Getting Started with NNS: Overview"

In NNS: Nonlinear Nonparametric Statistics

#| label: setup
#| include: true
# Prereqs (uncomment if needed):
# install.packages("NNS")
# install.packages(c("data.table","xts","zoo","Rfast"))

suppressPackageStartupMessages({
  library(NNS)
  library(data.table)
})
set.seed(42)

data.table::setDTthreads(1L)
options(mc.cores = 1)
RcppParallel::setThreadOptions(numThreads = 1)
Sys.setenv("OMP_THREAD_LIMIT" = 1)

Orientation

Goal. A complete, hands‑on curriculum for Nonlinear Nonparametric Statistics (NNS) using partial moments. Each section blends narrative intuition, precise math, and executable code.

Structure.

1. Foundations — partial moments & variance decomposition
2. Descriptive & distributional tools
3. Dependence & nonlinear association
4. Hypothesis testing & ANOVA (LPM‑CDF)
5. Regression, boosting, stacking & causality
6. Time series & forecasting
7. Simulation (max‑entropy), Monte Carlo & risk‑neutral rescaling
8. Portfolio & stochastic dominance

Notation. For a random variable $X$ and threshold/target $t$, the population $n$‑th partial moments are defined as

$$ \operatorname{LPM}(n,t,X) = \int_{-\infty}^{t} (t-x)^{n} \, dF_X(x), \qquad \operatorname{UPM}(n,t,X) = \int_{t}^{\infty} (x-t)^{n} \, dF_X(x). $$

The empirical estimators replace $F_X$ with the empirical CDF $\hat F_n$ (or, equivalently, use indicator functions):

$$ \widehat{\operatorname{LPM}}n(t;X) = \frac{1}{n} \sum{i=1}^n (t-x_i)^n \, \mathbf{1}{{x_i \le t}}, \qquad \widehat{\operatorname{UPM}}_n(t;X) = \frac{1}{n} \sum{i=1}^n (x_i-t)^n \, \mathbf{1}_{{x_i > t}}. $$

These correspond to integrals over the measurable subsets ${X \le t}$ and ${X > t}$ in a $\sigma$‑algebra; the empirical sums are discrete analogues of Lebesgue integrals.

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1. Foundations — Partial Moments & Variance Decomposition

1.1 Why partial moments

• Classical variance treats upside and downside symmetrically. Partial moments separate them, allowing asymmetric risk/reward analysis around a chosen target $t$ (often the mean or a benchmark).
• At $t=\mu_X$: $$ \operatorname{Var}(X) = \operatorname{UPM}(2,\mu_X,X) + \operatorname{LPM}(2,\mu_X,X)\quad\text{(exact empirical identity)}. $$ This is not the same as splitting conditional variances around a threshold; partial moments use a global reference, preserving the between‑group contribution.

1.2 Core functions and headers

• LPM(degree, target, variable)
• UPM(degree, target, variable)
• LPM.ratio(degree = 0, target, variable) (empirical CDF when degree=0)
• UPM.ratio(degree = 0, target, variable)
• LPM.VaR(p, degree, variable) (quantiles via partial‑moment CDFs)

1.3 Code: variance decomposition & CDF

# Normal sample
y <- rnorm(3000)
mu <- mean(y)
L2 <- LPM(2, mu, y); U2 <- UPM(2, mu, y)
cat(sprintf("LPM2 + UPM2 = %.6f vs var(y)=%.6f\n", (L2+U2)*(length(y) / (length(y) - 1)), var(y)))

# Empirical CDF via LPM.ratio(0, t, x)
for (t in c(-1,0,1)) {
  cdf_lpm <- LPM.ratio(0, t, y)
  cat(sprintf("CDF at t=%+.1f : LPM.ratio=%.4f | empirical=%.4f\n", t, cdf_lpm, mean(y<=t)))
}

# Asymmetry on a skewed distribution
z <- rexp(3000)-1; mu_z <- mean(z)
cat(sprintf("Skewed z: LPM2=%.4f, UPM2=%.4f (expect imbalance)\n", LPM(2,mu_z,z), UPM(2,mu_z,z)))

Interpretation. The equality LPM2 + UPM2 == var(x) (Bessel adjustment used) holds because deviations are measured against the global mean. LPM.ratio(0, t, x) constructs an empirical CDF directly from partial‑moment counts.

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2. Descriptive & Distributional Tools

2.1 Higher moments from partial moments

Define asymmetric analogues of skewness/kurtosis using $\operatorname{UPM}_3$, $\operatorname{LPM}_3$ (and degree 4), yielding robust tail diagnostics without parametric assumptions.

