Type:	Package
Title:	Time Series Forecasting Auxiliary Functions
Version:	1.0.0
Maintainer:	Alexios Galanos <alexios@4dscape.com>
Description:	A suite of auxiliary functions that enhance time series estimation and forecasting, including a robust anomaly detection routine based on Chen and Liu (1993) <doi:10.2307/2290724> (imported and wrapped from the 'tsoutliers' package), utilities for managing calendar and time conversions, performance metrics to assess both point forecasts and distributional predictions, advanced simulation by allowing the generation of time series components—such as trend, seasonal, ARMA, irregular, and anomalies—in a modular fashion based on the innovations form of the state space model and a number of transformation methods including Box-Cox, Logit, 'Softplus-Logit' and Sigmoid.
Depends:	R (≥ 4.1.0), tsmethods
Imports:	methods, zoo, xts, lubridate, car, Rdpack, scoringRules, stlplus, tsoutliers, forecast, data.table
RdMacros:	Rdpack
License:	GPL-2
Encoding:	UTF-8
BugReports:	https://github.com/tsmodels/tsaux/issues
URL:	https://github.com/tsmodels/tsaux
RoxygenNote:	7.3.2
Suggests:	knitr, kableExtra, rmarkdown, testthat (≥ 3.0.0)
VignetteBuilder:	knitr
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2025-03-29 19:39:16 UTC; alexios
Author:	Alexios Galanos [aut, cre, cph]
Repository:	CRAN
Date/Publication:	2025-03-31 17:30:01 UTC

tsaux: Time Series Forecasting Auxiliary Functions

Description

A suite of auxiliary functions that enhance time series estimation and forecasting, including a robust anomaly detection routine based on Chen and Liu (1993) doi:10.2307/2290724 (imported and wrapped from the 'tsoutliers' package), utilities for managing calendar and time conversions, performance metrics to assess both point forecasts and distributional predictions, advanced simulation by allowing the generation of time series components—such as trend, seasonal, ARMA, irregular, and anomalies—in a modular fashion based on the innovations form of the state space model and a number of transformation methods including Box-Cox, Logit, 'Softplus-Logit' and Sigmoid.

Author(s)

Maintainer: Alexios Galanos alexios@4dscape.com (ORCID) [copyright holder]

See Also

Useful links:

• https://github.com/tsmodels/tsaux

• Report bugs at https://github.com/tsmodels/tsaux/issues

Anomaly Component

Description

Anomaly Component

Usage

add_anomaly(x, ...)

## S3 method for class 'issm.component'
add_anomaly(x, time = NULL, delta = 0, ratio = 0.5, ...)

Arguments

Value

An object of class issm.component updated with the anomaly component.

ARMA Component

Description

ARMA Component

Usage

add_arma(x, ...)

## S3 method for class 'issm.component'
add_arma(x, order = c(0, 0), ar = 0, ma = 0, mu = 0, ...)

Arguments

Value

An object of class issm.component updated with the ARMA component.

Custom Component

Description

Custom Component

Usage

add_custom(x, ...)

## S3 method for class 'issm.component'
add_custom(x, custom = NULL, ...)

Arguments

Value

An object of class issm.component updated with the custom components.

Polynomial Trend Component

Description

Polynomial Trend Component

Usage

add_polynomial(x, ...)

## S3 method for class 'issm.component'
add_polynomial(
  x,
  order = 1,
  alpha = 0.1,
  beta = 0.01,
  phi = 1,
  l0 = 100,
  b0 = 1,
  ...
)

Arguments

Value

An object of class issm.component updated with the polynomial trend component.

Regressor Component

Description

Regressor Component

Usage

add_regressor(x, ...)

## S3 method for class 'issm.component'
add_regressor(x, xreg = NULL, pars = NULL, ...)

Arguments

Value

An object of class issm.component updated with the regressor components.

Seasonal Trend Component

Description

Seasonal Trend Component

Usage

add_seasonal(x, ...)

## S3 method for class 'issm.component'
add_seasonal(
  x,
  frequency = 12,
  gamma = 0.01,
  s0 = NULL,
  init_harmonics = frequency/2,
  normalized_seasonality = TRUE,
  init_scale = 1,
  ...
)

Arguments

Value

An object of class issm.component updated with the seasonal component.

Transform

Description

Transform

Usage

add_transform(x, ...)

