Foundations

Scientific relationships · 23 / 23

The relationship type describes the connection. Its linked statements, sources and conditions define the scope.

1. Variance, covariance and correlation → Expected value of a discrete random variable
   Uses definition
   Relationship sources · 1

   • STAT 414, Lesson 18: Covariance and correlation ↗

   Linked statements

   • For square-integrable real X,Y, Var X=E[(X−EX)²] and Cov(X,Y)=E[(X−EX)(Y−EY)]. If both variances are positive, Corr=Cov/√(Var X Var Y) lies in [−1,1]. Independence gives zero covariance; its converse generally fails.
2. Variance, covariance and correlation → Conditional probability and independence
   Uses definition
   Relationship sources · 1

   • STAT 414, Lesson 18: Covariance and correlation ↗

   Linked statements

   • For square-integrable real X,Y, Var X=E[(X−EX)²] and Cov(X,Y)=E[(X−EX)(Y−EY)]. If both variances are positive, Corr=Cov/√(Var X Var Y) lies in [−1,1]. Independence gives zero covariance; its converse generally fails.
3. Law of total variance → Variance, covariance and correlation
   Uses definition
   Relationship sources · 1

   • Introduction to Probability, second edition: selected summary material ↗

   Linked statements

   • For square-integrable X, Var(X)=E[Var(X|Y)]+Var(E[X|Y]). The terms separate average within-condition variation from variation of the conditional mean.
4. Moment generating functions → Variance, covariance and correlation
   Uses definition
   Relationship sources · 1

   • STAT 414, Lesson 9: Moment generating functions ↗

   Linked statements

   • M_X(t)=E[eᵗˣ] is an MGF when it is finite on an open interval about zero. Its derivatives at zero yield moments; in particular EX=M′(0) and Var X=M″(0)−[M′(0)]².
5. Estimator bias and mean squared error → Variance, covariance and correlation
   Uses definition
   Relationship sources · 1

   • STATS 200, Lecture 12: Parametric models and estimation ↗

   Linked statements

   • For estimator T of θ, bias is EθT−θ and MSE is Eθ[(T−θ)²]=Varθ(T)+bias². Unbiasedness means zero bias at every allowed θ and does not itself imply a small error or consistency.
6. Linear regression coefficients and covariance → Variance, covariance and correlation
   Uses definition
   Relationship sources · 1

   • STATS 200, Lecture 25: The linear model ↗

   Linked statements

   • For fixed full-column-rank X and Y=Xβ+ε with Eε=0 and Cov ε=σ²I, least squares gives β̂=(XᵀX)⁻¹XᵀY, Eβ̂=β and Cov β̂=σ²(XᵀX)⁻¹. Independent normal errors additionally make β̂ exactly multivariate normal.
7. Residual diagnostics and model adequacy → Variance, covariance and correlation
   Uses definition
   Relationship sources · 1

   • e-Handbook of Statistical Methods, 4.4.4: Model validation ↗

   Linked statements

   • Residuals eᵢ=yᵢ−f(xᵢ;β̂) compare observed and fitted responses. Patterns against predictors, fitted values or run order can expose functional misspecification, changing variance or dependence. A high R² does not certify model adequacy.
8. Homogeneous Poisson process → Variance, covariance and correlation
   Uses definition
   Relationship sources · 1

   • Introduction to Probability, second edition: selected summary material ↗

   Linked statements

   • A rate-λ Poisson counting process starts at zero, has independent increments and N(t+s)−N(s) distributed Poisson(λt). Successive interarrival times are independent exponentials of rate λ; EN(t)=Var N(t)=λt.
9. Standard Brownian motion → Variance, covariance and correlation
   Uses definition
   Relationship sources · 1

   • Course 275B (2012), Lecture 17: Brownian motion ↗

   Linked statements

   • Standard Brownian motion has B(0)=0, almost surely continuous paths and independent increments with B(t)−B(s)∼N(0,t−s) for 0≤s<t. It is a mean-zero Gaussian process with Cov(B(s),B(t))=min(s,t).
10. Chi-square distribution → Variance, covariance and correlation
    Uses definition
    Relationship sources · 1

