A er the ﬁ rst experiment (Fejfar, Weinlichová, Šťastný, 2009) it seems that the whole concept of the **clustering** algorithm is correct but that we have to perform **time** **series** **clustering** on much larger data- set to obtain more accurate results and to ﬁ nd the correlation between conﬁ gured parameters and re- sults more precisely. The second requirement arose in a need for a well-deﬁ ned evaluation of results. It seems useful to use sound recordings as instances of **time** **series** again. There are many recordings to use in digital libraries, many interesting features and patterns can be found in this area. We are search- ing for recordings with the similar development of information density in this experiment. It can be used for musical form investigation, cover songs detection and many others applications.

Our idea is about recording fixed amount of subsequence **time** **series** data in a buffer and analyze it **time** by **time**. It this way the new data will be assessed with the previous one and then they will merge together in order to reach the best results. In the other hand, they can upgrade each other if they cannot merge. While data produce forever, we can keep clusters in a file. Here, the daily periodicity is obvious, and a more careful inspection reveals very similar patterns. The rest of this paper is organized as follows. In part 2, we clarified some main definitions and background in pattern recognition and subsequence **time** **series** **clustering**. Section 3 includes the related works according to this paper. Decision and framework will be provided in section 4 and 5 respectively and , we conclude all the proposed concepts in this paper as a glance at the end.

The first three methods were used directly or modified for **time** **series** **clustering**. Partition **clustering** aims to separate set of objects into consistent group. At first, the objects will be placed randomly and later transferred into another cluster until being positioned in an almost similar group while for hierarchical **clustering**, each object is defined as a single group. Then, each object (group) will be merged to form a new one. The merging process continues until only one group is left.

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This dissertation addresses four items not extensively researched in the subsequence- based **clustering** literature. The first is the production of a comprehensive end-to-end **clustering** methodology which allows for full motif discovery as well as quick subsequence cluster membership updates in the presence of incremental loads/data streaming. Estimation of stochastic processes, and use of goodness of fit measures, provides a novel way for quick membership evaluation in the case of incremental loads. The extension of the univariate notion of motif to a truly multivariate notion, allows for interplay between variables in a **time** **series** not captured by univariate compilation methods. Finally artificial fracturing methods allow greedy subsequence-based approaches to be used on a wider variety of data, including cyclical signaled **time** **series** data sets in which no noise periods exist.

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Many factors can affect sea lice infestation levels, including the fact that cages may be stocked and harvested at different moments, treatments may be applied differentially between cages, and with varying effect, among other factors. Furthermore, events in one cage or group of cages can impact other cages differentially and in a way that cannot be measured with sim- ple **time**-point comparisons, but which require a more flexible way of understanding sea-lice patterns of infection. This has led to the need for better analytical methods by which to under- stand how sea lice populations may vary synchronously within a farm to facilitate better man- agement and evaluation of sea lice prevention and control efforts. This study makes use of **time**-**series** **clustering** strategies to discern common patterns in sea lice cage count data within three Norwegian farms. The general objective is to identify sets of cages that form homogenous groups, based on the sea lice counts, throughout a production cycle. Here we overcome the limitations of point-by-point **clustering** using **time** **series** **clustering** that takes into account the entire set of sea-lice **time** **series** counts for each cage. When such **clustering** is present, it may be that the commonalities in patterns is due to factors such as management practices or inter- nal transmission between cages. It may also be the case that common sources of external infes- tation are impacting on sets of cages, causing the sea lice dynamics within these cages to follow similar patterns. Typically the types of **clustering** algorithm used here are designed as explor- atory tools and do not allow for formal statistical inference regarding the causal mechanisms generating common patterns. However, in addition to their descriptive value they can generate hypotheses that can be explored using more traditional statistical methods.

