Getting Started with NNS: Overview
Fred Viole
# Prereqs (uncomment if needed):
# install.packages("NNS")
# install.packages(c("data.table","xts","zoo","Rfast"))
suppressPackageStartupMessages({
library(NNS)
library(data.table)
})
set.seed(42)
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.
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)))
## LPM2 + UPM2 = 1.011889 vs var(y)=1.011889
# 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)))
}
## CDF at t=-1.0 : LPM.ratio=0.1633 | empirical=0.1633
## CDF at t=+0.0 : LPM.ratio=0.5043 | empirical=0.5043
## CDF at t=+1.0 : LPM.ratio=0.8480 | empirical=0.8480
# 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)))
## Skewed z: LPM2=0.2780, UPM2=0.7682 (expect imbalance)
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.
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)))
## Pearson r = 0.0006
cat(sprintf("NNS.dep = %.4f\n", NNS.dep(x,y)$Dependence))
## NNS.dep = 0.7097
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
## $cupm
## a b c
## a 0.17384174 0.05668152 0.10450858
## b 0.05668152 0.05566363 0.04414923
## c 0.10450858 0.04414923 0.07529373
##
## $dupm
## a b c
## a 0.0000000000 0.05675501 0.0005598221
## b 0.0143108307 0.00000000 0.0036839026
## c 0.0004239566 0.04430691 0.0000000000
##
## $dlpm
## a b c
## a 0.0000000000 0.014310831 0.0004239566
## b 0.0567550147 0.000000000 0.0443069142
## c 0.0005598221 0.003683903 0.0000000000
##
## $clpm
## a b c
## a 0.16803827 0.014485430 0.102709867
## b 0.01448543 0.037120650 0.003051617
## c 0.10270987 0.003051617 0.074865823
##
## $cov.matrix
## a b c
## a 0.3418800141 0.0001011068 0.206234664
## b 0.0001011068 0.0927842833 -0.000789973
## c 0.2062346637 -0.0007899730 0.150159552
cop <- NNS.copula(X, continuous=TRUE, plot=FALSE)
cop
## [1] 0.5692785
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)
## $Control
## [1] -0.02110331
##
## $Treatment
## [1] 0.4020782
##
## $Grand_Statistic
## [1] 0.1904875
##
## $Control_CDF
## [1] 0.6311761
##
## $Treatment_CDF
## [1] 0.3869042
##
## $Certainty
## [1] 0.3904966
##
## $Effect_Size_LB
## 2.5%
## 0.1379073
##
## $Effect_Size_UB
## 97.5%
## 0.7182389
##
## $Confidence_Level
## [1] 0.95
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)
## Certainty
## 0.4870367
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. 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)
## $R2
## [1] 0.9311519
##
## $SE
## [1] 0.2026925
##
## $Prediction.Accuracy
## NULL
##
## $equation
## NULL
##
## $x.star
## NULL
##
## $derivative
## Coefficient X.Lower.Range X.Upper.Range
## 1: 1.2434405 -1.99750091 -1.73845327
## 2: 1.0742631 -1.73845327 -1.44607946
## 3: -1.7569574 -1.44607946 -1.15089951
## 4: -2.8177481 -1.15089951 -0.92639490
## 5: -2.7344640 -0.92639490 -0.71353077
## 6: -0.9100977 -0.71353077 -0.48670931
## 7: 0.9413676 -0.48670931 -0.26368123
## 8: 3.1046330 -0.26368123 -0.07056066
## 9: 2.3015348 -0.07056066 0.09052852
## 10: 3.1725079 0.09052852 0.25161770
## 11: 1.3891814 0.25161770 0.45309215
## 12: -1.2585111 0.45309215 0.66464355
## 13: -2.5182434 0.66464355 1.01253792
## 14: -2.9012891 1.01253792 1.23520889
## 15: -0.7677850 1.23520889 1.53007549
## 16: 1.8706437 1.53007549 1.63683301
## 17: 2.2033668 1.63683301 1.84911541
## 18: 2.5081594 1.84911541 1.97707911
##
## $Point.est
## NULL
##
## $pred.int
## NULL
##
## $regression.points
## x y
## 1: -1.99750091 0.30256920
## 2: -1.73845327 0.62467954
## 3: -1.44607946 0.93876593
## 4: -1.15089951 0.42014733
## 5: -0.92639490 -0.21245010
## 6: -0.71353077 -0.79451941
## 