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Generic Subsequence Matching Framework:

Modularity, Flexibility, Efficiency

David Novak, Petr Volny, and Pavel Zezula

Masaryk University, Brno, Czech Republic

[david.novak|xvolny1|zezula]@fi.muni.cz

Abstract. Subsequence matching has appeared to be an ideal approach

for solving many problems related to the fields of data mining and sim-

ilarity retrieval. It has been shown that almost any data class (audio,

image, biometrics, signals) is or can be represented by some kind of time

series or string of symbols, which can be seen as an input for various

subsequence matching approaches. The variety of data types, specific

tasks and their partial or full solutions is so wide that the choice, im-

plementation and parametrization of a suitable solution for a given task

might be complicated and time-consuming; a possibly fruitful combina-

tion of fragments from different research areas may not be obvious nor

easy to realize. The leading authors of this field also mention the imple-

mentation bias that makes difficult a proper comparison of competing

approaches. Therefore we present a new generic Subsequence Matching

Framework (SMF) that tries to overcome the aforementioned problems

by a uniform frame that simplifies and speeds up the design, development

and evaluation of subsequence matching related systems. We identify

several relatively separate subtasks solved differently over the literature

and SMF enables to combine them in straightforward manner achieving

new quality and efficiency. This framework can be used in many appli-

cation domains and its components can be reused effectively. Its strictly

modular architecture and openness enables also involvement of efficient

solutions from different fields, for instance efficient metric-based indexes.

This is an extended version of a paper published on DEXA 2012.

1 Introduction

Majority of the data being produced in current digital era is in the form of time

series or can be transformed into sequences of numbers. This concept is very

natural and ubiquitous: audio signals, various biometric data, image features,

economic data, etc. are often viewed as time series and need to be also organized

and searched in this way.

One of the key research issues drawing a lot of attention during the last two

decades is the subsequence matching problem, which can be basically formulated

as follows: Given a query sequence, find the best-matching subsequence from

the sequences in the database. Depending on the specific data and application,

this general problem has many variants – query sequences of fixed or variable

arXiv:1206.2510v1 [cs.MM] 12 Jun 2012

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size, data-specific definition of sequence matching, requirement of dynamic time

warping, etc. Therefore, the effort in this research area resulted in many ap-

proaches and techniques – both, very general and those focusing on a specific

fragment of this complex problem.

The leading authors in this field identified two main problems that limit the

comparability and cooperation potential of various approaches: the data bias (al-

gorithms are often evaluated on heterogeneous datasets) and the implementation

bias (the implementation of the specific technique can strongly influence exper-

iment results) [1]. The effort to overcome the data bias is expressed by founding

a common set of data collections [2] which is publicly available and that should

be used by any consequent research in this area. However, the implementation

bias lingers, which also obstructs a straightforward combination of compatible

approaches whose interconnection could be very efficient.

Analysis of this situation brought us to conclusion that there is a need for

a unified environment for developing, prototyping, testing, and combination of

subsequence matching approaches. In this paper we propose such generic subse-

quence matching framework (SMF), namely:

– we overview and decompose the state-of-the-art approaches and techniques

for subsequence matching and we identify several common sub-problems that

these approaches deal with in various ways (Section 2);

– we propose general architecture of SMF that should fulfill our targets and

we describe our implementation of the framework (Section 3); power of SMF

is demonstrated by elegant realization of several variants of fundamental

subsequence matching algorithms;

– we describe real applications with different requirements for subsequence

matching that can be simply implemented with the aid of our framework

(Section 4).

The paper concludes in Section 5 by future directions that cover possible per-

formance boost enabled by a straightforward cooperation of our framework with

advanced distance-based indexing and searching technologies [3,4].

2Time Series Processing Overview

The field opening paper by Faloutsos et al. [5] introduced a subsequence matching

application model that has been used ever since only with smaller modifications.

The model can be summarized in the following four steps that should be adopted

by a subsequence matching application:

slicing of the time series sequences (both data and query) into shorter subse-

quences (of a fixed length),

transforming each subsequence into lower dimension,

indexing the subsequences in a multi-dimensional index structure,

searching in the index with a distance measure that obeys the lower bounding

lemma on the transformed data.

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Fig.1. Basic hierarchy of representations. Those with asterisk after the name allows

lower bounding.

