Chapter 37
Correlation and Covariance — Shared Coherence

Flow Algebra

flow "Correlation and Covariance — Shared Coherence"

# ---------- CONCEPT ----------
# Covariance and correlation describe how two random variables share coherence — how their fluctuations align or oppose.
# Covariance measures the joint strength and direction of variation; correlation normalises this measure, revealing the degree of alignment independent of scale.
# Together, they form the geometry of dependency in probability space — the structure of shared flow between distributions.

# ---------- STATES ----------
state CoherenceSystem(C):
  RandomVariables:
    description: "X, Y — measurable distributed states on (Ω, 𝒜, P)."
  JointDistribution:
    description: "P_{X,Y}(x,y) — joint probability describing combined behaviour."
  MarginalDistributions:
    description: "P_X(x), P_Y(y) — individual distributions of X and Y."
  Covariance:
    description: "Cov(X,Y) = E[(X−E[X])(Y−E[Y])] — shared variation."
  Correlation:
    description: "ρ(X,Y) = Cov(X,Y)/(σ_X σ_Y) — normalised coherence measure."
  CovarianceMatrix:
    description: "Σ = E[(X−μ)(X−μ)^T] — multidimensional generalisation of joint coherence."
  Independence:
    description: "X ⟂ Y ⇔ P_{X,Y} = P_X P_Y ⇒ Cov(X,Y)=0 (for linear dependence)."

# ---------- PARAMETERS ----------
params:
  integrability: "E[X²], E[Y²] < ∞"
  centering: "μ_X=E[X], μ_Y=E[Y]"
  variances: "σ_X²=Var(X), σ_Y²=Var(Y)"
  closure: true

# ---------- INVARIANTS ----------
invariants:
  - "Cov(X,Y) = E[XY] − E[X]E[Y]."
  - "Var(X) = Cov(X,X)."
  - "Cov(aX+b, cY+d) = ac Cov(X,Y)."
  - "Var(aX+b) = a²Var(X)."
  - "ρ(X,Y) = Cov(X,Y)/(σ_X σ_Y)."
  - "−1 ≤ ρ(X,Y) ≤ 1."
  - "If ρ(X,Y)=0 ⇒ uncorrelated (no linear relation)."
  - "For jointly Gaussian X,Y: ρ=0 ⇔ independence."
  - "Covariance matrix Σ symmetric, positive semidefinite."
  - "Eigenvectors of Σ define principal directions of variance (PCA basis)."
  - "Total variance (trace Σ) = sum of eigenvalue variances."

# ---------- TRANSFORMS ----------
transforms:
  ComputeCovariance(X,Y):
    rule: "Cov(X,Y) = E[(X−E[X])(Y−E[Y])]."
  ComputeCorrelation(X,Y):
    rule: "ρ(X,Y) = Cov(X,Y)/(σ_X σ_Y)."
  Standardise(X):
    rule: "Z_X = (X−E[X])/σ_X (z-score normalisation)."
  CovarianceMatrixFromData(X):
    rule: "Σ = (1/n) Σ_i (x_i − μ)(x_i − μ)^T."
  CorrelationMatrixFromCov(Σ):
    rule: "R_{ij} = Σ_{ij}/√(Σ_{ii}Σ_{jj})."
  AffineTransform(X):
    rule: "Y = A X + b ⇒ Σ_Y = A Σ_X A^T."
  CovarianceOfSum(X,Y):
    rule: "Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y)."
  ConditionalCovariance(X|Y):
    rule: "Σ_{X|Y} = Σ_{XX} − Σ_{XY}Σ_{YY}^{−1}Σ_{YX}."
  PCA_Decomposition(Σ):
    rule: "Σ = PΛP^T ⇒ decorrelate via Z = P^T(X−μ)."

# ---------- MONITORS ----------
monitors:
  SymmetryCheck:
    measure: "Cov(X,Y)−Cov(Y,X)"
    meaning: "Zero ⇒ symmetry holds."
  CorrelationBounds:
    measure: "ρ(X,Y)"
    condition: "−1 ≤ ρ ≤ 1."
  IndependenceTest:
    meaning: "Check if P_{X,Y} = P_X P_Y ⇒ Cov=0."
  PositiveSemidefiniteCheck:
    measure: "v^T Σ v"
    condition: "≥0 for all v (covariance matrix valid)."
  PCA_VarianceCapture:
    meaning: "Fraction of total variance captured by first k eigenmodes."
  MulticollinearityCheck:
    meaning: "Detect strong correlations (|ρ|≈1) among variables."

# ---------- EQUATIONS ----------
equations:
  - "Cov(X,Y) = E[(X−E[X])(Y−E[Y])]"
  - "Cov(X,Y) = E[XY] − E[X]E[Y]"
  - "Var(X) = Cov(X,X)"
  - "ρ(X,Y) = \\frac{Cov(X,Y)}{σ_X σ_Y}, \\quad −1 ≤ ρ ≤ 1"
  - "Σ = E[(X−μ)(X−μ)^T]"
  - "Var(X+Y) = Var(X)+Var(Y)+2Cov(X,Y)"
  - "Σ_Y = AΣ_XA^T"
  - "Σ_{X|Y} = Σ_{XX} − Σ_{XY}Σ_{YY}^{−1}Σ_{YX}"
  - "R_{ij} = \\frac{Σ_{ij}}{\\sqrt{Σ_{ii}Σ_{jj}}}"
  - "Eigendecomposition: Σ = PΛP^T, Λ=diag(λ_1,…,λ_d), λ_i≥0"

# ---------- NOTES ----------
notes:
  - "Covariance measures joint deviation — the shared movement of two variables relative to their means."
  - "Correlation rescales covariance into a dimensionless measure of alignment — removing dependence on magnitude or units."
  - "Positive covariance/correlation: variables rise and fall together; negative: one rises as the other falls."
  - "The covariance matrix generalises this structure — its off-diagonal entries encode shared coherence between dimensions."
  - "Decorrelating via PCA rotates the system to orthogonal axes, eliminating shared covariance — independent modes of variation."
  - "Covariance quantifies the strength of shared fluctuation; correlation quantifies its purity — how perfectly aligned those fluctuations are."
  - "In Flow Algebra, covariance represents the coupling of coherence between flows, while correlation expresses the normalised symmetry of that coupling — shared coherence as measurable alignment in the probability field."