TopoCast: Topology-Aware Forecast Evaluation

• TopoCast is a topology-aware evaluation framework that embeds time series via Takens delay embedding to reveal hidden structures.
• It computes four topological fidelity measures from persistent homology—loop count, dominant cycle strength, total persistence, and persistence entropy—to assess forecasts.
• By integrating a phase-aware temporal co-localization term, TopoCast produces a Localized Topological Fidelity Score (LTFS) that detects over-smoothing, phase shifts, and frequency distortions.

Searching arXiv for TopoCast and related forecasting-evaluation papers. TopoCast is a topology-aware evaluation framework for time series forecasting that targets a specific deficiency in conventional forecast assessment: pointwise error metrics such as Mean Squared Error quantify numerical proximity but do not measure whether a forecast preserves the dynamical structure of the signal. The framework reconstructs forecast and ground-truth sequences in phase space using Takens delay embedding, summarizes their recurrent geometry with persistent homology, derives four topological fidelity measures from persistence diagrams, and aggregates them into a Topological Fidelity Score (TFS). It then adds a phase-aware temporal co-localization term, dominant cycle overlap, yielding the Localized Topological Fidelity Score (LTFS), which is intended to expose over-smoothing, phase shifts, frequency distortions, and temporally displaced oscillatory structure that can remain invisible under standard error criteria (Weerasekara et al., 24 Jun 2026).

1. Conceptual scope and problem formulation

TopoCast is situated within time series forecasting evaluation rather than forecasting model design. Its motivating claim is that numerical accuracy and structural fidelity are not equivalent. A forecast may track a trend closely in the pointwise sense while flattening peaks, shifting oscillations in time, altering periodicity, or introducing cyclic behavior absent from the target signal. The framework therefore treats the preservation of recurrent dynamics, oscillatory behavior, and phase alignment as evaluative objects in their own right (Weerasekara et al., 24 Jun 2026).

The failure modes emphasized in the formulation are over-smoothing, phase shifts or temporal lag, frequency distortion, and spurious oscillations or loop injection. In the paper’s framing, these are not merely aesthetic discrepancies; they represent distortions of the underlying dynamical system. This is why the framework compares forecast and target not only as scalar sequences, but as geometric objects in reconstructed state space.

A central implication is that model rankings induced by MSE-like criteria need not coincide with rankings induced by structural criteria. This suggests that benchmark evaluation in structurally sensitive domains can be incomplete when limited to pointwise losses.

2. Phase-space reconstruction and persistent-homology pipeline

TopoCast begins from the observation that a raw one-dimensional time series has trivial topology. It therefore lifts the signal into a higher-dimensional representation using Takens delay embedding. For each channel $x_i(t)$, with embedding dimension $m=3$ and delay $\tau=2$, the embedding is

$$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$

with

$N = T - (m-1)\tau = T - 4.$

For multivariate time series with $C$ channels, the channel-wise embeddings are concatenated column-wise into a joint point cloud $\mathcal{X} \in \mathbb{R}^{N \times mC}$. This reconstruction is performed independently for the forecast $\hat{y}$ and the ground truth $y$ (Weerasekara et al., 24 Jun 2026).

The reconstructed point clouds are then analyzed via a Vietoris–Rips filtration built with Euclidean distances. As the scale parameter $\varepsilon$ increases, simplicial complexes evolve and topological features appear and disappear. TopoCast focuses on $m=3$0, the first homology group, because loops in phase space correspond naturally to periodic or oscillatory behavior in time series. Each $m=3$1 feature is represented by a birth–death interval

$m=3$2

where $m=3$3 is the persistence or lifetime. Features with extremely small lifetime, $m=3$4, are treated as noise and discarded.

The persistence diagram is the primary topological summary. In this representation, periodic signals can manifest as closed loops in phase space, so recurrent temporal structure is translated into measurable topological structure.

3. Topological fidelity measures and the Topological Fidelity Score

From each $m=3$5 persistence diagram, TopoCast computes four scalar descriptors for both forecast and ground truth. These descriptors summarize complementary aspects of cyclic structure rather than a single diagram distance.

Descriptor	Symbol	Role
Loop count	$m=3$6	Number of significant loops
Dominant cycle strength	$m=3$7	Persistence of the strongest cycle
Total persistence	$m=3$8	Total amount of cyclic structure
Persistence entropy	$m=3$9	Complexity of the lifetime distribution

Each forecast descriptor is compared to its ground-truth counterpart through a bounded log-ratio score in $\tau=2$0. The four components are

$\tau=2$1

$\tau=2$2

$\tau=2$3

$\tau=2$4

These are aggregated with the geometric mean to form the Topological Fidelity Score,

$\tau=2$5

The use of the geometric mean is explicit and consequential: if any one structural component collapses, the combined score drops sharply. In effect, TFS is designed as a multiplicative fidelity criterion over loop count, dominant cyclic strength, aggregate topological energy, and persistence-diagram complexity (Weerasekara et al., 24 Jun 2026).

4. Temporal localization and the Localized Topological Fidelity Score

Diagram-level comparison alone does not specify when a topological feature occurs. A forecast can preserve the correct number of loops and similar persistence values while placing the dominant oscillatory pattern at the wrong time. TopoCast addresses this limitation with dominant cycle overlap.

