Linear Model

Definition

A class of problems where the Minimum Variance Unbiased Estimator always exist and is readily available to us.

Linearity is actually present (or at least approximately) for many real world problems.

$x=H\Theta +w$ Where $w[n]\sim N(w;0,{\sigma }^2)$, we have $N$ observations and the parameter vector has $d$ elements.

Link to original

---

Linear Mapping General Linear Model The Estimator for the Linear Model Linear Model with Non-White Noise

Some Linear Models

In the models that follows $x$ is our observation, $t$ is time and $w$ is the observation noise having a normal distribution, $N(w;0.{\sigma }^2)$

Constant plus noise:

• $x=A+w$, for example DC voltage with noise Linear trend:
• $x=At+B+w$ Linear trend plus seasonal variation:
• $x=At=B\sin (\omega t)+w$ General:
• $x=A{g}_1(t)+B{g}_2(t)+C{g}_3(t)+w$, where $\{g_i\}$ are arbitrary functions that cover all the above examples.