A New Use of Group Representation Theory in Statistics

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2 The (imprecise) Math Problem symmetrize f(x, y) Given a function of two variables, f(x, y), symmetrize it to produce a new function f(x, y) that satisfies the following: f(x, y) = f( x, y x) = f(y, x) = f(x y, y) = f( y, x y) = f(y x, x) /

3 The (imprecise) Math Problem symmetrize f(x, y) Given a function of two variables, f(x, y), symmetrize it to produce a new function f(x, y) that satisfies the following: f(x, y) = f( x, y x) = f(y, x) = f(x y, y) = f( y, x y) = f(y x, x) ( ) ( ) f(x, y) = f(x, 1 0 y) f(x, y) = 0 1 f(x 1 1 y, y) 0 1 ( ) ( ) f(x, y) = f( x, 1 0 y x) f(x, y) = 1 1 f( y, 0 1 x y) 1 1 ( ) ( ) f(x, y) = f(y, 0 1 x) f(x, y) = 1 0 f(y 1 1 x, x) /

4 Stationarity A Statistical Notion Definition (third-order stationarity) A mean zero sequence of random variables {X t } is called third-order stationary if C(x, y) := E(X t, X t+x X t+y ) is the same for all t. Example (identity induced by σ = (12)) C(x, y) = E[X t X t+x X t+y ] (12) = E[X t+x X t X t+y ] t t x = E[X t X t x X t x+y ] = C( x, y x) 3 /

5 Identifying a Group Representation Define τ : R 2 R 3 as τ(a, b) (0, a, b). Let σ be a permutation in S 3. Define ψ : R 3 R 2 as ψ(a, b, c) (b a, c a). Example (σ = (12) S 3 ) (x, y) τ (0, x, y) σ (x, 0, y) ψ ( x, y x) ( 1 ) This leads to a three-way correspondence among permutations in S 3, matrices in GL 2 (R), and symmetries imposed on f(x, y). Example (σ = (1234) S 4 ) (x, y, z) τ (0, x, y, z) σ (z, 0, x, y) ψ ( z, x z, y z) /

6 Identifying a Group Representation Define τ : R 2 R 3 as τ(a, b) (0, a, b). Let σ be a permutation in S 3. Define ψ : R 3 R 2 as ψ(a, b, c) (b a, c a). Example (σ = (12) S 3 ) (x, y) τ (0, x, y) σ (x, 0, y) ψ ( x, y x) ( 1 ) This leads to a three-way correspondence among permutations in S 3, matrices in GL 2 (R), and symmetries imposed on f(x, y). Example (σ = (1234) S 4 ) (x, y, z) τ (0, x, y, z) σ (z, 0, x, y) ψ ( z, x z, y z) /

7 Spectral Density and its Estimator Definition (third-order spectral density) Let {X t } be third-order stationary. The 3 rd -order spectral density is f(u, v) = 1 (2π) 2 C(x, y)e ixu iyv x= y= f(u, v) inherits symmetries induced by C(x, y). Definition (third-order spectral density estimate) Let x 1, x 2,..., x n be an observation of a 3 rd -order stationary time series. f(u, v) = 1 n n (2π) 2 κ(x, y)ĉ(x, y)e ixu iyv where Ĉ(x, y) = Ê[X tx t+x X t+y ]. x=1 y=1 5 /

8 Spectral Density and its Estimator Definition (third-order spectral density) Let {X t } be third-order stationary. The 3 rd -order spectral density is f(u, v) = 1 (2π) 2 C(x, y)e ixu iyv x= y= f(u, v) inherits symmetries induced by C(x, y). Definition (third-order spectral density estimate) Let x 1, x 2,..., x n be an observation of a 3 rd -order stationary time series. f(u, v) = 1 n n (2π) 2 κ(x, y)ĉ(x, y)e ixu iyv where Ĉ(x, y) = Ê[X tx t+x X t+y ]. x=1 y=1 5 /

9 Symmetries of C(x, y) and f(u, v) 6 /

10 Summarizing the motivation... For a given function κ(x, y), construct a symmetrized form, κ(x, y), with the same symmetries as C(x, y). Before the group representation link k(x, y) was constructed as k(x, y) = k(x)k(y)k(x y) where k( ) is any even function. This is too restrictive 7 /

11 The representation Theorem (Berg 2008) The mapping ρ : S n GL n 1 (R), previously described, is a faithful group representation. Corollary (symmetrizing f(x, y)) Let h(x 1,..., x 6 ) be a symmetric function of its six arguments. f(x, y) = h(f(x, y), f( x, y x), f(y, x), f(x y, y), f( y, x y), f(y x, x)) 8 /

12 Example Symmetrizing the upside-down bowl Let f(x, y) = (1 x 2 y 2 ) + upside-down bowl h = x i, h = max(x i ), h = min(x i ) 9 /

13 Generalizing the Optimal Kernel The Subba-Rao-Gabr optimal kernel is { 3 ( ( Λ opt (ω 1, ω 2 ) = π π ω ω2 2 + ω )) 1ω 2, if ω ω2 2 + ω 1ω 2 π 2 0, otherwise Define Theorem (Berg 2008) k 1 Λ opt (ω) = α 1 β i=1 ω 2 i + i<j ω i ω j Let Λ(ω) be any nonnegative kernel that integrates to one and satisfies all the necessary symmetries. Also assume ωj 2 Λ(ω) dω = ωj 2 Λ opt (ω) dω R k 1 R k 1 for j = 1,..., n 1. Then Λ L2 Λ opt L /

14 The End...Thank You! /