Functional summary statistics

Question 1

What is the first order moment?

Answer 1

It is the intensity measure μ(A) = 𝔼[N(A)].

Question 2

Campbell’s formula

Answer 2

Suppose the intensity measure μ has an intensity function ρ. For h : R^d → [0, ∞),
𝔼[ ∑ _{u ∈ X} h(u) ] = ∫ _{R^d} h(u) ρ(u) du.

Question 3

Explain non-parametric estimators

Answer 3

They are estimators requiring no model assumptions. They are different for the homogeneous and inhomogeneous case. We assume they are over a bounded observation window W .

Homogeneous:
The natural (and unbiased) estimator for ρ, which is actually also the MLE, is
ρ' = n(x ∩ W ) / |W |.

Inhomogeneous:
For k a pdf on R^d, called the kernel, and a edge correction factor c_w(v) = ∫_ {W} k(u - v) du, the estimator is
∑_{v ∈ X ∩ W} k(u - v) / c_w(v).

Question 4

List and define the different kernels

Answer 4

Uniform kernel:
k(u) = 1(|u| < 1 / 2)

Gauss kernel:
k(u) = 1 / (√2π)^d exp(−∥u∥^2 / 2).

Epinechnikov (2D) kernel:
k((u_1, u_2)) = e(u_1) e(u_2),
where e(u) = 3 / 4 (1 − |u|)1{|u| ≤ 1}.

Question 5

What is bandwidth?

Answer 5

It can be considered as the spread parameter, a larger bandwidth spreads the influence of event points over a greater distance, but is also more likely to experience edge effects close to the study region boundaries.

In practice, the bandwidth b is more important than the shape of the kernel k.

Question 6

What is the second order factorial moment measure?

Answer 6

The second order factorial moment measure α^{(2)} is defined for C ⊆ R^d × R^d, by
α^{(2)} (C ) = 𝔼[∑_{u,v ∈X}^≠ 1{ (u, v) ∈ C }], .

It has a second order product density ρ^{(2)} if
α^{(2)} (C ) =∫_C ρ^{(2)} (u, v ) du dv.

Question 7

State some properties of the second order moment

Answer 7

Cov(N(A), N(B)) = α^{(2)} (A × B) + μ(A ∩ B) − μ(A)μ(B).

If A and B are disjoint,
Cov(N(A), N(B)) = α^{(2)} (A × B) − μ(A)μ(B).

Var(N(A)) = α^{(2)} (A × A) + μ(A) − μ(A)2.

Question 8

What is the second order Campbell formula?

Answer 8

For h : R^d × R^d → [0, ∞),
𝔼[ ∑_{u,v ∈X}^≠ h(u, v )] =∫∫ h(u, v )ρ^{(2)} (u, v) du dv.

Question 9

What is the second order product density for Poisson?

Answer 9

ρ^{(2)}(u, v ) = ρ(u) ρ(v).

Question 10

What is the pair correlation function?

Answer 10

The pair correlation function is defined as
g (u, v) = ρ^{(2)} (u, v) / ρ(u)ρ(v),
where a / 0 = 0 for all a ≥ 0.

Poisson process: g = 1.
g > 1: More clustered than Poisson.
g < 1: More regular than Poisson.

If X is both stationary and isotropic, then g (u, v ) = g (u − v) = g (∥u − v ∥), thus depends only on the distance between u and v.

Question 11

What is Ripley’s K function?

Answer 11

Assume X is stationary with constant intensity ρ > 0. For r > 0,
Ripley’s K -function is given by
K(r) = 1 / ρ^2|A| 𝔼[ ∑ _{u∈X∩A} ∑ _{v ∈X{u}} 1{∥u − v ∥ ≤ r } ].

Question 12

Interpretation of K function in terms of clustering/regular.

Answer 12

For a Poisson process,
K(r) = |b(0, r)| = r^d ω_d ,
where ω_d = π^{d/2} / Γ(1 + d/2) is the volume of the unit ball in R^d.
If g > 1 (clustering): K(r) > r^d ω_d
If g < 1 (repulsion): K(r) < r^d ω_d

Question 13

Interpretation of the K function.

Answer 13

K(r) can be interpreted as the expected number of points in a ball around 0, conditionally on 0 being a point of the point process.

Question 14

What is Besag’s L function?

Answer 14

Besag’s L-function is given by
L(r) = (K(r) / ω_d )^{1 / d}.

Question 15

Intepretation of the L function in terms of clustering/regular

Answer 15

Poisson process: L(r) = r.
If g > 1 (clustering): L(r) > r.
If g < 1 (repulsion): L(r) < r.

Question 16

What is the non-parametric estimation of the K function?

Answer 16

An unbiased estimator:
ˆK(r) = 1 / ρ^2 ∑ _{u∈x_W} ∑ _{v ∈x_W \{u}} 1{∥v − u∥ ≤ r } / |W ∩ (W − v + u)|

Question 17

What is the non-parametric estimation of the L function?

Answer 17

ˆL(r ) = (ˆK(r) / ω_d)^{1 / d}.

Question 18

What is second-order intensity reweighted stationary?

Answer 18

A point process is said to be s.o.i.r.s. , if for any region A with 0 < |A| < ∞ the measure
K(B) = 1 / |A| 𝔼[ ∑_{u∈X} ∑_{v ∈X\{u}} 1{u ∈ A, v − u ∈ B} / ρ(u)ρ(v) ],
for B ∈ B, does not depend on A. In this case K(B) defines a measure called the second order reduced moment measure.

