FrechetDist.java

/*
 * Class:        FrechetDist
 * Description:  Fréchet distribution
 * Environment:  Java
 * Software:     SSJ 
 * Copyright (C) 2001  Pierre L'Ecuyer and Universite de Montreal
 * Organization: DIRO, Universite de Montreal
 * @author       
 * @since
 *
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *
 */
package umontreal.ssj.probdist;
 
import umontreal.ssj.util.*;
import optimization.*;
import umontreal.ssj.functions.MathFunction;

public class FrechetDist extends ContinuousDistribution {
   private double delta;
   private double beta;
   private double alpha;
 
   private static class Optim implements Lmder_fcn {
      protected double[] x;
      protected int n;
 
      public Optim(double[] x, int n) {
         this.n = n;
         this.x = x;
      }
 
      public void fcn(int m, int n, double[] par, double[] fvec, double[][] fjac, int iflag[]) {
         if (par[1] <= 0.0 || par[2] <= 0.0) {
            final double BIG = 1.0e100;
            fvec[1] = BIG;
            fvec[2] = BIG;
            fvec[3] = BIG;
            return;
         }
 
         double sum1, sum2, sumb, sum4, sum5;
         double z, w, v;
         double alpha = par[1];
         double beta = par[2];
         double mu = par[3];
 
         if (iflag[1] == 1) {
            sum1 = sum2 = sumb = sum4 = sum5 = 0;
            for (int i = 0; i < n; i++) {
               z = (x[i] - mu) / beta;
               sum1 += 1.0 / z;
               v = Math.pow(z, -alpha);
               sum2 += v / z;
               sumb += v;
               w = Math.log(z);
               sum4 += w;
               sum5 += v * w;
            }
 
            fvec[2] = sumb - n; // eq. for beta
            fvec[3] = (alpha + 1) * sum1 - alpha * sum2; // eq. for mu
            fvec[1] = n / alpha + sum5 - sum4; // eq. for alpha
 
         } else if (iflag[1] == 2) {
            throw new IllegalArgumentException("iflag = 2");
            // The 3 X 3 Jacobian must be calculated and put in fjac
         }
      }
   }
 
   private static class Function implements MathFunction {
      private int n;
      private double[] x;
      private double delta;
      public double sumxi;
      public double dif;
 
      public Function(double[] y, int n, double delta) {
         this.n = n;
         this.x = y;
         this.delta = delta;
         double xmin = Double.MAX_VALUE;
         for (int i = 0; i < n; i++) {
            if ((y[i] < xmin) && (y[i] > delta))
               xmin = y[i];
         }
         dif = xmin - delta;
      }
 
      public double evaluate(double alpha) {
         if (alpha <= 0.0)
            return 1.0e100;
         double v, w;
         double sum1 = 0, sum2 = 0, sum3 = 0;
         for (int i = 0; i < n; i++) {
            if (x[i] <= delta)
               continue;
            v = Math.log(x[i] - delta);
            w = Math.pow(dif / (x[i] - delta), alpha);
            sum1 += v;
            sum2 += w;
            sum3 += v * w;
         }
 
         sum1 /= n;
         sumxi = sum2 / n;
         return 1 / alpha + sum3 / sum2 - sum1;
      }
   }

   public FrechetDist(double alpha) {
      setParams(alpha, 1.0, 0.0);
   }

   public FrechetDist(double alpha, double beta, double delta) {
      setParams(alpha, beta, delta);
   }
 
   public double density(double x) {
      return density(alpha, beta, delta, x);
   }
 
   public double cdf(double x) {
      return cdf(alpha, beta, delta, x);
   }
 
   public double barF(double x) {
      return barF(alpha, beta, delta, x);
   }
 
   public double inverseF(double u) {
      return inverseF(alpha, beta, delta, u);
   }
 
   public double getMean() {
      return getMean(alpha, beta, delta);
   }
 
   public double getVariance() {
      return getVariance(alpha, beta, delta);
   }
 
   public double getStandardDeviation() {
      return getStandardDeviation(alpha, beta, delta);
   }

