Algorithmically Random Point Processes

Jan Reimann

When is a countable set of points ∈ ℝn random?

How “random”?

What measure?

Hauksson-Shearer-Yang catalog of southern CA earthquakes 1981-2011

Point Processes

• Random closed set (RACS): random variable

• Borel structure:

- Fell topology generated by

• Simple point process: RACS that is a.s. locally finite.

X : Ω → F(E)E locally compact, second countable Hausdorff spaceF(E) closed subsets of E

FG = F F : F G = G E openFK = F F : F K = K E compact

Algorithmically RACS

• Find a mapping and “push forward” ML-randomness [BBCDW].

• Or: Given a “nice” basis for E, one can define an effective basis for and develop ML-tests [Axon].

f : AN → F (E) (A finite)

F(E)

FBi1∪···∪BikBj1∩···∩Bjl

Choquet’s Theorem

• How to specify measures on ?

• Any RACS X comes with a capacity functional given by

• Any such functional is - monotone - subadditive - upper semicontinuous: - is completely alternating

• Choquet: Any T with these properties is the capacity of some RACS.

F(E)

TX : K(E) → [0, 1]

TX(K) = PX(FK) = P[X ∩ K = ∅]

for K1 ⊇ K2 ⊇ K3 ⊇, T(Ki) → T(⋂

Ki)

Examples

• Poisson process

• Generalized Poisson process

• ETAS model: (conditional) intensity function

Tμ(K) = 1 – e–μ(K) (μ Borel, non-atomic on E)

λ(t, x, y,m) = s(m)

⎡

⎣μ(x, y) +∑

k : tk<tκ(mk)g(t – tk)f(x – xk, y – yk;mk)

⎤

⎦

(non-stationary Poisson process)

TΠ(K) = 1 – e–λm(K) (m Lebesgue measure on Rn)

Examples

• Poisson process

• Generalized Poisson process

• ETAS model: (conditional) intensity function

µ is called the intensity measure of the process

Tμ(K) = 1 – e–μ(K) (μ Borel, non-atomic on E)

λ(t, x, y,m) = s(m)

⎡

⎣μ(x, y) +∑

k : tk<tκ(mk)g(t – tk)f(x – xk, y – yk;mk)

⎤

⎦

(non-stationary Poisson process)

TΠ(K) = 1 – e–λm(K) (m Lebesgue measure on Rn)

Questions

• Given a locally finite closed set, can we describe the RACS for which it is algorithmically random?

- How do computational and geometric properties interact?

• What is the relation between an algorithmically RACS and the randomness of its points?

• Is there an analog of Schnorr’s Theorem?

Randomness of Sets vs Randomness of Points

• Axon: For the 1-dimensional Poisson process Πℝ on ℝ, if X⊆ℝ is algorithmically Πℝ-random, then

• Axon & R.; Rute: This holds for generalized Poisson processes Πμ, too.

x ∈ X ⇒ x is ML-random.

Randomness of Sets vs Randomness of Points

• Let . Let , and Then X is algorithmically Πℝ-random with intensity λ (computable) iff

- Every finite collection of is mutually ML-random relative to , and

- The sequence is random with respect to the distribution induced by

Nk = |i : xi ∈ (k, k+ ҍ]|

xi = xi − [xi]

xi

(Nk) ∈ NN

P[Nk = n] = e−λ · λnn!

(Nk)

A similar (albeit technically more difficult) characterization is possible for ℝn.

X = x1 < x2 < x3 < . . . ⊂ R

Dynamics and Point Processes

• Often a point process is assumed to be generated as an orbit of a dynamical system .

• What can an (observed) orbit tell about the underlying measure?

(Ω, T, μ)

Measures as Fractals

• Multifractal spectra try to capture these variations by studying

(1) the local scaling behavior of a measure at a given point,

(2) the global (average) scaling behavior of balls.

