Stats 3B

Correlation(X,Y), Corr(X,Y)

Cov(X,Y) / signa(X) sigma(Y)

Variance of X based on Covariance

Covariance Function, C(h)

Covariance Vector

Covariance Matrix

Simple Kriging Prediction (BLUP Simple Kriging, Best Linear unbiased predictor)

Simple Kriging system equation, weights equation

Or

Simple Kriging Variance

Simple Kriging confidence interval

Does Simple Kriging Variance depend on observed values Z. True or False

False

Semigram conversion to Covariance function C

Simple Kriging form, Z(s)

Where m is a known constant

What two assumptions are made for simple kriging

Known constant mean and Known constant variance

Steps for a simple kriging question and ordinary kriging

1. Solve System Equation for weights λ(s₀)

2. Find predictior, Z^(s₀)

3. Find variance

4. Confidence intervals

The kriging variance is the prediction error variance not the variance of s₀

True

What is the interpolation property in kriging?

At an observed location, kriging reproduces the observed value exactly. If s_0=s_i, then \hat Z(s_0)=Z(s_i). The kriging weights become \lambda_i=1 and all others are 0, while the kriging variance becomes 0. Therefore kriging is an exact interpolator (without measurement error/nugget effect).

For non Gaussian processes is the kriging confidence intervals exact

False

How does the kriging hierarchy work? I.e what kriging is a special case of what kriging

Simple kriging is a special case of ordinary as ordinary assumes that the mean is constant but unknown and simple kriging assumes that the mean is know

Ordinary kriging is a special case of kriging where the number of basis functions, p=1, and that basis function f₁(s) = 1

Universal

| k=0, f₁(s) = 1

Ordinary

| mean is know

Simple kriging

By this simple kriging is a special case of universal where p = 1, f₁(s) = 1 and the mean is known

What is different with filtered kriging

Filtered kriging adds tilda² to the covariance matrix, Σ

Simple kriging

What’s different with ordinary kriging

The mean is a constant but it is unknown

Ordinary kriging system equation

The mean μ could be written as μ(s₀) but it is constant across all observations

Ordinary Kriging prediction

Ordinary Kriging Variance

The ordinary kriging variance σ²_OK is always at least as large as the simple kriging variance σ²_SK for the same data and prediction

True

What unbiasedness constraint is added in ordinary kriging?

Filtering kriging predicts the underlying process Y(s) not the noisy observation Z(s)

True, when doing filtering it it is Y hat (s₀) not Z hat s₀

Set up of the universal kriging system

where f_i(s) are basis functions

F matrix, for universal kriging

F(s₀) matrix

Universal Kriging system equation

Universal kriging variance using covariance function

What makes universal kriging unique compared to the other two?

That you don’t have to know the covariance function and it can be evaluated with the semivariogram

Universal kriging variance for semivariogram version

Predictive distribution and condition

Underlying process must be Gaussian

Conditional independence 3 functions X independent Y | Z

Conditional Partition where A and B are partition of the node set, A is the smaller set removed

Is X ⟂ Y | Z symmetric such that Y ⟂ X | Z

True, independence is symmetrical

Conditional distributions X_i | X_-i

Markov Properties

What is the hierarchical structure of the Markov properties

If you have the one, you have the smaller ones.

I.e if you have local, local implies pairwise

If X is a GMRF, does X satisfies all 3 markov condtions

True

Precision Matrix of Disconnected graphs, A and B

Unweighted Diagonal and Adjacency Matrices

Weighted Laplacian, diagonal and adjacency matrix

Where w_ij > 0 describes the strength of the conditional dependence between neighbouring nodes

Weighted laplacian to precision matrix

Where M is any positive definite matrix and theta 1 and theta 2 are > 0

Unweighted laplacian

Definition of White Noise

Conditions for weakly stationary

Autocovariance function γ(τ )

Autocorrelation function

MA process order q, mean zero

Beta_0 is the coefficient of Z_t, but is often set to 1

MA process order q, mean non zero

Is MA(q) a weakly stationary process

True

Autocovariance function of MA(q)

The auto correlation function is found by just diving by tilda(0)

AR(p) process

Backwards shift operator version of AR(p)

Backwards shift operator definition

Backwards shift version of MA(q)

ARMA definition

What does a causal ARMA process mean and what is it?

The ARMA process can be written as a function of it’s past shocks, MA(inf) representation and is found if the absolute value roots of AR part of the ARMA are greater than 1

What does Invertible ARMA mean?

The ARMA process can be written as a product of the observed values X_t, written as a AR(inf) and the absolutely value of roots of the MA part are greater than 1

ARIMA formula

Random field weakly stationary

Covariate notation Between C(s,t) and C(h)

A Gaussian process with finite second moment that is either strictly stationary or weakly stationary implies the other

True

Kriging distribution for all kriging types

Mean and covariance completely determine all finite-dimensional distributions

True

General isotropy definition

Condition of isotropy if Z_t is a weakly stationary process

The covariance function only depends on h via its norm ||h||

Three key conditions that the covariance function C(h) of a weakly stationary process must satisfy

Semivariogram formula

2 conditions of semivariograms

And conditional negative semi definite

Operations with semivariograms

If γ is a semivariogram than these are all also semivariograms

What is a nuggest

If a semivariogram has a discontinuity at 0, the size of the jump is called nugget

Covariance function definition

a function is a covariance function if it is symmetric and positive semi definite

Semivariogram conditional negative semidefinite

Brownian motion definition

Initial conditions for solving AR(2)

Real roots for AR(2) YW, b² -4ac > 0

AR(2) YW b² -4ac = 0

Complex roots AR(2) YW

Where roots are

Conditions for the roots in AR(2)