iprior: Regression using priors with Fisher information covariance kernels.

Description

The iprior package provides methods to perform and analyse I-prior regression models. Estimation is done either via direct optimisation of the log-likelihood or an EM algorithm.

Author(s)

Maintainer: Haziq Jamil

Contributors:

• Wicher Bergsma

Author(s)

Maintainer: Haziq Jamil haziq.jamil@gmail.com

See Also

https://phd.haziqj.ml/

Accessor functions for ipriorMod objects.

Description

Accessor functions for ipriorMod objects.

Usage

get_intercept(object)

get_y(object)

get_size(object, units = "kB", standard = "SI")

get_hyp(object)

get_lambda(object)

get_psi(object)

get_lengthscale(object)

get_hurst(object)

get_offset(object)

get_degree(object)

get_se(object)

get_kernels(object)

get_kern_matrix(object, theta = NULL, newdata)

get_prederror(object, error.type = c("RMSE", "MSE"))

get_estl(object)

get_method(object)

get_convergence(object)

get_niter(object)

get_time(object)

get_theta(object)

Arguments

Functions

• get_intercept(): Obtain the intercept.

• get_y(): Obtain the response variables.

• get_size(): Obtain the object size of the I-prior model.

• get_hyp(): Obtain the hyerparameters of the model (both estimated and fixed ones).

• get_lambda(): Obtain the scale parameters used.

• get_psi(): Obtain the error precision.

• get_lengthscale(): Obtain the lengthscale for the SE kernels used.

• get_hurst(): Obtain the Hurst coefficient of the fBm kernels used.

• get_offset(): Obtain the offset parameters for the polynomial kernels used.

• get_degree(): Obtain the degree of the polynomial kernels used.

• get_se(): Obtain the standard errors of the estimated hyperparameters.

• get_kernels(): Obtain the kernels used.

• get_kern_matrix(): Obtain the kernel matrix of the I-prior model.

• get_prederror(): Obtain the training mean squared error.

• get_estl(): Obtain information on which hyperparameters were estimated and which were fixed.

• get_method(): Obtain the estimation method used.

• get_convergence(): Obtain the convergence information.

• get_niter(): Obtain the number of iterations performed.

• get_time(): Obtain the time taken to complete the estimation procedure.

• get_theta(): Extract the theta value at convergence. Note that this is on an unrestricted scale (see the vignette for details).

Convert difftime class into time class

Description

Convert difftime class into time class

Usage

as.time(x)

Arguments

Value

A time object which contains the time difference and units.

Check the structure of the hyperparameters of an I-prior model

Description

Check the structure of the hyperparameters of an I-prior model

Usage

check_theta(object)

Arguments

Value

A printout of the structure of the hyperparameters.

Cut a numeric vector to a certain number of decimal places

Description

Cut a numeric vector to a certain number of decimal places

Usage

decimal_place(x, k = 2)

dec_plac(x, k = 2)

Arguments

Value

A character vector with the correct number of decimal places.

Examples

decimal_place(pi, 3)
decimal_place(c(exp(1), pi, sqrt(2)), 4)

Eigen decomposition of a matrix in C++.

Description

Returns the eigenvalues and eigenvectors of a matrix X.

Usage

eigenCpp(X)

Arguments

Details

A fast implementation of eigen for symmetric, positive-definite matrices. This helps speed up the I-prior EM algorithm.

Multiplying a symmetric matrix by itself in C++.

Description

Returns the square of a symmetric matrix X.

Usage

fastSquare(X)

Arguments

Details

A fast implementation of X^2 for symmetric matrices. This helps speed up the I-prior EM algorithm.

Computing a quadratic matrix form in C++.

Description

Returns XdiagyXT.

Usage

fastVDiag(X, y)

Arguments

Details

A fast implementation of XdiagyXT. This helps speed up the I-prior EM algorithm.

Generate simulated data for multilevel models

Description

Generate simulated data for multilevel models

Usage

gen_multilevel(
  n = 25,
  m = 6,
  sigma_e = 2,
  sigma_u0 = 2,
  sigma_u1 = 2,
  sigma_u01 = -2,
  beta0 = 0,
  beta1 = 2,
  x.jitter = 0.5,
  seed = NULL
)

Arguments

Value

A dataframe containing the response variable y, the unidimensional explanatory variables X, and the levels/groups (factors).

