/* *
* Title:
* Description:
* Copyright: Copyright (c)
* Company:
* @author
* @version 1.0
*/
package umontreal.iro.lecuyer.probdist;
import java.util.Arrays;
/**
* Extends the class {@link ContinuousDistribution} for
* the triangular distribution with domain [a, b] and mode
* (or shape parameter) m, where
* a <= m <= b.
* The density function is
*
*
*
*
* f (x) =
* 2(x - a)/[(b - a)(m - a)]
* for a <= x <= m,
* f (x) =
* 2(b - x)/[(b - a)(b - m)]
* for m <= x <= b,
* f (x) =
* 0
* elsewhere,
*
*
* the distribution function is
*
*
*
*
* F(x) =
* 0
* for x < a,
* F(x) =
* (x - a)2/[(b - a)(m - a)]
* if a <= x <= m,
* F(x) =
* 1 - (b - x)2/[(b - a)(b - m)]
* if m <= x <= b,
* F(x) =
* 1
* for x > b,
*
*
* and the inverse distribution function is given by
*
*
*
*
* F-1(u) =
* a + ((b - a)(m - a)u)1/2
* if 0 <= u <= (m - a)/(b - a),
* F-1(u) =
* b - ((b - a)(b - m)(1 - u))1/2
* if (m - a)/(b - a <= u <= 1.
*
*
*
*/
public class TriangularDist extends ContinuousDistribution {
private double a;
private double b;
private double m;
/**
* Constructs a TriangularDist object with default parameters
* a = 0, b = 1, and m = 0.5.
*
*/
public TriangularDist() {
setParams (0.0, 1.0, 0.5);
}
/**
* Constructs a TriangularDist object with parameters a = 0 ,
* b = 1 and m = m.
*
*/
public TriangularDist (double m) {
setParams (0.0, 1.0, m);
}
/**
* Constructs a TriangularDist object with
* parameters a, b and m.
*
*/
public TriangularDist (double a, double b, double m) {
setParams (a, b, m);
}
public double density (double x) {
return density (a, b, m, x);
}
public double cdf (double x) {
return cdf (a, b, m, x);
}
public double barF (double x) {
return barF (a, b, m, x);
}
public double inverseF (double u){
return inverseF (a, b, m, u);
}
public double getMean() {
return TriangularDist.getMean (a, b, m);
}
public double getVariance() {
return TriangularDist.getVariance (a, b, m);
}
public double getStandardDeviation() {
return TriangularDist.getStandardDeviation (a, b, m);
}
/**
* Computes the density function.
*
*/
public static double density (double a, double b, double m, double x) {
if (m < a || m > b)
throw new IllegalArgumentException ("m is not in [a,b]");
if (x < a || x > b)
return 0.0;
else if (x <= m && m != a)
return 2.0*(x - a)/((b - a)*(m - a));
else
return 2.0*(b - x)/((b - a)*(b - m));
}
/**
* Computes the distribution function.
*
*/
public static double cdf (double a, double b, double m, double x) {
if (m < a || m > b)
throw new IllegalArgumentException ("m is not in [a,b]");
if (x <= a)
return 0.0;
else if (x <= m && m != a)
return (x - a)*(x - a)/((b - a)*(m - a));
else if (x < b)
return 1.0 - (b - x)*(b - x)/((b - a)*(b - m));
else
return 1.0;
}
/**
* Computes the complementary distribution function.
*
*/
public static double barF (double a, double b, double m, double x) {
if (m < a || m > b)
throw new IllegalArgumentException ("m is not in [a,b]");
if (x <= a)
return 1.0;
else if (x <= m && m != a)
return 1.0 - (x - a)*(x - a)/((b - a)*(m - a));
else if (x < b)
return (b - x)*(b - x)/((b - a)*(b - m));
else
return 0.0;
}
/**
* Computes the inverse distribution function.
*
*/
public static double inverseF (double a, double b, double m, double u) {
if (m < a || m > b)
throw new IllegalArgumentException ("m is not in [a,b]");
if (u < 0.0 || u > 1.0)
throw new IllegalArgumentException ("u is not in [0,1]");
if (u <= 0.0)
return a;
if (u >= 1.0)
return b;
// the code is taken and adapted from unuran
// file /distributions/c_triangular_gen.c
double h = (m - a)/(b - a);
return u <= h && m != a ? a + Math.sqrt ((b - a)*(m - a)*u)
: b - Math.sqrt ((b - a)*(b - m)*(1 - u));
}
/**
* Estimates the parameter m of the triangular distribution using the
* maximum likelihood method, from the n observations x[i],
*
* i = 0, 1,…, n - 1. The estimated parameter is returned in a one-element
* array: [hat(m)]. See.
