Piecewise distribution boundaries
Generate a sample data set and create a
paretotails object by fitting a piecewise distribution with Pareto tails to the generated data. Find the boundary points between segments in a
paretotails object by using the object function
Generate a sample data set containing 20% outliers.
rng('default'); % For reproducibility left_tail = -exprnd(1,100,1); right_tail = exprnd(5,100,1); center = randn(800,1); x = [left_tail;center;right_tail];
paretotails object by fitting a piecewise distribution to
x. Specify the boundaries of the tails using the lower and upper tail cumulative probabilities so that a fitted object consists of the empirical distribution for the middle 80% of the data set and generalized Pareto distributions (GPDs) for the lower and upper 10% of the data set.
pd = paretotails(x,0.1,0.9)
pd = Piecewise distribution with 3 segments -Inf < x < -1.33251 (0 < p < 0.1): lower tail, GPD(-0.0063504,0.567017) -1.33251 < x < 1.80149 (0.1 < p < 0.9): interpolated empirical cdf 1.80149 < x < Inf (0.9 < p < 1): upper tail, GPD(0.24874,3.00974)
Return the boundary values between the piecewise segments by using the
[p,q] = boundary(pd)
p = 2×1 0.1000 0.9000
q = 2×1 -1.3325 1.8015
The values in
p are the cumulative probabilities at the boundaries, and the values in
q are the corresponding quantiles.
Plot the cdf of the
paretotails object and mark the boundary points on the figure.
xi = sort(x); plot(xi,cdf(pd,xi)) hold on plot(q,p,'ro') legend('Pareto Tails Object','Boundary Points','Location','best') hold off
pd— Piecewise distribution with Pareto tails
Piecewise distribution with Pareto tails, specified as a
j— Boundary index
Boundary index indicating which boundary to return, specified as a positive integer.
p— Cumulative probability at boundary
Cumulative probability at each boundary, returned as a numeric vector of
q— Quantile at boundary
Quantile at each boundary, returned as a numeric vector.