TimeVaryingResult¶

class TimeVaryingResult(*args)¶

Estimation result class for a non stationary GEV or GPD model.

Parameters:

factoryDistributionFactory: Parent distribution factory.
data2-d sequence of float: Sample drawn from $Z_t$ .
parameterFunctionFunction: The function $t \mapsto \vect{\theta}(t)$ .
timeStamps2-d sequence of float: Values of $t$ .
parameterDistributionDistribution: The distribution of $\vect{\beta}$ .
normalizationFunctionLinearFunction: The normalization function $t \mapsto \tau(t)$ .
llhfloat: Maximum log-likelihood.

See also

GeneralizedExtremeValueFactory

Notes

This class is created by the method buildTimeVarying() of the classes GeneralizedExtremeValueFactory and GeneralizedParetoFactory.

Let $Z_t$ be a non stationary random variable which follows a GEV distribution or whose excesses above $u$ follow a GPD. The parameters of the GEV or GPD model depend on $t$ :

$Z_t & \sim \mbox{GEV}(\mu(t), \sigma(t), \xi(t))\\ Z_t & \sim \mbox{GPD}(\sigma(t), \xi(t), u)$

For the GPD, the threshold $u$ is assumed to be known.

We denote by $(z_{t_1}, \dots, z_{t_n})$ the values of $Z_t$ on the time stamps $(t_1, \dots, t_n)$ .

For numerical reasons, the time stamps have been normalized using the linear function:

$\tau(t) = \dfrac{t-c}{d}$

Let $\vect{\theta} = (\theta_1, \dots, \theta_p)$ be the set of parameters $(\mu, \sigma, \xi)$ for the GEV model and $(\sigma, \xi)$ for the GPD. Each component $\theta_q$ can be written as a function of $t$ :

$\theta_q(t) = h_q\left(\sum_{i=1}^{d_{\theta_q}} \beta_i^{\theta_q} \varphi_i^{\theta_q}(\tau(t))\right)$

where:

$d_{\theta_q}$ is the size of the functional basis involved in the modelling of $\theta_q$ ,
$h_q: \Rset \mapsto \Rset$ is usually referred to as the inverse-link function of the component $\theta_q$ ,
each $\varphi_i^{\theta_q}$ is a scalar function $\Rset \mapsto \Rset$ ,
each $\beta_i^{j} \in \Rset$ .

We denote by $\vect{\beta} = (\beta_1^{\theta_1}, \dots, \beta_{d_{\theta_1}}^{\theta_1}, \dots, \beta_1^{\theta_p}, \dots, \beta_{d_{\theta_p}}^{\theta_p})$ the complete vector of parameters.

The estimator of the vector $\vect{\beta}$ maximizes the likelihood of the Parent distribution (ie the distribution of $Z_t$ ).

Methods

`drawDiagnosticPlot`()	Draw the 4 usual diagnostic plots.
`drawParameterFunction`([parameterIndex])	Draw the parameter function.
`drawQuantileFunction`(p)	Draw the quantile function.
`getClassName`()	Accessor to the object's name.
`getDistribution`(t)	Accessor to the Parent distribution at a given time.
`getLogLikelihood`()	Optimal log-likelihood value accessor.
`getName`()	Accessor to the object's name.
`getNormalizationFunction`()	Normalizing function accessor.
`getOptimalParameter`()	Optimal parameter accessor.
`getParameterDistribution`()	Accessor to the distribution of $\vect{\beta}$ .
`getParameterFunction`()	Parameter function accessor.
`getTimeGrid`()	Accessor to the time grid.
`hasName`()	Test if the object is named.
`setLogLikelihood`(logLikelihood)	Optimal log-likelihood value accessor.
`setName`(name)	Accessor to the object's name.
`setParameterDistribution`(parameterDistribution)	Accessor to the distribution of of $\vect{\beta}$ .

__init__(*args)¶

drawDiagnosticPlot()¶

Draw the 4 usual diagnostic plots.

Returns:

gridGridLayout

Returns a grid of 4 graphs:

the QQ-plot,
the PP-plot,
the return level graph (with confidence lines),
the density graph.

