1. Introduction
State of the art motif discovery. Over the last decade, data series^{1}^{1}1If the dimension that imposes the ordering of the series is time, then we talk about time series. However, a series can also be defined through other measures (e.g., angle in radial profiles in astronomy, mass in mass spectroscopy, position in genome sequences, etc.). Throughout this paper, we will use the terms time series, data series, and sequence interchangeably. motif discovery has emerged as perhaps the most used primitive for data series data mining, and it has many applications to a wide variety of domains (Whitney et al., 1998; Yankov et al., ), including classification, clustering, and rule discovery. More recently, there has been substantial progress on the scalability of motif discovery, and now massive datasets can be routinely searched on conventional hardware (Whitney et al., 1998). The stateofthe art algorithm (et al., ) only requires the user to set a single parameter, which is the desired length of the motifs. Moreover, the motif mining is supported by the Matrix profile output, which is a meta data series storing the znormalized Euclidean distance between each subsequence and its nearest neighbor. The Matrix profile does not exclusively provide the motif, i.e., the subsequence pair with the smallest distance, but also permits to rank and filter out the other pairs, giving also a convenient and graphical representation of their occurrences and proximity. In order to categorize motifs, we call the subsequences, with the smallest best match distances, top motif pairs.
Motif discovery of different lengths. Exact Motif discovery has merely become a single input parameter problem, namely the length of the patterns we want to mine. Unfortunately, this technique comes with an important lack. It does not provide an effective solution for trying several motif length in a range. If one has no cues about an effective fixed length, the simplest solution would be to run the algorithm over all lengths in the range and rank the various motifs discovered, picking eventually the patterns, which contain the desired insight. Clearly, this possibility is not optimal for at least two reasons; the scalability, since finding motif of one fixed length takes time, and also because it does not provide an effective way to compare motifs of different lengths. In this work, we demonstrate the solution to this problem, we recently introduced in (Linardi et al., 2018), to mine Motif discovery of variable lengths. In our contribution we propose VALMOD , the first approach for mining top motif pairs of variable length, which is up to orders of magnitude faster/more scalable than the alternatives that have been proposed in the literature.
In order to show the superiority of variablelength motif discovery, consider the following example. In Figure 1 (left) swe depict a snippet of an Electrocardiogram (ECG) recording in (a), paired with its Matrix profile, computed with fixed subsequence length: in (b). Note that each value in the Matrix profile corresponds to a point in the data, which is the representative starting point of a subsequence of length . Hence, given a data series of length , a Matrix profile records distances, avoiding trivial matches (Linardi et al., 2018). In Figure 1.(c) we plot the Index profile, which contains the offsets of the best matches.
Looking at the Matrix profile in this example, we note four deep valleys, which suggest the presence of very close matches, namely the motifs. Starting from the Matrix profile, it suffices to follow the dotted lines upwards, in order to detect the motifs, and downwards for finding the position of each subsequence best match. Despite the motifs (heartbeats) are easily detectable to the naked eye, since the snippet is relatively short, the highlighted motifs in Figure 1.(a) (red/orange subsequences), just report the second half of a ventricular contraction, giving thus a partial and unsatisfactory result.
In the next section we present the complete details of the VALMOD algorithm.
2. VALMOD Motif Management
VALMOD algorithm As previously introduced, our algorithm, VALMOD (Variable Length Motif Discovery), given a data series , starts by computing the Matrix profile using the smallest subsequence length, namely , within a specified input range . The key idea of our approach is to minimize the work that needs to be done for succeeding subsequence lengths (, , , ). To explain the main components and the idea of our algorithm we present a short example in Figure 2.
We start to consider the data series in (a) (snippet of ECG recording). To compute the Matrix profile, VALMOD considers all the contiguous subsequences of length , computing for each one the Distance profile in time. This latter, contains the znormalized Euclidean distance between a subsequence and all the other in . In Figure 2.(a) we report a distance profile for the subsequence (the subscript denotes offset=160 and length=600). The minimum distance of each distance profile is a point of the Matrix profile.