In the original work [5], this approach was demonstrated on a subsequence

matching algorithm that used the sliding window approach to slice the indexed

data and disjoint window for the query. The Discrete Fourier Transformation

(DFT) was used for dimensionality reduction and the data was indexed using

the minimum bounding rectangles in R-tree [6]. The Euclidean distance was used

for searching since it satisfies the lower bounding lemma on data transformed

by DFT.

Many works followed this model contributing to it by using other time series

representations, more sophisticated distance functions and corresponding lower

bounding mechanisms. In the rest of this section, we provide a brief overview of

the key approaches in this field.

2.1 Data Representation

The way in which the data is represented may be crucial for both efficiency

and effectiveness of the whole system. Naturally, one can work directly with

the original data, but this raw data is typically very large and it would require

great computing infrastructure to process such data fast enough. Therefore, a

lot of effort has been aimed at finding the best possible approximation of the

time series that would reduce the dimensionality of the data on one hand and

preserve the main features of the data on the other.

Figure 1 depicts the most significant approaches used for reducing time series

data representation: Discrete Fourier Transformation (DFT) [5], Discrete Co-

sine Transform (DCT) [7], Chebyshev Polynomials (CHEB) [8], Discrete Wavelet

Transform (DWT) [9], Piecewise Aggregate Approximation (PAA) [10], Single

Value Decomposition (SVD) [7], Adaptive Piecewise Constant Approximation

(APCA) [11]. On the higher level, we distinguish two groups of approaches: data

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Fig.2. Basic hierarchy of distance functions.

adaptive and non-data adaptive. An important feature for any representation is

the ability to be indexed because proper indexing can dramatically improve per-

formance of the whole retrieval process. Members of the latter group sometimes

struggle with indexing [12]. Another desirable property of the data representa-

tion is the possibility to perform lower bounding, which can again improve the

performance of the subsequence matching process.

2.2Distance computation

The notion of similarity between two sequences is typically expressed by a dis-

tance measure (distance function). The most frequently used functions are de-

picted in Figure 2. The lock-step measures, like Euclidean distance, are usually

relatively cheap to compute but they lack the robustness against even the basic

data transformations. On the other hand, more sophisticated dynamic program-

ming methods like Dynamic Time Warping (DTW) allow shifts on the time axis

and serve better to applications like speech recognition or query by humming.

It is also important to pair the data representation and the distance function

wisely in order to satisfy the lower bounding lemma [5]. Distance functions can

also differ by their input domain. Some functions are defined on continuous do-

mains (real numbers) and some work on strings (sequences of symbols from a

finite alphabet). The most common distance measures are: Euclidean Distance

(ED) [5], Dynamic Time Warping (DTW) [13,14,15], Edit Distance with Real

Penalty (ERP) [16], Edit Distance on Real Sequence (EDR) [17], Longest Com-

mon Subsequence (LCSS) [18].

2.3 Subsequence Matching Approaches

In order to build a subsequence matching application, number of questions has

to be answered: What kind of data will be used and what is its dimensionality?

What is the volume of the data? Do we need the warping ability? Are approxi-

mate answers acceptable? etc. On the basis of the answers, one can make proper

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design decisions like what representation and distance measure to use, which

storage index to employ, etc.

The above mentioned work by Faloutsos et al. [5] encouraged many new

approaches that followed his work. Moon et al. [19] suggested dual approach

for slicing and indexing sequences to the one used in [5]. This DualMatch uses

the sliding windows for queries and disjoint windows for data sequences to re-

duce the number of windows that are indexed. DualMatch was followed by the

generalization of windows creation method called GeneralMatch [20]. Another

significant leap forward was made by the effort of Keogh et al. in their work

about exact indexing of Dynamic Time Warping [15]. They introduced a sim-

ilarity measure that is relatively easy to compute and it lower-bounds the ex-

pensive DTW function. This approach was further enhanced by improving I/O

part of the subsequence matching process using Deferred Group Subsequence

Retrieval introduced in [21]. Moreover, many new distance functions and data

representations were introduced as we outlined in the previous sections.

