Author: Allison C Fialkowski
Date: 2018-06-28

# Comparison of Simulated Distribution to Theoretical Distribution or Empirical Data

#### Allison C Fialkowski

#### 2018-06-28

Headrick and Kowalchuk (2007) outlined a general method for comparing a simulated distribution \(\Large Y\) to a given theoretical distribution \(\Large Y^*\). Note that these could easily be modified for comparison to an empirical vector of data:

1. Obtain the standardized cumulants (skewness, kurtosis, fifth, and sixth) for \(\Large Y^*\). This can be done using `calc_theory` along with either the distribution name (plus up to 4 parameters) or the pdf fx (plus support bounds). In the case of an empirical vector of data, use `calc_moments` or `calc_fisherk`.

2. Obtain the constants for \(\Large Y\). This can be done using `find_constants` or by simulating the distribution with `nonnormvar1`.

3. Determine whether these constants produce a valid power method pdf. The results of `find_constants` or `nonnormvar1` indicate whether the constants yield an invalid or valid pdf. The constants may also be checked using `pdf_check`. If the constants generate an invalid pdf, the user should check if the kurtosis falls above the lower bound (using `calc_lower_skurt`). If yes, a vector of sixth cumulant correction values should be used in `find_constants` or `nonnormvar1` to find the smallest correction that produces valid pdf constants.

4. Select a critical value from \(\Large Y^*\), i.e. \(\Large y^*\) such that \(\Large Pr(Y^* \ge y^*) = \alpha\). This can be done using the appropriate quantile function and \(\Large 1 - \alpha\) value (i.e. `qexp(1 - 0.05)`).

5. Solve \(\Large m_{2}^{1/2} * p(z') + m_{1} - y^* = 0\) for \(\Large z'\), where \(\Large m_{1}\) and \(\Large m_{2}\) are the 1st and 2nd moments of \(\Large Y^*\).

6. Calculate \(\Large 1 - \Phi(z')\), the corresponding probability for the approximation \(\Large Y\) to \(\Large Y^*\) (i.e. \(\Large 1 - \Phi(z') = 0.05\)) and compare to the target value \(\Large \alpha\).

7. Plot a parametric graph of \(\Large Y^*\) and \(\Large Y\). This can be done with a set of constants using `plot_pdf_theory` (`overlay` = TRUE) or with a simulated vector of data using `plot_sim_pdf_theory` (`overlay` = TRUE). If comparing to an empirical vector of data, use `plot_pdf_ext` or `plot_sim_pdf_ext`.

## Example

Use these steps to compare a simulated exponential(mean = 2) variable to the theoretical exponential(mean = 2) density. (Note that the `printr` package is invoked to display the tables.)

### Step 1: Obtain the standardized cumulants

In R, the exponential parameter is `rate <- 1/mean`.

    library("SimMultiCorrData")
    library("printr")
    stcums <- calc_theory(Dist = "Exponential", params = 0.5)

### Step 2: Simulate the variable

Note that `calc_theory` returns the standard deviation, not the variance. The simulation functions require variance as the input.

    H_exp <- nonnormvar1("Polynomial", means = stcums[1], vars = stcums[2]^2,
                         skews = stcums[3], skurts = stcums[4],
                         fifths = stcums[5], sixths = stcums[6], n = 10000,
                         seed = 1234)

    ## Constants: Distribution  1
    ##
    ## Constants calculation time: 0 minutes
    ## Total Simulation time: 0 minutes

Look at the power method constants.

    as.matrix(H_exp$constants, nrow = 1, ncol = 6, byrow = TRUE)

c0 | c1 | c2 | c3 | c4 | c5
--- | --- | --- | --- | --- | ---
-0.3077396 | 0.8005605 | 0.318764 | 0.0335001 | -0.0036748 | 0.0001587

Look at a summary of the target distribution.

    as.matrix(round(H_exp$summary_targetcont[, c("Distribution", "mean", "sd",
                                                 "skew", "skurtosis", "fifth",
                                                 "sixth")], 5), nrow = 1, ncol = 7,
              byrow = TRUE)

| Distribution | mean | sd | skew | skurtosis | fifth | sixth
--- | --- | --- | --- | --- | --- | --- | ---
mean | 1 | 2 | 2 | 2 | 6 | 24 | 120

Compare to a summary of the simulated distribution.

    as.matrix(round(H_exp$summary_continuous[, c("Distribution", "mean", "sd",
                                                 "skew", "skurtosis", "fifth",
                                                 "sixth")], 5), nrow = 1, ncol = 7,
              byrow = TRUE)

| Distribution | mean | sd | skew | skurtosis | fifth | sixth
--- | --- | --- | --- | --- | --- | --- | ---
X1 | 1 | 1.99987 | 2.0024 | 2.03382 | 6.18067 | 23.74145 | 100.3358

### Step 3: Determine if the constants generate a valid power method pdf

    H_exp$valid.pdf

    ## [1] "TRUE"

### Step 4: Select a critical value

Let \(\Large \alpha = 0.05\).

    y_star <- qexp(1 - 0.05, rate = 0.5) # note that rate = 1/mean
    y_star

    ## [1] 5.991465

### Step 5: Solve for \(\Large z'\)

Since the exponential(2) distribution has a mean and standard deviation equal to 2, solve \(\Large 2 * p(z') + 2 - y_star = 0\) for \(\Large z'\). Here, \(\Large p(z') = c0 + c1 * z' + c2 * z'^2 + c3 * z'^3 + c4 * z'^4 + c5 * z'^5\).

    f_exp <- function(z, c, y) {
      return(2 * (c[1] + c[2] * z + c[3] * z^2 + c[4] * z^3 + c[5] * z^4 +
                    c[6] * z^5) + 2 - y)
    }

    z_prime <- uniroot(f_exp, interval = c(-1e06, 1e06),
                       c = as.numeric(H_exp$constants), y = y_star)$root
    z_prime

    ## [1] 1.644926

### Step 6: Calculate \(\Large \Phi(z')\)

    1 - pnorm(z_prime)

    ## [1] 0.04999249

This is approximately equal to the \(\Large \alpha\) value of 0.05, indicating the method provides a good approximation to the actual distribution.

### Step 7: Plot graphs

    plot_sim_pdf_theory(sim_y = H_exp$continuous_variable[, 1],
                        Dist = "Exponential", params = 0.5)

We can also plot the empirical cdf and show the cumulative probability up to y_star.

    plot_sim_cdf(sim_y = H_exp$continuous_variable[, 1], calc_cprob = TRUE,
                 delta = y_star)

### Calculate descriptive statistics.

    as.matrix(t(stats_pdf(c = H_exp$constants[1, ], method = "Polynomial",
                        alpha = 0.025, mu = stcums[1], sigma = stcums[2])))

trimmed_mean | median | mode | max_height
--- | --- | --- | ---
1.858381 | 1.384521 | 0.104872 | 1.094213

## References

Headrick, T C, and R K Kowalchuk. 2007. “The Power Method Transformation: Its Probability Density Function, Distribution Function, and Its Further Use for Fitting Data.” Journal of Statistical Computation and Simulation 77: 229–49. doi:10.1080/10629360600605065.