Order-preserving Pattern Research Articles

Order-Preserving Pattern Matching with Partition

Order-preserving pattern matching, which considers the relative orders of strings, can be applied to time-series data analysis. To perform a more meaningful analysis of time-series data, approximate criteria for the order-isomorphism are necessary, considering diverse types of errors. In this paper, we introduce a novel approximation criterion for the order-isomorphism, called the partitioned order-isomorphism. We then propose an efficient O(n+sort(m))-time algorithm for the order-preserving pattern matching problem considering the criterion of partition. A comparative experiment demonstrates that the proposed algorithm is more effective than the exact order-preserving pattern matching algorithm.

Serial and parallel algorithms for order-preserving pattern matching based on the duel-and-sweep paradigm

Serial and parallel algorithms for order-preserving pattern matching based on the duel-and-sweep paradigm

Co-occurrence Order-preserving Pattern Mining with Keypoint Alignment for Time Series

Recently, order-preserving pattern (OPP) mining has been proposed to discover some patterns, which can be seen as trend changes in time series. Although existing OPP mining algorithms have achieved satisfactory performance, they discover all frequent patterns. However, in some cases, users focus on a particular trend and its associated trends. To efficiently discover trend information related to a specific prefix pattern, this article addresses the issue of co-occurrence OPP mining (COP) and proposes an algorithm named COP-Miner to discover COPs from historical time series. COP-Miner consists of three parts: extracting keypoints, preparation stage, and iteratively calculating supports and mining frequent COPs. Extracting keypoints is used to obtain local extreme points of patterns and time series. The preparation stage is designed to prepare for the first round of mining, which contains four steps: obtaining the suffix OPP of the keypoint sub-time series, calculating the occurrences of the suffix OPP, verifying the occurrences of the keypoint sub-time series, and calculating the occurrences of all fusion patterns of the keypoint sub-time series. To further improve the efficiency of support calculation, we propose a support calculation method with an ending strategy that uses the occurrences of prefix and suffix patterns to calculate the occurrences of superpatterns. Experimental results indicate that COP-Miner outperforms the other competing algorithms in running time and scalability. Moreover, COPs with keypoint alignment yield better prediction performance.

COPP-Miner: Top-k Contrast Order-Preserving Pattern Mining for Time Series Classification

COPP-Miner: Top-k Contrast Order-Preserving Pattern Mining for Time Series Classification

OPF-Miner: Order-Preserving Pattern Mining With Forgetting Mechanism for Time Series

OPF-Miner: Order-Preserving Pattern Mining With Forgetting Mechanism for Time Series

OPR-Miner: Order-Preserving Rule Mining for Time Series

Discovering frequent trends in time series is a critical task in data mining. Recently, order-preserving matching was proposed to find all occurrences of a pattern in a time series, where the pattern is a relative order (regarded as a trend) and an occurrence is a sub-time series whose relative order coincides with the pattern. Inspired by the order-preserving matching, the existing order-preserving pattern (OPP) mining algorithm employs order-preserving matching to calculate the support, which leads to low efficiency. To address this deficiency, this paper proposes an algorithm called efficient frequent OPP miner (EFO-Miner) to find all frequent OPPs. EFO-Miner is composed of four parts: a pattern fusion strategy to generate candidate patterns, a matching process for the results of sub-patterns to calculate the support of super-patterns, a screening strategy to dynamically reduce the size of prefix and suffix arrays, and a pruning strategy to further dynamically prune candidate patterns. Moreover, this paper explores the order-preserving rule (OPR) mining and proposes an algorithm called OPR-Miner to discover strong rules from all frequent OPPs using EFO-Miner. Experimental results verify that OPR-Miner gives better performance than other competitive algorithms. More importantly, clustering and classification experiments further validate that OPR-Miner achieves good performance.

Order-preserving pattern matching with scaling

Given a text T and a pattern P, the order-preserving pattern matching (OPPM for short) problem is to find all substrings of T which have the same relative orders as P. Recently, approximate OPPM that allows errors have been studied such as OPPM with k-mismatches. In this paper we define the OPPM with scaling which is a novel criteria for approximate OPPM by considering the relative orders of cusps and the scale of lengths of strings between them. Also we present an algorithm to solve the OPPM problem with scaling in O(n+mlog⁡m) time, which is the same time bound as the best known exact OPPM algorithm.

An Improved Order-Preserving Pattern Matching Algorithm Using Fingerprints

Two strings of the same length are order isomorphic if their relative orders are the same. The order-preserving pattern matching problem is to find all substrings of text T that are order isomorphic to pattern P when T(|T|=n) and P(|P|=m) are given. An O(mn+nqlogq+q!)-time algorithm using the O(m+q!) space for the order-preserving pattern matching problem has been proposed utilizing fingerprints of q-grams based on the factorial number system and the bad character heuristic. In this paper, we propose an O(mn+2q)-time algorithm using the O(m+2q) space for the order-preserving pattern matching problem, but utilizing fingerprints of q-grams converted to binary numbers. A comparative experiment using three types of time series data demonstrates that the proposed algorithm is faster than existing algorithms because it reduces the number of order isomorphism tests.

