# FuzzyResampling

The goal of FuzzyResampling, a library written in R, is to provide additional resampling procedures, apart from the classical bootstrap (i.e. Efron’s approach, see (Efron and Tibshirani 1994)), for fuzzy data. In the classical approach, secondary samples are drawing with replacement from the initial sample. Therefore most of these bootstrap samples contain repeated values. Moreover, if the size of the primary sample is small then all secondary samples consist of only a few distinct values, which is a serious disadvantage.

To overcome this problem, special resampling algorithms for fuzzy data were introduced (see (Grzegorzewski, Hryniewicz, and Romaniuk 2019, 2020a, 2020b; Grzegorzewski and Romaniuk 2022; Romaniuk and Hryniewicz 2019)). These methods randomly create values that are “similar” to values from the initial sample, but not exactly the same. During the creation process, some of the characteristics of the initial fuzzy values are kept (e.g., the value, the width, etc.). It was shown that these algorithms provide serious advantages in some statistical areas (like standard error estimation or hypothesis testing) if they are compared with the classical approach. For detailed information concerning the theoretical foundations and practical applications of these resampling methods please see the above-mentioned references.

The initial sample (in the form of a vector or a matrix) should consist of triangular or trapezoidal fuzzy numbers.

Some additional procedures related to these resampling methods are also provided, like calculation of the Bertoluzza et al.’s distance (aka the mid/spread distance, see (Bertoluzza, Corral, and Salas 1995)), estimation of the p-value of the one-sample bootstrapped test for the mean (see (Lubiano et al. 2016)), and estimation of the standard error or the mean-squared error for the mean (see (Grzegorzewski and Romaniuk 2021)). Additionally, there are procedures which randomly generate trapezoidal fuzzy numbers using some well-known statistical distributions (see (Grzegorzewski, Hryniewicz, and Romaniuk 2020a)).

The following procedures are available in the library:

• Resampling procedures:

• ClassicalBootstrap - classical approach based on Efron’s method,
• VAMethod - resampling method which preserves the value and ambiguity (see (Grzegorzewski, Hryniewicz, and Romaniuk 2020a)),
• EWMethod - resampling method which preserves the expected value and width (see (Grzegorzewski, Hryniewicz, and Romaniuk 2020b)),
• VAAMethod - resampling method which preserves the value, left-hand and right-hand ambiguities (see (Grzegorzewski and Romaniuk 2022)),
• VAFMethod - resampling method which preserves the value, ambiguity and fuzziness (see (Grzegorzewski, Hryniewicz, and Romaniuk 2020a)),
• DMethod - resampling method which preserves the left end of the cores and increments (see (Romaniuk and Hryniewicz 2019)),
• WMethod - resampling method which uses the special w density to “smooth” the output fuzzy value (see (Romaniuk and Hryniewicz 2019)).
• Random generation of the initial samples:

• GeneratorNU - generation of the initial sample using the normal and uniform distributions (see (Grzegorzewski, Hryniewicz, and Romaniuk 2020a)),
• GeneratorNExpUU - generation of the initial sample using the normal, exponential and uniform distributions (see (Grzegorzewski, Hryniewicz, and Romaniuk 2020a)),
• GeneratorFuzzyNumbers - generation of the initial sample using various random distributions.
• Applications of the bootstrapped samples:

• OneSampleCTest - estimation of the p-value of the one-sample test for the mean (see (Lubiano et al. 2016)),
• TwoSampleCTest - estimation of the p-value of the two-sample test for the mean (see (Lubiano et al. 2016)),
• SEResamplingMean - estimation of the standard error or the mean-squared error for the mean (see (Grzegorzewski and Romaniuk 2021)).
• Calculation of the characteristics of fuzzy numbers:

• CalculateFuzziness - calculate the fuzziness of fuzzy number (see (Grzegorzewski, Hryniewicz, and Romaniuk 2019)),
• CalculateWidth - calculate the width of fuzzy number (see (Grzegorzewski and Romaniuk 2022)),
• CalculateAmbiguity - calculate the ambiguity of fuzzy number (see (Grzegorzewski, Hryniewicz, and Romaniuk 2019)),
• CalculateAmbiguityL - calculate the left-hand ambiguity of fuzzy number (see (Grzegorzewski and Romaniuk 2022)),
• CalculateAmbiguityR - calculate the right-hand ambiguity of fuzzy number (see (Grzegorzewski and Romaniuk 2022)),
• CalculateValue - calculate the value of fuzzy number (see (Grzegorzewski and Romaniuk 2022)),
• CalculateExpValue - calculate the expected value of fuzzy number (see (Grzegorzewski and Romaniuk 2022)).

