In many emerging applications, data streams are monitored in a network environment. Due to limited communication bandwidth and other resource constraints, a critical and practical demand is to online compress data streams continuously with quality guarantee. Although many data compression and digital signal processing methods have been developed to reduce data volume, their super-linear time and more-than-constant space complexity prevents them from being applied directly on data streams, particularly over resource-constrained sensor networks. In this paper, we tackle the problem of online quality guaranteed compression of data streams using fast linear approximation (i.e., using line segments to approximate a time series). Technically, we address two versions of the problem which explore quality guarantees in different forms. We develop online algorithms with linear time complexity and constant cost in space. Our algorithms are optimal in the sense they generate the minimum number of segments that approximate a time series with the required quality guarantee. To meet the resource constraints in sensor networks, we also develop a fast algorithm which creates connecting segments with very simple computation. The low cost nature of our methods leads to a unique edge on the applications of massive and fast streaming environment, low bandwidth networks, and heavily constrained nodes in computational power. We implement and evaluate our methods in the application of an acoustic wireless sensor network.
@cite_0 , an amnesic function is introduced to give weights to different points in the time series. The PLA-SegmentBound problem is discussed in the context of Unrestricted Window with Absolute Amnesic (UAA) problem, but complete solutions to this problem are not provided in @cite_0 .
@cite_0
2008348094 2140916841 2115366404 1552695456
Dynamic time warping (DTW), which finds the minimum path by providing non-linear alignments between two time series, has been widely used as a distance measure for time series classification and clustering. However, DTW does not account for the relative importance regarding the phase difference between a reference point and a testing point. This may lead to misclassification especially in applications where the shape similarity between two sequences is a major consideration for an accurate recognition. Therefore, we propose a novel distance measure, called a weighted DTW (WDTW), which is a penalty-based DTW. Our approach penalizes points with higher phase difference between a reference point and a testing point in order to prevent minimum distance distortion caused by outliers. The rationale underlying the proposed distance measure is demonstrated with some illustrative examples. A new weight function, called the modified logistic weight function (MLWF), is also proposed to systematically assign weights as a function of the phase difference between a reference point and a testing point. By applying different weights to adjacent points, the proposed algorithm can enhance the detection of similarity between two time series. We show that some popular distance measures such as DTW and Euclidean distance are special cases of our proposed WDTW measure. We extend the proposed idea to other variants of DTW such as derivative dynamic time warping (DDTW) and propose the weighted version of DDTW. We have compared the performances of our proposed procedures with other popular approaches using public data sets available through the UCR Time Series Data Mining Archive for both time series classification and clustering problems. The experimental results indicate that the proposed approaches can achieve improved accuracy for time series classification and clustering problems. We present polylogarithmic approximations for the R|prec|Cmax and R|prec|∑jwjCj problems, when the precedence constraints are “treelike” – i.e., when the undirected graph underlying the precedences is a forest. We also obtain improved bounds for the weighted completion time and flow time for the case of chains with restricted assignment – this generalizes the job shop problem to these objective functions. We use the same lower bound of “congestion+dilation”, as in other job shop scheduling approaches. The first step in our algorithm for the R|prec|Cmax problem with treelike precedences involves using the algorithm of Lenstra, Shmoys and Tardos to obtain a processor assignment with the congestion + dilation value within a constant factor of the optimal. We then show how to generalize the random delays technique of Leighton, Maggs and Rao to the case of trees. For the weighted completion time, we show a certain type of reduction to the makespan problem, which dovetails well with the lower bound we employ for the makespan problem. For the special case of chains, we show a dependent rounding technique which leads to improved bounds on the weighted completion time and new bicriteria bounds for the flow time. The past decade has seen a wealth of research on time series representations, because the manipulation, storage, and indexing of large volumes of raw time series data is impractical. The vast majority of research has concentrated on representations that are calculated in batch mode and represent each value with approximately equal fidelity. However, the increasing deployment of mobile devices and real time sensors has brought home the need for representations that can be incrementally updated, and can approximate the data with fidelity proportional to its age. The latter property allows us to answer queries about the recent past with greater precision, since in many domains recent information is more useful than older information. We call such representations amnesic. While there has been previous work on amnesic representations, the class of amnesic functions possible was dictated by the representation itself. We introduce a novel representation of time series that can represent arbitrary, user-specified amnesic functions. For example, a meteorologist may decide that data that is twice as old can tolerate twice as much error, and thus, specify a linear amnesic function. In contrast, an econometrist might opt for an exponential amnesic function. We propose online algorithms for our representation, and discuss their properties. Finally, we perform an extensive empirical evaluation on 40 datasets, and show that our approach can efficiently maintain a high quality amnesic approximation. Similarity-based search over time-series databases has been a hot research topic for a long history, which is widely used in many applications, including multimedia retrieval, data mining, web search and retrieval, and so on. However, due to high dimensionality (i.e. length) of the time series, the similarity search over directly indexed time series usually encounters a serious problem, known as the "dimensionality curse". Thus, many dimensionality reduction techniques are proposed to break such curse by reducing the dimensionality of time series. Among all the proposed methods, only Piecewise Linear Approximation (PLA) does not have indexing mechanisms to support similarity queries, which prevents it from efficiently searching over very large time-series databases. Our initial studies on the effectiveness of different reduction methods, however, show that PLA performs no worse than others. Motivated by this, in this paper, we re-investigate PLA for approximating and indexing time series. Specifically, we propose a novel distance function in the reduced PLA-space, and prove that this function indeed results in a lower bound of the Euclidean distance between the original time series, which can lead to no false dismissals during the similarity search. As a second step, we develop an effective approach to index these lower bounds to improve the search efficiency. Our extensive experiments over a wide spectrum of real and synthetic data sets have demonstrated the efficiency and effectiveness of PLA together with the newly proposed lower bound distance, in terms of both pruning power and wall clock time, compared with two state-of-the-art reduction methods, Adaptive Piecewise Constant Approximation (APCA) and Chebyshev Polynomials (CP).