QuantileConfidence¶

class QuantileConfidence(*args)¶

Estimate confidence intervals of a quantile.

Refer to Estimation of a quantile bound.

Warning

This class is experimental and likely to be modified in future releases. To use it, import the openturns.experimental submodule.

Parameters:

alphafloat: Quantile level
betafloat, optional: Confidence level. Default value is 0.95

Methods

`computeAsymptoticBilateralConfidenceInterval`(sample)	Evaluate asymptotic bilateral bounds of a quantile.
`computeAsymptoticBilateralRank`(size)	Evaluate the asymptotic bilateral rank of a quantile.
`computeBilateralConfidenceInterval`(sample)	Evaluate a bilateral confidence interval of a quantile.
`computeBilateralMinimumSampleSize`()	Evaluate the minimum size of the sample for the bilateral case.
`computeBilateralRank`(size)	Evaluate the bilateral rank of a quantile.
`computeUnilateralConfidenceInterval`(sample)	Evaluate an unilateral confidence interval of a quantile.
`computeUnilateralMinimumSampleSize`([rank, tail])	Evaluate the minimum sample size for the unilateral case.
`computeUnilateralRank`(size[, tail])	Evaluate an unilateral rank of a quantile.
`getAlpha`()	Quantile level accessor.
`getBeta`()	Confidence level accessor.
`getClassName`()	Accessor to the object's name.
`getName`()	Accessor to the object's name.
`hasName`()	Test if the object is named.
`setAlpha`(alpha)	Quantile level accessor.
`setBeta`(beta)	Confidence level accessor.
`setName`(name)	Accessor to the object's name.

Notes

This class estimates bounds of the quantile of level $\alpha \in [0,1]$ of the random variable $X$ with a confidence greater than $\beta$ using a given order statistics.

Let $x_{\alpha}$ be the unknown quantile of level $\alpha$ of the random variable $X$ of dimension 1. Let $(X_1, \dots, X_\sampleSize)$ be a sample of independent and identically distributed variables according to $X$ .

The bounds of the interval are computed from order statistics that we now introduce. Let $X_{(k)}$ be the $k$ -th order statistics of $(X_1, \dots, X_\sampleSize)$ which means that $X_{(k)}$ is the $k$ -th maximum of $(X_1, \dots, X_\sampleSize)$ for $1 \leq k \leq \sampleSize$ .

For example, $X_{(1)} = \min (X_1, \dots, X_\sampleSize)$ is the minimum and $X_{(\sampleSize)} = \max (X_1, \dots, X_\sampleSize)$ is the maximum.

We have:

$X_{(1)} \leq X_{(2)} \leq \dots \leq X_{(\sampleSize)}$

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> sample = ot.Gumbel().getSample(100)
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> ci = algo.computeUnilateralConfidenceInterval(sample)

__init__(*args)¶

computeAsymptoticBilateralConfidenceInterval(sample)¶

Evaluate asymptotic bilateral bounds of a quantile.

The asymptotic bounds are given by the order statistics:

$[X_{(k_1)}, X_{(k_2)}]$

so that:

$\lim_{n \rightarrow \infty} \Prob{X_{(k_1)} \leq x_{\alpha} \leq X_{(k_2)}} \geq \beta$

where the ranks $0 \leq k_1 \leq k_2 \leq n - 1$ are given by computeAsymptoticBilateralRank().

Parameters:

sample2-d sequence of float: Sample of the variable $X$

Returns:

ciInterval: Quantile confidence interval

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> sample = ot.Gumbel().getSample(60)
>>> ci = algo.computeAsymptoticBilateralConfidenceInterval(sample)

computeAsymptoticBilateralRank(size)¶

Evaluate the asymptotic bilateral rank of a quantile.

This method computes two integers $(k_1, k_2)$ such that:

$\lim_{n \rightarrow \infty} \Prob{X_{(k_1)} \leq x_\alpha \leq X_{(k_2)}} = \beta, 0 \leq k_1 \leq k_2 \leq n - 1$

In other words, the interval $\left[X_{(k_1)}, X_{(k_2)} \right]$ is an asymptotic confidence interval for $x_\alpha$ with asymptotic confidence $\beta$ . The asymptotic bilateral ranks $k_1, k_2 \in \llbracket 0, n - 1 \rrbracket$ are estimated from:

$k_1 & = \left\lfloor n \alpha - \sqrt{n} z_{1-\beta/2} \sqrt{\alpha (1 - \alpha)} \right\rfloor - 1 \\ k_2 & = \left\lfloor n \alpha + \sqrt{n} z_{1-\beta/2} \sqrt{\alpha (1 - \alpha)} \right\rfloor - 1$

with $z_{1-\beta/2}$ the standard Gaussian quantile of order $1-\beta/2$ , see [delmas2006] proposition 12.2.13 page 257.