Examples

gen_multilevel()

Generate simulated data for smoothing models

Description

Generate simulated data for smoothing models

Usage

gen_smooth(n = 150, xlim = c(0.2, 4.6), x.jitter = 0.65, seed = NULL)

Arguments

Value

A dataframe containing the response variable y and unidimensional explanatory variable X.

Examples

gen_smooth(10)

Emulate ggplot2 default colour palette

Description

Emulate ggplot2 default colour palette. ipriorColPal and ggColPal are DEPRECATED.

Usage

gg_colour_hue(x, h = c(0, 360) + 15, c = 100, l = 65)

gg_color_hue(x, h = c(0, 360) + 15, c = 100, l = 65)

gg_col_hue(x, h = c(0, 360) + 15, c = 100, l = 65)

ipriorColPal(x)

ggColPal(x)

Arguments

Details

This is the default colour scale for categorical variables in ggplot2. It maps each level to an evenly spaced hue on the colour wheel. It does not generate colour-blind safe palettes.

ipriorColPal() used to provide the colour palette for the iprior package, but this has been changed ggplot2's colour palette instead.

High school and beyond dataset

Description

A national longitudinal survey of of students from public and private high schools in the United States, with information such as students' cognitive and non-cognitive skills, high school experiences, work experiences and future plans collected.

Usage

hsb

Format

A data frame of 7185 observations on 3 variables.

mathach

Math achievement.

ses

Socio-Economic status.

schoolid

Categorical variable indicating the school the student went to. Treated as factor.

Source

High School and Beyond, 1980: A Longitudinal Survey of Students in the United States (ICPSR 7896)

References

Rabe-Hesketh, S., & Skrondal, A. (2008). Multilevel and longitudinal modeling using Stata. STATA press.

Raudenbush, S. W. (2004). HLM 6: Hierarchical linear and nonlinear modeling. Scientific Software International.

Raudenbush, S. W., & Bryk, A. S. (2002). Hierarchical linear models: Applications and data analysis methods (Vol. 1). Sage.

Examples

data(hsb)
str(hsb)

High school and beyond dataset

Description

Smaller subset of hsb.

Usage

hsbsmall

Format

A data frame of 661 observations on 3 variables.

mathach

Math achievement.

ses

Socio-Economic status.

schoolid

Categorical variable indicating the school the student went to. Treated as factor.

Details

A random subset of size 16 out of the original 160 groups.

Examples

data(hsbsmall)
str(hsbsmall)

Fit an I-prior regression model

Description

A function to perform regression using I-priors. The I-prior model parameters may be estimated in a number of ways: direct minimisation of the marginal deviance, EM algorithm, fixed hyperparameters, or using a Nystrom kernel approximation.

Usage

## Default S3 method:
iprior(
  y,
  ...,
  kernel = "linear",
  method = "direct",
  control = list(),
  interactions = NULL,
  est.lambda = TRUE,
  est.hurst = FALSE,
  est.lengthscale = FALSE,
  est.offset = FALSE,
  est.psi = TRUE,
  fixed.hyp = NULL,
  lambda = 1,
  psi = 1,
  nystrom = FALSE,
  nys.seed = NULL,
  model = list(),
  train.samp,
  test.samp,
  intercept
)

## S3 method for class 'formula'
iprior(
  formula,
  data,
  kernel = "linear",
  one.lam = FALSE,
  method = "direct",
  control = list(),
  est.lambda = TRUE,
  est.hurst = FALSE,
  est.lengthscale = FALSE,
  est.offset = FALSE,
  est.psi = TRUE,
  fixed.hyp = NULL,
  lambda = 1,
  psi = 1,
  nystrom = FALSE,
  nys.seed = NULL,
  model = list(),
  train.samp,
  test.samp,
  intercept,
  ...
)

## S3 method for class 'ipriorKernel'
iprior(object, method = "direct", control = list(), ...)

## S3 method for class 'ipriorMod'
iprior(object, method = NULL, control = list(), iter.update = 100, ...)

Arguments

Details

The iprior() function is able to take formula based input and non-formula. When not using formula, the syntax is as per the default S3 method. That is, the response variable is the vector y, and any explanatory variables should follow this, and separated by commas.

As described here, the model can be loaded first into an ipriorKernel object, and then passed to the iprior() function to perform the estimation.

