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 Viole ovvo.financial.systems@gmail.com

Other contributors:

• Roberto Spadim [contributor]

See Also

Useful links:

• Report bugs at https://github.com/OVVO-Financial/NNS/issues

Co-Lower Partial Moment (Lower Left Quadrant 4)

Description

This function generates a co-lower partial moment for between two equal length variables for any degree or target.

Usage

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

Arguments

Value

Co-LPM of two variables

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

Co-Upper Partial Moment (Upper Right Quadrant 1)

Description

This function generates a co-upper partial moment between two equal length variables for any degree or target.

Usage

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

Arguments

Value

Co-UPM of two variables

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

Divergent-Lower Partial Moment (Lower Right Quadrant 3)

Description

This function generates a divergent lower partial moment between two equal length variables for any degree or target.

Usage

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

Arguments

Value

Divergent LPM of two variables

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

Divergent-Upper Partial Moment (Upper Left Quadrant 2)

Description

This function generates a divergent upper partial moment between two equal length variables for any degree or target.

Usage

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

Arguments

Value

Divergent UPM of two variables

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

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

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

Value

Returns a numeric value representing the point at which "percentile" of the area of x 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-tail
LPM.VaR(0.05, 0, x)

## End(Not run)

Lower Partial Moment RATIO

Description

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

Usage

LPM.ratio(degree, target, variable)

Arguments

Value

Standardized 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)

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: 
## Empirical CDF (degree = 0)
lpm_cdf <- LPM.ratio(0, sort(x), x)
plot(sort(x), lpm_cdf)

## Continuous CDF (degree = 1)
lpm_cdf_1 <- LPM.ratio(1, sort(x), x)
plot(sort(x), lpm_cdf_1)

## Joint CDF
x <- rnorm(5000) ; y <- rnorm(5000)
plot3d(x, y, Co.LPM(0, sort(x), sort(y), x, y), col = "blue", xlab = "X", ylab = "Y",
zlab = "Probability", box = FALSE)

## End(Not run)

NNS ANOVA

Description

Analysis of variance (ANOVA) based on lower partial moment CDFs for multiple variables, evaluated at multiple quantiles (or means only). Returns a degree of certainty to whether the population distributions (or sample means) are identical, not a p-value.

Usage

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

Arguments

Value

Returns the following:

• "Control Mean" control mean.

• "Treatment Mean" treatment mean.

• "Grand Mean" mean of means.

• "Control CDF" CDF of the control from the grand mean.

• "Treatment CDF" CDF of the treatment from the grand mean.

• "Certainty" the certainty of the same population statistic.

• "Lower Bound Effect" and "Upper Bound Effect" the effect size of the treatment for the specified confidence interval.

• "Robust Certainty Estimate" and "Lower 95 CI", "Upper 95 CI" are the robust certainty estimate and its 95 percent confidence interval after permutations if robust = TRUE.

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 size
set.seed(123)
x <- rnorm(100) ; y <- rnorm(100)
NNS.ANOVA(control = x, treatment = y)

### Two variable analysis with no control variable
A <- cbind(x, y)
NNS.ANOVA(A)

### Medians test
NNS.ANOVA(A, means.only = TRUE, medians = TRUE)

### Multiple variable analysis with no control variable
set.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 list
x <- 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

Value

Returns a vector of forecasts of length (h) if no pred.int specified. Else, returns a data.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 lags
NNS.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 lag
NNS.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 given seasonal.factor vector in NNS.ARMA. Minimum sum of squared errors (forecast-actual) is used to determine optimum across all NNS.ARMA methods.

Usage

NNS.ARMA.optim(
  variable,
  h = NULL,
  training.set = NULL,
  seasonal.factor,
  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

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 which NNS.ARMA method was used.

• $shrink whether to use the shrink parameter in NNS.ARMA.

• $nns.regress whether to smooth the variable via NNS.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 the NNS.ARMA with the derived parameters.

• $errors a vector of model errors from internal calibration.

• $results a vector of length h.

• $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 maximum seasonal.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 ratios LPM.ratio, and resulting survival, hazard and cumulative hazard functions.

