| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.ScalarAtomic |
| import OAI.Probability.SKBarriers.Scalar.PartitionChain |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem quantileCDF_bounds (k : ℕ) (Q : Fin (k+1) → ℝ) (s : ℝ) : |
| quantileCDF k Q s ∈ Icc (0:ℝ) 1 := by |
| rw [quantileCDF_formula] |
| constructor |
| · exact Finset.sum_nonneg (fun _ _ => by split_ifs <;> positivity) |
| · rw [← uniformAtom_sum k] |
| exact Finset.sum_le_sum (fun _ _ => by split_ifs <;> first | exact le_rfl | positivity) |
|
|
| theorem quantileCDF_monotone (k : ℕ) (Q : Fin (k+1) → ℝ) : Monotone (quantileCDF k Q) := by |
| intro s t hst |
| simp only [quantileCDF_formula] |
| apply Finset.sum_le_sum |
| intro j _ |
| by_cases hj : Q j ≤ s |
| · simp only [ite_eq_left hj,ite_eq_left (hj.trans hst),le_refl] |
| · simp only [ite_eq_right hj] |
| split_ifs <;> positivity |
|
|
| theorem monotone_cons_zero_quantiles {k : ℕ} (Q : Fin (k+1) → ℝ) |
| (hQ : Monotone Q) (h0 : 0 ≤ Q 0) : Monotone (Fin.cons (α := fun _ => ℝ) 0 Q) := by |
| apply Fin.monotone_iff_le_succ.mpr |
| intro i |
| refine Fin.cases ?_ (fun j => ?_) i |
| · exact h0 |
| · exact hQ (Fin.castSucc_le_succ j) |
|
|
| theorem quantile_knots_gap (k : ℕ) (Q : Fin (k+1) → ℝ) (i : Fin (k+1)) : |
| Fin.cons (α := fun _ => ℝ) 0 Q i.succ-Fin.cons (α := fun _ => ℝ) 0 Q i.castSucc=cumulativeGapMap k Q i := by |
| refine Fin.cases ?_ (fun i => ?_) i |
| · simp |
| · rfl |
|
|
| theorem quantileCDF_on_interval {k : ℕ} (Q : Fin (k+1) → ℝ) (hQ : Monotone Q) |
| (i : Fin (k+1)) (z : ℝ) (hz : z ∈ Ico (Fin.cons (α := fun _ => ℝ) 0 Q i.castSucc) (Q i)) : |
| quantileCDF k Q z=quantileMass k i := by |
| have he (j : Fin (k+1)) : Q j ≤ z ↔ j < i := by |
| constructor |
| · intro hj |
| by_contra hji |
| have H := hQ (le_of_not_gt hji) |
| linarith [hz.2] |
| · intro hji |
| cases i using Fin.cases with |
| | zero => simp at hji |
| | succ l => |
| have hj : j ≤ l.castSucc := by simpa only [Fin.le_def,Fin.lt_def,Fin.val_castSucc] using (Nat.le_of_lt_succ hji) |
| exact (hQ hj).trans hz.1 |
| rw [quantileCDF_formula] |
| simp_rw [he] |
| rw [← Finset.sum_filter] |
| have hs : (Finset.univ.filter (fun j : Fin (k+1) => j < i))=Finset.Iio i := by ext j; simp |
| rw [hs,Finset.sum_const,Fin.card_Iio] |
| simp only [nsmul_eq_mul,quantileMass,div_eq_mul_inv] |
|
|
| theorem quantile_prefix_chain {k : ℕ} (Q : Fin (k+1) → ℝ) |
| (hQ : Monotone Q) (h0 : 0 ≤ Q 0) : |
| ∃ l : List (ℝ × ℝ≥0), |
| (chainDuration l:ℝ)=Q (Fin.last k) ∧ |
| (∀ p ∈ l, p.1 ∈ Icc (0:ℝ) 1) ∧ |
| TimeChainModels (quantileCDF k Q) 0 l ∧ |
| ∀ β f, scalarTimeChain β l f=scalarHierarchy (k+1) (quantileMass k) |
| (fun i => β*Real.sqrt (cumulativeGapMap k Q i)) f := by |
| let q := Fin.cons (α := fun _ => ℝ) 0 Q |
| let hq := monotone_cons_zero_quantiles Q hQ h0 |
| let l := partitionTimeChain (k+1) q hq (quantileMass k) |
| refine ⟨l,?_,?_,?_,?_⟩ |
| · have H := partitionTimeChain_duration (k+1) q hq (quantileMass k) |
| simpa only [q,← Fin.succ_last,Fin.cons_succ,Fin.cons_zero,sub_zero] using H |
| · intro p hp |
| obtain ⟨i,rfl⟩ := List.mem_ofFn.mp hp |
| change quantileMass k i ∈ Icc (0:ℝ) 1 |
| constructor |
| · exact div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _) |
| · apply (div_le_one (by positivity : (0:ℝ)<(k+1:ℕ))).mpr |
| exact_mod_cast i.isLt.le |
| · change TimeChainModels (quantileCDF k Q) (q 0) (partitionTimeChain (k+1) q hq (quantileMass k)) |
| apply partitionTimeChain_models |
| intro i z hz |
| exact quantileCDF_on_interval Q hQ i z hz |
| · intro β f |
| dsimp only [l,partitionTimeChain] |
| rw [scalarTimeChain_ofFn] |
| congr 2 |
| funext i |
| simp only [NNReal.coe_mk, q] |
| rw [quantile_knots_gap] |
| end SK.Analytic |
|
|
| end |
| end |
|
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| end OAI |
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