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Z-Statistic Method for Distribution Selection

Source: R/select-zstatistic.R
select_zstatistic.Rd

Selects the best-fit distribution by comparing a bias-corrected Z-statistic for the sample L-kurtosis (\(\tau_4\)) against the theoretical L-moments for a set of candidate distributions. The distribution with the smallest absolute Z-statistic is selected.

For NS-FFA: To select a distribution for a nonstationary model, include the observation years (ns_years) and the nonstationary model structure (ns_structure). Then, this method will detrend the original, nonstationary data internally using the data_decomposition() function prior to distribution selection.

Usage

select_zstatistic(data, ns_years = NULL, ns_structure = NULL, samples = 10000L)

Arguments

data

Numeric vector of observed annual maximum series values. Must be strictly positive, finite, and not missing.

ns_years

For NS-FFA only: Numeric vector of observation years corresponding to data. Must be the same length as data and strictly increasing.

ns_structure

For NS-FFA only: Named list indicating which distribution parameters are modeled as nonstationary. Must contain two logical scalars:

  • location: If TRUE, the location parameter has a linear temporal trend.

  • scale: If TRUE, the scale parameter has a linear temporal trend.

samples

Integer scalar. The number of bootstrap samples. Default is 10000.

Value

A list with the results of distribution selection:

  • method: "Z-selection".

  • decomposed_data: The detrended dataset used to compute the L-moments. For S-FFA, this is the data argument. For NS-FFA, it is output of data_decomposition().

  • metrics: List of computed Z-statistics for each candidate distribution.

  • recommendation: Name of the distribution with the smallest Z-statistic.

  • reg_params: Kappa distribution parameters for the data.

  • reg_bias_t4: Bias of the L-kurtosis from the bootstrap.

  • reg_std_t4: Standard deviation of the L-kurtosis from the bootstrap.

  • log_params: Kappa distribution parameters for the log-transformed data.

  • log_bias_t4: Bias of the L-kurtosis from the bootstrap using log_params.

  • log_std_t4: Standard deviation of the L-kurtosis from the bootstrap using log_params.

Details

First, this method fits a four-parameter Kappa distribution to both the original and log-transformed data. Then, bootstrapping is used to estimate the bias and variance of the L-kurtosis. These values, along with the difference between the sample and theoretical L-kurtosis, are used to compute the Z-statistic for each distribution.

References

Hosking, J.R.M. & Wallis, J.R., 1997. Regional frequency analysis: an approach based on L-Moments. Cambridge University Press, New York, USA.

See also

select_ldistance(), select_lkurtosis(), fit_lmoments_kappa(), utils_quantiles(), plot_lmom_diagram()

Examples

data <- rnorm(n = 100, mean = 100, sd = 10)
select_zstatistic(data)
#> $method
#> [1] "Z-statistic"
#> 
#> $decomposed_data
#>   [1] 108.46015 107.24958 101.17380 115.17878 100.48625  93.96213  84.40808
#>   [8]  94.33495 104.48243  92.09158  84.75779 106.39260  88.13787 102.65497
#>  [15]  89.25667 111.51548 123.18602 109.23480 108.39826  95.96181  93.45547
#>  [22]  98.95412  81.54312 100.81003 105.16527 107.33136 105.38797  95.60419
#>  [29] 102.40490 104.49495  78.84581 119.48319 106.81061  91.73682 107.50677
#>  [36] 101.43231  94.81273  93.39844  99.84991 105.99509 105.41103 112.38625
#>  [43] 109.05877 108.87397 106.26721 109.05813 106.73623 101.05394  99.82153
#>  [50]  89.31985 107.93599 100.47284 121.40463  95.28911  87.67504  95.45363
#>  [57]  92.68710 104.28608  97.24062 117.85477 128.66187 115.44697  81.82928
#>  [64] 105.55251 104.54792  98.46370 101.69646 107.14534  89.82659  96.33812
#>  [71]  99.35319 103.36448 102.99345 108.05845 115.88706 101.88099  90.27166
#>  [78]  90.94695 100.33448 104.28727 101.15612  77.60913 109.71337  90.89458
#>  [85]  96.01451  90.94567  96.34790 104.43368 103.14414 101.89257  91.76334
#>  [92] 106.41107 105.11925  91.95652 109.68356 105.88925 105.38664  98.80253
#>  [99] 117.03117  99.89794
#> 
#> $metrics
#> $metrics$GEV
#> [1] -1.573403
#> 
#> $metrics$GLO
#> [1] 0.1706968
#> 
#> $metrics$GNO
#> [1] -1.056235
#> 
#> $metrics$PE3
#> [1] -1.06734
#> 
#> $metrics$LP3
#> [1] -1.013761
#> 
#> $metrics$WEI
#> [1] -1.393823
#> 
#> 
#> $recommendation
#> [1] "GLO"
#> 
#> $reg_params
#> [1] 101.12117933   5.78188429   0.06658877  -0.78841285
#> 
#> $reg_bias_t4
#> [1] 0.0002458638
#> 
#> $reg_std_t4
#> [1] 0.03594635
#> 
#> $log_params
#> [1]  4.61509327  0.05887982  0.13361006 -0.73988063
#> 
#> $log_bias_t4
#> [1] -0.0002018031
#> 
#> $log_std_t4
#> [1] 0.03820009
#>