Getting Started with NNS: Clustering and Regression"

In NNS: Nonlinear Nonparametric Statistics

knitr::opts_chunk$set(echo = TRUE)
library(NNS)
library(data.table)
data.table::setDTthreads(1L)
options(mc.cores = 1)
RcppParallel::setThreadOptions(numThreads = 1)
Sys.setenv("OMP_THREAD_LIMIT" = 1)

library(NNS)
library(data.table)
require(knitr)
require(rgl)

Clustering and Regression

Below are some examples demonstrating unsupervised learning with NNS clustering and nonlinear regression using the resulting clusters. As always, for a more thorough description and definition, please view the References.

NNS Partitioning NNS.part()

NNS.part is both a partitional and hierarchical clustering method. NNS iteratively partitions the joint distribution into partial moment quadrants, and then assigns a quadrant identification (1:4) at each partition.

NNS.part returns a data.table of observations along with their final quadrant identification. It also returns the regression points, which are the quadrant means used in NNS.reg.

x = seq(-5, 5, .05); y = x ^ 3

for(i in 1 : 4){NNS.part(x, y, order = i, Voronoi = TRUE, obs.req = 0)}

X-only Partitioning

NNS.part offers a partitioning based on $x$ values only NNS.part(x, y, type = "XONLY", ...), using the entire bandwidth in its regression point derivation, and shares the same limit condition as partitioning via both $x$ and $y$ values.

for(i in 1 : 4){NNS.part(x, y, order = i, type = "XONLY", Voronoi = TRUE)}

Note the partition identifications are limited to 1's and 2's (left and right of the partition respectively), not the 4 values per the $x$ and $y$ partitioning.

NNS.part(x,y,order = 4, type = "XONLY")

Clusters Used in Regression

The right column of plots shows the corresponding regression (plus endpoints and central point) for the order of NNS partitioning. ```r,results='hide'} for(i in 1 : 3){NNS.part(x, y, order = i, obs.req = 0, Voronoi = TRUE, type = "XONLY") ; NNS.reg(x, y, order = i, ncores = 1)}

# NNS Regression `NNS.reg()`
**`NNS.reg`** can fit any $f(x)$, for both uni- and multivariate cases.  **`NNS.reg`** returns a self-evident list of values provided below.

