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Type:

Package

Title:

Nonlinear Nonparametric Statistics

Version:

11.6.3

Date:

2025-11-26

Maintainer:

Fred Viole <ovvo.financial.systems@gmail.com>

Description:

NNS (Nonlinear Nonparametric Statistics) leverages partial moments – the fundamental elements of variance that asymptotically approximate the area under f(x) – to provide a robust foundation for nonlinear analysis while maintaining linear equivalences. NNS delivers a comprehensive suite of advanced statistical techniques, including: Numerical integration, Numerical differentiation, Clustering, Correlation, Dependence, Causal analysis, ANOVA, Regression, Classification, Seasonality, Autoregressive modeling, Normalization, Stochastic dominance and Advanced Monte Carlo sampling. All routines based on: Viole, F. and Nawrocki, D. (2013), Nonlinear Nonparametric Statistics: Using Partial Moments (ISBN: 1490523995).

License:

GPL-3

BugReports:

https://github.com/OVVO-Financial/NNS/issues

Depends:

R (≥ 3.6.0)

Imports:

data.table, doParallel, foreach, quantmod, Rcpp, RcppParallel,Rfast, rgl, xts, zoo

Suggests:

knitr, rmarkdown, testthat (≥ 3.0.0)

VignetteBuilder:

knitr

LinkingTo:

Rcpp, RcppParallel

SystemRequirements:

GNU make

Config/testthat/edition:

RoxygenNote:

7.2.3

Encoding:

UTF-8

NeedsCompilation:

yes

Packaged:

2025-11-26 14:56:56 UTC; fredv

Author:

Fred Viole [aut, cre], Roberto Spadim [ctb]

Repository:

CRAN

Date/Publication:

2025-11-28 13:30:09 UTC

NNS: Nonlinear Nonparametric Statistics

Description

Nonlinear nonparametric statistics using partial moments. Partial moments are the elements of variance and asymptotically approximate the area of f(x). These robust statistics provide the basis for nonlinear analysis while retaining linear equivalences. NNS offers: Numerical integration, Numerical differentiation, Clustering, Correlation, Dependence, Causal analysis, ANOVA, Regression, Classification, Seasonality, Autoregressive modeling, Normalization and Stochastic dominance. All routines based on: Viole, F. and Nawrocki, D. (2013), Nonlinear Nonparametric Statistics: Using Partial Moments (ISBN: 1490523995).

Author(s)

Maintainer: Fred Violeovvo.financial.systems@gmail.com

Other contributors:

Roberto Spadim [contributor]

Co‑Lower Partial Moment

Description

Computes the co‑lower partial moment (lower‑left quadrant 4) between twoequal‑length numeric vectors at any degree and target.

Usage

Co.LPM(degree_lpm, x, y, target_x, target_y)

Arguments

degree_lpm

numeric; degree = 0 gives frequency, degree = 1 gives area.

x

numeric vector of observations.

y

numeric vector of the same length as x.

target_x

numeric vector; thresholds for x (defaults to mean(x)).

target_y

numeric vector; thresholds for y (defaults to mean(y)).

Value

Numeric vector of co‑LPM values.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. & Nawrocki, D. (2013) *Nonlinear Nonparametric Statistics: Using Partial Moments* (ISBN:1490523995)

Examples

  set.seed(123)  x <- rnorm(100); y <- rnorm(100)  Co.LPM(0, x, y, mean(x), mean(y))

Co‑Lower Partial Moment nD

Description

This function generates an n‑dimensional co‑lower partial moment (n >= 2) for any degree or target.

Usage

Co.LPM_nD(data, target, degree = 0, norm = TRUE)

Arguments

data

A numeric matrix with observations in rows and variables in columns.

target

A numeric vector, length equal to ncol(data).

degree

numeric; degree for lower deviations (0 = frequency, 1 = area).

norm

logical; ifTRUE (default) normalize to the maximum observed value (→ [0,1]), otherwise return the raw moment.

Value

Numeric; the n‑dimensional co‑lower partial moment.

Examples

## Not run: mat <- matrix(rnorm(200), ncol = 4)Co.LPM_nD(mat, rep(0, ncol(mat)), degree = 1, norm = FALSE)## End(Not run)

Co‑Upper Partial Moment

Description

Computes the co‑upper partial moment (upper‑right quadrant 1) between twoequal‑length numeric vectors at any degree and target.

Usage

Co.UPM(degree_upm, x, y, target_x, target_y)

Arguments

degree_upm

numeric; degree = 0 gives frequency, degree = 1 gives area.

x

numeric vector of observations.

y

numeric vector of the same length as x.

target_x

numeric vector; thresholds for x (defaults to mean(x)).

target_y

numeric vector; thresholds for y (defaults to mean(y)).

Value

Numeric vector of co‑UPM values.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. & Nawrocki, D. (2013) *Nonlinear Nonparametric Statistics: Using Partial Moments* (ISBN:1490523995)

Examples

  set.seed(123)  x <- rnorm(100); y <- rnorm(100)  Co.UPM(0, x, y, mean(x), mean(y))

Co‑Upper Partial Moment nD

Description

This function generates an n‑dimensional co‑upper partial moment (n >= 2) for any degree or target.

Usage

Co.UPM_nD(data, target, degree = 0, norm = TRUE)

Arguments

data

A numeric matrix with observations in rows and variables in columns.

target

A numeric vector, length equal to ncol(data).

degree

numeric; degree for upper deviations (0 = frequency, 1 = area).

norm

logical; ifTRUE (default) normalize to the maximum observed value (→ [0,1]), otherwise return the raw moment.

Value

Numeric; the n‑dimensional co‑upper partial moment.

Examples

## Not run: mat <- matrix(rnorm(200), ncol = 4)Co.UPM_nD(mat, rep(0, ncol(mat)), degree = 1, norm = FALSE)## End(Not run)

Divergent‑Lower Partial Moment

Description

Computes the divergent lower partial moment (lower‑right quadrant 3)between two equal‑length numeric vectors.

Usage

D.LPM(degree_lpm, degree_upm, x, y, target_x, target_y)

Arguments

degree_lpm

numeric; LPM degree = 0 gives frequency, = 1 gives area.

degree_upm

numeric; UPM degree = 0 gives frequency, = 1 gives area.

x

numeric vector of observations.

y

numeric vector of the same length as x.

target_x

numeric vector; thresholds for x (defaults to mean(x)).

target_y

numeric vector; thresholds for y (defaults to mean(y)).

Value

Numeric vector of divergent LPM values.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. & Nawrocki, D. (2013) *Nonlinear Nonparametric Statistics: Using Partial Moments* (ISBN:1490523995)

Examples

  set.seed(123)  x <- rnorm(100); y <- rnorm(100)  D.LPM(0, 0, x, y, mean(x), mean(y))

Divergent‑Upper Partial Moment

Description

Computes the divergent upper partial moment (upper‑left quadrant 2)between two equal‑length numeric vectors.

Usage

D.UPM(degree_lpm, degree_upm, x, y, target_x, target_y)

Arguments

degree_lpm

numeric; LPM degree = 0 gives frequency, = 1 gives area.

degree_upm

numeric; UPM degree = 0 gives frequency, = 1 gives area.

x

numeric vector of observations.

y

numeric vector of the same length as x.

target_x

numeric vector; thresholds for x (defaults to mean(x)).

target_y

numeric vector; thresholds for y (defaults to mean(y)).

Value

Numeric vector of divergent UPM values.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. & Nawrocki, D. (2013) *Nonlinear Nonparametric Statistics: Using Partial Moments* (ISBN:1490523995)

Examples

  set.seed(123)  x <- rnorm(100); y <- rnorm(100)  D.UPM(0, 0, x, y, mean(x), mean(y))

Divergent Partial Moment nD

Description

This function generates the aggregate n‑dimensional divergent partial moment (n >= 2) for any degree or target.

Usage

DPM_nD(data, target, degree = 0, norm = TRUE)

Arguments

data

A numeric matrix with observations in rows and variables in columns.

target

A numeric vector, length equal to ncol(data).

degree

numeric; degree for upper deviations (0 = frequency, 1 = area).

norm

logical; ifTRUE (default) normalize to the maximum observed value (→ [0,1]), otherwise return the raw moment.

Value

Numeric; the n‑dimensional co‑upper partial moment.

Examples

## Not run: mat <- matrix(rnorm(200), ncol = 4)DPM_nD(mat, rep(0, ncol(mat)), degree = 1, norm = FALSE)## End(Not run)

Lower Partial Moment

Description

This function generates a univariate lower partial moment for any degree or target.

Usage

LPM(degree, target, variable, excess_ret = FALSE)

Arguments

degree

numeric;(degree = 0) is frequency,(degree = 1) is area.

target

numeric; Set totarget = mean(variable) for classical equivalences, but does not have to be. (Vectorized)

variable

a numeric vector.data.frame orlist type objects are not permissible.

excess_ret

logical;FALSE (default)

Value

LPM of variable

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

set.seed(123)x <- rnorm(100)LPM(0, mean(x), x)

LPM VaR

Description

Generates a value at risk (VaR) quantile based on the Lower Partial Moment ratio.

Usage

LPM.VaR(percentile, degree, x)

Arguments

percentile

numeric [0, 1]; The percentile for left-tail VaR (vectorized).

degree

integer;(degree = 0) for discrete distributions,(degree = 1) for continuous distributions.

x

a numeric vector.

Value

Returns a numeric value representing the point at which"percentile" of the area ofx is below.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: set.seed(123)x <- rnorm(100)## For 5th percentile, left-tailLPM.VaR(0.05, 0, x)## End(Not run)

Lower Partial Moment Ratio

Description

This function generates a standardized univariate lower partial momentof any non‑negative degree for a given target.

Usage

LPM.ratio(degree, target, variable)

Arguments

degree

numeric; degree = 0 gives frequency (CDF), degree = 1 gives area.

target

numeric vector; threshold(s). Defaults to mean(variable).

variable

numeric vector or data‑frame column to evaluate.

Value

Numeric vector of standardized lower partial moments.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. & Nawrocki, D. (2013) *Nonlinear Nonparametric Statistics: Using Partial Moments* (ISBN:1490523995)

Viole, F. (2017) Continuous CDFs and ANOVA with NNS.doi:10.2139/ssrn.3007373

Examples

  set.seed(123)  x <- rnorm(100)  LPM.ratio(0, mean(x), x)## Not run:   plot(sort(x), LPM.ratio(0, sort(x), x))  plot(sort(x), LPM.ratio(1, sort(x), x))## End(Not run)

NNS ANOVA: Nonparametric Analysis of Variance

Description

Performs a distribution-free ANOVA using partial-moment statistics to assessdifferences between control and treatment groups. Depending on the setting ofmeans.only, the procedure tests either differences in central tendency(means or medians) or differences across the full empirical distributions.

