umontreal.iro.lecuyer.probdist
Class HypoExponentialDistQuick

java.lang.Object
  umontreal.iro.lecuyer.probdist.ContinuousDistribution
      umontreal.iro.lecuyer.probdist.HypoExponentialDist
          umontreal.iro.lecuyer.probdist.HypoExponentialDistQuick

All Implemented Interfaces:

Distribution

public class HypoExponentialDistQuick
extends HypoExponentialDist

This class is a subclass of HypoExponentialDist and also implements the hypoexponential distribution. It uses different algorithms to compute the probabilities. The formula for the complementary distribution is mathematically equivalent to (see and)

bar(F)(x) = P[X₁ + ^... + X_k > x] = ∑_i=1^ke^-λ_ix∏_{j=1, j≠i}^kλ_j/(λ_j-λ_i).

The expression is much faster to compute than the matrix exponential formula, but it becomes numerically unstable when k gets large and/or the differences between the λ_i are too small, because it is an alternating sum with relatively large terms of similar size. When the λ_i are close, many of the factors λ_j - λ_i in are small, and the effect of this is amplified when k is large. This gives rise to large terms of opposite sign in the sum and the formula becomes unstable due to subtractive cancellation. For example, with the computations done in standard 64-bit floating-point arithmetic, if the λ_i are regularly spaced with differences of λ_i+1 - λ_i = 0.1 for all i, the formula breaks down already for k = 15, while if the differences λ_i+1 - λ_i = 3, it gives a few decimal digits of precision for k up to = 300.

The formula for the density is mathematically equivalent to the much faster formula

f (x) = ∑_i=1^kλ_ie^-λ_ix∏_{j=1, j≠i}^kλ_j/(λ_j-λ_i),

which is also numerically unstable when k gets large and/or the differences between the λ_i are too small.

Field Summary

Fields inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

decPrec

Constructor Summary

HypoExponentialDistQuick(double[] lambda)
Constructs a HypoExponentialDistQuick object, with rates λ_i = lambda[i - 1], i = 1,…, k.

Method Summary

double	barF(double x) Returns the complementary distribution function.
static double	barF(double[] lambda, double x) Computes the complementary distribution bar(F)(x), with λi = lambda[i - 1], i = 1,…, k.
double	cdf(double x) Returns the distribution function F(x).
static double	cdf(double[] lambda, double x) Computes the distribution function F(x), with λi = lambda[i - 1], i = 1,…, k.
double	density(double x) Returns f (x), the density evaluated at x.
static double	density(double[] lambda, double x) Computes the density function f (x), with λi = lambda[i - 1], i = 1,…, k.
double	inverseF(double u) Returns the inverse distribution function x = F-1(u).
static double	inverseF(double[] lambda, double u) Computes the inverse distribution function F-1(u), with λi = lambda[i - 1], i = 1,…, k.
void	setLambda(double[] lambda) Sets the values λi =lambda[i - 1], i = 1,…, k for this object.
String	toString()

Methods inherited from class umontreal.iro.lecuyer.probdist.HypoExponentialDist

cdf2, getLambda, getMean, getMean, getParams, getStandardDeviation, getStandardDeviation, getVariance, getVariance

Methods inherited from class umontreal.iro.lecuyer.probdist.ContinuousDistribution

getXinf, getXsup, inverseBisection, inverseBrent, setXinf, setXsup

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Detail

HypoExponentialDistQuick

public HypoExponentialDistQuick(double[] lambda)

Constructs a HypoExponentialDistQuick object, with rates λ_i = lambda[i - 1], i = 1,…, k.

Parameters:

lambda - rates of the hypoexponential distribution

Method Detail

density

public double density(double x)

Description copied from class: ContinuousDistribution

Returns f (x), the density evaluated at x.

Overrides:

density in class HypoExponentialDist

Parameters:

x - value at which the density is evaluated

Returns:

density function evaluated at x

cdf

public double cdf(double x)

Description copied from interface: Distribution

Returns the distribution function F(x).

Specified by:

cdf in interface Distribution

Overrides:

cdf in class HypoExponentialDist

Parameters:

x - value at which the distribution function is evaluated

Returns:

distribution function evaluated at x

barF

public double barF(double x)

Description copied from class: ContinuousDistribution

Returns the complementary distribution function. The default implementation computes bar(F)(x) = 1 - F(x).

Specified by:

barF in interface Distribution

Overrides:

barF in class HypoExponentialDist

Parameters:

x - value at which the complementary distribution function is evaluated

Returns:

complementary distribution function evaluated at x

inverseF

public double inverseF(double u)

Description copied from class: ContinuousDistribution

Returns the inverse distribution function x = F^-1(u). Restrictions: u∈[0, 1].

Specified by:

inverseF in interface Distribution

Overrides:

inverseF in class HypoExponentialDist

Parameters:

u - value at which the inverse distribution function is evaluated

Returns:

the inverse distribution function evaluated at u

density

public static double density(double[] lambda,
                             double x)

Computes the density function f (x), with λ_i = lambda[i - 1], i = 1,…, k.

Parameters:

lambda - rates of the hypoexponential distribution

x - value at which the density is evaluated

Returns:

density at x

cdf

public static double cdf(double[] lambda,
                         double x)

Computes the distribution function F(x), with λ_i = lambda[i - 1], i = 1,…, k.

Parameters:

lambda - rates of the hypoexponential distribution

x - value at which the distribution is evaluated

Returns:

value of distribution at x

barF

public static double barF(double[] lambda,
                          double x)

Computes the complementary distribution bar(F)(x), with λ_i = lambda[i - 1], i = 1,…, k.

Parameters:

lambda - rates of the hypoexponential distribution

x - value at which the complementary distribution is evaluated

Returns:

value of complementary distribution at x

inverseF

public static double inverseF(double[] lambda,
                              double u)

Computes the inverse distribution function F^-1(u), with λ_i = lambda[i - 1], i = 1,…, k.

Parameters:

lambda - rates of the hypoexponential distribution

u - value at which the inverse distribution is evaluated

Returns:

inverse distribution at u

setLambda

public void setLambda(double[] lambda)

Description copied from class: HypoExponentialDist

Sets the values λ_i =lambda[i - 1], i = 1,…, k for this object.

Overrides:

setLambda in class HypoExponentialDist

toString

public String toString()

Overrides:

toString in class HypoExponentialDist