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# Compare Empirical Data to Distributions

Source: `R/utils-distribution-comparison.R`
`tidy_distribution_comparison.Rd`

Compare some empirical data set against different distributions to help find the distribution that could be the best fit.

## Usage

    tidy_distribution_comparison(
      .x,
      .distribution_type = "continuous",
      .round_to_place = 3
    )

## Arguments

.x

  The data set being passed to the function

.distribution_type

  What kind of data is it, can be one of `continuous` or `discrete`

.round_to_place

  How many decimal places should the parameter estimates be rounded off to for distibution construction. The default is 3

## Value

An invisible list object. A tibble is printed.

## Details

The purpose of this function is to take some data set provided and to try to find a distribution that may fit the best. A parameter of `.distribution_type` must be set to either `continuous` or `discrete` in order for this the function to try the appropriate types of distributions.

The following distributions are used:

Continuous:

- tidy_beta

- tidy_cauchy

- tidy_exponential

- tidy_gamma

- tidy_logistic

- tidy_lognormal

- tidy_normal

- tidy_pareto

- tidy_uniform

- tidy_weibull

Discrete:

- tidy_binomial

- tidy_geometric

- tidy_hypergeometric

- tidy_poisson

The function itself returns a list output of tibbles. Here are the tibbles that are returned:

- comparison_tbl

- deviance_tbl

- total_deviance_tbl

- aic_tbl

- kolmogorov_smirnov_tbl

- multi_metric_tbl

The `comparison_tbl` is a long `tibble` that lists the values of the `density` function against the given data.

The `deviance_tbl` and the `total_deviance_tbl` just give the simple difference from the actual density to the estimated density for the given estimated distribution.

The `aic_tbl` will provide the `AIC` for a `lm` model of the estimated density against the emprical density.

The `kolmogorov_smirnov_tbl` for now provides a `two.sided` estimate of the `ks.test` of the estimated density against the empirical.

The `multi_metric_tbl` will summarise all of these metrics into a single tibble.

## Author

Steven P. Sanderson II, MPH

## Examples

    xc <- mtcars$mpg
    output_c <- tidy_distribution_comparison(xc, "continuous")
    #> For the beta distribution, its mean 'mu' should be 0 < mu < 1. The data will
    #> therefore be scaled to enforce this.

    xd <- trunc(xc)
    output_d <- tidy_distribution_comparison(xd, "discrete")