Usage

NNS.ANOVA(  control,  treatment,  means.only = FALSE,  medians = FALSE,  confidence.interval = 0.95,  tails = "Both",  pairwise = FALSE,  plot = TRUE,  robust = FALSE)

Arguments

control

Numeric vector of control group observations

treatment

Numeric vector of treatment group observations

means.only

Logical;FALSE (default) uses full distribution analysis. SetTRUE for mean-only comparison

medians

Logical;FALSE (default) uses means. SetTRUE for median-based analysis

confidence.interval

Numeric [0,1]; confidence level for effect size bounds (e.g., 0.95)

tails

Character; specifies CI tail(s): "both", "left", or "right"

pairwise

logical;FALSE (default) Returns pairwise certainty tests when set topairwise = TRUE.

plot

Logical;TRUE (default) generates distribution plot

robust

logical;FALSE (default) Generates 100 independent random permutations to test results, and returns / plots 95 percent confidence intervals along with robust central tendency of all results for pairwise analysis only.

Details

The key output is theCertainty metric, a calibrated probability in[0, 1] representing the likelihood that the groups being compared arethe *same* with respect to the chosen comparison mode:

Ifmeans.only = TRUE:Certainty is the probability thatthe groupmeans (or medians, ifmedians = TRUE) are the same.
Ifmeans.only = FALSE:Certainty is the probability thatthe twoentire distributions are the same.

This makesCertainty the conceptual inverse of a classical p-value.A *low* Certainty (e.g., < 0.10) indicates strong evidence of difference,while a *high* Certainty (e.g., > 0.90) indicates strong evidence of similarity.

Value

Returns a list containing:

Control_Statistic: Mean/median of control group
Treatment_Statistic: Mean/median of treatment group
Grand_Statistic: Grand mean/median
Control_CDF: CDF value at grand statistic (control)
Treatment_CDF: CDF value at grand statistic (treatment)
Certainty: Probability that the groups are thesame(means-only or full distribution depending onmeans.only).
Effect_Size_LB: Lower bound of treatment effect (if CI requested)
Effect_Size_UB: Upper bound of treatment effect (if CI requested)
Confidence_Level: Confidence level used (if CI requested)

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Viole, F. (2017) "Continuous CDFs and ANOVA with NNS"doi:10.2139/ssrn.3007373

Examples

 ## Not run: ### Binary analysis and effect sizeset.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.ANOVA(control = x, treatment = y)### Two variable analysis with no control variableA <- cbind(x, y)NNS.ANOVA(A)### Medians testNNS.ANOVA(A, means.only = TRUE, medians = TRUE)### Multiple variable analysis with no control variableset.seed(123)x <- rnorm(100) ; y <- rnorm(100) ; z <- rnorm(100)A <- cbind(x, y, z)NNS.ANOVA(A)### Different length vectors used in a listx <- rnorm(30) ; y <- rnorm(40) ; z <- rnorm(50)A <- list(x, y, z)NNS.ANOVA(A)## End(Not run)

NNS ARMA

Description

Autoregressive model incorporating nonlinear regressions of component series.

Usage

NNS.ARMA(  variable,  h = 1,  training.set = NULL,  seasonal.factor = TRUE,  weights = NULL,  best.periods = 1,  modulo = NULL,  mod.only = TRUE,  negative.values = FALSE,  method = "nonlin",  dynamic = FALSE,  shrink = FALSE,  plot = TRUE,  seasonal.plot = TRUE,  pred.int = NULL)

Arguments

variable

a numeric vector.

h

integer; 1 (default) Number of periods to forecast.

training.set

numeric;NULL (default) Sets the number of variable observations

(variable[1 : training.set]) to monitor performance of forecast over in-sample range.

seasonal.factor

logical or integer(s);TRUE (default) Automatically selects the best seasonal lag from the seasonality test. To use weighted average of all seasonal lags set to(seasonal.factor = FALSE). Otherwise, directly input known frequency integer lag to use, i.e.(seasonal.factor = 12) for monthly data. Multiple frequency integers can also be used, i.e.(seasonal.factor = c(12, 24, 36))

weights

numeric or"equal";NULL (default) sets the weights of theseasonal.factor vector when specified as integers. If(weights = NULL) eachseasonal.factor is weighted on itsNNS.seas result and number of observations it contains, else an"equal" weight is used.

best.periods

integer; [2] (default) used in conjunction with(seasonal.factor = FALSE), uses thebest.periods number of detected seasonal lags instead ofALL lags when(seasonal.factor = FALSE, best.periods = NULL).

modulo

integer(s); NULL (default) Used to find the nearest multiple(s) in the reported seasonal period.

mod.only

logical;TRUE (default) Limits the number of seasonal periods returned to the specifiedmodulo.

negative.values

logical;FALSE (default) If the variable can be negative, set to(negative.values = TRUE). If there are negative values within the variable,negative.values will automatically be detected.

method

options: ("lin", "nonlin", "both", "means");"nonlin" (default) To select the regression type of the component series, select(method = "both") where both linear and nonlinear estimates are generated. To use a nonlinear regression, set to(method = "nonlin"); to use a linear regression set to(method = "lin"). Means for each subset are returned with(method = "means").

dynamic

logical;FALSE (default) To update the seasonal factor with each forecast point, set to(dynamic = TRUE). The default is(dynamic = FALSE) to retain the original seasonal factor from the inputted variable for all ensuingh.

shrink

logical;FALSE (default) Ensembles forecasts withmethod = "means".

plot

logical;TRUE (default) Returns the plot of all periods exhibiting seasonality and thevariable level reference in upper panel. Lower panel returns original data and forecast.

seasonal.plot

logical;TRUE (default) Adds the seasonality plot above the forecast. Will be set toFALSE if no seasonality is detected orseasonal.factor is set to an integer value.

pred.int

numeric [0, 1];NULL (default) Plots and returns the associated prediction intervals for the final estimate. Constructed using the maximum entropy bootstrapNNS.meboot on the final estimates.

Value

Returns a vector of forecasts of length(h) if nopred.int specified. Else, returns adata.table with the forecasts as well as lower and upper prediction intervals per forecast point.

Note

For monthly data series, increased accuracy may be realized from forcing seasonal factors to multiples of 12. For example, if the best periods reported are: {37, 47, 71, 73} use(seasonal.factor = c(36, 48, 72)).

(seasonal.factor = FALSE) can be a very computationally expensive exercise due to the number of seasonal periods detected.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Viole, F. (2019) "Forecasting Using NNS"doi:10.2139/ssrn.3382300

Examples

## Nonlinear NNS.ARMA using AirPassengers monthly data and 12 period lag## Not run: NNS.ARMA(AirPassengers, h = 45, training.set = 100, seasonal.factor = 12, method = "nonlin")## Linear NNS.ARMA using AirPassengers monthly data and 12, 24, and 36 period lagsNNS.ARMA(AirPassengers, h = 45, training.set = 120, seasonal.factor = c(12, 24, 36), method = "lin")## Nonlinear NNS.ARMA using AirPassengers monthly data and 2 best periods lagNNS.ARMA(AirPassengers, h = 45, training.set = 120, seasonal.factor = FALSE, best.periods = 2)## End(Not run)

NNS ARMA Optimizer

Description

Wrapper function for optimizing any combination of a givenseasonal.factor vector inNNS.ARMA. Minimum sum of squared errors (forecast-actual) is used to determine optimum across allNNS.ARMA methods.

Usage

NNS.ARMA.optim(  variable,  h = NULL,  training.set = NULL,  seasonal.factor,  lin.only = FALSE,  negative.values = FALSE,  obj.fn = expression(mean((predicted - actual)^2)/(NNS::Co.LPM(1, predicted, actual,    target_x = mean(predicted), target_y = mean(actual)) + NNS::Co.UPM(1, predicted,    actual, target_x = mean(predicted), target_y = mean(actual)))),  objective = "min",  linear.approximation = TRUE,  ncores = NULL,  pred.int = 0.95,  print.trace = TRUE,  plot = FALSE)

Arguments

variable

a numeric vector.

h

integer;NULL (default) Number of periods to forecast out of sample. IfNULL,h = length(variable) - training.set.

training.set

integer;NULL (default) Sets the number of variable observations as the training set. SeeNote below for recommended uses.

seasonal.factor

integers; Multiple frequency integers considered forNNS.ARMA model, i.e.(seasonal.factor = c(12, 24, 36)).

lin.only

logical;FALSE (default) For fast optimization of the linear regression method. More robust thanlin.only = TRUE.

negative.values

logical;FALSE (default) If the variable can be negative, set to(negative.values = TRUE). It will automatically select(negative.values = TRUE) if the minimum value of thevariable is negative.

obj.fn

expression;expression(cor(predicted, actual, method = "spearman") / sum((predicted - actual)^2)) (default) Rank correlation / sum of squared errors is the default objective function. Anyexpression(...) using the specific termspredicted andactual can be used.

objective

options: ("min", "max")"max" (default) Select whether to minimize or maximize the objective functionobj.fn.

linear.approximation

logical;TRUE (default) Uses the best linear output fromNNS.reg to generate a nonlinear and mixture regression for comparison.FALSE is a more exhaustive search over the objective space.

ncores

integer; value specifying the number of cores to be used in the parallelized procedure. If NULL (default), the number of cores to be used is equal to the number of cores of the machine - 1.

pred.int

numeric [0, 1]; 0.95 (default) Returns the associated prediction intervals for the final estimate. Constructed using the maximum entropy bootstrapNNS.meboot on the final estimates.

print.trace

logical;TRUE (default) Prints current iteration information. Suggested as backup in case of error, best parameters to that point still known and copyable!

plot

logical;FALSE (default)

Value

Returns a list containing:

$period a vector of optimal seasonal periods
$weights the optimal weights of each seasonal period between an equal weight or NULL weighting
$obj.fn the objective function value
$method the method identifying whichNNS.ARMA method was used.
$shrink whether to use theshrink parameter inNNS.ARMA.
$nns.regress whether to smooth the variable viaNNS.reg before forecasting.
$bias.shift a numerical result of the overall bias of the optimum objective function result. To be added to the final result when using theNNS.ARMA with the derived parameters.
$errors a vector of model errors from internal calibration.
$results a vector of lengthh.
$lower.pred.int a vector of lower prediction intervals per forecast point.
$upper.pred.int a vector of upper prediction intervals per forecast point.

Note

Typically,(training.set = 0.8 * length(variable)) is used for optimization. Smaller samples could use(training.set = 0.9 * length(variable)) (or larger) in order to preserve information.
The number of combinations will grow prohibitively large, they should be kept as small as possible.seasonal.factor containing an element too large will result in an error. Please reduce the maximumseasonal.factor.
Set(ncores = 1) if routine is used within a parallel architecture.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Nonlinear NNS.ARMA period optimization using 2 yearly lags on AirPassengers monthly data## Not run: nns.optims <- NNS.ARMA.optim(AirPassengers[1:132], training.set = 120,seasonal.factor = seq(12, 24, 6))## To predict out of sample using best parameters:NNS.ARMA.optim(AirPassengers[1:132], h = 12, seasonal.factor = seq(12, 24, 6))## Incorporate any objective function from external packages (such as \code{Metrics::mape})NNS.ARMA.optim(AirPassengers[1:132], h = 12, seasonal.factor = seq(12, 24, 6),obj.fn = expression(Metrics::mape(actual, predicted)), objective = "min")## End(Not run)

NNS CDF

Description

This function generates an empirical CDF using partial moment ratiosLPM.ratio, and resulting survival, hazard and cumulative hazard functions.