Question 19

What is the empty space function and its use?

Answer 19

Assume X is stationary with constant intensity ρ > 0. Let r > 0. The empty space function is given by

F (r ) = P(d(0, X) ≤ r ) = P(X ∩ b(0, r ) ≠ ∅),

where d(0, X) denotes the distance from the origin to the nearest point in X.

Question 20

What is the nearest neighbor distribution function and its use?

Answer 20

Assume X is stationary with constant intensity ρ > 0. Let A ⊆ Rd with 0 < |A| < ∞. The nearest neighbor distribution function is defined by
G(r) = 𝔼[ ∑_{u∈X∩A} 1{d(u, X \ {u}) ≤ r }] / ρ|A|.
G (r) can be interpreted as the probability that a ”typical point” in X has another point of X within distance r.

Question 21

What is the J function?

Answer 21

Assume X is stationary with constant intensity ρ > 0. The J-function is given (for F (r ) < 1) by
J(r) = 1 − G(r) / 1 − F(r) .
Using the interpretation of the F - and G -function, we get
J(r) = P((X \ {0}) ∩ b(0, r) = ∅|0 ∈ X) / P(X ∩ b(0, r) = ∅) .

Question 22

Edge correction factors/methods

Answer 22

Edge correction factor,
|1 / |W ∩ W_{η−ξ}|

A simpler alternative, minus sampling, for r > 0, let
W_r = {ξ ∈ W : b(ξ, r) ⊆ W }
where these points are then used for estimating the K function. Minus sampling discards data from the outer region of the observation window.

Minus sampling doesn’t use all points, whereas edge correction does. However, given sufficient data, one may prefer the reduced-sample estimate if very large weights 1/|W ∩Wη−ξ | are given to pairs of points.

Question 23

How is model checking performed using functional summary statistics?

Answer 23

Suppose x is an observed point pattern.
Suppose we have fitted a point process model to the data.
We compute a non-parametric functional summary T_0(r) from our data (we omit the ”hat” over T ).
We compare this to the theoretical value T(r) under the fitted model.
Then we use envelopes to check if the model is good.

Question 24

What is a pointwise envelope?

Answer 24

Simulate n i.i.d. point patterns from the fitted model and calculate the functional summary statistics T_1(r ), . . . , T_n(r ). Fix r > 0. If the fitted model is correct, (T_0(r ), . . . , T_n(r )) is exchangeable (the joint distribution is preserved under permutation). Thus, (if the probability of ties is zero),
P(T_0(r) ≤ min_{i=1,...,n} T_i(r) or T_0(r) ≥ max_{i=1,...,n} T_i(r) ) = 2 / n + 1
If n = 39, then with probability 95% we expect T_0(r ) to fall between the smallest and largest value of T_1(r ), … , T_n(r).
The interval from min_{i=1,...,39} T_i(r) to max_{i=1,...,39} T_i(r) is called a 95% envelope. Since r was fixed, it is a pointwise envelope.

Question 25

What is a global rank envelope and when to use it?

Answer 25

Decide on a set I = {r_1, . . . , r_N} of relevant r -values. Assume for simplicity that there are no ties between the values T_i(r_j). Define the global rank of T_i as `R_i = max{k | min_{i′=0,...,n}^k T_i′(r) ≤ T_i(r) ≤ max_{i′=0,...,n}^k T_i′(r) for all r ∈ I}`. Here min^k and max^k denotes the k'th smallest and k'th largest value, respectively. We reject that it is a good model if p+ ≤ α, where α is the significance level. We have no evidence for rejection if p− > α. If α ∈ [p−, p+), we do not know whether or not to reject. Better to use than a point-wise envelope, as it covers more r values.

Question 26

Why use global rank envelope?

Answer 26

The pointwise envelope test only allows us to use one r -value for the test. If more r -values are considered: multiple testing. Typically, a good choice of r is not known in advance. Makes more sense to consider a test based on several r -values. Requires a much wider 95% envelope in order to obtain a test of significance level 5%.

Question 27

Give an overview of the results in summary statistics

Answer 27

▶ The summary statistics are mainly estimates of: ▶ First order moments (intensity). ▶ Second order moments, in particular the pair correlation function g or number of close point pairs as given by K and L. ▶ Interpoint distances as described by F , G , and J). ▶ Results: ▶ Unbiased estimators in many cases. ▶ Relation between g and K (assuming s.o.i.r.s.). ▶ Independent thinning results for ρ, ρ^(2), g , K, K , L. ▶ Poisson case: closed form expressions for ”everything”, i.e. ρ, ρ^(2), g , K, K , L, F , G , J. ▶ Use: Preliminary analysis, summarizing useful info from data, model fitting (minimum contrast), model checking.

Question 28

Why use functional summary statistics?

Answer 28

Usually, a data analysis starts with a computation of some non-parametric summary statistics from the data to get an idea about the distribution. Functional summaries give more refined insights than single numbers. It can also be used for model checking.

Question 29

What are some important considerations for using functional summary statistics?

Answer 29

In principle any function, such as K , L, g , F , G , J could be used. Its theoretical expression under the given model should be known. It should reflect important aspects of the data. It should be well-defined (e.g. F and G are only defined for stationary processes). Estimation formulas should be available or we may use a grid search. If we use one functional summary statistic for fitting a model, we should use others for model checking!