Usage

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

Arguments

Value

Returns:

• "Function" a data.table containing the observations and resulting CDF of the variable.

• "target.value" value from the target 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 CDF
x <- rnorm(5000) ; y <- rnorm(5000)
A <- cbind(x,y)

NNS.CDF(A, 0)

## Joint CDF with target
NNS.CDF(A, 0, target = c(0,0))

## 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

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

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 routine NNS.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

Value

• ensemble average observation over all replicates as a vector.

• replicates maximum entropy bootstrap replicates as a list for each rho.

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 AirPassengers
MC_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

Details

The function applies NNS.SD.efficient.set iteratively, peeling off the SD-efficient set at each step if it meets or exceeds min_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 meets the min_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): If dendrogram = TRUE, an hclust 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 cluster
results <- NNS.SD.cluster(data = A, degree = 1, min_cluster = 2)
print(results$Clusters)

# Produce a dendrogram as well
results_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

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

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

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

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

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 incorporating NNS.ARMA estimates of variables into NNS.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

Value

Returns the following matrices of forecasted variables:

• "interpolated_and_extrapolated" Returns a data.frame of the linear interpolated and NNS.ARMA extrapolated values to replace NA values in the original variables argument. This is required for working with variables containing different frequencies, e.g. where NA 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 univariate NNS.ARMA forecasts.

• "multivariate" Returns the multi-variate NNS.reg forecasts.

• "ensemble" Returns the ensemble of both "univariate" and "multivariate" forecasts.

Note

• "Error in { : task xx failed -}" should be re-run with NNS.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 regression NNS.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

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 as Metrics::mape) via NNS.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)

 ## 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)

Arguments

Value

Returns the directional causation (x —> y) or (y —> x) and net quantity of association. For causal matrix, directional causation is returned as ([column variable] —> [row variable]). Negative numbers represent causal direction attributed to [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: 
## x causes y...
set.seed(123)
x <- rnorm(1000) ; y <- x ^ 2
NNS.caus(x, y, tau = "cs")

## Causal matrix without per factor causation
NNS.caus(iris, tau = 0)

## Causal matrix with per factor causation
NNS.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

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 0
NNS.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

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 Matrix
x <- 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 steps h, in the finite step method.

Usage

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

Arguments

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) / x
NNS.diff(f, 4.1)

## End(Not run)

NNS Distance

Description

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

Usage

NNS.distance(rpm, dist.estimate, k, class)

Arguments

Value

Returns sum of weighted distances.

NNS gravity

Description

Alternative central tendency measure more robust to outliers.

Usage

NNS.gravity(x, discrete = FALSE)

Arguments

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 from meboot https://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

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

Vectorized rho and drift parameters will not vectorize both simultaneously. Also, do not specify target_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 AirPassengers
boots <- NNS.meboot(AirPassengers, reps = 100, rho = 0, xmin = 0)

# Verify correlation of replicates ensemble to original
cor(boots["ensemble",]$ensemble, AirPassengers, method = "spearman")

# Plot all replicates
matplot(boots["replicates",]$replicates , type = 'l')

# Plot ensemble
lines(boots["ensemble",]$ensemble, lwd = 3)


### Vectorized drift with a single rho
boots <- 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 drift
boots <- 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 scale
boots <- 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

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

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

Value

Returns a data.frame of normalized values.

Note

Unequal vectors provided in a list will only generate linear=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 lengths

vec1 <- 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 autoregression NNS.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

Value

Returns the following matrices of forecasted variables:

• "interpolated_and_extrapolated" Returns a data.frame of the linear interpolated and NNS.ARMA extrapolated values to replace NA values in the original variables argument. This is required for working with variables containing different frequencies, e.g. where NA 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 univariate NNS.ARMA forecasts.

• "multivariate" Returns the multi-variate NNS.reg forecasts.

• "ensemble" Returns the ensemble of both "univariate" and "multivariate" forecasts.