Header. NNS.moments(x)

M <- NNS.moments(y)
M

2.2 Mode estimation (no bin‑or‑bandwidth angst)

Header. NNS.mode(x)

set.seed(23)
multimodal <- c(rnorm(1500,-2,.5), rnorm(1500,2,.5))
NNS.mode(multimodal,multi = TRUE)

2.3 CDF tables via LPM ratios

qgrid <- quantile(z, probs = seq(0.05,0.95,by=0.1))
CDF_tbl <- data.table(threshold = as.numeric(qgrid), CDF = sapply(qgrid, function(q) LPM.ratio(0,q,z)))
CDF_tbl

---

3. Dependence & Nonlinear Association

3.1 Why move beyond Pearson $r$

Pearson captures linear monotone relationships. Many structures (U‑shapes, saturation, asymmetric tails) produce near‑zero $r$ despite strong dependence. Partial‑moment dependence metrics respond to such structure.

Headers.

• Co.LPM(degree, target, x, y) / Co.UPM(...) (co‑partial moments)
• PM.matrix(l_degree, u_degree, target=NULL, variable, pop_adj=TRUE)
• NNS.dep(x, y) (scalar dependence coefficient)
• NNS.copula(X, target=NULL, continuous=TRUE, plot=FALSE, independence.overlay=FALSE)

3.2 Code: nonlinear dependence

set.seed(1)
x <- runif(2000,-1,1)
y <- x^2 + rnorm(2000, sd=.05)
cat(sprintf("Pearson r = %.4f\n", cor(x,y)))
cat(sprintf("NNS.dep  = %.4f\n", NNS.dep(x,y)$Dependence))

X <- data.frame(a=x, b=y, c=x*y + rnorm(2000, sd=.05))
pm <- PM.matrix(1, 1, target = "means", variable=X, pop_adj=TRUE)
pm

cop <- NNS.copula(X, continuous=TRUE, plot=FALSE)
cop

3.3 Code: copula

# Data
set.seed(123); x = rnorm(100); y = rnorm(100); z = expand.grid(x, y)

# Plot
rgl::plot3d(z[,1], z[,2], Co.LPM(0, z[,1], z[,2], z[,1], z[,2]), col = "red")

# Uniform values
u_x = LPM.ratio(0, x, x); u_y = LPM.ratio(0, y, y); z = expand.grid(u_x, u_y)

# Plot
rgl::plot3d(z[,1], z[,2], Co.LPM(0, z[,1], z[,2], z[,1], z[,2]), col = "blue")

Interpretation. NNS.dep remains high for curved relationships; PM.matrix collects co‑partial moments across variables; NNS.copula summarizes higher‑dimensional dependence using partial‑moment ratios. Copulas are returned and evaluated via Co.LPM functions.

---

4. Hypothesis Testing & ANOVA (LPM‑CDF)

4.1 Concept

Instead of distributional assumptions, compare groups via LPM‑based CDFs. Output is a degree of certainty (not a p‑value) for equality of populations or means.

Header.

• NNS.ANOVA(control, treatment, means.only=FALSE, medians=FALSE, confidence.interval=.95, tails=c("Both","left","right"), pairwise=FALSE, plot=TRUE, robust=FALSE)

4.2 Code: two‑sample & multi‑group

ctrl <- rnorm(200, 0, 1)
trt  <- rnorm(180, 0.35, 1.2)
NNS.ANOVA(control=ctrl, treatment=trt, means.only=FALSE, plot=FALSE)

A <- list(g1=rnorm(150,0.0,1.1), g2=rnorm(150,0.2,1.0), g3=rnorm(150,-0.1,0.9))
NNS.ANOVA(control=A, means.only=TRUE, plot=FALSE)

Math sketch. For each quantile/threshold $t$, compare CDFs built from LPM.ratio(0, t, •) (possibly with one‑sided tails). Aggregate across $t$ to a certainty score.

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5. Regression, Boosting, Stacking & Causality

5.1 Philosophy

NNS.reg learns partitioned relationships using partial‑moment weights — linear where appropriate, nonlinear where needed — avoiding fragile global parametric forms.

Headers.

