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.. _qqplot_graph:
QQ-plot
-------
The Quantile - Quantile - Plot (QQ Plot) enables to validate whether
two given samples of data
are drawn from the same continuous distribution of dimension 1.
We denote by :math:`\left\{ x_1,\ldots,x_{\sampleSize} \right\}`
and :math:`\left\{ y_1,\ldots,y_{\sampleSize} \right\}` two given samples of dimension 1.
A QQ-Plot is based on the comparison of some empirical quantiles. Let :math:`q_{X}(\alpha)`
be the quantile of order
:math:`\alpha` of the distribution :math:`F`, with :math:`\alpha \in (0, 1)`. It is defined by:
.. math::
\begin{aligned}
q_{X}(\alpha) = \inf \{ x \in \Rset \, |\, F(x) \geq \alpha \}
\end{aligned}
The empirical quantile of order :math:`\alpha` built on the sample is defined by:
.. math::
\begin{aligned}
\widehat{q}_{X}(\alpha) = x_{([\sampleSize \alpha]+1)}
\end{aligned}
where :math:`[\sampleSize\alpha]` denotes the integral part of :math:`\sampleSize \alpha`
and :math:`\left\{ x_{(1)},\ldots,x_{(\sampleSize)} \right\}` is the sample sorted in ascended order:
.. math::
x_{(1)} \leq \dots \leq x_{(\sampleSize)}
Thus, the :math:`j^\textrm{th}` smallest value of the sample
:math:`x_{(j)}` is an estimate :math:`\widehat{q}_{X}(\alpha)` of the
:math:`\alpha`-quantile where :math:`\alpha = (j-1)/\sampleSize`, for :math:`1 < j \leq \sampleSize`.
The QQ-plot draws the couples of empirical quantiles of the same order from both samples:
:math:`(x_{(j)}, y_{(j)})_{1 < j \leq \sampleSize}`.
If both samples follow the same distribution, then the points should be close to the diagonal.
The following figure illustrates a QQ-plot with two
samples of size :math:`\sampleSize=50`. In this example, the
points remain close to the diagonal and the hypothesis
“Both samples are drawn from the same distribution” does not seem false,
even if a more quantitative analysis should be
carried out to confirm this.
.. plot::
import openturns as ot
from openturns.viewer import View
ot.RandomGenerator.SetSeed(0)
distribution = ot.Normal(3.0, 2.0)
sample = distribution.getSample(150)
graph = ot.VisualTest.DrawQQplot(sample, distribution)
View(graph)
.. plot::
import openturns as ot
from openturns.viewer import View
ot.RandomGenerator.SetSeed(0)
distribution = ot.Normal(3.0, 3.0)
distribution2 = ot.Normal(2.0, 1.0)
sample = distribution.getSample(150)
graph = ot.VisualTest.DrawQQplot(sample, distribution2)
View(graph)
In this second example, the two samples clearly arise from two different
distributions.
.. topic:: API:
- See :py:func:`~openturns.VisualTest.DrawQQplot` to draw a QQ plot
.. topic:: Examples:
- See :doc:`/auto_data_analysis/statistical_tests/plot_qqplot_graph`
.. topic:: References:
- [saporta1990]_
- [dixon1983]_
- [nisthandbook]_
- [dagostino1986]_
- [bhattacharyya1997]_
- [sprent2001]_
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