GeneralizedParetoFactory¶

(Source code, png)

class GeneralizedParetoFactory(*args)¶

Generalized Pareto factory.

See also

DistributionFactory, GeneralizedPareto

Notes

Several estimators are available to build a GeneralizedPareto distribution from a scalar sample (see [matthys2003] for the theory).

The chosen algorithm depends on the size of the sample compared to the ResourceMap key GeneralizedParetoFactory-SmallSize:

When the size of the sample is small the probability weighted method of moments is used, with a fallback on the exponential regression method
When the size of the sample is large the exponential regression method is used, with a fallback on the probability weighted moment method

Moments based estimator:

Lets denote:

$\displaystyle \overline{x}_n = \frac{1}{n} \sum_{i=1}^n x_i$ the empirical mean of the sample,
$\displaystyle s_n^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \overline{x}_n)^2$ its empirical variance.

Then we estimate $(\hat{\sigma}_n, \hat{\xi}_n, \hat{u}_n)$ using:

(1)¶ $\begin{eqnarray*} \hat{u}_n &= x_{(1)} - \frac{x_{(1)}}{2 + n} \\ \hat{\xi}_n &= -\dfrac{1}{2}\left(\dfrac{(\overline{x}_n - \hat{u}_n)^2}{s_n^2}-1\right) \\ \hat{\sigma}_n &= \dfrac{(\overline{x}_n- \hat{u}_n)}{2}\left(\dfrac{(\overline{x}_n- \hat{u}_n)^2}{s_n^2}+1\right) \end{eqnarray*}$

This estimator is well-defined only if $\hat{\xi}>-1/4$ , otherwise the second moment does not exist.

Probability weighted moments based estimator:

Lets denote:

$\left(x_{(i)}\right)_{i\in\{1,\dots,n\}}$ the sample sorted in ascending order
$m=\dfrac{1}{n}\sum_{i=1}^n\left(1-\dfrac{i-7/20}{n}\right)x_{(i)}$
$\rho=\dfrac{m}{\overline{x}_n}$

Then we estimate $(\hat{\sigma}, \hat{\xi}, \hat{u})$ using:

(2)¶ $\begin{eqnarray*} \hat{u}_n &= x_{(1)} - \frac{x_{(1)}}{2 + n}\\ \hat{\xi}_n &= \dfrac{1-4\rho}{1-2\rho} \\ \hat{\sigma}_n &= \dfrac{2(\overline{x}_n- \hat{u}_n)}{1-2\rho} \end{eqnarray*}$

This estimator is well-defined only if $\hat{\xi}_n>-1/2$ , otherwise the first moment does not exist.

Maximum likelihood based estimator:

These estimators are not yet implemented.

For a given $u < x_{(1,n)}$ , we get $(\hat{\sigma}_n(u), \hat{\xi}_n(u)$ by maximizing the likelihood of the sample $\ell(\sigma(u), \xi(u),u)$ :

$(\hat{\sigma}_n(u), \hat{\xi}_n(u)) = \argmax_{\sigma, \xi} \ell(\sigma(u), \xi(u)| x_1, \dots, x_n, u)$

The threshold $u$ is obtained by maximizing the optimal likelihood $\ell(\hat{\sigma}_n(u), \hat{\xi}_n(u), u)$

$\hat{u}_n = \argmax_{u} \ell(\hat{\sigma}_n(u), \hat{\xi}_n(u),u) \text{ under the constraint } u < x_{(1,n)}$

The initial point of the optimisation problem is $u_0 = x_{(1,n)} - |x_{(1,n)}|/(2+n)$ .

