Past papers

mathematically criteria for weakly stationary process

C(h) must be even meaning

C(h) = C(-h) for all h

secondly the function C(S, S+h) must be non neg definite, meaning,

sum(n,j=1)sum(n,k=1) ajakC(Sj,Sk) > 0

for any scalar and location

equation to make semi variogram from cov function

γ(h) = C(0) − C(h)

from a cov function what are the

partial sill

sill

range

nugget

• limiting value of cov function as h → o from the right

• value of cov function at h=0

• min distance h at which the cov function is exactly 0 (often infinte as doesnt happen)

• sill minus partial sill

can the short range spatial correlation be identified in empirical semi varigrams

no because it is masked by the long range trend

what can you examine instead to see short range spatial correlation

examine RESIDUALS from a model on a empirical variogram, with 95% monte carlo intervals.

looking at the residuals removes the linear spatial trend so presence or absence of short range spatial correlation will be evident

how can i check if data is isotropic.

compute directional empirical semi variogram for the residuals from linear model trend

if four directions look similar then isotropy can be assumed.

what is a short guildline or steps of predicting geostatistical data

1. fit regular grid of predict locations over location

2. fit spatial cor model (with linear trend of not) to the data, estimating the mean and cov parameters

3. from these parameters sue kriging to predict at these locations

4. quantify uncertainty in these predictions (se maybe)

local indicator of spatial association version of Morans stat.

Ii = n(Zi − Z¯) ∑ wij (Zj − Z¯) / ∑(Zi − Z¯)²

what does this measure

strength of spatial correlation between i and its neighbours

if area i is an outlier then Ii will be a negative

Geary’s C stat =

C = (n-1)∑∑wij(Zi - Zj)² / 2(∑∑wij) ∑(Zi - Z¯)²

how do we interprete Georys C stat

near 0 / very small then strong positive correlation

near 1 then independent

more than 1 or very large then uncorrelated or negativly correlated with neighbours

what is a good way to choose a model for areal data

DIC

What DIC is best

minimum deviance info criterion value

so small DIC better trade off between goodness of fit and model simplicity

why is DIC good for areal data

prediction is not our primary goal, rather its to understant spatial structure

so fit of observed data is crucial

LOOK AT QUESTION 2bii paper 2021

in a CAR model what does wij=0 mean

then zi and zj are conditionally independent

definition of completely spatially random (whats this got to do with spatial dependency)

Points are randomly scattered across the study region

independence in space

definition of a regular process (whats this got to do with spatial dependency)

points try and stay away from each other

negative spatial dependency

Clustered process (whats this got to do with spatial dependency)

points clustered together in groups

corresponds to positive spatial dependency

Ripleys K function equation

K( r ) = 2𝜋 ∫ t p(t) dt

where p(t) is the pair correlation function

what is K under CSR

𝜋r²

if K( r ) is ≥ 2𝜋r²

if K( r ) is ≤ 2𝜋r² then

• clustard process

• regular process

describe how Monte Carlo envelopes could be constructed for plot of r (distance) vs Kobs( r ) - Kpoi( r )

• generate large number of simulated point process pattern from homogeneous poisson process (model corresponding or CSR)

• for each compute K function and

  K( r ) - 𝜋r²

• each distance r compute 2.5th and 97.5th percentile of the dis of 100 values of k( r ) - 𝜋r²

• forming MC envelopes of distance r

In a regression model using spatial data, what is the impact on parameter estimates and confidence intervals if spatial correlation is wrongly ignored (i.e., assuming independence when there is actually spatial dependence)?