Parameters:

sizeint: Sample size

Returns:

ranksIndices of size 2: Pair of lower and upper ranks $(k_1, k_2)$ with $0 \leq k_1 \leq k_2 \leq n-1$ .

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> k1, k2 = algo.computeAsymptoticBilateralRank(100)
>>> print(k1, k2)
0 8

computeBilateralConfidenceInterval(sample)¶

Evaluate a bilateral confidence interval of a quantile.

The confidence interval for the quantile of level $\alpha$ is given by the order statistics:

$[X_{(k_1)}, X_{(k_2)}]$

so that:

$\Prob{X_{(k_1)} \leq x_{\alpha} \leq X_{(k_2)}} \geq \beta$

where the ranks $0 \leq k_1 \leq k_2 \leq n - 1$ are given by computeBilateralRank().

Parameters:

sample2-d sequence of float: Sample of the variable $X$

Returns:

ciInterval: Quantile confidence interval

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> sample = ot.Gumbel().getSample(60)
>>> ci = algo.computeBilateralConfidenceInterval(sample)

computeBilateralMinimumSampleSize()¶

Evaluate the minimum size of the sample for the bilateral case.

The minimal bilateral sample size is the smallest integer $n$ such that:

$\Prob{X_{(1)} \leq x_{\alpha} \leq X_{(n)}} \geq \beta.$

In other words, this is the minimum sample size such that the interval $\left[X_{(1)}, X_{(n)} \right]$ is a confidence interval of the quantile $x_\alpha$ with confidence $\beta$ . The solution of this problem is the minimum value of $n$ such that:

$1-\alpha^n-(1-\alpha)^n \geq \beta$

The sample size $n$ is searched inside the bounds given by the inequality:

$\left\lfloor \frac{\log(1 - \beta)}{\log(\gamma)} \right\rfloor \leq n \leq \left\lceil \frac{\log(\frac{1-\beta}{2})}{\log(\gamma)} \right\rceil$

where $\gamma = \max(\alpha, 1- \alpha)$ .

Returns:

sizeint: Minimum sample size $n$

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> size = algo.computeBilateralMinimumSampleSize()
>>> print(size)
59

computeBilateralRank(size)¶

Evaluate the bilateral rank of a quantile.

The bilateral ranks $k_1, k_2$ are the integers minimizing the probability:

$\argmin_{0 \leq k_1 \leq k_2 \leq n - 1 \, | \, \Prob{k_1 < X \leq k_2} \geq \beta} \Prob{k_1 < X \leq k_2}$

where $X \sim \cB(n, \alpha)$ is the binomial distribution with parameters $n$ and $\alpha$ .

so that:

$\Prob{X_{(k_1)} \leq x_{\alpha} \leq X_{(k_2)}} \geq \beta.$

These ranks exist only if:

$1 - \alpha^n - (1-\alpha)^n \ge \beta.$

Parameters:

sizeint: Sample size $n$

Returns:

ranksIndices of size 2: Pair of lower and upper ranks $(k_1, k_2)$ such that $0 \leq k_1 \leq k_2 \leq n - 1$ .

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> k1, k2 = algo.computeBilateralRank(100)
>>> print((k1, k2))
(1, 10)

computeUnilateralConfidenceInterval(sample, tail=False)¶

Evaluate an unilateral confidence interval of a quantile.

The lower tail confidence interval is given by the order statistics:

$[X_{(k_{low})}, +\infty[$

so that:

$\Prob{X_{(k_{low})} \leq x_{\alpha}} \geq \beta.$

and the upper tail confidence interval is given by the order statistics:

$]-\infty, X_{(k_{up})}]$

so that:

$\Prob{x_{\alpha} \leq X_{(k_{up})}} \geq \beta.$

where the lower or upper ranks $0 \leq k_{low} \leq n - 1, 0 \leq k_{up} \leq n - 1$ are given by computeUnilateralRank().

Parameters:

sample2-d sequence of float: Quantile level
tailbool, optional: True indicates the interval is bounded by a lower value. False indicates the interval is bounded by an upper value. Default value is False.

