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7d9f027 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import Mathlib
import OAI.Probability.SKRatio.Quantization.BinFunctional
namespace OAI
noncomputable section
open scoped BigOperators Matrix
open MeasureTheory ProbabilityTheory Filter Real
namespace SKRatio.Bins
open Planted Scalar SKRatioClock.Regression
variable {α : Type*} [Fintype α] [DecidableEq α] {n : ℕ}
attribute [local instance] Classical.propDecidable
def siteMoment (p f : Fin n → ℝ) : ℝ := (∑ i, p i*f i)/sqrt (n:ℝ)
lemma siteMoment_bin {σ : Fin n → α} (hcount : ∀ a, 0<count σ a)
(z : Parameters α) (u : Directions σ) (f : α → ℝ) :
siteMoment (vectorOf z u) (fun i => f (σ i)) = moment (binT σ) (paramB z) f := by
simpa only [siteMoment,Fintype.card_fin] using vectorOf_moment hcount z u f
lemma momentProduct (p f q k : Fin n → ℝ) :
siteMoment p f*siteMoment q k=(∑ i, p i*f i)*(∑ i, q i*k i)/(n:ℝ) := by
unfold siteMoment
rw [div_mul_div_comm,←pow_two,sq_sqrt (Nat.cast_nonneg n)]
def correctedForm (β : ℝ) (H p : Fin n → ℝ) : ℝ :=
siteMoment p H*siteMoment p (fun i => v (H i))+
siteMoment p (fun _ => 1)*siteMoment p (fun i => H i*v (H i))-
β^2*siteMoment p (fun _ => 1)*siteMoment p (fun i => v (H i))+
2*β^2*siteMoment p (fun _ => 1)*siteMoment p (fun i => m (H i)*v (H i))+
diagonalForm (fun _ _ => β^2/(n:ℝ)) compactF (fun i => compactWeight (H i)) p+
weightedForm (fun _ _ => β^2/(n:ℝ)) compactK (fun i => compactWeight (H i)) p
lemma deterministic_corrected (β : ℝ) (H p : Fin n → ℝ) :
deterministicForm β H p = correctedForm β H p+
(β^2-(∑ i, H i)/(n:ℝ))*siteMoment p (fun _ => 1)*siteMoment p (fun i => v (H i)) := by
unfold deterministicForm correctedForm rankForm
simp only [mul_assoc]
simp only [←mul_assoc,momentProduct,mul_one,Finset.sum_mul]
ring
lemma diagonalForm_bin {σ : Fin n → α} (hcount : ∀ a, 0<count σ a)
(β : ℝ) (h : α → ℝ) (z : Parameters α) (u : Directions σ) :
diagonalForm (fun _ _ => β^2/(n:ℝ)) compactF
(fun i => compactWeight (h (σ i))) (vectorOf z u) =
β^2*(∑ a, ∑ d, binT σ d^2*(v (h d)^2/(w (h a)+w (h d)))*
((paramB z a)^2+(paramR z a)^2)) := by
let p := vectorOf z u
let K : α → α → ℝ := fun a d => v (h d)^2/(w (h a)+w (h d))
have he : diagonalForm (fun _ _ => β^2/(n:ℝ)) compactF
(fun i => compactWeight (h (σ i))) p =
β^2/(n:ℝ)*(∑ i, ∑ j, p i^2*(1:ℝ)*K (σ i) (σ j)) := by
simp only [diagonalForm,compactF_weight,Finset.sum_mul,Finset.mul_sum,K,mul_one]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
rw [he,double_weighted σ (fun i => p i^2) (fun _ => 1) K]
simp only [p,vectorOf_sq_sum hcount,sum_const_bin σ (fun _ => 1),mul_one,
binT_sq,Fintype.card_fin,Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
apply Finset.sum_congr rfl
intro d _
dsimp [K]
ring
lemma weightedForm_bin {σ : Fin n → α} (hcount : ∀ a, 0<count σ a)
(β : ℝ) (h : α → ℝ) (z : Parameters α) (u : Directions σ) :
weightedForm (fun _ _ => β^2/(n:ℝ)) compactK
(fun i => compactWeight (h (σ i))) (vectorOf z u) =
β^2*(∑ a, ∑ d, binT σ a*paramB z a*kernel (h a) (h d)*(binT σ d*paramB z d)) := by
let p := vectorOf z u
have he : weightedForm (fun _ _ => β^2/(n:ℝ)) compactK
(fun i => compactWeight (h (σ i))) p =
β^2/(n:ℝ)*(∑ i, ∑ j, p i*p j*kernel (h (σ i)) (h (σ j))) := by
simp only [weightedForm,quadratic,compactK_weight,Finset.mul_sum]
apply Finset.sum_congr rfl
intro i _
apply Finset.sum_congr rfl
intro j _
ring
rw [he,double_weighted σ p p (fun a d => kernel (h a) (h d))]
simp only [p,vectorOf_sum hcount,binT,Fintype.card_fin,Finset.mul_sum]
apply Finset.sum_congr rfl
intro a _
apply Finset.sum_congr rfl
intro d _
rw [show sqrt (count σ a:ℝ)/sqrt (n:ℝ)*paramB z a*kernel (h a) (h d)*
(sqrt (count σ d:ℝ)/sqrt (n:ℝ)*paramB z d) =
(sqrt (count σ a:ℝ)*paramB z a*sqrt (count σ d:ℝ)*paramB z d*kernel (h a) (h d))/
(sqrt (n:ℝ))^2 by ring,
sq_sqrt (Nat.cast_nonneg n)]
ring
theorem correctedForm_bin {σ : Fin n → α} (hcount : ∀ a, 0<count σ a)
(β : ℝ) (h : α → ℝ) (z : Parameters α) (u : Directions σ) :
correctedForm β (fun i => h (σ i)) (vectorOf z u) = finiteA β (binT σ) h z := by
unfold correctedForm
rw [diagonalForm_bin hcount,weightedForm_bin hcount]
simp only [siteMoment_bin hcount z u h,siteMoment_bin hcount z u (fun a => v (h a)),
siteMoment_bin hcount z u (fun _ => 1),siteMoment_bin hcount z u (fun a => h a*v (h a)),
siteMoment_bin hcount z u (fun a => m (h a)*v (h a))]
rfl
end SKRatio.Bins
end
end OAI
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