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In [2] authors proposed a **time**-**series** data mining is to try to extract all meaningful knowledge from the shape of data. Even if humans have a natural capacity to perform these tasks, it remains a complex problem for computers. In this article authors intended to provide a survey of the techniques applied for **time**-**series** data mining. In [3] authors proposed a method for **clustering** of **time** **series** based on their structural characteristics. Unlike other alternatives, this method does not cluster point values using a distance metric, rather it clusters based on global features extracted from the **time** **series**. In [4] authors illustrated the boxplot is a very popular graphical tool to visualize the distribution of continuous uni-modal data. It shows information about the location, spread, skewness as well as the tails of the data. However, when the data are skewed, usually many points exceed the whiskers and are often erroneously declared as outliers. In [5] authors showed that binary relevance-based methods have much to offer, especially in terms of scalability to large datasets. The authors exemplified this with a novel chaining method that can model label correlations while maintaining acceptable computational complexity.In [6] authors to provided a timely review on this area with emphasis on state-of-the-art multi-label learning algorithms. Firstly, fundamentals on multi-label learning including formal definition and evaluation metrics are given. Secondly and primarily, eight representative multi-label learning algorithms are scrutinized under common notations with relevant analyses and discussions. In [7] authors considered the **time** **series** are ubiquitous, and a measure to assess their similarity is a core part of many computational systems. In particular, the similarity measure is the most essential ingredient of **time** **series** **clustering** and classification systems. Because of this importance, countless approaches to estimate **time** **series** similarity have been proposed. However, there is a lack of comparative studies using empirical, rigorous, quantitative, and large-scale assessment strategies.

a nearest-neighbor network is , which is rather high. As a result, the authors attempt to reduce the search area by data preclustering (using k-Means) and limit the search to each cluster only to reduce the creation network. However, generating the network itself remains costly, rendering it inapplicable in large datasets. Additionally, the solution to the challenge of generating the prototypes via k-Means when the triangle is used as a distance measure is unclear. In this study, the low quality problem in existing works is addressed by the proposal of a new Two-step **Time** **series** **Clustering** (TTC) algorithm, which has a reasonable complexity. In the first step of the model, all the **time** **series** data are segmented into sub clusters. Each sub cluster is represented by a prototype generated based on the **time** **series** affinity factor. In the second step, the prototypes are combined to construct the ultimate clusters. To evaluate the accuracy of the proposed model, TTC is tested extensively using published **time** **series** datasets from diverse domains. This model is shown to be more accurate than any of the existing works and overcomes the limitations of conventional **clustering** algorithms in determining the clusters of **time** **series** data that are similar in shape. With TTC, the **clustering** of **time** **series** data based on similarity in shape does not require calculation of the exact distances among all the **time** **series** data in a dataset; instead, accurate clusters can be obtained using prototypes of similar **time** **series** data.

Results: We present a generative model-based Bayesian hierarchical **clustering** algorithm for microarray **time** **series** that employs Gaussian process regression to capture the structure of the data. By using a mixture model likelihood, our method permits a small proportion of the data to be modelled as outlier measurements, and adopts an empirical Bayes approach which uses replicate observations to inform a prior distribution of the noise variance. The method automatically learns the optimum number of clusters and can incorporate non-uniformly sampled **time** points. Using a wide variety of experimental data sets, we show that our algorithm consistently yields higher quality and more biologically meaningful clusters than current state-of-the-art methodologies. We highlight the importance of modelling outlier values by demonstrating that noisy genes can be grouped with other genes of similar biological function. We demonstrate the importance of including replicate information, which we find enables the discrimination of additional distinct expression profiles.

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**time** **series** used the static length of intervals, i.e., the same length of intervals. The drawback of the static length of intervals is that the historical data are roughly put into intervals, even if the variance of the historical data is not high. In this paper, we present a new method for forecasting enrolments based on Fuzzy **Time** **Series** and K-Mean **clustering**(FTS-KM). To verify the effectiveness of the proposed model, the empirical data for the enrolments of the University of Alabama are illustrated, and the experimental results show that the proposed model outperforms those of previous some forecasting models with various orders and different interval lengths.

Adjusted Rand index (ARI) is a measure of agreement between two partitions: one is the **clustering** result and the other is the standard partition. The value of ARI va- ries from 0 to 1 and higher value means that the cluster- ing result is more similar to the standard partitions. Sup- pose T is the true **clustering** of a gene expression data set based on domain knowledge and C a **clustering** result given by some **clustering** algorithm. Let a denote the number of gene pairs belonging to the same cluster in both T and C, b is the number of pairs belonging to the same cluster in T but to different clusters in C, c is the number of pairs belonging to different clusters in T but to the same cluster in C and d is the number of pairs be- longing to different clusters in both T and C.

