| 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 |
|
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| end OAI |
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