    output_c
    #> $comparison_tbl
    #> # A tibble: 352 × 8
    #>    sim_number     x     y    dx       dy     p     q dist_type
    #>    <fct>      <int> <dbl> <dbl>    <dbl> <dbl> <dbl> <fct>
    #>  1 1              1  21    2.97 0.000114 0.625  10.4 Empirical
    #>  2 1              2  21    4.21 0.000455 0.625  10.4 Empirical
    #>  3 1              3  22.8  5.44 0.00142  0.781  13.3 Empirical
    #>  4 1              4  21.4  6.68 0.00355  0.688  14.3 Empirical
    #>  5 1              5  18.7  7.92 0.00721  0.469  14.7 Empirical
    #>  6 1              6  18.1  9.16 0.0124   0.438  15   Empirical
    #>  7 1              7  14.3 10.4  0.0192   0.125  15.2 Empirical
    #>  8 1              8  24.4 11.6  0.0281   0.812  15.2 Empirical
    #>  9 1              9  22.8 12.9  0.0395   0.781  15.5 Empirical
    #> 10 1             10  19.2 14.1  0.0516   0.531  15.8 Empirical
    #> # ℹ 342 more rows
    #>
    #> $deviance_tbl
    #> # A tibble: 352 × 2
    #>    name                        value
    #>    <chr>                       <dbl>
    #>  1 Empirical                  0.451
    #>  2 Beta c(1.107, 1.577, 0)    0.287
    #>  3 Cauchy c(19.2, 7.375)     -0.0169
    #>  4 Exponential c(0.05)        0.133
    #>  5 Gamma c(11.47, 1.752)     -0.254
    #>  6 Logistic c(20.091, 3.27)  -0.0146
    #>  7 Lognormal c(2.958, 0.293)  0.315
    #>  8 Pareto c(10.4, 1.624)      0.412
    #>  9 Uniform c(8.341, 31.841)   0.143
    #> 10 Weibull c(3.579, 22.288)  -0.201
    #> # ℹ 342 more rows
    #>
    #> $total_deviance_tbl
    #> # A tibble: 10 × 2
    #>    dist_with_params          abs_tot_deviance
    #>    <chr>                                <dbl>
    #>  1 Beta c(1.107, 1.577, 0)              0.640
    #>  2 Lognormal c(2.958, 0.293)            0.734
    #>  3 Gaussian c(20.091, 5.932)            1.39
    #>  4 Cauchy c(19.2, 7.375)                1.56
    #>  5 Logistic c(20.091, 3.27)             2.79
    #>  6 Uniform c(8.341, 31.841)             2.99
    #>  7 Weibull c(3.579, 22.288)             3.34
    #>  8 Pareto c(10.4, 1.624)                3.77
    #>  9 Gamma c(11.47, 1.752)                4.06
    #> 10 Exponential c(0.05)                  7.08
    #>
    #> $aic_tbl
    #> # A tibble: 10 × 3
    #>    dist_type                 aic_value abs_aic
    #>    <fct>                         <dbl>   <dbl>
    #>  1 Beta c(1.107, 1.577, 0)       -22.4    22.4
    #>  2 Pareto c(10.4, 1.624)          85.6    85.6
    #>  3 Gamma c(11.47, 1.752)        -155.    155.
    #>  4 Logistic c(20.091, 3.27)     -161.    161.
    #>  5 Gaussian c(20.091, 5.932)    -172.    172.
    #>  6 Weibull c(3.579, 22.288)     -181.    181.
    #>  7 Cauchy c(19.2, 7.375)        -189.    189.
    #>  8 Exponential c(0.05)          -204.    204.
    #>  9 Lognormal c(2.958, 0.293)    -207.    207.
    #> 10 Uniform c(8.341, 31.841)     -208.    208.
    #>
    #> $kolmogorov_smirnov_tbl
    #> # A tibble: 10 × 6
    #>    dist_type              ks_statistic ks_pvalue ks_method alternative dist_char
    #>    <fct>                         <dbl>     <dbl> <chr>     <chr>       <chr>
    #>  1 Beta c(1.107, 1.577, …        0.75   0.000500 Monte-Ca… two-sided   Beta c(1…
    #>  2 Cauchy c(19.2, 7.375)         0.469  0.00100  Monte-Ca… two-sided   Cauchy c…
    #>  3 Exponential c(0.05)           0.5    0.00100  Monte-Ca… two-sided   Exponent…
    #>  4 Gamma c(11.47, 1.752)         0.156  0.839    Monte-Ca… two-sided   Gamma c(…
    #>  5 Logistic c(20.091, 3.…        0.125  0.970    Monte-Ca… two-sided   Logistic…
    #>  6 Lognormal c(2.958, 0.…        0.219  0.444    Monte-Ca… two-sided   Lognorma…
    #>  7 Pareto c(10.4, 1.624)         0.844  0.000500 Monte-Ca… two-sided   Pareto c…
    #>  8 Uniform c(8.341, 31.8…        0.25   0.269    Monte-Ca… two-sided   Uniform …
    #>  9 Weibull c(3.579, 22.2…        0.188  0.646    Monte-Ca… two-sided   Weibull …
    #> 10 Gaussian c(20.091, 5.…        0.188  0.658    Monte-Ca… two-sided   Gaussian…
    #>
    #> $multi_metric_tbl
    #> # A tibble: 10 × 8
    #>    dist_type abs_tot_deviance aic_value abs_aic ks_statistic ks_pvalue ks_method
    #>    <fct>                <dbl>     <dbl>   <dbl>        <dbl>     <dbl> <chr>
    #>  1 Beta c(1…            0.640     -22.4    22.4        0.75   0.000500 Monte-Ca…
    #>  2 Lognorma…            0.734    -207.    207.         0.219  0.444    Monte-Ca…
    #>  3 Gaussian…            1.39     -172.    172.         0.188  0.658    Monte-Ca…
    #>  4 Cauchy c…            1.56     -189.    189.         0.469  0.00100  Monte-Ca…
    #>  5 Logistic…            2.79     -161.    161.         0.125  0.970    Monte-Ca…
    #>  6 Uniform …            2.99     -208.    208.         0.25   0.269    Monte-Ca…
    #>  7 Weibull …            3.34     -181.    181.         0.188  0.646    Monte-Ca…
    #>  8 Pareto c…            3.77       85.6    85.6        0.844  0.000500 Monte-Ca…
    #>  9 Gamma c(…            4.06     -155.    155.         0.156  0.839    Monte-Ca…
    #> 10 Exponent…            7.08     -204.    204.         0.5    0.00100  Monte-Ca…
    #> # ℹ 1 more variable: alternative <chr>
    #>
    #> attr(,".x")
    #>  [1] 21.0 21.0 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 17.8 16.4 17.3 15.2 10.4
    #> [16] 10.4 14.7 32.4 30.4 33.9 21.5 15.5 15.2 13.3 19.2 27.3 26.0 30.4 15.8 19.7
    #> [31] 15.0 21.4
    #> attr(,".n")
    #> [1] 32