Algorithmically Random Point Processes
Transcript of Algorithmically Random Point Processes
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