Neighbourhoods of Randomness and Independence

Transcription

1 Neighbourhoods of Randomness and Independence C.T.J. Dodson School of Mathematics, Manchester University Augment information geometric measures in spaces of distributions, via explicit geometric representations of neighbourhoods for these important states for stochastic processes: randomness, independence, uniformity. Significant theoretically because very general, and practically because topological so stable under perturbations.

2 Proposition Every neighbourhood of a random process contains a neighbourhood of stochastic processes subordinate to gamma distributions. Proof Dodson and Matsuzoe provided affine immersion in R 3 for G, manifold of gamma pdfs. Natural coordinates (µ = α/β, α) Then G is the graph of the affine immersion {h, ξ} where ξ is a transversal vector field along h h : G R 3 : ( ) µ α µ α log Γ(α) α log µ Submanifold of exponential pdfs is curve, ξ =. (, ) R 3 : µ {µ,, log µ }

3 Tubular neighbourhood in R 3.6 cos θ.6 µ cos θ {µ,.6 sin θ, log µ} + µ + µ θ [, π) contains all immersions for small enough perturbations of exponential distributions. This tubular neighbourhood intersects with the gamma manifold immersion to yield the required neighbourhood of gamma distributions.

4 µ 3 - α 3 Gamma manifold affine immersion in natural coordinates µ = α/β, α as a surface in R 3. Tubular neighbourhood surrounds all exponential distributions these lie on the curve α = in the surface.

5 α 3 3 Continuous image of gamma manifold affine immersion as surface in R 3 using standard coordinates. β - - Tubular neighbourhood surrounds all exponential distributions these lie on the curve α = in the surface.

6 Log-gamma manifold L and neighbourhoods of uniformity The pdfs for random variable N [, ] given by g(n, µ, β) = for µ > and β >. β µ N ( β µ )β (log N )β Γ(β) Uniform distribution lim g(n, µ, ) = g(n,, ) =. µ

7 Log-gamma pdfs g(n; µ, β), N [, ], with central mean < N >=.5, and β =.5,,, 5. Cases β < correspond in gamma distributions to clustering in an underlying spatial process; conversely, β > corresponds to dispersion and greater evenness than random. g(n; µ, β).5 β = β = β = β = N

8 Family of log-gamma pdfs p ν (N), N [, ] with mean ν N.6.8 Uniform distribution for ν = : p (N) =, N ie random (Poisson) case; clustering ν < ; smoothed ν >.

9 Log-gamma manifold L L isometric with gamma manifold, G. Hence, immersion of G in R 3 above, represents also the log-gamma manifold L. Then, since the isometry sends the exponential distribution to the uniform distribution on [, ], we obtain a general deduction Proposition Every neighbourhood of the uniform distribution contains a neighbourhood of log-gamma distributions. Equivalently, Proposition Every neighbourhood of a uniform stochastic process contains a neighbourhood of stochastic processes subordinate to log-gamma distributions.

10 Freund manifold F and neighbourhoods of independence Let F be the manifold of Freund bivariate mixture exponential distributions, so with positive parameters α i, β i, F {f f(x, y; α, β, α, β ) = { α β e β y (α +α β )x for x < y α β e β x (α +α β )y for y x }. Submanifold F F : α = α, β = β The pdfs are of form : f(x, y; α, β ) = { α β e β y ( α β )x for < x < y α β e β x ( α β )y for < y < x Parameters α, β >. The covariance, correlation coefficient and marginal pdfs, for X and Y are given by : Cov(X, Y ) = ( α ), β ρ(x, Y ) = α 3 α + β,

11 f X (x) = ( ) α α β f Y (y) = ( ) α α β with x, y β e β x + β e β y + ( α β α β ( α β α β ) ) ( α ) e α x ( α ) e α y Independence case ρ(x, Y ) = = σ when α = β F an exponential family with parameters (α, β ) and potential function ψ = log(α β )

12 Proposition In the affine embedding of the Freund submanifold F in R 3, a tubular neighbourhood of the curve α = β will contain all immersions of bivariate exponential processes sufficiently close to the case of independence.

13 α - - β Affine immersion in natural coordinates (α, β ) as a surface in R 3 for the Freund submanifold F. Tubular neighbourhood surrounds the curve (α = β in the surface) consisting of all bivariate distributions having identical exponential marginals and zero covariance.

14 Neighbourhoods of independence for Gaussian processes Proposition Let N be the bivariate Gaussian manifold with the Fisher metric g and the exponential connection (). Denote by (θ i ) a natural coordinate system. Then N can be realized in R 6 by the affine immersion {f, ξ} f : Θ R 6 : [ [ ] ] θi θ i, ξ = ϕ(θ) with potential function ϕ(θ) = log( π ) ( θ θ 3 θ θ θ + θ ) θ 5.,

15 Proposition The bivariate Gaussian 5-manifold admits a -dimensional submanifold through which can be provided a neighbourhood of bivariate distributions having identical Gaussian marginals and zero covariance containing the independent case. Outline proof Bivariate Gaussians with zero means (µ = µ = ) and identical standard deviation σ = σ = σ is represented by the surface N R 3 : (θ, θ ) (θ, θ, ϕ(θ)), ξ = (,, ). where (θ, θ ) = ( σ, σ ) = σ σ ϕ(θ) = log( π σ).

16 Affine immersion for Gaussian processes Submanifold of the independent case with zero means and identical standard deviation is curve (, ) R 3 : ( ) (,, log( π σ)) σ σ Tubular neighbourhood represents departures from independence.

17 Proposition In the affine immersion as a surface in R 3 for the bivariate Gaussian distributions with zero means and identical standard deviation σ, a tubular neighbourhood of the case of zero covariance will contain all immersions of bivariate Gaussian processes sufficiently close to the independence case. Figure shows affine embedding of -submanifold as a surface in R 3, and an R 3 -tubular neighbourhood, of the curve σ = in the surface.

18 Affine immersion in natural coordinates (θ, θ ) = ( σ, σ ) as surface in R3 for bivariate Gaussian distributions with zero means and identical standard deviation σ. Tubular neighbourhood surrounds curve (σ = in the surface) representing all bivariate distributions having identical Gaussian marginals and zero covariance. Tubular neighourhood contains all small departures from independence. θ θ.5

19 Continuous image of the affine immersion as a surface in R 3 using standard coordinates for the bivariate Gaussian distributions with zero means and identical standard deviation σ. Tubular neighbourhood surrounds curve of independent states, (σ =,) and contains all small departures from independence. σ σ.5

20 Results Explicit representations in R 3 of tubular neighbourhoods that provide the following: A. All processes sufficiently close to a Poisson process B. All processes sufficiently close to a uniform process C. All bivariate processes sufficiently close to the independent bivariate Poisson 3 process D. All bivariate processes sufficiently close to the independent bivariate Gaussian process In an information geometric immersion Via unique associated exponential process 3 Marginals having same mean Marginals both N(, σ)