Quant: Hierarchical Interval Quantiles

• Quant is a hierarchical, interval-based feature extraction method that recursively partitions time series to compute quantiles, capturing local and multiscale distributional properties.
• It leverages dyadic interval decomposition and multiple representations (raw, differences, DFT) to build fixed, compressive feature maps for efficient analysis.
• With linear-time computation and robust accuracy, Quant is integral to scalable stand-alone and ensemble meta-learning frameworks in time series classification.

Quant (Hierarchical Interval Quantiles) is a hierarchical, interval-based feature extraction methodology for time series, centered on capturing both local and multiscale distributional characteristics via fast computation of quantiles over dyadic intervals and their representations. Originating as a statistical approach for nonparametric hierarchical Bayesian quantile modeling (Bornn et al., 2016), the Quant methodology has also been adopted as a scalable interval-feature extraction and classification primitive in modern time series analysis—particularly when efficient, large-scale operation is required, as in recent meta-learning and ensemble architectures (Maniar, 7 Dec 2025). Quant constructs a fixed, compressive feature map of the input series by recursively partitioning it into non-overlapping intervals and computing multiple quantiles per interval, across several series-level transformations. This approach enables efficient learning in both stand-alone and ensemble contexts, with strong empirical results and straightforward implementation.

1. Formal Definition and Feature Map Construction

Given a univariate time series $x = (x_1,\ldots,x_T) \in \mathbb{R}^T$, Quant requires the following configuration parameters: hierarchy depth $L \in \mathbb{N}$, divisor $v \in \mathbb{N}$ (quantile granularity, typically $v=4$), and a set of $R$ series-level transformations (default: $R=4$, using the original series, first difference, second difference, and DFT magnitude).

At each hierarchy level $\ell = 1,\ldots,L$, define interval size $\Delta_\ell = 2^{\ell-1}$ and partition the time series into $N_\ell = \lfloor T/\Delta_\ell \rfloor$ non-overlapping intervals. For each interval $I_{i,\ell}^{(r)}$ of length $\Delta_\ell$ in representation $r$, compute

• $$k_\ell = 1 + \left\lfloor \frac{\Delta_\ell -1}{v} \right\rfloor$$ quantiles at probability positions $p_j = \frac{j}{k_\ell+1},\ j=1,\ldots,k_\ell$,
• the interval mean $\mu_{i,\ell}^{(r)}$,
• augmented features

$$\phi_{i,\ell}^{(r),j} = \begin{cases} Q_{i,\ell}^{(r)}(p_j), & \text{if } j \text{ odd} \ Q_{i,\ell}^{(r)}(p_j) - \mu_{i,\ell}^{(r)}, & \text{if } j \text{ even} \end{cases}$$

where $Q_{i,\ell}^{(r)}(p_j)$ is the $p_j$-quantile of $I_{i,\ell}^{(r)}$. The final Quant feature vector $f(x) \in \mathbb{R}^{d_Q}$ is the concatenation over all representations, levels, intervals, and quantiles.

2. Algorithmic Structure and Pseudocode

Quant's extraction pipeline is described via the following steps (Maniar, 7 Dec 2025):

F = [] for r in R: if r == "orig": x_r = x elif r == "diff1": x_r = smoothed_first_difference(x) elif r == "diff2": x_r = second_difference(x) elif r == "DFT": x_r = abs(FFT(x)) for l in range(1, L+1): Delta = 2**(l-1) N_int = T // Delta k = 1 + (Delta - 1) // v for i in range(N_int): I = x_r[i*Delta:(i+1)*Delta] mu = mean(I) for j in range(1, k+1): p_j = j/(k+1) q = Quantile(I, p_j) if j % 2 == 0: q -= mu F.append(q) f = concatenate(F)

This structure ensures that feature computation is both time- and memory-efficient, leveraging non-overlapping ("fixed-interval") dyadic partitioning for speed and simplicity.

3. Dyadic Hierarchical Decomposition

The representation is strictly non-overlapping: for each level $\ell$, the series is divided into $\lfloor T/2^{\ell-1} \rfloor$ intervals of length $2^{\ell-1}$. This non-redundant arrangement—sometimes termed "fixed-interval"—permits highly efficient extraction in both computation and storage. For $L=6$ (the default setting), intervals range from length 1 (pointwise) to 32, covering a multiscale spectrum of the temporal structure. If $T$ is not divisible by $\Delta_\ell$, trailing elements are omitted for that level.