• NNS.reg(x, y, order=NULL, smooth=TRUE, ncores=1, ...) → $Fitted.xy, $Point.est, …
• NNS.boost(IVs.train, DV.train, IVs.test, epochs, learner.trials, status, balance, type, folds) - NNS.stack(IVs.train, DV.train, IVs.test, type, balance, ncores, folds) - NNS.caus(x, y) (directional causality score via conditional dependence)

5.2 Code: classification via regression + ensembles

# Example 1: Nonlinear regression
set.seed(123)
x_train <- runif(200, -2, 2)
y_train <- sin(pi * x_train) + rnorm(200, sd = 0.2)

x_test <- seq(-2, 2, length.out = 100)

NNS.reg(x = data.frame(x = x_train), y = y_train, order = NULL)

# Simple train/test for boosting & stacking
test.set = 141:150

boost <- NNS.boost(IVs.train = iris[-test.set, 1:4], 
              DV.train = iris[-test.set, 5],
              IVs.test = iris[test.set, 1:4],
              epochs = 10, learner.trials = 10, 
              status = FALSE, balance = TRUE,
              type = "CLASS", folds = 5)


mean(boost$results == as.numeric(iris[test.set,5]))
[1] 1


boost$feature.weights; boost$feature.frequency

stacked <- NNS.stack(IVs.train = iris[-test.set, 1:4], 
                     DV.train = iris[-test.set, 5],
                     IVs.test = iris[test.set, 1:4],
                     type = "CLASS", balance = TRUE,
                     ncores = 1, folds = 1)
mean(stacked$stack == as.numeric(iris[test.set,5]))
[1] 1

5.3 Code: directional causality

NNS.caus(mtcars$hp,  mtcars$mpg)  # hp -> mpg
NNS.caus(mtcars$mpg, mtcars$hp)   # hp -> mpg

Interpretation. Examine asymmetry in scores to infer direction. The method conditions partial‑moment dependence on candidate drivers.

---

6. Time Series & Forecasting

Headers.

• NNS.ARMA
• NNS.ARMA.optim
• NNS.seas
• NNS.VAR

# Univariate nonlinear ARMA
z <- as.numeric(scale(sin(1:480/8) + rnorm(480, sd=.35)))

# Seasonality detection (prints a summary)
NNS.seas(z, plot = FALSE)

# Validate seasonal periods
NNS.ARMA.optim(z, h=48, seasonal.factor = NNS.seas(z, plot = FALSE)$periods, plot = TRUE, ncores = 1)

Notes. NNS seasonality uses coefficient of variation instead of ACF/PACFs, and NNS ARMA blends multiple seasonal periods into the linear or nonlinear regression forecasts.

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7. Simulation, Bootstrap & Risk‑Neutral Rescaling

7.1 Maximum entropy bootstrap (shape‑preserving)

Header.

• NNS.meboot(x, reps=999, rho=NULL, type="spearman", drift=TRUE, ...)

x_ts <- cumsum(rnorm(350, sd=.7))
mb <- NNS.meboot(x_ts, reps=5, rho = 1)
dim(mb["replicates", ]$replicates)

7.2 Monte Carlo over the full correlation space

Header.

• NNS.MC(x, reps=30, lower_rho=-1, upper_rho=1, by=.01, exp=1, type="spearman", ...)

mc <- NNS.MC(x_ts, reps=5, lower_rho=-1, upper_rho=1, by=.5, exp=1)
length(mc$ensemble); head(names(mc$replicates),5)

7.3 Risk‑neutral rescale (pricing context)

Header.

• NNS.rescale(x, a, b, method=c("minmax","riskneutral"), T=NULL, type=c("Terminal","Discounted"))

px <- 100 + cumsum(rnorm(260, sd = 1))
rn <- NNS.rescale(px, a=100, b=0.03, method="riskneutral", T=1, type="Terminal")
c( target = 100*exp(0.03*1), mean_rn = mean(rn) )

Interpretation. riskneutral shifts the mean to match $S_0 e^{rT}$ (Terminal) or $S_0$ (Discounted), preserving distributional shape.

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8. Portfolio & Stochastic Dominance

Stochastic dominance orders uncertain prospects for broad classes of risk‑averse utilities; partial moments supply practical, nonparametric estimators.

Headers. - NNS.FSD.uni(x, y) - NNS.SSD.uni(x, y) - NNS.TSD.uni(x, y) - NNS.SD.cluster(R) - NNS.SD.efficient.set(R)

RA <- rnorm(240, 0.005, 0.03)
RB <- rnorm(240, 0.003, 0.02)
RC <- rnorm(240, 0.006, 0.04)

NNS.FSD.uni(RA, RB)
NNS.SSD.uni(RA, RB)
NNS.TSD.uni(RA, RB)

Rmat <- cbind(A=RA, B=RB, C=RC)
try(NNS.SD.cluster(Rmat, degree = 1))
try(NNS.SD.efficient.set(Rmat, degree = 1))

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Appendix A — Measure‑theoretic sketch (why partial moments are rigorous)