•For many measures the two aspects are closely related → Multifractal Formalism

Local: Pointwise Dimension

• Pointwise (local) dimension of μ at x:(If the limit does not exist work with lim inf and lim sup.)

• Thm: For μ computable and x μ-random,

:çY MJNδ→

MPH ç#Y δMPH δ

EJN) Y :çY

Local scaling behavior at x

Global: Generalized Renyi Dimensions

• Let B(x,ε) be the N-dimensional ε-ball around x.

• For -∞ < q < ∞, q ≠ 1, let

• For integer q ≥ 2, θ(q)/q-1 is also called the correlation dimension of order q.

θR MJNε→

MPH[∫

ç#Y εRoEçY]

MPH ε

θ measures the average scaling of the q-th moment of μ(B(x,ε))

Multifractal Formalism: from global to local

• μ satisfies the (strong) multifractal formalism if holds whenever f(y) > 0, where

Multifractal spectrum

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Multifractal Formalism: from global to local

• μ satisfies the (strong) multifractal formalism if holds whenever f(y) > 0, where Legendre transform

Multifractal spectrum

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A Universal Multifractal

• Furthermore, the multifractal spectrum of any other (computable) measure can be gauged against the spectrum of the universal semimeasure M*.

• Thm: If μ is computable, thenM* pointwise dimension

dimH fμ(y) = dimH

x : dimH x

dimμ x= y

.

A Universal Multifractal

• Furthermore, the multifractal spectrum of any other (computable) measure can be gauged against the spectrum of the universal semimeasure M*.

• Thm: If μ is computable, then

Billingsley dimension

M* pointwise dimension

dimH fμ(y) = dimH

x : dimH x

dimμ x= y

.

Estimation and Stability of Multifractal Spectra

• Grassberger-Procaccia: Cμ(ε) := Probability two random, independent points x, y are no more than distance ε apart. By Fubini’s Theorem,

• If we have only finitely many observations x1, ..., xn, this suggests using as an estimator of Cμ(ε).

• This estimator is consistent if the xi are chosen i.i.d. according to µ.

$çε ç× ç\Y Z ∥Y o Z∥ ≤ ε^ ∫

ç#Y εEçY θ

$O ε ∑O

J∑

KJ \∥YJoYK∥≤ε^(O)

(Similar estimators exist for higher moments.)

Information Distance

• In practice, the GP estimator often suffers from a boundary effect (earthquake catalogs).

• Idea: replace the use of the Euclidean distance in the GP-algorithm by an information distance (conditional to the boundary, if applicable).

• One such distance is based on Kolmogorov complexity [Bennet et al]:

• One can prove that this is a consistent estimator, too.

EC(σ,τ) = C(στ) - minC(σ),C(τ)

Practical Issues

• For applications, we have to approximate C with

• compressors (Lempel-Ziv etc.)

• string complexity functions (Lempel-Ziv, Ehrenfeucht-Mycielski, Becher-Heiber)

• The consistency results still seem to go through if we work with a normal compressor (Cilibrasi-Vitanyi):

(1) Idempotency: C(aa) = C(a)

(2) Monotonicity: C(ab) ≥ C(a)

(3) Symmetry: C(ab) = C(ba)

(4) Distributivity: C(ab) + C(c) ≤ C(ac) + C(bc)

Practical Issues

• For applications, we have to approximate C with

• compressors (Lempel-Ziv etc.)

• string complexity functions (Lempel-Ziv, Ehrenfeucht-Mycielski, Becher-Heiber)

• The consistency results still seem to go through if we work with a normal compressor (Cilibrasi-Vitanyi):

(1) Idempotency: C(aa) = C(a)

(2) Monotonicity: C(ab) ≥ C(a)

(3) Symmetry: C(ab) = C(ba)

(4) Distributivity: C(ab) + C(c) ≤ C(ac) + C(bc)

Problem: For the popular compression algorithms it has not been formally

established yet that they are normal