Time series prediction is of great significance in many applications and has attracted extensive attention from the data mining community. Existing work suggests that for many problems, the shape in the current time series may correlate an upcoming shape in the same or another series. Therefore, it is a promising strategy to associate two recurring patterns as a rule's antecedent and consequent: the occurrence of the antecedent can foretell the occurrence of the consequent, and the learned shape of consequent will give accurate predictions. Earlier work employs symbolization methods, but the symbolized representation maintains too little information of the original series to mine valid rules. The state-of-the-art work, though directly manipulating the series, fails to segment the series precisely for seeking antecedents consequents, resulting in inaccurate rules in common scenarios. In this paper, we propose a novel motif-based rule discovery method, which utilizes motif discovery to accurately extract frequently occurring consecutive subsequences, i.e. motifs, as antecedents consequents. It then investigates the underlying relationships between motifs by matching motifs as rule candidates and ranking them based on the similarities. Experimental results on real open datasets show that the proposed approach outperforms the baseline method by 23.9 . Furthermore, it extends the applicability from single time series to multiple ones.
The state-of-the-art work (Y15) directly manipulates the real-valued series @cite_4 . Their method is found on the assumption that a rule is contained in a subsequence, which splits a subsequence into the rule's antecedent and consequent. Usually, there is an interval between a rule's antecedent consequent, and the splitting method will append the extra series to the antecedent consequent. The complicated diversity of the intervals could result in rules with bad prediction performance. Besides, the splitting method cannot be applied to discover rules from two series either, since a motif is a subsequence in a single series.
@cite_4
151863654 2026924428 2135625884 2950700385
We consider the problem of finding rules relating patterns in a time series to other patterns in that series, or patterns in one series to patterns in another series. A simple example is a rule such as "a period of low telephone call activity is usually followed by a sharp rise in call volume". Examples of rules relating two or more time series are "if the Microsoft stock price goes up and Intel falls, then IBM goes up the next day," and "if Microsoft goes up strongly for one day, then declines strongly on the next day, and on the same days Intel stays about level, then IBM stays about level." Our emphasis is in the discovery of local patterns in multivariate time series, in contrast to traditional time series analysis which largely focuses on global models. Thus, we search for rules whose conditions refer to patterns in time series. However, we do not want to define beforehand which patterns are to be used; rather, we want the patterns to be formed from the data in the context of rule discovery. We describe adaptive methods for finding rules of the above type from time-series data. The methods are based on discretizing the sequence by methods resembling vector quantization. We first form subsequences by sliding a window through the time series, and then cluster these subsequences by using a suitable measure of time-series similarity. The discretized version of the time series is obtained by taking the cluster identifiers corresponding to the subsequence. Once the time-series is discretized, we use simple rule finding methods to obtain rules from the sequence. We present empirical results on the behavior of the method. where (an)n ENo is some sequence of nonnegative numbers, (Sn),nENo is the sequence of partial sums, S0 = 0, Sn = XflXk, of another sequence (Xk)kEN of i.i.d. random variables, and A c R is a fixed Borel set such as [0,1] or [0, oo). Examples of such convolution series are subordinated distributions (f=0Oan = 1) which arise as distributions of random sums, and harmonic and ordinary renewal measures (a0 = 0, an = 1 n for all n C N in the first, an = 1 for all n C NO in the second case). These examples are in turn essential for the analysis of the large time behaviour of diverse applied models such as branching and queueing processes, they are also of interest in connection with representation theorems such as the Levy representation of infinitely divisible distributions. A traditional approach to such problems is via regular variation: If the underlying random variables are nonnegative we can use Laplace transforms and the related Abelian and Tauberian theorems [see, e.g., Stam (1973) in the context of subordination and Feller (1971, XIV.3) in connection with renewal theory; Embrechts, Maejima, and Omey (1984) is a recent treatment of generalized renewal measures along these lines]. The approach of the present paper is based on the Wiener-Levy-Gel'fand theorem and has occasionally been called the Banach algebra method. In Gruibel (1983) we gave a new variant of this method for the special case of lattice distributions, showing that by using the appropriate Banach algebras of sequences, arbitrarily fine expansions are possible under certain assumptions on the higher-order differences of (P(X1 = n))fnEN. Here we give a corresponding treatment of nonlattice distributions. We restrict ourselves to an analogue of first-order differences and obtain a number of theorems which perhaps are described best as next-term results. To explain this let us consider a special case in more detail. 1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. Topic models, such as Latent Dirichlet Allocation (LDA), posit that documents are drawn from admixtures of distributions over words, known as topics. The inference problem of recovering topics from admixtures, is NP-hard. Assuming separability, a strong assumption, [4] gave the first provable algorithm for inference. For LDA model, [6] gave a provable algorithm using tensor-methods. But [4,6] do not learn topic vectors with bounded @math error (a natural measure for probability vectors). Our aim is to develop a model which makes intuitive and empirically supported assumptions and to design an algorithm with natural, simple components such as SVD, which provably solves the inference problem for the model with bounded @math error. A topic in LDA and other models is essentially characterized by a group of co-occurring words. Motivated by this, we introduce topic specific Catchwords, group of words which occur with strictly greater frequency in a topic than any other topic individually and are required to have high frequency together rather than individually. A major contribution of the paper is to show that under this more realistic assumption, which is empirically verified on real corpora, a singular value decomposition (SVD) based algorithm with a crucial pre-processing step of thresholding, can provably recover the topics from a collection of documents drawn from Dominant admixtures. Dominant admixtures are convex combination of distributions in which one distribution has a significantly higher contribution than others. Apart from the simplicity of the algorithm, the sample complexity has near optimal dependence on @math , the lowest probability that a topic is dominant, and is better than [4]. Empirical evidence shows that on several real world corpora, both Catchwords and Dominant admixture assumptions hold and the proposed algorithm substantially outperforms the state of the art [5].