Usage

NNS.CDF(variable, degree = 0, target = NULL, type = "CDF", plot = TRUE)

Arguments

variable

a numeric vector or data.frame of >= 2 variables for joint CDF.

degree

numeric;(degree = 0) (default) is frequency,(degree = 1) is area.

target

numeric;NULL (default) Must lie within support of each variable.

type

options("CDF", "survival", "hazard", "cumulative hazard");"CDF" (default) Selects type of function to return for bi-variate analysis. Multivariate analysis is restricted to"CDF".

plot

logical; plots CDF.

Value

Returns:

"Function" a data.table containing the observations and resulting CDF of the variable.
"target.value" value from thetarget argument.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Viole, F. (2017) "Continuous CDFs and ANOVA with NNS"doi:10.2139/ssrn.3007373

Examples

## Not run: set.seed(123)x <- rnorm(100)NNS.CDF(x)## Empirical CDF (degree = 0)NNS.CDF(x)## Continuous CDF (degree = 1)NNS.CDF(x, 1)## Joint CDFx <- rnorm(5000) ; y <- rnorm(5000)A <- cbind(x,y)NNS.CDF(A, 0)## Joint CDF with targetNNS.CDF(A, 0, target = rep(0, ncol(A)))## End(Not run)

NNS FSD Test

Description

Bi-directional test of first degree stochastic dominance using lower partial moments.

Usage

NNS.FSD(x, y, type = "discrete", plot = TRUE)

Arguments

x

a numeric vector.

y

a numeric vector.

type

options: ("discrete", "continuous");"discrete" (default) selects the type of CDF.

plot

logical;TRUE (default) plots the FSD test.

Value

Returns one of the following FSD results:"X FSD Y","Y FSD X", or"NO FSD EXISTS".

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012.

Viole, F. (2017) "A Note on Stochastic Dominance."doi:10.2139/ssrn.3002675.

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.FSD(x, y)## End(Not run)

NNS FSD Test uni-directional

Description

Uni-directional test of first degree stochastic dominance using lower partial moments used in SD Efficient Set routine.

Usage

NNS.FSD.uni(x, y, type = "discrete")

Arguments

x

a numeric vector.

y

a numeric vector.

type

options: ("discrete", "continuous");"discrete" (default) selects the type of CDF.

Value

Returns (1) if"X FSD Y", else (0).

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012

Viole, F. (2017) "A Note on Stochastic Dominance."doi:10.2139/ssrn.3002675

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.FSD.uni(x, y)## End(Not run)

NNS Monte Carlo Sampling

Description

Monte Carlo sampling from the maximum entropy bootstrap routineNNS.meboot, ensuring the replicates are sampled from the full [-1,1] correlation space.

Usage

NNS.MC(  x,  reps = 30,  lower_rho = -1,  upper_rho = 1,  by = 0.01,  exp = 1,  type = "spearman",  drift = TRUE,  target_drift = NULL,  target_drift_scale = NULL,  xmin = NULL,  xmax = NULL,  ...)

Arguments

x

vector of data.

reps

numeric; number of replicates to generate,30 default.

lower_rho

numeric[-1,1];.01 default will set thefrom argument inseq(from, to, by).

upper_rho

numeric[-1,1];.01 default will set theto argument inseq(from, to, by).

by

numeric;.01 default will set theby argument inseq(-1, 1, step).

exp

numeric;1 default will exponentially weight maximum rho value ifexp > 1. Shrinks values towardsupper_rho.

type

options("spearman", "pearson", "NNScor", "NNSdep");type = "spearman"(default) dependence metric desired.

drift

logical;drift = TRUE (default) preserves the drift of the original series.

target_drift

numerical;target_drift = NULL (default) Specifies the desired drift whendrift = TRUE, i.e. a risk-free rate of return.

target_drift_scale

numerical; instead of calculating atarget_drift, provide a scalar to the existing drift whendrift = TRUE.

xmin

numeric; the lower limit for the left tail.

xmax

numeric; the upper limit for the right tail.

...

possible additional arguments to be passed toNNS.meboot.

Value

ensemble average observation over all replicates as a vector.
replicates maximum entropy bootstrap replicates as a list for eachrho.

References

Vinod, H.D. and Viole, F. (2020) Arbitrary Spearman's Rank Correlations in Maximum Entropy Bootstrap and Improved Monte Carlo Simulations.doi:10.2139/ssrn.3621614

Examples

## Not run: # To generate a set of MC sampled time-series to AirPassengersMC_samples <- NNS.MC(AirPassengers, reps = 10, lower_rho = -1, upper_rho = 1, by = .5, xmin = 0)## End(Not run)

NNS SD-based Clustering

Description

Clusters a set of variables by iteratively extracting Stochastic Dominance (SD)-efficient sets,subject to a minimum cluster size.

Usage

NNS.SD.cluster(  data,  degree = 1,  type = "discrete",  min_cluster = 1,  dendrogram = FALSE)

Arguments

data

A numeric matrix or data frame of variables to be clustered.

degree

Numeric options: (1, 2, 3). Degree of stochastic dominance test.

type

Character, either"discrete" (default) or"continuous"; specifies the type of CDF.

min_cluster

Integer. The minimum number of elements required for a valid cluster.

dendrogram

Logical;FALSE (default). IfTRUE, a dendrogram is produced based on a simple "distance" measure between clusters.

Details

The function appliesNNS.SD.efficient.set iteratively, peeling off the SD-efficient set at each stepif it meets or exceedsmin_cluster in size, until no more subsets can be extracted or all variables are exhausted.Variables in each SD-efficient set form a cluster, with any remaining variables aggregated into the final cluster if it meetsthemin_cluster threshold.

Value

A list with the following components:

Clusters: A named list of cluster memberships where each element is the set of variable names belonging to that cluster.
Dendrogram (optional): Ifdendrogram = TRUE, anhclust object is also returned.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012.

Viole, F. (2017) "A Note on Stochastic Dominance."doi:10.2139/ssrn.3002675

Examples

## Not run: set.seed(123)x <- rnorm(100)y <- rnorm(100)z <- rnorm(100)A <- cbind(x, y, z)# Perform SD-based clustering (degree 1), requiring at least 2 elements per clusterresults <- NNS.SD.cluster(data = A, degree = 1, min_cluster = 2)print(results$Clusters)# Produce a dendrogram as wellresults_with_dendro <- NNS.SD.cluster(data = A, degree = 1, min_cluster = 2, dendrogram = TRUE)## End(Not run)

NNS SD Efficient Set

Description

Determines the set of stochastic dominant variables for various degrees.

Usage

NNS.SD.efficient.set(x, degree, type = "discrete", status = TRUE)

Arguments

x

a numeric matrix or data frame.

degree

numeric options: (1, 2, 3); Degree of stochastic dominance test from (1, 2 or 3).

type

options: ("discrete", "continuous");"discrete" (default) selects the type of CDF.

status

logical;TRUE (default) Prints status update message in console.

Value

Returns set of stochastic dominant variable names.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012.

Viole, F. (2017) "A Note on Stochastic Dominance."doi:10.2139/ssrn.3002675

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y<-rnorm(100) ; z<-rnorm(100)A <- cbind(x, y, z)NNS.SD.efficient.set(A, 1)## End(Not run)

NNS SSD Test

Description

Bi-directional test of second degree stochastic dominance using lower partial moments.

Usage

NNS.SSD(x, y, plot = TRUE)

Arguments

x

a numeric vector.

y

a numeric vector.

plot

logical;TRUE (default) plots the SSD test.

Value

Returns one of the following SSD results:"X SSD Y","Y SSD X", or"NO SSD EXISTS".

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012.

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.SSD(x, y)## End(Not run)

NNS SSD Test uni-directional

Description

Uni-directional test of second degree stochastic dominance using lower partial moments used in SD Efficient Set routine.

Usage

NNS.SSD.uni(x, y)

Arguments

x

a numeric vector.

y

a numeric vector.

Value

Returns (1) if"X SSD Y", else (0).

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012.

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.SSD.uni(x, y)## End(Not run)

NNS TSD Test

Description

Bi-directional test of third degree stochastic dominance using lower partial moments.

Usage

NNS.TSD(x, y, plot = TRUE)

Arguments

x

a numeric vector.

y

a numeric vector.

plot

logical;TRUE (default) plots the TSD test.

Value

Returns one of the following TSD results:"X TSD Y","Y TSD X", or"NO TSD EXISTS".

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012.

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.TSD(x, y)## End(Not run)

NNS TSD Test uni-directional

Description

Uni-directional test of third degree stochastic dominance using lower partial moments used in SD Efficient Set routine.

Usage

NNS.TSD.uni(x, y)

Arguments

x

a numeric vector.

y

a numeric vector.

Value

Returns (1) if"X TSD Y", else (0).

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2016) "LPM Density Functions for the Computation of the SD Efficient Set." Journal of Mathematical Finance, 6, 105-126.doi:10.4236/jmf.2016.61012.

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.TSD.uni(x, y)## End(Not run)

NNS VAR

Description

Nonparametric vector autoregressive model incorporatingNNS.ARMA estimates of variables intoNNS.reg for a multi-variate time-series forecast.

Usage

NNS.VAR(  variables,  h,  tau = 1,  dim.red.method = "cor",  naive.weights = TRUE,  obj.fn = expression(mean((predicted - actual)^2)/(NNS::Co.LPM(1, predicted, actual,    target_x = mean(predicted), target_y = mean(actual)) + NNS::Co.UPM(1, predicted,    actual, target_x = mean(predicted), target_y = mean(actual)))),  objective = "min",  status = TRUE,  ncores = NULL,  nowcast = FALSE)

Arguments

variables

a numeric matrix or data.frame of contemporaneous time-series to forecast.

h

integer; 1 (default) Number of periods to forecast.(h = 0) will return just the interpolated and extrapolated values.

tau

positive integer [ > 0]; 1 (default) Number of lagged observations to consider for the time-series data. Vector for single lag for each respective variable or list for multiple lags per each variable.

dim.red.method

options: ("cor", "NNS.dep", "NNS.caus", "all") method for reducing regressors viaNNS.stack.(dim.red.method = "cor") (default) uses standard linear correlation for dimension reduction in the lagged variable matrix.(dim.red.method = "NNS.dep") usesNNS.dep for nonlinear dependence weights, while(dim.red.method = "NNS.caus") usesNNS.caus for causal weights.(dim.red.method = "all") averages all methods for further feature engineering.

naive.weights

logical;TRUE (default) Equal weights applied to univariate and multivariate outputs in ensemble.FALSE will apply weights based on the number of relevant variables detected.

obj.fn

expression;expression(mean((predicted - actual)^2)) / (Sum of NNS Co-partial moments) (default) MSE / co-movements is the default objective function. Anyexpression(...) using the specific termspredicted andactual can be used.

objective

options: ("min", "max")"min" (default) Select whether to minimize or maximize the objective functionobj.fn.

status

logical;TRUE (default) Prints status update message in console.

ncores

integer; value specifying the number of cores to be used in the parallelized subroutineNNS.ARMA.optim. If NULL (default), the number of cores to be used is equal to the number of cores of the machine - 1.

nowcast

logical;FALSE (default) internal call forNNS.nowcast.

