Inverse transform sampling

For any random variable $X\backslash in\backslash mathbb\; R$, the random variable $F\_X^\{-1\}(U)$ has the same distribution as $X$, where $F\_X^\{-1\}$ is the generalized inverse of the cumulative distribution function $F\_X$ of $X$ and $U$ is uniform on $[0,1]$.{{cite book | last1 = McNeil | first1 = Alexander J. | last2 = Frey | first2 = Rüdiger | last3 = Embrechts | first3 = Paul | title = Quantitative risk management | date=2005 | series=Princeton Series in Finance | publisher=Princeton University Press, Princeton, NJ | page=186 | isbn=0-691-12255-5}}

For continuous random variables, the inverse probability integral transform is indeed the inverse of the probability integral transform, which states that for a continuous random variable $X$ with cumulative distribution function $F\_X$, the random variable $U=F\_X(X)$ is uniform on $[0,1]$.

File:InverseFunc.png

From $U\; \backslash sim\; \backslash mathrm\{Unif\}[0,1]$, we want to generate $X$ with CDF $F\_X(x).$ We assume $F\_X(x)$ to be a continuous, strictly increasing function, which provides good intuition.

We want to see if we can find some strictly monotone transformation $T:[0,1]\backslash mapsto\; \backslash mathbb\{R\}$, such that $T(U)\backslash overset\{d\}\{=\}X$. We will have

$F\_X(x)=\backslash Pr(X\backslash leq\; x)=\backslash Pr(T(U)\backslash leq\; x)\; =\; \backslash Pr(U\backslash leq\; T^\{-1\}(x))=T^\{-1\}(x),\; \backslash text\{\; for\; \}\; x\backslash in\; \backslash mathbb\{R\},$

where the last step used that $\backslash Pr(U\; \backslash leq\; y)\; =\; y$ when $U$ is uniform on $[0,1]$.

So we got $F\_X$ to be the inverse function of $T$, or, equivalently $T(u)=F\_X^\{-1\}(u),\; u\backslash in\; [0,1].$

Therefore, we can generate $X$ from $F\_X^\{-1\}(U).$

File:Generalized inversion method.svg

File:Inverse Transform Sampling Example.gif

The problem that the inverse transform sampling method solves is as follows:

• Let $X$ be a random variable whose distribution can be described by the cumulative distribution function $F\_X$.
• We want to generate values of $X$ which are distributed according to this distribution.

The inverse transform sampling method works as follows:

1. Generate a random number $u$ from the standard uniform distribution in the interval $[0,1]$, i.e. from $U\; \backslash sim\; \backslash mathrm\{Unif\}[0,1].$
2. Find the generalized inverse of the desired CDF, i.e. $F\_X^\{-1\}(u)$.
3. Compute $X\text{'}(u)=F\_X^\{-1\}(u)$. The computed random variable $X\text{'}(U)$ has distribution $F\_X$ and thereby the same law as $X$.

Expressed differently, given a cumulative distribution function $F\_X$ and a uniform variable $U\backslash in[0,1]$, the random variable $X\; =\; F\_X^\{-1\}(U)$ has the distribution $F\_X$.

In the continuous case, a treatment of such inverse functions as objects satisfying differential equations can be given.{{cite journal | last1 = Steinbrecher | first1 = György | last2 = Shaw | first2 = William T. | title = Quantile mechanics | journal = European Journal of Applied Mathematics | date = 19 March 2008 | volume = 19 | issue = 2 | doi = 10.1017/S0956792508007341| s2cid = 6899308 }} Some such differential equations admit explicit power series solutions, despite their non-linearity.{{Cite journal |last1=Arridge |first1=Simon |last2=Maass |first2=Peter |last3=Öktem |first3=Ozan |last4=Schönlieb |first4=Carola-Bibiane |title=Solving inverse problems using data-driven models |journal=Acta Numerica |year=2019 |language=en |volume=28 |pages=1–174 |doi=10.1017/S0962492919000059 |s2cid=197480023 |issn=0962-4929|doi-access=free }}

• As an example, suppose we have a random variable $U\; \backslash sim\; \backslash mathrm\{Unif\}(0,1)$ and a cumulative distribution function

: $$

\begin{align}

F(x)=1-\exp(-\sqrt{x})

\end{align}

: In order to perform an inversion we want to solve for $F(F^\{-1\}(u))=u$

: $$

\begin{align}

F(F^{-1}(u))&=u \\

1-\exp\left(-\sqrt{F^{-1}(u)}\right) &= u \\

F^{-1}(u) &= (-\log(1-u))^2 \\

&= (\log(1-u))^2

\end{align}

: From here we would perform steps one, two and three.