7: -0.48670931 -1.00094909
## 8: -0.26368123 -0.79099767
## 9: -0.07056066 -0.19142920
## 10: 0.09052852 0.17932316
## 11: 0.25161770 0.69037987
## 12: 0.45309215 0.97026443
## 13: 0.66464355 0.70402465
## 14: 1.01253792 -0.17205805
## 15: 1.23520889 -0.81809091
## 16: 1.53007549 -1.04448505
## 17: 1.63683301 -0.84477976
## 18: 1.84911541 -0.37704378
## 19: 1.97707911 -0.05609043
##
## $Fitted.xy
## x.x y y.hat NNS.ID gradient residuals
## 1: -0.8496899 -0.5969396 -0.4221971 q11222 -2.7344640 0.174742446
## 2: 1.1532205 -0.4116052 -0.5802190 q21222 -2.9012891 -0.168613758
## 3: -0.3640923 -0.9595645 -0.8855214 q12122 0.9413676 0.074043066
## 4: 1.5320696 -1.0644376 -1.0407547 q22122 1.8706437 0.023682815
## 5: 1.7618691 -0.8705785 -0.5692793 q22212 2.2033668 0.301299171
## ---
## 196: -0.1338692 -0.3949338 -0.3879789 q12221 3.1046330 0.006954855
## 197: -0.3726696 -0.5476828 -0.8935958 q12122 0.9413676 -0.345913048
## 198: 0.6369213 0.6387223 0.7389134 q21211 -1.2585111 0.100191135
## 199: -1.3906135 0.9457286 0.8413147 q11122 -1.7569574 -0.104413995
## 200: 0.2914682 1.0429566 0.7457395 q21112 1.3891814 -0.297217109
## standard.errors
## 1: 0.2830616
## 2: 0.1682392
## 3: 0.2096766
## 4: 0.1380874
## 5: 0.2328510
## ---
## 196: 0.1281159
## 197: 0.2096766
## 198: 0.2474949
## 199: 0.2329677
## 200: 0.2435048
# Simple train/test for boosting & stacking
test.set = 141:150
boost <- NNS.boost(IVs.train = iris[-test.set, 1:4],
DV.train = iris[-test.set, 5],
IVs.test = iris[test.set, 1:4],
epochs = 10, learner.trials = 10,
status = FALSE, balance = TRUE,
type = "CLASS", folds = 1)
mean(boost$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$mpg) # hp -> mpg
## Causation.x.given.y Causation.y.given.x C(x--->y)
## 0.2607148 0.3863580 0.3933374
NNS.caus(mtcars$mpg, mtcars$hp) # mpg -> hp
## Causation.x.given.y Causation.y.given.x C(y--->x)
## 0.3863580 0.2607148 0.3933374
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)
## $all.periods
## Period Coefficient.of.Variation Variable.Coefficient.of.Variation
## 1: 99 5.122054e-01 7.866136e+16
## 2: 147 5.256021e-01 7.866136e+16
## 3: 100 5.598477e-01 7.866136e+16
## 4: 146 5.618687e-01 7.866136e+16
## 5: 199 5.766158e-01 7.866136e+16
## ---
## 235: 235 2.273832e+02 7.866136e+16
## 236: 30 4.402306e+02 7.866136e+16
## 237: 11 4.879713e+02 7.866136e+16
## 238: 46 5.854636e+02 7.866136e+16
## 239: 1 2.621230e+16 7.866136e+16
##
## $best.period
## Period
## 99
##
## $periods
## [1] 99 147 100 146 199 98 97 200 49 198 48 96 145 196 194 193 195 197
## [19] 144 50 192 149 176 53 191 148 95 156 190 106 150 202 239 52 143 201
## [37] 105 142 189 209 158 238 157 211 160 155 107 212 177 104 210 188 154 208
## [55] 126 163 152 103 207 127 172 117 159 128 206 109 175 224 179 47 170 221
## [73] 227 171 86 85 130 54 222 129 236 178 153 141 237 108 203 181 34 101
## [91] 180 226 161 94 87 213 162 43 115 113 166 102 80 59 204 66 40 135
## [109] 61 217 81 90 131 56 44 223 83 60 132 84 74 93 88 45 229 167
## [127] 173 133 122 116 151 70 174 187 51 112 219 228 121 136 232 120 37 169
## [145] 25 182 22 14 186 82 118 89 184 39 72 67 234 64 18 140 91 216
## [163] 57 124 233 168 32 16 134 73 75 92 139 36 41 220 230 58 123 69
## [181] 38 76 28 164 26 138 225 125 165 185 68 55 65 35 2 183 6 42
## [199] 78 215 137 21 8 20 15 62 7 231 79 13 205 24 17 119 12 31
## [217] 214 63 33 27 111 3 9 19 10 218 114 4 23 5 110 71 77 29
## [235] 235 30 11 46 1
# Validate seasonal periods
NNS.ARMA.optim(z, h=48, seasonal.factor = NNS.seas(z, plot = FALSE)$periods, plot = TRUE, ncores = 1)
