NNS
Nonlinear Nonparametric Statistics

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Getting Started with NNS: Overview"

Getting Started with NNS: Overview"
In NNS: Nonlinear Nonparametric Statistics 

# Prereqs (uncomment if needed):
# install.packages("NNS")

library(NNS)

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. Normalization & Rescaling 5. Hypothesis testing, ANOVA & Stochastic Superiority 6. Regression, boosting, stacking & causality 7. Time series & forecasting 8. Simulation (max‑entropy) & Monte Carlo 9. 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.

1. Foundations — Partial Moments & Variance Decomposition

1.1 Why partial moments

1.2 Core functions and headers

1.3 Code: variance decomposition & CDF

set.seed(42)

# 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.

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.

M <- NNS.moments(y)
M

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

Header. 

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

Headers.

qgrid <- LPM.VaR(seq(0.05,0.95,.1),0,z) # equivalent to quantile(z,probs = seq(0.05,0.95,by=0.1))
CDF_tbl <- data.frame(threshold = as.numeric(qgrid), CDF = LPM.ratio(0,qgrid,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.

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. Normalization and Rescaling

NNS provides two main tools for scaling data while preserving rank structure and distributional shape. Both operate via deterministic affine transformations.

4.1 Normalization

NNS.norm() rescales variables to a common magnitude while preserving distributional structure. The method can be linear (all variables forced to have the same mean) or nonlinear (using dependence weights to produce a more nuanced scaling). In the nonlinear case, the degree of association between variables influences the final normalized values.

Header. 

A <- rnorm(100, mean = 0, sd = 1)
B <- rnorm(100, mean = 0, sd = 5)
C <- rnorm(100, mean = 10, sd = 1)
D <- rnorm(100, mean = 10, sd = 10)

X <- data.frame(A, B, C, D)

# Linear scaling
lin_norm <- NNS.norm(X, linear = TRUE, chart.type=NULL, location=NULL)

Interpretation. NNS.norm() brings variables to a common scale without distorting their distributional shape. Linear mode equalizes means; nonlinear mode additionally weights each variable by its dependence with others, so more correlated variables exert greater influence on the final scaling.

4.2 Risk‑neutral rescale (pricing context)

NNS.rescale() performs one‑dimensional affine transformations.

Header. 

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.

5. Hypothesis Testing, ANOVA & Stochastic Superiority

5.1 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.

5.2 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.

5.3 Stochastic Superiority

Stochastic superiority asks a different question than equality of means or equality of distributions. Rather than testing whether two samples came from the same population, or whether they share the same mean or median, stochastic superiority measures the probability that a random draw from one distribution exceeds a random draw from another.

For two random variables (X) and (Y), the stochastic superiority probability is:

[ P(X > Y) ]

and with ties accounted for, the tie-adjusted stochastic superiority measure is:

[ P^* = P(X > Y) + \frac{1}{2} P(X = Y) ]

A value of (P^* = 0.5) indicates no directional advantage, values above (0.5) favor (X), and values below (0.5) favor (Y).

This differs from stochastic dominance. Stochastic superiority is a pairwise exceedance probability, while stochastic dominance requires one distribution to be preferred to another over the entire shared support.

Below is an example comparing two distributions with unequal means.

set.seed(123)
x = rnorm(1000, mean = 0, sd = 1)
y = rnorm(1000, mean = 1, sd = 1)

NNS.SS(x, y)

Since (y) was generated with a higher mean, the stochastic superiority probability for (x) relative to (y) should be less than (0.5), indicating that a draw from (x) is less likely to exceed a draw from (y).

We can also obtain confidence intervals for the tie-adjusted superiority probability using maximum entropy bootstrap replicates.

NNS.SS(x, y, confidence.interval = TRUE, reps = 999, ci = 0.95)[1:5]

$p_gt
[1] 0.233915

$p_tie
[1] 0

$p_star
[1] 0.233915

$lower
[1] 0.2105631

$upper
[1] 0.2537789

This provides an interpretable effect size for directional comparison between two distributions without requiring identical distributions or equal variances.

For discrete variables, ties may occur with positive probability, and the reported p_tie and p_star values reflect that adjustment explicitly.

set.seed(123)
x = sample(1:5, 100, replace = TRUE)
y = sample(1:5, 100, replace = TRUE)

NNS.SS(x, y)

6. Regression, Boosting, Stacking & Causality

6.1 Philosophy

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

Headers.

6.2 Code: classification via regression + ensembles

# Example 1: Nonlinear regression
set.seed(123)
x_train <- runif(1000, -2, 2)
y_train <- sin(pi * x_train) + rnorm(1000, sd = 0.2)

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

NNS.reg(x = x_train, y = y_train, order = NULL, point.est = x_test)

# 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

6.3 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.

7. Time Series & Forecasting

Headers.

# Univariate nonlinear ARMA
set.seed(42)
z <- as.numeric(scale(sin(1:480/8) + rnorm(480, sd=.35)))

# Seasonality detection (prints a summary)
seasonal_period <- NNS.seas(z, plot = FALSE)
head(seasonal_period$all.periods)

# Validate seasonal periods
NNS.ARMA.optim(z, h = 48, seasonal.factor = seasonal_period$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.

8. Simulation & Bootstrap & Risk‑Neutral Rescaling

8.1 Maximum entropy bootstrap (shape‑preserving)

Header. 

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

8.2 Monte Carlo over the full correlation space

Header. 

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

head(mc$replicates$`rho = 0`)

9. Portfolio & Stochastic Dominance

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

Headers. 

set.seed(42)
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))

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.

Appendix B — Quick Reference (Grouped by Topic)

Overall Theory

1. Partial Moments & Ratios

2. Descriptive Statistics & Distributions

See NNS Vignette: Getting Started with NNS: Partial Moments

3. Dependence & Association

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

4. Normalization & Rescaling

See NNS Vignette: Getting Started with NNS: Normalization and Rescaling

5. Hypothesis Testing

See NNS Vignette: Getting Started with NNS: Comparing Distributions

6. Regression, Classification & Causality

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

\medskip

See NNS Vignette: Getting Started with NNS: Classification

7. Differentiation & Slope Measures

8. Time Series & Forecasting

See NNS Vignette: Getting Started with NNS: Forecasting

9. Simulation & Bootstrap

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

10. Portfolio Analysis & Stochastic Dominance

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



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NNS documentation built on July 1, 2026, 1:08 a.m.

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