Source-linked AI summary
Robust Model Order Selection via Dithered Differential Step-Down Thresholding
Aleksandr Kharin
TL;DR
The paper addresses model order selection when single-threshold detection is vulnerable to extreme noise excursions and false alarms from lowered thresholds. It proposes dithered differential step-down thresholding with a clustering-based stopping rule, and proves asymptotic exact order recovery while maintaining robustness when the floor threshold is lowered or extreme noise excursions occur.
Problem
Single-threshold detection in stationary noise is sensitive to extreme noise excursions, especially when lowering the detection threshold to capture weak deterministic components.
Method
The method combines differential step-down thresholding, randomized grid dithering, and a stopping rule that detects clustered noise extrema.
Results
The proposed algorithm achieves asymptotic exact order recovery and yields zero false alarms under the stated threshold condition.
Takeaways & Limitations
The algorithm remains robust when the floor detection threshold is lowered or floor-threshold exceedances occur.
Takeaways & Limitations
The analysis assumes stationary noise with Leadbetter’s D(un) mixing condition and, in the observation model, deterministic shifts with nonnegative values.
Abstract
from arXiv · showhide
This paper studies the problem of model order selection in stationary noise. Single-threshold detection is sensitive to extreme noise excursions, particularly when the detection threshold is lowered to capture weak deterministic components. To augment single-threshold detection, we propose a differential step-down thresholding algorithm. We use a threshold grid in this algorithm. To overcome the threshold grid misalignment error induced by grid evaluation, we utilize randomized grid dithering. Using extreme value theory, we show the clustering of the noise extrema. The stopping rule of the proposed algorithm detects this clustering and stops the algorithm to prevent false alarms. By analytically bounding the threshold grid misalignment error, we prove that our algorithm achieves asymptotic exact order recovery under the 0-1 loss function. Moreover, the proposed algorithm remains robust even if the detection threshold in the original single-threshold algorithm is lowered or extreme noise excursions occur.
1 Introduction
The paper addresses model order selection when single-threshold detection is vulnerable to extreme noise excursions and false alarms after lowering thresholds for weak deterministic components. It proposes differential step-down thresholding with randomized grid dithering and a stopping rule designed to separate deterministic components from clustered noise extrema.
- Single-threshold detection is sensitive to extreme noise excursions, while lowering its threshold to capture weak deterministic components introduces false alarms.
- The proposed differential step-down algorithm evaluates decreasing thresholds and uses differences between sequential model-order estimates to detect clustered noise extrema.
- Under assumed detectability conditions, deterministic components separate while upper noise extrema cluster densely, with expected spacing tending to zero for sub-Gaussian noise.
- Randomized grid dithering bounds threshold grid misalignment error and supports asymptotic exact order recovery under the proposed stopping rule.
2 Problem Formulation and Algorithm
The paper formulates model order selection on a search grid containing deterministic shifts and stationary noise, then defines a dithered threshold-grid procedure for estimating the order. The algorithm uses a floor threshold, randomized grid alignment, sequential count differences, and a stopping-based model-order estimate.
- 2.1 Observation Model and Risk: The observation model maps data onto M grid statistics V_i = d_i(Θ0) + ε_i, combining deterministic shifts with noise.
- 2.1 Observation Model and Risk: The parameters Θ0 are continuous, deterministic shifts satisfy d_i(Θ0) ≥ 0, and the noise satisfies Leadbetter’s D(un) mixing condition.
- 2.1 Observation Model and Risk: The search grid is partitioned into ν0 deterministic statistics with positive shifts and M −ν0 noise statistics with zero shifts, under a sparse-model assumption.
- 2.1 Observation Model and Risk: At threshold T, the estimator counts grid statistics satisfying V_i ≥ T, and consistency is evaluated using 0-1 loss under a high-dimensional scaling regime.
- 2.2 Algorithm: The floor detection threshold controls family-wise error over the search grid and is primarily set to the minimax detection threshold, though lowered floors are also considered.
- 2.2 Algorithm: Randomized grid dithering draws ξ ∼ Unif(0, ∆T) independently of the observed data to randomize threshold-grid alignment.
- 2.2 Algorithm: The algorithm chooses Tmax a priori, keeps thresholds at or above Tfloor, and computes sequential estimate differences over adjacent threshold intervals.
- 2.2 Algorithm: The final model-order estimate is ˆνDSDT = ˆν(Tkstop−1), using the estimate immediately before the stopping index.
3 Theoretical Analysis
The analysis establishes exact order recovery under detectability and separation conditions, while showing how randomized dithering controls threshold-grid misalignment. It also characterizes noise-extrema spacing and the algorithm’s behavior under lowered floor thresholds and deterministic grids.
- Detectability conditions: Exact asymptotic recovery requires detectable, separated deterministic shifts, and identical shifts cannot be resolved.The weakest shift must exceed the floor threshold by a diverging margin, while consecutive shifts require diverging separation.
- Extrema asymptotics: For sub-Gaussian noise, upper-extrema spacings vanish asymptotically; local correlation further clusters extrema and shrinks their spacings.Under local anti-clustering, normalized spacings converge to Exp(1), with expected spacing tending to zero.
- Stopping rule: The stopping rule detects clustered noise extrema while remaining unmet across separated deterministic components.When consecutive deterministic statistics are separated by more than the threshold step, an interval of width ∆T covers at most one component.
- Deterministic-grid limitation: A deterministic threshold grid can become inconsistent when the floor threshold is lowered and noise exceeds that floor with probability tending to one.In this regime, the expected risk does not tend to zero even when deterministic components remain separated from noise.
- Randomized grid: Randomized grid dithering decouples threshold alignment from noise extrema and bounds the misalignment probability by E[S]/∆T.The independent uniform dither enables an analytical bound on grid misalignment.
- Consistency results: Under the stated assumptions, the dithered algorithm achieves asymptotic exact order recovery and can produce zero false alarms below the floor threshold.When the largest noise statistic is below Tfloor, the grid descends to Tfloor and stops without false alarms.
4 Conclusion
The paper studies a differential step-down thresholding algorithm for model order selection.
- The proposed method targets model order selection using differential step-down thresholding.