## [1] "CURRNET METHOD: lin"
## [1] "COPY LATEST PARAMETERS DIRECTLY FOR NNS.ARMA() IF ERROR:"
## [1] "NNS.ARMA(... method = 'lin' , seasonal.factor = c( 52 ) ...)"
## [1] "CURRENT lin OBJECTIVE FUNCTION = 0.449145584053097"
## [1] "NNS.ARMA(... method = 'lin' , seasonal.factor = c( 52, 49 ) ...)"
## [1] "CURRENT lin OBJECTIVE FUNCTION = 0.364719193840196"
## [1] "NNS.ARMA(... method = 'lin' , seasonal.factor = c( 52, 49, 50 ) ...)"
## [1] "CURRENT lin OBJECTIVE FUNCTION = 0.303033712560494"
## [1] "BEST method = 'lin', seasonal.factor = c( 52, 49, 50 )"
## [1] "BEST lin OBJECTIVE FUNCTION = 0.303033712560494"
## [1] "CURRNET METHOD: nonlin"
## [1] "COPY LATEST PARAMETERS DIRECTLY FOR NNS.ARMA() IF ERROR:"
## [1] "NNS.ARMA(... method = 'nonlin' , seasonal.factor = c( 52, 49, 50 ) ...)"
## [1] "CURRENT nonlin OBJECTIVE FUNCTION = 1.58051085217018"
## [1] "BEST method = 'nonlin' PATH MEMBER = c( 52, 49, 50 )"
## [1] "BEST nonlin OBJECTIVE FUNCTION = 1.58051085217018"
## [1] "CURRNET METHOD: both"
## [1] "COPY LATEST PARAMETERS DIRECTLY FOR NNS.ARMA() IF ERROR:"
## [1] "NNS.ARMA(... method = 'both' , seasonal.factor = c( 52, 49, 50 ) ...)"
## [1] "CURRENT both OBJECTIVE FUNCTION = 0.447117273188387"
## [1] "BEST method = 'both' PATH MEMBER = c( 52, 49, 50 )"
## [1] "BEST both OBJECTIVE FUNCTION = 0.447117273188387"
## $periods
## [1] 52 49 50
##
## $weights
## NULL
##
## $obj.fn
## [1] 0.3030337
##
## $method
## [1] "lin"
##
## $shrink
## [1] FALSE
##
## $nns.regress
## [1] FALSE
##
## $bias.shift
## [1] 0.1079018
##
## $errors
## [1] 0.06841911 -0.23650978 0.23306891 -0.31474170 -0.16347937 0.56078801
## [7] -0.19340157 -0.54788961 -0.34463351 -0.04971714 0.81131522 -1.04034772
## [13] -0.01124973 0.18532001 0.42228850 0.77875534 0.21204992 0.75989291
## [19] 0.03648050 -0.12410190 0.78169808 -0.37190642 0.04673305 0.22143951
## [25] 0.21784535 -0.36207177 0.06303110 0.27494889 0.61355674 -0.36588877
## [31] -0.53670212 -0.59710016 -0.33562214 0.52319489 -0.28558752 -0.06318330
## [37] 0.46174079 0.85423779 -0.17957169 0.88745345 -0.22575406 -0.65533631
## [43] -0.50769155 -0.18710610 -0.19702948 -0.61676209 -0.64456532 -0.60764796
## [49] 0.39155766 -0.99138140 -0.58599672 -0.41332955 -0.35110299 -0.31785231
## [55] -0.33368188 -0.79321483 -0.67548303 -0.29994123 -1.40951519 -0.23496159
## [61] -0.11326961 0.93761236 -1.12638974 0.56134385 -0.82647659 -0.15698867
## [67] -0.66092883 -0.23941287 -0.11793511 -0.13131032 0.23980082 0.11145491
## [73] -0.29324462 0.20996125 1.18368703 0.39817389 0.11233666 0.18104853
## [79] -0.34704039 1.00778283 -0.12855809 -0.44890273 -0.16127326 0.23878907
## [85] 0.08958084 -0.42127816 0.83025782 0.21535622 0.18499525 0.55580864
## [91] -0.63033063 -1.18279040 0.01593275 -0.38943895 -0.73303803 0.24461725
##
## $results
## [1] -0.48668691 -1.12897072 -1.19746701 -1.08366883 -1.17430589 -1.09503747
## [7] -1.27716033 -1.53097703 -1.12872199 -1.25223633 -1.03806114 -1.03996796
## [13] -0.86012134 -0.85334076 -1.29205205 -0.41549038 -0.43645317 -0.70613650
## [19] -0.12979825 -0.20838439 -0.43674783 0.04939343 0.21128735 0.41848130
## [25] 0.58665881 0.65204679 1.11856457 0.81855013 1.33909393 1.17188036
## [31] 1.53195554 1.09195702 1.71916462 1.49064664 1.62340003 1.71600851
## [37] 1.54806110 1.34095102 1.16533567 1.08567248 0.73267472 0.82949748
## [43] 0.65297434 0.09082443 0.55476313 0.53988329 -0.09198186 -0.18380226
##
## $lower.pred.int
## [1] -1.51339152 -2.15567532 -2.22417162 -2.11037344 -2.20101050 -2.12174208
## [7] -2.30386494 -2.55768164 -2.15542660 -2.27894094 -2.06476575 -2.06667256
## [13] -1.88682595 -1.88004537 -2.31875666 -1.44219499 -1.46315778 -1.73284111
## [19] -1.15650286 -1.23508900 -1.46345244 -0.97731118 -0.81541725 -0.60822331
## [25] -0.44004579 -0.37465782 0.09185996 -0.20815448 0.31238932 0.14517575
## [31] 0.50525093 0.06525242 0.69246002 0.46394203 0.59669542 0.68930391
## [37] 0.52135649 0.31424642 0.13863106 0.05896787 -0.29402989 -0.19720713
## [43] -0.37373026 -0.93588018 -0.47194148 -0.48682132 -1.11868647 -1.21050687
##
## $upper.pred.int
## [1] 0.54001770 -0.10226611 -0.17076240 -0.05696423 -0.14760128 -0.06833286
## [7] -0.25045572 -0.50427242 -0.10201738 -0.22553172 -0.01135653 -0.01326335
## [13] 0.16658327 0.17336385 -0.26534744 0.61121423 0.59025144 0.32056811
## [19] 0.89690636 0.81832022 0.58995678 1.07609804 1.23799196 1.44518591
## [25] 1.61336342 1.67875140 2.14526918 1.84525474 2.36579853 2.19858497
## [31] 2.55866015 2.11866163 2.74586923 2.51735125 2.65010464 2.74271312
## [37] 2.57476571 2.36765563 2.19204028 2.11237709 1.75937933 1.85620209
## [43] 1.67967895 1.11752903 1.58146774 1.56658789 0.93472275 0.84290235
Notes. NNS seasonality uses coefficient of variation
instead of ACF/PACFs, and NNS ARMA blends multiple seasonal periods into
the linear or nonlinear regression forecasts.
7. Simulation, Bootstrap & Risk‑Neutral Rescaling
7.1 Maximum entropy bootstrap (shape‑preserving)
Header.
NNS.meboot(x, reps=999, rho=NULL, type="spearman", drift=TRUE, ...)
x_ts <- cumsum(rnorm(350, sd=.7))
mb <- NNS.meboot(x_ts, reps=5, rho = 1)
dim(mb["replicates", ]$replicates)
## [1] 350 5
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)
## [1] 350
## [1] "rho = 1" "rho = 0.5" "rho = 0" "rho = -0.5" "rho = -1"
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) )
## target mean_rn
## 103.0455 103.0455
Interpretation. riskneutral shifts the
mean to match \(S_0 e^{rT}\) (Terminal)
or \(S_0\) (Discounted), preserving
distributional shape.
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)
## [1] 0
## [1] 0
## [1] 0
Rmat <- cbind(A=RA, B=RB, C=RC)
try(NNS.SD.cluster(Rmat, degree = 1))
## $Clusters
## $Clusters$Cluster_1
## [1] "C" "A" "B"
try(NNS.SD.efficient.set(Rmat, degree = 1))
## Checking 1 of 2Checking 2 of 2
## [1] "C" "A" "B"
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)
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
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
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.