gaussian process

Let \(X_t\) be a random variable indexed by \(t \in T\). Think of \(t\) as being time. Stochastic processes are sets of random variables of the form

\[\{ X_t : t \in T \}.\]

For actual data we can only ever have one observation of each \(X_t\), which is why more assumptions are helpful.

Special Cases

• If \(X_t\) is vector valued then this is called a random field.
• If a linear combination of samples from a stochastic process is jointly normal then this is called a Gaussian process, or Gaussian random field for the multivariate case.
• If \(T\) is discrete and the value of \(X_{t+1}\) only depends on \(X_t\) then the process is a discrete time Markov chain.