Understanding the sample space and probability distribution of a Gaussian process

Question

I'm trying to unpack the mathematical definition of a Gaussian process, by applying it to an example problem of modelling the height $h(t)$ of the tide at time $t$.

Let's begin with two definitions of a Gaussian process:

From Wikipedia:

A Gaussian process is a stochastic process $X_t$ such that for every finite set of indices $t_1,\ldots,t_k$ in the index set $T$, $$(X_{t_1},\ldots,X_{t_k})$$ is a multivariate Gaussian random variable.

From Gaussian Processes for Machine Learning by Rasmussen & Williams:

Definition 2.1 A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution.

A Gaussian process is completely specified by its mean function and covariance function: $$m(\mathbf{x})=\mathbb{E}[f(\mathbf{x})],$$ $$k(\mathbf{x},\mathbf{x}')=\mathbb{E}[(f(\mathbf{x})-m(\mathbf{x}))(f(\mathbf{x}')-m(\mathbf{x}'))].$$

From the definition of a stochastic process, for each $t$, $X_t$ is a measurable function from a probability space to the reals: $$X_t:(\Omega,\mathcal{F},P)\rightarrow R$$ It seems to me that taking $k=1$ in the multivariate Gaussian condition above, immediately implies that the probability distribution $P$ is Gaussian. Moreover, doesn't this require that $P$ has mean $m(t_k)$ and variance $k(t_k,t_k)$? But wouldn't this require $P$ to depend on time?

Now let's say I want to model the height $h(t)$ of the tide at some time $t$. Then $X_t$ would represent the height at time $t$. The sample space $\Omega$ would be the real line, and $\sigma$-algebra $\mathcal{F}$ the Borel sets. But what is the probability measure $P$? I believe this must be a Gaussian distribution, and the kernel $k$ gives me the variance. But what do I take as the mean?

Answer 1

Your description here is accurate. I think the problem is with the notation provided in Rasmussen & Williams which I think swaps out $t$ for $\mathbf{x}$ and $X$ for $f$

Consider that your GP doesn't need to be stationary, so $P$ depending on time is not a problem: that is, $X_t$ and $X_{t'}$ do not need to follow the same distribution. In particular, what you say about $m(t_k)$ and $k(t_k,t_k)$ is right on, it absolutely depends on time.

For modelling tide, take a look at section 2.7 of Rasmussen. If you have prior knowledge of how the tide works, ie. its mean and variance as a function of $t$, then you can incorporate that into the description of a gaussian process. For instance,

$h(t)\sim \mathcal{N}(\alpha(t), \beta(t))$

A more familiar example for me would be $d(t)$, "how far am I from my bed at time $t$". For all $t$ corresponding to nighttime, $\alpha=0$ and the variance is very tight. For all $t$ corresponding to daytime, the mean and variance would change dramatically.