Covariance matrix #
noncomputable def
ProbabilityTheory.covInnerBilin
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
(μ : MeasureTheory.Measure E)
:
Covariance of a measure on an inner product space, as a continuous bilinear form.
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
ProbabilityTheory.covInnerBilin_eq_covarianceBilin
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
(x y : E)
:
((covInnerBilin μ) x) y = ((covarianceBilin μ) ((InnerProductSpace.toDualMap ℝ E) x)) ((InnerProductSpace.toDualMap ℝ E) y)
theorem
ProbabilityTheory.covInnerBilin_apply
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[CompleteSpace E]
[MeasureTheory.IsFiniteMeasure μ]
(h : MeasureTheory.MemLp id 2 μ)
(x y : E)
:
theorem
ProbabilityTheory.IsGaussian.covInnerBilin_apply
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[IsGaussian μ]
[SecondCountableTopology E]
[CompleteSpace E]
(x y : E)
:
theorem
ProbabilityTheory.covInnerBilin_comm
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[MeasureTheory.IsFiniteMeasure μ]
(h : MeasureTheory.MemLp id 2 μ)
(x y : E)
:
theorem
ProbabilityTheory.covInnerBilin_self
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[CompleteSpace E]
[MeasureTheory.IsFiniteMeasure μ]
(h : MeasureTheory.MemLp id 2 μ)
(x : E)
:
theorem
ProbabilityTheory.covInnerBilin_self_nonneg
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[CompleteSpace E]
[MeasureTheory.IsFiniteMeasure μ]
(h : MeasureTheory.MemLp id 2 μ)
(x : E)
:
theorem
ProbabilityTheory.covInnerBilin_map
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[FiniteDimensional ℝ E]
{F : Type u_2}
[NormedAddCommGroup F]
[InnerProductSpace ℝ F]
[MeasurableSpace F]
[BorelSpace F]
[CompleteSpace E]
[CompleteSpace F]
[MeasureTheory.IsFiniteMeasure μ]
(h : MeasureTheory.MemLp id 2 μ)
(L : E →L[ℝ] F)
(u v : F)
:
((covInnerBilin (MeasureTheory.Measure.map (⇑L) μ)) u) v = ((covInnerBilin μ) ((ContinuousLinearMap.adjoint L) u)) ((ContinuousLinearMap.adjoint L) v)
noncomputable def
ProbabilityTheory.covMatrix
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
[FiniteDimensional ℝ E]
(μ : MeasureTheory.Measure E)
:
Matrix (Fin (Module.finrank ℝ E)) (Fin (Module.finrank ℝ E)) ℝ
Covariance matrix of a measure on a finite dimensional inner product space.
Equations
- ProbabilityTheory.covMatrix μ = Matrix.of fun (i j : Fin (Module.finrank ℝ E)) => ((ProbabilityTheory.covInnerBilin μ) ((stdOrthonormalBasis ℝ E) i)) ((stdOrthonormalBasis ℝ E) j)
Instances For
theorem
ProbabilityTheory.covMatrix_mulVec
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[FiniteDimensional ℝ E]
(x : Fin (Module.finrank ℝ E) → ℝ)
:
(covMatrix μ).mulVec x = fun (i : Fin (Module.finrank ℝ E)) =>
((covInnerBilin μ) ((stdOrthonormalBasis ℝ E) i)) (∑ j : Fin (Module.finrank ℝ E), x j • (stdOrthonormalBasis ℝ E) j)
theorem
ProbabilityTheory.dotProduct_covMatrix_mulVec
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[FiniteDimensional ℝ E]
(x y : Fin (Module.finrank ℝ E) → ℝ)
:
x ⬝ᵥ (covMatrix μ).mulVec y = ((covInnerBilin μ) (∑ j : Fin (Module.finrank ℝ E), x j • (stdOrthonormalBasis ℝ E) j))
(∑ j : Fin (Module.finrank ℝ E), y j • (stdOrthonormalBasis ℝ E) j)
theorem
ProbabilityTheory.posSemidef_covMatrix
{E : Type u_1}
[NormedAddCommGroup E]
[InnerProductSpace ℝ E]
[MeasurableSpace E]
[BorelSpace E]
{μ : MeasureTheory.Measure E}
[FiniteDimensional ℝ E]
[IsGaussian μ]
:
(covMatrix μ).PosSemidef