Integrime

Integrime is a theoretical construct in time-series analysis that describes a family of integral operators designed to transform irregular or sparse temporal data into smooth, continuous representations. The concept emphasizes the accumulation of information over temporal intervals and is intended to facilitate comparisons and fusion of measurements collected at nonuniform times.

Formally, given a time-series f(t) defined on a subset of the real line, an integrime operator I_gamma is defined by I_gamma f (t) = ∫ f(s) gamma(t−s) ds, where gamma is a temporally local kernel chosen to reflect the desired windowing behavior. Special cases include standard convolution with a fixed kernel and adaptive kernels that adjust to sampling density. Adaptive integrime selects gamma based on local data density or confidence estimates, while normalized integrime rescales by the integral of gamma to preserve scale across time. The framework subsumes certain smoothing techniques and generalizes moving-average constructs.

Variants of integrime include adaptive and normalized forms, with the former aiming to tailor the window to data availability and the latter ensuring consistent amplitude across time. The choice of gamma and Normalization can influence bias, variance, and interpretability of the resulting representation.

Applications and limitations: Integrime operators are used in sensor data fusion, neuroscience signal processing, economic time series, and environmental monitoring to reduce irregularity, stabilize features, and enable derivative-like analyses on irregular data. However, kernel choice, boundary effects, and computational cost are important considerations; mis-specification can introduce bias or oversmoothing, and interpretability depends on the selected gamma and windowing strategy.

History: The term "integrime" appears in theoretical discussions and pedagogical contexts in the 2020s as a way to describe integral-based time representations; it is not yet a universally standardized term across all subfields. See also: convolution, time-series analysis, kernel methods.