TSP instance generator (for testing/examples)

Description

Adapted from stats::optim(). Check their documentation / examples for details.

Usage

TSP.dist(x, mydist)

Arguments

Bootstrap the sampling distribution of the mean

Description

Bootstraps the sampling distribution of the means for a given vector of observations

Usage

boot_sdm(x, boot.R = 999, ncpus = 1, seed = NULL)

Arguments

Value

vector of bootstrap estimates of the sample mean

References

• A.C. Davison, D.V. Hinkley: Bootstrap methods and their application. Cambridge University Press (1997)

• F. Campelo, F. Takahashi: Sample size estimation for power and accuracy in the experimental comparison of algorithms. Journal of Heuristics 25(2):305-338, 2019.

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

Examples

x <- rnorm(15, mean = 4, sd = 1)
my.sdm <- boot_sdm(x)
hist(my.sdm, breaks = 30)
qqnorm(my.sdm, pch = 20)

x <- runif(12)
my.sdm <- boot_sdm(x)
qqnorm(my.sdm, pch = 20)

# Convergence of the SDM to a Normal distribution as sample size is increased
X <- rchisq(1000, df = 3)
x1 <- rchisq(10, df = 3)
x2 <- rchisq(20, df = 3)
x3 <- rchisq(40, df = 3)
par(mfrow = c(2, 2))
plot(density(X), main = "Estimated pop distribution");
hist(boot_sdm(x1), breaks = 25, main = "SDM, n = 10")
hist(boot_sdm(x2), breaks = 25, main = "SDM, n = 20")
hist(boot_sdm(x3), breaks = 25, main = "SDM, n = 40")
par(mfrow = c(1, 1))

Calculates number of instances for the comparison of multiple algorithms

Description

Calculates either the number of instances, or the power(s) of the comparisons of multiple algorithms.

Usage

calc_instances(
  ncomparisons,
  d,
  ninstances = NULL,
  power = NULL,
  sig.level = 0.05,
  alternative.side = "two.sided",
  test = "t.test",
  power.target = "mean"
)

Arguments

Details

The main use of this routine uses the closed formula of the t-test to calculate the number of instances required for the comparison of pairs of algorithms, given a desired power and standardized effect size of interest. Significance levels of each comparison are adjusted using Holm's step-down correction (the default). The routine also takes into account whether the desired statistical power refers to the mean power (the default), median, or worst-case (which is equivalent to designing the experiment for the more widely-known Bonferroni correction). See the reference by ⁠Campelo and Wanner⁠ for details.

Value

a list object containing the following items:

• ninstances - number of instances

• power - the power of the comparison

• d - the effect size

• sig.level - significance level

• alternative.side - type of alternative hypothesis

• test - type of test

Sample Sizes for Nonparametric Methods

If the parameter test is set to either Wilcoxon or Binomial, this routine approximates the number of instances using the ARE of these tests in relation to the paired t.test, using the formulas (see reference by ⁠Campelo and Takahashi⁠ for details):

n.wilcox = n.ttest / 0.86 = 1.163 * n.ttest

n.binom = n.ttest / 0.637 = 1.570 * n.ttest

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

References

• P. Mathews. Sample size calculations: Practical methods for engineers and scientists. Mathews Malnar and Bailey, 2010.

• F. Campelo, F. Takahashi: Sample size estimation for power and accuracy in the experimental comparison of algorithms. Journal of Heuristics 25(2):305-338, 2019.

• F. Campelo, E. Wanner: Sample size calculations for the experimental comparison of multiple algorithms on multiple problem instances. Submitted, Journal of Heuristics, 2019.

