MuRAL-CPD: Active Learning CPD

• MuRAL-CPD is a semi-supervised change point detection framework that combines active learning and multiresolution wavelet analysis for user-aligned temporal segmentation.
• It leverages a multilevel discrete wavelet decomposition to extract features across scales, enabling accurate change detection with minimal supervision.
• Empirical validation demonstrates that MuRAL-CPD efficiently tunes feature weights and thresholds to outperform prior methods on diverse real-world datasets.

MuRAL-CPD is a semi-supervised change point detection (CPD) framework designed for time series analysis where the aim is to identify temporal indices at which the statistical properties of the observed process shift. The method introduces active learning into a multiresolution wavelet-based backbone, enabling iterative human-in-the-loop supervision that aligns the detector’s output with task-specific, user-defined notions of change. By leveraging a multilevel discrete wavelet decomposition (MDWD) and user-queried feedback, MuRAL-CPD achieves high accuracy and interpretability with minimal supervision, outperforming or matching prior semi-supervised CPD approaches across diverse real-world datasets (Bertolasi et al., 28 Jan 2026).

1. Problem Formulation

Change point detection (CPD) in a time series $x \in \mathbb{R}^{d \times n}$ consists of estimating a set of change indices

$\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$

such that for each CP $\tau_j$, the data distribution changes,

$$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$

with $\phi_j \neq \phi_{j+1}$ being unknown. MuRAL-CPD adopts a semi-supervised paradigm: the user can annotate short temporal intervals $W_i$ providing binary labels (no CP, contains CP). This labeled dataset $\mathcal{D}_S$ steers optimization, ensuring the detector’s working definition of “change” coincides with the user’s preference.

2. Multiresolution Feature Extraction

The core of MuRAL-CPD’s architecture is the Multilevel Discrete Wavelet Decomposition (MDWD) using Daubechies-2 filters. For a $K$-level decomposition, the process iteratively computes

$$x_{l,k} = l \circledast x_{l,k-1}, \quad x_{h,k} = h \circledast x_{l,k-1}$$

for $k=1, \dots, K$, where $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$0 are low-pass/high-pass filters; each stage down-samples by factor 2. This yields a set of subbands $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$1, providing a multiscale representation.

Within each subband $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$2, for window size $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$3 at time $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$4, consider left/right windows: $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$5 For each, a Normal Discrepancy score is calculated: $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$6 where $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$7 are the sample covariances of the sliding window and its two halves. Each $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$8 is resampled via Fourier interpolation to length $\{\tau_1, \dots, \tau_N\} \subset \{1, \dots, n\}$9, yielding aligned features $\tau_j$0 for subsequent aggregation.

3. Active Learning and Query Strategy

MuRAL-CPD implements an active query loop, maintaining:

• $\tau_j$1: Unlabeled indices (initially all $\tau_j$2)
• $\tau_j$3: Labeled change points (user-annotated)
• $\tau_j$4: Nonnegative weights for each feature scale
• $\tau_j$5: Detection threshold

At each of $\tau_j$6 iterations:

1. Compute current scalar score:

$\tau_j$7

where $\tau_j$8 is a peak-prominence transform that subtracts the background from each value.

2. Select two maximally uncertain, unqueried indices relative to $\tau_j$9:

$$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$0

1. For each $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$1, define local window $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$2, query the user for true change points within $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$3, add new labels to $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$4, and remove $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$5 from $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$6.
2. Re-optimize $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$7 by minimizing the surrogate loss:

$$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$8

using standard F1-score, to maximize correspondence with user labels.

1. Update the score function $$X_i \sim \begin{cases} \phi_j, & i<\tau_j \ \phi_{j+1}, & i \ge \tau_j \end{cases}$$9 and repeat.

The initial threshold $\phi_j \neq \phi_{j+1}$0 is selected by the curvature (“elbow”) heuristic: for sorted, normalized $\phi_j \neq \phi_{j+1}$1, $\phi_j \neq \phi_{j+1}$2, where

$\phi_j \neq \phi_{j+1}$3

and set $\phi_j \neq \phi_{j+1}$4, where $\phi_j \neq \phi_{j+1}$5 is the piecewise-linear curve of the scores.

