09-EFT.WP.Core.Density v1.0 | Appendix C — Kernel and Window Library (Density Volume)

Home ／ Docs-Technical WhitePaper ／ 09-EFT.WP.Core.Density v1.0

Appendix C — Kernel and Window Library (Density Volume)

I. Scope and Unified Notation

• This appendix standardizes kernel functions K(u) (for kde_h(x), see S92-5, S92-6) and discrete spectral windows w[n] (for S_xx(f), see S92-8, S92-9): definitions, common properties, and quick-reference constants. All symbols follow the volume-wide conventions and registry.
• Kernel convention (1D primitive)
  ∫ K(u) du = 1, K(u) ≥ 0. Symmetric kernels satisfy K(u) = K(-u). The second raw moment is mu2(K) = ( ∫ u^2 K(u) du ), and roughness is R(K) = ( ∫ K(u)^2 du ). In d-dimensional isotropic form,
  kde_h(x) = ( 1 / (N h^d) ) * ∑ K( ||x - x_i|| / h ); for anisotropy see Section III.
• Window convention (length N, sampling rate fs)
  w[n] is defined for n = 0,1,...,N-1. Window power U_w = ( 1 / N ) * ∑_0^{N-1} w[n]^2 (see S92-9); equivalent noise bandwidth ENBW_Hz = fs * ( ∑ w[n]^2 ) / ( ∑ w[n] )^2 (see S92-8); coherent gain CG = ( ∑ w[n] ) / N. In “bin” units, ENBW_bins = U_w / CG^2.

II. Kernel Catalogue (1D, symmetric, normalized)

1. Gaussian (unbounded support)
   • Definition: K(u) = ( 1 / sqrt(2*pi) ) * exp( - u^2 / 2 )
   • Support: u ∈ (-∞, +∞); maximum smoothness
   • Constants: mu2(K) = 1, R(K) = 1 / ( 2 * sqrt(pi) ) ≈ 0.282094
   • Note: neutral baseline in AMISE analysis; Scott bandwidth in d dims in Section III.
2. Epanechnikov (compact support)
   • Definition: K(u) = ( 3 / 4 ) * ( 1 - u^2 ) * 1{|u| ≤ 1}
   • Constants: mu2(K) = 1/5 = 0.2, R(K) = 3/5 = 0.6
   • Note: AMISE-optimal among compact, symmetric, second-order kernels with fixed mu2.
3. Uniform / Rectangular (compact support)
   • Definition: K(u) = ( 1 / 2 ) * 1{|u| ≤ 1}
   • Constants: mu2(K) = 1/3 ≈ 0.333333, R(K) = 1/2 = 0.5
   • Note: boundary-friendly interpretation; least smooth estimates.
4. Triangular (compact support)
   • Definition: K(u) = ( 1 - |u| ) * 1{|u| ≤ 1}
   • Constants: mu2(K) = 1/6 ≈ 0.166667, R(K) = 2/3 ≈ 0.666667
5. Biweight / Quartic (compact support)
   • Definition: K(u) = ( 15 / 16 ) * ( 1 - u^2 )^2 * 1{|u| ≤ 1}
   • Constants: mu2(K) = 1/7 ≈ 0.142857, R(K) = 5/7 ≈ 0.714286
6. Triweight (compact support)
   • Definition: K(u) = ( 35 / 32 ) * ( 1 - u^2 )^3 * 1{|u| ≤ 1}
   • Constants: mu2(K) = 1/9 ≈ 0.111111, R(K) = 350/429 ≈ 0.815851
7. Cosine (compact support)
   • Definition: K(u) = ( pi / 4 ) * cos( ( pi / 2 ) * u ) * 1{|u| ≤ 1}
   • Constants: mu2(K) ≈ 0.18943, R(K) ≈ 0.61685
8. Laplace / Double-Exponential (unbounded support)
   • Definition: K(u) = ( 1 / 2 ) * exp( - |u| )
   • Constants: mu2(K) = 2, R(K) = 1/4 = 0.25
   • Note: heavier tails; AMISE suggests shrinking h to temper variance.

• Quick AMISE guide (1D, twice-differentiable target)
  AMISE(h) ≈ ( R(K) / ( N * h ) ) + ( ( mu2(K)^2 / 4 ) * h^4 * R( f'' ) ) * c_dim, with c_dim = 1 (1D) and R(f'') = ( ∫ ( f''(x) )^2 dx ). The optimal h* balances the two terms (see Chapter 4 and Mx-93).