   public static double density(double alpha, double beta, double delta, double x) {
      if (beta <= 0)
         throw new IllegalArgumentException("beta <= 0");
      if (alpha <= 0)
         throw new IllegalArgumentException("alpha <= 0");
      final double z = (x - delta) / beta;
      if (z <= 0.0)
         return 0.0;
      double t = Math.pow(z, -alpha);
      return alpha * t * Math.exp(-t) / (z * beta);
   }

   public static double cdf(double alpha, double beta, double delta, double x) {
      if (beta <= 0)
         throw new IllegalArgumentException("beta <= 0");
      if (alpha <= 0)
         throw new IllegalArgumentException("alpha <= 0");
      final double z = (x - delta) / beta;
      if (z <= 0.0)
         return 0.0;
      double t = Math.pow(z, -alpha);
      return Math.exp(-t);
   }

   public static double barF(double alpha, double beta, double delta, double x) {
      if (beta <= 0)
         throw new IllegalArgumentException("beta <= 0");
      if (alpha <= 0)
         throw new IllegalArgumentException("alpha <= 0");
      final double z = (x - delta) / beta;
      if (z <= 0.0)
         return 1.0;
      double t = Math.pow(z, -alpha);
      return -Math.expm1(-t);
   }

   public static double inverseF(double alpha, double beta, double delta, double u) {
      if (u < 0.0 || u > 1.0)
         throw new IllegalArgumentException("u not in [0, 1]");
      if (beta <= 0)
         throw new IllegalArgumentException("beta <= 0");
      if (alpha <= 0)
         throw new IllegalArgumentException("alpha <= 0");
      if (u >= 1.0)
         return Double.POSITIVE_INFINITY;
      if (u <= 0.0)
         return delta;
      double t = Math.pow(-Math.log(u), 1.0 / alpha);
      if (t <= Double.MIN_NORMAL)
         return Double.MAX_VALUE;
      return delta + beta / t;
   }

   public static double[] getMLE(double[] x, int n, double delta) {
      if (n <= 1)
         throw new IllegalArgumentException("n <= 1");
 
      Function func = new Function(x, n, delta);
      double a = 1e-4;
      double b = 1.0e12;
      double alpha = RootFinder.brentDekker(a, b, func, 1e-12);
      double par[] = new double[2];
      par[0] = alpha;
      par[1] = func.dif * Math.pow(func.sumxi, -1.0 / alpha);
      return par;
   }

   public static FrechetDist getInstanceFromMLE(double[] x, int n, double delta) {
      double par[] = getMLE(x, n, delta);
      return new FrechetDist(par[0], par[1], delta);
   }

   public static double getMean(double alpha, double beta, double delta) {
      if (beta <= 0)
         throw new IllegalArgumentException("beta <= 0");
      if (alpha <= 1)
         throw new IllegalArgumentException("alpha <= 1");
      double t = Num.lnGamma(1.0 - 1.0 / alpha);
      return delta + beta * Math.exp(t);
   }

   public static double getVariance(double alpha, double beta, double delta) {
      if (beta <= 0)
         throw new IllegalArgumentException("beta <= 0");
      if (alpha <= 2)
         throw new IllegalArgumentException("alpha <= 2");
      double t = Num.lnGamma(1.0 - 1.0 / alpha);
      double mu = Math.exp(t);
      double v = Math.exp(Num.lnGamma(1.0 - 2.0 / alpha));
      return beta * beta * (v - mu * mu);
   }

   public static double getStandardDeviation(double alpha, double beta, double delta) {
      return Math.sqrt(getVariance(alpha, beta, delta));
   }

   public double getAlpha() {
      return alpha;
   }

   public double getBeta() {
      return beta;
   }

   public double getDelta() {
      return delta;
   }

   public void setParams(double alpha, double beta, double delta) {
      if (beta <= 0)
         throw new IllegalArgumentException("beta <= 0");
      if (alpha <= 0)
         throw new IllegalArgumentException("alpha <= 0");
      this.delta = delta;
      this.beta = beta;
      this.alpha = alpha;
   }

   public double[] getParams() {
      double[] retour = { alpha, beta, delta };
      return retour;
   }

   public String toString() {
      return getClass().getSimpleName() + " : alpha = " + alpha + ", beta = " + beta + ", delta = " + delta;
   }
 
}