Value

Returns the following matrices of forecasted variables:

"interpolated_and_extrapolated" Returns adata.frame of the linear interpolated andNNS.ARMA extrapolated values to replaceNA values in the originalvariables argument. This is required for working with variables containing different frequencies, e.g. whereNA would be reported for intra-quarterly data when indexed with monthly periods.
"relevant_variables" Returns the relevant variables from the dimension reduction step.
"univariate" Returns the univariateNNS.ARMA forecasts.
"multivariate" Returns the multi-variateNNS.reg forecasts.
"ensemble" Returns the ensemble of both"univariate" and"multivariate" forecasts.

Note

"Error in { : task xx failed -}" should be re-run withNNS.VAR(..., ncores = 1).
Not recommended for factor variables, even after transformed to numeric.NNS.reg is better suited for factor or binary regressor extrapolation.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Viole, F. (2019) "Multi-variate Time-Series Forecasting: Nonparametric Vector Autoregression Using NNS"doi:10.2139/ssrn.3489550

Viole, F. (2020) "NOWCASTING with NNS"doi:10.2139/ssrn.3589816

Viole, F. (2019) "Forecasting Using NNS"doi:10.2139/ssrn.3382300

Vinod, H. and Viole, F. (2017) "Nonparametric Regression Using Clusters"doi:10.1007/s10614-017-9713-5

Vinod, H. and Viole, F. (2018) "Clustering and Curve Fitting by Line Segments"doi:10.20944/preprints201801.0090.v1

Examples

 ## Not run:  #################################################### ### Standard Nonparametric Vector Autoregression ### #################################################### set.seed(123) x <- rnorm(100) ; y <- rnorm(100) ; z <- rnorm(100) A <- cbind(x = x, y = y, z = z) ### Using lags 1:4 for each variable NNS.VAR(A, h = 12, tau = 4, status = TRUE) ### Using lag 1 for variable 1, lag 3 for variable 2 and lag 3 for variable 3 NNS.VAR(A, h = 12, tau = c(1,3,3), status = TRUE) ### Using lags c(1,2,3) for variables 1 and 3, while using lags c(4,5,6) for variable 2 NNS.VAR(A, h = 12, tau = list(c(1,2,3), c(4,5,6), c(1,2,3)), status = TRUE) ### PREDICTION INTERVALS # Store NNS.VAR output nns_estimate <- NNS.VAR(A, h = 12, tau = 4, status = TRUE) # Create bootstrap replicates using NNS.meboot replicates <- NNS.meboot(nns_estimate$ensemble[,1], rho = seq(-1,1,.25))["replicates",] replicates <- do.call(cbind, replicates) # Apply UPM.VaR and LPM.VaR for desired prediction interval...95 percent illustrated # Tail percentage used in first argument per {LPM.VaR} and {UPM.VaR} functions lower_CIs <- apply(replicates, 1, function(z) LPM.VaR(0.025, 0, z)) upper_CIs <- apply(replicates, 1, function(z) UPM.VaR(0.025, 0, z)) # View results cbind(nns_estimate$ensemble[,1], lower_CIs, upper_CIs) ######################################### ### NOWCASTING with Mixed Frequencies ### ######################################### library(Quandl) econ_variables <- Quandl(c("FRED/GDPC1", "FRED/UNRATE", "FRED/CPIAUCSL"),type = 'ts',                          order = "asc", collapse = "monthly", start_date = "2000-01-01") ### Note the missing values that need to be imputed head(econ_variables) tail(econ_variables) NNS.VAR(econ_variables, h = 12, tau = 12, status = TRUE) ## End(Not run)

NNS Boost

Description

Ensemble method for classification using the NNS multivariate regressionNNS.reg as the base learner instead of trees.

Usage

NNS.boost(  IVs.train,  DV.train,  IVs.test = NULL,  type = NULL,  depth = NULL,  learner.trials = 100,  epochs = NULL,  CV.size = NULL,  balance = FALSE,  ts.test = NULL,  folds = 5,  threshold = NULL,  obj.fn = expression(sum((predicted - actual)^2)),  objective = "min",  extreme = FALSE,  features.only = FALSE,  feature.importance = TRUE,  pred.int = NULL,  status = TRUE)

Arguments

IVs.train

a matrix or data frame of variables of numeric or factor data types.

DV.train

a numeric or factor vector with compatible dimensions to(IVs.train).

IVs.test

a matrix or data frame of variables of numeric or factor data types with compatible dimensions to(IVs.train). If NULL, will use(IVs.train) as default.

type

depth

options: (integer, NULL, "max");(depth = NULL)(default) Specifies theorder parameter in theNNS.reg routine, assigning a number of splits in the regressors, analogous to tree depth.

learner.trials

integer; 100 (default) Sets the number of trials to obtain an accuracythreshold level. If the number of all possible feature combinations is less than selected value, the minimum of the two values will be used.

epochs

integer;2*length(DV.train) (default) Total number of feature combinations to run.

CV.size

numeric [0, 1];NULL (default) Sets the cross-validation size. Defaults to a random value between 0.2 and 0.33 for a random sampling of the training set.

balance

logical;FALSE (default) Uses both up and down sampling to balance the classes.type="CLASS" required.

ts.test

integer; NULL (default) Sets the length of the test set for time-series data; typically2*h parameter value fromNNS.ARMA or double known periods to forecast.

folds

integer; 5 (default) Sets the number offolds in theNNS.stack procedure for optimaln.best parameter.

threshold

numeric;NULL (default) Sets theobj.fn threshold to keep feature combinations.

obj.fn

expression;expression( sum((predicted - actual)^2) ) (default) Sum of squared errors is the default objective function. Anyexpression(...) using the specific termspredicted andactual can be used. Automatically selects an accuracy measure when(type = "CLASS").

objective

options: ("min", "max")"max" (default) Select whether to minimize or maximize the objective functionobj.fn.

extreme

logical;FALSE (default) Uses the maximum (minimum)threshold obtained from thelearner.trials, rather than the upper (lower) quintile level for maximization (minimization)objective.

features.only

logical;FALSE (default) Returns only the final feature loadings along with the final feature frequencies.

feature.importance

logical;TRUE (default) Plots the frequency of features used in the final estimate.

pred.int

numeric [0,1];NULL (default) Returns the associated prediction intervals for the final estimate.

status

logical;TRUE (default) Prints status update message in console.

Value

Returns a vector of fitted values for the dependent variable test set$results, prediction intervals$pred.int, and the final feature loadings$feature.weights, along with final feature frequencies$feature.frequency.

Note

Like a logistic regression, the(type = "CLASS") setting is not necessary for target variable of two classes e.g. [0, 1]. The response variable base category should be 1 for classification problems.
Incorporate any objective function from external packages (such asMetrics::mape) viaNNS.boost(..., obj.fn = expression(Metrics::mape(actual, predicted)), objective = "min")

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. (2016) "Classification Using NNS Clustering Analysis"doi:10.2139/ssrn.2864711

Examples

 ## Using 'iris' dataset where test set [IVs.test] is 'iris' rows 141:150. ## Not run:  a <- NNS.boost(iris[1:140, 1:4], iris[1:140, 5], IVs.test = iris[141:150, 1:4], epochs = 100, learner.trials = 100, type = "CLASS", depth = NULL, balance = TRUE) ## Test accuracy mean(a$results == as.numeric(iris[141:150, 5])) ## End(Not run)

NNS Causation

Description

Returns the causality from observational data between two variables.

Usage

NNS.caus(  x,  y = NULL,  factor.2.dummy = FALSE,  tau = 0,  plot = FALSE,  p.value = FALSE,  nperm = 100L,  permute = c("y", "x", "both"),  seed = NULL,  conf.int = 0.95)

Arguments

x

a numeric vector, matrix or data frame.

y

NULL (default) or a numeric vector with compatible dimensions tox.

factor.2.dummy

logical;FALSE (default) Automatically augments variable matrix with numerical dummy variables based on the levels of factors. Includes dependent variabley.

tau

options: ("cs", "ts", integer); 0 (default) Number of lagged observations to consider (for time series data). Otherwise, set(tau = "cs") for cross-sectional data.(tau = "ts") automatically selects the lag of the time series data, while(tau = [integer]) specifies a time series lag.

plot

logical;FALSE (default) Plots the raw variables, tau normalized, and cross-normalized variables.

p.value

logical;FALSE (default) IfTRUE, runs a permutation test to compute empirical p-values for the signed causation from x -> y.

nperm

integer; number of permutations to use whenp.value = TRUE. Default 100.

permute

one of "both", "y", or "x"; which variable(s) to shuffle when constructing the null distribution.

seed

optional integer seed for reproducibility of the permutation test.

conf.int

numeric; 0.95 (default) confidence level for the partial-moment based interval computed on the permutation null distribution.

Value

Ifp.value=FALSE returns the original causation vector of length 3 (directional given/received and net), named either "C(x—>y)" or "C(y—>x)" in the third slot. Ifp.value=TRUE returns a list with components:*causation: the original causation vector as above.*p.value: a list with empirical two-sided and one-sided p-values (x_causes_y, y_causes_x), the null distribution, the observed signed statistic, and metadata (permute, nperm).Ifp.value=TRUE for a matrix, the function returns a list with components:*causality: the causality matrix.*lower_CI: matrix of lower confidence bounds (partial-moment based).*upper_CI: matrix of upper confidence bounds (partial-moment based).*p.value: matrix of empirical two-sided p-values.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: ## x causes y...set.seed(123)x <- rnorm(1000) ; y <- x ^ 2NNS.caus(x, y, tau = "cs")## Causal matrix without per factor causationNNS.caus(iris, tau = 0)## Causal matrix with per factor causationNNS.caus(iris, factor.2.dummy = TRUE, tau = 0)## End(Not run)

NNS Co-Partial Moments Higher Dimension Dependence

Description

Determines higher dimension dependence coefficients based on co-partial moment matrices ratios.

Usage

NNS.copula(  X,  target = NULL,  continuous = TRUE,  plot = FALSE,  independence.overlay = FALSE)

Arguments

X

a numeric matrix or data frame.

target

numeric; Typically the mean of Variable X for classical statistics equivalences, but does not have to be. (Vectorized)(target = NULL) (default) will set the target as the mean of every variable.

continuous

logical;TRUE (default) Generates a continuous measure using degree 1PM.matrix, while discreteFALSE uses degree 0PM.matrix.

plot

logical;FALSE (default) Generates a 3d scatter plot with regression points.

independence.overlay

logical;FALSE (default) Creates and overlays independentCo.LPM andCo.UPM regions to visually reference the difference in dependence from the data.frame of variables being analyzed. Under independence, the light green and red shaded areas would be occupied by green and red data points respectively.

Value

Returns a multivariate dependence value [0,1].

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. (2016) "Beyond Correlation: Using the Elements of Variance for Conditional Means and Probabilities"doi:10.2139/ssrn.2745308.

Examples

## Not run: set.seed(123)x <- rnorm(1000) ; y <- rnorm(1000) ; z <- rnorm(1000)A <- data.frame(x, y, z)NNS.copula(A, target = colMeans(A), plot = TRUE, independence.overlay = TRUE)### Target 0NNS.copula(A, target = rep(0, ncol(A)), plot = TRUE, independence.overlay = TRUE)## End(Not run)

NNS Dependence

Description

Returns the dependence and nonlinear correlation between two variables based on higher order partial moment matrices measured by frequency or area.

