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| In [[statistics]], the '''probability integral transform''' or '''transformation''' relates to the result that data values that are modelled as being [[random variable]]s from any given [[continuous distribution]] can be converted to random variables having a [[uniform distribution]].<ref name=Dodge/> This holds exactly provided that the distribution being used is the true distribution of the random variables; if the distribution is one fitted to the data the result will hold approximately in large samples.
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| The result is sometimes modified or extended so that the result of the transformation is a standard distribution other than the uniform distribution, such as the [[exponential distribution]].
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| ==Applications==
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| One use for the probability integral transform in statistical [[data analysis]] is to provide the basis for testing whether a set of observations can reasonably be modelled as arising from a specified distribution. Specifically, the probability integral transform is applied to construct an equivalent set of values, and a test is then made of whether a uniform distribution is appropriate for the constructed dataset. Examples of this are [[P-P plot]]s and [[Kolmogorov-Smirnov test]]s.
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| A second use for the transformation is in the theory related to [[Copula (statistics)|copulas]] which are a means of both defining and working with distributions for statistically dependent multivariate data. Here the problem of defining or manipulating a [[joint probability distribution]] for a set of random variables is simplified or reduced in apparent complexity by applying the probability integral transform to each of the components and then working with a joint distribution for which the marginal variables have uniform distributions.
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| A third use is based on applying the inverse of the probability integral transform to convert random variables from a uniform distribution to have a selected distribution: this is known as [[inverse transform sampling]].
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| ==Examples==
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| Suppose that a random variable ''X'' has a [[continuous distribution]] for which the [[cumulative distribution function]] is ''F''<sub>''X''</sub>. Then the random variable ''Y'' defined as
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| :<math>Y=F_X(X) \,,</math> | |
| has a uniform distribution.<ref name=Dodge>Dodge, Y. (2003) ''The Oxford Dictionary of Statistical Terms'', OUP. ISBN 0-19-920613-9</ref>
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| For an illustrative example, let ''X'' be a random variable with a standard normal distribution N(0,1) where <math>\operatorname{erf}(),</math> is the [[error function]]. Then its CDF is
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| :<math>\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-t^2/2} \, dt
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| = \frac12\Big[\, 1 + \operatorname{erf}\Big(\frac{x}{\sqrt{2}}\Big)\,\Big],\quad x\in\mathbb{R}.
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| \,</math>
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| Then the new random variable ''Y'', defined by ''Y''=Φ(''X''), is uniformly distributed.
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| If X has an [[exponential distribution]] with unit mean, then
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| :<math>F(x)=1-\exp(-x),</math> | |
| and the immediate result of the probability integral transform is that
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| :<math>Y=1-\exp(-X)</math>
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| has a uniform distribution. However, the symmetry of the uniform distribution can then be used to show that
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| :<math>Y'=\exp(-X)</math>
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| also has a uniform distribution.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Probability Integral Transform}}
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| [[Category:Theory of probability distributions]]
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