The point estimate of the regression coefficient (e.g., β1\beta_1β1​) remains unbiased, but the confidence interval will be too narrow. This is because assuming independence incorrectly treats the observations as if they provide more information than they actually do. In reality, spatially correlated data are not fully independent, so the true uncertainty is underestimated, leading to overconfident (too precise) inference.

symbols for

partial sill

nugget

range parameter

• (σ² , τ² , φ^ ) easy mate

2 assumptions about geo stat model with (expo, sph,…) cov model

assumes spatial correlation structure in the residuals is weakly stationary and isotropic

assumption the mean model assumes

Z ~ N(B0 +B1X, ..)

assumes the cov is linearly related to the responce variable

spacial correlation range =

min distance at which cor is exactly 0

partial spatial correlation range

min distance at which cor falls to 0.05

how do we compute the signal noise ratio

partial sill (σ²) = spatial signal (variance due to spatial correlation)

nugget τ² = noise (random measure error or microvariation)

thus the ratio is

σ²/ τ² = 10 for example

then spatial correlated variation in the DATA is 10 times more likely then in the random measurement error

how can we check that Σ(θ) captures the spatial correlation in the data

• check residuals from model for presence of spatial autocorrelation using variogram analysis

• raw data are correlated

• compute innovation residuals

  u=L^-1𝜀

• where u = (u(s1),…,u(s))

• and L^-1 comes from

  Σ(θ) = LL^T

• if model has appropriate spatial correlation structure then the innovation residuals will be independent

• binned empirical semi variogram for u’s to asses the presence or absence of spatial cor using MC envelopes generated under the assumption of independence

youve got this queen

when choosing a covariance model i geo stats how can we pick one if

• our goal is to estimate B1

• our goes is to predict unmeasured loactions

• model fit stat such as AIC or BIC

• predictive measures of model accuracy such as leave one out cross validation

what much we have in our geostat model to make a prediction of our process

we need the covariates measured at all prediction locations.

basically if Z ~ N(B0 +B1X, ..) then we need X estimated for all locations not just the diagonal

in a proper Car model then what is

p =

τ² =

describes the amount of spatial dependence in the data

describes amount of spatial variation in the data

what happens to a proper car model when

p=0

the process becomes indep in space (because the mean goes to zero) meaning zi doesnt depend on any of its neighbours however the variance is still effected on the number of neighbours

this aint vey legit is it

what model do we thus prefer

the Leroux CAR model

as this has a P in the variance

what does Leroux CAR model simplify to when p = 0

N(0,τ²)

the equation to compute the partial correlation implied by the ICAR model for Zr,Zs

Corr[Zr, Zs|Z−rs] = wrs/sqrt{(∑_kwrk)(∑_kwsk)}

hypothesises of the quadrats test

H0: Point process is consistent with CSR

H1: Point process is not consistent with CRS

test stat for Quadrats test

X² = ∑^r ∑^c {(n_ij - n^-) / n^-}

where we have r rows and c collumns in our grid and ∑^r ∑^cn_ij = n

and n^- = n/rc

what do we compare this test stat to

X² ~ approx X²_rc-1 if H0 if true

if the test stat is rejected and

• higher then the dis

• lower than the dis

then what does this tell us

we have clustering

we have regularity

what process does complete spatial randomness follow

homogeneous poisson process

equation linking first order intensity function and the expected number of points in a domain

E[Z(D)] = ∫_A λ(s) ds

difference in range between autocov functions of

expo and spherical

does this difference matter

range for expo model is infinte but for spherical model its finite. this doesnt matter in practical terms as for the expo model the range can get arbitary close to 0 at a finite range

check ∑(ϴ) is should be modelled as an isotropic process then we what??

compute residuals,

r = z - B^0 + B^1X (or what ever is our mean in our Z ~ normal model) ~ N(0, ∑(ϴ) )

where B^0 and B^1 are estimated by ordinary least square.

then compare semi-variograms in different directions. If they look similar, the data is isotropic; if not, it's anisotropic

whats one disadvange of AIC

AIC is an asymptotic criterion and thus becomes less reliable as the data set size decreases.

C(h)={ 0​ if h>0

1. if h=0​

what does this tell us

values at different spatial locations have zero correlation and ones at the same have variance 1

morans I equation less the o

negative spatial autocorrelation

whats one small rule for a valid precision matrix

it must be symmetric

under a homogeneous poisson process

p( r ) =

and why

= 1 for all r

as points are uniformly and independently distributed