Effective and Efficient Variable-Length Data Series Analytics
EEffective and EfficientVariable-Length Data Series Analytics
Michele LinardiSupervised by: Themis Palpanas
LIPADE, Universit ´e de Paris [email protected]
ABSTRACT
In the last twenty years, data series similarity search hasemerged as a fundamental operation at the core of severalanalysis tasks and applications related to data series collec-tions. Many solutions to different mining problems work bymeans of similarity search. In this regard, all the proposedsolutions require the prior knowledge of the series length onwhich similarity search is performed. In several cases, thechoice of the length is critical and sensibly influences thequality of the expected outcome. Unfortunately, the obvi-ous brute-force solution, which provides an outcome for alllengths within a given range is computationally untenable.In this Ph.D. work, we present the first solutions that inher-ently support scalable and variable-length similarity searchin data series, applied to sequence/subsequences matching,motif and discord discovery problems. The experimental re-sults show that our approaches are up to orders of magnitudefaster than the alternatives. They also demonstrate that wecan remove the unrealistic constraint of performing analyt-ics using a predefined length, leading to more intuitive andactionable results, which would have otherwise been missed.
1. INTRODUCTION
Data series (i.e., ordered sequences of points) are oneof the most common data types , present in almost ev-ery scientific and social domain (such as meteorology, as-tronomy, chemistry, medicine, neuroscience, finance, agri-culture, entomology, sociology, smart cities, marketing, op-eration health monitoring, human action recognition andothers) [20].Once the data series have been collected, the domain ex-perts face the arduous tasks of processing and analyzingthem [30, 6] in order to gain insights, e.g., by identifying sim-ilar patterns, and performing classification, or clustering. Acore operation that is part of all these analysis tasks is sim-ilarity search, which has attracted lots of attention becauseof its importance [2]. Nevertheless, all existing scalable andindex-based similarity search techniques are restricted inthat they only support queries of a fixed length, and they If the dimension that imposes the ordering of the sequenceis time then we talk about time series. Though, a seriescan also be defined over other measures (e.g., angle in radialprofiles in astronomy, mass in mass spectroscopy in physics,position in genome sequences in biology, etc.). We use theterms data series , time series , and sequence interchangeably. Paper published in the Ph.D. Workshop - VLDB Conference 2019 require that this length is chosen at index construction [10,24, 1, 25, 28, 29, 26, 21, 11]. The same observation holdsfor techniques proposed to discover motifs [12] and discords(i.e., anomalous subsequences) [27]: they all assume a fixedsequence length, which has to be predefined.Evidently, this is a constraint that penalizes the flexibilityneeded by analysts, who often times need to analyze pat-terns of slightly different lengths (within a given data seriescollection) [7, 8, 5, 17, 16]. For example, in the
SENTINEL-2 mission data, oceanographers are interested in searchingfor similar coral bleaching patterns of different lengths; atAirbus engineers need to perform similarity search queriesfor patterns of variable length when studying aircraft take-offs and landings [19]; and in neuroscience, analysts need tosearch in Electroencephalogram (EEG) recordings for CyclicAlternating Patterns (CAP) of different lengths (duration),in order to get insights about brain activity during sleep [22].In our work, we focus on three core problems that arebased on similarity search: subsequence matching, and mo-tif and discord discovery, organized under the ULISSE andMAD methods:1. ULISSE (ULtra compact Index for variable-length Sim-ilarity SEarch in data series) is the first indexing techniquethat supports variable-length subsequence matching for nonZ-normalized and Z-normalized data series [15, 13, 14].2. MAD (Motif and Discord discovery framework) im-plements two novel algorithms for variable-length motif anddiscord discovery in large data series [17, 4, 16].
2. VARIABLE-LENGTH ANALYTICS
In this section, we describe our proposed approaches tothe aforementioned problems. In the next part we describethe notions and the elements used in our solutions.
Preliminaries.
