0
$\begingroup$

I'm reading a really nice discussion of Kalman filters An Introduction to the Kalman Filter G. Welch, G. Bishop (SIGGRAPH Course 8, 2001) and as I usually do I'm trying to understand the math by "reading around" some of the notation I don't understand or remember, or just "getting it" from context. This usually works for me well enough but this time I'd really like to make sure since I'm going to implement a Kalman filter from scratch by reading about it here and then doing it.

I'm going to reproduce a small section here then ask my question(s):

4.1 The Discrete Kalman Filter

This section describes the filter in its original formulation (Kalman 1960) where the measurements occur and the state is estimated at discrete points in time.

4.1.1 The Process to be Estimated

The Kalman filter addresses the general problem of trying to estimate the state $x \in \Re^n$ of a discrete-time controlled process that is governed by the linear stochastic difference equation

$$x_k = Ax_{k-1}+Bu_k+w_{k-1},$$

with a measurement $z \in \Re^m$ that is

$$z_k=H x_k + v_k.$$

The random variables $w_k$ and $v_k$ represent the process and measurement noise (respectively). They are assumed to be independent (of each other), white, and with normal probability distributions

$$p(w) \sim N(0, Q),$$ $$p(v) \sim N(0,R).$$

In practice, the process noise covariance Q and measurement noise covariance R matrices might change with each time step or measurement, however here we assume they are constant.

Q1: Does $x \in \Re^n$ mean for my practical purposes that $x$ represent an array of real numbers with length $n$? I know the notation is from set theory, but for scripting purposes, would $\Re^n$ suggest 1D array of floats of length $n$?

Q2: Does $p(v) \sim N(0, R)$ mean that the probability distribution of possible values of $w$ is a normal distribution (thus the "$N$") with a centroid at zero and a standard deviation of $Q$? Why the use of "$\sim$" instead of "$=$"? Is $N$ not normalized?

$\endgroup$
0
$\begingroup$

I'll re-post the answer as an answer, since SE comments can be considered ephemeral.

Q1: Does $x \in \Re^n$ mean for my practical purposes that $x$ represent an array of real numbers with length $n$? I know the notation is from set theory, but for scripting purposes, would $\Re^n$ suggest 1D array of floats of length $n$?

YES, per @MauroALLEGRANZA's comment.

Q2: Does $p(v) \sim N(0, R)$ mean that the probability distribution of possible values of $w$ is a normal distribution (thus the "$N$") with a centroid at zero and a standard deviation of $Q$? Why the use of "$\sim$" instead of "$=$"? Is $N$ not normalized?

Here $N$ is a Multivariate normal distribution per @MikeMathMan's comment, what I would loosely call an $n$-dimensional Gaussian with $R$ being an array of $\sigma^2$ values, the squares of the standard deviations.

Per @MikeMathMan's comment the tilde "$\sim$" used as its own symbol (rather than on top of something else) is described in the Wolfram MathWorld article on Tilde, where item #4 applies here:

The tilde is sometimes used as its own symbol.

  1. In asymptotic notation, f∼phi is used to mean that f/phi->1.

  2. Physicists and astronomers use same notation to mean "f is of the same order of magnitude as phi."

  3. In set theory, x∼y means that there is an equivalence relation between x and y.

  4. In statistics, the tilde is frequently used to mean "has the distribution (of)," for instance, X∼N(0,1) means "the stochastic (random) variable X has the distribution N(0,1) (the standard normal distribution). If X and Y are stochastic variables then X∼Y means "X has the same distribution as Y.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.