Lecture 2

Question 1

What is a Gaussian process (in words)

Answer 1

A Gaussian process is an infinite dimensional stochastic function all
of whose marginal, conditional and joint distributions are Gaussian.

Question 2

What is the pdf of a Gaussian distribution

Answer 2

1/{sqrt{2 pi} sigma} exp(-1/2 * (x- mu)^2/sigma^2)

Question 3

What is a strictly stationary process

Answer 3

• A strictly stationary process has the same statistical properties everywhere.
• Denote the process by Y .
• Then the distribution of Y (x) is the same as for Y (x + h)

Question 4

How can we work around the restrictions of stationarity

Answer 4

Stationarity appears to place great restrictions on our mean function
restricting us to constant mean processes.
But we can write
Y (x) = µ(x) + Z(x)
where Z(x) is a zero mean process
and µ(x) is deterministic

Question 5

What is a weakly stationary process

Answer 5

For a weakly stationary process the first and second order moments
are the same everywhere
So
Cov(x1, x2) = Cov(x1 + h, x2 + h)
and the covariance simply depends on distance.
This is also known as second order stationarity

Question 6

Does weak stationarity imply strict stationarity

Answer 6

It does for Gaussian process, but only for GPs. In general the converse always applys

Question 7

Intrinsic stationarity

Answer 7

Assume a constant mean process then
E(Y (x + h) − Y (x))^2 = Var(Y (x + h) − Y (x)) = 2γ(h)
If this only depends on h then the process is said to have intrinsic
stationarity

Question 8

Prove the covariance function of a weakly stationary process is positive definite

Answer 8

For a weakly stationary process Y we have for all ai

Var[Sum(aiY(xi))] >=0

= Sum(aiaj Cov(Y(xi), Y(xj))

= Sum(aiaj c(|xi-xj|))

Question 9

Bochners Theorem

Answer 9

Bochner’s theorem states that c(h)is positive definite if and only if

c(h) = Integral exp(iw^(T_h))G(dw)

Question 10

if G(dw) = g(w) dw what is the spectral density

Answer 10

Then g(w)/c(0) is the spectral density

Question 11

What are the implications of Bochners theorem (3 of them)

Answer 11

• We can find out if a proposed covariance function is positive definite
• We can generate neew covariance functions
• We can work in the Fourier domain if we want

Question 12

Isotropy

Answer 12

If the covariance depends only on distance (not direction) the process is isotropic

Question 13

What is special about the covariance function of an isotropic process

Answer 13

It’s Univariate

Question 14

Separability

Answer 14

A process is separable if
C((x1, y1)^T, (x2, y2)^T) = C_x(x1-x2).C_y(y1=y2)

The corellation structure int he x direction doesn’t change in the y direction