Efficient Regression Models for Scan Statistics

Gazi Abdur Rakib; Tristan Ashton; Ryan A. Loomis; Brian S. Mason; Eric J. Murphy; Ci Xue; Jeff M. Phillips

stat.MEAug 23, 2026

Efficient Regression Models for Scan Statistics

Authors: Gazi Abdur Rakib, Tristan Ashton, Ryan A. Loomis, Brian S. Mason, Eric J. Murphy, Ci Xue, Jeff M. Phillips

Organizations: University of Utah & CosmicAI · National Radio Astronomy Observatory & CosmicAI

Abstract

We introduce a new class of regression models for scan statistics on real-valued signals. These allow for improved fitting of non-stationary signals to contrast with the interval anomalies identified by the scan statistics. Our models can represent generalized likelihood ratio statistics. While these methods naively require $O(n^4)$ for a length $n$ signal, we provide algorithmic improvements which lead to linear time algorithms (with assumptions on max interval width). Our methods, especially ones based on Nadaraya-Watson kernel regression, are demonstrated as especially effective in detecting both synthetically planted anomalies, and for identifying a real ``platforming'' issue in interferometric astronomy.

Figures & tables

Model	Solve	Update	Evaluate	Total	Improved Total	Truncated Total
$\mathcal{F}_{0}$	$O(n)$	$O(1)$	$O(1)$	$O(n^{2}w)$	$O(nw)$	$O(nw)$
$\mathcal{F}_{d}$	$O(nd^{2})$	$O(d^{3})$	$O(d)$	$O(n^{2}wd^{3})$	$O(nwd^{3})$	$O(nwd^{3})$
$\mathcal{F}_{\textsc{krr}}$	$O(n^{3})$	$O(n^{3})$	$O(n)$	$O(n^{4}w)$	$O(n^{4}w)$	$O(n^{4}w)$
$\mathcal{F}_{\textsc{nwkr}}$	$O(1)$	$O(1)$	$O(n)$	$O(n^{3}w)$	$O(n^{2}w)$	$O(nwr)$

Table 1: Runtime Comparison of models in Algorithm 1 for a sequence of length

n

, with max interval size

w

, and kernels truncated at radius

O(r)

. Improved results derived in Section 3.2 .

Panel of Figure 1 — Figure 1: Example synthetic signal. Anomaly width at $5\%$ and SNR= $2.5\!\times\!\sigma$ .

Panel of Figure 2 — Figure 2: Mean localization score comparison on quadratic trend data by varying SNR or anomaly width.

Panel of Figure 3 — Figure 3: Mean localization score comparison on AR(2) data by varying SNR or anomaly width.

Figure 4: Comparison of NWKR with other baselines on quadratic trend data in a varying SNR (left) and anomaly width (right).

Panel of Figure 5 — Figure 5: Log-log plots for runtime scaling on signal as a function of signal length $n$ (left), $w$ (middle), and truncation range $r$ (right). Otherwise parameters fixed ( $n=10,000$ , $w=200$ , $r=300$ )

Method	time (ms)
$\mathcal{F}_{0}$	17
$\mathcal{F}_{\text{NWKR}}$ Laplace	69
$\mathcal{F}_{\text{NWKR}}$ Gaussian	70
$\mathcal{F}_{1}$ Poly (deg 1)	392
$\mathcal{F}_{2}$ Poly (deg 2)	472
$\mathcal{F}_{\text{NWKR}}$ Gaussian (naive)	759,396

Table 2: Median runtime at

n=1000

Figure 6: ALMA calibration signal with platforming.

Figure 7: Performance analysis of different methods on variable IOU settings on ALMA dataset

Figure 8: Performance analysis of different methods on variable score settings on ALMA dataset

Method	TP	FP	TN	FN	Accuracy	Precision	Recall	F1
$\mathcal{F}_{0}$ Mean	118	2,561	36,089	113	0.9312	0.0440	0.5108	0.0811
$\mathcal{F}_{1}$ Poly (deg 1)	154	3,980	34,670	77	0.8957	0.0373	0.6667	0.0706
$\mathcal{F}_{2}$ Poly (deg 2)	174	2,932	35,718	57	0.9231	0.0560	0.7532	0.1043
$\mathcal{F}_{\text{KR}}$ Gaussian	231	191	38,459	0	0.9951	0.5474	1.0000	0.7075
$\mathcal{F}_{\text{KR}}$ Laplace	231	277	38,373	0	0.9929	0.4547	1.0000	0.6252
CAPA	31	554	38,096	200	0.9806	0.0530	0.1342	0.0760

Table 3: Performance metrics on the large ALMA dataset (

N=38{,}881

) at

\tau_{S}=0.3

Method	Precision	Recall	F1	FN rate	FP rate
Data reducer (a person)	0.3665	0.5108	0.4268	0.4892	0.0053
Existing ALMA heuristic	0.7018	0.4233	0.5281	0.5767	0.0013
Our $\mathcal{F}_{\textsc{nwkr}}$ Gaussian	0.5474	1.0000	0.7075	0	0.0049