Examples

# Calculate sample size for mean-case power
K      <- 10   # number of comparisons
alpha  <- 0.05 # significance level
power  <- 0.9  # desired power
d      <- 0.5  # MRES

out <- calc_instances(K, d,
                      power     = power,
                      sig.level = alpha)

# Plot power of each comparison to detect differences of magnitude d
plot(1:K, out$power,
     type = "b", pch = 20, las = 1, ylim = c(0, 1), xlab = "comparison",
     ylab = "power", xaxs = "i", xlim = c(0, 11))
grid(11, NA)
points(c(0, K+1), c(power, power), type = "l", col = 2, lty = 2, lwd = .5)
text(1, 0.93, sprintf("Mean power = %2.2f for N = %d",
                     out$mean.power, out$ninstances), adj = 0)

# Check sample size if planning for Wilcoxon tests:
calc_instances(K, d,
               power     = power,
               sig.level = alpha,
               test = "wilcoxon")$ninstances


# Calculate power profile for predefined sample size
N <- 45
out2 <- calc_instances(K, d, ninstances = N, sig.level = alpha)

points(1:K, out2$power, type = "b", pch = 19, col = 3)
text(6, .7, sprintf("Mean power = %2.2f for N = %d",
                     out2$mean.power, out2$ninstances), adj = 0)

# Sample size for worst-case (Bonferroni) power of 0.8, using Wilcoxon
out3 <- calc_instances(K, d, power = 0.9, sig.level = alpha,
                       test = "wilcoxon", power.target = "worst.case")
out3$ninstances

# For median power:
out4 <- calc_instances(K, d, power = 0.9, sig.level = alpha,
                       test = "wilcoxon", power.target = "median")
out4$ninstances
out4$power

Determine sample sizes for a set of algorithms on a single problem instance

Description

Iteratively calculates the required sample sizes for K algorithms on a given problem instance, so that the standard errors of the estimates of the pairwise differences in performance is controlled at a predefined level.

Usage

calc_nreps(
  instance,
  algorithms,
  se.max,
  dif = "simple",
  comparisons = "all.vs.all",
  method = "param",
  nstart = 20,
  nmax = 1000,
  seed = NULL,
  boot.R = 499,
  ncpus = 1,
  force.balanced = FALSE,
  load.folder = NA,
  save.folder = NA
)

Arguments

Value

a list object containing the following items:

• instance - alias for the problem instance considered

• Xk - list of observed performance values for all algorithms

• Nk - vector of sample sizes generated for each algorithm

• Diffk - data frame with point estimates, standard errors and other information for all algorithm pairs of interest

• seed - seed used for the PRNG

• dif - type of difference used

• method - method used ("param" / "boot")

• comparisons - type of pairings ("all.vs.all" / "all.vs.first")

Instance

Parameter instance must be a named list containing all relevant parameters that define the problem instance. This list must contain at least the field instance$FUN, with the name of the function implementing the problem instance, that is, a routine that calculates y = f(x). If the instance requires additional parameters, these must also be provided as named fields.

Algorithms

Object algorithms is a list in which each component is a named list containing all relevant parameters that define an algorithm to be applied for solving the problem instance. In what follows algorithm[[k]] refers to any algorithm specified in the algorithms list.

algorithm[[k]] must contain an algorithm[[k]]$FUN field, which is a character object with the name of the function that calls the algorithm; as well as any other elements/parameters that algorithm[[k]]$FUN requires (e.g., stop criteria, operator names and parameters, etc.).

The function defined by the routine algorithm[[k]]$FUN must have the following structure: supposing that the list in algorithm[[k]] has fields algorithm[[k]]$FUN = "myalgo", algorithm[[k]]$par1 = "a" and algorithm$par2 = 5, then:

         myalgo <- function(par1, par2, instance, ...){
               # do stuff
               # ...
               return(results)
         }
   

That is, it must be able to run if called as:

         # remove '$FUN' and '$alias' fields from list of arguments
         # and include the problem definition as field 'instance'
         myargs          <- algorithm[names(algorithm) != "FUN"]
         myargs          <- myargs[names(myargs) != "alias"]
         myargs$instance <- instance

         # call function
         do.call(algorithm$FUN,
                 args = myargs)
   

The algorithm$FUN routine must return a list containing (at least) the performance value of the final solution obtained, in a field named value (e.g., result$value) after a given run.

