Chapter 27 Appendix: Notation and Function Reference
This appendix consolidates the notation used across the book and maps each object to its primary R implementation pattern. It is intended as a quick lookup for readers moving between theoretical sections and code-first workflows.
Unless otherwise noted, all functions in this appendix come from the NNS package. In executable code, load the package once and then call functions directly as shown in the table entries below.
27.1 Core directional operators and partial moments
| Symbol | Definition | Interpretation | R function / pattern |
|---|---|---|---|
| \(x^+\) | \(\max(x,0)\) | Positive-part operator | pmax(x, 0) |
| \((X-t)^+\) | \(\max(X-t,0)\) | Deviation above benchmark \(t\) | internal to UPM(...) |
| \((t-X)^+\) | \(\max(t-X,0)\) | Deviation below benchmark \(t\) | internal to LPM(...) |
| \(L_r(t;X)\) | \(E[(t-X)_+^r]\) | Lower partial moment, degree \(r\) | LPM(r, t, X) |
| \(U_r(t;X)\) | \(E[(X-t)_+^r]\) | Upper partial moment, degree \(r\) | UPM(r, t, X) |
| \(L_r/(L_r+U_r)\) | Degree-\(r\) lower ratio | Directional CDF-style probability below \(t\) | LPM.ratio(r, t, X) |
| \(U_r/(L_r+U_r)\) | Degree-\(r\) upper ratio | Directional probability above \(t\) | UPM.ratio(r, t, X) |
27.2 Co-partial moments, dependence, and causation
| Symbol | Definition / role | R function |
|---|---|---|
| \(CoLPM, CoUPM\) | Concordant lower/upper co-partial moments | Co.LPM(...), Co.UPM(...) |
| \(DLPM, DUPM\) | Divergent lower/upper co-partial moments | D.LPM(...), D.UPM(...) |
| \(NNS.dep(X,Y)\) | Global nonlinear dependence measure | NNS.dep(x, y) |
| \(NNS.copula(X,Y)\) | Nonparametric dependence geometry / copula view | NNS.copula(x, y) |
| \(NNS.caus(X,Y)\) | Directional causation diagnostic | NNS.caus(x, y) |
27.3 Distribution comparison, dominance, and interval objects
| Symbol | Definition / role | R function |
|---|---|---|
| \(F^{(0)}(t)\) | Degree-zero empirical CDF (step measure) | ecdf(x)(t) or LPM.ratio(0, t, x) |
| \(F^{(1)}(t)\) | Degree-one continuous CDF-style ratio | LPM.ratio(1, t, x) |
| \(p = P(X' > Y')\) | Directional exceedance probability for pairwise comparison | estimated by cross-sample indicator averages |
| \(\text{Certainty}_{\text{ANOVA}}\) | NNS ANOVA agreement certainty from CDF benchmark deviations (\(1\) = strongest agreement) | NNS.ANOVA(...) |
| FSD / SSD / TSD | First-, second-, third-order stochastic dominance | NNS.FSD(...), NNS.SSD(...), NNS.TSD(...) |
| \(Q^-_{d}(\alpha)\) | Lower degree-\(d\) quantile | LPM.VaR(alpha, degree = d, x) |
| \(Q^+_{d}(\alpha)\) | Upper degree-\(d\) quantile | UPM.VaR(alpha, degree = d, x) |
| PI\(_{1-\alpha}\) | Prediction interval \([Q^-_d(\alpha/2),Q^+_d(\alpha/2)]\) | LPM.VaR(...) + UPM.VaR(...) |
LPM.VaR(percentile, degree, variable)
Lower-tail threshold operator obtained by inverting the degree-specific lower partial-moment probability representation.
Interpretation by degree:
degree = 0: empirical-CDF lower quantile,degree = 1: severity-weighted lower threshold based on directional magnitude,degree = 2: extreme-deviation-sensitive lower threshold. In finance, the degree-zero case is commonly called VaR, but the operator is more general than that label.
UPM.VaR(percentile, degree, variable)
Upper-tail analog of LPM.VaR, used for right-tail threshold selection and interval construction.
