paramtm

Paramtm is a term used in some technical writings to denote a parameterized temporal model. It is not a standard term in established textbooks, and its usage can vary between authors. In this context, a paramtm denotes a model in which the state or output depends on time and one or more parameters, typically written as x(t; p). The parameter vector p may be constant, piecewise constant, or itself a function of time.

Definition and notation: A paramtm can be viewed as a mapping F: T × P → Y, where T is the time domain, P is the parameter space, and Y is the outcome space. The observed or evolved quantity is described by y = F(t, p) or by a dynamic form such as x' = f(x, t; p). When p varies with time, one writes p = p(t), yielding an explicitly time-varying model.

Relation to parametric models: Unlike static parametric models with fixed parameters, a paramtm emphasizes the time dimension in the parameterization. This makes it a convenient framework for describing non-stationary systems, adaptive settings, or scenarios where structural changes are captured by evolving parameters.

Applications: Paramtms are used in control systems with time-varying gains, signal processing with evolving filter coefficients, epidemiological models with changing transmission rates, and economic models incorporating policy shifts or regime changes.

Variants and practice: In discrete time, t takes values in integers; in continuous time, t is real-valued. Notation and rigor depend on the source, with paramtm often implemented within differential equations, state-space representations, or regression-type models that incorporate p(t).

Limitations: The lack of a universal definition means precise meaning is source-dependent; explicit specification of F, p(t), and the domain is essential.