Exponential regression based estimator:

Lets denote:

$y_{i}=i\log\left(\dfrac{x_{(n-i)}-x_{(1)}}{x_{(n-i-1)}-x_{(1)}}\right)$ for $i\in\{1,n-3\}$

Then we estimate $(\hat{\sigma}, \hat{\xi}, \hat{u})$ using:

(3)¶ $\hat{\xi} &= \xi^* \\ \hat{\sigma} &= \dfrac{2(\overline{x}_n- \hat{u}_n)}{1-2\rho} \\ \hat{u} &= x_{(1)} - \frac{x_{(1)}}{2 + n}$

Where $\xi^*$ maximizes:

(4)¶ $\sum_{i=1}^{n-2}\log\left(\dfrac{1-(j/n)^{\xi}}{\xi}\right)-\dfrac{1-(j/n)^{\xi}}{\xi}y_i$

under the constraint $-1 \leq \xi \leq 1$ .

The following ResourceMap entries can be used to tweak the parameters of the optimization solver involved in the different estimators:

GeneralizedParetoFactory-DefaultOptimizationAlgorithm
GeneralizedParetoFactory-MaximumEvaluationNumber
GeneralizedParetoFactory-MaximumAbsoluteError
GeneralizedParetoFactory-MaximumRelativeError
GeneralizedParetoFactory-MaximumObjectiveError
GeneralizedParetoFactory-MaximumConstraintError

Methods

`build`(*args)	Build the distribution.
`buildAsGeneralizedPareto`(*args)	Build the distribution as a GeneralizedPareto type.
`buildEstimator`(*args)	Build the distribution and the parameter distribution.
`buildMethodOfExponentialRegression`(sample)	Build the distribution based on the exponential regression estimator.
`buildMethodOfMoments`(sample)	Build the distribution based on the method of moments estimator.
`buildMethodOfProbabilityWeightedMoments`(sample)	Build the distribution based on the probability weighted moments estimator.
`drawMeanResidualLife`(sample)	Draw the mean residual life plot.
`getBootstrapSize`()	Accessor to the bootstrap size.
`getClassName`()	Accessor to the object's name.
`getId`()	Accessor to the object's id.
`getName`()	Accessor to the object's name.
`getOptimizationAlgorithm`()	Accessor to the solver.
`getShadowedId`()	Accessor to the object's shadowed id.
`getVisibility`()	Accessor to the object's visibility state.
`hasName`()	Test if the object is named.
`hasVisibleName`()	Test if the object has a distinguishable name.
`setBootstrapSize`(bootstrapSize)	Accessor to the bootstrap size.
`setName`(name)	Accessor to the object's name.
`setOptimizationAlgorithm`(solver)	Accessor to the solver.
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

__init__(*args)¶

build(*args)¶

Build the distribution.

Available usages:

build()

build(sample)

build(param)

Parameters:

sample2-d sequence of float, of dimension 1: The sample from which the distribution parameters are estimated.
paramsequence of float: The parameters of the GeneralizedPareto.

Returns:

distDistribution: The built distribution.

Notes

In the first usage, the default GeneralizedPareto distribution is built.

In the second usage, the parameters are evaluated according the following strategy:

If the sample size is less or equal to GeneralizedParetoFactory-SmallSize from ResourceMap, then the method of probability weighted moments is used. If it fails, the method of exponential regression is used.
Otherwise, the first method tried is the method of exponential regression, then the method of probability weighted moments if the first one fails.

In the third usage, a GeneralizedPareto distribution corresponding to the given parameters is built.

buildAsGeneralizedPareto(*args)¶

Build the distribution as a GeneralizedPareto type.

Available usages:

build()

build(sample)

build(param)

Parameters:

sample2-d sequence of float, of dimension 1: The sample from which the distribution parameters are estimated.
paramsequence of float,: A vector of parameters of the GeneralizedPareto.

Returns:

distGeneralizedPareto

The estimated distribution as a GeneralizedPareto.

In the first usage, the default GeneralizedPareto distribution is built.

buildEstimator(*args)¶

Build the distribution and the parameter distribution.

Parameters:

sample2-d sequence of float: Data.
parametersDistributionParameters: Optional, the parametrization.

Returns:

resDistDistributionFactoryResult: The results.

Notes

According to the way the native parameters of the distribution are estimated, the parameters distribution differs:

Moments method: the asymptotic parameters distribution is normal and estimated by Bootstrap on the initial data;

Maximum likelihood method with a regular model: the asymptotic parameters distribution is normal and its covariance matrix is the inverse Fisher information matrix;

Other methods: the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting (see KernelSmoothing).