Bayesian optimization (implemented via Mango) tunes $\phi_j \neq \phi_{j+1}$6, triggered after the first 10 queries and every 2 queries afterward.

4. Complete Algorithmic Workflow

The MuRAL-CPD pipeline consists of the following stages:

1. Receive input time series $\phi_j \neq \phi_{j+1}$7.
2. Apply $\phi_j \neq \phi_{j+1}$8-level MDWD, yielding subbands $\phi_j \neq \phi_{j+1}$9.
3. Compute disparity features $W_i$0 for each subband and upsample to $W_i$1.
4. Aggregate features with nonnegative weights: $W_i$2.
5. Initialize $W_i$3; set threshold $W_i$4 via the curvature elbow method.
6. For up to $W_i$5 active queries:
   • Identify two uncertain points.
   • Obtain user feedback on local windows.
   • Update labeled/unlabeled sets.
   • Re-optimize the feature weights and threshold.
   • Recompute detection scores.
7. Output predicted change points: $W_i$6.

Key tunable hyperparameters are the weight vector $W_i$7 (by scale), the decision threshold $W_i$8, number of wavelet levels $W_i$9, window size $\mathcal{D}_S$0, and query window half-width $\mathcal{D}_S$1. Bayesian optimization operates in a search space of size 5000 with up to 50 function evaluations per cycle.

5. Empirical Validation

MuRAL-CPD was evaluated against semi-supervised and unsupervised baselines on various real-world datasets:

Dataset	Key Settings ($\mathcal{D}_S$2, $\mathcal{D}_S$3, $\mathcal{D}_S$4)	F1 ($\mathcal{D}_S$5 std) after $\mathcal{D}_S$6 queries	ICPD Baseline
BabyECG	5, 15, 15	$\mathcal{D}_S$7 (50)	$\mathcal{D}_S$8
Honeybee Dance	5, 30, 15	$\mathcal{D}_S$9 (30)	$K$0
UCI-HAR	2, 12, 8	$K$1 (100)	$K$2
USC-HAD	6, 100, 100	$K$3 (0 to 50)	--

Datasets include infant heart-rate (BabyECG), 3D bee flight trajectories (Honeybee), multi-sensor human activity recognition (UCI-HAR, USC-HAD). Precision, recall, and F1 are measured within a tolerance window $K$4.

Ablation studies on Honeybee Dance reveal that threshold initialization by the elbow rule accelerates convergence (early F1 $K$5 at 5 queries vs $K$6 for max initialization), batching queries two-at-a-time improves stability, and a warm-up phase before optimization is beneficial for recall and early F1.

6. Interpretability and User Alignment

MuRAL-CPD’s design permits user-guided adjustment of sensitivity to different temporal scales by re-weighting $K$7: larger values heighten response to subbands depicting either abrupt or gradual changes. The peak-prominence transform $K$8 yields well-separated score peaks, clarifying which regions exceed threshold and thus enhancing interpretational transparency.

Active learning queries are confined to small windows, minimizing required user labeling per iteration. Empirical studies indicate that after few feedback rounds, MuRAL-CPD rapidly eliminates spurious detections and conforms its output to the desired “meaningful change” for the application (e.g., major shifts in heart rate versus minor fluctuations).

7. Comparative Performance and Limitations

On all tested datasets and across multiple query budgets, MuRAL-CPD consistently matches or surpasses the performance of ICPD (a semi-supervised one-class SVM on TIRE embeddings), especially in low-supervision regimes. Notably, in the USC-HAD dataset, the F1-score of MuRAL-CPD increases from approximately $K$9 (unsupervised) to $$x_{l,k} = l \circledast x_{l,k-1}, \quad x_{h,k} = h \circledast x_{l,k-1}$$0 (after 50 queries), with precision surging after threshold re-estimation and recall improving subsequently.

A plausible implication is that MuRAL-CPD’s scaling and hyperparameter tuning mechanisms allow it to adapt more efficiently to user-specific definitions of change with less annotation effort than direct classifier-based approaches. However, effectiveness may depend on the informativeness of the initial active queries and appropriateness of wavelet decomposition levels for the domain context (Bertolasi et al., 28 Jan 2026).