## S3 method for class 'issm.component'
add_transform(x, method = "box-cox", lambda = 1, lower = 0, upper = 1, ...)

Arguments

Details

The inverse transform is applied to the simulated series. Valid methods are the “box-cox”, “logit”, “softplus-logit” and “sigmoid” transforms.

Value

An object of class issm.component updated with the transformation.

Anomaly Creation

Description

Creates specific types of anomalies given a series.

Usage

additive_outlier(y, time = 1, parameter = 0.5, add = TRUE)

temporary_change(y, time = 1, parameter = 0.5, alpha = 0.7, add = TRUE)

level_shift(y, time = 1, parameter = 0.5, add = TRUE)

Arguments

Details

These functions allow the generation of anomalies and may be chained together.

Value

Either the contaminated series else a matrix of the anomaly.

Author(s)

Alexios Galanos for this wrapper function.

Automatic Cleaning of Outliers and Temporary Changes

Description

A wrapper function for tso from the tsoutliers package. Takes as input a univariate xts object and returns a series decontaminated from outliers and temporary changes.

Usage

auto_clean(
  y,
  frequency = 1,
  lambda = NULL,
  types = c("AO", "TC"),
  stlm_opts = list(etsmodel = "AAN"),
  auto_arima_opts = list(max.p = 1, max.q = 1, d = 1, allowdrift = FALSE),
  method = c("sequential", "full"),
  ...
)

Arguments

Details

Calls the auto_regressors function to obtain the matrix of regressors and coefficients which are then used to decontaminate the series. If lambda is not NULL, the series is first transformed to perform the decontamination and then back transformed afterwards.

Value

A xts vector.

Author(s)

Alexios Galanos for this wrapper function.
Rob Hyndman for the forecast package.
Javier López-de-Lacalle for the tsoutliers package.

Automatic Detection of Outliers, Trends Breaks and Temporary Changes

Description

A wrapper function for function tso from the tsoutliers package. Takes as input a univariate xts object and returns a list with an xts object with any identified outliers, trend breaks and/or temporary changes to be used as regressors during estimation as well initial coefficients (see details).

Usage

auto_regressors(
  y,
  frequency = 1,
  lambda = NULL,
  forc_dates = NULL,
  sampling = NULL,
  h = 0,
  stlm_opts = list(etsmodel = "AAN"),
  auto_arima_opts = list(max.p = 1, max.q = 1, d = 1, allowdrift = FALSE),
  return_table = FALSE,
  method = c("sequential", "full"),
  ...
)

Arguments

Details

For generating future values of the identified outliers, the filter function is used with additive outliers having a filter value of 0, trend changes a value of 1, and temporary changes have value between 0 and 1. For the sequential method, the routine first interpolates any missing values, followed by an optional Box Cox transformation, and then elimination (and identification) of any outliers during the first pass. The cleaned series is then run through an stl filter (if any frequency is greater than 1) in order to deseasonalize the data (with multiple seasonality supported), after which the deseasonalized series is passed to the tso function where any additive outliers (AO), temporary shifts (TC) or level shift (LS) are identified. Additive outliers from this stage are added to any identified outliers from the initial stage. For each regressor, initial parameter values are returned together with the regressor matrix which should be passed to the estimation routine. This is critically important since in the absence of good parameter scaling, initial values are key to good convergence. Care should be taken with regards to any automatic Box Cox parameter estimation. In the presence of large outliers or level shifts, this is likely to be badly estimated which is why we do not allow automatic calculation of this, but instead place the burden on the user to decide what is a reasonable value (if any). If a Box Cox transformation is used in the estimation routine, then it is important to use the same lambda parameter in this function in order to get sensible results. Again, avoid automatic Box Cox calculations throughout when you suspect significant contamination of the series by outliers and breaks. For the full method, the series is directly passed to the tso function of the tsoutliers package. Finally, it should be noted that this function is still experimental, and may change in the future.

Value

A list with an xts outlier matrix (if any where identified) as well as a vector of initial parameter for use in the initialization of the optimizer.

Author(s)

Alexios Galanos for this wrapper function.
Rob Hyndman for the forecast package.
Javier Lopez-de-Lacalle for the tsoutliers package.

