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.. _pearson_coefficient:
Pearson correlation coefficient
-------------------------------
Pearson’s correlation coefficient :math:`\rho_{U,V}` aims to measure
the strength of a linear relationship between two random variables
:math:`U` and :math:`V`. It is defined as follows:
.. math::
\begin{aligned}
\rho_{U,V} = \frac{\displaystyle \Cov{U,V}}{\sigma_U \sigma_V}
\end{aligned}
where
:math:`\Cov{U,V} = \Expect{ \left( U - m_U \right) \left( V - m_V \right) }`,
:math:`m_U= \Expect{U}`, :math:`m_V= \Expect{V}`,
:math:`\sigma_U= \sqrt{\Var{U}}` and :math:`\sigma_V= \sqrt{\Var{V}}`.
If we have a sample made up of a set of :math:`N` pairs
:math:`\left\{ (u_1,v_1),(u_2,v_2),\ldots,(u_N,v_N) \right\}`, Pearson’s
correlation coefficient can be estimated using the formula:
.. math::
\begin{aligned}
\widehat{\rho}_{U,V} = \frac{ \displaystyle \sum_{i=1}^N \left( u_i - \overline{u} \right) \left( v_i - \overline{v} \right) }{ \sqrt{\displaystyle \sum_{i=1}^N \left( u_i - \overline{u} \right)^2 \left( v_i - \overline{v} \right)^2} }
\end{aligned}
where :math:`\overline{u}` and :math:`\overline{v}` represent the
empirical means of the samples :math:`(u_1,\ldots,u_N)` and
:math:`(v_1,\ldots,v_N)`.
Pearson’s correlation coefficient takes values between -1 and 1. The
closer its absolute value is to 1, the stronger the indication is that a
linear relationship exists between variables :math:`U` and :math:`V`.
The sign of Pearson’s coefficient indicates if the two variables
increase or decrease in the same direction (positive coefficient) or in
opposite directions (negative coefficient). We note that a correlation
coefficient equal to 0 does not necessarily imply the independence of
variables :math:`U` and :math:`V`: this property is in fact
theoretically guaranteed only if :math:`U` and :math:`V` both follow a
Normal distribution. In all other cases, there are two possible
situations in the event of a zero Pearson’s correlation coefficient:
- the variables :math:`U` and :math:`V` are in fact independent,
- or a non-linear relationship exists between :math:`U` and :math:`V`.
.. plot::
import openturns as ot
from openturns.viewer import View
N = 20
ot.RandomGenerator.SetSeed(10)
x = ot.Uniform(0.0, 10.0).getSample(N)
f = ot.SymbolicFunction(['x'], ['5*x+10'])
y = f(x) + ot.Normal(0.0, 5.0).getSample(N)
graph = f.draw(0.0, 10.0)
graph.setTitle('A linear relationship exists between U and V:\n Pearson\'s coefficient is a relevant measure of dependency')
graph.setXTitle('u')
graph.setYTitle('v')
cloud = ot.Cloud(x, y)
cloud.setPointStyle('circle')
cloud.setColor('orange')
graph.add(cloud)
View(graph)
.. plot::
import openturns as ot
from openturns.viewer import View
N = 20
ot.RandomGenerator.SetSeed(10)
x = ot.Uniform(0.0, 10.0).getSample(N)
f = ot.SymbolicFunction(['x'], ['x^2'])
y = f(x) + ot.Normal(0.0, 5.0).getSample(N)
graph = f.draw(0.0, 10.0)
graph.setTitle('There is a strong, non-linear relationship between U and V:\n Pearson\'s coefficient is not a relevant measure of dependency')
graph.setXTitle('u')
graph.setYTitle('v')
cloud = ot.Cloud(x, y)
cloud.setPointStyle('circle')
cloud.setColor('orange')
graph.add(cloud)
View(graph)
.. plot::
import openturns as ot
from openturns.viewer import View
N = 20
ot.RandomGenerator.SetSeed(10)
x = ot.Uniform(0.0, 10.0).getSample(N)
f = ot.SymbolicFunction(['x'], ['5'])
y = ot.Uniform(0.0, 10.0).getSample(N)
graph = f.draw(0.0, 10.0)
graph.setTitle('Pearson\'s coefficient estimate is quite close to zero\nbecause U and V are independent')
graph.setXTitle('u')
graph.setYTitle('v')
cloud = ot.Cloud(x, y)
cloud.setPointStyle('circle')
cloud.setColor('orange')
graph.add(cloud)
View(graph)
.. plot::
import openturns as ot
from openturns.viewer import View
N = 20
ot.RandomGenerator.SetSeed(10)
x = ot.Uniform(0.0, 10.0).getSample(N)
f = ot.SymbolicFunction(['x'], ['30*sin(x)'])
y = f(x) + ot.Normal(0.0, 5.0).getSample(N)
graph = f.draw(0.0, 10.0)
graph.setTitle('Pearson\'s coefficient estimate is quite close to zero\neven though U and V are not independent')
graph.setXTitle('u')
graph.setYTitle('v')
cloud = ot.Cloud(x, y)
cloud.setPointStyle('circle')
cloud.setColor('orange')
graph.add(cloud)
View(graph)
The estimate :math:`\widehat{\rho}` of Pearson’s correlation
coefficient is sometimes denoted by :math:`r`.
.. topic:: API:
- See method :py:meth:`~openturns.CorrelationAnalysis.computeLinearCorrelation`
- See method :py:meth:`~openturns.Sample.computeLinearCorrelation`
.. topic:: Examples:
- See :doc:`/auto_data_analysis/manage_data_and_samples/plot_sample_correlation`
.. topic:: References:
- [saporta1990]_
- [dixon1983]_
- [nisthandbook]_
- [dagostino1986]_
- [bhattacharyya1997]_
- [sprent2001]_
- [burnham2002]_
|
