Figure 5.9 and 5.10 indicate that results were not degraded using DTW-aided MDPLA for any of the 40 dataset/dissimilarity measure pairs. This is especially remarkable when it is compared to any other method that targets to make DTW better by eliminating pathological warpings. For example, application of R-K band [10]
degrades the results for 6 of the 20 datasets compared to unconstrained DTW for the same 1-NN test [77] using the original data. This definitely suggests that a properly selected local dissimilarity measure can prevent pathological warping as effective as any constraint.
In Section 5.1, it was demonstrated by a theoretical example how MDPLA can help to avoid unwanted alignments, now it is demonstrated empirically. The best way to determine whether a warping path is pathological or not is to inspect it manually, which would be intractable for the UCR datasets. However, considering the fact that all the previously reviewed constraints assume that each non-diagonal movement facilitates unwanted alignments and thus such movements must be minimized, it is possible to define Tightness of Warpingfor local dissimilarity measuredm and datasetA:
T oW(A, dm) =
P(|A|2)
i=1 li
P(|A|2 )
i=1 max(mi, ni)
, (5.4)
where|A|denotes the number of time series in datasetAin which all time series are compared to each other with DTW using local dissimilarity measuredm. Warping path lengths (li) resulted from these comparisons are summarized and divided by the sum of the minimum warping path lengths. Minimum warping path length for theithcomparison equals the length of the diagonal of the warping matrix, i.e. the maximum of the lengths of time series used in theithcomparison (mi,ni).
T oW(A, dx) = 1 means that all the warping path lies on the diagonal, while T oW(A, dy) = 3indicates that local dissimilarity measuredy on the average triples the warping path in length and it most probably introduces pathological warpings.
UsingT oW, it is possible to compare local dissimilarity measures from warping point of view. All requirements are that the same constraints (local or global, penalties, etc.) and the same time series (or representation) shall be used. These requirements were fulfilled in Section 5.2.1 when MDPLA performance was assessed using the segmentations given by MPLA and SPLA. Table 5.2 contains theT oW values calculated using these segmentations. Warping path reduction was also
Table 5.2. Effect of the applied local dissimilarity measure on the length of the warping path. Reduction shows how much the warping path was reduced (plus), or enlarged (minus) in percentage term. 100% would show that the warping path is reduced to its minimum, i.e. it is always located on the diagonal.
Dataset
Segment number trained for MPLA Segment number trained for SPLA Tightness of
Syn Ctrl 1.439 1.199 54.68% 1.176 1.066 62.31%
Gun-Point 1.163 1.107 34.73% 1.224 1.224 0.00%
CBF 1.420 1.261 37.93% 1.186 1.110 40.92%
Face(all) 1.301 1.153 49.15% 1.171 1.161 6.10%
OSULeaf 1.334 1.142 57.63% 1.205 1.196 4.10%
SwedishLeaf 1.327 1.184 43.66% 1.177 1.138 21.89%
50Words 1.427 1.274 35.87% 1.287 1.282 2.03%
Trace 1.292 1.275 5.77% 1.266 1.266 0.00%
TwoPatterns 1.264 1.216 18.00% 1.178 1.168 5.99%
Wafer 1.368 1.335 8.82% 1.412 1.350 15.07%
Face(four) 1.277 1.140 49.50% 1.163 1.163 0.00%
lightning-2 1.221 1.155 29.87% 1.309 1.265 14.15%
lightning-7 1.300 1.227 24.32% 1.221 1.169 23.31%
ECG 1.296 1.193 34.85% 1.165 1.150 8.79%
Adiac 1.108 1.089 17.26% 1.159 1.108 32.12%
Yoga 1.266 1.200 24.69% 1.228 1.222 2.61%
Fish 1.125 1.117 6.34% 1.213 1.168 21.31%
Beef 1.329 1.169 48.57% 1.152 1.152 0.00%
Coffee 1.370 1.096 74.04% 1.077 1.077 0.00%
OliveOil 1.028 1.027 1.93% 1.121 1.121 0.00%
calculated in percentage term, i.e. 100%and−100%would show when the warping path was reduced to the diagonal or when it was doubled, respectively. Looking at the table, it can be seen that application ofdM DP LA as local dissimilarity measure really shortened the warping path in most cases — and what is also important — the warping path was never enlarged.
5.4 Conclusion
Finding a suitable time series representation alongside the matching dissimilarity measure has always been — and most probably always will be — a crucial task in time series data mining. While in many cases it is tempting to use DTW with any of the off-the-self distance measures created for a given representation, local dissimilarity measures consider one feature only often leaves place for improvement.