Usage

NNS.dep(x, y = NULL, asym = FALSE, p.value = FALSE, print.map = FALSE)

Arguments

x

a numeric vector, matrix or data frame.

y

NULL (default) or a numeric vector with compatible dimensions tox.

asym

logical;FALSE (default) Allows for asymmetrical dependencies.

p.value

logical;FALSE (default) Generates 100 independent random permutations to test results against and plots 95 percent confidence intervals along with all results.

print.map

logical;FALSE (default) Plots quadrant means, or p-value replicates.

Value

Returns the bi-variate"Correlation" and"Dependence" or correlation / dependence matrix for matrix input.

Note

For asymmetrical(asym = TRUE) matrices, directional dependence is returned as ([column variable] —> [row variable]).

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.dep(x, y)## Correlation / Dependence Matrixx <- rnorm(100) ; y <- rnorm(100) ; z <- rnorm(100)B <- cbind(x, y, z)NNS.dep(B)## End(Not run)

NNS Numerical Differentiation

Description

Determines numerical derivative of a given univariate function using projected secant lines on the y-axis. These projected points infer finite stepsh, in the finite step method.

Usage

NNS.diff(f, point, h = 0.1, tol = 1e-10, digits = 12, print.trace = FALSE)

Arguments

f

an expression or call or a formula with no lhs.

point

numeric; Point to be evaluated for derivative of a given functionf.

h

numeric [0, ...]; Initial step for secant projection. Defaults to(h = 0.1).

tol

numeric; Sets the tolerance for the stopping condition of the inferredh. Defaults to(tol = 1e-10).

digits

numeric; Sets the number of digits specification of the output. Defaults to(digits = 12).

print.trace

logical;FALSE (default) Displays each iteration, lower y-intercept, upper y-intercept and inferredh.

Value

Returns a matrix of values, intercepts, derivatives, inferred step sizes for multiple methods of estimation.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: f <- function(x) sin(x) / xNNS.diff(f, 4.1)## End(Not run)

NNS Distance

Description

Internal kernel function for NNS multivariate regressionNNS.reg parallel instances.

Usage

NNS.distance(rpm, dist.estimate, k = "all", class = NULL)

Arguments

rpm

REGRESSION.POINT.MATRIX fromNNS.reg

dist.estimate

Vector to generate distances from.

k

n.best fromNNS.reg

class

if classification problem.

Value

Returns sum of weighted distances.

NNS gravity

Description

Alternative central tendency measure more robust to outliers.

Usage

NNS.gravity(x, discrete = FALSE)

Arguments

x

vector of data.

discrete

logical;FALSE (default) for discrete distributions.

Value

Returns a numeric value representing the central tendency of the distribution.

Author(s)

Fred Viole, OVVO Financial Systems

Examples

## Not run: set.seed(123)x <- rnorm(100)NNS.gravity(x)## End(Not run)

NNS meboot

Description

Adapted maximum entropy bootstrap routine frommeboothttps://cran.r-project.org/package=meboot.

Usage

NNS.meboot(  x,  reps = 999,  rho = NULL,  type = "spearman",  drift = TRUE,  target_drift = NULL,  target_drift_scale = NULL,  trim = 0.1,  xmin = NULL,  xmax = NULL,  reachbnd = TRUE,  expand.sd = TRUE,  force.clt = TRUE,  scl.adjustment = FALSE,  sym = FALSE,  elaps = FALSE,  digits = 6,  colsubj,  coldata,  coltimes,  ...)

Arguments

x

vector of data.

reps

numeric; number of replicates to generate.

rho

numeric [-1,1] (vectorized); Arho must be provided, otherwise a blank list will be returned.

type

options("spearman", "pearson", "NNScor", "NNSdep");type = "spearman"(default) dependence metric desired.

drift

logical;drift = TRUE (default) preserves the drift of the original series.

target_drift

numerical;target_drift = NULL (default) Specifies the desired drift whendrift = TRUE, i.e. a risk-free rate of return.

target_drift_scale

numerical; instead of calculating atarget_drift, provide a scalar to the existing drift whendrift = TRUE.

trim

numeric [0,1]; The mean trimming proportion, defaults totrim = 0.1.

xmin

numeric; the lower limit for the left tail.

xmax

numeric; the upper limit for the right tail.

reachbnd

logical; IfTRUE potentially reached bounds (xmin = smallest value - trimmed mean andxmax = largest value + trimmed mean) are given when the random draw happens to be equal to 0 and 1, respectively.

expand.sd

logical; IfTRUE the standard deviation in the ensemble is expanded. Seeexpand.sd inmeboot::meboot.

force.clt

logical; IfTRUE the ensemble is forced to satisfy the central limit theorem. Seeforce.clt inmeboot::meboot.

scl.adjustment

logical; IfTRUE scale adjustment is performed to ensure that the population variance of the transformed series equals the variance of the data.

sym

logical; IfTRUE an adjustment is performed to ensure that the ME density is symmetric.

elaps

logical; IfTRUE elapsed time during computations is displayed.

digits

integer; 6 (default) number of digits to round output to.

colsubj

numeric; the column inx that contains the individual index. It is ignored if the input datax is not apdata.frame object.

coldata

numeric; the column inx that contains the data of the variable to create the ensemble. It is ignored if the input datax is not apdata.frame object.

coltimes

numeric; an optional argument indicating the column that contains the times at which the observations for each individual are observed. It is ignored if the input dataxis not apdata.frame object.

...

possible argumentfiv to be passed toexpand.sd.

Value

Returns the following row names in a matrix:

x original data provided as input.
replicates maximum entropy bootstrap replicates.
ensemble average observation over all replicates.
xx sorted order stats (xx[1] is minimum value).
z class intervals limits.
dv deviations of consecutive data values.
dvtrim trimmed mean of dv.
xmin data minimum for ensemble=xx[1]-dvtrim.
xmax data x maximum for ensemble=xx[n]+dvtrim.
desintxb desired interval means.
ordxx ordered x values.
kappa scale adjustment to the variance of ME density.
elaps elapsed time.

Note

Vectorizedrho anddrift parameters will not vectorize both simultaneously. Also, do not specifytarget_drift = NULL.

References

Vinod, H.D. and Viole, F. (2020) Arbitrary Spearman's Rank Correlations in Maximum Entropy Bootstrap and Improved Monte Carlo Simulations.doi:10.2139/ssrn.3621614
Vinod, H.D. (2013), Maximum Entropy Bootstrap Algorithm Enhancements.doi:10.2139/ssrn.2285041
Vinod, H.D. (2006), Maximum Entropy Ensembles for Time Series Inference in Economics,Journal of Asian Economics,17(6), pp. 955-978.
Vinod, H.D. (2004), Ranking mutual funds using unconventional utility theory and stochastic dominance,Journal of Empirical Finance,11(3), pp. 353-377.

Examples

## Not run: # To generate an orthogonal rank correlated time-series to AirPassengersboots <- NNS.meboot(AirPassengers, reps = 100, rho = 0, xmin = 0)# Verify correlation of replicates ensemble to originalcor(boots["ensemble",]$ensemble, AirPassengers, method = "spearman")# Plot all replicatesmatplot(boots["replicates",]$replicates , type = 'l')# Plot ensemblelines(boots["ensemble",]$ensemble, lwd = 3)# Plot originallines(1:length(AirPassengers), AirPassengers, lwd = 3, col = "red")### Vectorized drift with a single rhoboots <- NNS.meboot(AirPassengers, reps = 10, rho = 0, xmin = 0, target_drift = c(1,7))matplot(do.call(cbind, boots["replicates", ]), type = "l")lines(1:length(AirPassengers), AirPassengers, lwd = 3, col = "red")### Vectorized rho with a single target driftboots <- NNS.meboot(AirPassengers, reps = 10, rho = c(0, .5, 1), xmin = 0, target_drift = 3)matplot(do.call(cbind, boots["replicates", ]), type = "l")lines(1:length(AirPassengers), AirPassengers, lwd = 3, col = "red")### Vectorized rho with a single target drift scaleboots <- NNS.meboot(AirPassengers, reps = 10, rho = c(0, .5, 1), xmin = 0, target_drift_scale = 0.5)matplot(do.call(cbind, boots["replicates", ]), type = "l")lines(1:length(AirPassengers), AirPassengers, lwd = 3, col = "red") ## End(Not run)

NNS mode

Description

Mode of a distribution, either continuous or discrete.

Usage

NNS.mode(x, discrete = FALSE, multi = TRUE)

Arguments

x

vector of data.

discrete

logical;FALSE (default) for discrete distributions.

multi

logical;TRUE (default) returns multiple mode values.

Value

Returns a numeric value representing the mode of the distribution.

Author(s)

Fred Viole, OVVO Financial Systems

Examples

## Not run: set.seed(123)x <- rnorm(100)NNS.mode(x)## End(Not run)

NNS moments

Description

This function returns the first 4 moments of the distribution.

Usage

NNS.moments(x, population = TRUE)

Arguments

x

a numeric vector.

population

logical;TRUE (default) Performs the population adjustment. Otherwise returns the sample statistic.

Value

Returns:

"$mean" mean of the distribution.
"$variance" variance of the distribution.
"$skewness" skewness of the distribution.
"$kurtosis" excess kurtosis of the distribution.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: set.seed(123)x <- rnorm(100)NNS.moments(x)## End(Not run)

NNS Normalization

Description

Normalizes a matrix of variables based on nonlinear scaling normalization method.

Usage

NNS.norm(X, linear = FALSE, chart.type = NULL, location = "topleft")

Arguments

X

a numeric matrix or data frame, or a list.

linear

logical;FALSE (default) Performs a linear scaling normalization, resulting in equal means for all variables.

chart.type

options: ("l", "b");NULL (default). Set(chart.type = "l") for line,(chart.type = "b") for boxplot.

location

Sets the legend location within the plot, per thex andy co-ordinates used in base graphicslegend.

Value

Returns adata.frame of normalized values.

Note

Unequal vectors provided in a list will only generatelinear=TRUE normalized values.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)A <- cbind(x, y)NNS.norm(A)### Normalize list of unequal vector lengthsvec1 <- c(1, 2, 3, 4, 5, 6, 7)vec2 <- c(10, 20, 30, 40, 50, 60)vec3 <- c(0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3)vec_list <- list(vec1, vec2, vec3)NNS.norm(vec_list)## End(Not run)

NNS Nowcast

Description

Wrapper function for NNS nowcasting method using the nonparametric vector autoregressionNNS.VAR, and Federal Reserve Nowcasting variables.

