Extended Score: Advanced Scoring Frameworks

• Extended Score is a comprehensive framework that generalizes traditional scoring functions by integrating high-precision temporal controls and semantic alignment across multimedia, forecasting, and anomaly detection.
• It employs innovative methods such as multivariate L²-scoring, level set evaluation, and combined histogram-based outlier detection to improve diagnostic robustness and predictive accuracy.
• The framework also advances music transcription and signal processing through precise micro-macro scheduling and adaptive denoising techniques, enabling scalable and interpretable applications.

The concept of "Extended Score" encompasses a range of advanced theoretical and practical developments in scoring, representation, and control frameworks across diverse fields, including multimedia scenario design, statistical forecast evaluation, signal processing, anomaly detection, and symbolic and temporal music transcription. In all cases, the “extended score” refers either to a generalization of traditional scoring functions (e.g., incorporating richer structure, precision, or semantic alignment) or to a framework in which control and synchronization are made more precise—often at multiple levels of abstraction.

1. Generalizations in Interactive Multimedia Scenario Design

The interactive score formalism was originally conceived for coordinating multimedia events such as sound, video, and lighting through explicit temporal constraints, hierarchies, and flexible scheduling. Traditional methods separated fixed timelines (macroform) and cue lists (microform), lacking temporal and structural interconnectedness. The extended interactive scores framework builds on this foundation by introducing:

• High-Precision Temporal Relations: Sub-second delays and sample-level control (e.g., 500 μs offsets for sound events), formally described by constraints such as

$$\begin{aligned} \text{start\_time} + \text{duration} & = \text{end\_time} \ \text{time\_of\_point}_1 + \Delta & = \text{time\_of\_point}_2 \end{aligned}$$

with $\Delta$ in seconds for macro controls and as fixed sample intervals for micro controls.

• Dataflow Relations: Modeling the movement and transformation of signal data between temporal objects using formal dataflow links, enabling complex sound synthesis or inter-object dependencies.

The practical implementation leverages the ntcc (Non-deterministic Timed Concurrent Constraint) model for macro scheduling and the Faust DSP language for sample-accurate micro controls, integrated within the Pure Data (Pd) environment. Jitter measurements (~500 μs at 85% CPU load) demonstrate robust temporal accuracy essential for live multimedia performances (Toro et al., 2015).

2. Extensions of Scoring Rules for Multivariate and Level Set Evaluation

In forecast evaluation and probability distribution assessment, traditional scoring functions such as the quadratic score or log score are extended via new frameworks:

• L²-Scoring Rule Framework: Scores of the form

$$S'(P_x, y; w, h) = \int_{\mathbb{R}^d} \big[(f_x * w)(z) - w(z-y)\big]^2 h(z) dz$$

generalize both quadratic scoring and multivariate CRPS (Continuous Ranked Probability Score). The smoothing function $w$ allows for construction of novel scores; for instance, using higher-order convolutions (e.g., lower partial moments) directly captures tail risks and other application-specific features (Meng et al., 2020).

• Level Set Scoring: For probabilistic anomaly detection or risk measurement, scoring functions over regions (or "level sets") are derived via layer-cake decompositions of the L²-score, providing measurement and calibration tools for density level sets, CDF level sets, and tail risk domains.

A Monte Carlo algorithm with error order $O(M^{-1/2})$ is used for efficient estimation, supporting applications such as forecast combining via quadratic programming, and CoVaR estimation in risk management.

3. Extended Scoring in Anomaly Detection: Modeling Feature Dependencies

The Extended Histogram-Based Outlier Score (EHBOS) enhances the classical HBOS method, which relies on independent univariate histograms, by introducing:

• Two-Dimensional Histogram Scoring: Joint density estimation for feature pairs

$$h_{jk}(x_{i,j}, x_{i,k}) = \frac{n \, \text{count}\big((x_{i,j}, x_{i,k}) \in \text{bin}_{bjk}\big)}{\text{bin\_area}_{bjk}}$$

with anomaly scores aggregated over all pairs,

$$s^{(2D)}_i = \sum_{j=1}^d \sum_{k=j+1}^d -\log(h_{jk}(x_{i,j}, x_{i,k}))$$

• Combined Scoring: Aggregation of 1D and 2D contributions:

$$S^{(EHBOS)}_i = \alpha\, s^{(1D)}_i + \beta\, s^{(2D)}_i$$

EHBOS demonstrates marked improvements over HBOS in datasets with relevant feature interactions, with ROC AUC gains of up to 0.85–0.91 in evaluation benchmarks (Islam, 8 Feb 2025).