Let a data series D = d ,..., d | D | be a se-quence of numbers d i ∈ R , where i ∈ N represents the posi-tion in D . We denote the length, or size of the data series D with | D | . The subsequence D s,(cid:96) = d s ,..., d s + (cid:96) − of length (cid:96) , is a contiguous subset of (cid:96) points of D starting at offset s ,where 1 ≤ s ≤ | D | and 1 ≤ (cid:96) ≤ | D | . A subsequence is itselfa data series. A data series collection, C , is a set of dataseries. We say that a data series D is Z-normalized, denoted D n , when its mean µ is 0 and its standard deviation σ is1. Z-normalization is an essential operation in several ap-plications, because it allows similarity search irrespective ofshifting and scaling [5]. The Piecewise Aggregate Approx-imation (PAA) of a data series D , P AA ( D ) = { p , ..., p w } , a r X i v : . [ c s . D B ] S e p epresents D in a w -dimensional space by means of w real-valued segments of length s , where the value of each segmentis the mean of the corresponding values of D [9]. We denotethe first k dimensions of P AA ( D ), ( k ≤ w ), as P AA ( D ) ,..,k .The iSAX representation of a data series D , denoted by iSAX ( D, w, | alphabet | ), is the representation of P AA ( D ) by w discrete coefficients, drawn from an alphabet of cardinal-ity | alphabet | [24]. The main idea of the iSAX represen-tation, is that the real-value space may be segmented by | alphabet | − | alphabet | regions, which arelabeled by discrete symbols (e.g., with | alphabet | = 4 theavailable labels may be { , , , } ). The subsequence matching problem is defined as follows:Given a data series collection C = { D , ..., D C } , a se-ries length range [ (cid:96) min , (cid:96) max ], a query data series Q , where (cid:96) min ≤ | Q | ≤ (cid:96) max , and k ∈ N , we want to find theset R = { D io,(cid:96) | D i ∈ C ∧ (cid:96) = | Q | ∧ ( (cid:96) + o − ≤ | D i |} ,where | R | = k . We require that ∀ D io,(cid:96) ∈ R (cid:64) D i (cid:48) o (cid:48) ,(cid:96) (cid:48) s.t.dist ( D i (cid:48) o (cid:48) ,(cid:96) (cid:48) , Q ) < dist ( D io,(cid:96) , Q ), where (cid:96) (cid:48) = | Q | , ( (cid:96) (cid:48) + o (cid:48) − ≤| D i (cid:48) | and D i (cid:48) ∈ C . We informally call R , the k nearest neigh-bors set of Q . Given two generic series of the same length,namely D and D (cid:48) the function dist ( D, D (cid:48) ) can be EuclideanDistance or Dynamic Time Warping.
Variable Length Subsequences.
In a data series, whenwe consider contiguous and overlapping subsequences of dif-ferent lengths within the range [ (cid:96) min , (cid:96) max ],we expect theoutcome as a bunch of similar series, whose differences areaffected by the misalignment and the different number ofpoints. Given a data series D , and a subsequence lengthrange [ (cid:96) min , (cid:96) max ], we define the master series as the sub-sequences of the form D i,min ( | D |− i +1 ,(cid:96) max ) , for each i suchthat 1 ≤ i ≤ | D | − ( (cid:96) min − ≤ (cid:96) min ≤ (cid:96) max ≤ | D | .We observe that for any master series of the form D i,(cid:96) (cid:48) , wehave that P AA ( D i,(cid:96) (cid:48) ) ,..,k = P AA ( D i,(cid:96) (cid:48)(cid:48) ) ,..,k holds for each (cid:96) (cid:48)(cid:48) such that (cid:96) (cid:48)(cid:48) ≥ (cid:96) min , (cid:96) (cid:48)(cid:48) ≤ (cid:96) (cid:48) ≤ (cid:96) max and (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) % k = 0.Therefore, by computing only the P AA of the master se-ries in D , we are able to represent the P AA prefix of any sub-sequence of D . When we zero-align the P AA summaries ofthe master series, we compute the minimum and maximum
P AA values (over all the subsequences) for each segment:this forms what we call an
Envelope . (When the length of amaster series is not a multiple of the
P AA segment length,we compute the
P AA coefficients of the longest prefix that ismultiple of a segment.) We call containment area the spacein between the segments that define the
Envelope . PAA Envelope.