If we focus on the performance side of the system, we have to employ en-

hancements like indexing, lower bounding, window size optimization, reducing

I/O operations or approximate queries. Lots of approaches for building sub-

sequence matching applications often use the very same techniques for solving

common sub-tasks included in the whole retrieval process and changes only some

parts with some novel approach. This leads to implementing the same parts of

the process like DFT or DWT repeatedly which leads to the phenomenon of the

implementation bias [1]. The modern subsequence matching approaches [21,15]

employ many smaller tasks in the retrieval process that solve sub-problems like

optimizing I/O operations. Implementations of routines that solve such sub-

problems should be reusable and employable in similar approaches. This led

us to think about the whole subsequence matching process as a chain of sub-

tasks, each solving a small part of the problem. We have observed that many of

the published approaches fit into this model and their novelty is often only in

reordering, changing or adding new subtask implementation into the chain.

3 Subsequence Matching Framework

In this section, we describe the general Subsequence Matching Framework (SMF)

that is currently available under GPL license at http://mufin.fi.muni.cz/

smf/. The framework can be perceived on the following two levels that should,

naturally, coincide:

– on the conceptual level, the framework is composed of mutually cooperating

modules, each of which solves a specific sub-task, and these modules are

cooperating within specific subsequence matching algorithms;

– on the implementation level, the framework defines the functionality of in-

dividual module types and their communication interfaces; a subsequence

matching algorithm is then implemented as a skeleton that combines mod-

ules in a specific way and this skeleton can be filled with actual module

implementations.

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Table 1. Notation used throughout this paper.

Symbol

S[k]

S[i : j]

S.len

S.id

S?.pid

S?.offset

D(Q,S)

Definition

the k-th value of the sequence S

subsequence of S from S[i] to S[j], inclusive

the length of sequence S

the unique identifier of sequence S

if S?is subsequence of S then S?.pid = S.id

if S?= S[i : j] then S?.offset = i and S?.len = j − i + 1

distance between two sequences Q and S

In Section 3.1, we describe the common sub-problems (sub-tasks) that we identi-

fied in the field and we define corresponding types of modules (conceptual level).

Further, in Section 3.2, we justify our approach by describing fundamental sub-

sequence algorithms in terms of our modules and we present a straightforward

implementation of these algorithms within SMF. Section 3.3 is devoted to details

about implementation of the framework including an example of a configuration

file by which one can create a brand new algorithm only by exchanging specific

modules in a text configuration file.

The key term in the whole framework is, naturally, a sequence. As we want

to keep the framework as general as possible, we do not lay practically any re-

strictions on the components of the sequence – it can be integers, real numbers,

vectors of numbers, or any more sophisticated structures. The sequence similar-

ity functions are defined relatively independently of specific sequence type (see

Section 3.3). In the following, we will use the notation summarized in Table 1.

3.1 Common Sub-problems: Modules in SMF

Studying the field of subsequence matching, we identified several common sub-

problems addressed by a number of approaches in some sense. Specifically, we can

see the following sub-tasks that correspond to isolated modules in our framework.

Data Representation (Data Transformer Module) The raw data se-

quences entering an application are often transformed into other representation

which can be motivated either by simple dimensionality reduction (DFT, DWT,

SVD, PAA) [5,9,7,10] or also by extracting some important characteristics that

should improve the effectiveness of the retrieval [22] (see Section 2.1 for details).

In either case, the general task can be defined simply as follows: Transform given

sequence S into another sequence S?. We will use the symbol in Figure 3 (a) for

this data transformer module. The following table summarizes information about

this module and gives a few examples of specific approaches implementing this

functionality.

data transformer transform sequence S into sequence S?

DFT apply the DFT on a sequence of real numbers S [5]

PAAapply the PAA on a sequence of real numbers S [10]

Landmarks extract landmarks from a sequence S [22]

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slicer

sequence

storage

data

transformer

distance

index

(a)(b) (c)(d) (e)

distance

function

Fig.3. Types of SMF modules and their notation.

Windows and Subwindows (Slicer Module) Majority of the subsequence

matching approaches partitions the data and/or query sequences into subse-

quences of, typically, fixed length (windows) [5,19,20,21]. Again, this task can

be isolated, well defined, and the implementation can be reused in many variants

of subsequence matching algorithms. Partitioning a sequence S, each resulting

subsequence S?= S[i : j] has S?.pid = S.id, S?.offset = i, and S?.len = j −i+1.