## Univariate:
```r
NNS.reg(x, y, ncores = 1)

Multivariate:

Multivariate regressions return a plot of $y$ and $\hat{y}$, as well as the regression points ($RPM) and partitions ($rhs.partitions) for each regressor.

f = function(x, y) x ^ 3 + 3 * y - y ^ 3 - 3 * x
y = x ; z <- expand.grid(x, y)
g = f(z[ , 1], z[ , 2])
NNS.reg(z, g, order = "max", plot = FALSE, ncores = 1)

Inter/Extrapolation

NNS.reg can inter- or extrapolate any point of interest. The NNS.reg(x, y, point.est = ...) parameter permits any sized data of similar dimensions to $x$ and called specifically with NNS.reg(...)$Point.est.

NNS Dimension Reduction Regression

NNS.reg also provides a dimension reduction regression by including a parameter NNS.reg(x, y, dim.red.method = "cor", ...). Reducing all regressors to a single dimension using the returned equation NNS.reg(..., dim.red.method = "cor", ...)$equation.

NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft", ncores = 1)$equation

a = NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", location = "topleft", ncores = 1, plot = FALSE)$equation

Thus, our model for this regression would be: $$Species = \frac{r round(a$Coefficient[1],3)Sepal.Length r round(a$Coefficient[2],3)Sepal.Width +r round(a$Coefficient[3],3)Petal.Length +r round(a$Coefficient[4],3)Petal.Width}{4} $$

Threshold

NNS.reg(x, y, dim.red.method = "cor", threshold = ...) offers a method of reducing regressors further by controlling the absolute value of required correlation.

NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, location = "topleft", ncores = 1)$equation

a = NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, location = "topleft", ncores = 1, plot = FALSE)$equation

Thus, our model for this further reduced dimension regression would be: $$Species = \frac{\: r round(a$Coefficient[1],3)Sepal.Length + r round(a$Coefficient[2],3)Sepal.Width +r round(a$Coefficient[3],3)Petal.Length +r round(a$Coefficient[4],3)Petal.Width}{3} $$

and the point.est = (...) operates in the same manner as the full regression above, again called with NNS.reg(...)$Point.est.

NNS.reg(iris[ , 1 : 4], iris[ , 5], dim.red.method = "cor", threshold = .75, point.est = iris[1 : 10, 1 : 4], location = "topleft", ncores = 1)$Point.est

Classification

For a classification problem, we simply set NNS.reg(x, y, type = "CLASS", ...).

NOTE: Base category of response variable should be 1, not 0 for classification problems.

NNS.reg(iris[ , 1 : 4], iris[ , 5], type = "CLASS", point.est = iris[1 : 10, 1 : 4], location = "topleft", ncores = 1)$Point.est

Cross-Validation NNS.stack()

The NNS.stack routine cross-validates for a given objective function the n.best parameter in the multivariate NNS.reg function as well as the threshold parameter in the dimension reduction NNS.reg version. NNS.stack can be used for classification:

NNS.stack(..., type = "CLASS", ...)

or continuous dependent variables:

NNS.stack(..., type = NULL, ...).

Any objective function obj.fn can be called using expression() with the terms predicted and actual, even from external packages such as Metrics.

NNS.stack(..., obj.fn = expression(Metrics::mape(actual, predicted)), objective = "min").

NNS.stack(IVs.train = iris[ , 1 : 4], 
          DV.train = iris[ , 5], 
          IVs.test = iris[1 : 10, 1 : 4],
          dim.red.method = "cor",
          obj.fn = expression( mean(round(predicted) == actual) ),
          objective = "max", type = "CLASS", 
          folds = 1, ncores = 1)

Folds Remaining = 0 
Current NNS.reg(... , threshold = 0.80 ) MAX Iterations Remaining = 1
Current NNS.reg(... , threshold = 0.40 ) MAX Iterations Remaining = 0
Current NNS.reg(. , n.best = 1 ) MAX Iterations Remaining = 12
Current NNS.reg(. , n.best = 2 ) MAX Iterations Remaining = 11
Current NNS.reg(. , n.best = 3 ) MAX Iterations Remaining = 10
Current NNS.reg(. , n.best = 4 ) MAX Iterations Remaining = 9
Current NNS.reg(. , n.best = 5 ) MAX Iterations Remaining = 8
$OBJfn.reg
[1] 0.9733333

$NNS.reg.n.best
[1] 1

$probability.threshold
[1] 0.547

$OBJfn.dim.red
[1] 0.9666667

$NNS.dim.red.threshold
[1] 0.8

$reg
 [1] 1 1 1 1 1 1 1 1 1 1

$reg.pred.int
NULL

$dim.red
 [1] 1 1 1 1 1 1 1 1 1 1

$dim.red.pred.int
NULL

$stack
 [1] 1 1 1 1 1 1 1 1 1 1

$pred.int
NULL

Increasing Dimensions

Given multicollinearity is not an issue for nonparametric regressions as it is for OLS, in the case of an ill-fit univariate model a better option may be to increase the dimensionality of regressors with a copy of itself and cross-validate the number of clusters n.best via:

NNS.stack(IVs.train = cbind(x, x), DV.train = y, method = 1, ...).

set.seed(123)
x = rnorm(100); y = rnorm(100)

nns.params = NNS.stack(IVs.train = cbind(x, x),
                        DV.train = y,
                        method = 1, ncores = 1)

set.seed(123)
x = rnorm(100); y = rnorm(100)

nns.params = list()
nns.params$NNS.reg.n.best = 100

NNS.reg(cbind(x, x), y, 
        n.best = nns.params$NNS.reg.n.best,
        point.est = cbind(x, x), 
