(a) The sequence for Dow Jones indices during 500 trading days. (b) The corresponding energy spectrum. 

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... number of DCT coefficients increases exponentially as the dimensionality increases. Clearly, computing all coefficients for possible selection is computationally prohibitive. Therefore, the DCT-based approaches choose and compute only the coefficients that are deemed the most representative. In practical OLAP systems, the data distribution should have a correlation among data items; that is, the frequency spectrum of the distribution should show large values in its low frequency coefficients and small values in its high frequency coefficients [8], [10]. In general, high- frequency coefficients are usually of less interest in OLAP, whereas low-frequency coefficients are of high interest, since they correspond to range-aggregation queries [20]. To select representative coefficients, several geometrical zonal sampling techniques such as triangular, spherical, rectangular, and reciprocal sampling have been proposed [21]. Lee et al. [10] extends the techniques to MDSs and shows that the reciprocal zonal sampling technique is able to select the most representative coefficients of DCT under the brown noise assumption. This sampling technique selects coefficients by the constraint f k ! u j Q d i 1⁄4 1 ð u i þ 1 Þ b g , where k ! u is a DCT coefficient located at ð u 1 ; u 2 ; . . . u d Þ , and u i is the frequency index in the i th dimension. Since only the most efficient coefficients need to be transformed, the execution time of DCT for data is reduced to O ð nk Þ , where k is the number of selected coefficients. Thus, the reciprocal zonal sampling technique improves the execution efficiency significantly. For range-sum queries, it is expensive to estimate the value of individual data points and then compute the aggregation. Lee et al. [10] introduced the integral approach to compute the aggregation efficiently. For a d -dimensional range-sum query, it costs O ð kd Þ to compute the aggregation if k coefficients are used. For example, the answer of the range-sum query requesting the sum of those cells located at be a estimated < u 1 < b and as c S < 1⁄4 u 2 R < c d R d a b in f b ð u an 1 ; u M 2 Þ du  N 1 du 2D 2 ; , data where cube f b can is defined in Section 2.2. Though not proper for estimating single data point [22], the DCT-based approaches are able to provide high-quality answers for range-sum queries due to the compensation of errors of some cell pairs. Because it does not require extra storage space for coefficient transforming, the DCT works well with small working space. However, the time complexity, estimated as O ð nB Þ , is deemed too high for the SCS applications to process the MDS blocks online. To resolve the issues of the space cost in DWT and the time cost in DCT, we propose the DAWA algorithm, which stands for an integrated algorithm of DCT for Data and DWT, to approximate the SCS. The DAWA algorithm is able to generate a snapshot from a PB in a small working buffer of only via O one ð p nB ffiffiffi N þ data B log scan, p ffiffiffiffi N with Þ , which a computational we discuss complexity further in Sections 3.2 and 3.3. The DAWA framework comprises two phases for generating snapshots. The first phase, called the DCT phase , partitions a PB into several subblocks, called DAWA cells, and then applies DCT to each DAWA cell based on the optimal set of coefficients selected by the reciprocal zonal sampling. The second phase, called the DWT phase , groups the DCT coefficients from each DAWA cell, whose time-to-frequency transformations are close to each other, into several isofrequency MDSs, denoted as IMS. As it will be shown in Section 3.1, the variance of data points in the same IMS is small, and the IMS can thus be further compressed by DWT efficiently. Also, several techniques, including null map, IMS smoothing, and global thresholding, have been developed based on the character- istics of IMS to improve the accuracy of the DAWA algorithm. The essence of DCT or DWT is based on the effectiveness of power concentration of the transformations. Basically, brown noise and random walks are the prevalent formats in real signals [23]. For brown noise, the energy is concentrated more in low frequencies, since the data points are more related to each other than those in white noise generated by the pure random process. Note that real signals have a skewed energy spectrum [8]. For example, stock movements and exchange rates can be successfully modeled as brown noise, which exhibit an energy spectrum O ð f À 2 Þ . Therefore, the DCT coefficients of such data, referred to as the amplitudes of the signals, can be modeled as O ð f À 1 Þ [8]; that is, the amplitude is inversely proportional to the frequency. A scenario of the brown noise assumption for cube streams is shown in Fig. 4, where it can be seen that the amplitudes are inversely approximately proportional to the frequencies. Also, the relationship between power density and frequencies can be explored mathematically to provide a theoretical foundation for zonal sampling. Consequently, we have the following ...

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... can be successfully modeled as brown noise, which exhibit an energy spectrum Oðf À2 Þ. Therefore, the DCT coefficients of such data, referred to as the amplitudes of the signals, can be modeled as Oðf À1 Þ [8]; that is, the amplitude is inversely proportional to the frequency. A scenario of the brown noise assumption for cube streams is shown in Fig. 4, where it can be seen that the amplitudes are inversely approximately proportional to the ...

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... shown in Fig. 14, the quality of answers from the DAWA algorithm is much better than that from DWT. The error rates of the answers from DAWA are all below 5 percent for both D-TPC and D-TEL, showing that DAWA works well under a small working buffer and is able to generate very small snapshots for the original data sets, with acceptable error rates. In ...