# Diagnostic Plots for Fitting Distributions

*Thomas Roh*

*December 17, 2017*

The `fitur`

package includes several tools for visually inspecting how good of a fit a distribution is. To start, fictional empirical data is generated below. Typically this would come from a *real-world* dataset such as the time it takes to serve a customer at a bank, the length of stay in an emergency department, or customer arrivals to a queue.

```
set.seed(438)
x <- rweibull(10000, shape = 5, scale = 1)
```

## Histogram

Below is a histogram showing the shape of the distribution and the y-axis has been set to show the probability density.

```
dt <- data.frame(x)
nbins <- 30
g <- ggplot(dt, aes(x)) +
geom_histogram(aes(y = ..density..),
bins = nbins, fill = NA, color = "black") +
theme_bw() +
theme(panel.grid = element_blank())
g
```

## Histogram vs Density Plot

Three distributions have been chosen below to test against the dataset. Using the `fit_univariate`

function, each of the distributions are fit to a *fitted* object. The first item in each of the *fits* is the probabilty density function. Each *fit* is overplotted onto the histogram to see which distribution fits best.

```
dists <- c('gamma', 'lnorm', 'weibull')
multipleFits <- lapply(dists, fit_univariate, x = x)
plot_density(x, multipleFits, 30)
```

## Q-Q Plot

The next plot used is the quantile-quantile plot. The `plot_qq`

function takes a numeric vector *x* of the empirical data and sorts them. A range of probabilities are computed and then used to compute comparable quantiles using the `q`

distribution function from the *fitted* objects. A good fit would closely align with the abline y = 0 + 1*x. Note: the q-q plot tends to be more sensitive around the “tails” of the distributions.

```
plot_qq(x, multipleFits) +
theme_bw() +
theme(panel.grid = element_blank())
```

## P-P Plot

The Percentile-Percentile plot rescales the input data to the interval (0, 1] and then calculates the theoretical percentiles to compare. The `plot_pp`

function takes the same inputs as the Q-Q Plot but it performs on rescaling of x and then computes the percentiles using the `p`

distribution of the *fitted* object. A good fit matches the abline y = 0 + 1*x. Note: The P-P plot tends to be more sensitive in the middle of the distribution.

```
plot_pp(x, multipleFits) +
theme_bw() +
theme(panel.grid = element_blank())
```