Parameter Estimation

Kalman, in all its variants, is classically viewed as a filter or state estimator. However, it is widely used, primarily in machine learning, to apply these techniques for estimating the parameters of a model (the meta-model):

$$\mathbf{y}_k = f(\mathbf{x_k}, \boldsymbol\beta)$$ (2.132)

where $\mathbf{y}_k$ are the system outputs, $\mathbf{x_k}$ are the inputs, and $f$ is a function based on the parameters $\boldsymbol\beta$ to be estimated. The concept of training, or fitting, the model consists of determining the parameters $\boldsymbol\beta$.

Kalman allows for the determination of parameters, which may be variable, of the model by using as the state to be estimated precisely $\boldsymbol\beta$, thereby obtaining an iterative system of the form

$$\left\{ \begin{array}{l} \boldsymbol\beta_{k+1} = \boldsymbo... ...{y}_k = f(\mathbf{x}_k, \boldsymbol\beta_k) \end{array}\right.$$ (2.133)

, where the optional noise $\mathbf{w}_k$ is used to model any variations of the model over time: the choice of the variance of $\mathbf{w}$ determines the responsiveness to changes in the model parameters.