Weighted Mean Frequencies (WMF)

Updated 1 July 2025

Weighted Mean Frequencies (WMF) are statistical tools that aggregate frequency measurements with assigned weights to optimize estimation accuracy.
They are applied across diverse fields—such as metrology, stochastic processes, and medical imaging—to enhance data reliability and interpretability.
Methodologies like inverse-variance weighting, hierarchical averaging, and graph-based selection allow WMF to efficiently handle uncertainties and heterogeneities.

Weighted Mean Frequencies (WMF) refers to a broad class of statistical and analytical constructions in which frequencies—arising in physical measurement, stochastic processes, or signal analysis—are aggregated using explicit weights to form summary metrics. The concept has a rigorous presence in the literature spanning experimental metrology, stochastic point process theory, systems of coupled oscillators, stochastic control, and, most recently, medical imaging and data science. WMF and related constructs, such as weighted mean marks and weighted ensemble frequencies, provide both improved estimators and deeper interpretations for systems marked by heterogeneity, uncertainty, or complex temporal dynamics.

1. Fundamental Definitions and Mathematical Formulation

The general form of a Weighted Mean Frequency is grounded in the idea of combining observed or computed frequencies $\{f_i\}$ with associated weights $\{w_i\}$ . The prototypical definition is: $WMF = \frac{\sum_{i=1}^n w_i f_i}{\sum_{i=1}^n w_i}$ Weights can represent statistical confidence (as in inverse-variance weighting), event intensities, function values, or energy contributions, depending on the domain. In time series and signal analysis, WMF often involves averaging frequency components weighted by their spectral energy: $WMF = \frac{\sum_{i=1}^n E_i f_i}{\sum_{i=1}^n E_i}$ where $E_i$ is the energy at frequency $f_i$ .

For marked point processes, the weighted mean mark (a generalization of WMF) is given by: $\mu_f^{(1)} = \frac{ \sum_{(t, y, z)\in\Phi} z f(y) }{ \sum_{(t, y, z)\in\Phi} z }$ with $z$ corresponding to the weight assigned to the mark $y$ at point $t$ (1210.1335).

In physical measurement aggregation, WMF arises through the weighted average: $\bar{f}_w = \frac{\sum_{i=1}^n p_i f_i}{p}, \quad p_i = \frac{1}{s_i^2}, \quad p = \sum_{i=1}^n p_i$ where $s_i$ is the standard deviation or uncertainty of the $i$ -th measurement (1110.6639).

2. Statistical and Theoretical Principles

WFM serves as a robust estimator in the synthesis of heterogeneous or noisy data, correcting for measurement uncertainty, inconsistent input data, or varying event intensities. The classical weighted average acknowledges the variability in reliability among source data, giving higher influence to more precise or representative observations.

In metrological applications, estimating the uncertainty of WMF is critical. The standard approaches include:

Classical uncertainty estimate: $\sigma_1 = 1/\sqrt{p}$
Scatter-based estimate: $\sigma_2 = \sigma_1 \sqrt{H/(n-1)}$ , with $H$ quantifying observed scatter.
Robust combined estimate: $\sigma_c = \sqrt{\sigma_1^2 + \sigma_2^2}$ , recommended for its ability to adapt to both consistent and highly discrepant inputs (1110.6639).

In non-ergodic stochastic systems (e.g., marked point processes), the proper definition of WMF must account for realization-to-realization variation. A "hierarchical" or two-stage mean first averages within each ergodic class and then across classes, yielding an interpretation as a typical or regime-averaged frequency (1210.1335).

3. Applications and Interpretations Across Domains

Weighted Mean Frequencies are foundational in numerous areas:

Metrology and Physical Measurement: Combining independent frequency measurements or standards where uncertainty estimates may be inconsistent or understated (1110.6639).
Marked Point Processes: WMF (as weighted mean mark) is crucial for summarizing spatial event distributions in ecology, finance, and geosciences, with explicit choices in weights reflecting scientific objectives such as ecosystem average, resource-weighted density, or transaction-volume-weighted average price (1210.1335).
Coupled Oscillators: Distinction between mean field frequency (entrainment frequency) and mean ensemble frequency (population-average frequency) is essential. In systems with heterogeneity or asymmetry, the observed WMF may not correspond to the mean or mode of the underlying distribution, and can even fall outside its support (1302.7164).
Stochastic Control and Mean-Field Systems: In optimal control of weighted mean-field systems, WMF appears in the coefficients of stochastic differential equations, representing collective, weighted state effects crucial for applications like insurance portfolio management (2208.11679).
Signal and Image Processing, Medical Imaging: In 4D Flow MRI, WMF is constructed as an energy-weighted average of strictly positive Fourier frequencies per voxel, robustly distinguishing pulsatile fluid domains from static tissue—aiding vessel segmentation beyond what is possible with traditional PC-MRA (2506.20614).