4. Computational Complexity and Scalability

Let $R$ be the number of representations, $T$ the length, $L$ the hierarchy depth, and $k_\ell = O(\Delta_\ell/v)$ quantiles per interval. The total feature dimension is $d_Q = O((R\,T\,L)/v)$. Feature extraction, using an $O(\Delta_\ell)$ quantile selection algorithm, runs in $O(R\,T\,L)$ time; naive methods or coarser quantile resolution push this to $O(R\,T\,L^2)$. Empirically, for $v=4$ and $L=6$, the implementation is linear-time in $T$ and well-suited for large-scale time series tasks, with per-series feature extraction and classifier training typically below $0.01$ and $0.005$ seconds per thousand series, respectively, on standard hardware (Maniar, 7 Dec 2025).

5. Hyperparameter Choices and Tuning

Quant exposes several critical hyperparameters:

• Depth $L$: Determines the range of intervals. Default $L=6$ is optimal for $T \geq 32$, but adjust downward for smaller $T$.
• Divisor $v$: Controls quantile granularity. Default $v=4$ balances expressiveness and speed; $v \in [2,8]$ is typical.
• Representations $R$: The default suite includes raw, first and second differences, and DFT magnitude. Reducing $R$ accelerates extraction at slight (1–2% absolute) accuracy cost.
• Classifier configuration: Reference implementation employs ExtraTrees with 200 trees, entropy split criterion, and $\text{max\_features}=0.1$ to prevent undersampling in high-dimensional feature spaces. Adjust $L$ or $v$ to modulate speed and information content.

Increasing $L$ beyond $\log_2 T$ is ineffective, and decreasing $v$ quadratically reduces feature count at higher hierarchy levels. Richer transformation sets or lower $v$ values capture finer distributional patterns at higher computational expense.

6. Empirical Results and Ensemble Integration

Quant achieves state-of-the-art accuracy among interval-feature extractors on the 142-dataset UCR archive, extracting all features in under 15 minutes total CPU time (approx. 6 seconds per dataset) (Maniar, 7 Dec 2025). On the MONSTER large-scale benchmark (training sets up to 1.17M), Quant yields mean accuracy 0.8252 stand-alone and is efficiently trainable for millions of instances within an hour.

In ensemble contexts, Quant is highly complementary to dictionary/convolution-kernel approaches (e.g., Hydra). Its features exhibit moderate feature-space correlation (0.46–0.71) and low error correlation (mean 0.42) with Hydra, allowing ensembles to systematically outperform individual classifiers. In such configurations, Quant+Hydra ensembles achieved mean accuracy gains from 0.8294 to as high as 0.8362. Asymmetric stacking of Quant features and out-of-fold Hydra logits via ExtraTrees captured 11% of the oracle ensemble gain, surpassing prediction-level fusion (weighted averaging +0.43 pp) and raw concatenation (+0.6 pp).

For scenarios with tight computational budgets, Quant's linear scaling and efficient implementation recommend it as a stand-alone method. For maximal accuracy, hybrid ensemble strategies with heterogeneous meta-learners (e.g., ExtraTrees on concatenated Quant+Hydra features/logits) are most effective; instance-wise oracle gain should be evaluated pre-ensemble, as improvements may be negligible when base classifier overlap is high.

7. Contexts, Significance, and Theoretical Properties

Quant in its feature-extraction form (Maniar, 7 Dec 2025) is distinct from but inspired by nonparametric hierarchical Bayesian quantile inference (Bornn et al., 2016), which applies geometric measure theory and the Hausdorff measure to directly specify priors on quantiles and enables hierarchical probabilistic shrinkage for grouped, censored, or noisy discrete data. Both approaches stress flexibility, scalability, and the ability to borrow strength across hierarchical structure. In practice, Quant’s frequentist coverage properties for interval-based estimates and its non-redundant, multiscale decomposition allow it to efficiently summarize the distributional landscape of time series inputs for classification—and to serve as a building block in broader ensemble and meta-learning frameworks.

Significant findings include robustness of coverage for quantiles (comparable or superior to classical or bootstrap intervals), scalability to large datasets, and interpretability of feature contributions at multiple scales. The ensemble results reveal the current meta-learning gap: much of the potential accuracy gain from classifier complementarity is left untapped, and improved meta-learning or stacking strategies could further advance generalization across diverse time series applications.

References: (Bornn et al., 2016, Maniar, 7 Dec 2025)