Note

Specific regressors include:

1. PAYEMS – Payroll Employment

2. JTSJOL – Job Openings

3. CPIAUCSL – Consumer Price Index

4. DGORDER – Durable Goods Orders

5. RSAFS – Retail Sales

6. UNRATE – Unemployment Rate

7. HOUST – Housing Starts

8. INDPRO – Industrial Production

9. DSPIC96 – Personal Income

10. BOPTEXP – Exports

11. BOPTIMP – Imports

12. TTLCONS – Construction Spending

13. IR – Import Price Index

14. CPILFESL – Core Consumer Price Index

15. PCEPILFE – Core PCE Price Index

16. PCEPI – PCE Price Index

17. PERMIT – Building Permits

18. TCU – Capacity Utilization Rate

19. BUSINV – Business Inventories

20. ULCNFB – Unit Labor Cost

21. IQ – Export Price Index

22. GACDISA066MSFRBNY – Empire State Mfg Index

23. GACDFSA066MSFRBPHI – Philadelphia Fed Mfg Index

24. PCEC96 – Real Consumption Spending

25. GDPC1 – Real Gross Domestic Product

26. ICSA – Weekly Unemployment Claims

27. DGS10 – 10-year Treasury rates

28. T10Y2Y – 2-10 year Treasury rate spread

29. WALCL – Total Assets

30. PALLFNFINDEXM – Global Price Index of All Commodities

31. FEDFUNDS – Federal Funds Effective Rate

32. PPIACO – Producer Price Index All Commodities

33. CIVPART – Labor Force Participation Rate

34. M2NS – M2 Money Supply

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, dependence NNS.dep, regression NNS.reg routines.

Usage

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

Arguments

Value

Returns:

• "dt" a data.table of x and y observations with their partition assignment "quadrant" in the 3rd column and their prior partition assignment "prior.quadrant" in the 4th column.

• "regression.points" the data.table of regression points for that given (order = ...).

• "order" the order 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 partitions
NNS.part(x, y, order = 1)$dt

## Regression points
NNS.part(x, y, order = 1)$regression.points

## Voronoi style plot
NNS.part(x, y, Voronoi = TRUE)

## Examine final counts by quadrant
DT <- NNS.part(x, y)$dt
DT[ , 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,
  stn = 0.95,
  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,
  noise.reduction = "off",
  dist = "L2",
  ncores = NULL,
  point.only = FALSE,
  multivariate.call = FALSE
)

Arguments

Value

UNIVARIATE REGRESSION RETURNS THE FOLLOWING VALUES:

• "R2" provides the goodness of fit;

• "SE" returns the overall standard error of the estimate between y and y.hat;

• "Prediction.Accuracy" returns the correct rounded "Point.est" used in classifications versus the categorical y;

• "derivative" for the coefficient of the x 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 a data.table of x, 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 a data.table consisting of regressor and its coefficient. Denominator is simply the length of all coefficients > 0, returned in last row of equation data.table.

• "x.star" returns the synthetic X* as a vector;

• "rhs.partitions" returns the partition points for each regressor x;

• "RPM" provides the Regression Point Matrix, the points for each x 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 a data.table of x,y, y.hat, gradient, and NNS.ID.

Note

• Please ensure point.est is of compatible dimensions to x, 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 using NNS.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

Examples

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

## Manual {order} selection
NNS.reg(x, y, order = 2)

## Maximum {order} selection
NNS.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

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 (\( S_0 e^(rT) \) if type = "Terminal", or \( S_0 \) if type = "Discounted").

Author(s)

Fred Viole, OVVO Financial Systems

Examples

## Not run: 
set.seed(123)
# Min-max scaling: a = lower limit, b = upper limit
x <- 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 105.13
prices <- 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 100
NNS.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

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 into NNS.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 models NNS.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,
  norm = NULL,
  method = c(1, 2),
  stack = TRUE,
  dim.red.method = "cor",
  pred.int = NULL,
  status = TRUE,
  ncores = NULL
)

Arguments

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 the NNS.reg multivariate regression. "SSE.reg" returns the SSE for the NNS.reg multivariate regression.

• "OBJfn.reg" returns the obj.fn for the NNS.reg regression.

• "NNS.dim.red.threshold" returns the optimum "threshold" from the NNS.reg dimension reduction regression.

• "OBJfn.dim.red" returns the obj.fn for the NNS.reg dimension reduction regression.

