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Getting Started with NNS: Normalization and Rescaling"
In NNS: Nonlinear Nonparametric Statistics
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 5)
suppressPackageStartupMessages(library(NNS))
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)
Overview
This vignette covers two related tools:
NNS.norm() for cross‑variable normalization when comparing multiple series.NNS.rescale() for single‑vector rescaling with either min‑max or risk‑neutral targets.
Both functions perform deterministic affine transformations that preserve rank structure while modifying scale.
NNS.norm(): Normalize Multiple Variables
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.
Mathematical Structure
Let (X) be an (n \times p) matrix of variables.
Step 1: Compute Mean Vector
[
m_j = \text{mean}(X_{\cdot j})
]
If any (m_j = 0), it is replaced with (10^{-10}) to prevent division by zero.
Step 2: Construct Mean Ratio Matrix
[
RG_{ij} = \frac{m_i}{m_j}
]
In R this corresponds to:
RG <- outer(m, 1 / m)
Step 3: Dependence Weight Matrix
If linear = FALSE:
- If number of variables (p < 10):
[
W = |\mathrm{cor}(X)|
]
- Otherwise:
[
W = |D| \quad \text{where } D = \text{NNS.dep}(X)\$Dependence
]
NNS.dep() returns a symmetric matrix of nonlinear dependence measures.
If linear = TRUE, the weighting effectively becomes:
[
W_{ij} = 1
]
Step 4: Scaling Factors
[
s_j = \frac{1}{p} \sum_{i=1}^{p} RG_{ij} W_{ij}
]
Each column is scaled:
[
X_{\cdot j}^{*} = s_j X_{\cdot j}
]
Linear Case Proof
If (W_{ij} = 1):
[
s_j = \frac{1}{p} \sum_{i=1}^{p} \frac{m_i}{m_j}
= \frac{\bar{m}}{m_j}
]
Then:
[
\text{mean}(X_{\cdot j}^{*}) = s_j m_j = \bar{m}
]
All variables share the same mean.
Nonlinear Case Interpretation
[
\text{mean}(X_{\cdot j}^{*})
=
\frac{1}{p}
\sum_{i=1}^{p}
m_i W_{ij}
]
Thus, the normalized mean becomes a dependence‑weighted average of original means. Variables more strongly dependent with higher‑mean variables scale upward more.
Examples
Basic Multivariate Example
This holds for any distribution type and can be applied to vectors of different lengths.
set.seed(123)
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)
head(lin_norm)
A Normalized B Normalized C Normalized D Normalized
[1,] -29.929719 31.889828 5.819152 1.4264014
[2,] -12.291609 -11.531393 5.396317 1.2388239
[3,] 83.235911 11.073887 4.643781 0.3078703
[4,] 3.765188 15.601030 5.029380 -0.2630481
[5,] 6.904039 42.717726 4.572611 2.8193657
[6,] 91.585447 2.021274 4.543080 6.6681079
# Verify means are equal
apply(lin_norm, 2, function(x) c(mean = mean(x), sd = sd(x)))
A Normalized B Normalized C Normalized D Normalized
mean 4.827727 4.827727 4.8277270 4.827727
sd 48.744888 43.407590 0.4531172 5.203436
Now compare with nonlinear scaling:
nonlin_norm <- NNS.norm(X, linear = FALSE, chart.type = NULL)
head(nonlin_norm)
A Normalized B Normalized C Normalized D Normalized
[1,] -2.7834653 0.32807768 3.178568 0.7439872
[2,] -1.1431202 -0.11863321 2.947605 0.6461499
[3,] 7.7409438 0.11392645 2.536550 0.1605800
[4,] 0.3501627 0.16050101 2.747174 -0.1372015
[5,] 0.6420759 0.43947344 2.497676 1.4705341
[6,] 8.5174510 0.02079456 2.481545 3.4779738
apply(nonlin_norm, 2, function(x) c(mean = mean(x), sd = sd(x)))
A Normalized B Normalized C Normalized D Normalized
mean 0.4489788 0.04966692 2.637026 2.518062
sd 4.5332769 0.44657066 0.247504 2.714025
Note that the means differ and the standard deviations are smaller than in the linear case, reflecting the dependence structure.
