Mittausfunktio

Mittausfunktio, or measurement function, is a concept in mathematical modelling and signal processing that describes how the internal state of a system is mapped to observable measurements. In a discrete-time state-space model, the system state x_k evolves according to a state-transition function, while the observation or measurement y_k is related to the state by the mittausfunktio h, often with added noise: y_k = h(x_k) + w_k. The function h can be linear, h(x) = Hx, or nonlinear, such as h(x) = [x1^2, sin(x2)]^T.

Properties commonly associated with the mittausfunktio include determinism given the state, and potential smoothness or differentiability. In many applications, h is assumed to be Lipschitz continuous to support the stability and convergence of estimation algorithms. The exact form of h depends on the sensing modality and the physical relationship between the state and the observed data. In some contexts, h may depend on auxiliary inputs or calibration parameters, but its essential role is to translate latent state variables into measurements that can be collected by sensors.

Applications of the mittausfunktio span state estimation, filtering, and sensor fusion. It is central to methods such as the Kalman filter and its nonlinear variants, where the predicted state is mapped into expected measurements for comparison with actual observations. Distinctions are often drawn between the mittausfunktio (observation model) and the state-transition function that governs state dynamics, with both together forming the broader state-space model. In Finnish usage, mittausfunktio is a key term in control theory, robotics, and related fields.