Why is the concept of a "state covariance matrix" necessary in estimation?

Question

I'm currently taking a course in optimal estimation (and it's still very early in the course).

Much of our work is based around the idea of a measurement model

$y=Hx + v$

This model assumes our measurement $y$ is a function of some "state" and measurement noise.

Now when attempting to estimate the state $x$ in a stochastic least squares sense, one of the goals is to minimize the trace of the state covariance matrix $P$.

My question revolves around the idea of a state covariance matrix. Why exactly does it make sense to assume such a thing exists? By doing so we making the assertion that our "state" is a random variable.

Let's say we are trying to estimate the resistance of an unknown resistor by inserting a known voltage and measuring the current. In this case, our "state" is the resistance. Why would it make sense to assume our resistance is a random variable?

Do we assume the state is a random variable since there might be uncertainty in our model?

Answer 1

A resistance isn’t a particularly dynamic state, should be an unknown constant, but you might have a bin of resistors where they might vary. Taking one “randomly” out the bin makes it a random variable. The bin of resistors will have a mean value, so perhaps that mean constitutes a state variable. Perhaps someone starts putting resistors in the bin from a different source where they tend to run high, so the mean changes.

You don’t “have” to assume that the state is random if the model isn’t really random. Later you will find out in your class that states do not have to be stochastic. The KF equations will work for deterministic states. The measurement does need to have noise. If it doesn’t have measurement noise, you use a state observer, not a KF.

A random state is actually very general and modeling errors are often successfully considered noise.

Typically we use states because they encapsulate all the previous states. You use the current state to move to the next state. you don’t need to consider any past states to go to the next state. Some people will say a state has memory. If the state is stochastic, it has the Markov property. Your measurement depends only on the current state.

The idea of randomness has many interpretations. It is part of what makes the topic interesting.