra cost at query time. Then when comparing two

time series, we take the minimum (maximum) that

corresponds to the same symbol of the two times

series to find the mutual minimum (maximum,

respectively) that corresponds to each symbol. These

minima and maxima are used to update the lookup

tables. The update process includes very few

addition operations (three for alphabet size= 3, for

instance), whose cost is very low compared with the

cost of computing the distance. So UMD is as fast as

MINDIST.

So, as we can see, the cost of UMD is a little bit

higher at the indexing phase, which is not important,

but it has the same complexity as MINDIST at the

retrieval phase

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Page 6

4 EXPERIMENTS

We conducted extensive experiments on the

proposed similarity distance. In our experiments we

tested UMD on all the 20 data sets available at UCR

(UCR Time Series datasets) and for all alphabet

sizes, which vary between 3 (the least possible size

that was used to test MINDIST) to 20 (the largest

possible alphabet size). The size of these data sets

varies between 28 (Coffee) and 6164 (wafer). The

length of the time series varies between 60

(Synthetic Control) and 637 (Lightning-2). So these

data sets are very diverse. In all our experiments we

took the Euclidean distance as the reference

distance, because this distance is widely used in the

time series data mining community (Keogh and

Kasetty 2002 and Reinert et al . 2000), even though

it is has a few inconveniences; it is sensitive to noise

and to shifts on the time axis. It is also applied to

series of identical lengths only (Megalooikonomou

et al . 2005).

4.1 Tightness

As mentioned in section 2.1, tightness of the

similarity distances enhances the search process,

because it minimizes the number of false alarms. As

a result, it decreases the post processing time.

123456

0

5

10

15

20

25

30

35

40

45

50

CBF FaceAll wafer TwoPatterns yoga SwedishLeaf

% of the Euclidean Distance

UMD

MINDIST

Figure 3: Comparison of the tightness of UMD with the

tightness of MINDIST in 6 data sets and for alphabet

size=3. The figure shows that UMD is tighter than

MINDIST.

We compared the tightness of UMD with the

tightness of MINDIST, for all the datasets and for all

values of the alphabet size. In all the experiments,

UMD was tighter than MINDIST. In Figure 3 and

Figure 4, we present some of the results we obtained

for alphabet size=3 and for alphabet size=10,

respectively. We chose to report these data sets

because they are the largest data sets in the UCR

archive, so these results are the most significant

statistically.

The experiments conducted on other data sets and

for all values of the alphabet size, in addition to the

other values of the alphabet size on the data sets

presented in Figures 3 and 4, all gave similar results.

123456

0

10

20

30

40

50

60

70

80

CBF FaceAll wafer TwoPatterns yoga SwedishLeaf

% of the Euclidean Distance

UMD

MINDIST

Figure 4: Comparison of the tightness of UMD with the

tightness of MINDIST in 6 data sets and for alphabet

size=10. The figure shows that UMD is tighter than

MINDIST.

4.2 Classification

Classification is one of the main tasks in time series

data mining. We tested the proposed similarity

distance in a classification task based on the first

nearest-neighbor rule on all the data sets available at

UCR. We used leaving-one-out cross validation.

In order to make a fair comparison, we used the

same compression ratio (the number of points used

to represent one segment in PAA) that was used to

test SAX with MINDIST (i.e.1 to 4).

4.2.1 The First Experiment

The first version of SAX used alphabet size that

varies between 3 and 10. In the first classification

experiment we conducted we used an alphabet size

that varies between 3 and 10. We tested all the data

sets in the UCR archive . We start by varying the

alphabet size between 3 and 10 on the training set of

each data set to find the optimal value of the

alphabet size of that data set; i.e. the value that

minimizes the classification error rate. Then we use

that optimal alphabet size on the testing set of that

data set. Table 3 shows the results of our

experiment (There is no training phase for the

Euclidian distance). The best result between UMD

and MINDIST is highlighted. The results show that

for this range of alphabet size UMD outperforms

MINDIST in 14 data sets, and MINDIST

outperforms UMD in 5 data sets, and in one case

they both give the same result.

hal-00690016, version 1 - 21 Apr 2012

Page 7

Table 3: The error rate of UMD and MINDIST for α (the

alphabet size) between 3 and 10. Column 2 shows the

error rate of the Euclidean distance

1-NN

Euclidean

Distance

UMD

(α between

3 and 10)