Initial Number of Observations

In the general case the initial number of observations per algorithm (nstart) should be relatively high. For the parametric case we recommend between 10 and 20 if outliers are not expected, or between 30 and 50 if that assumption cannot be made. For the bootstrap approach we recommend using at least 20. However, if some distributional assumptions can be made - particularly low skewness of the population of algorithm results on the test instances), then nstart can in principle be as small as 5 (if the output of the algorithms were known to be normal, it could be 1).

In general, higher sample sizes are the price to pay for abandoning distributional assumptions. Use lower values of nstart with caution.

Pairwise Differences

Parameter dif informs the type of difference in performance to be used for the estimation (\mu_a and \mu_b represent the mean performance of any two algorithms on the test instance, and mu represents the grand mean of all algorithms given in algorithms):

• If dif == "perc" and comparisons == "all.vs.first", the estimated quantity is \phi_{1b} = (\mu_1 - \mu_b) / \mu_1 = 1 - (\mu_b / \mu_1).

• If dif == "perc" and comparisons == "all.vs.all", the estimated quantity is \phi_{ab} = (\mu_a - \mu_b) / \mu.

• If dif == "simple" it estimates \mu_a - \mu_b.

Author(s)

Felipe Campelo (fcampelo@gmail.com)

References

• F. Campelo, F. Takahashi: Sample size estimation for power and accuracy in the experimental comparison of algorithms. Journal of Heuristics 25(2):305-338, 2019.

• P. Mathews. Sample size calculations: Practical methods for engineers and scientists. Mathews Malnar and Bailey, 2010.

• A.C. Davison, D.V. Hinkley: Bootstrap methods and their application. Cambridge University Press (1997)

• E.C. Fieller: Some problems in interval estimation. Journal of the Royal Statistical Society. Series B (Methodological) 16(2), 175–185 (1954)

• V. Franz: Ratios: A short guide to confidence limits and proper use (2007). https://arxiv.org/pdf/0710.2024v1.pdf

• D.C. Montgomery, C.G. Runger: Applied Statistics and Probability for Engineers, 6th ed. Wiley (2013)

Examples

# Example using dummy algorithms and instances. See ?dummyalgo for details.
# We generate dummy algorithms with true means 15, 10, 30, 15, 20; and true
# standard deviations 2, 4, 6, 8, 10.
algorithms <- mapply(FUN = function(i, m, s){
                          list(FUN   = "dummyalgo",
                               alias = paste0("algo", i),
                               distribution.fun  = "rnorm",
                               distribution.pars = list(mean = m, sd = s))},
                     i = c(alg1 = 1, alg2 = 2, alg3 = 3, alg4 = 4, alg5 = 5),
                     m = c(15, 10, 30, 15, 20),
                     s = c(2, 4, 6, 8, 10),
                     SIMPLIFY = FALSE)

# Make a dummy instance with a centered (zero-mean) exponential distribution:
instance = list(FUN = "dummyinstance", distr = "rexp", rate = 5, bias = -1/5)

# Explicitate all other parameters (just this one time:
# most have reasonable default values)
myreps <- calc_nreps(instance   = instance,
                      algorithms = algorithms,
                      se.max     = 0.05,          # desired (max) standard error
                      dif        = "perc",        # type of difference
                      comparisons = "all.vs.all", # differences to consider
                      method     = "param",       # method ("param", "boot")
                      nstart     = 15,            # initial number of samples
                      nmax       = 1000,          # maximum allowed sample size
                      seed       = 1234,          # seed for PRNG
                      boot.R     = 499,           # number of bootstrap resamples (unused)
                      ncpus      = 1,             # number of cores to use
                      force.balanced = FALSE,     # force balanced sampling?
                      load.folder   = NA,         # file to load results from
                      save.folder = NA)         # folder to save results
summary(myreps)
plot(myreps)

Calculates the standard error for simple and percent differences

Description

Calculates the sample standard error for the estimator differences between multiple algorithms on a given instance.

