| import Mathlib |
| import OAI.Probability.SKBarriers.Hierarchy.WeightedAugment |
| import OAI.Probability.SKBarriers.Hierarchy.TimeChainOn |
| import OAI.Probability.SKBarriers.Hierarchy.ConstantWeightStats |
| import OAI.Probability.SKBarriers.Hierarchy.ConstantWeightMean |
| import OAI.Probability.SKBarriers.Hierarchy.WeightedListMean |
| import OAI.Probability.SKBarriers.Locking.NarrowRetainedStats |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| structure NarrowTimeBlocks (α : ℝ → ℝ) (r h q : ℝ) where |
| before : TimeChainOn α 0 (r-h) |
| left : TimeChainOn α (r-h) r |
| right : TimeChainOn α r (r+h) |
| after : TimeChainOn α (r+h) q |
| tail : TimeChainOn α q 1 |
|
|
| namespace NarrowTimeBlocks |
| variable {α : ℝ → ℝ} {r h q : ℝ} (B : NarrowTimeBlocks α r h q) |
|
|
| def common : TimeChainOn α 0 r := B.before.append B.left |
| def middle : TimeChainOn α r q := B.right.append B.after |
| def front : TimeChainOn α 0 q := B.common.append B.middle |
| def full : TimeChainOn α 0 1 := B.front.append B.tail |
|
|
| def b : List (ℝ × ℝ) := rawTimeChain B.before.chain |
| def l : List (ℝ × ℝ) := rawTimeChain B.left.chain |
| def j : List (ℝ × ℝ) := rawTimeChain B.right.chain |
| def k : List (ℝ × ℝ) := rawTimeChain B.after.chain |
| def t : List (ℝ × ℝ) := rawTimeChain B.tail.chain |
|
|
| def c (a : ℝ) : List (ℝ × (ℝ × ℝ)) := zeroWeightChain B.b++constantWeightChain (a/h) B.l |
| def v (θ : ℝ) : List (ℝ × (ℝ × ℝ)) := constantWeightChain (θ/h) B.j++zeroWeightChain B.k |
| def w (a : ℝ) : List (ℝ × (ℝ × ℝ)) := constantWeightChain (-a/h) B.j++zeroWeightChain B.k |
|
|
| @[simp] theorem c_underlying (a : ℝ) : weightedUnderlying (B.c a)=B.b++B.l := by |
| simp only [c,weightedUnderlying_append,weightedUnderlying_zeroWeight,constantWeightChain_underlying] |
| @[simp] theorem v_underlying (θ : ℝ) : weightedUnderlying (B.v θ)=B.j++B.k := by |
| simp only [v,weightedUnderlying_append,weightedUnderlying_zeroWeight,constantWeightChain_underlying] |
| @[simp] theorem w_underlying (a : ℝ) : weightedUnderlying (B.w a)=B.j++B.k := by |
| simp only [w,weightedUnderlying_append,weightedUnderlying_zeroWeight,constantWeightChain_underlying] |
|
|
| theorem common_raw (a : ℝ) : rawTimeChain B.common.chain=weightedUnderlying (B.c a) := by |
| simp only [common,TimeChainOn.append,rawTimeChain_append,c_underlying,b,l] |
| theorem middle_raw (a : ℝ) : rawTimeChain B.middle.chain=weightedUnderlying (B.w a) := by |
| simp only [middle,TimeChainOn.append,rawTimeChain_append,w_underlying,j,k] |
| theorem front_raw (a : ℝ) : rawTimeChain B.front.chain=weightedUnderlying (B.c a)++weightedUnderlying (B.w a) := by |
| rw [front,TimeChainOn.append,rawTimeChain_append,B.common_raw a,B.middle_raw a] |
| theorem full_raw (a : ℝ) : rawTimeChain B.full.chain=weightedUnderlying (B.c a)++weightedUnderlying (B.w a)++B.t := by |
| rw [full,TimeChainOn.append,rawTimeChain_append,B.front_raw a]; rfl |
|
|
| @[simp] theorem l_variance : rawVariance B.l=h := by |
| rw [l,rawVariance_rawTimeChain,B.left.duration]; ring |
| @[simp] theorem j_variance : rawVariance B.j=h := by |
| rw [j,rawVariance_rawTimeChain,B.right.duration]; ring |
| @[simp] theorem common_variance (a : ℝ) : rawVariance (weightedUnderlying (B.c a))=r := by |
| rw [← B.common_raw a,rawVariance_rawTimeChain,B.common.duration,sub_zero] |
| @[simp] theorem front_variance (a : ℝ) : rawVariance (weightedUnderlying (B.c a)++weightedUnderlying (B.w a))=q := by |
| rw [← B.front_raw a,rawVariance_rawTimeChain,B.front.duration,sub_zero] |
| @[simp] theorem full_variance (a : ℝ) : rawVariance (weightedUnderlying (B.c a)++weightedUnderlying (B.w a)++B.t)=1 := by |
| rw [← B.full_raw a,rawVariance_rawTimeChain,B.full.duration,sub_zero] |
|
|
| theorem cross_stats (hh : h≠0) (a θ : ℝ) : |
| weightedCross (B.c a)=a ∧ weightedCross (B.v θ)=θ ∧ weightedCross (B.w a)=-a := by |
| simp only [c,v,w,weightedCross_append,weightedCross_zeroWeight,constantWeightChain_cross,B.l_variance,B.j_variance,zero_add,add_zero] |
| field_simp |
| simp |
|
|
| theorem variance_stats (hh : h≠0) (a θ : ℝ) : |
| narrowVariance (B.c a) (B.v θ) (B.w a)=(2*a^2+θ^2)/h := by |
| simp only [narrowVariance,c,v,w,weightedVariance_append,weightedVariance_zeroWeight,constantWeightChain_variance,B.l_variance,B.j_variance,zero_add,add_zero] |
| field_simp |
| ring |
|
|
| theorem absCross_stats (hh : 0<h) {a θ : ℝ} (ha : 0 ≤ a) (hθ : 0 ≤ θ) : |
| narrowAbsCross (B.c a) (B.v θ) (B.w a)=5*a+θ := by |
| simp only [narrowAbsCross,c,v,w,weightedAbsCross_append,weightedAbsCross_zeroWeight,constantWeightChain_absCross,B.l_variance,B.j_variance,zero_add,add_zero] |
| rw [abs_of_nonneg (div_nonneg ha hh.le),abs_of_nonneg (div_nonneg hθ hh.le),abs_div,abs_neg,abs_of_nonneg ha,abs_of_pos hh] |
| field_simp |
| ring |
|
|
| theorem v_mean (θ : ℝ) : weightedMean (B.v θ)=(θ/h)*rawArea B.j := by |
| simp [v] |
|
|
| end NarrowTimeBlocks |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
