====== QQ plots ======

A quantile-quantile plot, or QQ plot, is a visual goodness-of-fit test. It checks if two distributions $F$ (distribution we're testing against) and $F_n$ (empirical distribution) are close. This is done by plotting $F^{-1}$ on the x-axis against $F_n^{-1}$ on the y-axis. If the points are close to the line $y=x$, the two distributions are close.

To plot a QQ plot:

  - Reorder the samples in increasing order. Denote the samples $X_i$ such that $X_i$ is the $i$th smallest sample.
  - Plot the points

$$(F^{-1}(\frac{1}{n}),X_{(1)}), (F^{-1}(\frac{2}{n}),X_{(2)}), ... , (F^{-1}(\frac{i}{n}),X_{(i)}), ... , (F^{-1}(\frac{n-1}{n}),X_{(n-1)})$$

($(F^{-1}(\frac{n}{n}),X_{(n)})$ is skipped because for a Gaussian cdf $F$, $F^{-1}(1) = \infty$. If $F^{-1}(1)$ is defined, the point can be included.)

To plot a QQ plot in MATLAB, use the //qqplot// command.

<code matlab>

pd = makedist(distname, Name1, Value1, Name2, Value2, ...)
qqplot(x, pd)

</code>

distname can be any family of distributions, including Normal, Gamma, Binomial, Beta, Uniform, Poisson, etc. Parameters of the distribution can be specified with the name/value pairs. See the [[https://www.mathworks.com/help/stats/makedist.html|MATLAB help page for makedist]] for more details.

x is the array of sample data.

===== Patterns of QQ plots =====

There are four recognizable patterns in QQ plots that gives information about the distribution of the sample. These plots were generated with [[kb:qq_plots_matlab|this MATLAB code]].

==== Heavy/fat tails ====

{{kb:probstat:t3_pdf.png?400|}}
{{kb:probstat:t3_heavytails.png?400|}}

==== Light/skinny tails ====

{{kb:probstat:unifm11_pdf.png?400|}}
{{kb:probstat:unifm11_lighttails.png?400|}}

==== Right skewed ====

{{kb:probstat:exp1_pdf.png?400|}}
{{kb:probstat:exp1_rightskewed.png?400|}}


==== Left skewed ====

{{kb:probstat:expm1_pdf.png?400|}}
{{kb:probstat:expm1_leftskewed.png?400|}}