138 lines
6.5 KiB
C++
138 lines
6.5 KiB
C++
/*
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* Copyright © 2004 Ondra Kamenik
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* Copyright © 2019 Dynare Team
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*
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* This file is part of Dynare.
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*
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* Dynare is free software: you can redistribute it and/or modify
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* it under the terms of the GNU General Public License as published by
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* the Free Software Foundation, either version 3 of the License, or
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* (at your option) any later version.
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*
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* Dynare is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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* GNU General Public License for more details.
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*
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* You should have received a copy of the GNU General Public License
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* along with Dynare. If not, see <https://www.gnu.org/licenses/>.
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*/
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// Moments of normal distribution.
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/* Here we calculate the higher order moments of normally distributed
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random vector u with means equal to zero and given
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variance-covariance matrix V, i.e. u↝𝒩(0,V). The moment
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generating function for such distribution is f(t)=e^½tᵀVt. If
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we derivate it w.r.t. t and unfold the higher dimensional tensors
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row-wise, we obtain terms like:
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∂f(t)/∂t = f(t)·Vt
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∂²f(t)/∂t² = f(t)·(Vt⊗v)
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∂³f(t)/∂t³ = f(t)·(Vt⊗Vt⊗Vt + P_?(v⊗Vt) + P_?(Vt⊗v) + v⊗Vt)
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∂⁴f(t)/∂t⁴ = f(t)·(Vt⊗Vt⊗Vt⊗Vt + S_?(v⊗Vt⊗Vt) + S_?(Vt⊗v⊗Vt) + S_?(Vt⊗Vt⊗v) + S_?(v⊗v))
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where v is vectorized V (v=vec(V)), and P_? is a suitable row permutation
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(corresponds to permutation of multidimensional indices) which permutes the
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tensor data, so that the index of a variable being derived would be the
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last. This ensures that all (permuted) tensors can be summed yielding a
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tensor whose indices have some order (in here we chose the order that more
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recent derivating variables are to the right). Finally, S_? is a suitable sum
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of various P_?.
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We are interested in S_? multiplying the Kronecker powers ⊗ⁿv. The S_? is a
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(possibly) multi-set of permutations of even order. Note that we know the
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number of permutations in S_?. The above formulas for f(t) derivatives are
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valid also for monomial u, and from literature we know that 2n-th moment is
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(2n!)/(n!2ⁿ)·σ². So there are (2n!)/(n!2ⁿ) permutations in S_?.
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In order to find the S_? permutation we need to define a couple of things.
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First we define a sort of equivalence between the permutations applicable to
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even number of indices. We write P₁ ≡ P₂ whenever P₁⁻¹∘P₂ permutes only whole
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pairs, or items within pairs, but not indices across the pairs. For instance
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the permutations (0,1,2,3) and (3,2,0,1) are equivalent, but (0,2,1,3) is
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not equivalent with the two. Clearly, the ≡ relationship is an equivalence.
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This allows to define a relation ⊑ between the permutation multi-sets S,
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which is basically the subset relation ⊆ but with respect to the equivalence
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relation ≡, more formally:
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S₁ ⊑ S₂ iff P ∈ S₁ ⇒ ∃Q ∈ S₂ : P ≡ Q
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This induces an equivalence S₁ ≡ S₂.
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Now let Fₙ denote a set of permutations on 2n indices which is maximal with
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respect to ⊑, and minimal with respect to ≡ (in other words, it contains
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everything up to the equivalence ≡). It is straightforward to calculate a
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number of permutations in Fₙ. This is a total number of all permutations
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of 2n divided by permutations of pairs divided by permutations within the
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pairs. This is (2n!)/(n!2ⁿ).
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We now prove that S_? ≡ Fₙ. Clearly S_? ⊑ Fₙ, since Fₙ is maximal. In order
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to prove that Fₙ ⊑ S_?, let us assert that for any permutation P and for
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any (semi-)positive definite matrix V we have
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P·S_?(⊗ⁿv)=S_?(⊗ⁿv). Below we show that there is a positive
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definite matrix V of some dimension that for any two permutation
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multi-sets S₁, S₂, we have
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S₁ ≢ S₂ ⇒ S₁(⊗ⁿv) ≠ S₂(⊗ⁿv)
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So it follows that for any permutation P, we have P·S_? ≡ S_?. For a purpose
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of contradiction let P ∈ Fₙ be a permutation which is not equivalent to any
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permutation from S_?. Since S_? is non-empty, let us pick P₀ ∈ S_?. Now
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assert that P₀⁻¹S_? ≢ P⁻¹S_? since the first contains an identity and the
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second does not contain a permutation equivalent to identity. Thus we have
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(P∘P₀⁻¹)S_? ≢ S_? which gives the contradiction and we have proved that
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Fₙ ⊑ S_?. Thus Fₙ ≡ S_?. Moreover, we know that S_? and Fₙ have the same
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number of permutations, hence the minimality of S_? with respect to ≡.
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Now it suffices to prove that there exists a positive definite V such that
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for any two permutation multi-sets S₁ and S₂ holds S₁ ≢ S₂ ⇒ S₁(⊗ⁿv) ≠ S₂⊗ⁿv.
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If V is a n×n matrix, then S₁ ≢ S₂ implies that there is
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identically nonzero polynomial of elements from V of order n over
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integers. If V=AᵀA then there is identically non-zero polynomial of
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elements from A of order 2n. This means, that we have to find n(n+1)/2
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tuple x of real numbers such that all identically non-zero polynomials p
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of order 2n over integers yield p(x)≠0.
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The x is constructed as follows: x_i = π^log(rᵢ), where rᵢ is i-th prime.
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Let us consider the monom x₁^j₁·…·xₖ^jₖ. When the monom is evaluated, we get
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π^{log(r₁^j₁)+…+log(rₖ^jₖ)}= π^log(r₁^j₁+…+rₖ^jₖ)
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Now it is easy to see that if an integer combination of such terms is zero,
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then the combination must be either trivial or sum to 0 and all monoms
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must be equal. Both cases imply a polynomial identically equal to zero. So,
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any non-trivial integer polynomial evaluated at x must be non-zero.
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So, having this result in hand, now it is straightforward to calculate
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higher moments of normal distribution. Here we define a container, which
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does the job. In its constructor, we simply calculate Kronecker powers of v
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and apply Fₙ to ⊗ⁿv. Fₙ is, in fact, a set of all equivalences in sense of
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class Equivalence over 2n elements, having n classes each of them having
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exactly 2 elements.
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*/
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#ifndef NORMAL_MOMENTS_HH
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#define NORMAL_MOMENTS_HH
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#include "t_container.hh"
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class UNormalMoments : public TensorContainer<URSingleTensor>
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{
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public:
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UNormalMoments(int maxdim, const TwoDMatrix& v);
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private:
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void generateMoments(int maxdim, const TwoDMatrix& v);
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static bool selectEquiv(const Equivalence& e);
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};
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class FNormalMoments : public TensorContainer<FRSingleTensor>
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{
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public:
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FNormalMoments(const UNormalMoments& moms);
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};
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#endif
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