The module will be denoted as in Figure 3 (b) and its description and specific

examples are as follows:

sequence slicer partition S into list of subsequences S?

disjoint slicer partition S disjointly into subsequences of length w [5]

sliding slicer

use sliding window of size w to partition S [5]

1,...,S?

n

Distance Functions (Distance Module) As analyzed in Section 2.2, there

emerged a high number of specific distance functions D that can be evaluated

between two sequences S and T. The intention of SMF is to partially separate

the distance functions from the data and to use the specific distance function as a

parameter of the algorithm (see Section 3.3 for details on implementation of the

this independence). Of course, it is the matter of configuration to use appropriate

function for respective data type, e.g. to preserve the lower bounding property.

The distance functions symbol is in Figure 3 (c) and it can be summarized as

follows:

distance function evaluate dissimilarity of sequences S, T

Lpmetricsevaluate distance Lpon equally long number sequences

DTW use DTW on any pair of number sequences S, T [13]

ERP calculate Edit distance with Real Penalty on S, T [16]

LB PAA, LB Keogh measures that lower bound the DTW [15]

Efficient Indexing (Distance Index Module) An efficient subsequence-

matching algorithm typically employs an index to efficiently evaluate distance-

based queries on the stored (sub-)sequences using the query-by-example paradigm

(QBE). Again, we see the choice of the specific index as a relatively separate com-

ponent of the whole algorithm and thus as an exchangeable module. Also, we see

a space for improvement in boosting the efficiency of this component in future.

We denote this module as in Figure 3 (d):

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distance index evaluate efficiently distance-based QBE queries

R-Tree familyindex sequences as n-dimensional spatial data

iSAX tree use a symbolic representation of the sequences [23,24]

metric indexes index and search the data according to mutual distances [3]

Efficient Aligning (Sequence Storage Module) The approaches that use

sequence slicing typically also need to store the original whole sequences. The

slice index (for window size w) returns a set of candidate subsequences S?,

S?.len = w each matching some query subsequence Q?such that Q?.len = w. If

the query sequence Q is actually longer than w, the subsequent task is to align Q

to corresponding subsequence S[i : (i+Q.len−1)] where i = S?.offset−Q?.offset

and S.id = S?.pid. To do this aligning for each S?in the candidate set may be

very demanding. For smaller datasets, this can be done in memory with no spe-

cial treatment, but more advanced approaches are profitable on disk [21]. We

will call this module sequence storage (Figure 3 (e)) and it is specified as follows:

sequence storage store sequences S and return S[i : j] for given S.id

hash mapbasic hash map evaluating queries one by one

deferred retrieval deferred group sequence retrieval (I/O efficient) [21]

3.2Subsequence Matching Strategies in SMF

Staying at the conceptual level, let us have a look at the whole subsequence

matching algorithms and their composition from individual modules introduced

above. As an example, we take again the fundamental algorithm [5] for general

subsequence matching of queries Q, Q.len ≥ w for an established window size

w. The schema of a slight modification of this algorithm is in Figure 4. The

solid lines correspond to data insertion and the dash lines (with italic labels)

correspond to the query processing.

A data sequence S is first partitioned by the sliding window approach (slid-

ing slicer module) into slices S?= S[i : (i + w − 1)], these are transformed

by Discrete Fourier Transformation (data transformer module DFT), and the

Minimum Bounding Rectangles (MBR) of these transformed slices are stored

in an R∗-tree storage (distance index module); the original sequences S is also

stored (whole sequence storage module). Processing a subsequence query, the

query sequence Q is partitioned using the disjoint slicer module, each slice

Q?= Q[i : (i + w − 1)] is transformed by DFT and it is searched within the

slice index (using L2distance or a simple binary function intersect). For each

of the returned candidate subsequences S?, a query-corresponding alignment

S[i : (i + Q.len − 1)] is retrieved from the whole storage (see above for details)

and the candidate set is refined using L2distance D(Q,S[i : (i + Q.len − 1)]).

Preserving the skeleton of an algorithm (module types and their cooperation),

one can substitute individual modules with other compatible modules obtaining

a different processing efficiency or even a fundamentally different algorithm. For

instance, swapping the sliding and disjoint slicer modules practically results in

the DualMatch approach [19]. Exchanging the R∗-tree index for a metric-based

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sliding

slicer

DFT

data sequence

data subseq.

query sequence

query subseq.

disjoint

slicer

store MBRs

R*-Tree:

slice

index

whole

sequence

storage

data sequence

candidate MBRs

align and refine

search slices

intersect

L2

DFT

Fig.4. Schema of the fundamental subsequence matching algorithm [5].

index like the M-Index [4] could possibly improve the efficiency for larger dimen-

sionality of the stored slices (especially if the M-Index could use approximate

evaluation strategy). In general, a fast and straightforward alternation of mod-

ules is very helpful when seeking the best solution for a particular task and data

collection and tuning the performance of this solution.