        residual.plot = TRUE,  
        ncores = 1, confidence.interval = .95)

Smoothing Option

Smoothness is not required for curve fitting, but the NNS.reg function offers an optional smoothed fit. This feature applies a smoothing spline to regression points generated internally using the partitioning method described earlier.

NNS.reg(x, y, smooth = TRUE)

Imputation

Imputation in NNS is a direct application of nearest neighbor regression. When values of $y$ are missing, we use the observed $(X,y)$ pairs as the training set and the predictors of the missing rows as point.est.

A key insight is that even in univariate regressions, NNS.reg benefits from the increasing dimensions trick: by duplicating the predictor into a multivariate form, e.g. cbind(x, x), the distance function underlying NNS.reg operates in a 2-D space. This sharpened distance metric allows a more robust donor selection, effectively turning univariate imputation into a special case of multivariate nearest neighbor regression.

For multivariate predictors, the same form applies directly — supply the full set of observed predictors in $x$, the observed responses in $y$, and the incomplete rows in point.est. With order = "max", n.best = 1, the imputation is always 1-NN donor-based: each missing $y$ is filled in by the response of its closest donor under the NNS hybrid distance. This ensures imputations remain strictly within the support of the observed data.

Categorical data is handled analogously, only requiring NNS.reg(..., type = "CLASS") in the procedure.

Univariate Imputation

set.seed(123)

# Univariate predictor with nonlinear signal
n <- 400
x <- sort(runif(n, -3, 3))
y <- sin(x) + 0.2 * x^2 + rnorm(n, 0, 0.25)

# Induce ~25% MCAR missingness in y
miss <- rbinom(n, 1, 0.25) == 1
y_mis <- y
y_mis[miss] <- NA

# ---- Increasing dimensions trick ----
# Duplicate x so the distance operates in a 2D space: cbind(x, x).
# This sharpens nearest-neighbor selection even in a nominally univariate setting.
x2_train <- cbind(x[!miss], x[!miss])
x2_miss  <- cbind(x[miss],  x[miss])

# 1-NN donor imputation with NNS.reg
y_hat_uni <- NNS::NNS.reg(
  x         = x2_train,             # predictors (duplicated x)
  y         = y[!miss],             # observed responses
  point.est = x2_miss,              # rows to impute
  order     = "max",                # dependence-maximizing order
  n.best    = 1,                    # 1-NN donor
  plot      = FALSE
)$Point.est

# Fill back
y_completed_uni <- y_mis
y_completed_uni[miss] <- y_hat_uni

# Plot observed vs imputed (NNS 1-NN)
plot(x, y, pch = 1, col = "steelblue", cex = 1.5, lwd = 2,
     xlab = "x", ylab = "y", main = "NNS 1-NN Imputation")
points(x[miss], y_hat_uni, col = "red", pch = 15, cex = 1.3)

legend("topleft",
       legend = c("Observed", "Imputed (NNS 1-NN)"),
       col    = c("steelblue", "red"),
       pch    = c(1, 15),
       pt.lwd = c(2, NA),
       bty    = "n")

{width="600" height="600"}

Multivariate Imputation

set.seed(123)

# Multivariate predictors with nonlinear & interaction structure
n <- 800
X <- cbind(
  x1 = rnorm(n),
  x2 = runif(n, -2, 2),
  x3 = rnorm(n, 0, 1)
)

f <- function(x1, x2, x3) 1.1*x1 - 0.8*x2 + 0.5*x3 + 0.6*x1*x2 - 0.4*x2*x3 + 0.3*sin(1.3*x1)
y <- f(X[,1], X[,2], X[,3]) + rnorm(n, 0, 0.4)

# Induce ~30% MCAR missingness in y
miss <- rbinom(n, 1, 0.30) == 1
y_mis <- y
y_mis[miss] <- NA

# Training (observed) vs rows to impute
X_obs <- X[!miss, , drop = FALSE]
y_obs <- y[!miss]
X_mis <- X[ miss, , drop = FALSE]

# 1-NN donor imputation with NNS.reg
y_hat_mv <- NNS::NNS.reg(
  x         = X_obs,     # all observed predictors
  y         = y_obs,     # observed responses
  point.est = X_mis,     # rows to impute
  order     = "max",     # dependence-maximizing order
  n.best    = 1,         # 1-NN donor
  plot      = FALSE
)$Point.est

# Completed vector
y_completed_mv <- y_mis
y_completed_mv[miss] <- y_hat_mv

# Plot observed vs imputed (multivariate, NNS 1-NN)
plot(seq_along(y), y, 
     pch = 1, col = "steelblue", cex = 1.5, lwd = 2,
     xlab = "Observation index", ylab = "y",
     main = "NNS 1-NN Multivariate Imputation")

# Overlay imputed values
points(which(miss), y_hat_mv, pch = 15, col = "red", cex = 1.2)

# Legend
legend("topleft",
       legend = c("Observed", "Imputed (NNS 1-NN)"),
       col    = c("steelblue", "red"),
       pch    = c(1, 15),
       pt.lwd = c(2, NA),
       bty    = "n")

{width="600" height="600"}

References

If the user is so motivated, detailed arguments further examples are provided within the following:

• Nonlinear Nonparametric Statistics: Using Partial Moments

• Deriving Nonlinear Correlation Coefficients from Partial Moments

• Nonparametric Regression Using Clusters

• Clustering and Curve Fitting by Line Segments

• Classification Using NNS Clustering Analysis

• Partitional Estimation Using Partial Moments

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NNS documentation built on Nov. 28, 2025, 5:09 p.m.