III. Multivariate Kernels and Bandwidth Matrices

• Isotropic: kde_h(x) = ( 1 / ( N * h^d ) ) * ∑ K( ||x - x_i|| / h ).
• Anisotropic (SPD bandwidth H):
  kde_H(x) = ( 1 / ( N * |H|^{1/2} ) ) * ∑ K( || H^{-1/2} * ( x - x_i ) || ).
• Scott rule: H_scott = h_scott^2 * Sigma, h_scott = ( 4 / ( d + 2 ) )^{ 1 / ( d + 4 ) } * N^{ -1 / ( d + 4 ) }, with Sigma the sample covariance.
• Silverman (1D): h_silverman = 0.9 * min( sigma, IQR / 1.34 ) * N^{-1/5}.
• Implementation note: in kde_build(..., kernel:str, h:float|H:matrix) report K(•), h, and the selection score CV(h) (see I90-2 and rule set III).

IV. Boundary and Grid-Aware Kernel Corrections

• Reflection (mirror at boundary x_b): x_ref = 2 * x_b - x_i; use K( ( x - x_i ) / h ) + K( ( x - x_ref ) / h ) with normalization preserved.
• Truncate–renormalize: integrate contributions only over D and renormalize with Z(x) = ( ∫_D K( ( x - u ) / h ) du ).
• Local linear kernels: add first-order terms to reduce boundary bias.
• Consistency with voxelization: when reconstructing from a grid rho_i, preserve mass_preserve = ( ∑ rho_i * V_i ) (see S92-11, Chapter 7).

V. Window Catalogue (for discrete spectral density)

1. Cosine-sum form and metrics
   w[n] = ∑_{k=0}^K a_k * cos( 2*pi*k*n / (N-1) ) (typical alternating signs a0 - a1 cos + a2 cos2 - ...).
   For large N: U_w ≈ a0^2 + 0.5 * ∑_{k=1}^K a_k^2, CG ≈ a0, ENBW_bins ≈ U_w / CG^2.
   ENBW_Hz = ENBW_bins * ( fs / N ).
   • Rectangular — w[n] = 1
     U_w = 1, CG = 1, ENBW_bins = 1.000; first sidelobe ≈ −13 dB.
   • Bartlett / Triangular — w[n] = 1 - | ( 2*n / (N-1) ) - 1 |
     U_w ≈ 1/3, CG ≈ 0.5, ENBW_bins ≈ 4/3 = 1.333; first sidelobe ≈ −26 dB.
   • Hann — w[n] = 0.5 - 0.5 * cos( 2*pi*n / (N-1) )
     U_w ≈ 0.375, CG ≈ 0.5, ENBW_bins ≈ 1.500; first sidelobe ≈ −31 dB. (Default trade-off; see Chapter 6).
   • Hamming — w[n] = 0.54 - 0.46 * cos( 2*pi*n / (N-1) )
     U_w ≈ 0.3974, CG ≈ 0.54, ENBW_bins ≈ 1.3628; first sidelobe ≈ −43 dB.
   • Blackman — w[n] = 0.42 - 0.5 * cos( 2*pi*n / (N-1) ) + 0.08 * cos( 4*pi*n / (N-1) )
     U_w ≈ 0.3046, CG ≈ 0.42, ENBW_bins ≈ 1.7268; first sidelobe ≈ −58 dB.
   • Blackman–Harris (4-term, 92 dB) —
     w[n] = a0 - a1 * cos( 2*pi*n / (N-1) ) + a2 * cos( 4*pi*n / (N-1) ) - a3 * cos( 6*pi*n / (N-1) )
     with a0=0.35875, a1=0.48829, a2=0.14128, a3=0.01168.
     U_w ≈ 0.25796, CG ≈ 0.35875, ENBW_bins ≈ 2.004; first sidelobe ≈ −92 dB.
   • Nuttall (4-term) — coefficients a0=0.355768, a1=0.487396, a2=0.144232, a3=0.012604 (same structure).
     U_w ≈ 0.25583, CG ≈ 0.355768, ENBW_bins ≈ 2.021; sidelobe suppression comparable to BH.
   • Flat-top (5-term, amplitude accuracy priority) —
     w[n] = a0 - a1 * cos( 2*pi*n / (N-1) ) + a2 * cos( 4*pi*n / (N-1) ) - a3 * cos( 6*pi*n / (N-1) ) + a4 * cos( 8*pi*n / (N-1) )
     typical a0=1.0, a1=1.93, a2=1.29, a3=0.388, a4=0.028.
     U_w ≈ 3.7702, CG ≈ 1.0, ENBW_bins ≈ 3.770; minimizes amplitude bias.
   • Kaiser (parameterized) —
     w[n] = I0( pi*beta*sqrt( 1 - x^2 ) ) / I0( pi*beta ), with x = ( 2*n / (N-1) ) - 1, I0 the zeroth-order modified Bessel.
     Empirical: beta=6 → ENBW_bins ≈ 1.9, beta=8 → ≈ 2.23, beta=10 → ≈ 2.5.
     Beta tunes sidelobe vs. main-lobe width continuously.
2. Implementation hint: in spectral_density(sig, method="welch", window="hann") report U_w, ENBW_Hz, and CG, and use the same window power normalization in the energy check (Mx-95, Chapter 6).