Returns:

ciInterval: Quantile confidence interval

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> sample = ot.Gumbel().getSample(100)
>>> ci = algo.computeUnilateralConfidenceInterval(sample)

computeUnilateralMinimumSampleSize(rank=0, tail=False)¶

Evaluate the minimum sample size for the unilateral case.

For the lower bound the minimal unilateral sample size is the smallest integer $n$ such that:

$\Prob{X_{(r)} \leq x_{\alpha}} \geq \beta.$

For the upper bound the minimal unilateral sample size is the smallest integer $n$ such that:

$\Prob{x_{\alpha} \leq X_{(n-r-1)}} \geq \beta.$

In the general case the minimum size of the sample is the smallest $n$ such that:

$F_{n,\alpha}(n-r-1) \geq \beta$

where $F_{n,\alpha}$ is the cumulative distribution function of the Binomial distribution with parameters $n$ and $\alpha$ .

When the rank $r$ is zero the solution is analytical.

For the lower bound the minimum sample size is the integer that satisfies:

$1 - (1 - \alpha)^n \geq \beta$

which is given by:

$n_{low} = \left\lceil \frac{\log(1-\beta)}{\log(1-\alpha)} \right\rceil.$
And for the upper bound the minimum sample size is the integer that satisfies:

$1 - \alpha^n \geq \beta$

which is given by:

$n_{up} = \left\lceil \frac{\log(1-\beta)}{\log(\alpha)} \right\rceil.$

Parameters:

rankint, optional: Rank of the quantile. Default value is 0.
tailbool, optional: True indicates the interval is bounded by a lower value. False indicates the interval is bounded by an upper value. Default value is False.

Returns:

sizeint: Minimum sample size

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> size = algo.computeUnilateralMinimumSampleSize(0)
>>> print(size)
59

computeUnilateralRank(size, tail=False)¶

Evaluate an unilateral rank of a quantile.

The lower tail rank $k_{low}$ is the largest integer $k$ such that:

$\Prob{X_{(k)} \leq x_{\alpha}} \geq \beta.$

In other words, the interval $\left[ X_{(k_{low})}, +\infty\right[$ is a unilateral confidence interval for the quantile $x_\alpha$ with confidence $\beta$ .

It is given by:

$k_{low} = \overline{F}^{-1}_{n, \alpha}(\beta)$

where $\overline{F}^{-1}_{n, \alpha}$ is the complementary quantile function of the binomial distribution with parameters $n$ and $\alpha$ .

The problem has a solution only if:

$1 - (1 - \alpha)^n \geq \beta.$

The upper tail rank $k_{up}$ is the smallest integer $k$ such that:

$\Prob{x_{\alpha} \leq X_{(k)}} \geq \beta.$

In other words, the interval $\left]-\infty, X_{(k_{up})}\right]$ is a unilateral confidence interval for the quantile $x_\alpha$ with confidence $\beta$ . The solution is:

$k_{up} = F^{-1}_{n, \alpha}(\beta)$

where $F^{-1}_{n, \alpha}$ is the quantile function of the cumulative distribution function of the binomial distribution with parameters $n$ and $\alpha$ . The problem has a solution only if:

$1 - \alpha^n \geq \beta.$

Parameters:

sizeint: Sample size
tailbool, optional: True indicates the interval is bounded by a lower value. False indicates the interval is bounded by an upper value. Default value is False.

Returns:

rankint: Rank $k \in \{0, \hdots, n-1\}$

Examples

>>> import openturns as ot
>>> import openturns.experimental as otexp
>>> alpha = 0.05
>>> beta = 0.95
>>> algo = otexp.QuantileConfidence(alpha, beta)
>>> rank = algo.computeUnilateralRank(100)
>>> print(rank)
9

getAlpha()¶

Quantile level accessor.

Returns:

alphafloat: Quantile level

getBeta()¶

Confidence level accessor.

Returns:

betafloat: Confidence level

getClassName()¶

Accessor to the object’s name.

Returns:

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

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

setAlpha(alpha)¶

Quantile level accessor.

Parameters:

alphafloat: Quantile level

setBeta(beta)¶

Confidence level accessor.

Parameters:

betafloat: Confidence level

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

Examples using the class¶

Estimate quantile confidence intervals from data

Estimate a confidence interval of a quantile

OpenTURNS

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

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QuantileConfidence¶

Examples using the class¶