We formalize the concept of the
Envelope ,introducing a new series representation. We denote by L and U the P AA coefficients, which delimit the lower and upperparts, respectively, of a containment area. Furthermore, weintroduce a parameter γ , which permits to select the numberof master series we represent by the Envelope . We refer to itusing the following signature: paaENV [ D,(cid:96) min ,(cid:96) max ,a,γ,s ] =[ L, U ]. It delimits the containment area generated by the
P AA coefficients of the master series.
Indexing the Envelopes.
Given a paaENV , wecan translate its
P AA extremes into the correspond-ing iSAX representation: uENV paaENV [ D,(cid:96)min,(cid:96)max,a,γ,s ] =[ iSAX ( L ) , iSAX ( U )], where iSAX ( L ) ( iSAX ( U )) is thevector of the minimum (maximum) P AA coefficients of allthe segments corresponding to the subsequences of D . The C u m u l a t i v e Q u e r y T i m e ( h o u r s ) γ = (% of ( l max - l min ))query answering disk i\oquery answering cpuIndexing (disk i\o + cpu time)050100150200 160 192 224 256 A v g E x a c t Q u e r y T i m e C P U + d i s k I / O ( S e c s ) Query length
0% 20%40% 60%80% 100% (a) (b) γ Figure 1: Query answering time performance, vary-ing γ on non Z-normalized data series. (a) ULISSE average query time (CPU + disk I/O). (b) ULISSE average query disk I/O time. (b) Comparison of
ULISSE to other techniques (cumulative indexing+ query answering time).
Envelope uENV represents the principal building block ofthe
ULISSE
Index. In details,
ULISSE is a tree structure,where each internal node stores the
Envelope uENV repre-senting all the sequences in the subtree rooted at that node.Leaf nodes contain several
Envelopes , which by constructionhave the same iSAX ( L ). On the contrary, their iSAX ( U )varies, since it get updated with every new insertion in thenode. Each Envelope in leaf nodes point the the representedsequences in the original data series collection.
Approximate Subsequence Matching.
Subsequencematching performed on
ULISSE index relies on the mindist
ULiSSE () lower bounding function to prune thesearch space. This allows to navigate the tree in order, vis-iting first the most promising nodes. As soon as a leaf nodeis discovered, we can load the raw data series pointed bythe
Envelopes in the leaf. Each time we compute the trueEuclidean or DTW distance between the series in a leaf, thebest-so-far distance (bsf) is updated, along with a vectorcontaining the k best matches, where k refers to the k near-est neighbors. Since priority is given to the most promisingnodes, we can terminate our visit, when at the end of a leafvisit the k bsf’s have not improved. Exact Subsequence Matching.
Note that the approxi-mate search described above may not visit leaves that con-tain answers better than the approximate answers alreadyidentified, and therefore, it will fail to produce exact, correctresults.The exact nearest neighbor search algorithm we proposefinds the k sequences with the absolute smallest distancesto the query. In this case, the search algorithm may visitseveral leaves: the process stops after it has either visited,or pruned (when the lower bounding distance to the node isgreater than the bsf) all the nodes of the index, guaranteeingthe correctness of the results. Experiments.
To evaluate
ULISSE , we used synthetic andreal data (but in the interest of space we only report resultswith the synthetic data). We record the average
CPU time , disk I/O (time to fetch data from disk (Total time - CPUtime)), for queries, extracted from the datasets with theaddition of Gaussian noise. We compare ULISSE with
UCRsuite [5] the non index-based state-of-the-art technique foranswering similarity search queries. Concerning the com-petitor indexing techniques, the state-of-the-art is the Com-pact Multi Resolution Index [7]
CMRI .In Figure 1, we present results for subsequence matchingqueries on
ULISSE when we vary γ , ranging from to itsmaximum value in this dataset, i.e., (cid:96) max − (cid:96) min . In Fig-ure 1, we report the results concerning non Z-normalized se-ries. We observe that grouping contiguous and overlappingubsequences under the same summarization ( Envelope ) byincreasing γ , affects positively the performance of index con-struction, as well as query answering.The latter may seem counterintuitive, since inserting moremaster series into a single Envelope is likely to generate largecontainment areas, which are not tight representations of thedata series. On the other hand, it leads to an overall numberof
Envelope that is several orders of magnitude smaller thanthe one for γ = 0%, where only a single master series isrepresented by each Envelope . Motif and Discord are data mining primitives that rep-resent frequent and rare (anomalous) patterns, respectively.Given a data series D, they are defined as follows: • Data series motif: D a,(cid:96) and D b,(cid:96) is a motif pair iff dist ( D a,(cid:96) , D b,(cid:96) ) ≤ dist ( D i,(cid:96) , D j,(cid:96) ) ∀ i, j ∈ [1 , , ..., | D | − (cid:96) + 1], where a (cid:54) = b and i (cid:54) = j , and dist is a functionthat computes the z-normalized Euclidean distance be-tween the input subsequences. • Data series discord: We call the k subsequences of D , with the k largest distances to their m th NearestNeighbor (according the Euclidean distance), the
Top-k m th discords. Variable length motif and discord discovery.
