probplot - Probability plots

Syntax

probplot(Y) probplot(distribution,Y) probplot(Y,cens,freq) probplot(ax,Y) probplot(...,'noref') probplot(ax,PD) probplot(ax,fun,params) h = probplot(...)

Description

probplot(Y) produces a normal probability plot comparing the distribution of the data Y to the normal distribution. Y can be a single vector, or a matrix with a separate sample in each column. The plot includes a reference line useful for judging whether the data follow a normal distribution.

probplot uses midpoint probability plotting positions. The i^th sorted value from a sample of size N is plotted against the midpoint in the jump of the empirical CDF on the y axis. With uncensored data, that midpoint is (i–0.5)/N. With censored data (see below), the y value is more complicated to compute.

probplot(distribution,Y) creates a probability plot for the distribution specified by distribution. Acceptable strings for distribution are:

'exponential' — Exponential probability plot (nonnegative values)
'extreme value' — Extreme value probability plot (all values)
'lognormal' — Lognormal probability plot (positive values)
'normal' — Normal probability plot (all values)
'rayleigh' — Rayleigh probability plot (positive values)
'weibull' — Weibull probability plot (positive values)

The y axis scale is based on the selected distribution. The x axis has a log scale for the Weibull and lognormal distributions, and a linear scale for the others.

Not all distributions are appropriate for all data sets, and probplot will error when asked to create a plot with a data set that is inappropriate for a specified distribution. Appropriate data ranges for each distribution are given parenthetically in the list above.

probplot(Y,cens,freq) or probplot(distname,Y,cens,freq) requires a vector Y. cens is a vector of the same size as Y and contains 1 for observations that are right-censored and 0 for observations that are observed exactly. freq is a vector of the same size as Y, containing integer frequencies for the corresponding elements in Y.

probplot(ax,Y) takes a handle ax to an existing probability plot, and adds additional lines for the samples in Y. ax is a handle for a set of axes.

probplot(...,'noref') omits the reference line.

probplot(ax,PD) takes a probability distribution object, PD, and adds a fitted line to the axes specified by ax to represent the probability distribution specified by PD. PD is a ProbDist object of the ProbDistUnivParam class or ProbDistUnivKernel class.

probplot(ax,fun,params) takes a function fun and a set of parameters, params, and adds fitted lines to the axes of an existing probability plot specified by ax. fun is a function handle to a cdf function, specified with @ (for example, @wblcdf). params is the set of parameters required to evaluate fun, and is specified as a cell array or vector. The function must accept a vector of X values as its first argument, then the optional parameters, and must return a vector of cdf values evaluated at X.

h = probplot(...) returns handles to the plotted lines.

Examples

Example 1

The following plot assesses two samples, one from a Weibull distribution and one from a Rayleigh distribution, to see if they may have come from a Weibull population.

x1 = wblrnd(3,3,100,1);
x2 = raylrnd(3,100,1);
probplot('weibull',[x1 x2])
legend('Weibull Sample','Rayleigh Sample','Location','NW')

Example 2

Consider the following data, with about 20% outliers:

left_tail = -exprnd(1,10,1);
right_tail = exprnd(5,10,1);
center = randn(80,1);
data = [left_tail;center;right_tail];

Neither a normal distribution nor a t distribution fits the tails very well:

probplot(data);
p = mle(data,'dist','tlo');
t = @(data,mu,sig,df)cdf('tlocationscale',data,mu,sig,df);
h = probplot(gca,t,p);
set(h,'color','r','linestyle','-')
title('{\bf Probability Plot}')
legend('Data','Normal','t','Location','NW')