VI. Shape Mapping and Selection Guidance

1. Shape isomorphisms: Uniform ↔ Rectangular, Triangular ↔ Bartlett, Cosine kernel ↔ Hann window, higher-order compact polynomial kernels ↔ polynomial cosine-sum windows (e.g., Blackman). Choosing shape-matched pairs maintains a coherent smoothing style across domains (density ↔ spectrum).
2. Defaults (cross-volume aligned):
   • For density estimation, prefer Epanechnikov or Biweight (compact support, AMISE-friendly).
   • For spectrum estimation, prefer Hann (balanced leakage vs. resolution); for amplitude-critical work use Flat-top.

VII. I90 Binding and Example Configurations

• KDE: kde_build(data, kernel="epanechnikov", h=..., rule="cv") -> PdfRef; evaluate with normalize=True, and record CV(h) plus kernel constants mu2(K)/R(K) (see I90-2).
• Spectrum: spectral_density(sig, method="welch", window="hann") -> SpecRef; then spec_to_energy(spec, band) for energy consistency (I90-6).
• Mandatory metadata: bind K, h, U_w, ENBW_Hz, CG as estimator attributes (I90-8).

VIII. Pre-Publication Self-Check (Kernels & Windows)

• check_dim: ensure kernel-density units match the input measure; PSD reports ENBW_Hz and bin width fs/N.
• Normalization: kernels must satisfy ∫ K du = 1; windows must use the same U_w in energy checks.
• Reporting: KDE must include K(•), h, and the selection criterion; PSD must include window, U_w, ENBW_Hz, CG.
• Boundaries: if D is bounded, declare the boundary-correction strategy.
• Arrival-time axis: where spectral/time-frequency metrics are tied to T_arr, provide both forms and record delta_form (see ruleset and Chapter 9).

IX. Quick Constants (Summary)

1. Kernel second moments and roughness
   • Gaussian: mu2=1, R=1/(2*sqrt(pi))≈0.282094
   • Epanechnikov: mu2=1/5, R=3/5
   • Uniform: mu2=1/3, R=1/2
   • Triangular: mu2=1/6, R=2/3
   • Biweight: mu2=1/7, R=5/7
   • Triweight: mu2=1/9, R=350/429≈0.815851
   • Cosine: mu2≈0.18943, R≈0.61685
   • Laplace: mu2=2, R=1/4
2. Common window ENBW_bins (approx.)
   Rectangular = 1.000, Bartlett = 1.333, Hann = 1.500, Hamming = 1.363, Blackman = 1.727, Blackman-Harris(4) = 2.004, Nuttall(4) = 2.021, Flat-top(5) = 3.770, Kaiser(beta=8) ≈ 2.23

X. Cross-References and Version Alignment

• Direct reuse with Chapter 4 (Mx-93 bandwidth selection), Chapter 6 PSD formalism (S92-8/S92-9, Mx-95), Chapter 7 conservation discretization (S92-11, Mx-96).
• Bound to interface layer I90-2/4/6/8; window and energy conventions match Core.Sea exactly. Where arrival-time metrics apply, adhere to the two-form disparity delta_form recording rule.