Table 4: Comparison with estimated existing ALMA False Positive and False Negative rates

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 9 — Figure 9: Synthetic dataset construction and anomaly injection

Figure 11: Different anomalies visualized in signal

Panel of Figure 12 — Figure 12: AR(2) dataset construction and anomaly injection

Figure 13: Different generated AR(2) signals

Figure 14: Different anomalies visualized in AR(2) signal

Method	0.50	0.55	0.60	0.65	0.70	0.75	0.80	0.85	0.90	0.95	1.00
$\mathcal{F}_{0}$ Mean	0.6000	0.5920	0.5920	0.5797	0.5797	0.5756	0.5546	0.5460	0.5106	0.2624	0.2182
$\mathcal{F}_{1}$ Poly (deg 1)	0.6649	0.6545	0.6510	0.6223	0.6150	0.6075	0.5611	0.5408	0.4854	0.2381	0.1888
$\mathcal{F}_{2}$ Poly (deg 2)	0.7172	0.7074	0.7041	0.6702	0.6596	0.6489	0.5810	0.5444	0.4699	0.1922	0.1727
$\mathcal{F}_{\textsc{nwkr}}$ Gaussian	0.9902	0.9902	0.9902	0.9902	0.9902	0.9892	0.9892	0.9892	0.9892	0.9870	0.9870
$\mathcal{F}_{\textsc{nwkr}}$ Laplace	0.9825	0.9825	0.9780	0.9689	0.9573	0.9526	0.9333	0.9057	0.8873	0.8627	0.8342
FS NWKR Gaussian	0.0339	0.0171	0.0171	0.0171	0.0171	0.0171	0.0171	0.0171	0.0171	0.0000	0.0000

Appendix

Table 5: F1 Score across IoU Thresholds

\tau_{I}

for each method, with

\tau_{S}=0.3

Method	TP	FP	FN	Accuracy	Precision	Recall	F1	Runtime (ms)
$\mathcal{F}_{0}$ Mean	99	14	132	0.708	0.8761	0.4286	0.5756	680.4
$\mathcal{F}_{1}$ Poly (deg 1)	113	28	118	0.708	0.8014	0.4892	0.6075	15,392.7
$\mathcal{F}_{2}$ Poly (deg 2)	122	23	109	0.736	0.8414	0.5281	0.6489	18,561.5
$\mathcal{F}_{\textsc{nwkr}}$ Gaussian	228	2	3	0.990	0.9913	0.9870	0.9892	223.0
$\mathcal{F}_{\textsc{nwkr}}$ Laplace	211	1	20	0.958	0.9953	0.9134	0.9526	214.1
FS NWKR Gaussian	2	1	229	0.540	0.6667	0.0087	0.0171	143.1

Appendix

Table 6: Performance metrics on sampled balanced dataset (

N=500

) at

\tau_{I}=0.75

\tau_{S}=0.3

	IoU $\geq$ 0.50				IoU $\geq$ 0.75				IoU $\geq$ 0.90
Method	$\tau_{S}^{*}$	Prec.	Rec.	F1	$\tau_{S}^{*}$	Prec.	Rec.	F1	$\tau_{S}^{*}$	Prec.	Rec.	F1
$\mathcal{F}_{0}$ Mean	0.20	0.690	0.558	0.617	0.25	0.766	0.480	0.590	0.25	0.728	0.394	0.511
$\mathcal{F}_{1}$ Poly (deg 1)	0.30	0.822	0.558	0.665	0.30	0.801	0.489	0.608	0.30	0.748	0.359	0.485
$\mathcal{F}_{2}$ Poly (deg 2)	0.25	0.833	0.649	0.730	0.25	0.808	0.546	0.651	0.25	0.730	0.351	0.474
$\mathcal{F}_{\text{KR}}$ Gaussian	0.35	0.996	0.987	0.991	0.35	0.996	0.987	0.991	0.35	0.996	0.987	0.991
$\mathcal{F}_{\text{KR}}$ Laplace	0.30	0.996	0.957	0.976	0.30	0.995	0.913	0.953	0.30	0.995	0.801	0.887

Appendix

Table 7: Summary of best F1 operating points per method across IoU thresholds. For each method, the optimal score threshold (

\tau_{S}^{*}

) is selected to maximize F1. Precision, Recall, and F1 are reported at that threshold.