Examples

library(xts)
set.seed(200)
y = cumprod(c(100,(1+rnorm(100,0.01, 0.04))))
y = xts(y, as.Date(1:101, origin = as.Date("2000-01-01")))
yclean = y
outlier1 = rep(0, 101)
outlier1[20] = 0.35
outlier2 = rep(0, 101)
outlier2[40] = 0.25
outlier2 = as.numeric(filter(outlier2, filter = 0.6, method = "recursive"))
y = y + y*xts(outlier1, index(y))
y = y + y*xts(outlier2, index(y))
# may need some tweaking of the tso options.
x = auto_regressors(y, frequency = 1, sampling = "days", h = 20,
check.rank = TRUE, discard.cval = 4)
head(x$xreg)
tail(x$xreg)
min(which(x$xreg[,1]==1))
min(which(x$xreg[,2]==1))
#plot(as.numeric(y), type = "l", ylab = "")
#lines(as.numeric(yclean) + (x$xreg %*% x$init)[1:101], col = 2)

Box-Cox transform specification

Description

Creates a specification for the Box Cox transformation.

Usage

box_cox(lambda = NA, lower = 0, upper = 1.5, multivariate = FALSE, ...)

Arguments

Details

The function returns a list of 2 functions called “transform” and “inverse” which can be called with a data object and a frequency to calculate the transformed values. The auto_lambda function uses the method of Guerrero(1993).

Value

A list with the transform and inverse functions.

Note

The returned transform function will take additional argument “frequency” which determines whether a series is seasonal or not. When estimating lambda (when setting this to NA), a series with frequency > 1 will first be de-seasonalized using an STL decomposition.

Author(s)

Alexios Galanos for the BoxCox function.
John Fox for the powerTransform function used in the multivariate case.

References

Box GE, Cox DR (1964). “An analysis of transformations.” Journal of the Royal Statistical Society Series B: Statistical Methodology, 26(2), 211–243.
Velilla S (1993). “A note on the multivariate Box–Cox transformation to normality.” Statistics & Probability Letters, 17(4), 259–263.
Guerrero VM (1993). “Time-series analysis supported by power transformations.” Journal of forecasting, 12(1), 37–48.

Examples

y = cumprod(c(1, 1 + rnorm(100,0.01, 0.005)))
B = box_cox(lambda = NA)
yt = B$transform(y, frequency = 1)
lambda = attr(yt,"lambda")
ye = B$inverse(yt, lambda)

End of Month Date

Description

Returns the last day of the month from a Date within the month.

Usage

calendar_eom(date, ...)

Arguments

Details

Given a Date (such as 2019-01-02), will return the last Date within that year month.

Value

Date object

Author(s)

Alexios Galanos

End of Quarter Date

Description

Returns the last day of the quarter from a Date.

Usage

calendar_eoq(date, ...)

Arguments

Details

Given a date (such as 2019-01-02), will return the last date within that year quarter.

Value

Date object

Author(s)

Alexios Galanos

End of Week Date

Description

Returns the last day of the week from a Date given a choice of week days.

Usage

calendar_eow(date, day = 7, ...)

Arguments

Details

Given a Date (such as 2019-01-02) and a day of 7, will return the Date for the Sunday at or immediately after that. The week starting day is Monday (1). A simple use case is when one wants to aggregate daily data to a regular weekly sequence.

Value

Date object

Author(s)

Alexios Galanos

End of Year Date

Description

Returns the last day of the year from a Date.

Usage

calendar_eoy(date, ...)

Arguments

Details

Given a date (such as 2019-01-02), will return the last date within that year.

Value

Date object

Author(s)

Alexios Galanos

Checks on regressor matrix.

Description

Used internally by other packages, these functions provides some commonly used validation checks on regressor matrices in both in and out of sample.

Usage

check_xreg(xreg, valid_index)

check_newxreg(newdata, xreg_names = NULL, h = 1, forc_dates = NULL)

Arguments

Value

Returns the xts input matrix if checks are passed else raises an error.

Fourier terms for modeling seasonality

Description

Returns a matrix containing terms from a Fourier series, up to order K

Usage

fourier_series(dates, period = NULL, K = NULL)

Arguments

Value

A matrix of size N (length of dates) by 2*K.

Generate Regular Interval Future Dates

Description

Generates regular interval future dates for use in forecast routine.