Usage

calc_se(
  Xk,
  dif = "simple",
  comparisons = "all.vs.all",
  method = "param",
  boot.R = 999
)

Arguments

Details

• If dif == "perc" it returns the standard errors for the sample estimates of pairs (mu2 - mu1) / mu, where mu1, mu2 are the means of the populations that generated sample vectors x1, x2, and

• If dif == "simple" it returns the SE for sample estimator of (mu2 - mu1)

Value

a list object containing the following items:

• Phi.est - estimated values of the statistic of interest for each pair of algorithms of interest (all pairs if comparisons == "all.vs.all", or all pairs containing the first algorithm if comparisons == "all.vs.first").

• se - standard error estimates

References

• F. Campelo, F. Takahashi: Sample size estimation for power and accuracy in the experimental comparison of algorithms. Journal of Heuristics 25(2):305-338, 2019.

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

Examples

# three vectors of normally distributed observations
set.seed(1234)
Xk <- list(rnorm(10, 5, 1),  # mean = 5, sd = 1,
           rnorm(20, 10, 2), # mean = 10, sd = 2,
           rnorm(50, 15, 5)) # mean = 15, sd = 3

calc_se(Xk, dif = "simple", comparisons = "all.vs.all", method = "param")
calc_se(Xk, dif = "simple", comparisons = "all.vs.all", method = "boot")

calc_se(Xk, dif = "perc", comparisons = "all.vs.first", method = "param")
calc_se(Xk, dif = "perc", comparisons = "all.vs.first", method = "boot")

calc_se(Xk, dif = "perc", comparisons = "all.vs.all", method = "param")
calc_se(Xk, dif = "perc", comparisons = "all.vs.all", method = "boot")

Consolidate results from partial files

Description

Consolidates results from a set of partial files (each generated by an individual call to calc_nreps()) into a single output structure, similar (but not identical) to the output of run_experiment(). This is useful e.g., to consolidate the results from interrupted experiments.

Usage

consolidate_partial_results(Configuration, folder = "./nreps_files")

Arguments

Value

a list object containing the following fields:

• data.raw - data frame containing all observations generated

• data.summary - data frame summarizing the experiment.

• N - number of instances sampled

• total.runs - total number of algorithm runs performed

• instances.sampled - names of the instances sampled

Dummy algorithm routine to test the sampling procedures

Description

This is a dummy algorithm routine to test the sampling procedures, in combination with dummyinstance(). dummyalgo() receives two parameters that determine the distribution of performances it will exhibit on a hypothetical problem class: distribution.fun (with the name of a random number generation function, e.g. rnorm, runif, rexp etc.); and distribution.pars, a named list of parameters to be passed on to distribution.fun. The third parameter is an instance object (see calc_nreps() for details), which is a named list with the following fields:

• FUN = "dummyinstance" - must always be "dummyinstance" (will be ignored otherwise).

• distr - the name of a random number generation function.

• ... - other named fields with parameters to be passed down to the function in distr.

Usage

dummyalgo(
  distribution.fun = "rnorm",
  distribution.pars = list(mean = 0, sd = 1),
  instance = list(FUN = "dummyinstance", distr = "rnorm", mean = 0, sd = 1)
)

Arguments

Details

distribution.fun and distribution.pars regulate the mean performance of the dummy algorithm on a given (hypothetical) problem class, and the between-instances variance of performance. The instance specification in instance regulates the within-instance variability of results. Ideally the distribution parameters passed to the instance should result in a within-instance distribution of values with zero mean, so that the mean of the values returned by dummyalgo is regulated only by distribution.fun and distribution.pars.

The value returned by dummyalgo is sampled as follows:

   offset <- do.call(distribution.fun, args = distribution.pars)
   y <- offset + do.call("dummyinstance", args = instance)

Value

a list object with a single field $value, containing a scalar numerical value distributed as described at the end of Details.

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

See Also

dummyinstance()

Examples

# Make a dummy instance with a centered (zero-mean) exponential distribution:
instance = list(FUN = "dummyinstance", distr = "rexp", rate = 5, bias = -1/5)

# Simulate a dummy algorithm that has a uniform distribution of expected
# performance values, between -25 and 50.
dummyalgo(distribution.fun = "runif",
          distribution.pars = list(min = -25, max = 50),
          instance = instance)

Dummy instance routine to test the sampling procedures

Description

This is a dummy instance routine to test the sampling procedures, in combination with dummyalgo(). dummyinstance() receives a parameter distr containing the name of a random number generation function (e.g. rnorm, runif, rexp etc.), plus a variable number of arguments to be passed down to the function in distr.