• BertoluzzaDistance - calculation of the Bertoluzza et al.’s distance (aka the mid/spread distance, see (Bertoluzza, Corral, and Salas 1995)),
• ComparisonOneSampleCTest - comparison of resampling methods based on percentage of rejections for the one-sample C-test (see (Grzegorzewski and Romaniuk 2022)),
• ComparisonSEMean - comparison of resampling methods based on the SE/MSE for the mean (see (Grzegorzewski and Romaniuk 2022)),
• ComparePowerOneSampleCTest - comparison of resampling methods based on percentage of rejections for the one-sample C-test (see (Grzegorzewski and Romaniuk 2022)).

## Installation

You can install the latest development version of FuzzyResampling with:

``````library(devtools)
install_github("mroman-ibs/FuzzyResampling")``````

You can install the latest stable version from CRAN with:

``install.packages("FuzzyResampling")``

## Examples

``````# set seed

set.seed(12345)

library(FuzzyResampling)

# prepare some fuzzy numbers

fuzzyValues <- matrix(c(0.25,0.5,1,1.25,0.75,1,1.5,2.2,-1,0,0,2),ncol = 4,byrow = TRUE)

fuzzyValues
#>       [,1] [,2] [,3] [,4]
#> [1,]  0.25  0.5  1.0 1.25
#> [2,]  0.75  1.0  1.5 2.20
#> [3,] -1.00  0.0  0.0 2.00

# seed PRNG

set.seed(12345)

# generate the secondary sample using the classical approach

ClassicalBootstrap(fuzzyValues)
#>       [,1] [,2] [,3] [,4]
#> [1,]  0.75    1  1.5  2.2
#> [2,] -1.00    0  0.0  2.0
#> [3,]  0.75    1  1.5  2.2

# generate the secondary sample using the VA method

VAMethod(fuzzyValues)
#>            [,1]       [,2]       [,3]     [,4]
#> [1,]  0.9141124  0.9179438  1.7262290 1.747542
#> [2,] -0.5303703  0.8901852  0.9132088 1.423582
#> [3,] -0.3356065 -0.3321967 -0.3321967 2.664393

# generate the secondary sample (6 fuzzy numbers) using the d-method

DMethod(fuzzyValues, b = 6)
#>       [,1] [,2] [,3] [,4]
#> [1,]  0.75  1.0  1.5 3.50
#> [2,]  0.00  1.0  1.5 1.75
#> [3,]  0.25  0.5  1.0 1.25
#> [4,]  0.75  1.0  1.0 3.00
#> [5,] -0.25  0.0  0.5 1.20
#> [6,] -0.50  0.5  1.0 1.25

# calculate the mid/spread distance between the first value
# (from the first row) and the second one (from the second row)

#>  0.6204837

# seed PRNG

set.seed(1234)

# calculate the p-value using the classical (i.e. Efron's) bootstrap
# for the one-sample test for the mean

OneSampleCTest(fuzzyValues, mu_0 = c(0,0.5,1,1.5))
#>  0.82

# calculate the p-value using the VA resampling method

OneSampleCTest(fuzzyValues, mu_0 = c(0,0.5,1,1.5),resamplingMethod = "VAMethod")
#>  0.91

# seed PRNG

set.seed(1234)

# calculate the p-value using the classical (i.e. Efron's) bootstrap
# for the two-sample test for the mean

TwoSampleCTest(fuzzyValues, fuzzyValues+0.1)
#>  0.86

# calculate the p-value using the VA resampling method

TwoSampleCTest(fuzzyValues, fuzzyValues+0.1,resamplingMethod = "VAMethod")
#>  0.95

# seed PRNG

set.seed(1234)

# calculate the SE of the mean using the classical (i.e. Efron's) bootstrap

SEResamplingMean(fuzzyValues)
#> \$mean
#>  0.0075000 0.5100000 0.8416667 1.8413333
#>
#> \$SE
#>  0.05487521

# calculate the SE of the mean using the VA resampling method

SEResamplingMean(fuzzyValues, resamplingMethod = "VAMethod")
#> \$mean
#>  -0.2846996  0.5985998  0.8490542  1.7328917
#>
#> \$SE
#>  0.05504196