    • STATS 200 (Autumn 2016), Lecture 3: Joint laws and sampling statistics ↗

    Linked statements

    • The sum of squares of k independent standard normal variables has chi-square law χ²ₖ=Gamma(k/2,1/2), with mean k and variance 2k.
11. Markov and Chebyshev probability bounds → Variance, covariance and correlation
    Uses definition
    Relationship sources · 1

    • Measure, Integration & Real Analysis, section 5B: Iterated integrals ↗

    Linked statements

    • For nonnegative X with finite mean and a>0, P(X≥a)≤E[X]/a. For finite-variance Y and t>0, P(|Y−E[Y]|≥t)≤Var(Y)/t², by applying the first bound to (Y−E[Y])².
12. Inverse calibration and fitted-parameter uncertainty → Variance, covariance and correlation
    Uses definition
    Relationship sources · 3

    • NIST/SEMATECH e-Handbook: Calibration of future measurements ↗
    • NIST/SEMATECH e-Handbook: Uncertainty of calibrated values ↗
    • NIST/SEMATECH e-Handbook: Propagation of calibration errors ↗

    Linked statements

    • For the valid linear calibration y=a+bx, an unknown input is estimated as x̂=(y₀−â)/b̂. Its uncertainty includes the future indication, fitted intercept/slope, their covariance, reference standards and applicable extra influences. First-order propagation uses gradient (1/b̂,−1/b̂,−x̂/b̂) with respect to (y₀,â,b̂).

    Intercept/slope dependence and any future-indication dependence must remain in the covariance matrix.

13. Measurement noise, correlation and averaging → Variance, covariance and correlation
    Derived from
    Relationship sources · 2

    • NIST TN 1297 (1994): Type A evaluation ↗
    • NIST TN 1297 (1994): Propagation law ↗

    Linked statements

    • For repeated indications, the mean has variance Var(x̄)=(1/n²)Σ_iΣ_j Cov(x_i,x_j). It reduces to σ²/n for independent equal-variance observations; shared offsets, temporal correlation and drift can prevent the usual square-root improvement from averaging.

    Mean variance is a double sum of sample covariance; the independent 1/N law is only a specialization.

14. Vector measurement uncertainty and a two-output example → Variance, covariance and correlation
    Derived from
    Relationship sources · 1

    • JCGM 102:2011: extension to multiple output quantities ↗

    Linked statements

    • For a differentiable real model Y=f(X), first-order output covariance is Σ_Y≈JΣ_XJᵀ, J_ai=∂f_a/∂x_i at the input estimate; the rule is exact for an affine model with finite input covariance. For impedance inferred from voltage/current amplitudes and phase, R=V cosφ/I and X_r=V sinφ/I share the same uncertain inputs, so their covariance must be retained as well as their separate uncertainties.

    An affine transformed vector has exactly JΣJᵀ covariance; nonlinear use requires an adequately controlled local approximation.

15. Experimental units and biological inference → Variance, covariance and correlation
    Related to
    Relationship sources · 2

    • Experimental Design Assistant, Experimental Unit ↗
    • Key Elements of a Well-Designed Experiment, randomization and blinding ↗

    Linked statements

    • The experimental unit is the entity independently assigned to a treatment in a specified design. Multiple observations from one treated unit need an appropriate within-unit analysis and do not automatically increase the number of independent treatment replicates. Defining units, randomization and masking helps relate biological variability and measurement variability to the intended inference.

    Dependence affects variance of an estimated response.

16. Quantitative inheritance and heritability scope → Variance, covariance and correlation
    Related to
    Relationship sources · 1

    • 5.1 Quantitative Traits and Heritability, slides 2–7 ↗

    Linked statements

    • In a specified population model, narrow-sense heritability is the ratio of additive genetic variance to phenotypic variance. Genetic effects, environment and their interactions contribute to trait variation. Heritability depends on the sampled population and environment; it is not the fraction of one individual's trait caused by genes.

    The variance decomposition requires stated covariance assumptions.