We pro-vide solutions to the following problems: • Variable-Length Motif Discovery: Given a data series D and a subsequence length-range [ (cid:96) min , ..., (cid:96) max ], wewant to find the data series motif pairs of all lengthsin [ (cid:96) min , ..., (cid:96) max ], occurring in D . • Variable-Length
Top-k m th Discord Discovery: Givena data series D , a subsequence length-range[ (cid:96) min , ..., (cid:96) max ] and the parameters a, b ∈ N + we wantto enumerate the Top-k m th discords for each k ∈{ , .., a } and each m ∈ { , .., b } , and for all lengthsin [ (cid:96) min , ..., (cid:96) max ], occurring in D . Fixed length motif and discord discovery.
The state-of-the art algorithm for fixed length motif and discord dis-covery [3] requires the user to define the length of the de-sired motif or discord. This mining operation is supportedby computation of the
Matrix profile , which is a meta dataseries storing the z-normalized Euclidean distance betweeneach subsequence and its nearest neighbor. The Matrix pro-file does not only derive the motif, but also ranks and filtersout the other pairs, giving also a convenient and graphicalrepresentation of their occurrences and proximity. Unfortu-nately, this technique comes with an important shortcoming:it does not provide an effective solution for trying several dif-ferent motif lengths. Therefore, the analyst is forced to runthe algorithm using all possible lengths in a range of interest,and rank the various motifs discovered, picking eventuallythe patterns that contain the desired insight. Clearly, thispossibility is not optimal for at least two reasons: the scal-ability, since finding motif of one fixed length takes O ( | D | )time, and also because it does not provide an effective wayto compare motifs of different lengths. MAD Framework.
Our framework for Variable LengthMotif and Discord Discovery (MAD) works by applying an incremental computing strategy, which aims to prune unnec-essary distance computations for larger motif and discord lengths. Hence, given a data series D , we compute the Ma-trix profile using the smallest subsequence length, namely (cid:96) min , within a specified input range [ (cid:96) min , (cid:96) max ]. The keyidea of our approach is to minimize the work that needsto be done for succeeding subsequence lengths ( (cid:96) min + 1, (cid:96) min + 2, . . . , (cid:96) max ). Matrix Profile Computation.
We start the computationof the Matrix profile, considering all the contiguous subse-quences of length (cid:96) min , computing for each one the
Distanceprofile in O ( | D | ) time. This latter is a vector that containsthe z-normalized Euclidean distances between a fixed sub-sequence and all the other in D (excluding trivial matches). Lower Bound Subsequences of Different Length.
Wemoreover introduce a new lower bounding distance [17],which lower bounds (is always smaller than) the true Eu-clidean distances between subsequences longer than (cid:96) min .We initially compute this lower bound using the true Eu-clidean distances computation of subsequences with length (cid:96) min . For the larger lengths, we update the lower bound,considering only the variation generated by the trailingpoints in the longer subsequences. This measure enjoys animportant property: if we rank the subsequences accordingto this measure, the same rank will be preserved along all thelower bound updates for the subsequences of greater length.We exploit this property, in order to prune computations.
Pruning the Search Space.