MINDIST

(α between

3 and 10)

Synthetic

Control

Gun-Point

0.12 0.007

0.033

0.087 0.213

0.233

CBF

0.148 0.131

0.104

Face (all)

0.286 0.306

0.319

OSULeaf

0.483 0.471

0.475

Swedish-

Leaf

50words

0.213

0.291

0.490

0.369 0.338

0.327

Trace

0.24 0.34

0.42

Two_

Patterns

Wafer

0.09 0.076

0.081

0.005 0.004

0.004

Face (four)

0.216 0.273

0.239

Lighting-2

0.246 0.230

0.213

Lighting-7

0.425 0.411

0.493

ECG200

0.12 0.11

0.09

Adiac

0.389 0.634

0.903

Yoga

0.170 0.193

0.199

Fish

0.217 0.366

0.514

Beef

0.467 0.367

0.533

Coffee

0.25 0.179

0.464

OliveOil 0.133

0.367

0.833

MEAN

0.234 0.265 0.348

STD 0.134

0.158 0.247

It is worth to mention that for the Euclidian

distance, there is no compression of information, so

in some cases it may give better results than

symbolic, compressed distances.

The average error of UMD over all the datasets

and for this range of alphabet size is smaller than

that of MINDIST. The standard deviation for UMD

is also smaller than that of MINDIST. The

significance of this statistical parameter is that when

the standard deviation is small, the similarity

distance is more robust, and can be applied to

different kinds of datasets

4.2.2 The Second Experiment

SAX appeared in two versions; in the first one the

average size varied in the interval (3:10), and in the

second one the alphabet size varied in the interval

(3:20). So we also conducted another experiment,

where the alphabet size ranged in the interval (3:20),

and on all the datasets in the UCR archive. We

proceeded in the same way; we start by varying the

alphabet size between 3 and 20 on the training set to

find the optimal value of the alphabet size of that

data set. Then we use this optimal alphabet size on

the testing set. The results of our second experiment

are shown in table 4 (We did not report the results of

the Euclidean distance, because they are the same as

in table 3). The results in table 4 show that UMD

outperforms MINDIST in 14 datasets and MINDIST

outperforms UMD in 4 datasets, and in 2 data sets

they both give the same results.

In this experiment too, the mean and the standard

deviation of UMD is better than that of MINDIST

The results of the two experiments show that the

general performance of UMD is better than that of

MINDIST

5 CONCLUSION AND

PERSPECTIVES

In this paper we presented a new similarity distance

to be used with the

approximation (SAX). The new distance UMD

improves the performance of SAX compared with

the original similarity distance MINDIST used with

SAX. We conducted several experiments of times

series data mining tasks. The results obtained show

that SAX with UMD gives better results than SAX

with MINDIST. Another interesting feature of the

new similarity distance is that it as fast to compute

as the original one

symbolic aggregate

hal-00690016, version 1 - 21 Apr 2012

Page 8

Table 4: The error rate of UMD and MINDIST for α (the

alphabet size) between 3 and 20.

UMD

(α between 3 and

20)

MINDIST

(α between 3 and

20)

Synthetic

Control

Gun-Point

0.003 0.023

0.127

0.073

0.305

0.475

0.253

0.334

0.35

0.066

0.004

0.239

0.148

0.425

0.13

0.867

0.181

0.263

0.433

0.143

0.833

0.284

0.14

CBF

0.054

Face (all)

0.305

OSULeaf

0.471

Swedish-

Leaf

50words

0.242

0.345

Trace

0.27

Two_

Patterns

Wafer

0.065

0.004

Face (four)

0.273

Lighting-2

0.229

Lighting-7

0.411

ECG200

0.11

Adiac

0.494

Yoga

0.172

Fish

0.257

Beef

0.333

Coffee

0.071

OliveOil

0.3

MEAN

0.227

STD

0.147 0.237

The future work can be improving this distance by

tracing other values in the original time series. This

can make the distance even tighter, which is what

we are working on now.

Another direction of future work focuses on

modifying SAX to benefit more from UMD. We

think this can be achieved by using a representation

method other than PAA. This may include more

calculations or more storage space at indexing time,

but it could give better results

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