TiVy: Scalable Time Series Summary

• TiVy is a method that clusters time-aligned subsequences to create compact, legible visual summaries using small multiples.
• It employs Locality-Sensitive Hashing and DTW-based hierarchical clustering with gap statistics to efficiently group similar patterns, achieving significant speed-ups.
• Empirical evaluations on synthetic, stock, and ECG datasets demonstrate TiVy’s effectiveness in revealing meaningful trends and reducing visual clutter.

Searching arXiv for the requested topic and directly related papers. TiVy is a method and system for turning a large collection of time series into a compact “visual summary” that remains readable as line charts. It addresses the standard tradeoff between many small multiples, which preserve clarity but do not scale, and a single overplotted superposition, which uses space efficiently but quickly becomes visually cluttered. TiVy summarizes multiple time series by clustering subsequences rather than whole series, requires those subsequences to be aligned in time, and represents the result as a small set of disjoint subsequence groups rendered as uncluttered superpositions with fewer small multiples (Chan et al., 25 Jul 2025).

1. Problem setting and design objective

Time series visualization commonly alternates between two classical extremes. One chart per series provides legibility but explodes in number when the collection is large. Superposing all series into one chart economizes space but produces a spaghetti plot even for modest counts. TiVy is explicitly positioned between these extremes: it selectively superposes time series that are visually similar and aligned in time, while juxtaposing those that are visually different.

The central design goal is to “cluster subsequences.” Within each time interval, TiVy superposes time series that have similar shape so that the superposition remains legible, and separates series with different shapes into different small multiples. The resulting summary is intended to preserve the overview function of a large-scale visualization while reducing both the number of charts and the overplotting within each chart.

A defining property of TiVy is that it does not cluster whole time series. Instead, it clusters subsequences of varying lengths, and those subsequences must occur in the same time interval. This time-aligned, subsequence-centric formulation is tailored to visual analysis questions such as identifying what happened around a particular period across many stocks or what kinds of beats exist around a particular ECG time window. A plausible implication is that TiVy is less concerned with global similarity across an entire series than with local shape regularities that are meaningful in a shared temporal context.

2. Formal model and optimization target

TiVy assumes a collection of univariate, regularly sampled time series,

$$T = t_1, t_2, \ldots, t_n,\quad t_i \in \mathbb{R},$$

with dataset

$\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$

Each series has the same length $n$. The time axis $I=\{1,\dots,n\}$ is partitioned into contiguous, fixed-size segments of length $l$,

$$T = s_1 \oplus s_2 \oplus \cdots \oplus s_{n'}, \quad n' = n/l,$$

where $s_i$ is the subsequence in interval $I_i$ and $\oplus$ is concatenation.

For each window $I_i$, TiVy clusters all segments from all series within that window into visually similar groups and assigns a symbol to each cluster. A time series is thereby rewritten as a symbolic sequence,

$\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$0

The input to the pattern-mining stage is a matrix $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$1 of size $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$2, one symbolic sequence per series. The output is a set of subsequence groups $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$3, where each group corresponds to a pattern over consecutive windows, has support at least $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$4, and groups those series subsequences that realize that pattern in that time interval.

The feasibility constraints are stated as

$\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$5

where $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$6 is the number of time series in group $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$7. Among feasible partitions, TiVy seeks one that covers the largest total number of time series segments and uses as few groups as possible. In the authors’ formulation, a “visual summary” is therefore a partition of the time-series–time plane into disjoint subsequence clusters that maximize coverage and minimize the number of clusters.

This optimization target distinguishes TiVy from conventional clustering pipelines. The object being summarized is not a set of whole-series clusters but a compact covering of local, time-aligned trends.

3. Symbolization, pattern mining, and group extraction

Algorithm 1 in TiVy consists of three stages: construct_sequences, prefix_scan, and extract_groups (Chan et al., 25 Jul 2025). The first stage converts real-valued time series into symbolic sequences. The second profiles and stores all frequent sequential patterns and their supports. The third greedily chooses a disjoint, high-coverage set of subsequence groups.

The symbolization stage begins with per-window segmentation. For each interval $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$8, the algorithm extracts the matrix $\mathcal{T} = \{T^{(1)}, \ldots, T^{(m)}\}.$9 of all segments $n$0. Direct hierarchical clustering with DTW on all segments is too slow, because DTW on two length-$n$1 sequences costs $n$2 and all-pairs hierarchical clustering is $n$3 per window. TiVy therefore uses Locality-Sensitive Hashing with a p-stable distribution as a coarse grouping step:

$n$4

where $n$5, $n$6, $n$7, and $n$8 is the bucket width. Because Euclidean distance upper-bounds DTW, this provides a coarse approximation that is safe for subsequent DTW-based refinement.