Method	TP	FP	TN	FN	Accuracy	Precision	Recall	F1	Mean (ms)	Median (ms)
$\mathcal{F}_{0}$ Mean	99	2,561	36,089	132	0.9307	0.0372	0.4286	0.0685	394.3	94.9
$\mathcal{F}_{1}$ Poly (deg 1)	113	3,980	34,670	118	0.8946	0.0276	0.4892	0.0523	9,200.6	2,213.6
$\mathcal{F}_{2}$ Poly (deg 2)	122	2,932	35,718	109	0.9218	0.0399	0.5281	0.0743	11,181.2	2,697.3
$\mathcal{F}_{\text{KR}}$ Gaussian	228	191	38,459	3	0.9950	0.5442	0.9870	0.7015	190.3	45.2
$\mathcal{F}_{\text{KR}}$ Laplace	211	277	38,373	20	0.9924	0.4324	0.9134	0.5869	185.5	45.1
CAPA	14	554	38,096	217	0.9802	0.0246	0.0606	0.0350	8.6	5.4

Appendix

Table 8: Performance metrics on the full ALMA dataset (

N=38{,}881

) at

\tau_{I}=0.75

\tau_{S}=0.3

$\tau_{S}$	TP	FP	FN	Accuracy	Precision	Recall	F1	FPR	FNR
0.10	228	6,166	3	0.8413	0.0357	0.9870	0.0688	0.1595	0.0130
0.11	228	4,544	3	0.8831	0.0478	0.9870	0.0911	0.1176	0.0130
0.12	228	3,365	3	0.9134	0.0635	0.9870	0.1192	0.0871	0.0130
0.13	228	2,542	3	0.9345	0.0823	0.9870	0.1519	0.0658	0.0130
0.14	228	1,957	3	0.9496	0.1043	0.9870	0.1887	0.0506	0.0130
0.15	228	1,507	3	0.9612	0.1314	0.9870	0.2319	0.0390	0.0130

Appendix

Table 9: Fine-grained score threshold grid search at

\tau_{I}=0.75

for

\mathcal{F}_{\textsc{nwkr}}

Gaussian on the full ALMA dataset (

N=38{,}881

Figure 15: Example spectra where $\mathcal{F}_{\textsc{nwkr}}$ methods outperform other baselines

Panel of Figure 16 — Figure 16: High-scoring rows produced from ALMA dataset

Panel of Figure 17 — Figure 17: Low-scoring rows produced from ALMA dataset

Figure 18: Scan statistic results on WBB solar radiation signal from 2024-12-10.

Time-Series Anomaly Detection

Time-Series Anomaly Detection identifies unusual patterns or events in sequential data, from sensor readings and network traffic to health and industrial telemetry.

See all 208 papers

Have an LLM Write Your Anomaly Detector: Autonomous Discovery of Compact, Interpretable Detectors for Time Series
Oct 1, 2026David BerghausInterpretable Anomaly Detection LLM-Based Program Synthesis
Lamarr Institute Germany · Fraunhofer IAIS Germany
Solve for the Hyperparameter, Skip the Search: Kolmogorov-Optimal Scaling Laws for Spline Regression
Jun 22, 2026Yong Yi Bay, Kathleen A. YearickModel Selection Scaling Laws
PhD, University of Illinois at Urbana-Champaign
Unsupervised Symbolic Anomaly Detection
Mar 18, 2026Md Maruf Hossain, Tim Katzke, Simon Klüttermann +1Unsupervised Anomaly Detection Anomaly Detection
TU Dortmund University, Dortmund, Germany · Research Center Trustworthy Data Science and Security, UA Ruhr, Germany

Oct 1, 2026J/K move · Enter open · S save

Have an LLM Write Your Anomaly Detector: Autonomous Discovery of Compact, Interpretable Detectors for Time Series

David Berghaus

Lamarr Institute Germany · Fraunhofer IAIS Germany

Interpretable Anomaly Detection LLM-Based Program Synthesis Unsupervised Anomaly Detection Time-Series Anomaly Detection Anomaly Detection

Open paper AskPDF arXiv

Time-series anomaly detection trades off predictive accuracy, computational efficiency, and interpretability. We use a large language model not as the detector but as the author of one: an autonomous research loop in which the model repeatedly edits a single short NumPy program under a leakage-free objective, keeping the best-scoring detector it finds. The loop discovers two compact detectors, one for univariate and one for multivariate series, that describe short windows by their local spectral features and compare them with the training-region distribution through a covariance-aware distance. On the TSB-AD benchmark these detectors lead the field across metrics, ahead of the strongest classical, deep, and foundation-model baselines including Time-RCD, yet they train no network and use no GPU, and the multivariate detector is faster than every similarly performing baseline. LLM-driven program search is thus a practical route to accurate, efficient, and transparent detectors.

Abstract

Time-Series Anomaly Detection

Explore similar work

Have an LLM Write Your Anomaly Detector: Autonomous Discovery of Compact, Interpretable Detectors for Time Series

Solve for the Hyperparameter, Skip the Search: Kolmogorov-Optimal Scaling Laws for Spline Regression

Unsupervised Symbolic Anomaly Detection