We moreover introduce a new lower bounding distance(Linardi et al., 2018), which lower bounds the true Euclidean distances between longer subsequences in the distance profiles. We initially compute this lower bound from scratch, using as a base the true Euclidean distances computation of subsequences with length 600. For the larger lengths, we update the lower bound, considering only the variation generated by the trailing points in the longer subsequences. This measure enjoys an important property: if we rank the subsequences according to this measure (ascending order), the same rank will be preserved along all the lower bound updates. We want to exploit this property, in order to prune computation. Hence, when the distance profiles are computed (in this example for length=600), we keep in memory the Euclidean distances, which have the smallest lower bounding distance (LB); this is done for each distance profile. We show in Figure 2.(b) how the algorithm proceeds for the length 601. Instead of computing from scratch the whole distance profiles, we consider just the elements we stored in the previous step. Here, each distance profile is denoted as partial distance profile. We proceed computing the true Euclidean distances of each partial distance profile, updating the relative LB (this result is depicted in Figure 2.(b). After this operation, we may have two cases: if in a new computed distance profile the minimum true distance (minDist) is shorter than the maximum lower bound (maxLB), we know that no elements, among those not computed, can be smaller than minDist. In this case a partial distance profile becomes a valid distance profile, as in the case of the subsequence . On the other hand, when maxLB is smaller than minDist, as in the case of subsequence , no true minimum distance is found within the distance profile. At the end of this process, we pick the minimum maxLB of all the nonvalid distance profile, which is denoted as minLBAbs. Hence, all the mindist in the valid (parital) distance profiles, smaller than minAbsLB are considered top motif distances. If no mindist are smaller than minAbsLB, we recompute only the distance profiles, which have the maxLB smaller than the smallest mindist found, since only those may contain better matches than the already computed ones. We keep extracting in this way, the top motifs of each length, until .
Experimental Evaluation. To benchmark VALMOD , we use several different datasets in (Linardi et al., 2018), comparing it with two types of algorithms. The first are two stateoftheart motif discovery algorithms, which receive a single subsequence length as input: QUICKMOTIF (Li et al., 2015) and STOMP (et al., ). In our experiments, they have been adapted to find all the motifs for a given subsequence length range. The other approach in the comparative analysis is MOEN (Mueen, ), which accepts a range of lengths as input, producing the best motif pair for each length. We report in Figure 3 a sample of the experiments we conducted (detailed experimental results on several datasets are reported elsewhere (Linardi et al., 2018)). Here, we show the results of VALMOD , which finds motifs in an Electrocardiogram recording (ECG) and in a data series representing celestial objects (ASTRO) (Linardi et al., 2018). We couple the VALMOD results with those of its competitors. In the plots, we report the total execution time of VALMOD , which includes all the operations performed by the algorithm (also the VALMAP computation introduced later), varying motif length ranges (Figure 3 (top)) and the size of the input data series, considering different prefix snippets (Figure 3 (bottom)).
From this experiment, we observe that VALMOD maintains a good and stable performance across datasets and parameter settings, quickly producing results, even in cases where the competitors do not terminate within a reasonable amount of time.
Rank Motif Pairs of Variable Lengths. Since we can discover motifs of different lengths, we propose a ranking method, suitable for comparing differentlength patterns. We aim to favor longer and similar sequences in the ranking process of matches that have different lengths. As a consequence, we factorize the Euclidean distance by the following quantity: , where is the length of the sequences. We call the new distance, length normalized distance (Linardi et al., 2018).
VALMAP. While the proposed motif rank weights the subsequences importance according to the ratio distancelength, we want to know also, whether and how the motif pairs changes, helping the user to extract the desired insights at the correct length. To that extent, we introduce a new metadata, called Variable Length Matrix Profile (VALMAP), maintaining the same logic and structure of the Matrix profile depicted in Figure 1
(top), with the difference that this new structure carries length normalized distances and it is coupled with a new vector called
Length profile, which contains the lengths of the subsequences. More formally, given a data series , and a range of subsequence lengths, whose extremes are denoted by and , we define VALMAP as a triple , where is the Matrix profile containing length normalized distances, whereas and are the relative Index and Length Profile. If we consider just a fixed length, VALMAP will coincide with the length normalized version of the Matrix profile, with a flat Length profile. This is basically the structure that VALMOD builds, considering subsequences of length . In the second stage, we can update VALMAP using the top motif pairs, computed for each length until . We thus consider each () top motif pairs, where are the subsequences offsets, their lengths and their length normalized Euclidean distance. Note that in a motif pair the right subsequence is the one with the absolute shortest distance to the one at the left. Hence, VALMAP , is updated with if , which was containing the distance between and its best match. If this update takes place, the Index and Length profile are respectively assigned with , the offset of the new best match, and the new length. The update operation takes place for each top motif pair of any length between and . Once the algorithms ends, VALMAP contains a picture of the motif pairs showing, at which length the last update takes place. If a motif pair is updated, this implies that a longer pattern represent a better match and thus it might reveal either a new event or the same event lasting longer.Example of VALMAP Expressiveness.
