TimeVaryingResult¶

class TimeVaryingResult(*args)¶

Distribution factory result for non stationary likelihood estimation.

We consider a non stationary GEV model to describe the distribution of $Z_t$ :

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

We have 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:

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

Each of $\mu(t), \sigma(t), \xi(t)$ has an expression in terms of a parameter vector and time functions:

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

where:

$h: \Rset \mapsto \Rset$ is usually referred to as the inverse-link function. The function $\theta(t)$ denotes either $\mu(t)$ , $\sigma(t)$ or $\xi(t)$ ,
each $\varphi_i^{\theta}$ is a scalar function $\Rset \mapsto \Rset$ ,
each $\beta_i^{j} \in \Rset$ .

We denote by $d_{\mu}$ , $d_{\sigma}$ and $d_{\xi}$ the size of the functional basis of $\mu$ , $\sigma$ and $\xi$ respectively. We denote by $\vect{\beta} = (\beta_1^{\mu}, \dots, \beta_{d_{\mu}}^{\mu}, \beta_1^{\sigma}, \dots, \beta_{d_{\sigma}}^{\sigma}, \beta_1^{\xi}, \dots, \beta_{d_{\xi}}^{\xi})$ the complete vector of parameters.

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

Parameters:

factoryDistributionFactory: Parent distribution factory.
data2-d sequence of float: Data.
parameterFunctionFunction: The function $\vect{\theta}$ .
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

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.
`getId`()	Accessor to the object's id.
`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.
`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.
`setLogLikelihood`(logLikelihood)	Optimal log-likelihood value accessor.
`setName`(name)	Accessor to the object's name.
`setParameterDistribution`(parameterDistribution)	Accessor to the distribution of of $\vect{\beta}$ .
`setShadowedId`(id)	Accessor to the object's shadowed id.
`setVisibility`(visible)	Accessor to the object's visibility state.

getTimeGrid

__init__(*args)¶

drawDiagnosticPlot()¶

Draw the 4 usual diagnostic plots.

If $Z_t$ is a non stationary GEV model defined as follows:

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

then the standardized variables $\hat{Z}_t$ defined by:

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

have the standard Gumbel distribution which is the GEV model with $(\mu, \sigma, \xi) = (0, 1, 0)$ .

Then, the 4 diagnostic graphs are built from the transformed data compared to the Gumbel model: the probability-probability plot, the quantile-quantile plot, the return level plot, the data histogram with the Gumbel distribution.

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

The probability-probability plot consists of the points:

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

The quantile-quantile plot consists of the points:

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

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 of the transformed data and $\hat{z}_{m}$ the $m$ -block return level of the Gumbel distribution.

Returns:

gridGridLayout

Returns a grid of 4 graphs:

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

drawParameterFunction(parameterIndex=0)¶

Draw the parameter function.

Parameters:

parameterIndexint,: The index $j$ specifying the marginal of $\vect{\theta}$ .

Returns:

graphGraph: Graph of $\theta_j(t)$ .

drawQuantileFunction(p)¶

Draw the quantile function.

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 $t$ .

Parameters:

pfloat: The quantile level.

Returns:

graphGraph: Quantile function graph.

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 $t$ .

getId()¶

Accessor to the object’s id.

Returns:

idint: Internal unique identifier.

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 $t \mapsto \vect{\theta}(t)$ .

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.

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 $\vect{\beta}$ .

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¶

Estimate a GEV on the Port Pirie sea-levels data

Estimate a GEV on the Fremantle sea-levels data

Estimate a GEV on race times data

OpenTURNS

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

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TimeVaryingResult¶

Examples using the class¶