The dominant cycle is defined as the longest-lived $\tau=2$6 feature. Using the representative cocycle returned by Ripser, the framework maps that feature back to the original time indices associated with the embedded points, producing temporal index sets for the forecast and the ground truth:

$\tau=2$7

Their temporal co-localization is then measured with a Jaccard overlap,

$\tau=2$8

An overlap of $\tau=2$9 indicates that the dominant oscillatory structure occurs at the same time steps; an overlap of $$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$0 indicates complete displacement. This construction makes TopoCast more than a persistence-diagram comparison framework, because it explicitly re-attaches topological features to temporal location (Weerasekara et al., 24 Jun 2026).

The Localized Topological Fidelity Score is then defined as

$$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$1

LTFS is therefore phase-aware. TFS measures whether the overall topology is preserved; overlap measures whether the dominant structure is temporally aligned. A forecast with high TFS but low overlap is structurally similar in aggregate yet mislocalized in time.

5. Experimental characterization of Transformer forecasters

The empirical study evaluates five Transformer-based architectures: Transformer, Informer, Autoformer, FEDformer, and PatchTST. Three multivariate benchmark datasets are used: ETTm2, with 7 variables and 15-minute sampling; Exchange, consisting of daily exchange rates for 8 currencies; and ILI, a weekly influenza-like illness dataset. The evaluation protocol fixes Takens parameters at $$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$2 and $$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$3, computes persistent homology with Ripser, samples 100 randomly selected non-overlapping test windows per setting for ETTm2 and Exchange and 50 for ILI, and evaluates horizons 48, 96, and 192 for ETTm2 and Exchange, and 36, 48, and 60 for ILI (Weerasekara et al., 24 Jun 2026).

A synthetic ECG experiment isolates several structural failure modes: smooth forecast, attenuated peaks, missing beat, and phase shift. The reported behavior is diagnostic. Smoothing collapses structural scores; attenuated peaks reduce dominant-cycle strength and total persistence; missing beats reduce loop count and overlap; and phase shift can leave TFS high while collapsing overlap and hence LTFS. This establishes a distinction between structural similarity in the persistence-diagram sense and temporal alignment in the phase-aware sense.

Across real datasets, the main finding is that pointwise error and topological fidelity often disagree. On ETTm2, models with competitive MSE can have poor topological fidelity. On Exchange, PatchTST often has the best MSE but the worst or near-worst LTFS. On ILI, good performance under standard metrics does not guarantee preservation of structure or temporal alignment.

The results are also interpreted as architecture-specific failure modes. Standard attention models such as Transformer and Informer tend to lose loops on periodic data such as ETTm2. Decomposition-based models such as Autoformer and FEDformer can inject loops on sparse or non-periodic data like Exchange, because their seasonal priors impose cyclic structure even when the signal does not support it. PatchTST shows strong loop injection on Exchange, especially at longer horizons. Temporal mislocalization is reported as pervasive: overlap values are often low even when TFS is moderate, particularly on Exchange, where temporal displacement becomes the dominant failure mode.

The framework is also compared with DTW, TDI, and Wasserstein distance $$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$4. The reported conclusion is not that LTFS replaces these metrics, but that it complements them by decomposing structural fidelity into interpretable topological components and by explicitly measuring temporal localization.

6. Limitations, applications, and terminological scope

The framework has several explicit limitations. It focuses primarily on $$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$5, so it captures loops but not higher-dimensional topological features. It uses fixed Takens parameters $$\mathbf{X}_i = \bigl[x_i(t),\; x_i(t+\tau),\; x_i(t+2\tau)\bigr], \quad t = 0, 1, \ldots, N-1$$6 rather than dataset-specific tuning. It evaluates windowed forecasts, so adaptation may be needed for very long-range or non-windowed settings. The overlap term depends on the quality of cocycle extraction, and the method is presented as an evaluation procedure rather than a training loss (Weerasekara et al., 24 Jun 2026).

Its intended application domain is correspondingly broad but structurally specific: medical signals such as ECG or patient monitoring, energy and sensor data with periodic regimes, financial and economic series, and any setting in which phase, recurrence, and oscillatory timing matter. In those settings, the framework is meant to support model comparison beyond MSE, detection of over-smoothing, detection of spurious periodicity, identification of phase-shift errors, and architecture selection matched to data characteristics.

The name “TopoCast” can be confused with unrelated “Topo-” systems in other literatures. It is distinct from TOPCAT Corner Plot, a desktop GUI feature for linked tabular-data visualization (Taylor, 2024); from the Topo-trigger stereo-trigger concept for imaging atmospheric Cherenkov telescopes (López-Coto et al., 2016); from TopoAct, a topological visualization system for deep-network activations (Rathore et al., 2019); from TopoMask, an instance-mask-based road-topology method (Kalfaoglu et al., 2023); and from TopoCap, a topology-agnostic video-to-animation framework (Pu et al., 10 Jun 2026). In the forecasting context, “TopoCast” refers specifically to the topological fidelity framework built on Takens embedding, persistent homology, TFS, and LTFS.

Taken as a whole, TopoCast formalizes a shift in forecasting evaluation from value-wise discrepancy to structural fidelity. Its central claim is that two forecasts can be numerically similar yet topologically different, and that the latter difference is operationally meaningful when recurrent dynamics, oscillatory structure, and phase alignment are integral to downstream interpretation.