This class implements the hypoexponential distribution, also called the generalized Erlang distribution. More...
Public Member Functions | |
| HypoExponentialDist (double[] lambda) | |
Constructs a HypoExponentialDist object, with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| double | density (double x) |
| double | cdf (double x) |
| Returns the distribution function \(F(x)\). More... | |
| double | barF (double x) |
| Returns \(\bar{F}(x) = 1 - F(x)\). More... | |
| double | inverseF (double u) |
| Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ). More... | |
| double | getMean () |
| Returns the mean of the distribution function. | |
| double | getVariance () |
| Returns the variance of the distribution function. | |
| double | getStandardDeviation () |
| Returns the standard deviation of the distribution function. | |
| double [] | getLambda () |
| Returns the values \(\lambda_i\) for this object. | |
| void | setLambda (double[] lambda) |
Sets the values \(\lambda_i = \)lambda[ \(i-1\)], \(i = 1,…,k\) for this object. | |
| double [] | getParams () |
| Same as getLambda. | |
| String | toString () |
Returns a String containing information about the current distribution. | |
 Public Member Functions inherited from ContinuousDistribution | |
| abstract double | density (double x) |
| Returns \(f(x)\), the density evaluated at \(x\). More... | |
| double | barF (double x) |
| Returns the complementary distribution function. More... | |
| double | inverseBrent (double a, double b, double u, double tol) |
| Computes the inverse distribution function \(x = F^{-1}(u)\), using the Brent-Dekker method. More... | |
| double | inverseBisection (double u) |
| Computes and returns the inverse distribution function \(x = F^{-1}(u)\), using bisection. More... | |
| double | inverseF (double u) |
| Returns the inverse distribution function \(x = F^{-1}(u)\). More... | |
| double | getMean () |
| Returns the mean. More... | |
| double | getVariance () |
| Returns the variance. More... | |
| double | getStandardDeviation () |
| Returns the standard deviation. More... | |
| double | getXinf () |
| Returns \(x_a\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
| double | getXsup () |
| Returns \(x_b\) such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
| void | setXinf (double xa) |
Sets the value \(x_a=\) xa, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
| void | setXsup (double xb) |
Sets the value \(x_b=\) xb, such that the probability density is 0 everywhere outside the interval \([x_a, x_b]\). More... | |
Static Public Member Functions | |
| static double | density (double[] lambda, double x) |
Computes the density function \(f(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| static double | cdf (double[] lambda, double x) |
Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| static double | cdf2 (double[] lambda, double x) |
Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| static double | barF (double[] lambda, double x) |
Computes the complementary distribution \(\bar{F}(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| static double | inverseF (double[] lambda, double u) |
Computes the inverse distribution function \(F^{-1}(u)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| static double | getMean (double[] lambda) |
Returns the mean, \(E[X] = \sum_{i=1}^k 1/\lambda_i\), of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| static double | getVariance (double[] lambda) |
Returns the variance, \(\mbox{Var}[X] = \sum_{i=1}^k 1/\lambda_i^2\), of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
| static double | getStandardDeviation (double[] lambda) |
Returns the standard deviation of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\). More... | |
Static Protected Member Functions | |
| static void | testLambda (double[] lambda) |
Protected Attributes | |
| double [] | m_lambda |
| Protected Attributes inherited from ContinuousDistribution | |
| double | supportA = Double.NEGATIVE_INFINITY |
| double | supportB = Double.POSITIVE_INFINITY |
Additional Inherited Members | |
| Public Attributes inherited from ContinuousDistribution | |
| int | decPrec = 15 |
| Static Protected Attributes inherited from ContinuousDistribution | |
| static final double | XBIG = 100.0 |
| static final double | XBIGM = 1000.0 |
| static final double [] | EPSARRAY |
Detailed Description
This class implements the hypoexponential distribution, also called the generalized Erlang distribution.
Let the \(X_j\), \(j=1,…,k\), be \(k\) independent exponential random variables with different rates \(\lambda_j\), i.e. assume that \(\lambda_j \neq\lambda_i\) for \(i \neq j\). Then the sum \(\sum_{j=1}^kX_j\) is called a hypoexponential random variable.