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space, $X: \Omega\to\mathbb{R}$ measurable. For any fixed $t\in\mathbb{R}$, the sets ${X\le t}$ and ${X>t}$ are in $\mathcal{F}$ because they are preimages of Borel sets. The population partial moments are

$$ \operatorname{LPM}(k,t,X) = \int_{-\infty}^{t} (t-x)^k\, dF_X(x), \qquad \operatorname{UPM}(k,t,X) = \int_{t}^{\infty} (x-t)^k\, dF_X(x). $$

The empirical versions correspond to replacing $F_X$ with the empirical measure $\mathbb{P}_n$ (or CDF $\hat F_n$):

$$ \widehat{\operatorname{LPM}}k(t;X) = \int{(-\infty,t]} (t-x)^k\, d\mathbb{P}n(x), \qquad \widehat{\operatorname{UPM}}_k(t;X) = \int{(t,\infty)} (x-t)^k\, d\mathbb{P}_n(x). $$

Centering at $t=\mu_X$ yields the variance decomposition identity in Section 1.

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Appendix B — Quick Reference (Grouped by Topic)

1. Partial Moments & Ratios

• LPM(degree, target, variable) — lower partial moment of order degree at target.
• UPM(degree, target, variable) — upper partial moment of order degree at target.
• LPM.ratio(degree, target, variable); UPM.ratio(...) — normalized shares; degree=0 gives CDF.
• LPM.VaR(p, degree, variable) — partial-moment quantile at probability p.
• Co.LPM(degree, target, x, y) — co-lower partial moment between two variables.
• Co.UPM(degree, target, x, y) — co-upper partial moment between two variables.
• D.LPM(degree, target, variable) — divergent lower partial moment (away from target).
• D.UPM(degree, target, variable) — divergent upper partial moment (away from target).
• NNS.CDF(x, target = NULL, points = NULL, plot = TRUE/FALSE) — CDF from partial moments.
• NNS.moments(x) — mean/var/skew/kurtosis via partial moments.

2. Descriptive Statistics & Distributions

• NNS.mode(x, multi=FALSE) — nonparametric mode(s).
• PM.matrix(l_degree, u_degree, target, variable, pop_adj) — co-/divergent partial-moment matrices.
• NNS.gravity(x, w = NULL) — partial-moment weighted location (gravity center).
• NNS.norm(x, method = "moment") — normalization retaining target moments.

See NNS Vignette: Getting Started with NNS: Partial Moments

3. Dependence & Association

• NNS.dep(x, y) — nonlinear dependence coefficient.
• NNS.copula(X, target, continuous, plot, independence.overlay) — dependence from co-partial moments.

See NNS Vignette: Getting Started with NNS: Correlation and Dependence

4. Hypothesis Testing

• NNS.ANOVA(control, treatment, ...) — certainty of equality (distributions or means).

See NNS Vignette: Getting Started with NNS: Comparing Distributions

5. Regression, Classification & Causality

• NNS.part(x, y, ...) — partition analysis for variable segmentation.
• NNS.reg(x, y, ...) — partition-based regression/classification ($Fitted.xy, $Point.est).
• NNS.boost(IVs, DV, ...), NNS.stack(IVs, DV, ...) — ensembles using NNS.reg base learners.
• NNS.caus(x, y) — directional causality score.

See NNS Vignette: Getting Started with NNS: Clustering and Regression

\medskip

See NNS Vignette: Getting Started with NNS: Classification

6. Differentiation & Slope Measures

• dy.dx(x, y) — numerical derivative of y with respect to x via partial moments.
• dy.d_(x, Y, var) — partial derivative of multivariate Y w.r.t. var.
• NNS.diff(x, y) — derivative via secant projections.

7. Time Series & Forecasting

• NNS.ARMA(...), NNS.ARMA.optim(...) — nonlinear ARMA modeling.
• NNS.seas(...) — detect seasonality.
• NNS.VAR(...) — nonlinear VAR modeling.
• NNS.nowcast(x, h, ...) — near-term nonlinear forecast.

See NNS Vignette: Getting Started with NNS: Forecasting

8. Simulation, Bootstrap & Rescaling

• NNS.meboot(...) — maximum entropy bootstrap.
• NNS.MC(...) — Monte Carlo over correlation space.
• NNS.rescale(...) — risk-neutral or min–max rescaling.

See NNS Vignette: Getting Started with NNS: Sampling and Simulation

9. Portfolio Analysis & Stochastic Dominance

• NNS.FSD.uni(x, y), NNS.SSD.uni(x, y), NNS.TSD.uni(x, y) — univariate stochastic dominance tests.
• NNS.SD.cluster(R), NNS.SD.efficient.set(R) — dominance-based portfolio sets.

For complete references, please see the Vignettes linked above and their specific referenced materials.

NNS documentation built on Nov. 5, 2025, 7:27 p.m.