Within each coarse bucket, TiVy samples one representative segment per bucket, performs agglomerative hierarchical clustering with DTW as the distance metric on the sampled representatives, and determines the number of clusters via Gap statistics with clustering strength parameter $n$9:

$I=\{1,\dots,n\}$0

Cluster labels inferred from the sampled representatives are then propagated to all segments in the corresponding bucket. DTW is used as the visual similarity metric because it permits warping along the time axis. For two sequences $I=\{1,\dots,n\}$1 and $I=\{1,\dots,n\}$2, the distance is

$I=\{1,\dots,n\}$3

with $I=\{1,\dots,n\}$4 a monotone, continuous warping path. TiVy uses the dtaidistance optimized CPU implementation for DTW; the experiments also consider SoftDTW on GPU and FastDTW, but report that CPU-optimized DTW plus LSH gives the best speed/quality tradeoff.

After clustering per window, each series becomes a symbolic sequence over time-local symbols. These symbols are local to a time window: cluster IDs are never repeated across windows. That structural property makes the sequential pattern-mining stage unusually simple. A pattern is a contiguous sequence of symbols over consecutive windows, and its support is the number of time series whose symbolic sequence contains that pattern in that exact temporal location. prefix_scan recursively scans the first column of the current submatrix, branches on distinct symbols, prunes any branch whose support falls below minsup, records the surviving prefix, and recurses on the remaining columns. Because the search proceeds left-to-right and symbols are unique to each timestep, TiVy avoids the projected-database machinery of generic sequential pattern miners such as PrefixSpan, BIDE, or VMSP.

The resulting dictionary $I=\{1,\dots,n\}$5 maps each pattern to the series segments that support it. Since patterns overlap heavily in both time and series membership, extract_groups applies a heuristic greedy strategy. It first removes patterns with support below minsup, then focuses on patterns of maximum length. Within a length tier it repeatedly samples a candidate pattern, checks whether selecting it would cause more than one peer pattern to drop below minsup, rejects it if it “kills” too many peers, and otherwise accepts it, removes its segments from all other patterns, and prunes the dictionary again. The procedure then moves to shorter patterns. This heuristic approximates the coverage-maximizing, group-minimizing optimization target by prioritizing long patterns while controlling destructive overlap.

4. Visual summary representation and interactive system

A TiVy summary is rendered as a grid of small multiples, where each chart corresponds to one group $I=\{1,\dots,n\}$6 and contains all subsequences in that group superposed. Since group members are shape-homogeneous by construction, superposition within a chart is intended to remain legible, while juxtaposition across charts highlights distinct patterns. The system thereby combines aggregated display with representative traces rather than plotting every line independently (Chan et al., 25 Jul 2025).

Three visual encodings define each chart. The first is a band graph with two uncertainty bands: an outer min–max band and an inner quantile band such as the 90% quantile range. The second is a density line chart based on a 2D density map over time-by-value bins, displayed through opacity or color intensity; this makes rendering cost less sensitive to the number of member series. The third is a center line defined by the medoid,

$I=\{1,\dots,n\}$7

where $I=\{1,\dots,n\}$8 is DTW and $I=\{1,\dots,n\}$9. Using the medoid avoids barycenter computations such as DTW Barycenter or SoftDTW barycenter, and line width encodes cluster size.

The implementation is a WebGL-based widget for Jupyter. Charts are arranged in a left-to-right, top-to-bottom grid. Optional grouping by an attribute, such as ECG class or industry sector, assigns shared color and vertical grouping to charts with the same label. Charts are also sorted by shape similarity by sampling one series from each group, hierarchically clustering those samples, and following the dendrogram leaf order.

Interaction includes pan and zoom, support and duration filtering, toggles between band graphs and density lines, show/hide control for center lines, and grouping by attributes. Rendering performance is supported by precomputing mesh vertices for bands, density polygons, and center lines and storing them in GPU buffers; switching views only changes which buffers are drawn. The implementation is open source at https://github.com/GromitC/TiVy.