Let the \(k\times k\) upper triangular bidiagonal matrix
\[ \tag{tail-hypomatrix} \mathbf{A}= \begin{pmatrix} -\lambda_1 & \lambda_1 & 0 & … & 0 \\ 0 & -\lambda_2 & \lambda_2 & … & 0 \\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & … & 0 & -\lambda_{k-1} & \lambda_{k-1} \\ 0 & … & 0 & 0 & -\lambda_k \end{pmatrix} \]
with \(\lambda_j\) the rates of the \(k\) exponential random variables; then the cumulative complementary probability of the hypoexponential distribution is given by [184], [116]
\[ \tag{tail-hypoexp} \bar{F}(x) = \mathbb P \left[X_1 + \cdots+ X_k > x \right] = \sum_{j=1}^k \left(e^{\mathbf{A}x}\right)_{1j}, \]
i.e., it is the sum of the elements of the first row of matrix \(e^{\mathbf{A}x}\). The density of the hypoexponential distribution is
\[ f(x) = \left(-e^{\mathbf{A}x}\mathbf{A}\right)_{1k} = \lambda_k \left(e^{\mathbf{A}x}\right)_{1k}, \tag{fhypoexp} \]
i.e., it is element \((1,k)\) of matrix \(-e^{\mathbf{A}x}\mathbf{A}\). The distribution function is as usual \(F(x) = 1 - \bar{F}(x)\).
See the class HypoExponentialDistQuick for alternative formulae for the probabilities.
Constructor & Destructor Documentation
◆ HypoExponentialDist()
| HypoExponentialDist | ( | double [] | lambda | ) |
Constructs a HypoExponentialDist object, with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
- Parameters
-
lambda rates of the hypoexponential distribution
Member Function Documentation
◆ barF() [1/2]
| double barF | ( | double | x | ) |
Returns \(\bar{F}(x) = 1 - F(x)\).
- Parameters
-
x value at which the complementary distribution function is evaluated
- Returns
- complementary distribution function evaluated at
x
Implements Distribution.
◆ barF() [2/2]
|
static |
Computes the complementary distribution \(\bar{F}(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
- Parameters
-
lambda rates of the hypoexponential distribution x value at which the complementary distribution is evaluated
- Returns
- complementary distribution at \(x\)
◆ cdf() [1/2]
| double cdf | ( | double | x | ) |
Returns the distribution function \(F(x)\).
- Parameters
-
x value at which the distribution function is evaluated
- Returns
- distribution function evaluated at
x
Implements Distribution.
◆ cdf() [2/2]
|
static |
Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
- Parameters
-
lambda rates of the hypoexponential distribution x value at which the distribution is evaluated
- Returns
- distribution at \(x\)
◆ cdf2()
|
static |
Computes the distribution function \(F(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
Returns \(1 - \)barF(lambda, x), which is much faster than cdf but loses precision in the lower tail.
- Parameters
-
lambda rates of the hypoexponential distribution x value at which the distribution is evaluated
- Returns
- distribution at \(x\)
◆ density()
|
static |
Computes the density function \(f(x)\), with \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
- Parameters
-
lambda rates of the hypoexponential distribution x value at which the density is evaluated
- Returns
- density at \(x\)
◆ getMean()
|
static |
Returns the mean, \(E[X] = \sum_{i=1}^k 1/\lambda_i\), of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
- Parameters
-
lambda rates of the hypoexponential distribution
- Returns
- mean of the hypoexponential distribution
◆ getStandardDeviation()
|
static |
Returns the standard deviation of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
- Parameters
-
lambda rates of the hypoexponential distribution
- Returns
- standard deviation of the hypoexponential distribution
◆ getVariance()
|
static |
Returns the variance, \(\mbox{Var}[X] = \sum_{i=1}^k 1/\lambda_i^2\), of the hypoexponential distribution with rates \(\lambda_i = \) lambda[ \(i-1\)], \(i = 1,…,k\).
- Parameters
-
lambda rates of the hypoexponential distribution
- Returns
- variance of the hypoexponential distribution
◆ inverseF() [1/2]
| double inverseF | ( | double | u | ) |
Returns the inverse distribution function \(F^{-1}(u)\), defined in ( inverseF ).
- Parameters
-
u value in the interval \((0,1)\) for which the inverse distribution function is evaluated
- Returns
- the inverse distribution function evaluated at
u
Implements Distribution.
◆ inverseF() [2/2]
|
static |
Computes the inverse distribution function \(F^{-1}(u)\), with \(\lambda_i = \) 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 \(u\)
The documentation for this class was generated from the following file:
- HypoExponentialDist.java