The presented novel mixed dissimilarity measure for piecewise linear approx-imation (MDPLA) combined two features of a PLA segment, its mean and its trend. MDPLA was validated on the datasets of the UCR time series classifica-tion/clustering repository and it was shown that considering both of these features
yields to superior results over classical measures that consider one feature only. In addition, MDPLA provides more advantages for practicing engineers who deal with PLA-based systems. In case of an existing PLA-based workflow, MDPLA does not require resegmentation and the non-warped version of MDPLA can excel warped versions of classical measures in many cases lowering runtime.
Finally, it was empirically demonstrated that MDPLA shortens the warping path of DTW by considering two different features for comparison and this way it lowers the chance of unwanted, pathological alignments often created by DTW when only one feature is taken into account.
Chapter 6
Summary
In the last decades, the availability of easily accessible computational resources and storage capacities has made it possible to apply classical data mining techniques on time series data. Such methods have rapidly gained popularity in engineering applications, thanks to their ease of use, effectiveness and the fact that several time independent problems can easily be formulated and solved as a time series related task.
Over the time, however, it became clear that dissimilarity measures inherited from classical data mining methods are not well suited for time series data because they do not take into account the nonlinear fluctuation of the time axis. To overcome this problem, elastic dissimilarity measures were proposed, however, the expected results still could not be achieved in several applications.
This thesis provides solutions for time series similarity problems that previously could not be handled with elastic dissimilarity measures. The presented dissimilarity measures allow correlation and process dynamics-based data mining of multivariate time series in an elastic manner, and it was also shown how global constraints of dynamic time warping can be omitted while pathological warpings are still avoided.
6.1 New scientific results
Thesis I By combining principal component analysis-based segmentation and dy-namic time warping, I defined a novel dissimilarity measure for multivariate time series that does not only fulfill expectations regarding similarity but also provides better results than the existing methods in case of multivariate time series with complex correlation structure.
(a) During comparison of multivariate time series, neither dissimilarity mea-sures derived from univarite time series comparison nor correlation-based methods consider the alternation of correlation structure alongside the shifts of the time axis. In order to address this problem of both methods, I proposed a novel dissimilarity measure: correlation-based time warping.
(b) I utilized principal component analysis to create homogeneous segments from correlation point of view. The dissimilarity of the segments was originated from the principal component analysis similarity factor and to make the comparison method resistant to fluctuation of the time axis, I applied dynamic time warping.
(c) Using freely available verification repositories, I proved that the presented algorithm provides better results than the currently used dissimilarity measures in case of multivariate time series with complex correlation structure.
Related publications: 1, 3, 8, 13
Thesis II To enable the comparison of multivariate time series based on their dynam-ics, I demonstrated that multivariate time series can be segmented according to the changes in process dynamics by replacing principal component analy-sis with its dynamic version in my correlation-based dynamic time warping algorithm.
(a) Classical time series comparison methods — including principal com-ponent analysis-based techniques — do not consider changes in process dynamics, i.e. they cannot be used to analyze process dynamics. To overcome this limitation, I replaced principal component analysis with dynamic principal component analysis in the algorithm I have defined in Thesis I and thus have put the focus on process dynamics.
(b) I proved that it is possible to segment multivariate time series according to its dynamics via the aforementioned enhancement. In this way, the modified version of the algorithm presented in Thesis I became capable to compare and data mine multivariate time series based on process dynamics.
Related publications: 2, 6, 11, 12, 13
Theis III Inspired by correlation-based dynamic time warping, I introduced a novel similarity measure for time series segmented using piecewise linear approxi-mation. The presented similarity measure can be combined with the classical, mean-based similarity measure to achieve better results than that of the existing methods. Moreover, using this combined similarity measure, I empirically proved that similarity measures considering multiple features shorten the warping path and thus they reduce the possibility of pathological warpings.
(a) Application of global constraints plays an important role during dynamic time warping because it considerably influences the results. I showed in case of piecewise linear approximated time series that by using a local dissimilarity measure incorporates multiple features, the length of the warping path significantly shortens and thus the possibility of pathological warping is reduced.
(b) In addition to the aforementioned, the introduced dissimilarity measure has another advantages. Utilizing the most popular time series verifica-tion repository, I showed that the introduced dissimilarity measure can significantly reduce the run-time or increase the precision without the need of changing the segmentation.
Related publications: 2, 4, 5, 7, 9, 10, 11, 12, 13