Value

An ipriorMod object. Several accessor functions have been written to obtain pertinent things from the ipriorMod object. The print() and summary() methods display the relevant model information.

Methods (by class)

• iprior(ipriorKernel): Takes in object of type ipriorKernel, a loaded and prepared I-prior model, and proceeds to estimate it.

• iprior(ipriorMod): Re-run or continue running the EM algorithm from last attained parameter values in object ipriorMod.

See Also

optim, update, check_theta, print, summary, plot, coef, sigma, fitted, predict, logLik, deviance.

Examples

# Formula based input
(mod.stackf <- iprior(stack.loss ~ Air.Flow + Water.Temp + Acid.Conc.,
                      data = stackloss))
mod.toothf <- iprior(len ~ supp * dose, data = ToothGrowth)
summary(mod.toothf)

# Non-formula based input
mod.stacknf <- iprior(y = stackloss$stack.loss,
                      Air.Flow = stackloss$Air.Flow,
                      Water.Temp = stackloss$Water.Temp,
                      Acid.Conc. = stackloss$Acid.Conc.)
mod.toothnf <- iprior(y = ToothGrowth$len, ToothGrowth$supp, ToothGrowth$dose,
                      interactions = "1:2")

# Formula based model option one.lam = TRUE
# Sets a single scale parameter for all variables
modf <- iprior(stack.loss ~ ., data = stackloss, one.lam = TRUE)
modnf <- iprior(y = stackloss$stack.loss, X = stackloss[1:3])
all.equal(coef(modnf), coef(modnf))  # both models are equivalent

# Fit models using different kernels
dat <- gen_smooth(n = 100)
mod <- iprior(y ~ X, dat, kernel = "fbm")  # Hurst = 0.5 (default)
mod <- iprior(y ~ X, dat, kernel = "poly3")  # polynomial degree 3

# Fit models using various estimation methods
mod1 <- iprior(y ~ X, dat)
mod2 <- iprior(y ~ X, dat, method = "em")
mod3 <- iprior(y ~ X, dat, method = "canonical")
mod4 <- iprior(y ~ X, dat, method = "mixed")
mod5 <- iprior(y ~ X, dat, method = "fixed", lambda = coef(mod1)[1],
               psi = coef(mod1)[2])
c(logLik(mod1), logLik(mod2), logLik(mod3), logLik(mod4),
  logLik(mod5))

## Not run: 

# For large data sets, it is worth trying the Nystrom method
mod <- iprior(y ~ X, gen_smooth(5000), kernel = "se", nystrom = 50,
              est.lengthscale = TRUE)  # a bit slow
plot_fitted(mod, ci = FALSE)

## End(Not run)

Perform a cross-validation experiment with the iprior function

Description

A convenience function to perform a k-fold cross-validation experiment and obtain mean squared error of prediction. Most of the arguments are similar to iprior() and kernL().

Usage

## Default S3 method:
iprior_cv(
  y,
  ...,
  folds = 2,
  par.cv = TRUE,
  kernel = "linear",
  method = "direct",
  control = list(),
  interactions = NULL,
  est.lambda = TRUE,
  est.hurst = FALSE,
  est.lengthscale = FALSE,
  est.offset = FALSE,
  est.psi = TRUE,
  fixed.hyp = NULL,
  lambda = 1,
  psi = 1,
  nystrom = FALSE,
  nys.seed = NULL
)

## S3 method for class 'formula'
iprior_cv(
  formula,
  data,
  folds = 2,
  one.lam = FALSE,
  par.cv = TRUE,
  kernel = "linear",
  method = "direct",
  control = list(),
  est.lambda = TRUE,
  est.hurst = FALSE,
  est.lengthscale = FALSE,
  est.offset = FALSE,
  est.psi = TRUE,
  fixed.hyp = NULL,
  lambda = 1,
  psi = 1,
  nystrom = FALSE,
  nys.seed = NULL,
  ...
)

Arguments

Details

Uses a multicore loop to fit the folds by default, set par.cv = FALSE to not use multithreading.

Value

An iprior_xv object containing a data frame of the cross-validated values such as the log-likelihood, training MSE and test MSE.