We live in an era of abundant data. This has necessitated the development of new and innovative statistical algorithms to get the most from experimental data. For example, faster algorithms make practical the analysis of larger genomic data sets, allowing us to extend the utility of cutting-edge statistical methods. We present a randomised algorithm that accelerates the **clustering** of **time** **series** data using the Bayesian Hierarchical **Clustering** (BHC) statistical method. BHC is a general method for **clustering** any discretely sampled **time** **series** data. In this paper we focus on a particular application to microarray gene expression data. We define and analyse the randomised algorithm, before presenting results on both synthetic and real biological data sets. We show that the randomised algorithm leads to substantial gains in speed with minimal loss in **clustering** quality. The randomised **time** **series** BHC algorithm is available as part of the R package BHC, which is available for download from Bioconductor (version 2.10 and above) via http://bioconductor.org/packages/2.10/ bioc/html/BHC.html. We have also made available a set of R scripts which can be used to reproduce the analyses carried out in this paper. These are available from the following URL. https://sites.google.com/site/randomisedbhc/.

**series** set, a pattern **time** **series** also represents a tech- nique template, which indicates a representative **time** **series** segment in a **time** **series**. For different practical applications, the corresponding technique templates usually represent the special meaning. The aim of PTSS is to obtain the **time** **series** segments which are similar to a certain pattern template in the given pattern **time** **series** set. In this paper, we mainly focus on the study of PTSS. In this study, by comparing PTSS with the proto- type-based **clustering** (PC), we find that PTSS may be interpreted as an inverse problem of PC. According to this interpretation, a generalized model, called Cluster- ing-Inverse model (CI-model), is proposed for PTSS. Then, the main components and processing operations of this model are discussed in detail. Furthermore, a de- tailed algorithm is presented to put this model into prac- tice. As the matching measure of **time** **series** is of funda- mental importance in segmentation, we also propose a new Perceptually Important Point (PIP) based Dynamic **Time** Warping (DTW) measure by integrating PIP iden- tification mechanism [12] with DTW measure [13]. The advantage of the proposed measure is that it combines the merits of both PIP mechanism and DTW simulta- neously. To investigate the performance of the proposed CI-model, we have applied the proposed algorithms to the real-world **time** **series**.

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Finally, we show that our methods can be implemented efficiently: they are at most quadratic in each of their arguments, and are linear (up to log terms) in some formulations. To test the empirical performance of our algorithms we evaluated them on both synthetic and real data. To reflect the generality of the suggested framework in the experimental setup, we had our synthetic data generated by stationary ergodic process distributions that do not belong to any “simpler” class of distributions, and in particular cannot be modelled as hidden Markov processes with countable sets of states. In the batch setting, the error rates of both methods go to zero with sequence length. In the online setting with new samples arriving at every **time** step, the error rate of the offline algorithm remains consistently high, whereas that of the online algorithm converges to zero. This demonstrates that, unlike the offline algorithm, the online algorithm is robust to “bad” sequences. To demonstrate the applicability of our work to real-world scenarios, we chose the problem of **clustering** motion-capture sequences of human locomotion. This application area has also been studied in the works (Li and Prakash, 2011) and (Jebara et al., 2007) that (to the best of our knowledge) constitute the state-of-the-art performance 1 on the data sets they consider, and against which we compare the performance of our methods. We obtained consistently better performance on the data sets involving motion that can be considered ergodic (walking, running), and competitive performance on those involving non-ergodic motions (single jumps).

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The accurate measurement and forecasting of the volatility of financial markets is crucial for the economy of Tanzania due to the fact that the country depends significantly on imports and that important reserves are held in foreign ex- change, especially in USD Moreover, there is an increasing amount of foreign investment in Tanzania. This paper aims at examining the volatility of exchange rate in Tanzania. To achieve this goal the empirical analysis involves ARCH/ GARCH models, so that to investigate the major volatility characteristics ac- companied with exchange volatility. In the same vein, the paper applies an EGARCH model to capture the asymmetry in volatility **clustering** and the leve- rage effect in exchange rate for the period spanning from January 4, 2009 to July 27, 2015. The empirical results suggest that the conditional variance or volatility is quite persistent for TZS/USD returns. In particular, the results show that ex- change rate behaviour in Tanzania is generally influenced by previous informa- tion about exchange rate. In other words, results suggest existence of conditional heteroscedasticity or volatility **clustering**. In this case, the paper concludes that the exchange rates volatility can be adequately modeled by the GARCH (1,1) model. The results of the Mean Absolute Error (MAE) and the Root Mean Square Error (RMSE) for the forecasted volatility show that GARCH (1,1) has a predictive power. However, the fact that GARCH (1,1) is symmetric, an asym- metric model, EGARCH estimation results suggest the presence of leverage ef- fect in the exchange rate volatility. The policy implication of these results is that, the fact that exchange rate forecasting is very important to gauge the benefits and cost of international trade, policy makers should be aware of the possible ef- fect of asymmetry when modeling volatility of an exchange rate **series**. In fact, there are plenty of practical applications of the results of this paper. Future re- search includes macroeconomic effect of exchange rate volatility in developing countries such as Tanzania. Variables such as interest rate, international re- serves, trade flows and openness may be considered.