Usage

NNS.nowcast(  h = 1,  additional.regressors = NULL,  additional.sources = NULL,  naive.weights = FALSE,  specific.regressors = NULL,  start.date = "2000-01-03",  keep.data = FALSE,  status = TRUE,  ncores = NULL)

Arguments

h

integer;(h = 1) (default) Number of periods to forecast.(h = 0) will return just the interpolated and extrapolated values up to the current month.

additional.regressors

character;NULL (default) add more regressors to the base model. The format must utilize thegetSymbols format for FRED data, else specify the source.

additional.sources

character;NULL (default) specify thesource argument pergetSymbols for eachadditional.regressors specified.

naive.weights

logical;TRUE Equal weights applied to univariate and multivariate outputs in ensemble.FALSE (default) will apply weights based on the number of relevant variables detected.

specific.regressors

integer;NULL (default) Select individual regressors from the base model per Viole (2020) listed in theNote below.

start.date

character;"2000-01-03" (default) Starting date for all data series download.

keep.data

logical;FALSE (default) Keeps downloaded variables in a new environmentNNSdata.

status

logical;TRUE (default) Prints status update message in console.

ncores

Value

Returns the following matrices of forecasted variables:

"interpolated_and_extrapolated" Returns adata.frame of the linear interpolated andNNS.ARMA extrapolated values to replaceNA values in the originalvariables argument. This is required for working with variables containing different frequencies, e.g. whereNA would be reported for intra-quarterly data when indexed with monthly periods.
"relevant_variables" Returns the relevant variables from the dimension reduction step.
"univariate" Returns the univariateNNS.ARMA forecasts.
"multivariate" Returns the multi-variateNNS.reg forecasts.
"ensemble" Returns the ensemble of both"univariate" and"multivariate" forecasts.

Note

Specific regressors include:

PAYEMS – Payroll Employment
JTSJOL – Job Openings
CPIAUCSL – Consumer Price Index
DGORDER – Durable Goods Orders
RSAFS – Retail Sales
UNRATE – Unemployment Rate
HOUST – Housing Starts
INDPRO – Industrial Production
DSPIC96 – Personal Income
BOPTEXP – Exports
BOPTIMP – Imports
TTLCONS – Construction Spending
IR – Import Price Index
CPILFESL – Core Consumer Price Index
PCEPILFE – Core PCE Price Index
PCEPI – PCE Price Index
PERMIT – Building Permits
TCU – Capacity Utilization Rate
BUSINV – Business Inventories
ULCNFB – Unit Labor Cost
IQ – Export Price Index
GACDISA066MSFRBNY – Empire State Mfg Index
GACDFSA066MSFRBPHI – Philadelphia Fed Mfg Index
PCEC96 – Real Consumption Spending
GDPC1 – Real Gross Domestic Product
ICSA – Weekly Unemployment Claims
DGS10 – 10-year Treasury rates
T10Y2Y – 2-10 year Treasury rate spread
WALCL – Total Assets
PALLFNFINDEXM – Global Price Index of All Commodities
FEDFUNDS – Federal Funds Effective Rate
PPIACO – Producer Price Index All Commodities
CIVPART – Labor Force Participation Rate
M2NS – M2 Money Supply
ADPMNUSNERNSA – ADP Payrolls

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Viole, F. (2019) "Multi-variate Time-Series Forecasting: Nonparametric Vector Autoregression Using NNS"doi:10.2139/ssrn.3489550

Viole, F. (2020) "NOWCASTING with NNS"doi:10.2139/ssrn.3589816

Examples

 ## Not run:  ## Interpolates / Extrapolates all variables to current month NNS.nowcast(h = 0)  ## Additional regressors and sources specified NNS.nowcast(h = 0, additional.regressors = c("SPY", "USO"),              additional.sources = c("yahoo", "yahoo"))                            ### PREDICTION INTERVALS  ## Store NNS.nowcast output nns_estimates <- NNS.nowcast(h = 12)             # Create bootstrap replicates using NNS.meboot (GDP Variable) gdp_replicates <- NNS.meboot(nns_estimates$ensemble$GDPC1,                               rho = seq(0,1,.25),                               reps = 100)["replicates",]                               replicates <- do.call(cbind, gdp_replicates)  # Apply UPM.VaR and LPM.VaR for desired prediction interval...95 percent illustrated # Tail percentage used in first argument per {LPM.VaR} and {UPM.VaR} functions lower_GDP_CIs <- apply(replicates, 1, function(z) LPM.VaR(0.025, 0, z)) upper_GDP_CIs <- apply(replicates, 1, function(z) UPM.VaR(0.025, 0, z))  # View results cbind(nns_estimates$ensemble$GDPC1, lower_GDP_CIs, upper_GDP_CIs) ## End(Not run)

NNS Partition Map

Description

Creates partitions based on partial moment quadrant centroids, iteratively assigning identifications to observations based on those quadrants (unsupervised partitional and hierarchial clustering method). Basis for correlation, dependenceNNS.dep, regressionNNS.reg routines.

Usage

NNS.part(  x,  y,  Voronoi = FALSE,  type = NULL,  order = NULL,  obs.req = 8,  min.obs.stop = TRUE,  noise.reduction = "off")

Arguments

x

a numeric vector.

y

a numeric vector with compatible dimensions tox.

Voronoi

logical;FALSE (default) Displays a Voronoi type diagram using partial moment quadrants.

type

NULL (default) Controls the partitioning basis. Set to(type = "XONLY") for X-axis based partitioning. Defaults toNULL for both X and Y-axis partitioning.

order

integer; Number of partial moment quadrants to be generated.(order = "max") will institute a perfect fit.

obs.req

integer; (8 default) Required observations per cluster where quadrants will not be further partitioned if observations are not greater than the entered value. Reduces minimum number of necessary observations in a quadrant to 1 when(obs.req = 1).

min.obs.stop

logical;TRUE (default) Stopping condition where quadrants will not be further partitioned if a single cluster contains less than the entered value ofobs.req.

noise.reduction

the method of determining regression points options for the dependent variabley: ("mean", "median", "mode", "off");(noise.reduction = "mean") uses means for partitions.(noise.reduction = "median") uses medians instead of means for partitions, while(noise.reduction = "mode") uses modes instead of means for partitions. Defaults to(noise.reduction = "off") where an overall central tendency measure is used, which is the default for the independent variablex.

Value

Returns:

"dt" adata.table ofx andy observations with their partition assignment"quadrant" in the 3rd column and their prior partition assignment"prior.quadrant" in the 4th column.
"regression.points" thedata.table of regression points for that given(order = ...).
"order" theorder of the final partition given"min.obs.stop" stopping condition.

Note

min.obs.stop = FALSE will not generate regression points due to unequal partitioning of quadrants from individual cluster observations.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.part(x, y)## Data.table of observations and partitionsNNS.part(x, y, order = 1)$dt## Regression pointsNNS.part(x, y, order = 1)$regression.points## Voronoi style plotNNS.part(x, y, Voronoi = TRUE)## Examine final counts by quadrantDT <- NNS.part(x, y)$dtDT[ , counts := .N, by = quadrant]DT## End(Not run)

NNS Regression

Description

Generates a nonlinear regression based on partial moment quadrant means.

Usage

NNS.reg(  x,  y,  factor.2.dummy = TRUE,  order = NULL,  dim.red.method = NULL,  tau = NULL,  type = NULL,  point.est = NULL,  location = "top",  return.values = TRUE,  plot = TRUE,  plot.regions = FALSE,  residual.plot = TRUE,  confidence.interval = NULL,  threshold = 0,  n.best = NULL,  smooth = FALSE,  noise.reduction = "off",  dist = "L2",  ncores = NULL,  point.only = FALSE,  multivariate.call = FALSE)

Arguments

x

a vector, matrix or data frame of variables of numeric or factor data types.

y

a numeric or factor vector with compatible dimensions tox.

factor.2.dummy

logical;TRUE (default) Automatically augments variable matrix with numerical dummy variables based on the levels of factors.

order

integer; Controls the number of partial moment quadrant means. Users are encouraged to try different(order = ...) integer settings with(noise.reduction = "off").(order = "max") will force a limit condition perfect fit.

dim.red.method

options: ("cor", "NNS.dep", "NNS.caus", "all", "equal",numeric vector, NULL) method for determining synthetic X* coefficients (per Dana and Dawes (2004)). Selection of a method automatically engages the dimension reduction regression. The default isNULL for full multivariate regression.(dim.red.method = "NNS.dep") usesNNS.dep for nonlinear dependence weights, while(dim.red.method = "NNS.caus") usesNNS.caus for causal weights.(dim.red.method = "cor") uses standard linear correlation for weights.(dim.red.method = "all") averages all methods for further feature engineering.(dim.red.method = "equal") uses unit weights. Alternatively, user can specify a numeric vector of coefficients.

tau

options("ts", NULL);NULL(default) To be used in conjunction with(dim.red.method = "NNS.caus") or(dim.red.method = "all"). If the regression is using time-series data, set(tau = "ts") for more accurate causal analysis.

type

NULL (default). To perform a classification, set to(type = "CLASS"). Like a logistic regression, it is not necessary for target variable of two classes e.g. [0, 1].

point.est

a numeric or factor vector with compatible dimensions tox. Returns the fitted valuey.hat for any value ofx.

location

Sets the legend location within the plot, per thex andy co-ordinates used in base graphicslegend.

return.values

logical;TRUE (default), set toFALSE in order to only display a regression plot and call values as needed.

plot

logical;TRUE (default) To plot regression.

plot.regions

logical;FALSE (default). Generates 3d regions associated with each regression point for multivariate regressions. Note, adds significant time to routine.

residual.plot

logical;TRUE (default) To ploty.hat andY.

confidence.interval

numeric [0, 1];NULL (default) Plots the associated confidence interval with the estimate and reports the standard error for each individual segment. Also applies the same level for the prediction intervals.

threshold

numeric [0, 1];(threshold = 0) (default) Sets the threshold for dimension reduction of independent variables when(dim.red.method) is notNULL.

n.best

integer;NULL (default) Sets the number of nearest regression points to use in weighting for multivariate regression atsqrt(# of regressors).(n.best = "all") will select and weight all generated regression points. Analogous tok in ak Nearest Neighbors algorithm. Different values ofn.best are tested using cross-validation inNNS.stack.

smooth

logical;FALSE (default) Applies a smoothing spline instead of local linear fit to regression points.

noise.reduction

the method of determining regression points options: ("mean", "median", "mode", "off"); In low signal:noise situations,(noise.reduction = "mean") uses means forNNS.dep restricted partitions,(noise.reduction = "median") uses medians instead of means forNNS.dep restricted partitions, while(noise.reduction = "mode") uses modes instead of means forNNS.dep restricted partitions.(noise.reduction = "off") uses an overall central tendency measure for partitions.

dist

options:("L1", "L2", "FACTOR") the method of distance calculation; Selects the distance calculation used.dist = "L2" (default) selects the Euclidean distance and(dist = "L1") selects the Manhattan distance;(dist = "FACTOR") uses a frequency.

ncores

integer; value specifying the number of cores to be used in the parallelized procedure. If NULL (default), the number of cores to be used is equal to the number of cores of the machine - 1.

point.only

Internal argument for abbreviated output.

multivariate.call

Internal argument for multivariate regressions.