• "probability.threshold" returns the optimum probability threshold for classification, else 0.5 when set to FALSE.

• "reg" returns NNS.reg output.

• "reg.pred.int" returns the prediction intervals for the regression output.

• "dim.red" returns NNS.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 as Metrics::mape) via NNS.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 using na.omit or complete.cases on the full dataset.

If error received:

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

reduce the CV.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")

 ## Using 'iris' dataset to determine [n.best] and [threshold] with no test set.
 NNS.stack(iris[ , 1:4], iris[ , 5], type = "CLASS")

 ## Selecting NNS.reg and dimension reduction techniques.
 NNS.stack(iris[1:140, 1:4], iris[1:140, 5], iris[141:150, 1:4], method = c(1, 2), type = "CLASS")
 
## End(Not run)

NNS Term Matrix

Description

Generates a term matrix for text classification use in NNS.reg.

Usage

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

Arguments

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 Example
require(RTextTools)
data(NYTimes)

### Concatenate Columns 3 and 4 containing text, with column 5 as DV
NYT <- 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

This function generates a co-partial moment matrix for the specified co-partial moment.

Usage

PM.matrix(LPM_degree, UPM_degree, target, variable, pop_adj)

Arguments

Value

Matrix of partial moment quadrant values (CUPM, DUPM, DLPM, CLPM), and overall covariance matrix. Uncalled quadrants will return a matrix of zeros.

Note

For divergent asymmetrical "D.LPM" and "D.UPM" matrices, matrix is D.LPM(column,row,...).

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) "Bayes' Theorem From Partial Moments" doi:10.2139/ssrn.3457377

Examples

set.seed(123)
x <- rnorm(100) ; y <- rnorm(100) ; z <- rnorm(100)
A <- cbind(x,y,z)
PM.matrix(LPM_degree = 1, UPM_degree = 1, variable = A, target = colMeans(A), pop_adj = TRUE)

## Use of vectorized numeric targets (target_x, target_y, target_z)
PM.matrix(LPM_degree = 1, UPM_degree = 1, target = c(0, 0.15, .25), variable = A, pop_adj = TRUE)

## Calling Individual Partial Moment Quadrants
cov.mtx <- PM.matrix(LPM_degree = 1, UPM_degree = 1, variable = A, target = colMeans(A), 
                     pop_adj = TRUE)
cov.mtx$cupm

## Full covariance matrix
cov.mtx$cov.matrix

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

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

Value

Returns a numeric value representing the point at which "percentile" of the area of x 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-tail
UPM.VaR(0.05, 0, x)

Upper Partial Moment RATIO

Description

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

Usage

UPM.ratio(degree, target, variable)

Arguments

Value

Standardized 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.ratio(0, mean(x), x)

## Joint Upper CDF
## Not run: 
x <- rnorm(5000) ; y <- rnorm(5000)
plot3d(x, y, Co.UPM(0, sort(x), sort(y), x, y), col = "blue", xlab = "X", ylab = "Y",
zlab = "Probability", box = FALSE)

## End(Not run)

Partial Derivative dy/d_[wrt]

Description

Returns the numerical partial derivative of y with respect to [wrt] any regressor for a point of interest. Finite difference method is used with NNS.reg estimates as f(x + h) and f(x - h) values.

Usage

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

Arguments

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 use eval.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 ^ 2
B <- cbind(x_1, x_2)

## To find derivatives of y wrt 1st regressor for specific points of both regressors
dy.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 CI
LPM.VaR(.025, 0, apd[,1]$First)

### Upper CI
UPM.VaR(.025, 0, apd[,1]$First)

## End(Not run)

Partial Derivative dy/dx

Description

Returns the numerical partial derivative of y wrt x for a point of interest.

Usage

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

Arguments

Value

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

Note

If a vector of derivatives is required, ensure (deriv.method = "FD").

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 derivative
dy.dx(x, y, eval.point = 1.75)[ , first.derivative]

# Second derivative
dy.dx(x, y, eval.point = 1.75)[ , second.derivative]

# Vector of derivatives
dy.dx(x, y, eval.point = c(1.75, 2.5))

## End(Not run)