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AHC can be defined as a bottom-up approach in hierarchical **clustering**. It started by a single object in a set and merge with another object that has a similar characteristic [5], [6]. This process would be continued until the termination condition achieved and produce a set of the cluster as the endpoint [20], [27]. The result also can be seen as a tree, which can be plotted, as a dendrogram. At each iteration, the two clusters would be measured by a distance. The cluster with the shortest distance would be merge together. In AHC, the use of distance function would be differentiating the different between AHC. Therefore, this paper would be compared to the performance of four- distance function, including Minkowski, Euclidean, standard DTW, and modified DTW as similarity measures.

The PMC is tested with a subset of the gene expression data of the malaria intraerythrocytic developmental cycle [2]. This subset was chosen, because a functional interpretation of the genes is known and can be used to assess the clusterings. It comprises 530 genes in 14 functional groups. The gene expression is measured in 48 TPs with 1 hour **time** diﬀerences. The data set contained 0.32% of missing data and had a low noise level, which has been veriﬁed through [2] and by visually plotting many of the functional groups.

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forecasting problems in which the historical data are represented by linguistic values. Ref. [1], [2] proposed the **time**-invariant FTS and the **time**-variant FTS model which use the max–min operations to forecast the enrolments of the University of Alabama. However, the main drawback of these methods is enormous computation load. Then, Ref. [3] proposed the first-order FTS model by introducing a more efficient arithmetic method. After that, FTS has been widely studied to improve the accuracy of

Assume that, applying the procedure described above, we have fitted a sum of exponentials to the **time** profile mea- sured for a certain gene. Finding similarities between the ex- pressions of this gene and another gene expressions reduces to a comparison between the two sets of the estimated pa- rameters. At this point, a large family of comparison criteria can be considered. For example, we can first compare the es- timated structure parameters, and if both model orders are the same we can further compare the gains and the **time** con- stants, respectively. Since generally a microarray data set con- tains measurements for thousands of genes, it is prohibitive to consider all possible pairs of genes for finding similarities. We decide that two di ﬀ erent genes share common regula- tion if the set of **time** constants is the same for both of them, and we cluster the genes together. Observe that the proposed similarity measure for genes ignores the gains. We do not know the true values of the **time** constants, and the esti- mated values are all di ﬀ erent with probability one. We model the **time** constants estimated for all genes from a microar- ray data set as outcomes of a Gaussian mixture model, and cluster them in N TC clusters with classification-expectation-

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[16] Wang, N.-Y, & Chen, S.-M. Temperature prediction and TAIFEX forecasting based on automatic **clustering** techniques and two-factors high-order fuzzy **time** **series**. Expert Systems with Applications, 36, 2143–2154, 2009. [17] Kuo, I. H., Horng, S.-J., Kao, T.-W., Lin, T.-L., Lee, C.- L., & Pan. An improved method for forecasting enrollments based on fuzzy **time** **series** and particle swarm optimization. Expert Systems with applications, 36, 6108–6117, 2009a.

variety of algorithms and data structures that are only defined for discrete data, including hashing, Markov models, and suffix trees (Lin et al., 2007). Symbolic Aggregation approXmation (or SAX (Lin et al., 2003)) is the first symbolic represen- tation for **time** **series** that allows for dimensional- ity reduction and indexing with a lower-bounding distance measure at the same **time**. The SAX al- gorithm first transforms the original **time**-**series** into a representation with a lower **time** resolution, using Piecewise Aggregate Approximation tech- nique (PAA, (Keogh et al., 2001)) as an interme- diate step. Then it quantizes the pitch space us- ing an alphabet, therefore transforms the entire **time**-**series** into a string. It has two parameters: a word size(w=desired length of the symbolic fea- ture vector ) and an alphabet size (a), the latter being used to divide the pitch space of the con- tour into a equiprobable parts assuming a Gaus- sian distribution of F0 values (the breakpoints are obtained from a lookup statistical table). After we obtain the breakpoints, each segment of the **time**- **series** can be assigned a letter based on the alpha- bet bin it is in. Figure 1 shows an example of SAX transformation of a **time**-**series** of length 128 into the string ’baabccbc’.