3.3Implementation of SMF

The SMF was not implemented from scratch but with the aid of framework

MESSIF [25]. The MESSIF is a collection of Java packages supporting mainly

development of metric-based search approaches. From MESSIF, the SMF uses

especially the following functionality:

– encapsulation of the concept of data objects and distances,

– implementation of the queries and query evaluation process,

– distance based indexes (building, querying),

– configuration and management of the algorithm via text config files.

Data Independence The sequence is in SMF handled very generally; it is

defined as an interface which requires that each specific sequence type (e.g. a

simple float sequence) must, among other, specify the distance between two

sequence components d(S[i],S?[j]). For number sequences, this distances could

be, naturally, absolute value of differences d(S[i],S?[j]) = |S[i] − S?[j]|, but one

can imagine complex sequence components, for instance vectors where d could

be an Lpmetric. Implementation of a sequence distance D(S,S?) (for instance,

DTW) then treats S and S?only as general sequences with component distance

d and, thus, this implementation can be independent of specific sequence type.

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data sequence

query sequence

sequence

storage

data sequence

candidate slices

align and refine

slicer

slicer

store slices

search slices

distance

index

distance f.

Fig.5. Skeleton of VariableQueryAlgorithm, a simple subsequence matching algo-

rithm with variable length query.

Module Implementation The SMF module types specified in Section 3.1 are

typically implemented as Java interfaces and the specific modules as Java classes.

The interface specifies prototypes of the methods (inputs and outputs) that the

specific module must provide.

Algorithm Implementation Realizing a specific algorithm, one must imple-

ment its skeleton – module types, their connections, and all algorithm-specific

operations. This skeleton is compiled within the SMF package together with

all the interfaces and module implementations. Then, the algorithm is instanti-

ated only by means of a text configuration file – module types required by the

algorithm skeleton are filled by specific modules.

Let us describe this principle on an example of a simple algorithm for general

subsequence matching with variable query length – see Figure 5 for the schema

of the skeleton. This schema uses two slicer modules, one distance index with a

distance function, and a sequence storage for the whole sequences (again, with a

distance function).

In order to run a specific algorithm, we have to instantiate these module types

by specific modules. Figure 6 shows a part of a SMF configuration file that starts

such algorithm. The syntax of these files is taken from the MESSIF framework

and it is quite intuitive. On the first two lines, the sliding slicer module is created

by an action called namedInstanceAdd which creates an instance slidingSlicer

of class smf.modules.slicer.SlidingSlicer (with parameter w); the disjoint

slicer is created accordingly. Then, the instance of distance index is created; it

is a self-standing algoritm, namely a metric index M-Index [4] (we assume that

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slidingSlicer = namedInstanceAdd

slidingSlicer.param.1 = smf.modules.slicer.SlidingSlicer(<w>)

disjointSlicer = namedInstanceAdd

disjointSlicer.param.1 = smf.modules.slicer.DisjointSlicer(<w>)

index = namedInstanceAdd

index.param.1 = smf.modules.index

.ApproximateAlgorithmDistanceIndex(mIndex)

seqStorage = namedInstanceAdd

seqStorage.param.1 = smf.modules.seqstorage.MemorySequenceStorage()

startSearchAlg = algorithmStart

startSearchAlg.param.1 = smf.algorithms.VariableQueryAlgorithm

startSearchAlg.param.2 = smf.sequence.impl.SequenceFloatL2

startSearchAlg.param.3 = seqStorage

startSearchAlg.param.4 = index

startSearchAlg.param.5 = slidingSlicer

startSearchAlg.param.6 = disjointSlicer

startSearchAlg.param.7 = <w>

Fig.6. Example of a SMF configuration file

the instance mIndex has been already created). In this example, the sequence

storage is instantiated as a simple memory storage (seqStorage).

Finally, the actual VariableQueryAlgorithm is started passing the created

module instances as parameters to the skeleton. The param.2 of this action spec-

ifies that this particular algorithm instance requires sequences of floating point

numbers and will compare them by Euclidean distance. Such SMF configuration

files are supported directly by the MESSIF framework that enables an efficient

management of the running algorithm.

