Sync Without Guesswork: Incomplete Time Series Alignment

Multivariate time series alignment is critical for ensuring coherent analysis across variables, but missing values and timestamp inconsistencies make this task highly challenging. Existing approaches often rely on prior imputation, which can introduce errors and lead to suboptimal alignments. To address these limitations, we propose a constraint-based alignment framework for incomplete multivariate time series that avoids imputation and ensures temporal and structural consistency. We further design efficient approximation algorithms to balance accuracy and scalability. Experiments on multiple real-world datasets demonstrate that our approach achieves superior alignment quality compared to existing methods under varying missing rates. Our contributions include: (1) formally defining incomplete multiple temporal data alignment problem; (2) proposing three approximation algorithms balancing accuracy and efficiency; and (3) validating our approach on diverse real-world datasets, where it consistently outperforms existing methods in alignment accuracy and the number of aligned tuples.

💡 Research Summary

The paper tackles the challenging problem of aligning multivariate time‑series that suffer simultaneously from missing values and timestamp inconsistencies. Conventional pipelines first impute missing entries (e.g., linear interpolation, model‑based filling) and then apply alignment methods such as DTW, CTW, or SAMC. This two‑step approach is fragile because imputation errors propagate to the alignment stage and because imputed values are estimated independently of the global alignment constraints, often resulting in temporally or statistically incoherent alignments.

To overcome these drawbacks, the authors formulate an “incomplete multivariate time‑series alignment” problem that does not require any prior imputation. The formulation is built on three explicit constraints: (1) a time constraint limiting the maximum pairwise timestamp difference within a tuple to a threshold θ, (2) a position constraint limiting the maximum index gap between series to β, and (3) a model‑consistency constraint that forces the average normalized absolute error of a learned multivariate time‑series model (e.g., MARSS) on the aligned tuples to stay below δ. Together, these constraints guarantee temporal coherence, positional compactness, and statistical fidelity to the underlying dynamics.

Each candidate tuple r is assigned a weight W(r) = k₁·p(r) + b·k₂·d(r) + c, where p(r)=λ·(λ−1)/2 counts the number of non‑missing value pairs (λ is the number of observed values in the tuple) and d(r) aggregates the absolute index differences across all series. The parameters k₁, k₂, b, and c are user‑tunable, allowing practitioners to balance completeness against positional compactness. The overall objective is to select a set of non‑conflicting tuples that maximizes the total weight while respecting the three constraints.

The algorithmic contribution consists of four increasingly scalable strategies:

1. Exact combinatorial search – enumerates all feasible subsets of the candidate set R_c (generated by pruning with the time and position constraints) and picks the weight‑maximizing solution. This method is exponential (worst‑case O(2^{|R_c|})) and serves as a performance upper bound for small‑scale problems.

2. Weighted k‑set packing approximation – reformulates the selection problem as a weighted k‑set packing instance. Using known approximation algorithms, the authors prove an approximation ratio ξ ≥ 3 /