5. Empirical evaluation and demonstrated use cases

The evaluation uses three datasets: synthetic data, stock time series, and ECG signals (Chan et al., 25 Jul 2025). The synthetic collection contains six canonical shape classes—cyclic, normal, increasing, decreasing, upward shift, and downward shift—with deviations in alignment and amplitude, together with datasets containing different durations and combinations across time. The stock dataset contains 4,470 daily adjusted NASDAQ stock prices from 2015–2016 with sector labels, normalized to zero mean and unit variance. The ECG dataset uses 100,000 heartbeat signals from the MIT-BIH Arrhythmia database, with each beat labeled into five AAMI groups: Normal (N), SVEB, VEB, Fusion (F), and Unknown (Q).

On synthetic data, TiVy is reported to cleanly separate the six shape classes into different charts, recover patterns with different durations and combinations, and achieve recall $l$0 across window sizes $l$1 for three datasets. For data with different pattern lengths and timings, the paper introduces optional adaptive window sizing through a grid search over partitions of base window size $l$2. Cluster quality is scored by

$l$3

where $l$4 is the inter-cluster distance, computed from DTW distances among medoids including time overlap, and $l$5 is the intra-cluster distance, computed from medoid-to-member distances. Higher $l$6 indicates better separation and cohesion, although the search can be expensive because it spans $l$7 partitions.

The ECG scenario is used to show hidden structure across 100,000 beats. The summary reveals long flat segments with dropped beats associated with junctional escape beats in the SVEB group and strong contraction beats followed by long pauses indicating premature ventricular contractions in the VEB group. Two independent cardiologists confirmed that these patterns correspond to clinically meaningful beat types. The system also supports filtering by cluster size, zooming into specific clusters, and grouping by AAMI class labels.

The stock market scenario focuses on trend exploration and portfolio construction. TiVy produces 23 subsequence clusters covering most of the 4,470 stock series across 2015–2016. The resulting summary reveals predominantly decreasing trends, as well as increasing and zig-zag patterns. Filtering to “V-shaped” patterns early in 2016 exposes a market-wide dip and bounce, interpreted in the paper as a global risk event rather than a company-specific event. Splitting clusters by sector shows that the “V-shaped” trend is concentrated in Industrial, Technology, Healthcare, and mutual funds.

In computational terms, the paper reports a significant speed-up of 1000X relative to straightforward DTW clustering in the symbolization step, reducing hours to seconds on the 100,000-series ECG dataset. The pattern-mining stage also completes in seconds even for 100,000 sequences and is reported to be markedly faster than general-purpose sequential mining algorithms.

6. Assumptions, limitations, and nomenclature

TiVy assumes regular sampling. Irregularly sampled series must be interpolated onto a common grid before DTW clustering. The method also depends on a fixed base window size $l$8. If $l$9 is too small, patterns are split into many short windows and the summary contains many small, redundant charts. If $$T = s_1 \oplus s_2 \oplus \cdots \oplus s_{n'}, \quad n' = n/l,$$0 is too large, short-lived patterns can be absorbed or missed. Adaptive window sizing mitigates this but is computationally expensive (Chan et al., 25 Jul 2025).

The clustering is constrained to fixed time intervals aligned to the global time axis. TiVy therefore does not search for patterns that occur at different absolute times but have similar shape, such as shifted seasonal motifs. This is an intentional restriction rather than an omission, because the method is designed for time-aligned visual comparison. The summary also enforces disjoint groups, so overlapping motif assignments are not represented. In addition, the current pipeline is univariate: it handles multiple univariate series rather than multivariate series with several dimensions per timestamp.

Parameter choice is explicitly presented as a tradeoff. Larger LSH bucket width $$T = s_1 \oplus s_2 \oplus \cdots \oplus s_{n'}, \quad n' = n/l,$$1 accelerates processing but can merge dissimilar shapes and increase clutter. Higher minsup yields faster execution and fewer groups but lower coverage. Smaller $$T = s_1 \oplus s_2 \oplus \cdots \oplus s_{n'}, \quad n' = n/l,$$2 reduces DTW cost per segment but can increase redundancy and weaken alignment over longer lags. The paper provides qualitative guidance through failure examples, but automatic parameter tuning is left open.

A nomenclature ambiguity also exists. In a separate 2026 paper, “TiVy” is used informally to denote a practical, accessible SOTA Vision Transformer stack derived from TuringViT rather than the time-series summarization method itself. That usage belongs to the framing of “TuringViT as a Blueprint for ‘TiVy’” and is conceptually unrelated to the original time series visual summary algorithm (Wu et al., 23 Jun 2026). The primary arXiv referent of TiVy remains the visualization system for scalable time series summarization (Chan et al., 25 Jul 2025).