# boundary

Piecewise distribution boundaries

## Description

example

[p,q] = boundary(pd) returns the boundary points between segments in pd, the piecewise distribution. p is a vector of the cumulative probabilities at the boundaries, and q is a vector of the corresponding quantiles.

[p,q] = boundary(pd,j) returns boundary values of the jth boundary.

## Examples

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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 boundary.

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];

Create a 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 boundary function.

[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

## Input Arguments

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Piecewise distribution with Pareto tails, specified as a paretotails object.

Boundary index indicating which boundary to return, specified as a positive integer.

Data Types: single | double

## Output Arguments

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Cumulative probability at each boundary, returned as a numeric vector of range (0,1) values.

Quantile at each boundary, returned as a numeric vector.

## Version History

Introduced in R2007a