Order-preserving pattern matching indeterminate strings

Given a pattern p of size m and a text t, the problem of order-preserving pattern matching (OPPM) is to find all substrings of t that satisfy one of the orderings defined by p. This problem has applications on time series analysis. However given its strict nature this model is unable to deal with indetermination, thus limiting its application to noisy time series. In this paper we introduce indeterminate characters to alleviate this limitation. We then propose two polynomial time algorithms. If the indetermination is limited to p confirming one occurrence can be computed in O(rmlg⁡r) time, where r is a bound on the number of uncertain characters per position. If the indetermination alternates, but does not occur at the same position in t and p, we present an algorithm that requires O(rm(m+log⁡r)) time. We also show that the general problem is NP-hard and provide a polynomial size boolean formula.

FleBiC: Learning classifiers from high-dimensional biomedical data using discriminative biclusters with non-constant patterns

The discovery of discriminative patterns from high-dimensional data offers the possibility to learn from informative subspaces and pattern-centric features, paving the way to associative classifiers. Despite the success achieved by associative classifiers, such as random forests or XGBoost, they generally neglect discriminative subspaces with non-constant coherencies. Research on biclustering has for two decades highlighted the role of non-constant patterns in biomedical domains, including additive and order-preserving patterns. Still, their relevance for classification remains unexplored.This work assesses the impact of discriminative patterns with varying coherence and quality on associative classification. A novel classifier, FleBiC, is proposed as a result. FleBiC extends pattern-based biclustering with principles to match observations against non-constant and noise-tolerant patterns, address generalization difficulties, minimize scarcity of matches, support class disjunctions, and offer statistical guarantees. Results on biological and clinical data highlight the role of non-constant patterns, specially order-preserving patterns, for improving the performance of state-of-the-art classifiers.

The order-preserving pattern matching problem in practice

Given a pattern x of length m and a text y of length n, both over a totally ordered alphabet, the order-preserving pattern matching (OPPM) problem consists in finding all substrings of the text with the same relative order as the pattern. The OPPM problem, which might be viewed as an approximate variant of the well-known exact pattern matching problem, has gained attention in recent years. This interesting problem finds applications in such diverse fields as time series analysis (as share prices on stock markets or weather data analysis) and musical melody matching, just to mention a few.In this paper we present two new filtering approaches which turn out to be much more effective in practice than the previously presented methods, reducing the number of false positives up to 99%. We also present a new efficient approach inspired by the well-known Skip Search algorithm for the exact string matching problem. It makes use of efficient SIMD SSE instructions for speeding up the searching phase. Experimental results show that our proposed algorithms are up to twice faster than previous solutions.

Z-함수를 이용한 순위패턴매칭과 순위다중패턴매칭 병렬계산

Disclosed is a z-function-based parallel computation method for order-preserving pattern matching and order-preserving multiple pattern matching. Provided is a z-function-based parallel computation method, which includes a step of calculating a z-function which is a maximum length at which a prefix of a suffix CC starting at each position BB with respect to a character string AA of length n coincides with a prefix of a character string S. According to an embodiment of the present invention, it is possible to provide a parallel algorithm using a z-function, which can solve order-preserving pattern matching using DD threads for an EE time.

EBIC: an evolutionary-based parallel biclustering algorithm for pattern discovery.

Biclustering algorithms are commonly used for gene expression data analysis. However, accurate identification of meaningful structures is very challenging and state-of-the-art methods are incapable of discovering with high accuracy different patterns of high biological relevance. In this paper, a novel biclustering algorithm based on evolutionary computation, a sub-field of artificial intelligence, is introduced. The method called EBIC aims to detect order-preserving patterns in complex data. EBIC is capable of discovering multiple complex patterns with unprecedented accuracy in real gene expression datasets. It is also one of the very few biclustering methods designed for parallel environments with multiple graphics processing units. We demonstrate that EBIC greatly outperforms state-of-the-art biclustering methods, in terms of recovery and relevance, on both synthetic and genetic datasets. EBIC also yields results over 12 times faster than the most accurate reference algorithms. EBIC source code is available on GitHub at https://github.com/EpistasisLab/ebic. Supplementary data are available at Bioinformatics online.