Notes

Let $Z_t$ be a non stationary random variable which follows a GEV distribution or whose excesses above $u$ follow a GPD. The parameters of the GEV or GPD model depend on $t$ :

$Z_t & \sim \mbox{GEV}(\mu(t), \sigma(t), \xi(t))\\ Z_t & \sim \mbox{GPD}(\sigma(t), \xi(t), u)$

Then, the standardized variables $\hat{Z}_t$ defined respectively by:

$\hat{Z}_t & = \dfrac{1}{\xi(t)} \log \left[1 + \xi(t)\left( \dfrac{Z_t-\mu(t)}{\sigma(t)} \right)\right] \\ \hat{Z}_t & = \dfrac{1}{\xi(t)} \log \left[1 + \xi(t)\left( \dfrac{Z_t-u}{\sigma(t)} \right)\right]$

respectively follows the standard Gumbel distribution (which is the GEV model with $(\mu, \sigma, \xi) = (0, 1, 0)$ ) or the standard Exponential distribution (which is the GPD with $(\sigma, \xi, u) = (1, 0, 0)$ ).

Then, the 4 usual diagnostic graphs are built from the transformed data compared to the Gumbel or Exponential model:

the probability-probability plot,
the quantile-quantile plot,
the return level plot,
the data histogram with the Gumbel/Exponential distribution.

If $(\hat{z}_{(1)}, \dots, \hat{z}_{(n)})$ denotes the ordered transformed data, the graphs are defined as follows.

The probability-probability plot consists of the points (respectively for the GEV and GPD models):

$\left\{ \left( i/(n+1),\exp (-\exp (-\hat{z}_{(i)})) \right), i=1, \dots , n\right\}\\ \left\{ \left( i/(n+1), 1- \exp(-\tilde{z}_{(i)}) \right), i=1, \dots , n\right\}$

The quantile-quantile plot consists of the points (respectively for the GEV and GPD models):

$\left\{ \left( \hat{z}_{(i)}, -\log(-\log(i/(n+1))) \right), i=1, \dots , n\right\} \\ \left\{ \left( \tilde{z}_{(i)}, -\log(1-i/(n+1)) \right), i=1, \dots , n\right\}$

The return level plot consists of the points:

$\left\{ \left( m, \hat{z}_m \right), m> 0 \right\}$

and the points:

$\left\{ \left( m, \hat{z}_{m}^{emp} \right), m> 0 \right\}$

where $\hat{z}_{m}^{emp}$ is the empirical $m$ -block return level (for the GEV model) or the $m$ -observation return level (for the GPD model) of the transformed data and $\hat{z}_{m}$ the same quantity evaluated from the Gumbel/Exponential distribution.

drawParameterFunction(parameterIndex=0)¶

Draw the parameter function.

Parameters:

parameterIndexint,: The index $q$ specifying the component of $\theta_q$ .

Returns:

graphGraph: Graph of $t \mapsto \theta_q(t)$ .

drawQuantileFunction(p)¶

Draw the quantile function.

Parameters:

pfloat: The quantile level.

Returns:

graphGraph: Quantile function graph.

Notes

The quantile function is the function $t \mapsto q_p(t)$ where $q_p(t)$ is the quantile of order $p$ of the Parent distribution at time $t$ .

getClassName()¶

Accessor to the object’s name.

Returns:

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

getDistribution(t)¶

Accessor to the Parent distribution at a given time.

Parameters:

tfloat: Time value.

Returns:

distributionDistribution: The Parent distribution at time t.

getLogLikelihood()¶

Optimal log-likelihood value accessor.

Returns:

llhfloat: Maximum log-likelihood.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getNormalizationFunction()¶

Normalizing function accessor.

Returns:

normalizeFunctionFunction: The function $t \mapsto \tau(t)$ .

getOptimalParameter()¶

Optimal parameter accessor.

Returns:

optimalParameterPoint: Optimal vector of parameters $\vect{\beta}$ .

getParameterDistribution()¶

Accessor to the distribution of $\vect{\beta}$ .

Returns:

parameterDistributionDistribution: The distribution of the estimator of $\vect{\beta}$ .

getParameterFunction()¶

Parameter function accessor.

Returns:

parameterFunctionFunction: The function $t \mapsto \vect{\theta}(t)$ .

getTimeGrid()¶

Accessor to the time grid.

Returns:

timeGridSample: Values of $t$ .

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

setLogLikelihood(logLikelihood)¶

Optimal log-likelihood value accessor.

Parameters:

llhfloat: Maximum log-likelihood.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setParameterDistribution(parameterDistribution)¶

Accessor to the distribution of of $\vect{\beta}$ .

Parameters:

parameterDistributionDistribution: The distribution of the estimator of $\vect{\beta}$ .

Examples using the class¶

Estimate a GEV on the Port Pirie sea-levels data

Estimate a GPD on the daily rainfall data

Estimate a GEV on race times data

Estimate a GEV on the Fremantle sea-levels data

OpenTURNS

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

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TimeVaryingResult¶

Examples using the class¶