Usage

future_dates(start, frequency, n = 1)

Arguments

Value

A Date vector

Author(s)

Alexios Galanos

Simulator Initializer

Description

Simulator Initializer

Usage

initialize_simulator(x, index = NULL, sampling = NULL, model = "issm", ...)

Arguments

Value

A object whose class depends on the type of model used.

Add Connected Line Segments to a Simulation Object

Description

Add Connected Line Segments to a Simulation Object

Usage

## S3 method for class 'issm.component'
lines(x, y = NULL, type = "l", ...)

Arguments

Details

Overlays the simulated series from the object (x), and is meant to be used when plotting different simulations from the same series for comparison.

Value

a line plot.

The logit transformation

Description

The logit transformation as an alternative to the Box Cox for bounded outcomes.

Usage

logit(lower = 0, upper = 1, ...)

Arguments

Value

A list with the transform and inverse functions.

Author(s)

Alexios Galanos

Forecast Performance Metrics

Description

Functions to calculate a number of performance metrics.

Usage

mape(actual, predicted)

bias(actual, predicted)

mslre(actual, predicted)

mase(actual, predicted, original_series = NULL, frequency = 1)

mis(actual, lower, upper, alpha)

wape(actual, predicted, weights)

wslre(actual, predicted, weights)

wse(actual, predicted, weights)

pinball(actual, distribution, alpha = 0.1)

crps(actual, distribution)

rmape(actual, predicted)

smape(actual, predicted)

msis(actual, lower, upper, original_series, frequency = 1, alpha)

Arguments

Details

The following performance metrics are implemented:

Mean Absolute Percentage Error (MAPE)

Measures the average percentage deviation of predictions from actual values.

MAPE = \frac{1}{n} \sum_{t=1}^{n} \left| \frac{A_t - P_t}{A_t} \right|

where A_t is the actual value and P_t is the predicted value.

Rescaled Mean Absolute Percentage Error (RMAPE)

A transformation of MAPE using a Box-Cox transformation for scale invariance (Swanson et al.).

Symmetric Mean Absolute Percentage Error (SMAPE)

An alternative to MAPE that symmetrizes the denominator.

SMAPE = \frac{2}{n} \sum_{t=1}^{n} \frac{|A_t - P_t|}{|A_t| + |P_t|}

Mean Absolute Scaled Error (MASE)

Compares the absolute error to the mean absolute error of a naive seasonal forecast.

MASE = \frac{\frac{1}{n} \sum_{t=1}^{n} |P_t - A_t|}{\frac{1}{N-s} \sum_{t=s+1}^{N} |A_t - A_{t-s}|}

where s is the seasonal period.

Mean Squared Logarithmic Relative Error (MSLRE)

Measures squared log relative errors to penalize large deviations.

MSLRE = \frac{1}{n} \sum_{t=1}^{n} \left( \log(1 + A_t) - \log(1 + P_t) \right)^2

Mean Interval Score (MIS)

Evaluates the accuracy of prediction intervals.

MIS = \frac{1}{n} \sum_{t=1}^{n} (U_t - L_t) + \frac{2}{\alpha} [(L_t - A_t) I(A_t < L_t) + (A_t - U_t) I(A_t > U_t)]

where L_t and U_t are the lower and upper bounds of the interval.

Mean Scaled Interval Score (MSIS)

A scaled version of MIS, dividing by the mean absolute seasonal error.

MSIS = \frac{1}{h} \sum_{t=1}^{h} \frac{(U_t - L_t) + \frac{2}{\alpha} [(L_t - A_t) I(A_t < L_t) + (A_t - U_t) I(A_t > U_t)]}{\frac{1}{N-s} \sum_{t=s+1}^{N} |A_t - A_{t-s}|}

Bias

Measures systematic overestimation or underestimation.

Bias = \frac{1}{n} \sum_{t=1}^{n} (P_t - A_t)

Weighted Absolute Percentage Error (WAPE)

A weighted version of MAPE.

WAPE = \sum_{t=1}^{n} \mathbf{w} \frac{|P_t - A_t|}{A_t}

where \mathbf{w} is the weight vector.

Weighted Squared Logarithmic Relative Error (WSLRE)

A weighted version of squared log relative errors.

WSLRE = \sum_{t=1}^{n} \mathbf{w} (\log(P_t / A_t))^2

Weighted Squared Error (WSE)

A weighted version of squared errors.