17. Phylogenetic covariance and comparative trait methods → Variance, covariance and correlation
    Related to
    Relationship sources · 1

    • Chapter4 sections4.2–4.3: Contrast Assumptions and Brownian Covariance Likelihood ↗

    Linked statements

    • Species traits can be correlated because related lineages share evolutionary history. Under an explicit Brownian trait model on a timed tree, shared branch history determines expected covariance; independent-contrast or likelihood methods account for that structure instead of treating every tip as an independent observation. Tree/model uncertainty and measurement error affect the resulting rate or association inference.

    Shared history creates off-diagonal covariance.

18. Principal components and representation assumptions → Variance, covariance and correlation
    Related to
    Relationship sources · 1

    • Understanding Machine Learning: From Theory to Algorithms ↗

    Linked statements

    • PCA finds an orthogonal low-rank subspace minimizing squared reconstruction error for centered observations. Leading covariance eigenvectors solve this objective; scale choices and outliers change the subspace. Large explained variance need not carry the target signal, and a low-dimensional representation does not establish causal factors. Identifiability can fail when relevant eigenvalues repeat.

    Covariance structure determines the selected directions.

19. Precision electroweak fits and nuisance parameters → Variance, covariance and correlation
    Related to
    Relationship sources · 1

    • Review of Particle Physics · Electroweak model and constraints on new physics ↗

    Linked statements

    • A precision Standard Model fit compares predicted electroweak observables with correlated measurements, jointly propagating input-parameter, systematic and missing-order uncertainties. The inspected PDG review uses a declared renormalization convention and assumptions such as lepton universality. A residual or an excluded input tests that specific model/fit construction; it is not automatically evidence for a unique new particle.

    Correlated likelihoods require the full covariance structure.

20. Reynolds stresses and turbulence closure → Variance, covariance and correlation
    Related to
    Relationship sources · 2

    • Reynolds averaging · Aerodynamics lecture 26 (2005) ↗
    • CFD validation assessment ↗

    Linked statements

    • For constant-density incompressible flow, averaging u=U+u′ adds the Reynolds stress R_ij=⟨u′_iu′_j⟩ to the mean momentum equation. Its transport introduces higher correlations, so averaging does not close the equations. Eddy-viscosity, Reynolds-stress and data-informed closures encode different assumptions; a fitted mean velocity alone does not validate every stress or transient.

    Velocity correlations are tensor covariance components.

21. Nonlinear correlated output propagation → Variance, covariance and correlation
    Related to
    Relationship sources · 1

    • JCGM 102:2011 · Multiple output quantities ↗

    Linked statements

    • For a vector measurand Y=f(X), propagating a joint input distribution retains output dependence and non-Gaussian shape. Local U_Y≈J U_X Jᵀ is only a linearization. For the explicit X∼N(0,u²), Y=(X,X²), Cov(Y₁,Y₂)=0 although the outputs are dependent; a covariance matrix alone need not specify their joint distribution.

    Zero covariance is weaker than independence.

22. Network covariance and common reference limits → Variance, covariance and correlation
    Related to
    Relationship sources · 2

    • JCGM GUM-6:2020 · Developing measurement models ↗
    • Statistical methods for next-generation clock networks (2025 project update) ↗

    Linked statements

    • For linear observations z=Aθ+ε with known positive-definite covariance Σ and full-column-rank A, generalized least squares yields θ̂=(AᵀΣ⁻¹A)⁻¹AᵀΣ⁻¹z. Shared reference, environmental and transfer errors create off-diagonal covariance; they do not average away as independent scatter. Fix an identifiable reference/gauge and model synchronization separately.

    Common uncertainty enters a joint covariance matrix.

23. First-order uncertainty propagation → Variance, covariance and correlation
    Uses definition
    Relationship sources · 1

    • Technical Note 1297, Appendix A: Law of Propagation of Uncertainty ↗

    Linked statements

    • For y = f(x₁,…,xₙ), first-order propagation weights input variances and covariances by the model's sensitivity coefficients. Correlations must be retained when the input estimates are correlated.

    The uncertainty rule uses the joint covariance of input quantities. Finite covariance in the declared measurement model; retain shared dependencies.