Normalize list of unequal vector lengths
set.seed(123)
vec1 <- rnorm(n = 10, mean = 0, sd = 1)
vec2 <- rnorm(n = 5, mean = 5, sd = 5)
vec3 <- rnorm(n = 8, mean = 10, sd = 10)
vec_list <- list(vec1, vec2, vec3)
NNS.norm(vec_list)
$`x_1 Normalized`
[1] 13.074058 -3.004912 -11.745878 25.406891 -4.647966 -5.481229 6.225165 5.920719 6.113733 9.640242
$`x_2 Normalized`
[1] 2.875960212 0.008876158 1.230826150 5.855582361 10.779166523
$`x_3 Normalized`
[1] 4.0749062 2.2395840 0.4067264 0.7457562 15.6445780 5.1941416 2.3326665 2.5622994
Quantile Normalization Comparison
Quantile normalization forces distributions to be identical. This is literally the opposite intended effect of NNS.norm, which preserves individual distribution shapes while aligning ranges. The quantile normalized series become identical in distribution, while the NNS methods retain the original patterns.
Practical Applications
Normalization eliminates the need for multiple y‑axis charts and prevents their misuse. By placing variables on the same axes with shared ranges, we enable more relevant conditional probability analyses. This technique, combined with time normalization, is used in NNS.caus() to identify causal relationships between variables.
NNS.rescale(): Distribution Rescaling
NNS.rescale() performs one‑dimensional affine transformations.
Function signature:
NNS.rescale(x, a, b, method = "minmax", T = NULL, type = "Terminal")
1) Min-Max Scaling
If method = "minmax":
[
x^{*}
=
a
+
(b - a)
\frac{x - \min(x)}
{\max(x) - \min(x)}
]
Properties:
- Preserves order
- Maps support to ([a,b])
- Linear transformation
Example
raw_vals <- c(-2.5, 0.2, 1.1, 3.7, 5.0)
scaled_minmax <- NNS.rescale(
x = raw_vals,
a = 5,
b = 10,
method = "minmax",
T = NULL,
type = "Terminal"
)
cbind(raw_vals, scaled_minmax)
range(scaled_minmax)
2) Risk-Neutral Scaling
If method = "riskneutral":
Let:
- ( S_0 = a )
- ( r = b )
- ( T ) = time horizon
Terminal Type
Target:
[
\mathbb{E}[S_T] = S_0 e^{rT}
]
Transformation form:
[
x^{*}
=
x
\cdot
\frac{S_0 e^{rT}}
{\text{mean}(x)}
]
This enforces the required expectation.
Discounted Type
Target:
[
\mathbb{E}[e^{-rT} S_T] = S_0
]
Equivalent to:
[
\mathbb{E}[S_T] = S_0 e^{rT}
]
but the returned series is scaled so that its discounted mean equals (S_0). In practice, the function applies the same multiplicative factor as above, because:
[
\text{mean}(e^{-rT} x^{}) = e^{-rT} \cdot \text{mean}(x^{}) = e^{-rT} \cdot S_0 e^{rT} = S_0.
]
Risk-Neutral Example
set.seed(123)
S0 <- 100
r <- 0.05
T <- 1
# Simulate a price path
prices <- S0 * exp(cumsum(rnorm(250, 0.0005, 0.02)))
rn_terminal <- NNS.rescale(
x = prices,
a = S0,
b = r,
method = "riskneutral",
T = T,
type = "Terminal"
)
c(
mean_original = mean(prices),
mean_rescaled = mean(rn_terminal),
target = S0 * exp(r * T)
)
mean_original mean_rescaled target
109.7019 105.1271 105.1271
Discounted Example
rn_discounted <- NNS.rescale(
x = prices,
a = S0,
b = r,
method = "riskneutral",
T = T,
type = "Discounted"
)
c(
mean_rescaled = mean(rn_discounted),
target_discounted_mean = S0
)
mean_rescaled target_discounted_mean
100 100
Conceptual Summary
NNS.norm()
- Multivariate
- Dependence‑aware scaling
- Equalizes means only in linear mode
- Preserves shape and order
NNS.rescale()
- Univariate
- Affine transformation
- Either range‑targeted or expectation‑targeted
- Preserves rank structure
Both functions maintain monotonicity and are therefore compatible with NNS copula and dependence modeling frameworks.