WSE = \sum_{t=1}^{n} \mathbf{w} \left( \frac{P_t}{A_t} \right)^2

Pinball Loss

A scoring rule used for quantile forecasts.

\text{Pinball} = \frac{1}{n} \sum_{t=1}^{n} \left[ \tau (A_t - Q^\tau_t) I(A_t \geq Q^\tau_t) + (1 - \tau) (Q^\tau_t - A_t) I(A_t < Q^\tau_t) \right]

where

Q^\tau_t

is the predicted quantile at level

\tau

.

Continuous Ranked Probability Score (CRPS)

A measure of probabilistic forecast accuracy.

CRPS = \frac{1}{n} \sum_{t=1}^{n} \int_{-\infty}^{\infty} (F_t(y) - I(y \geq A_t))^2 dy

where F_t(y) is the cumulative forecast distribution.

Value

A numeric value.

Note

The RMAPE is the rescaled measure for MAPE based on the paper by Swanson et al.

Author(s)

Alexios Galanos

References

Tofallis C (2015). “A better measure of relative prediction accuracy for model selection and model estimation.” Journal of the Operational Research Society, 66(8), 1352–1362.
Hyndman RJ, Koehler AB (2006). “Another look at measures of forecast accuracy.” International journal of forecasting, 22(4), 679–688.
Gneiting T, Raftery AE, Westveld III AH, Goldman T (2005). “Calibrated probabilistic forecasting using ensemble model output statistics and minimum CRPS estimation.” Monthly weather review, 133(5), 1098–1118.
Gneiting T, Raftery AE (2007). “Strictly proper scoring rules, prediction, and estimation.” Journal of the American statistical Association, 102(477), 359–378.
Swanson DA, Tayman J, Bryan TM (2011). “MAPE-R: a rescaled measure of accuracy for cross-sectional subnational population forecasts.” Journal of Population Research, 28, 225–243.

Ensemble Setup

Description

Ensemble Setup

Usage

mixture_modelspec(...)

Arguments

Details

The function performs certain checks on the inputs to ensure they conform to the simulation models in the package and are of the same length.

Value

A object of class tssim.mixture ready for ensembling,

Plot Simulation Object

Description

Plot Simulation Object

Usage

## S3 method for class 'issm.component'
plot(x, y = c("simulated", "components"), ...)

Arguments

Value

a plot of the simulated series or multiple plots of the simulation components.

POSIXct Processing

Description

Ceiling, Floor and Other operations on a POSIXct object

Usage

process_time(x, second_precision = 3600, method = ceiling, ...)

Arguments

Value

POSIXct object

Author(s)

Alexios Galanos

Examples

# end of hour
process_time(as.POSIXct('2022-08-03 03:00:01', tz = 'UTC'), 3600, method = ceiling)
# start of hour
process_time(as.POSIXct('2022-08-03 03:00:01', tz = 'UTC'), 3600, method = floor)
# end of minute
process_time(as.POSIXct('2022-08-03 03:00:01', tz = 'UTC'), 60, method = ceiling)

Infers the sampling frequency of a time series

Description

Given either a vector of time indices or an xts object will infer the sampling frequency.

Usage

sampling_frequency(x)

Arguments

Value

the sampling period (character).

Examples

w <- sampling_frequency(seq(as.Date("2010-01-01"), as.Date("2011-01-01"), by="weeks"))
m <- sampling_frequency(seq(as.POSIXct("2010-01-01 12:00:00"),
as.POSIXct("2010-01-02 12:00:00"), by="15 mins"))

Sampling frequency sequence

Description

Given a sampling period, the function will return the proportion of units of that period in secs, mins, hours, days, weeks, months and years, but will return NA for periods of higher frequency i.e. for a period of days it will return NA for secs, mins and hours. The function serves as a helper for seasonal periodicity calculations.

Usage

sampling_sequence(period)

Arguments

Value

A named numeric vector.

Author(s)

Alexios Galanos

Examples

w <- sampling_sequence(sampling_frequency(seq(as.Date("2010-01-01"),
as.Date("2011-01-01"), by="weeks")))
m <- sampling_sequence(sampling_frequency(seq(as.POSIXct("2010-01-01 12:00:00"),
as.POSIXct("2010-01-02 12:00:00"), by="15 mins")))

Seasonal Dummies

Description

Creates a matrix of seasonal dummies.

