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)

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.

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

## $mean
## [1] -0.0114498
## 
## $variance
## [1] 1.011552
## 
## $skewness
## [1] -0.007412142
## 
## $kurtosis
## [1] 0.06723772

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)

## [1] -2.049405  1.987674

2.3 CDF tables via LPM ratios

Headers.

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)
UPM.VaR(p, degree, variable)

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.table(threshold = as.numeric(qgrid), CDF = LPM.ratio(0,qgrid,z))
CDF_tbl

##       threshold   CDF
##           <num> <num>
##  1: -0.94052127  0.05
##  2: -0.83748109  0.15
##  3: -0.71317882  0.25
##  4: -0.57443327  0.35
##  5: -0.41017671  0.45
##  6: -0.20424962  0.55
##  7:  0.06850182  0.65
##  8:  0.41462712  0.75
##  9:  0.94307172  0.85
## 10:  2.09633977  0.95

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
##           <num>         <num>         <num>
##  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
##           <num>       <num>
##  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
##           <num>      <num>      <num> <char>      <num>        <num>
##   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
##                <num>
##   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 = 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

## Causation.x.given.y Causation.y.given.x           C(x--->y) 
##           0.2607148           0.3863580           0.3933374

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

## 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)
seasonal_period <- NNS.seas(z, plot = FALSE)
head(seasonal_period$all.periods)

##   Period Coefficient.of.Variation Variable.Coefficient.of.Variation
## 1     99                0.5122054                      1.168502e+17
## 2    147                0.5256021                      1.168502e+17
## 3    100                0.5598477                      1.168502e+17
## 4    146                0.5618687                      1.168502e+17
## 5    199                0.5766158                      1.168502e+17
## 6     98                0.5801409                      1.168502e+17

# Validate seasonal periods
NNS.ARMA.optim(z, h=48, seasonal.factor = seasonal_period$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 = 1.58350102101444"
## [1] "BEST method = 'nonlin' PATH MEMBER = c( 52, 49, 50 )"
## [1] "BEST nonlin OBJECTIVE FUNCTION = 1.58350102101444"
## [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 = 0.44738816518873"
## [1] "BEST method = 'both' PATH MEMBER = c( 52, 49, 50 )"
## [1] "BEST both OBJECTIVE FUNCTION = 0.44738816518873"

## $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$ensemble); names(mc$replicates)

## [1] 350

## [1] "rho = 1"    "rho = 0.5"  "rho = 0"    "rho = -0.5" "rho = -1"

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

##      Replicate 1 Replicate 2 Replicate 3 Replicate 4 Replicate 5
## [1,]   -6.097304   -0.246557   -4.427608   -2.153337   -5.050326
## [2,]   -6.582158    0.343863   -2.445919   -3.175586   -4.777579
## [3,]   -4.393280    0.548543   -1.822378   -2.637809   -4.708166
## [4,]   -6.134514   -0.584076   -6.920453   -1.800864   -4.861689
## [5,]   -5.270871    1.095869   -1.964841   -1.204533   -1.626527
## [6,]   -5.627466    1.128039   -3.351027   -2.736647   -0.587572

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

NNS.SSD.uni(RA, RB)

## [1] 0

NNS.TSD.uni(RA, RB)

## [1] 0

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

## $Clusters
## $Clusters$Cluster_1
## [1] "A" "B" "C"

try(NNS.SD.efficient.set(Rmat, degree = 1))

## Checking 1 of 2Checking 2 of 2

## [1] "A" "B" "C"

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

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

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 NNS.reg.
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.

Getting Started with NNS: Overview

Fred Viole

Orientation

1. Foundations — Partial Moments & Variance Decomposition

1.1 Why partial moments

1.2 Core functions and headers

1.3 Code: variance decomposition & CDF

2. Descriptive & Distributional Tools

2.1 Higher moments from partial moments

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

2.3 CDF tables via LPM ratios

3. Dependence & Nonlinear Association

3.1 Why move beyond Pearson \(r\)

3.2 Code: nonlinear dependence

3.3 Code: copula

4. Hypothesis Testing & ANOVA (LPM‑CDF)

4.1 Concept

4.2 Code: two‑sample & multi‑group

5. Regression, Boosting, Stacking & Causality

5.1 Philosophy

5.2 Code: classification via regression + ensembles

5.3 Code: directional causality

6. Time Series & Forecasting

7. Simulation, Bootstrap & Risk‑Neutral Rescaling

7.1 Maximum entropy bootstrap (shape‑preserving)

7.2 Monte Carlo over the full correlation space

7.3 Risk‑neutral rescale (pricing context)

8. Portfolio & Stochastic Dominance

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

Appendix B — Quick Reference (Grouped by Topic)

1. Partial Moments & Ratios

2. Descriptive Statistics & Distributions

3. Dependence & Association

4. Hypothesis Testing

5. Regression, Classification & Causality

6. Differentiation & Slope Measures

7. Time Series & Forecasting

8. Simulation, Bootstrap & Rescaling

9. Portfolio Analysis & Stochastic Dominance