set.seed(123)
x <- rnorm(1000, 5, 2)
y <- rgamma(1000, 3, 1)
# Combine variables
X <- cbind(x, y)
# NNS normalization
X_norm_lin <- NNS.norm(X, linear = TRUE)
X_norm_nonlin <- NNS.norm(X, linear = FALSE)
# Standard min-max normalization
minmax <- function(v) (v - min(v)) / (max(v) - min(v))
X_minmax <- apply(X, 2, minmax)
par(mfrow = c(2,2))
steelblue_alpha <- rgb(1,0,0,0.4)
red_alpha <- rgb(0,0,1,0.4)
# Breaks for original data
br_orig <- pretty(range(c(x, y)), n = 15)
# Original variables
hist(x,
col = steelblue_alpha,
breaks = br_orig,
main = "Original Variables",
xlab = "")
hist(y,
col = red_alpha,
breaks = br_orig,
add = TRUE)
# Breaks for NNS normalized variables
br_norm <- pretty(range(c(X_norm_lin[,1], X_norm_lin[,2])), n = 15)
# NNS normalized
hist(X_norm_lin[,1],
col = steelblue_alpha,
breaks = br_norm,
main = "NNS.norm(..., Linear=TRUE)",
xlab = "")
hist(X_norm_lin[,2],
col = red_alpha,
breaks = br_norm,
add = TRUE)
# Breaks for NNS normalized variables
br_norm <- pretty(range(c(X_norm_nonlin[,1], X_norm_nonlin[,2])), n = 15)
# NNS normalized
hist(X_norm_nonlin[,1],
col = steelblue_alpha,
breaks = br_norm,
main = "NNS.norm(..., Linear=FALSE)",
xlab = "")
hist(X_norm_nonlin[,2],
col = red_alpha,
breaks = br_norm,
add = TRUE)
# Breaks for min-max normalized variables
br_minmax <- pretty(range(c(X_minmax[,1], X_minmax[,2])), n = 15)
# Standard min-max normalization
hist(X_minmax[,1],
col = steelblue_alpha,
breaks = br_minmax,
main = "Standard Min-Max",
xlab = "")
hist(X_minmax[,2],
col = red_alpha,
breaks = br_minmax,
add = TRUE)
References
If the user is so motivated, detailed arguments further examples are provided within the following:
Try the NNS package in your browser
Any scripts or data that you put into this service are public.
NNS documentation built on April 10, 2026, 9:10 a.m.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
knitr::opts_chunk$set(collapse = TRUE, comment = "#>", fig.width = 7, fig.height = 5) suppressPackageStartupMessages(library(NNS)) 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)
Overview
This vignette covers two related tools:
NNS.norm()for cross‑variable normalization when comparing multiple series.NNS.rescale()for single‑vector rescaling with either min‑max or risk‑neutral targets.
Both functions perform deterministic affine transformations that preserve rank structure while modifying scale.
NNS.norm(): Normalize Multiple Variables
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.
Mathematical Structure
Let (X) be an (n \times p) matrix of variables.
Step 1: Compute Mean Vector
[ m_j = \text{mean}(X_{\cdot j}) ]
If any (m_j = 0), it is replaced with (10^{-10}) to prevent division by zero.
Step 2: Construct Mean Ratio Matrix
[ RG_{ij} = \frac{m_i}{m_j} ]
In R this corresponds to:
RG <- outer(m, 1 / m)
Step 3: Dependence Weight Matrix
If linear = FALSE:
- If number of variables (p < 10): [ W = |\mathrm{cor}(X)| ]
- Otherwise:
[
W = |D| \quad \text{where } D = \text{NNS.dep}(X)\$Dependence
]
NNS.dep()returns a symmetric matrix of nonlinear dependence measures.
If linear = TRUE, the weighting effectively becomes:
[ W_{ij} = 1 ]
Step 4: Scaling Factors
[ s_j = \frac{1}{p} \sum_{i=1}^{p} RG_{ij} W_{ij} ]
Each column is scaled:
[ X_{\cdot j}^{*} = s_j X_{\cdot j} ]
Linear Case Proof
If (W_{ij} = 1):
[ s_j = \frac{1}{p} \sum_{i=1}^{p} \frac{m_i}{m_j} = \frac{\bar{m}}{m_j} ]
Then:
[ \text{mean}(X_{\cdot j}^{*}) = s_j m_j = \bar{m} ]
All variables share the same mean.