If another set of parameters is specified, the native parameters distribution is first estimated and the new distribution is determined from it:

if the native parameters distribution is normal and the transformation regular at the estimated parameters values: the asymptotic parameters distribution is normal and its covariance matrix determined from the inverse Fisher information matrix of the native parameters and the transformation;

in the other cases, the asymptotic parameters distribution is estimated by Bootstrap on the initial data and kernel fitting.

buildMethodOfExponentialRegression(sample)¶

Build the distribution based on the exponential regression estimator.

Parameters:

sample2-d sequence of float, of dimension 1: The sample from which the distribution parameters are estimated.

buildMethodOfMoments(sample)¶

Build the distribution based on the method of moments estimator.

Parameters:

sample2-d sequence of float, of dimension 1: The sample from which the distribution parameters are estimated.

buildMethodOfProbabilityWeightedMoments(sample)¶

Build the distribution based on the probability weighted moments estimator.

Parameters:

sample2-d sequence of float, of dimension 1: The sample from which the distribution parameters are estimated.

drawMeanResidualLife(sample)¶

Draw the mean residual life plot.

Let $X$ a random variable defined so the excess of a threshold $u_s$ follow a Generalized Pareto distribution $GPD(\xi, \sigma_s)$ . The mean of excesses of $X$ for $u>u_s$ is

$M_n(u) = \Expect{X-u|X>u} = \frac{\sigma_s+\xi_u}{1-\xi}$

So for all $u>u_s$ M_n(u) is linear wrt. $u$ . The threshold $u_s$ is the smallest value of $u$ for which the curve is linear.

The quantity $M_n(u)$ is estimater par the empirical estimator of the mean:

$m_n(u) = \frac{1}{n} \sum_{i=1}^n (X_i - u) 1_{X_i \ge u} = \frac{1}{n} \sum_{i=1}^n X_i 1_{X_i \ge u} - u$

The quantity $M_n(u)$ is asymptotically Gaussian with mean $m_n(u)$ and variance $m_n(u) (1 - m_n(u))/n$

Parameters:

sample2-d sequence of float, of dimension 1: The sample from which the distribution parameters are estimated.

Returns:

graphGraph: The graph of $m_n(u)$ and its confidence interval.

Notes

The confidence level can be set using the ResourceMap key GeneralizedParetoFactory-MeanResidualLifeConfidenceLevel The number of threshold points in the graph can be set with the key GeneralizedParetoFactory-MeanResidualLifePointNumber.

getBootstrapSize()¶

Accessor to the bootstrap size.

Returns:

sizeinteger: Size of the bootstrap.

getClassName()¶

Accessor to the object’s name.

Returns:

class_namestr: The object class name (object.__class__.__name__).

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOptimizationAlgorithm()¶

Accessor to the solver.

Returns:

solverOptimizationAlgorithm: The solver used for numerical optimization of the likelihood.

getShadowedId()¶

Accessor to the object’s shadowed id.

Returns:

idint: Internal unique identifier.

getVisibility()¶

Accessor to the object’s visibility state.

Returns:

visiblebool: Visibility flag.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

hasVisibleName()¶

Test if the object has a distinguishable name.

Returns:

hasVisibleNamebool: True if the name is not empty and not the default one.

setBootstrapSize(bootstrapSize)¶

Accessor to the bootstrap size.

Parameters:

sizeinteger: The size of the bootstrap.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setOptimizationAlgorithm(solver)¶

Accessor to the solver.

Parameters:

solverOptimizationAlgorithm: The solver used for numerical optimization of the likelihood.

setShadowedId(id)¶

Accessor to the object’s shadowed id.

Parameters:

idint: Internal unique identifier.

setVisibility(visible)¶

Accessor to the object’s visibility state.

Parameters:

visiblebool: Visibility flag.

Examples using the class¶

Fit an extreme value distribution

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An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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Examples using the class¶