topic Quantiles for Observations from a Triangular Distribution in Statistical Procedures

Quantiles for Observations from a Triangular Distribution

PaulN — Mon, 21 Nov 2022 22:56:27 GMT

The QUANTILE function lists several distributions but the TRIANGULAR distribution is not one of them. The RAND function can compute random variables from a triangular distribution. Suppose I have observations from a triangular distribution, how do I compute their quantiles? It's easy for distributions that the QUANTILE allows, but the triangle distribution isn't one of them. Any code is appreciated.

Re: Quantiles for Observations from a Triangular Distribution

Ksharp — Tue, 22 Nov 2022 03:11:09 GMT

Simulation and PROC UNIVARIATE?

or Calling @Rick_SAS

Re: Quantiles for Observations from a Triangular Distribution

mkeintz — Tue, 22 Nov 2022 04:43:09 GMT

@PaulN wrote:

The QUANTILE function lists several distributions but the TRIANGULAR distribution is not one of them. The RAND function can compute random variables from a triangular distribution. Suppose I have observations from a triangular distribution, how do I compute their quantiles? It's easy for distributions that the QUANTILE allows, but the triangle distribution isn't one of them. Any code is appreciated.

Do you know the parameters of the trangular distribution? I.e. the minimum (a), maximum (b) and mode (c)?

If so, then according to https://github.com/distributions-io/triangular-quantile the formula for quantiles (q) of a triangular distribution is:

  if p < (c-a)/(b-a) then q= a + sqrt((b-a)*(c-a)*p) ;
  else q= b - sqrt((b-a)*(b-c)*(1-p)) ;

where

p is the cumulative distribution level for which you want a quantile
and
(c-a)/(b-a) is the cumulative distribution at the mode value c (i.e. F(c) )

Re: Quantiles for Observations from a Triangular Distribution

PaulN — Tue, 22 Nov 2022 16:03:03 GMT

I do know the minimum (a), maximum (b), and mode (c).

minimum = 0

maximum = 1

mode = 0.2

Height of triangle = 2.

I'll use your code and see what I get. I'll compare it to my hand computations. Thank you for the help.

Re: Quantiles for Observations from a Triangular Distribution

PaulN — Thu, 01 Dec 2022 21:57:19 GMT

Here's the triangle distribution that I'm attempting to get SAS to compute quantiles. When I do it by hand I get the following but when I run the code it doesn't match. Any suggestions?

data triangle;
input x;
datalines;
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
;
run;

data triangle_percentile;
set triangle;

/*a=minimum, b=maximum, c=mode*/
a=0;
b=1;
c=0.2;

if x <= (c-a)/(b-a) then
q=a + sqrt((b-a)*(c-a)*x);
else
q=b - sqrt((b-a)*(b-c)*(1-x));
run;

Re: Quantiles for Observations from a Triangular Distribution

PaulN — Thu, 01 Dec 2022 23:00:31 GMT

Here's what I came up with. It seems to work.

data triangle_percentile;
set triangle;

/*a=minimum, b=maximum, c=mode h =height*/

/*using the 1/2*base*height for a triangle */

/*and slope = (y2-y1)/(x2-x1), which is either */

/* h/(c-a) for x<=c or h/(b-c) for x > c */

/*the negative sign for the slope for the latter */

/is the minus sign after the 1. */
a=0;
b=1;
c=0.2;

h=2;

if x <= c then
q= 0.5*(x-a)*(h/(c-a))*x;
else
q=1 - 0.5*(b-x)*(h/(b-c))*(1-x);
run;

Re: Quantiles for Observations from a Triangular Distribution

FreelanceReinh — Fri, 02 Dec 2022 13:04:01 GMT

@PaulN wrote:

Here's what I came up with. It seems to work.

data triangle_percentile;
set triangle;

/*a=minimum, b=maximum, c=mode h =height*/

/*using the 1/2*base*height for a triangle */

/*and slope = (y2-y1)/(x2-x1), which is either */

/* h/(c-a) for x<=c or h/(b-c) for x > c */

/*the negative sign for the slope for the latter */

/is the minus sign after the 1. */
a=0;
b=1;
c=0.2;

h=2;

if x <= c then
q= 0.5*(x-a)*(h/(c-a))*x;
else
q=1 - 0.5*(b-x)*(h/(b-c))*(1-x);
run;

Hello @PaulN,

Your algorithm correctly computes the values of the cumulative distribution function (CDF) from given quantiles. (For general values of a and b the formulas should read

q = 0.5*(x-a)*(h/(c-a))*(x-a)

and

q = 1 - 0.5*(b-x)*(h/(b-c))*(b-x)

respectively, but since a=0 and b=1 in your example, you get the correct results.)

But you set out to compute quantiles (from cumulative probabilities) like SAS's QUANTILE function -- the inverse function of the CDF. For that purpose @mkeintz's algorithm is correct.

For example, to answer the typical question "What is the 95% quantile of that triangular distribution?", his formula yields the 0.8 in the left column of your table from the 0.95 (=95%) in the right column:

1 - sqrt((1-0)*(1-0.2)*(1-0.95)) = 0.8

(mathematically; in SAS under Windows the result deviates from 0.8 by a tiny rounding error due to numeric representation issues).

Re: Quantiles for Observations from a Triangular Distribution

PaulN — Fri, 02 Dec 2022 14:01:47 GMT

Thanks for the clarification. I should have stated my problem differently. Your help is appreciated.

Re: Quantiles for Observations from a Triangular Distribution

FreelanceReinh — Fri, 02 Dec 2022 14:31:18 GMT

You're welcome. So, you really wanted to extend the CDF function to the triangular distribution, not the QUANTILE function mentioned in your initial post. (Otherwise mkeintz's reply, not my clarification, should be marked as the accepted solution.) Indeed, rereading your question "Suppose I have observations from a triangular distribution, how do I compute their quantiles?" it sounds like you start with observations x between a and b and want to compute cumulative probabilities p (which you denote with q), not vice versa.