Usage

dummyinstance(distr, ..., bias = 0)

Arguments

Value

a single numeric value sampled from the desired distribution.

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

See Also

dummyalgo()

Examples

dummyinstance(distr = "rnorm", mean = 10, sd = 1)

# Make a centered (zero-mean) exponential distribution:
lambda = 4

# 10000 observations
set.seed(1234)
y <- numeric(10000)
for (i in 1:10000) y[i] <- dummyinstance(distr = "rexp", rate = lambda,
                                         bias = -1/lambda)
mean(y)
hist(y, breaks = 50, xlim = c(-0.5, 2.5))

Simulated annealing (for testing/examples)

Description

Adapted from stats::optim(). Check their documentation / examples for details.

Usage

example_SANN(Temp, budget, instance)

Arguments

Examples

## Not run: 
instance <- list(FUN = "TSP.dist", mydist = datasets::eurodist)

example_SANN(Temp = 2000, budget = 10000, instance = instance)

## End(Not run)

Run an algorithm on a problem.

Description

Call algorithm routine for the solution of a problem instance

Usage

get_observations(algo, instance, n = 1)

Arguments

Value

vector of observed performance values

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

See Also

calc_nreps()

Examples

# Make a dummy instance with a centered (zero-mean) exponential distribution:
instance <- list(FUN = "dummyinstance", distr = "rexp", rate = 5, bias = -1/5)

# Simulate a dummy algorithm that has a uniform distribution of expected
# performance values, between -25 and 50.
algorithm <- list(FUN = "dummyalgo",
                 distribution.fun = "runif",
                 distribution.pars = list(min = -25, max = 50))
x <- get_observations(algorithm, instance, n = 1000)
hist(x)

plot.CAISEr

Description

S3 method for plotting CAISEr objects output by run_experiment().

Usage

## S3 method for class 'CAISEr'
plot(
  x,
  y = NULL,
  ...,
  latex = FALSE,
  reorder = FALSE,
  show.text = TRUE,
  digits = 3,
  layout = NULL
)

Arguments

Value

list containing (1) a list of of ggplot2 objects generated, and (2) a list of data frames used for the creation of the plots.

plot.nreps

Description

S3 method for plotting nreps objects output by calc_nreps()).

Usage

## S3 method for class 'nreps'
plot(
  x,
  y = NULL,
  ...,
  instance.name = NULL,
  latex = FALSE,
  show.SE = TRUE,
  show.CI = TRUE,
  sig.level = 0.05,
  show.text = TRUE
)

Arguments

Value

ggplot object (invisibly)

print.CAISEr

Description

S3 method for printing CAISEr objects (the output of run_experiment()).

Usage

## S3 method for class 'CAISEr'
print(x, ..., echo = TRUE, digits = 4, right = TRUE, breakrows = FALSE)

Arguments

Value

data frame object containing the summary table (invisibly)

Examples

# Example using four dummy algorithms and 100 dummy instances.
# See [dummyalgo()] and [dummyinstance()] for details.
# Generating 4 dummy algorithms here, with means 15, 10, 30, 15 and standard
# deviations 2, 4, 6, 8.
algorithms <- mapply(FUN = function(i, m, s){
  list(FUN   = "dummyalgo",
       alias = paste0("algo", i),
       distribution.fun  = "rnorm",
       distribution.pars = list(mean = m, sd = s))},
  i = c(alg1 = 1, alg2 = 2, alg3 = 3, alg4 = 4),
  m = c(15, 10, 30, 15),
  s = c(2, 4, 6, 8),
  SIMPLIFY = FALSE)