Nonlinear Case Interpretation
[ \text{mean}(X_{\cdot j}^{*}) = \frac{1}{p} \sum_{i=1}^{p} m_i W_{ij} ]
Thus, the normalized mean becomes a dependence‑weighted average of original means. Variables more strongly dependent with higher‑mean variables scale upward more.
Examples
Basic Multivariate Example
This holds for any distribution type and can be applied to vectors of different lengths.
set.seed(123) 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) head(lin_norm) A Normalized B Normalized C Normalized D Normalized [1,] -29.929719 31.889828 5.819152 1.4264014 [2,] -12.291609 -11.531393 5.396317 1.2388239 [3,] 83.235911 11.073887 4.643781 0.3078703 [4,] 3.765188 15.601030 5.029380 -0.2630481 [5,] 6.904039 42.717726 4.572611 2.8193657 [6,] 91.585447 2.021274 4.543080 6.6681079 # Verify means are equal apply(lin_norm, 2, function(x) c(mean = mean(x), sd = sd(x))) A Normalized B Normalized C Normalized D Normalized mean 4.827727 4.827727 4.8277270 4.827727 sd 48.744888 43.407590 0.4531172 5.203436
Now compare with nonlinear scaling:
nonlin_norm <- NNS.norm(X, linear = FALSE, chart.type = NULL) head(nonlin_norm) A Normalized B Normalized C Normalized D Normalized [1,] -2.7834653 0.32807768 3.178568 0.7439872 [2,] -1.1431202 -0.11863321 2.947605 0.6461499 [3,] 7.7409438 0.11392645 2.536550 0.1605800 [4,] 0.3501627 0.16050101 2.747174 -0.1372015 [5,] 0.6420759 0.43947344 2.497676 1.4705341 [6,] 8.5174510 0.02079456 2.481545 3.4779738 apply(nonlin_norm, 2, function(x) c(mean = mean(x), sd = sd(x))) A Normalized B Normalized C Normalized D Normalized mean 0.4489788 0.04966692 2.637026 2.518062 sd 4.5332769 0.44657066 0.247504 2.714025
Note that the means differ and the standard deviations are smaller than in the linear case, reflecting the dependence structure.
Normalize list of unequal vector lengths
set.seed(123) vec1 <- rnorm(n = 10, mean = 0, sd = 1) vec2 <- rnorm(n = 5, mean = 5, sd = 5) vec3 <- rnorm(n = 8, mean = 10, sd = 10) vec_list <- list(vec1, vec2, vec3) NNS.norm(vec_list) $`x_1 Normalized` [1] 13.074058 -3.004912 -11.745878 25.406891 -4.647966 -5.481229 6.225165 5.920719 6.113733 9.640242 $`x_2 Normalized` [1] 2.875960212 0.008876158 1.230826150 5.855582361 10.779166523 $`x_3 Normalized` [1] 4.0749062 2.2395840 0.4067264 0.7457562 15.6445780 5.1941416 2.3326665 2.5622994
Quantile Normalization Comparison
Quantile normalization forces distributions to be identical. This is literally the opposite intended effect of NNS.norm, which preserves individual distribution shapes while aligning ranges. The quantile normalized series become identical in distribution, while the NNS methods retain the original patterns.
Practical Applications
Normalization eliminates the need for multiple y‑axis charts and prevents their misuse. By placing variables on the same axes with shared ranges, we enable more relevant conditional probability analyses. This technique, combined with time normalization, is used in NNS.caus() to identify causal relationships between variables.
NNS.rescale(): Distribution Rescaling
NNS.rescale() performs one‑dimensional affine transformations.
Function signature:
NNS.rescale(x, a, b, method = "minmax", T = NULL, type = "Terminal")
1) Min-Max Scaling
If method = "minmax":
[ x^{*} = a + (b - a) \frac{x - \min(x)} {\max(x) - \min(x)} ]
Properties:
- Preserves order
- Maps support to ([a,b])
- Linear transformation
Example
raw_vals <- c(-2.5, 0.2, 1.1, 3.7, 5.0) scaled_minmax <- NNS.rescale( x = raw_vals, a = 5, b = 10, method = "minmax", T = NULL, type = "Terminal" ) cbind(raw_vals, scaled_minmax) range(scaled_minmax)
2) Risk-Neutral Scaling
If method = "riskneutral":
Let:
- ( S_0 = a )
- ( r = b )
- ( T ) = time horizon
Terminal Type
Target:
[ \mathbb{E}[S_T] = S_0 e^{rT} ]
Transformation form:
[ x^{*} = x \cdot \frac{S_0 e^{rT}} {\text{mean}(x)} ]
This enforces the required expectation.