Examples

## Not run: 

# 5-fold CV experiment
(mod.cv <- iprior_cv(y ~ X, gen_smooth(100), kernel = "se", folds = 5))

# LOOCV experiment
(mod.cv <- iprior_cv(y ~ X, gen_smooth(100), kernel = "se", folds = Inf))

# Can also get root MSE
print(mod.cv, "RMSE")

## End(Not run)

Test iprior objects

Description

Test whether an object is an ipriorMod, ipriorKernel, or either object with Nystrom method enabled.

Usage

is.ipriorMod(x)

is.ipriorKernel(x)

is.nystrom(x)

Arguments

Value

Logical.

Test kernel attributes

Description

Test whether an object uses a specific type of kernel.

Usage

is.kern_linear(x)

is.kern_canonical(x)

is.kern_fbm(x)

is.kern_pearson(x)

is.kern_se(x)

is.kern_poly(x)

Arguments

Value

Logical.

Load the kernel matrices for I-prior models

Description

Load the kernel matrices for I-prior models

Usage

kernL(
  y,
  ...,
  kernel = "linear",
  interactions = NULL,
  est.lambda = TRUE,
  est.hurst = FALSE,
  est.lengthscale = FALSE,
  est.offset = FALSE,
  est.psi = TRUE,
  fixed.hyp = NULL,
  lambda = 1,
  psi = 1,
  nystrom = FALSE,
  nys.seed = NULL,
  model = list(),
  train.samp,
  test.samp,
  intercept
)

## S3 method for class 'formula'
kernL(
  formula,
  data,
  kernel = "linear",
  one.lam = FALSE,
  est.lambda = TRUE,
  est.hurst = FALSE,
  est.lengthscale = FALSE,
  est.offset = FALSE,
  est.psi = TRUE,
  fixed.hyp = NULL,
  lambda = 1,
  psi = 1,
  nystrom = FALSE,
  nys.seed = NULL,
  model = list(),
  train.samp,
  test.samp,
  intercept,
  ...
)

Arguments

Value

An ipriorKernel object which contains the relevant material to be passed to the iprior function for model fitting.

See Also

iprior

Examples

str(ToothGrowth)
(mod <- kernL(y = ToothGrowth$len,
               supp = ToothGrowth$supp,
               dose = ToothGrowth$dose,
               interactions = "1:2"))
kernL(len ~ supp * dose, data = ToothGrowth)  # equivalent formula call

# Choosing different kernels
str(stackloss)
kernL(stack.loss ~ ., stackloss, kernel = "fbm")  # all fBm kernels
kernL(stack.loss ~ ., stackloss, kernel = "FBm")  # cApS dOn't MatTeR
kernL(stack.loss ~ ., stackloss,
       kernel = c("linear", "se", "poly3"))  # different kernels

# Sometimes the print output is too long, can use str() options here
print(mod, strict.width = "cut", width = 30)

Reproducing kernels for the I-prior package

Description

The kernel functions used in this package are:

• The (canonical) linear kernel

• The fractional Brownian motion (fBm) kernel with Hurst index \gamma

• The Pearson kernel

• The (scaled) d-degree polynomial kernel with offset c

• The squared exponential (SE) kernel with lengthscale l

Usage

kern_canonical(x, y = NULL, centre = TRUE)

kern_linear(x, y = NULL, centre = TRUE)

kern_pearson(x, y = NULL)

kern_fbm(x, y = NULL, gamma = 0.5, centre = TRUE)

kern_se(x, y = NULL, l = 1, centre = FALSE)

kern_poly(x, y = NULL, c = 0, d = 2, lam.poly = 1, centre = TRUE)

Arguments

Details

The Pearson kernel is used for nominal-type variables, and thus factor-type variables are treated with the Pearson kernel automatically when fitting I-prior models. The other kernels are for continuous variables, and each emits different properties of functions.

The linear kernel is used for "straight-line" functions. In addition, if squared, cubic, or higher order terms are to be modelled, then the polynomial kernel is suitable for this purpose. For smoothing models, the fBm kernel is preferred, although the SE kernel may be used as well.

Value

A matrix whose [i, j] entries are given by h(\code{x[i]}, \code{y[j]}), with h being the appropriate kernel function. The matrix has dimensions m by n according to the lengths of y and x respectively. When a single argument x is supplied, then y is taken to be equal to x, and a symmetric n by n matrix is returned.