# Generate 100 dummy instances with centered exponential distributions
instances <- lapply(1:100,
                    function(i) {rate <- runif(1, 1, 10)
                                 list(FUN   = "dummyinstance",
                                      alias = paste0("Inst.", i),
                                      distr = "rexp", rate = rate,
                                      bias  = -1 / rate)})

my.results <- run_experiment(instances, algorithms,
                             d = 1, se.max = .1,
                             power = .9, sig.level = .05,
                             power.target = "mean",
                             dif = "perc", comparisons = "all.vs.all",
                             seed = 1234, ncpus = 1)
my.results

Run a full experiment for comparing multiple algorithms using multiple instances

Description

Design and run a full experiment - calculate the required number of instances, run the algorithms on each problem instance using the iterative approach based on optimal sample size ratios, and return the results of the experiment. This routine builds upon calc_instances() and calc_nreps(), so refer to the documentation of these two functions for details.

Usage

run_experiment(
  instances,
  algorithms,
  d,
  se.max,
  power = 0.8,
  sig.level = 0.05,
  power.target = "mean",
  dif = "simple",
  comparisons = "all.vs.all",
  alternative = "two.sided",
  test = "t.test",
  method = "param",
  nstart = 20,
  nmax = 100 * length(algorithms),
  force.balanced = FALSE,
  ncpus = 2,
  boot.R = 499,
  seed = NULL,
  save.partial.results = NA,
  load.partial.results = NA,
  save.final.result = NA
)

Arguments

Value

a list object containing the following fields:

• Configuration - the full input configuration (for reproducibility)

• data.raw - data frame containing all observations generated

• data.summary - data frame summarizing the experiment.

• N - number of instances sampled

• N.star - number of instances required

• total.runs - total number of algorithm runs performed

• instances.sampled - names of the instances sampled

• Underpowered - flag: TRUE if N < N.star

Instance List

Parameter instances must contain a list of instance objects, where each field is itself a list, as defined in the documentation of function calc_nreps(). In short, each element of instances is an instance, i.e., a named list containing all relevant parameters that define the problem instance. This list must contain at least the field instance$FUN, with the name of the problem instance function, that is, a routine that calculates y = f(x). If the instance requires additional parameters, these must also be provided as named fields. An additional field, "instance$alias", can be used to provide the instance with a unique identifier (e.g., when using an instance generator).

Algorithm List

Object algorithms is a list in which each component is a named list containing all relevant parameters that define an algorithm to be applied for solving the problem instance. In what follows algorithms[[k]] refers to any algorithm specified in the algorithms list.

algorithms[[k]] must contain an algorithms[[k]]$FUN field, which is a character object with the name of the function that calls the algorithm; as well as any other elements/parameters that algorithms[[k]]$FUN requires (e.g., stop criteria, operator names and parameters, etc.).

The function defined by the routine algorithms[[k]]$FUN must have the following structure: supposing that the list in algorithms[[k]] has fields algorithm[[k]]$FUN = "myalgo", algorithms[[k]]$par1 = "a" and algorithms[[k]]$par2 = 5, then:

         myalgo <- function(par1, par2, instance, ...){
               #
               # <do stuff>
               #
               return(results)
         }
   

That is, it must be able to run if called as:

         # remove '$FUN' and '$alias' field from list of arguments
         # and include the problem definition as field 'instance'
         myargs          <- algorithm[names(algorithm) != "FUN"]
         myargs          <- myargs[names(myargs) != "alias"]
         myargs$instance <- instance

         # call function
         do.call(algorithm$FUN,
                 args = myargs)
   

The algorithm$FUN routine must return a list containing (at least) the performance value of the final solution obtained, in a field named value (e.g., result$value) after a given run. In general it is easier to write a small wrapper function around existing implementations.

Initial Number of Observations

In the general case the initial number of observations / algorithm / instance (nstart) should be relatively high. For the parametric case we recommend 10~15 if outliers are not expected, and 30~40 (at least) if that assumption cannot be made. For the bootstrap approach we recommend using at least 15 or 20. However, if some distributional assumptions can be made - particularly low skewness of the population of algorithm results on the test instances), then nstart can in principle be as small as 5 (if the output of the algorithm were known to be normal, it could be 1).