27.4 Directional Decision Regions Crosswalk (Classical → NNS)
To maintain continuity with Chapter 22’s directional decision-region framing, the table below maps common classical statistics and procedures to their directional NNS counterparts.
| Classical statistic / workflow | Typical classical role | Directional NNS counterpart | Reference chapter | Notes |
|---|---|---|---|---|
| Pearson correlation | Linear association summary | NNS.dep(x, y) |
Chapter 10 | Captures nonlinear and asymmetric dependence, not only linear co-movement. |
| Parametric VaR / empirical quantile VaR | Tail-loss thresholding | LPM.VaR(alpha, degree, x) (degree-dependent) |
Chapter 16 | Degree controls sensitivity to tail severity beyond degree-0 quantiles. |
| Upper-tail quantile threshold | Right-tail risk/opportunity cutoff | UPM.VaR(alpha, degree, x) (degree-dependent) |
Chapter 16 | Upper-tail analog to LPM.VaR for asymmetric interval construction. |
| Classical ANOVA (mean-comparison test) | Group-level location comparison | NNS.ANOVA(...) (degree-dependent CDF benchmarking) |
Chapter 14 | Agreement certainty is benchmarked through directional CDF-style deviations. |
| Linear Granger-style directional inference | Lead-lag direction under linear structure | NNS.caus(x, y) |
Chapter 13 | Directionality can be nonlinear and state dependent. |
| Copula / Joint Tail Dependency | Joint probability of concurrent outcomes | Co.LPM(degree, target.x, target.y, x, y) / Co.UPM(degree, target.x, target.y, x, y) |
Chapter 4 | Co.LPM captures concurrent downside structure; Co.UPM is the upper-tail counterpart for joint directional events. |
| Mean-variance interval heuristics | Uncertainty bands under Gaussian assumptions | LPM.VaR(...) + UPM.VaR(...) (degree-dependent bounds) |
Chapters 16–17 | Produces directional prediction intervals without normality assumptions. |
27.5 Regression and forecasting workflow objects
| Symbol | Definition / role | R function | Reference chapter |
|---|---|---|---|
| \(\hat y = \hat E[Y\mid X]\) | NNS conditional mean estimate | NNS.reg(x, y) |
Chapter 21 |
| Residual local distribution | Partition-level error distribution | via NNS.reg(...)$Fitted.xy$residuals outputs |
Chapter 21 |
| \(\widehat{PI}(x_0)\) | Conditional prediction interval at \(x_0\) | NNS.reg(..., point.est = x0, confidence.interval = ...) |
Chapter 15 |
| Regime-specific directional dependence | Time-local / state-local dependence | NNS.dep(...) on rolling/segmented windows |
Chapter 10 |
27.6 A.3 Technical Note: Adaptive Order and Consistency Conditions
Chapter 18 established two core consistency conditions for recursive mean-split regression:
- Shrinking cell diameter at each target location so local bias vanishes,
- Growing cell occupancy so local sample averages stabilize.
When order = NULL, the implementation determines effective recursion depth per regressor from directional dependence with the response (NNS.dep-style logic). This modifies how quickly local cells contract across predictors, but it does not alter the fundamental structure of the consistency argument.
High-dependence predictors are allocated deeper partitioning, so their local diameters contract faster in regions where signal is strong. Low-dependence predictors are partitioned more conservatively, preserving broader local averaging where aggressive refinement would primarily amplify noise. Occupancy control remains enforced through the minimum cell-size rule.
The resulting estimator is therefore locally adaptive in rate:
- in high-signal regions, convergence tracks a faster path closer to an oracle fixed-order choice for that local structure;
- in low-signal regions, the estimator intentionally retains coarser cells, exchanging some local bias for improved stability.
A concise takeaway is:
Under the Chapter 18 regularity conditions (shrinking local diameters and diverging local occupancy), dependence-driven order allocation preserves the same bias–variance decomposition used for consistency arguments, while allowing refinement to concentrate where dependence signal is stronger.
The key practical point is that internal adaptive order selection keeps the same consistency checklist for users—control occupancy and ensure progressive local refinement—while often improving finite-sample stability by avoiding unnecessary depth in weak-signal coordinates.