Value

UNIVARIATE REGRESSION RETURNS THE FOLLOWING VALUES:

"R2" provides the goodness of fit;
"SE" returns the overall standard error of the estimate betweeny andy.hat;
"Prediction.Accuracy" returns the correct rounded"Point.est" used in classifications versus the categoricaly;
"derivative" for the coefficient of thex and its applicable range;
"Point.est" for the predicted value generated;
"pred.int" lower and upper prediction intervals for the"Point.est" returned using the"confidence.interval" provided;
"regression.points" provides the points used in the regression equation for the given order of partitions;
"Fitted.xy" returns adata.table ofx,y,y.hat,resid,NNS.ID,gradient;

MULTIVARIATE REGRESSION RETURNS THE FOLLOWING VALUES:

"R2" provides the goodness of fit;
"equation" returns the numerator of the synthetic X* dimension reduction equation as adata.table consisting of regressor and its coefficient. Denominator is simply the length of all coefficients > 0, returned in last row ofequationdata.table.
"x.star" returns the synthetic X* as a vector;
"rhs.partitions" returns the partition points for each regressorx;
"RPM" provides the Regression Point Matrix, the points for eachx used in the regression equation for the given order of partitions;
"Point.est" returns the predicted value generated;
"pred.int" lower and upper prediction intervals for the"Point.est" returned using the"confidence.interval" provided;
"Fitted.xy" returns adata.table ofx,y,y.hat,gradient, andNNS.ID.

Note

Please ensurepoint.est is of compatible dimensions tox, error message will ensue if not compatible.
Like a logistic regression, the(type = "CLASS") setting is not necessary for target variable of two classes e.g. [0, 1]. The response variable base category should be 1 for classification problems.
For low signal:noise instances, increasing the dimension may yield better results usingNNS.stack(cbind(x,x), y, method = 1, ...).

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Vinod, H. and Viole, F. (2017) "Nonparametric Regression Using Clusters"doi:10.1007/s10614-017-9713-5

Vinod, H. and Viole, F. (2018) "Clustering and Curve Fitting by Line Segments"doi:10.20944/preprints201801.0090.v1

Viole, F. (2020) "Partitional Estimation Using Partial Moments"doi:10.2139/ssrn.3592491

Dana, J., and Dawes, R. M. (2004). The Superiority of Simple Alternatives to Regression for Social Science Predictions. Journal of Educational and Behavioral Statistics, 29(3), 317–331.

Examples

## Not run: set.seed(123)x <- rnorm(100) ; y <- rnorm(100)NNS.reg(x, y)## Manual {order} selectionNNS.reg(x, y, order = 2)## Maximum {order} selectionNNS.reg(x, y, order = "max")## x-only paritioning (Univariate only)NNS.reg(x, y, type = "XONLY")## For Multiple Regression:x <- cbind(rnorm(100), rnorm(100), rnorm(100)) ; y <- rnorm(100)NNS.reg(x, y, point.est = c(.25, .5, .75))## For Multiple Regression based on Synthetic X* (Dimension Reduction):x <- cbind(rnorm(100), rnorm(100), rnorm(100)) ; y <- rnorm(100)NNS.reg(x, y, point.est = c(.25, .5, .75), dim.red.method = "cor", ncores = 1)## IRIS dataset examples:# Dimension Reduction:NNS.reg(iris[,1:4], iris[,5], dim.red.method = "cor", order = 5, ncores = 1)# Dimension Reduction using causal weights:NNS.reg(iris[,1:4], iris[,5], dim.red.method = "NNS.caus", order = 5, ncores = 1)# Multiple Regression:NNS.reg(iris[,1:4], iris[,5], order = 2, noise.reduction = "off")# Classification:NNS.reg(iris[,1:4], iris[,5], point.est = iris[1:10, 1:4], type = "CLASS")$Point.est## To call fitted values:x <- rnorm(100) ; y <- rnorm(100)NNS.reg(x, y)$Fitted## To call partial derivative (univariate regression only):NNS.reg(x, y)$derivative## End(Not run)

NNS rescale

Description

Rescale a vector using either min-max scaling or risk-neutral adjustment.

Usage

NNS.rescale(x, a, b, method = "minmax", T = NULL, type = "Terminal")

Arguments

x

numeric vector; data to rescale (e.g., terminal prices for risk-neutral method).

a

numeric; defines the scaling target:- Formethod = "minmax": the lower limit of the output range (e.g., 5 to scale to [5, b]).- Formethod = "riskneutral": the initial price $ S_0 $ (must be positive, e.g., 100), used to set the target mean.

b

numeric; defines the scaling range or rate:- Formethod = "minmax": the upper limit of the output range (e.g., 10 to scale to [a, 10]).- Formethod = "riskneutral": the risk-free rate $ r $ (e.g., 0.05), used with $ T $ to adjust the mean.

method

character; scaling method:"minmax" (default) for min-max scaling, or"riskneutral" for risk-neutral adjustment.

T

numeric; time to maturity in years (required formethod = "riskneutral", ignored otherwise; e.g., 1). Default is NULL.

type

character; formethod = "riskneutral":"Terminal" (default) or"Discounted" (mean = $ S_0 $).

Value

Returns a rescaled distribution:- For"minmax": values scaled linearly to the range[a, b].- For"riskneutral": values scaled multiplicatively to a risk-neutral mean ($ S_0 e^(rT) $ iftype = "Terminal", or $ S_0 $ iftype = "Discounted").

Author(s)

Fred Viole, OVVO Financial Systems

Examples

## Not run: set.seed(123)# Min-max scaling: a = lower limit, b = upper limitx <- rnorm(100)NNS.rescale(x, a = 5, b = 10, method = "minmax")  # Scales to [5, 10]# Risk-neutral scaling (Terminal): a = S_0, b = r  # Mean approx 105.13prices <- 100 * exp(cumsum(rnorm(100, 0.001, 0.02)))NNS.rescale(prices, a = 100, b = 0.05, method = "riskneutral", T = 1, type = "Terminal")# Risk-neutral scaling (Discounted): a = S_0, b = r  # Mean approx 100NNS.rescale(prices, a = 100, b = 0.05, method = "riskneutral", T = 1, type = "Discounted")## End(Not run)

NNS Seasonality Test

Description

Seasonality test based on the coefficient of variation for the variable and lagged component series. A result of 1 signifies no seasonality present.

Usage

NNS.seas(variable, modulo = NULL, mod.only = TRUE, plot = TRUE)

Arguments

variable

a numeric vector.

modulo

integer(s); NULL (default) Used to find the nearest multiple(s) in the reported seasonal period.

mod.only

logical;TRUE (default) Limits the number of seasonal periods returned to the specifiedmodulo.

plot

logical;TRUE (default) Returns the plot of all periods exhibiting seasonality and the variable level reference.

Value

Returns a matrix of all periods exhibiting less coefficient of variation than the variable with"all.periods"; and the single period exhibiting the least coefficient of variation versus the variable with"best.period"; as well as a vector of"periods" for easy call intoNNS.ARMA.optim. If no seasonality is detected,NNS.seas will return ("No Seasonality Detected").

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: set.seed(123)x <- rnorm(100)## To call strongest period based on coefficient of variation:NNS.seas(x, plot = FALSE)$best.period## Using modulos for logical seasonal inference:NNS.seas(x, modulo = c(2,3,5,7), plot = FALSE)## End(Not run)

NNS Stack

Description

Prediction model using the predictions of the NNS base modelsNNS.reg as features (i.e. meta-features) for the stacked model.

Usage

NNS.stack(  IVs.train,  DV.train,  IVs.test = NULL,  type = NULL,  obj.fn = expression(sum((predicted - actual)^2)),  objective = "min",  optimize.threshold = TRUE,  dist = "L2",  CV.size = NULL,  balance = FALSE,  ts.test = NULL,  folds = 5,  order = NULL,  method = c(1, 2),  stack = TRUE,  dim.red.method = "cor",  pred.int = NULL,  status = TRUE,  ncores = NULL)

Arguments

IVs.train

a vector, matrix or data frame of variables of numeric or factor data types.

DV.train

a numeric or factor vector with compatible dimensions to(IVs.train).

IVs.test

a vector, matrix or data frame of variables of numeric or factor data types with compatible dimensions to(IVs.train). If NULL, will use(IVs.train) as default.

type

NULL (default). To perform a classification of discrete integer classes from factor target variable(DV.train) with a base category of 1, set to(type = "CLASS"), else for continuous(DV.train) set to(type = NULL). Like a logistic regression, this setting is not necessary for target variable of two classes e.g. [0, 1].

obj.fn

expression;expression(sum((predicted - actual)^2)) (default) Sum of squared errors is the default objective function. Anyexpression() using the specific termspredicted andactual can be used.

objective

options: ("min", "max")"min" (default) Select whether to minimize or maximize the objective functionobj.fn.

optimize.threshold

logical;TRUE (default) Will optimize the probability threshold value for rounding in classification problems. IfFALSE, returns 0.5.

dist

options:("L1", "L2", "DTW", "FACTOR") the method of distance calculation; Selects the distance calculation used.dist = "L2" (default) selects the Euclidean distance and(dist = "L1") selects the Manhattan distance;(dist = "DTW") selects the dynamic time warping distance;(dist = "FACTOR") uses a frequency.

CV.size

numeric [0, 1];NULL (default) Sets the cross-validation size if(IVs.test = NULL). Defaults to a random value between 0.2 and 0.33 for a random sampling of the training set.

balance

logical;FALSE (default) Uses both up and down sampling to balance the classes.type="CLASS" required.

ts.test

integer; NULL (default) Sets the length of the test set for time-series data; typically2*h parameter value fromNNS.ARMA or double known periods to forecast.

folds

integer;folds = 5 (default) Select the number of cross-validation folds.

order

options: (integer, "max", NULL);NULL (default) Sets the order forNNS.reg, where(order = "max") is the k-nearest neighbors equivalent, which is suggested for mixed continuous and discrete (unordered, ordered) data.

method

numeric options: (1, 2); Select the NNS method to include in stack.(method = 1) selectsNNS.reg;(method = 2) selectsNNS.reg dimension reduction regression. Defaults tomethod = c(1, 2), which will reduce the dimension first, then find the optimaln.best.

stack

logical;TRUE (default) Uses dimension reduction output inn.best optimization, otherwise performs both analyses independently.

dim.red.method

options: ("cor", "NNS.dep", "NNS.caus", "equal", "all") method for determining synthetic X* coefficients.(dim.red.method = "cor") uses standard linear correlation for weights.(dim.red.method = "NNS.dep") (default) usesNNS.dep for nonlinear dependence weights, while(dim.red.method = "NNS.caus") usesNNS.caus for causal weights.(dim.red.method = "all") averages all methods for further feature engineering.

pred.int

numeric [0,1];NULL (default) Returns the associated prediction intervals with eachmethod.

status

logical;TRUE (default) Prints status update message in console.

ncores

integer; value specifying the number of cores to be used in the parallelized subroutineNNS.reg. If NULL (default), the number of cores to be used is equal to the number of cores of the machine - 1.

Value

Returns a vector of fitted values for the dependent variable test set for all models.

"NNS.reg.n.best" returns the optimum"n.best" parameter for theNNS.reg multivariate regression."SSE.reg" returns the SSE for theNNS.reg multivariate regression.
"OBJfn.reg" returns theobj.fn for theNNS.reg regression.
"NNS.dim.red.threshold" returns the optimum"threshold" from theNNS.reg dimension reduction regression.
"OBJfn.dim.red" returns theobj.fn for theNNS.reg dimension reduction regression.
"probability.threshold" returns the optimum probability threshold for classification, else 0.5 when set toFALSE.
"reg" returnsNNS.reg output.
"reg.pred.int" returns the prediction intervals for the regression output.
"dim.red" returnsNNS.reg dimension reduction regression output.
"dim.red.pred.int" returns the prediction intervals for the dimension reduction regression output.
"stack" returns the output of the stacked model.
"pred.int" returns the prediction intervals for the stacked model.