In general, higher sample sizes are the price to pay for abandoning distributional assumptions. Use lower values of nstart with caution.

Pairwise Differences

Parameter dif informs the type of difference in performance to be used for the estimation (\mu_a and \mu_b represent the mean performance of any two algorithms on the test instance, and mu represents the grand mean of all algorithms given in algorithms):

• If dif == "perc" and comparisons == "all.vs.first", the estimated quantity is: \phi_{1b} = (\mu_1 - \mu_b) / \mu_1 = 1 - (\mu_b / \mu_1).

• If dif == "perc" and comparisons == "all.vs.all", the estimated quantity is: \phi_{ab} = (\mu_a - \mu_b) / \mu.

• If dif == "simple" it estimates \mu_a - \mu_b.

Sample Sizes for Nonparametric Methods

If the parameter “ is set to either Wilcoxon or 'Binomial', this routine approximates the number of instances using the ARE of these tests in relation to the paired t.test:

• n.wilcox = n.ttest / 0.86 = 1.163 * n.ttest

• n.binom = n.ttest / 0.637 = 1.570 * n.ttest

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

References

• F. Campelo, F. Takahashi: Sample size estimation for power and accuracy in the experimental comparison of algorithms. Journal of Heuristics 25(2):305-338, 2019.

• P. Mathews. Sample size calculations: Practical methods for engineers and scientists. Mathews Malnar and Bailey, 2010.

• A.C. Davison, D.V. Hinkley: Bootstrap methods and their application. Cambridge University Press (1997)

• E.C. Fieller: Some problems in interval estimation. Journal of the Royal Statistical Society. Series B (Methodological) 16(2), 175–185 (1954)

• V. Franz: Ratios: A short guide to confidence limits and proper use (2007). https://arxiv.org/pdf/0710.2024v1.pdf

• D.C. Montgomery, C.G. Runger: Applied Statistics and Probability for Engineers, 6th ed. Wiley (2013)

• D.J. Sheskin: Handbook of Parametric and Nonparametric Statistical Procedures, 4th ed., Chapman & Hall/CRC, 1996.

Examples

# Example using four dummy algorithms and 100 dummy instances.
# See [dummyalgo()] and [dummyinstance()] for details.
# Generating 4 dummy algorithms here, with means 15, 10, 30, 15 and standard
# deviations 2, 4, 6, 8.
algorithms <- mapply(FUN = function(i, m, s){
  list(FUN   = "dummyalgo",
       alias = paste0("algo", i),
       distribution.fun  = "rnorm",
       distribution.pars = list(mean = m, sd = s))},
  i = c(alg1 = 1, alg2 = 2, alg3 = 3, alg4 = 4),
  m = c(15, 10, 30, 15),
  s = c(2, 4, 6, 8),
  SIMPLIFY = FALSE)

# Generate 100 dummy instances with centered exponential distributions
instances <- lapply(1:100,
                    function(i) {rate <- runif(1, 1, 10)
                                 list(FUN   = "dummyinstance",
                                      alias = paste0("Inst.", i),
                                      distr = "rexp", rate = rate,
                                      bias  = -1 / rate)})

my.results <- run_experiment(instances, algorithms,
                             d = .5, se.max = .1,
                             power = .9, sig.level = .05,
                             power.target = "mean",
                             dif = "perc", comparisons = "all.vs.all",
                             ncpus = 1, seed = 1234)

# Take a look at the results
summary(my.results)
plot(my.results)

Bootstrap standard errors

Description

Calculates the standard errors of a given statistic using bootstrap

Usage

se_boot(Xk, dif = "simple", comparisons = "all.vs.all", boot.R = 999, ...)

Arguments

Value

Data frame containing, for each pair of interest, the estimated difference (column "Phi") and the sample standard error (column "SE")

References

• A.C. Davison, D.V. Hinkley: Bootstrap methods and their application. Cambridge University Press (1997)

• F. Campelo, F. Takahashi: Sample size estimation for power and accuracy in the experimental comparison of algorithms. Journal of Heuristics 25(2):305-338, 2019.