The matrix has a "kernel" attribute indicating which type of kernel function was called.

References

https://phd.haziqj.ml/intro/

Examples

kern_linear(1:3)
kern_fbm(1:5, 1:3, gamma = 0.7)

Obtain the log-likelihood and deviance of an I-prior model

Description

This function calculates the log-likelihood value or deviance (twice the negative log-likelihood) for I-prior models. It works for both ipriorMod and ipriorKernel class objects.

Usage

## S3 method for class 'ipriorMod'
logLik(object, theta = NULL, ...)

## S3 method for class 'ipriorMod'
deviance(object, theta = NULL, ...)

## S3 method for class 'ipriorKernel'
logLik(object, theta = NULL, ...)

## S3 method for class 'ipriorKernel'
deviance(object, theta = NULL, ...)

Arguments

Details

For ipriorKernel objects, the log-likelihood or deviance is calculated at the default parameter values: scale parameters and error precision are equal to one, while hyperparameters of the kernels (e.g. Hurst index, lengthscale, etc.) are the default values (see here for details) or ones that has been specified. For ipriorMod objects, the log-likelihood or deviance is calculated at the last obtained value from the estimation method.

For both types of objects, it is possible to supply parameter values at which to calculate the log-likelihood/deviance. This makes estimating an I-prior model more flexible, by first loading the variables into an ipriorKernel object, and then using an optimiser such as optim. Parameters have been transformed so that they can be optimised unconstrained.

See Also

check_theta.

Plots for I-prior models

Description

There are three types of plots that are currently written in the package:

plot_fitted

Plot the fitted regression line with credibility bands.

plot_predict

Plot residuals against fitted values.

plot_iter

Plot the progression of the log-likelihood value over time.

The S3 method plot for class ipriorMod currently returns plot_fitted.

Usage

## S3 method for class 'ipriorMod'
plot(x, ...)

plot_resid(x)

plot_fitted_multilevel(
  x,
  X.var = 1,
  grp.var = 1,
  facet = c(2, 3),
  cred.bands = TRUE,
  show.legend = TRUE,
  show.points = TRUE,
  x.lab = NULL,
  y.lab = NULL,
  grp.lab = NULL,
  extrapolate = FALSE
)

plot_fitted(x, X.var = 1, cred.bands = TRUE, size = 1, linetype = "solid")

plot_iter(x, niter.plot = NULL, lab.pos = c("up", "down"))

plot_ppc(x, draws = 100)

Arguments

Details

See ggplot2 documentation for the plotting parameters.

Air pollution and mortality

Description

Data on the relation between weather, socioeconomic, and air pollution variables and mortality rates in 60 Standard Metropolitan Statistical Areas (SMSAs) of the USA, for the years 1959-1961.

Usage

pollution

Format

A data frame of 16 observations on 16 variables.

Mortality

Total age-adjusted mortality rate per 100,000.

Rain

Mean annual precipitation in inches.

Humid

Mean annual precipitation in inches.

JanTemp

Mean annual precipitation in inches.

JulTemp

Mean annual precipitation in inches.

Over65

Mean annual precipitation in inches.

Popn

Mean annual precipitation in inches.

Educ

Mean annual precipitation in inches.

Hous

Mean annual precipitation in inches.

Dens

Mean annual precipitation in inches.

NonW

Mean annual precipitation in inches.

WhiteCol

Mean annual precipitation in inches.

Poor

Mean annual precipitation in inches.

HC

Mean annual precipitation in inches.

NOx

Mean annual precipitation in inches.

SO2

Mean annual precipitation in inches.

References

McDonald, G. C. and Schwing, R. C. (1973). Instabilities of regression estimates relating air pollution to mortality. Technometrics, 15(3):463-481.

Examples

data(pollution)
str(pollution)

Obtain predicted values from ipriorMod objects

Description

Obtain predicted values from ipriorMod objects

Usage

## S3 method for class 'ipriorMod'
fitted(object, intervals = FALSE, alpha = 0.05, ...)

## S3 method for class 'ipriorMod'
predict(
  object,
  newdata = list(),
  y.test = NULL,
  intervals = FALSE,
  alpha = 0.05,
  ...
)

## S3 method for class 'ipriorPredict'
print(x, rows = 10, dp = 3, ...)

Arguments

Value

A list of class ipriorPredict containing the fitted values, residuals (observed minus fitted), the training mean squared error, and the lower and upper intervals (if called).

