09-EFT.WP.Core.Density v1.0 | Appendix D — Formula Compendium and Sketch Derivations

Appendix D — Formula Compendium and Sketch Derivations

I. How to Use This Table and the Numbering Scheme

II. Axioms and Measure Normalization (P91 series)

  1. P91-1 (measure explicitness)
    ∫ p(x) dx = 1 ; ( ∫_V rho(x,t) dV ) defines a physical total. dx, dV are base measure elements.
  2. P91-2 (unit/dim conservation)
    dim( ∫ rho dV ) = dim(rho) * dim(V) ; never mix dimensionless p(x) with dimensional rho(x,t).
  3. P91-3 (normalization discipline)
    In discrete settings sum_i p_i = 1 ; histogram density p_hat = count / ( N * Delta ) (see S92-10).
  4. Derivation highlights
    • Leverages additivity and linearity of Lebesgue integrals.
    • Uses dimensional homogeneity via check_dim(expr) to verify conservation.

III. Continuity and Conservation (S92-1, S92-2)

  1. S92-1 (continuity with sources/sinks)
    ∂_t rho(x,t) + ∇•J(x,t) = s(x,t)
  2. S92-2 (total-balance criterion)
    d/dt ( ∫V rho dV ) = - ( ∮{∂V} J•n dS ) + ( ∫_V s dV )
  3. Derivation highlights
    • Integrate S92-1 over V and apply the divergence theorem ( ∫_V ∇•J dV ) = ( ∮_{∂V} J•n dS ).
    • With s=0 and J•n|_{∂V}=0, the total M = ( ∫_V rho dV ) is constant in time.

IV. Probability Density, Likelihood, and Fisher Information (S92-3, S92-4)

  1. S92-3 (likelihood for independent samples)
    L(theta) = ∏_{i=1}^N p( x_i | theta )
  2. S92-4 (Fisher information)
    I_F(theta) = E[ ( ∂_theta log p(X|theta) )^T ( ∂_theta log p(X|theta) ) ]
  3. Derivation highlights
    • ∂_theta log L = ∑ ∂_theta log p(x_i|theta) ; the MLE satisfies ∂_theta log L = 0.
    • For regular families, I_F = - E[ ∂^2_{theta} log p ]; variance lower bounds follow S92-17.

V. Kernel Density Estimation and Error (S92-5, S92-6)

  1. S92-5 (1D KDE)
    kde_h(x) = ( 1 / ( N h ) ) * ∑_{i=1}^N K( ( x - x_i ) / h )
  2. S92-6 (mean integrated squared error, MISE)
    MISE(h) = Var_term(h) + Bias_term(h)
    AMISE(h) ≈ ( R(K) / ( N h ) ) + ( ( mu2(K)^2 / 4 ) * h^4 * R( f'' ) )
  3. Derivation highlights
    • E[ kde_h(x) ] = ( K_h * f )(x), so Bias ≈ ( mu2(K) * h^2 / 2 ) * f''(x).
    • Var[ kde_h(x) ] ≈ ( f(x) * R(K) ) / ( N h ).
    • Setting d(AMISE)/dh = 0 yields the optimal rate h* ∝ N^{-1/5} in 1D.

VI. Spatial / Spatio-temporal Intensity (S92-7)

  1. S92-7 (intensity to counts)
    Lambda(A) = ( ∫_A lambda(x) dV ), with N(A) ~ Poisson( Lambda(A) ) for Poisson processes.
  2. Hawkes intensity (see Chapter 5)
    lambda(t) = mu + ( ∑_k H(t - t_k) ), where the kernel H(•) satisfies stability ( ∫ H dt ) < 1.
  3. Derivation highlights
    • Campbell’s theorem: E[ N(A) ] = Lambda(A).
    • Hawkes mean intensity: E[lambda] = mu / ( 1 - ∫ H ).

VII. Spectral Density and Window Normalization (S92-8, S92-9)

  1. S92-8 (equivalent noise bandwidth)
    ENBW_Hz = fs * ( ∑{n=0}^{N-1} w[n]^2 ) / ( ∑{n=0}^{N-1} w[n] )^2
  2. S92-9 (window power coefficient)
    U_w = ( 1 / N ) * ∑_{n=0}^{N-1} w[n]^2
  3. Energy consistency (one-sided PSD, empirical)
    ∫_{0}^{fs/2} S_xx(f) df ≈ var(x) under proper window normalization and de-meaning.
  4. Derivation highlights
    • Welch’s method maps in-window variance to PSD via U_w and ENBW_Hz.
    • For one-sided spectra, do not double the DC and Nyquist bins (volume-wide rule).

VIII. Discretization and Histogram Conservation (S92-10, S92-11)

  1. S92-10 (histogram density)
    p_hat[j] = count[j] / ( N * Delta )
  2. S92-11 (voxelized total conservation)
    mass_preserve = ( ∑_i rho_i * V_i ) (should equal the numerical approximation of the continuous integral).
  3. Derivation highlights
    • ∑_j p_hat[j] * Delta = 1.
    • Grid refinement/coarsening must preserve ∑ rho_i V_i (see Mx-96).

IX. Regularization and Maximum Entropy (S92-12, S92-13)

  1. S92-12 (variational sparse–smooth hybrid)
    argmin_rho || A rho - y ||_2^2 + lambda * || ∇rho ||_1
  2. S92-13 (MaxEnt solution with moment constraints)
    p*(x) ∝ exp( ∑_{k=1}^m lambda_k * phi_k(x) )
  3. Derivation highlights
    • S92-12 uses first-order differences to approximate ∇rho; ADMM or PDHG are typical solvers.
    • For S92-13, construct the Lagrangian L = -∫ p log p dx + ∑ lambda_k ( ∫ p phi_k dx - c_k ) + alpha ( ∫ p dx - 1 ) and take the variational derivative w.r.t. p to obtain the exponential family.

X. Normalization & Calibration (S92-14, S92-15) and the Two Arrival-Time Forms

  1. S92-14 (standardization)
    z = ( x - mu_x ) / sigma_x
  2. S92-15 (change of variables with Jacobian)
    p_Y(y) = p_X( x(y) ) * | det( ∂x / ∂y ) |
  3. Two arrival-time forms (cross-volume anchor; see Core.Sea Chapter 8)
    T_arr = ( 1 / c_ref ) * ( ∫{gamma(ell)} n_eff d ell )
    T_arr = ( ∫{gamma(ell)} ( n_eff / c_ref ) d ell )
    delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |
  4. Derivation highlights
    • S92-15 follows from probability conservation P( x ∈ dx ) = P( y ∈ dy ).
    • delta_form records implementation-disparity; it must be propagated through the dataflow (see Mx-98).

XI. Uncertainty and Information Bounds (S92-16, S92-17) and Propagation

  1. S92-16 (Fisher information, matrix form)
    I_F(theta) = E[ ( ∂_theta log p )^T ( ∂_theta log p ) ]
  2. S92-17 (Cramer–Rao lower bound)
    cov( theta_hat ) ≥ I_F(theta)^{-1}
  3. Propagation and combination (metrological)
    u_c^2 = ∑i ( c_i^2 * u(x_i)^2 ) + 2 * ∑{i<j} c_i c_j * cov(x_i,x_j)
    U = k * u_c
  4. Derivation highlights
    • From zero-mean score and the information identity we obtain S92-17.
    • The linearized Delta method for g(x) yields sensitivity coefficients c_i = ∂g/∂x_i.

XII. Quick Constants and Equivalences (Kernels / Windows)

XIII. Workflow Hooks (Mx-9 and I90-)**

XIV. Pre-Publication Consistency Checklist