4. Practical Computation and Robust Estimation Techniques

WMF computation must adapt to data structure and context:

Inverse-variance weighting: Recommended for combining independent measurements with known uncertainties. The robust combined uncertainty estimator, $\sigma_c$ , is preferred in contexts where input uncertainties may be unreliable or data are particularly scant or discrepant (1110.6639).
Hierarchical averaging: Necessary in non-ergodic or highly non-stationary systems. A first-stage average within homogeneous regimes, followed by an equal-weighted average across regimes, ensures that the estimator represents a typical system, not one dominated by high-intensity or high-frequency subpopulations (1210.1335).
Graph-based selection: In robust harmonic ENF estimation, weights may incorporate signal-to-noise measures, and harmonics are selected via maximum weight clique solution to exclude corrupted components, leading to direct calculation of WMF over the selected subset (2011.03414).
Functional and distribution-based weights: In the context of maximum weighted likelihood estimation, weights may be assigned based on data-dependent relevance, with probabilistic interpretations leading to Lehmer or Hölder mean generalizations—unifying statistical estimation and WMF computation (2305.18366).

5. Quantitative and Theoretical Implications

WMF estimators possess favorable statistical properties when constructed with attention to variance minimization and robustness:

Variance-optimal weighting: For independent, unbiased estimators, inverse-variance weighting yields the minimum variance for the WMF (1210.1335).
Moment and deviation control: WMF of rare events or frequencies can be rigorously bounded using weighted summability conditions, providing precise tail, moment, and exponential moment estimates for process excursions in probability theory (2204.04369).
Operator extensions and inequalities: WMF concepts extend to operator settings, with refined inequalities bounding the WMF between tight combinations of arithmetic and geometric means, applicable to matrix and quantum systems (2001.01345).

6. Impact in Data-Driven and Biomedical Applications

Recent advances demonstrate the critical practical impact of WMF:

4D Flow MRI Segmentation: WMF, as a temporal Fourier energy-weighted feature, substantially improves vessel segmentation quality, outperforming PC-MRA in both threshold-based and deep learning (U-Net) frameworks, with reported IoU and Dice increases of $0.12$ and $0.13$ respectively (2506.20614).
Pulsatility Markers: The WMF feature can localize regions of physiologically relevant, temporally varying flow, serving as a generalizable biomarker or auxiliary input across cardiac and neurovascular domains (2506.20614).
Generalizability and Interpretability: WMF's construction based on rigorous statistical principles permits its transferability and interpretability across application domains, enhancing reliability and supporting downstream inference and decision-making.

7. Summary Table: Representative WMF Use Cases and Methods

Application	Domain	Weight Definition	Key Impact/Interpretation
Frequency standards	Metrology	Inverse-variance	Robust aggregation of inconsistent measures
VWAP	Finance	Trade volume	Price weighted by transaction size
MRI segmentation	Medical imaging	Spectral energy (Fourier)	Segregates pulsatile flow in vessels
Point processes	Ecology/geoscience	Resource/event intensity	Mean trait per unit resource or event
ENF estimation	Forensics/signal	SNR, clique selection	Reliable aggregate frequency in noisy signals
Controlled SDEs	Stochastic control	Agent-specific (e.g., capital/liability)	Captures weighted collective effect

Weighted Mean Frequencies thus offer a mathematically rigorous, context-sensitive, and computationally adaptable methodology for extracting interpretable summary statistics from complex measured, simulated, or inferred data, across a wide spectrum of scientific and engineering fields.

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References (9)

Intrinsically Weighted Means of Marked Point Processes (2012)

On computation of a common mean (2011)

Mean Field and Mean Ensemble Frequencies of a System of Coupled Oscillators (2013)

Stochastic maximum principle for weighted mean-field system (2022)

Weighted Mean Frequencies: a handcraft Fourier feature for 4D Flow MRI segmentation (2025)

Robust ENF Estimation Based on Harmonic Enhancement and Maximum Weight Clique (2020)

Using maximum weighted likelihood to derive Lehmer and Hölder mean families (2023)

Moment estimates in the first Borel-Cantelli Lemma with applications to mean deviation frequencies (2022)

Refined inequalities on the weighted logarithmic mean (2020)