• As another example, we use the exponential distribution with $F\_X(x)=1-e^\{-\backslash lambda\; x\}$ for x ≥ 0 (and 0 otherwise). By solving y=F(x) we obtain the inverse function

: $x\; =\; F^\{-1\}(y)\; =\; -\backslash frac\{1\}\{\backslash lambda\}\backslash ln(1-y).$

:It means that if we draw some $y\_0$from a $U\; \backslash sim\; \backslash mathrm\{Unif\}(0,1)$ and compute $x\_0\; =\; F\_X^\{-1\}(y\_0)\; =\; -\backslash frac\{1\}\{\backslash lambda\}\backslash ln(1-y\_0),$ This $x\_0$ has exponential distribution.

: The idea is illustrated in the following graph:

: File:Inverse transformation method for exponential distribution.jpg

: Note that the distribution does not change if we start with 1-y instead of y. For computational purposes, it therefore suffices to generate random numbers y in [0, 1] and then simply calculate

: $x\; =\; F^\{-1\}(y)\; =\; -\backslash frac\{1\}\{\backslash lambda\}\backslash ln(y).$

Let $F$ be a cumulative distribution function, and let $F^\{-1\}$ be its generalized inverse function (using the infimum because CDFs are weakly monotonic and right-continuous):{{cite book |author=Luc Devroye |title=Non-Uniform Random Variate Generation |publisher=Springer-Verlag |place=New York |year=1986 |chapter=Section 2.2. Inversion by numerical solution of F(X) = U |chapter-url=http://luc.devroye.org/chapter_two.pdf}}

:

Claim: If $U$ is a uniform random variable on $[0,1]$ then $F^\{-1\}(U)$ has $F$ as its CDF.

Proof:

:$$

\begin{align}

& \Pr(F^{-1}(U) \leq x) \\

& {} = \Pr(U \leq F(x)) \quad &(F\text{ is right-continuous, so }\{u:F^{-1}(u)\le x\}=\{u:u\le F(x)\}) \\

& {} = F(x) \quad &(\text{because }\Pr(U \leq u) = u,\text{ when } U \text{ is uniform on }[0,1]) \\

\end{align}

Inverse transform sampling can be simply extended to cases of truncated distributions on the interval $(a,b]$ without the cost of rejection sampling: the same algorithm can be followed, but instead of generating a random number $u$ uniformly distributed between 0 and 1, generate $u$ uniformly distributed between $F(a)$ and $F(b)$, and then again take $F^\{-1\}(u)$.

In order to obtain a large number of samples, one needs to perform the same number of inversions of the distribution.

One possible way to reduce the number of inversions while obtaining a large number of samples is the application of the so-called Stochastic Collocation Monte Carlo sampler (SCMC sampler) within a polynomial chaos expansion framework. This allows us to generate any number of Monte Carlo samples with only a few inversions of the original distribution with independent samples of a variable for which the inversions are analytically available, for example the standard normal variable.L.A. Grzelak, J.A.S. Witteveen, M. Suarez, and C.W. Oosterlee. The stochastic collocation Monte Carlo sampler: Highly efficient sampling from “expensive” distributions. https://ssrn.com/abstract=2529691

There are software implementations available for applying the inverse sampling method by using numerical approximations of the inverse in the case that it is not available in closed form. For example, an approximation of the inverse can be computed if the user provides some information about the distributions such as the PDF

{{cite journal | last1 = Derflinger | first1 = Gerhard | last2 = Hörmann | first2 = Wolfgang | last3 = Leydold | first3 = Josef | title = Random variate generation by numerical inversion when only the density is known | journal = ACM Transactions on Modeling and Computer Simulation | date = 2010 | volume = 20 | issue = 4 | doi = 10.1145/945511.945517| url = https://epub.wu.ac.at/664/1/document.pdf }} or the CDF.

• C library UNU.RAN {{cite web | url=https://statmath.wu.ac.at/unuran/index.html | title=UNU.RAN - Universal Non-Uniform RANdom number generators }}
• R library Runuran {{cite web | url=https://cran.r-project.org/package=Runuran | title=Runuran: R Interface to the 'UNU.RAN' Random Variate Generators | date=17 January 2023 }}
• Python subpackage sampling in scipy.stats{{cite web | url=https://docs.scipy.org/doc/scipy/reference/stats.sampling.html | title=Random Number Generators (Scipy.stats.sampling) — SciPy v1.12.0 Manual }}

{{cite book | last1 = Baumgarten | first1 = Christoph | last2 = Patel | first2 = Tirth | chapter = Automatic random variate generation in Python | title = Proceedings of the 21st Python in Science Conference | date = 2022 | pages = 46–51 | doi = 10.25080/majora-212e5952-007}}

• Probability integral transform
• Copula, defined by means of probability integral transform.
• Quantile function, for the explicit construction of inverse CDFs.
• Inverse distribution function for a precise mathematical definition for distributions with discrete components.
• Rejection sampling is another common technique to generate random variates that does not rely on inversion of the CDF.

{{DEFAULTSORT:Inverse Transform Sampling}}

Category:Monte Carlo methods

Category:Non-uniform random numbers