4. Temporal and Symbolic Extensions in Music Transcription and Tracking

Music transcription and score following systems expand the classical notion of score to include both symbolic and temporal alignment:

• Time-Aligned Score Generation: The transcription output jointly encodes onset, offset, pitch, and discrete note value tokens:

$y_i = (\hat{t}_i, p_i, \tilde{t}_i, v_i)$

with note value $v_i$ typically quantized to sixteenth-note units. End-to-end Transformer architectures are trained directly on these quadruple-sequence tokens.

• Pseudo-Labeling for Rhythm: When datasets lack ground-truth note values, detected beats are subdivided, onsets and offsets are quantized to the sixteenth grid, and note values are computed as grid differences. This approach enables rhythm-aware transcription models.
• Symbol-Level Score Following: Real-time tracking systems first transcribe live audio to note events. These are then aligned via customized Online Time Warping (OLTW), which uses a pairwise distance

$\text{pd} = e^P + c \cdot e^T$

where

$e^T = |\hat{b}^T_j - b^T_j|$

and $e^P$ reflects pitch errors. The system maintains high robustness and precision in score-position reporting (Kim et al., 18 Feb 2025, Peter et al., 8 May 2025).

5. Semantic and Structural Robustness in Document Parsing

Within generative document parsing, conventional metrics often misjudge semantic equivalence when output formats diverge. The SCORE framework corrects for this by:

• Adjusted Edit Distance: Using word-weighted or fuzzy element alignment,

$$\text{NED}_{adj}(s,g) = \max\left\{\text{NED}(s,g), \frac{\sum_k W_k}{W_{total}}\right\}$$

to tolerate reordered, flattened, or hierarchical markup.

• Token-Level Diagnostics: Differentiating omissions from hallucinations:

$$\text{TokensFound}(s,g) = \frac{\sum_{t} \min(\text{freq}_s(t),\, \text{freq}_g(t))}{\sum_{t}\text{freq}_g(t)}$$

$$\text{TokensAdded}(s,g) = \frac{\sum_{t} \max(0,\, \text{freq}_s(t) - \text{freq}_g(t))}{\sum_{t}\text{freq}_s(t)}$$

• Spatial and Semantic Table Analysis: Robust F-measure and index accuracy with spatial tolerance, normalization to format-agnostic representations.

SCORE corrects 12–25% ranking distortions found in standard metrics on ambiguous tables and achieves F1 up to 0.93 in generative-only settings, elucidating both semantic diversity and system behavior (Li et al., 16 Sep 2025).

6. Extended Score in Signal Processing and Denoising

Score-based frameworks for signal denoising generalize standard denoising score matching (DSM) to operate directly on noisy data, as in Corruption2Self (C2S):

• Generalized DSM (GDSM) Loss:

$$J(\theta) = \mathbb{E}\left[\| \gamma(t, \sigma_{target}) h_\theta(X_t, t) + \delta(t,\sigma_{target}) X_t - X_{t,data}\|^2 \right]$$

with adaptive noise level reparameterization.

• Detail Refinement and Multi-Contrast Extension: Predicts $E[X_{t, target}|X_t]$ to maintain fine spatial details and leverages complementary contrasts in MRI by feature fusion.

Quantitative results show state-of-the-art self-supervised denoising and competitive performance with supervised MRI restoration (Tu et al., 8 May 2025).

7. Implications for Future Research and Application

Across these domains, extended score frameworks suggest unified principles:

• Rigorous Multiscale Evaluation: By integrating micro-temporal, macro-temporal, and semantic levels within a single system, scenario designers and forecasters can maintain both high precision and interpretive flexibility.
• Interpretable Diagnostics and Scalability: By exposing representational diversity and feature dependencies, these extended scoring systems enable more robust, fair, and interpretable benchmarking—particularly in generative modeling, risk analysis, anomaly detection, or multimedia control.
• Potential Extensions: Research may explore further generalizations, such as adaptive hierarchical scoring, domain-specific normalization, or multi-modal integration (e.g., extending time-aligned scores to video or document evaluation to multi-modal outputs).

A plausible implication is that extended score frameworks, by embracing structural diversity while enforcing rigor, enable practitioners to design systems and evaluation protocols that capture the true semantic intent and fidelity of complex outputs, regardless of format or underlying representation.