On Representations of Ternary Order Relations in Numeric Strings

Order-preserving matching is a string matching problem of two numeric strings where the relative orders of consecutive substrings are matched instead of the characters themselves. The order relation between two characters is a ternary relation (>, <, $$=$$ ) rather than a binary relation (>, <), but it was not sufficiently studied in previous works [Cho et al. (Fast order-preserving pattern matching, 2013), Kim et al. (Theor Comput Sci 525:68—79, 2014), Kubica et al. (Inform Process Lett 113:430–433, 2013)]. In this paper, we extend the representations of order relations by Kim et al. (Theor Comput Sci 525:68—79, 2014) to ternary order relations, and prove the equivalence of those representations. With our extensions, the time complexities of order-preserving matching in binary order relations can be achieved in ternary order relations as well.

Sequential Prediction Over Hierarchical Structures

We study sequential compound decision problems in the context of sequential prediction of real valued sequences. In particular, we consider finite state (FS) predictors that are constructed based on a hierarchical structure, such as the order preserving patterns of the sequence history. We define hierarchical equivalence classes by tying certain models at a hierarchy level in a recursive manner in order to mitigate undertraining problems. These equivalence classes defined on a hierarchical structure are then used to construct a super exponential number of sequential FS predictors based on their combinations and permutations. We then introduce truly sequential algorithms with computational complexity only linear in the pattern length that $1)$ asymptotically achieve the performance of the best FS predictor or the best linear combination of all the FS predictors in an individual sequence manner without any stochastic assumptions over any data length $n$ under a wide range of loss functions; $2)$ achieve the mean square error of the best linear combination of all FS filters or predictors in the steady-state for certain nonstationary models. We illustrate the superior convergence and tracking capabilities of our algorithm with respect to several state-of-the-art methods in the literature through simulations over synthetic and real benchmark data.

순위다중패턴매칭을 위한 해싱기반 알고리즘

순위패턴매칭문제는 텍스트 T와 패턴 P가 주어질 때, P와 각 문자들의 순위가 동일한 순서로 나타나는 T의 모든 부분문자열을 찾는 문제이다. 순위패턴매칭문제는 주가지수분석과 음악의 유사성분석과 같이 문자 자체를 비교하는 것보다 값의 변화순서가 중요한 분야에서 연구가 진행되었다. 순위다중패턴매칭문제는 텍스트 T와 여러 개의 패턴들로 이루어진 패턴집합 ℙ가 주어질 때, ℙ에 속한 패턴과 각 문자들의 순위가 동일한 순서로 나타나는 T의 모든 부분문자열을 찾는 문제이다. 본 논문에서는 순위다중 패턴매칭문제를 해결하는 해싱기반 알고리즘을 제시한다.

Generalized pattern matching and periodicity under substring consistent equivalence relations

Let ≈ be a substring consistent equivalence relation (SCER) on strings such that for any two strings x,y, x≈y implies that (1) |x|=|y| and (2) x[i..j]≈y[i..j] for all 0≤i≤j<|x|. Examples of SCER are parameterized pattern matching and order-preserving pattern matching. We present a generalized and efficient algorithm for pattern matching with SCER ≈. Also, we show analogues of Fine and Wilf's periodicity lemma hold for SCER.

Order-preserving pattern matching with k mismatches

We consider a generalization of the recently introduced order-preserving pattern matching. The difference between the standard pattern matching and the order-preserving variant is that instead of looking for an exact copy of the pattern, we only require that the relative order between the elements is the same. In our generalization, we additionally allow up to k mismatches between the pattern of length m and the text of length n, and the goal is to construct an efficient algorithm for small values of k. Our solution detects an order-preserving occurrence with up to k mismatches in O(n(log⁡log⁡m+klog⁡log⁡k)) time.

Order-preserving indexing

Kubica et al. [33] and Kim et al. [29] introduced order-preserving pattern matching: for a given text the goal is to find its factors having the same ‘shape’ as a given pattern. Known results include a linear-time algorithm for this problem (in case of polynomially-bounded alphabet) and a generalization to multiple patterns. We propose an index that enables order-preserving pattern matching queries in time proportional to pattern length. The index can be constructed in O(nlog⁡log⁡n) expected time or in O(nlog2⁡log⁡n/log⁡log⁡log⁡n) worst-case time. It is an incomplete order-preserving suffix tree which may miss a single edge label at each branching node. For most applications such incomplete suffix trees provide the same functional power as the complete ones. We show a number of their applications, including computation of longest common factors, longest previously occurring factors and squares in a string in the order-preserving setting. We also give an O(nlog⁡n)-time algorithm constructing complete order-preserving suffix trees.

Order preserving pattern matching revisited

In this paper, we study the order preserving pattern matching (OPPM) problem, which is a very recent variant of the classic pattern matching problem. We revisit this variant, present a new interesting pattern matching algorithm and for the first time consider string regularities from this new perspective.