# calculate the MSE of the given mean using the classical (i.e. Efron's) bootstrap

SEResamplingMean(fuzzyValues, trueMean = c(0,0.5,1,2))
#> \$mean
#>  0.0 0.5 1.0 2.0
#>
#> \$SE
#>  0.02721175

# calculate the MSE of the given mean using the VA resampling method

SEResamplingMean(fuzzyValues, resamplingMethod = "VAMethod", trueMean = c(0,0.5,1,2))
#> \$mean
#>  0.0 0.5 1.0 2.0
#>
#> \$SE
#>  0.03119963

# seed PRNG

set.seed(1234)

# generate 10 trapezoidal fuzzy numbers using the normal and uniform distributions

GeneratorNU(10, 0,1,1,2)
#>             [,1]        [,2]        [,3]       [,4]
#>  [1,] -2.6303454 -1.52367820 -0.75097427 -0.6034145
#>  [2,] -1.3180763 -0.02526413  0.54261591  1.1619891
#>  [3,]  0.3017466  0.92539517  1.38911338  2.8236569
#>  [4,] -3.6293320 -2.38569362 -1.83839083 -0.8292990
#>  [5,] -0.4492152  0.21032515  0.61022090  0.9162188
#>  [6,] -1.3085376 -0.30454266  1.26572653  2.2735935
#>  [7,] -2.4546266 -1.10043751 -0.37349192  0.6144299
#>  [8,] -2.4312725 -1.46129002 -0.28782204  1.2145784
#>  [9,] -1.8836547 -1.39579705  0.42769842  0.7769981
#> [10,] -2.4667277 -0.93580809 -0.08268549  1.6140993

# calculate the ambiguity for the whole matrix

CalculateAmbiguity(fuzzyValues)
#>  0.3333333 0.4083333 0.5000000``````
``````# help concerning the VA method

?VAMethod``````
Bertoluzza, Carlo, Norberto Corral, and Antonia Salas. 1995. “On a New Class of Distances Between Fuzzy Numbers.” Mathware and Soft Computing 2 (2): 71–84. https://eudml.org/doc/39054.
Efron, B., and R. J. Tibshirani. 1994. An Introduction to the Bootstrap. Chapman; Hall/CRC.
Grzegorzewski, P., O. Hryniewicz, and M. Romaniuk. 2019. “Flexible Bootstrap Based on the Canonical Representation of Fuzzy Numbers.” In Proceedings of EUSFLAT 2019. Atlantis Press. https://doi.org/10.2991/eusflat-19.2019.68.
———. 2020a. “Flexible Bootstrap for Fuzzy Data Based on the Canonical Representation.” International Journal of Computational Intelligence Systems 13: 1650–62. https://doi.org/10.2991/ijcis.d.201012.003.
———. 2020b. “Flexible Resampling for Fuzzy Data.” International Journal of Applied Mathematics and Computer Science 30: 281–97. https://doi.org/10.34768/amcs-2020-0022.
Grzegorzewski, P., and M. Romaniuk. 2021. “Epistemic Bootstrap for Fuzzy Data.” In Joint Proceedings of IFSA-EUSFLAT-AGOP 2021 Conferences, 538–45. Atlantis Press.
———. 2022. “Bootstrap Methods for Fuzzy Data.” In Uncertainty and Imprecision in Decision Making and Decision Support: New Advances, Challenges, and Perspectives, edited by Krassimir T. Atanassov, Vassia Atanassova, Janusz Kacprzyk, Andrzej Kałuszko, Maciej Krawczak, Jan W. Owsiński, Sotir S. Sotirov, Evdokia Sotirova, Eulalia Szmidt, and Sławomir Zadrożny, 28–47. Cham: Springer International Publishing.
Lubiano, M. A., M. Montenegro, B. Sinova, S. De la Rosa de Sáa, and M. A. Gil. 2016. “Hypothesis Testing for Means in Connection with Fuzzy Rating Scale-Based Data: Algorithms and Applications.” European Journal of Operational Research 251: 918–29. https://doi.org/10.1016/j.ejor.2015.11.016.
Romaniuk, M., and O. Hryniewicz. 2019. “Interval-Based, Nonparametric Approach for Resampling of Fuzzy Numbers.” Soft Computing 23: 5883–903. https://doi.org/10.1007/s00500-018-3251-5.