*
* @param x the list of observations used to evaluate parameters
*
* @param n the number of observations used to evaluate parameters
*
* @param a lower limit of range
*
* @param b upper limit of range
*
* @return returns the parameter [m]
*
*/
public static double[] getMLE (double[] x, int n, double a, double b) {
if (n <= 0)
throw new IllegalArgumentException ("n <= 0");
double[] Y = new double[n]; // sorted x[i]
System.arraycopy (x, 0, Y, 0, n);
Arrays.sort (Y);
int rmax = -1;
double prodmax = -1.0e300;
final double ba = b - a;
double z;
int i;
for (int r = 0; r < n; r++) {
z = (Y[r] - a) / ba;
if ((z <= (double)r/n) || (z >= (double)(r + 1)/n))
continue; // MLE cannot be there
double prod = 1.0;
double d = Y[r] - a;
for (i = 0; i < r; i++)
prod *= (Y[i] - a)/d;
d = b - Y[r];
for (i = r+1; i < n; i++)
prod *= (b - Y[i])/d;
if (prod > prodmax) {
prodmax = prod;
rmax = r;
}
}
if (rmax < 0)
throw new UnsupportedOperationException (
" data cannot fit a triangular distribution");
double[] param = new double[1];
param[0] = Y[rmax];
return param;
}
/**
* Creates a new instance of a triangular distribution with parameters
* a and b. m is estimated using the maximum
* likelihood method based on the n observations
* x[i],
* i = 0, 1,…, n - 1.
*
* @param x the list of observations used to evaluate parameters
*
* @param n the number of observations used to evaluate parameters
*
* @param a lower limit of range
*
* @param b upper limit of range
*
*
*/
public static TriangularDist getInstanceFromMLE (double[] x, int n,
double a, double b) {
double param[] = getMLE (x, n, a, b);
return new TriangularDist (a, b, param[0]);
}
/**
* Computes and returns the mean
* E[X] = (a + b + m)/3
* of the triangular distribution with parameters a, b, m.
*
* @return the mean of the triangular distribution
*
*/
public static double getMean (double a, double b, double m) {
if ((a == 0.0 && b == 1.0) && (m < 0 || m > 1))
throw new IllegalArgumentException ("m is not in [0,1]");
else if (m < a || m > b)
throw new IllegalArgumentException ("m is not in [a,b]");
return ((a + b + m) / 3.0);
}
/**
* Computes and returns the variance
*
* Var[X] = (a2 + b2 + m2 - ab - am - bm)/18
* of the triangular distribution with parameters a, b, m.
*
* @return the variance of the triangular distribution
*
*/
public static double getVariance (double a, double b, double m) {
if ((a == 0.0 && b == 1.0) && (m < 0 || m > 1))
throw new IllegalArgumentException ("m is not in [0,1]");
else if (m < a || m > b)
throw new IllegalArgumentException ("m is not in [a,b]");
return ((a * a + b * b + m * m - a * b - a * m - b * m) / 18.0);
}
/**
* Computes and returns the standard deviation
* of the triangular distribution with parameters a, b, m.
*
* @return the standard deviation of the triangular distribution
*
*/
public static double getStandardDeviation (double a, double b, double m) {
return Math.sqrt (TriangularDist.getVariance (a, b, m));
}
/**
* Returns the value of a for this object.
*
*/
public double getA() {
return a;
}
/**
* Returns the value of b for this object.
*
*/
public double getB() {
return b;
}
/**
* Returns the value of m for this object.
*
*/
public double getM() {
return m;
}
/**
* Sets the value of the parameters a, b and m for this object.
*
*/
public void setParams (double a, double b, double m) {
if ((a == 0.0 && b == 1.0) && (m < 0 || m > 1))
throw new IllegalArgumentException ("m is not in [0,1]");
else if (a >= b)
throw new IllegalArgumentException ("a >= b");
else if (m < a || m > b)
throw new IllegalArgumentException ("m is not in [a,b]");
this.a = a;
this.b = b;
this.m = m;
supportA = a;
supportB = b;
}
/**
* Return a table containing the parameters of the current distribution.
* This table is put in regular order: [a, b, m].
*
*/
public double[] getParams () {
double[] retour = {a, b, m};
return retour;
}
/**
* Returns a String containing information about the current distribution.
*
*/
public String toString () {
return getClass().getSimpleName() + " : a = " + a + ", b = " + b + ", m = " + m;
}
}
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