In order to show the expressiveness of VALMAP , we ran VALMOD on the ECG data snippet previously considered, showing the VALMAP structure in Figure 1 (right). We use the following input parameter: and . We note that VALMAP reports the motif with the shortest length normalized distance of length 56, which is the same partial event detected by the Matrix profile in the fixed length case, at the top of the picture.
If we look at the Length profile in Figure 1.(f), we observe that, at an earlier time than the discovered motifs pair, a sequence of contiguous updates took place, as we reported. The subsequences concerned have distances almost as short as the one of the best motifs in VALMAP , thus, remaining longer and possibly valid matches.
In Figure 1.(d) we depict and highlight the motif pair of length 400. Immediately, we can note that, the subsequences in red, which compose this motif, are a better representation of a recurrent heartbeat. In fact, the two typical components (Artia and Ventricles contract) are correctly detected.
3. System Description
We now describe the architecture of our system, depicted also in Figure 4. The input is represented by a data series of interest. As a starting point, the user has the possibility to inspect the data and also setting the desired parameter (lengths range [,]). Afterwards, she can run the VALMOD algorithm, which is a part of the system backend we implemented in C. Once terminated, VALMOD outputs the VALMAP metadata. This latter is thus sent to the frontend, implemented in Python. Here, the user can interact with the system analyzing the showcased elements, such as:

the checkpoints of the VALMAP, namely all the updates occurred from the length till the desired length, selected with a dedicated slider.

all the top motifs of variable length, which VALMAP reports.

expand a selected motif pair to the relative Motif Set, containing all the similar subsequences of the pair in the data.
In Figure 5 we show a screenshot of the VALMAP analysis in our demonstration.
4. Demonstration
We now present the scenarios proposed to the audience. Need for Variable Length Motifs. We will showcase variable length motif discovery using VALMOD on different real datasets (Linardi et al., 2018), including ECG and ASTRO, as well as datasets coming from the domains of Entomology and Seismology. In these two particular cases, the user can understand the importance of using variable length motif detection (with the support of VALMAP ), in order to identify patterns of interesting behavior exhibiting themselves as sequences of different lengths.
Traditional Motif discovery VS VALMOD.
In this scenario, we will challenge the user to find the motifs without having any knowledge of their lengths, just by inspecting the data themselves When this takes place, the user can experience the VALMOD support in finding motif pairs that can be of variable length, understanding the quantity and quality of the insights that are not achievable with a simple raw data visual analysis.
VALMOD VS Competitors. In this scenario, the user can compare VALMOD to alternative approaches used for motif discovery. Specifically the audience will note the performance improvement, concerning fixed and variable length motif discovery, and the increased expressiveness provided by VALMAP .
5. Conclusions
In this work, we present VALMOD , a system that can efficiently find data series motif of variable length.
As opposed to the other approaches, our framework provides a new meta dataseries
(VALMAP ), which ranks motif pairs of variable length, using a new length normalized distance.
Our system provides enriched insights, which help to detect not only the correct resolution (length) of an interesting event, but also the occurrences of repeated patterns with different meanings, which are typical in numerous domains.
References
 [1] Matrix profile I: all pairs similarity joins for time series: A unifying view that includes motifs, discords and shapelets. In IEEE , ICDM 2016, Cited by: §2.
 [2] Matrix profile II: exploiting a novel algorithm and gpus to break the one hundred million barrier for time series motifs and joins. In IEEE, ICDM 2016, Cited by: §1.
 Cited by: §2.
 Matrix profile x: valmod  scalable discovery of variablelength motifs in data series. Cited by: §1, §1, §2, §2, §2, §4.
 [5] Enumeration of time series motifs of all lengths. In ICDM, 2013, Cited by: §2.
 Reliability of scoring respiratory disturbance indices and sleep staging. Sleep. Cited by: §1.
 [7] Detecting time series motifs under uniform scaling. In KDD 2007, Cited by: §1.
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