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

Examples

# three vectors of normally distributed observations
set.seed(1234)
Xk <- list(rnorm(10, 5, 1),  # mean = 5, sd = 1,
           rnorm(20, 10, 2), # mean = 10, sd = 2,
           rnorm(20, 15, 5)) # mean = 15, sd = 3

se_boot(Xk, dif = "simple", comparisons = "all.vs.all")
se_boot(Xk, dif = "perc", comparisons = "all.vs.first")
se_boot(Xk, dif = "perc", comparisons = "all.vs.all")

Parametric standard errors

Description

Calculates the standard errors of a given statistic using parametric formulas

Usage

se_param(Xk, dif = "simple", comparisons = "all.vs.all", ...)

Arguments

Value

Data frame containing, for each pair of interest, the estimated difference (column "Phi") and the sample standard error (column "SE")

References

• E.C. Fieller: Some problems in interval estimation. Journal of the Royal Statistical Society. Series B (Methodological) 16(2), 175–185 (1954)

• V. Franz: Ratios: A short guide to confidence limits and proper use (2007). https://arxiv.org/pdf/0710.2024v1.pdf

• D.C. Montgomery, C.G. Runger: Applied Statistics and Probability for Engineers, 6th ed. Wiley (2013)

• F. Campelo, F. Takahashi: Sample size estimation for power and accuracy in the experimental comparison of algorithms. Journal of Heuristics 25(2):305-338, 2019.

Author(s)

Felipe Campelo (fcampelo@ufmg.br, f.campelo@aston.ac.uk)

Examples

# three vectors of normally distributed observations
set.seed(1234)
Xk <- list(rnorm(10, 5, 1),  # mean = 5, sd = 1,
           rnorm(20, 10, 2), # mean = 10, sd = 2,
           rnorm(20, 15, 5)) # mean = 15, sd = 3

se_param(Xk, dif = "simple", comparisons = "all.vs.all")
se_param(Xk, dif = "perc", comparisons = "all.vs.first")
se_param(Xk, dif = "perc", comparisons = "all.vs.all")

summary.CAISEr

Description

S3 method for summarizing CAISEr objects output by run_experiment()). Input parameters test, alternative and sig.level can be used to override the ones used in the call to run_experiment().

Usage

## S3 method for class 'CAISEr'
summary(object, test = NULL, alternative = NULL, sig.level = NULL, ...)

Arguments

Value

A list object is returned invisibly, containing the details of all tests performed as well as information on the total number of runs dedicated to each algorithm.

Examples

# Example using four dummy algorithms and 100 dummy instances.
# See [dummyalgo()] and [dummyinstance()] for details.
# Generating 4 dummy algorithms here, with means 15, 10, 30, 15 and standard
# deviations 2, 4, 6, 8.
algorithms <- mapply(FUN = function(i, m, s){
  list(FUN   = "dummyalgo",
       alias = paste0("algo", i),
       distribution.fun  = "rnorm",
       distribution.pars = list(mean = m, sd = s))},
  i = c(alg1 = 1, alg2 = 2, alg3 = 3, alg4 = 4),
  m = c(15, 10, 30, 15),
  s = c(2, 4, 6, 8),
  SIMPLIFY = FALSE)

# Generate 100 dummy instances with centered exponential distributions
instances <- lapply(1:100,
                    function(i) {rate <- runif(1, 1, 10)
                                 list(FUN   = "dummyinstance",
                                      alias = paste0("Inst.", i),
                                      distr = "rexp", rate = rate,
                                      bias  = -1 / rate)})

my.results <- run_experiment(instances, algorithms,
                             d = 1, se.max = .1,
                             power = .9, sig.level = .05,
                             power.target = "mean",
                             dif = "perc", comparisons = "all.vs.all",
                             seed = 1234, ncpus = 1)
summary(my.results)

# You can override some defaults if you want:
summary(my.results, test = "wilcoxon")

summary.nreps

Description

S3 method for summarizing nreps objects output by calc_nreps()).

Usage

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

Arguments