Once we compute motif anddiscords, with length greater than (cid:96) min , instead of comput-ing from scratch each distance profile, we update the truedistances (in constant time) of the subsequences that havethe p smallest lower bounding distances (computed in theprevious step). These distances form what we call partialdistance profile . In each partial distance profile, we also up-date the lower bound. After this operation, we may have twocases: if in a new computed distance profile the minimumtrue distance ( minDist ) is shorter than the maximum lowerbound ( maxLB ), we know that no distances among thosenot computed can be smaller than minDist. In this case, apartial distance profile becomes a valid distance profile . Onthe other hand, when maxLB is smaller than minDist , thislatter is not guaranteed to be the nearest neighbor distance.For discord discovery, we need to test this condition for the m smallest true distances in the partial distance profile. Inthis case a valid (partial) distance profile must contain thetrue m th best match distances, which are smaller than, orequal to maxLB . Exact Motif and Discord Discovery.
Once the partialdistance profiles are computed, we pick the absolute smallestlower bounding value from all the non-valid distance profiles,namely minLBAbs (if any). Therefore, the global minimum(true) distance of all the valid (partial) distance profiles,which is smaller than minLBAbs is guaranteed to be thedistance between the motif pair subsequences. Symmetri-cally, we consider the valid (partial) distance profiles to findthe true m th best match distances, which are the greatestnearest neighbor distances that are larger than maxLBAbs .This latter is the largest lower bounding distance of the non-valid distance profiles.In the motif discovery task, if no nearest neighbor dis-tance is smaller than minLBAbs , we recompute only thedistance profiles that have the maxLB distance smaller thanthe smallest true distance computed.On the other hand, for discord discovery, if no true near-est neighbor distances are found we need to iterate the non
100 150 200 400 600 T i m e ( H o u r s ) Subsequence length range
ECG
100 150 200 400 600
Subsequence length range
ASTRO
Time out after 24h
Figure 2: Time over motif length ranges (default (cid:96) min = , data series length= . EMG ASTRO T i m e ( H o u r s ) (a) (b) m (Top-1 m th discords) Dataset GrammarVizMAD (Discord discovery)DAD Time out after 48h k (Top-k 1 st discords) Dataset Figure 3: (a)
T op − m th discords discovery, and (b) T op − k st discords discovery time performance. valid (partial) distance profiles, which contain the maxLB distance greater than the largest m th best match distance.We keep extracting in this manner the motif and the dis-cord subsequences of each length, until (cid:96) max . Motif Discovery Experimental Evaluation.
To bench-mark the MAD framework, we used several differentreal datasets. Concerning the motif discovery problem,the competitors we considered are: QUICKMOTIF [12],STOMP [3], and MOEN [18]. We report in Figure 2 a sam-ple of the experiments we conducted (detailed experimen-tal results on several datasets are reported elsewhere [17]).Here, we show the results of MAD, which finds motifs indifferent real datasets. In the plots, we report the totalexecution time varying motif length ranges. From this ex-periment, we observe that VALMOD maintains a good andstable performance across datasets and parameter settings,quickly producing results, even in cases where the competi-tors do not terminate within a reasonable amount of time.
Discord Discovery Experimental Evaluation.
Weidentify two state-of-the-art competitors to compare to ourapproach, the Motif And Discord (MAD) framework. Thefirst one, DAD (Disk aware discord discovery) [27], imple-ments an algorithm suitable to enumerate the fixed-length
T op − m th discords. The second approach, Grammar-Viz [23], is the most recent technique, which discovers Top-k1 st discords. In Figure 3.(a) we report the results of T op − m th discord discovery, varying m . We note that MAD grace-fully scales over the number of discords to enumerate andis up to one order of magnitude faster than DAD. In Fig-ure 3.(b), we show the result of T op − k st discords dis-covery. Once again, MAD scales better over the numberof discovered discords, as its execution time remains almostconstant. A different trend is observed for GrammarViz,whose performance significantly deteriorates as k increases.
3. CONCLUSIONS
Even though much effort has been dedicated for develop-ping techniques for data series analytics, existing solutionsfor subsequence matching, motif and discord discovery arelimited to fixed length queries/results. In this Ph.D. work,we propose the first scalable solutions to the variable-lengthversion of these problems:
ULISSE is the first index thatsupports variable-length subsequence matching over both Z-normalized and non Z-normalized sequences [15, 13, 14],while MAD is the first framework that implements variable-length motif and discord discovery [17, 4, 16].
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