4 Use Cases

One of the motivations for building the SMF framework were several applications

that all need subsequence matching but are relatively heterogeneous. Let us

demonstrate three such applications that differ in data type, distance function,

query specification, and requirements for indexing efficiency; all of them can be

straightforwardly implemented using SMF.

4.1General Subsequence Matching System

The first demonstration is a general subsequence matching system for queries

with variable length. The demo is built on a small collection of simple time-series

data and it uses the algorithm described in Section 3.3 (Figures 5 and 6). Figure 7

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Fig.7. General subsequence matching system with variable query length.

shows screenshot of the GUI of this demo: The user can specify a subsequence

(offset and width) of the query sequence and after clicking “Find similar sub-

sequences”, the most similar ones (according to Euclidean distance) are located

(their distances are above the answer series). The demo is publicly available at

http://mufin.fi.muni.cz/subseq/. The characteristics of this simple demo

can be summarized as follows:

data type

similarity measure Euclidean distance

query type

requirements

series of real numbers

subsequence query with variable length (longer than w)

no additional requirements

4.2Gait Recognition

The biometrics form a wide application area that often manages data in the

form of sequences. Among others, recognition of people according to their gait

characteristics is currently a on the increase. There are several fundamental

approaches to this problem and one of them works directly with 3D coordinates

of anatomical landmarks of human body. The feasibility of this approach is

growing as the hardware that can extract such 3D positions is more and more

mature and available [26]. The left image in Figure 8 sketches the principle.

The trajectories are further processed and the development of mutual distances

between various anatomical landmarks is studied [27] – see time-series in the

right part of Figure 8. Currently, the SMF is used for research in this area:

data type

similarity measure various Lpmetrics, DTW, special measures

query type subsequence query with fixed or variable length

requirements data and similarity flexibility, high performance in future

series of real numbers or of number vectors

4.3Unsupervised Spoken Term Detection

The Automatic Speech Recognition (ASR) is an extremely important area for

human-computer interaction and the fundamental problem in this field is Spoken

Term Detection (STD). Besides well-developed complex approaches based on

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anatomical landmarks

trajectories

x

y

z

Fig.8. Gait recognition based on 3D coordinates of anatomical landmarks.

specific language models, the pattern recognition based on unsupervised methods,

that focus on situations when no linguistic corpus is available, represent quite

recent stream of research [28]. One of the studied approaches uses posteriorgram

templates [29] extracted using a phonetic recognizer. The DTW measure is often

used with this data [29], but we would like to evaluate other options like ERD

and ERP comparing the results. Again, the SMF is an ideal platform for this

research and future applications:

data type

similarity measure DTW, ERD, ERP; special distances between components

query typesubsequence query with variable length

requirements data and similarity flexibility, high performance in future

series of probabilities or of probability vectors

5 Conclusions and Future Work

The data in the form of sequences are all around us in various forms and exten-

sive volumes. The research in the area of subsequence matching has been very

intensive resulting in many partial or full solutions in various sub-areas of the

field. In this work, we identified several sub-tasks that circulate over the field and

are tackled within various subsequence matching approaches and algorithms.

We present a generic subsequence matching framework (SMF) that brings

the option of choosing freely among the existing partial solutions and combin-

ing them in order to achieve ideal solutions for heterogeneous requirements of

different applications. Also, this framework overcomes the often mentioned im-

plementation bias present in the field and it enables a straightforward utilization

of techniques from different areas, for instance advanced metric indexes. We de-

scribe SMF on conceptual and implementation levels and present several exam-

ples and demonstration applications from diverse fields. The SMF is available

under GPL license at http://mufin.fi.muni.cz/smf/.

The architecture of the SMF framework is strictly modular and thus one of

natural directions of future development is implementation of other modules.

Also, we will develop SMF according to requirements emerging from continu-

ous research streams that utilize SMF. Finally and most importantly, we would

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like to contribute to the efficiency of the subsequence matching systems by in-

volvement of advanced metric indexes. We believe in a positive impact of such

cooperation of these two research fields that were so far evolving relatively sep-

arately.

Acknowledgments

This work was supported by national research projects GACR 103/10/0886, and

GACR P202/10/P220.

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