Note

Incorporate any objective function from external packages (such asMetrics::mape) viaNNS.stack(..., obj.fn = expression(Metrics::mape(actual, predicted)), objective = "min")
Like a logistic regression, the(type = "CLASS") setting is not necessary for target variable of two classes e.g. [0, 1]. The response variable base category should be 1 for multiple class problems.
Missing data should be handled prior as well usingna.omit orcomplete.cases on the full dataset.

If error received:

"Error in is.data.frame(x) : object 'RP' not found"

reduce theCV.size.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. (2016) "Classification Using NNS Clustering Analysis"doi:10.2139/ssrn.2864711

Examples

 ## Using 'iris' dataset where test set [IVs.test] is 'iris' rows 141:150. ## Not run:  NNS.stack(iris[1:140, 1:4], iris[1:140, 5], IVs.test = iris[141:150, 1:4], type = "CLASS",  balance = TRUE) ## Using 'iris' dataset to determine [n.best] and [threshold] with no test set. NNS.stack(iris[ , 1:4], iris[ , 5], type = "CLASS") ## End(Not run)

NNS Term Matrix

Description

Generates a term matrix for text classification use inNNS.reg.

Usage

NNS.term.matrix(x, oos = NULL)

Arguments

x

mixed data.frame; character/numeric; A two column dataset should be used. Concatenate text from original sources to comply with format. Also note the possibility of factors in"DV", so"as.numeric(as.character(...))" is used to avoid issues.

oos

mixed data.frame; character/numeric; Out-of-sample text dataset to be classified.

Value

Returns the text as independent variables"IV" and the classification as the dependent variable"DV". Out-of-sample independent variables are returned with"OOS".

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

## Not run: x <- data.frame(cbind(c("sunny", "rainy"), c(1, -1)))NNS.term.matrix(x)### Concatenate Text with space separator, cbind with "DV"x <- data.frame(cbind(c("sunny", "rainy"), c("windy", "cloudy"), c(1, -1)))x <- data.frame(cbind(paste(x[ , 1], x[ , 2], sep = " "), as.numeric(as.character(x[ , 3]))))NNS.term.matrix(x)### NYT Examplerequire(RTextTools)data(NYTimes)### Concatenate Columns 3 and 4 containing text, with column 5 as DVNYT <- data.frame(cbind(paste(NYTimes[ , 3], NYTimes[ , 4], sep = " "),                     as.numeric(as.character(NYTimes[ , 5]))))NNS.term.matrix(NYT)## End(Not run)

Partial Moment Matrix

Description

Builds a list containing CUPM, DUPM, DLPM, CLPM and the overall covariance matrix.

Usage

PM.matrix(LPM_degree, UPM_degree, target, variable, pop_adj, norm = FALSE)

Arguments

LPM_degree

numeric; lower partial moment degree (0 = freq, 1 = area).

UPM_degree

numeric; upper partial moment degree (0 = freq, 1 = area).

target

numeric vector; thresholds for each column (defaults to colMeans).

variable

numeric matrix or data.frame.

pop_adj

logical; TRUE adjusts population vs. sample moments.

norm

logical; default FALSE. If TRUE, each quadrant matrix is cell-wise normalized so their sum is 1 at each (i,j).

Details

Partial Moment Matrix

Value

A list: $cupm, $dupm, $dlpm, $clpm, $cov.matrix.

Examples

set.seed(123)A <- cbind(rnorm(100), rnorm(100), rnorm(100))PM.matrix(1, 1, NULL, A, TRUE)          # uses norm = FALSE by defaultPM.matrix(1, 1, NULL, A, TRUE, TRUE)    # enable normalization

Upper Partial Moment

Description

This function generates a univariate upper partial moment for any degree or target.

Usage

UPM(degree, target, variable, excess_ret = FALSE)

Arguments

degree

numeric;(degree = 0) is frequency,(degree = 1) is area.

target

numeric; Set totarget = mean(variable) for classical equivalences, but does not have to be. (Vectorized)#' @param variable a numeric vector.data.frame orlist type objects are not permissible.

variable

a numeric vector.data.frame orlist type objects are not permissible.

excess_ret

logical;FALSE (default)

Value

UPM of variable

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

set.seed(123)x <- rnorm(100)UPM(0, mean(x), x)

UPM VaR

Description

Generates an upside value at risk (VaR) quantile based on the Upper Partial Moment ratio

Usage

UPM.VaR(percentile, degree, x)

Arguments

percentile

numeric [0, 1]; The percentile for right-tail VaR (vectorized).

degree

integer;(degree = 0) for discrete distributions,(degree = 1) for continuous distributions.

x

a numeric vector.

Value

Returns a numeric value representing the point at which"percentile" of the area ofx is above.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Examples

set.seed(123)x <- rnorm(100)## For 5th percentile, right-tailUPM.VaR(0.05, 0, x)

Upper Partial Moment Ratio

Description

This function generates a standardized univariate upper partial momentof any non‑negative degree for a given target.

Usage

UPM.ratio(degree, target, variable)

Arguments

degree

numeric; degree = 0 gives frequency, degree = 1 gives area.

target

numeric vector; threshold(s). Defaults to mean(variable).

variable

numeric vector or data‑frame column to evaluate.

Value

Numeric vector of standardized upper partial moments.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. & Nawrocki, D. (2013) *Nonlinear Nonparametric Statistics: Using Partial Moments* (ISBN:1490523995)

Examples

  set.seed(123)  x <- rnorm(100)  UPM.ratio(0, mean(x), x)## Not run:   plot3d(x, y, Co.UPM(0, sort(x), sort(y), x, y), …)## End(Not run)

Partial Derivative dy/d_[wrt]

Description

Returns the numerical partial derivative ofy with respect to [wrt] any regressor for a point of interest. Finite difference method is used withNNS.reg estimates asf(x + h) andf(x - h) values.

Usage

dy.d_(x, y, wrt, eval.points = "obs", mixed = FALSE, messages = TRUE)

Arguments

x

a numeric matrix or data frame.

y

a numeric vector with compatible dimensions tox.

wrt

integer; Selects the regressor to differentiate with respect to (vectorized).

eval.points

numeric or options: ("obs", "apd", "mean", "median", "last"); Regressor points to be evaluated.

Numeric values must be in matrix or data.frame form to be evaluated for each regressor, otherwise, a vector of points will evaluate only at thewrt regressor. See examples for use cases.
Set to(eval.points = "obs") (default) to find the average partial derivative at every observation of the variable with respect tofor specific tuples of given observations.
Set to(eval.points = "apd") to find the average partial derivative at every observation of the variable with respect toover the entire distribution of other regressors.
Set to(eval.points = "mean") to find the partial derivative at the mean of value of every variable.
Set to(eval.points = "median") to find the partial derivative at the median value of every variable.
Set to(eval.points = "last") to find the partial derivative at the last observation of every value (relevant for time-series data).

mixed

logical;FALSE (default) If mixed derivative is to be evaluated, set(mixed = TRUE).

messages

logical;TRUE (default) Prints status messages.

Value

Returns column-wise matrix of wrt regressors:

dy.d_(...)[, wrt]$First the 1st derivative
dy.d_(...)[, wrt]$Second the 2nd derivative
dy.d_(...)[, wrt]$Mixed the mixed derivative (for two independent variables only).

Note

For binary regressors, it is suggested to useeval.points = seq(0, 1, .05) for a better resolution around the midpoint.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Vinod, H. and Viole, F. (2020) "Comparing Old and New Partial Derivative Estimates from Nonlinear Nonparametric Regressions"doi:10.2139/ssrn.3681104

Examples

## Not run: set.seed(123) ; x_1 <- runif(1000) ; x_2 <- runif(1000) ; y <- x_1 ^ 2 * x_2 ^ 2B <- cbind(x_1, x_2)## To find derivatives of y wrt 1st regressor for specific points of both regressorsdy.d_(B, y, wrt = 1, eval.points = t(c(.5, 1)))## To find average partial derivative of y wrt 1st regressor,only supply 1 value in [eval.points], or a vector of [eval.points]:dy.d_(B, y, wrt = 1, eval.points = .5)dy.d_(B, y, wrt = 1, eval.points = fivenum(B[,1]))## To find average partial derivative of y wrt 1st regressor,for every observation of 1st regressor:apd <- dy.d_(B, y, wrt = 1, eval.points = "apd")plot(B[,1], apd[,1]$First)## 95% Confidence Interval to test if 0 is within### Lower CILPM.VaR(.025, 0, apd[,1]$First)### Upper CIUPM.VaR(.025, 0, apd[,1]$First)## End(Not run)

Partial Derivative dy/dx

Description

Returns the numerical partial derivative ofy wrtx for a point of interest.

Usage

dy.dx(x, y, eval.point = NULL)

Arguments

x

a numeric vector.

y

a numeric vector.

eval.point

numeric or ("overall");x point to be evaluated, must be provided. Defaults to(eval.point = NULL). Set to(eval.point = "overall") to find an overall partial derivative estimate (1st derivative only).

Value

Returns adata.table of eval.point along with both 1st and 2nd derivative.

Author(s)

Fred Viole, OVVO Financial Systems

References

Viole, F. and Nawrocki, D. (2013) "Nonlinear Nonparametric Statistics: Using Partial Moments" (ISBN: 1490523995)

Vinod, H. and Viole, F. (2017) "Nonparametric Regression Using Clusters"doi:10.1007/s10614-017-9713-5

Examples

## Not run: x <- seq(0, 2 * pi, pi / 100) ; y <- sin(x)dy.dx(x, y, eval.point = 1.75)# First derivativedy.dx(x, y, eval.point = 1.75)[ , first.derivative]# Second derivativedy.dx(x, y, eval.point = 1.75)[ , second.derivative]# Vector of derivativesdy.dx(x, y, eval.point = c(1.75, 2.5))## End(Not run)

Movatterモバイル変換

NNS: Nonlinear Nonparametric Statistics

Description

Author(s)

See Also

Co‑Lower Partial Moment

Description

Usage

Arguments

Value

Author(s)

References

Examples

Co‑Lower Partial Moment nD

Description

Usage

Arguments

Value

Examples

Co‑Upper Partial Moment

Description

Usage

Arguments

Value

Author(s)

References

Examples

Co‑Upper Partial Moment nD

Description

Usage

Arguments

Value

Examples

Divergent‑Lower Partial Moment

Description

Usage

Arguments

Value

Author(s)

References

Examples

Divergent‑Upper Partial Moment

Description

Usage

Arguments

Value

Author(s)

References

Examples

Divergent Partial Moment nD

Description

Usage

Arguments

Value

Examples

Lower Partial Moment

Description

Usage

Arguments

Value

Author(s)

References

Examples

LPM VaR

Description

Usage

Arguments

Value

Author(s)

References

Examples

Lower Partial Moment Ratio

Description

Usage

Arguments

Value

Author(s)

References

Examples

NNS ANOVA: Nonparametric Analysis of Variance