Usage

seasonal_dummies(y = NULL, n = nrow(y), seasons = 12)

Arguments

Details

Generates seasons-1 dummy variables.

Value

Either a matrix (if y is missing or y is not an xts vector) or an xts matrix (when y is an xts vector).

Author(s)

Alexios Galanos

Examples

head(seasonal_dummies(n=100, seasons=12))

Simple Seasonality Test

Description

Checks for the presence of seasonality based on the QS test of Gomez and Maravall (1996).

Usage

seasonality_test(x, frequency = NULL)

Arguments

Details

Given the identified frequency of the xts vector (using the sampling_frequency), the function checks for seasonality at that frequency. The frequency can be overridden by directly supplying a frequency argument, in which case y does not need to be a xts vector.

Value

Logical.

Author(s)

Alexios Galanos

References

Gómez V, Maravall A (1995). Programs TRAMO and SEATS. European University Institute, Florence.

The sigmoid transformation

Description

The sigmoid function is a smooth, S-shaped function that maps any real-valued input into a bounded interval, typically (0,1) . It is widely used in probability modeling, logistic regression, and neural networks as an activation function.

Usage

sigmoid(lower = 0, upper = 1, ...)

Arguments

Value

A list with the transform and inverse functions.

Author(s)

Alexios Galanos

Examples

y = cumprod(c(1, 1 + rnorm(100,0.01, 0.005)))
B = sigmoid()
yt = B$transform(y)
ye = B$inverse(yt)

The softplus logit transformation

Description

The softplus logit transformation is an alternative to the logit transform for bounded outcomes with positive output.

Usage

softlogit(lower = 0, upper = 1, ...)

Arguments

Value

A list with the transform and inverse functions.

Author(s)

Alexios Galanos

Examples

y = cumprod(c(1, 1 + rnorm(100,0.01, 0.005)))
B = softlogit(lower = 0,  upper = 15)
yt = B$transform(y)
ye = B$inverse(yt)

Generate Train/Test Splits

Description

Generates train/test splits given a vector of dates and other options

Usage

time_splits(
  x,
  start = x[1],
  test_length = 1,
  by = test_length,
  window_size = NULL,
  calendar_end = NULL,
  complete_index = TRUE,
  ...
)

Arguments

Value

A list with each slot having the training dates and test dates

Note

For months, quarters and years this will split into the end date of these. For splitting into mins or hours, x must also have this resolution else will throw an error. Additionally, the strict requirement of regularly spaced time is required (no gaps).

Author(s)

Alexios Galanos

State Decomposition

Description

State Decomposition

Usage

## S3 method for class 'issm.component'
tsdecompose(object, ...)

Arguments

Details

Creates a simplified decomposition of the states and aligns their time indices so that the sum up to the simulated component per period.

Value

A matrix of the simplified state decomposition.

Ensembling of Simulations

Description

Ensembling of Simulations

Usage

## S3 method for class 'tssim.mixture'
tsensemble(object, weights = NULL, difference = TRUE, ...)

Arguments

Details

When mixing dynamics for the same series, and when series are not stationary, differences should be used. In that case the rate of change transformation is applied to each simulated series and then weighted by the weights matrix. Since the weights matrix will have one more row than is required (the first row), this can be used to choose how the initial level is generated. For instance, if we want to use the level of the first simulated series, then the first row would have a 1 on the first column and zeros in the rest. For aggregating series, difference should be set to FALSE since we are looking at summation of data (under the assumption of flow variables). In this case, the p matrix is usually static by column (i.e. the same weights).

Value

A vector of the simulated series.

Linear Time Series Filter

Description

Estimates a simple linear time series model with trend, seasonal and regressors.

Usage

tslinear(y, trend = FALSE, seasonal = FALSE, xreg = NULL, frequency = 1, ...)

Arguments

Value

An object of class “tslinear” which also inherits “lm”.

Author(s)

Alexios Galanos

General transformation function

Description

Includes the Box Cox, logit, softplus-logit and sigmoid transforms. Returns a list of functions for the transform and its inverse.

Usage

tstransform(method = "box-cox", lambda = NULL, lower = 0, upper = 1, ...)

Arguments

Value

A list with the transform and inverse functions.

Author(s)

Alexios Galanos