Discounted Type
Target:
[ \mathbb{E}[e^{-rT} S_T] = S_0 ]
Equivalent to:
[ \mathbb{E}[S_T] = S_0 e^{rT} ]
but the returned series is scaled so that its discounted mean equals (S_0). In practice, the function applies the same multiplicative factor as above, because:
[ \text{mean}(e^{-rT} x^{}) = e^{-rT} \cdot \text{mean}(x^{}) = e^{-rT} \cdot S_0 e^{rT} = S_0. ]
Risk-Neutral Example
set.seed(123) S0 <- 100 r <- 0.05 T <- 1 # Simulate a price path prices <- S0 * exp(cumsum(rnorm(250, 0.0005, 0.02))) rn_terminal <- NNS.rescale( x = prices, a = S0, b = r, method = "riskneutral", T = T, type = "Terminal" ) c( mean_original = mean(prices), mean_rescaled = mean(rn_terminal), target = S0 * exp(r * T) ) mean_original mean_rescaled target 109.7019 105.1271 105.1271
Discounted Example
rn_discounted <- NNS.rescale( x = prices, a = S0, b = r, method = "riskneutral", T = T, type = "Discounted" ) c( mean_rescaled = mean(rn_discounted), target_discounted_mean = S0 ) mean_rescaled target_discounted_mean 100 100
Conceptual Summary
NNS.norm()
- Multivariate
- Dependence‑aware scaling
- Equalizes means only in linear mode
- Preserves shape and order
NNS.rescale()
- Univariate
- Affine transformation
- Either range‑targeted or expectation‑targeted
- Preserves rank structure
Both functions maintain monotonicity and are therefore compatible with NNS copula and dependence modeling frameworks.
set.seed(123) x <- rnorm(1000, 5, 2) y <- rgamma(1000, 3, 1) # Combine variables X <- cbind(x, y) # NNS normalization X_norm_lin <- NNS.norm(X, linear = TRUE) X_norm_nonlin <- NNS.norm(X, linear = FALSE) # Standard min-max normalization minmax <- function(v) (v - min(v)) / (max(v) - min(v)) X_minmax <- apply(X, 2, minmax)
par(mfrow = c(2,2)) steelblue_alpha <- rgb(1,0,0,0.4) red_alpha <- rgb(0,0,1,0.4) # Breaks for original data br_orig <- pretty(range(c(x, y)), n = 15) # Original variables hist(x, col = steelblue_alpha, breaks = br_orig, main = "Original Variables", xlab = "") hist(y, col = red_alpha, breaks = br_orig, add = TRUE) # Breaks for NNS normalized variables br_norm <- pretty(range(c(X_norm_lin[,1], X_norm_lin[,2])), n = 15) # NNS normalized hist(X_norm_lin[,1], col = steelblue_alpha, breaks = br_norm, main = "NNS.norm(..., Linear=TRUE)", xlab = "") hist(X_norm_lin[,2], col = red_alpha, breaks = br_norm, add = TRUE) # Breaks for NNS normalized variables br_norm <- pretty(range(c(X_norm_nonlin[,1], X_norm_nonlin[,2])), n = 15) # NNS normalized hist(X_norm_nonlin[,1], col = steelblue_alpha, breaks = br_norm, main = "NNS.norm(..., Linear=FALSE)", xlab = "") hist(X_norm_nonlin[,2], col = red_alpha, breaks = br_norm, add = TRUE) # Breaks for min-max normalized variables br_minmax <- pretty(range(c(X_minmax[,1], X_minmax[,2])), n = 15) # Standard min-max normalization hist(X_minmax[,1], col = steelblue_alpha, breaks = br_minmax, main = "Standard Min-Max", xlab = "") hist(X_minmax[,2], col = red_alpha, breaks = br_minmax, add = TRUE)
References
If the user is so motivated, detailed arguments further examples are provided within the following:
Try the NNS package in your browser
Any scripts or data that you put into this service are public.
R Package Documentation
Browse R Packages
We want your feedback!
Note that we can't provide technical support on individual packages. You should contact the package authors for that.
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