Examples

dat <- gen_smooth(20)
mod <- iprior(y ~ ., dat, kernel = "se")
fitted(mod)
fitted(mod, intervals = TRUE)
predict(mod, gen_smooth(5))

with(dat, mod <<- iprior(y, X, kernel = "poly"))
newdat <- gen_smooth(30)
mod.pred <- predict(mod, list(newdat$X), y.test = newdat$y, intervals = TRUE)
str(mod.pred)
print(mod.pred, row = 5)

Obtain the standard deviation of the residuals 'sigma'

Description

Extract the standard deviation of the residuals. For I-prior models, this is sigma = 1 / sqrt(psi).

Usage

## S3 method for class 'ipriorMod'
sigma(object, ...)

Arguments

Print and summary method for I-prior models

Description

Print and summary method for I-prior models

Usage

## S3 method for class 'ipriorMod'
print(x, digits = 5, ...)

## S3 method for class 'ipriorMod'
summary(object, ...)

Arguments

Results of I-prior cross-validation experiment on Tecator data set

Description

Results of I-prior cross-validation experiment on Tecator data set

Usage

tecator.cv

Format

Results from iprior_cv cross validation experiment. This is a list of seven, with each component bearing the results for the linear, quadratic, cubic, fBm-0.5, fBm-MLE and SE I-prior models. The seventh is a summarised table of the results.

Details

For the fBm and SE kernels, it seems numerical issues arise when using a direct optimisation approach. Terminating the algorithm early (say using a relaxed stopping criterion) seems to help.

Examples

# Results from the six experiments
print(tecator.cv[[1]], "RMSE")
print(tecator.cv[[2]], "RMSE")
print(tecator.cv[[3]], "RMSE")
print(tecator.cv[[4]], "RMSE")
print(tecator.cv[[5]], "RMSE")
print(tecator.cv[[6]], "RMSE")

# Summary of results
print(tecator.cv[[7]])

## Not run: 

# Prepare data set
data(tecator, package = "caret")
endpoints <- as.data.frame(endpoints)
colnames(endpoints) <- c("water", "fat", "protein")
absorp <- -t(diff(t(absorp)))  # this takes first differences using diff()
fat <- endpoints$fat

# Here is the code to replicate the results
mod1.cv <- iprior_cv(fat, absorp, folds = Inf)
mod2.cv <- iprior_cv(fat, absorp, folds = Inf, kernel = "poly2",
                     est.offset = TRUE)
mod3.cv <- iprior_cv(fat, absorp, folds = Inf, kernel = "poly3",
                     est.offset = TRUE)
mod4.cv <- iprior_cv(fat, absorp, method = "em", folds = Inf, kernel = "fbm",
                     control = list(stop.crit = 1e-2))
mod5.cv <- iprior_cv(fat, absorp, folds = Inf, kernel = "fbm",
                     est.hurst = TRUE, control = list(stop.crit = 1e-2))
mod6.cv <- iprior_cv(fat, absorp, folds = Inf, kernel = "se",
                     est.lengthscale = TRUE, control = list(stop.crit = 1e-2))

tecator_res_cv <- function(mod) {
  res <- as.numeric(apply(mod$res[, -1], 2, mean))  # Calculate RMSE
  c("Training RMSE" = res[1], "Test RMSE" = res[2])
}

tecator_tab_cv <- function() {
  tab <- t(sapply(list(mod1.cv, mod2.cv, mod3.cv, mod4.cv, mod5.cv, mod6.cv),
                  tecator_res_cv))
  rownames(tab) <- c("Linear", "Quadratic", "Cubic", "fBm-0.5", "fBm-MLE",
                     "SE-MLE")
  tab
}

tecator.cv <- list(
  "linear"   = mod1.cv,
  "qudratic" = mod2.cv,
  "cubic"    = mod3.cv,
  "fbm-0.5"  = mod4.cv,
  "fbm-MLE"  = mod5.cv,
  "SE"       = mod6.cv,
  "summary"  = tecator_tab_cv()
)

## End(Not run)

Update an I-prior model

Description

Update an I-prior model

Usage

## S3 method for class 'ipriorMod'
update(object, method = NULL, control = list(), iter.update = 100, ...)

Arguments