This class is a subclass of HypoExponentialDist and also implements the hypoexponential distribution. More...
Public Member Functions | |
| HypoExponentialDistQuick (double[] lambda) | |
Constructs a HypoExponentialDistQuick 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... | |
| void | setLambda (double[] lambda) |
| String | toString () |
 Public Member Functions inherited from HypoExponentialDist | |
| 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 | 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 Public Member Functions inherited from HypoExponentialDist | |
| 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... | |
Additional Inherited Members | |
| Public Attributes inherited from ContinuousDistribution | |
| int | decPrec = 15 |
| Static Protected Member Functions inherited from HypoExponentialDist | |
| static void | testLambda (double[] lambda) |
| Protected Attributes inherited from HypoExponentialDist | |
| double [] | m_lambda |
| Protected Attributes inherited from ContinuousDistribution | |
| double | supportA = Double.NEGATIVE_INFINITY |
| double | supportB = Double.POSITIVE_INFINITY |
| 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 is a subclass of HypoExponentialDist and also implements the hypoexponential distribution.
It uses different algorithms to compute the probabilities. The formula ( tail-hypoexp ) for the complementary distribution is mathematically equivalent to (see [209] (page 299) and [68] (Appendix B))
\[ \bar{F}(x) = \mathbb P\left[X_1 + \cdots+ X_k > x \right] = \sum_{i=1}^k e^{-\lambda_i x} \prod_{\substack {j=1\\j\not i}}^k \frac{\lambda_j}{\lambda_j - \lambda_i}. \tag{convolution-hypo} \]
The expression ( convolution-hypo ) is much faster to compute than the matrix exponential formula ( tail-hypoexp ), but it becomes numerically unstable when \(k\) gets large and/or the differences between the \(\lambda_i\) are too small, because it is an alternating sum with relatively large terms of similar size. When the \(\lambda_i\) are close, many of the factors \(\lambda_j - \lambda_i\) in ( convolution-hypo ) 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 \(\lambda_i\) are regularly spaced with differences of \(\lambda_{i+1} - \lambda_i = 0.1\) for all \(i\), the formula ( convolution-hypo ) breaks down already for \(k \approx15\), while if the differences \(\lambda_{i+1} - \lambda_i = 3\), it gives a few decimal digits of precision for \(k\) up to \(\approx300\).
The formula ( fhypoexp ) for the density is mathematically equivalent to the much faster formula
\[ f(x) = \sum_{i=1}^k\lambda_i e^{-\lambda_i x} \prod_{\substack {j=1\\j\not i}}^k \frac{\lambda_j}{\lambda_j - \lambda_i}, \tag{fhypoexp2} \]
which is also numerically unstable when \(k\) gets large and/or the differences between the \(\lambda_i\) are too small.
Constructor & Destructor Documentation
◆ HypoExponentialDistQuick()
| HypoExponentialDistQuick | ( | double [] | lambda | ) |
Constructs a HypoExponentialDistQuick 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
- value of 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
- value of 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 \(x\)
◆ 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:
- HypoExponentialDistQuick.java
