Q-function: Difference between revisions
cumulative, not above the mean |
Adding/improving reference(s) |
||
(47 intermediate revisions by 28 users not shown) | |||
Line 1:
{{Short description|Statistics function}}
{{For|the phase-space function representing a quantum state|Husimi Q representation}}
[[Image:Q-function.png|thumb|right|400px|A plot of the Q-function.]]
In [[statistics]], the '''Q-function''' is the [[Cumulative distribution function#Complementary cumulative distribution function (tail distribution)|tail distribution function]] of the [[Normal distribution#Standard normal distribution|standard normal distribution]].<ref>
If <math>Y</math> is a Gaussian random variable with mean <math>\mu</math> and variance <math>\sigma^2</math>, then <math>X = \frac{Y-\mu}{\sigma}</math> is [[Normal distribution#Standard normal distribution|standard normal]] and
Line 8 ⟶ 10:
where <math>x = \frac{y-\mu}{\sigma}</math>.
Other definitions of the ''Q''-function, all of which are simple transformations of the normal [[cumulative distribution function]], are also used occasionally.<ref>[http://mathworld.wolfram.com/NormalDistributionFunction.html Normal Distribution Function
Because of its relation to the [[cumulative distribution function]] of the normal distribution, the ''Q''-function can also be expressed in terms of the [[error function]], which is an important function in applied mathematics and physics.
Line 21 ⟶ 23:
:<math>Q(x) = 1 - Q(-x) = 1 - \Phi(x)\,\!,</math>
where <math>\Phi(x)</math> is the [[Standard normal distribution#Cumulative distribution function|cumulative distribution function of the standard normal Gaussian distribution]].
The ''Q''-function can be expressed in terms of the [[error function]], or the complementary error function, as<ref name="jo"/>
Line 33 ⟶ 35:
</math>
An alternative form of the ''Q''-function known as Craig's formula, after its discoverer, is expressed as:<ref>
:<math>Q(x) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} \right) d\theta.</math>
Line 39 ⟶ 41:
This expression is valid only for positive values of ''x'', but it can be used in conjunction with ''Q''(''x'') = 1 − ''Q''(−''x'') to obtain ''Q''(''x'') for negative values. This form is advantageous in that the range of integration is fixed and finite.
Craig's formula was later extended by Behnad (2020)<ref>{{cite journal |doi=10.1109/TCOMM.2020.2986209 |title=A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis|journal=IEEE Transactions on Communications |volume=68|issue=7|pages=4117–4125|year=2020|last1=Behnad|first1=Aydin|s2cid=216500014}}</ref> for the ''Q''-function of the sum of two non-negative variables, as follows:
:[[File:Q function complex plot plotted with Mathematica 13.1 ComplexPlot3D.svg|alt=the Q-function plotted in the complex plane|thumb|the Q-function plotted in the complex plane]]<math>Q(x+y) = \frac{1}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{2 \sin^2 \theta} - \frac{y^2}{2 \cos^2 \theta} \right) d\theta, \quad x,y \geqslant 0 .</math>
==Bounds and approximations==
*The ''Q''-function is not an [[elementary function]]. However, it can be upper and lower bounded as,<ref name = "Gordon">{{Cite journal |doi = |title = Values of Mills’ ratio of area to bounding ordinate and of the normal probability integral for large values of the argument| journal = Ann. Math. Stat.|volume = 12|issue = |pages = 364-366|year = 1941|last = Gordon|first = R.D.}}</ref><ref name = "Borjesson">{{Cite journal |doi = 10.1109/TCOM.1979.1094433|title = Simple Approximations of the Error Function Q(x) for Communications Applications|journal = IEEE Transactions on Communications|volume = 27|issue = 3|pages = 639–643|year = 1979|last1 = Borjesson|first1 = P.|last2 = Sundberg|first2 = C.-E.}}</ref>
::<math>\left (\frac{x}{1+x^2} \right ) \phi(x) < Q(x) < \frac{\phi(x)}{x}, \qquad x>0,</math>
:where <math>\phi(x)</math> is the density function of the standard normal distribution, and the bounds become increasingly tight for large ''x''
:Using the [[integration by substitution|substitution]] ''v'' =''u''<sup>2</sup>/2, the upper bound is derived as follows:
Line 55 ⟶ 62:
:Solving for ''Q''(''x'') provides the lower bound.
:The [[geometric mean]] of the upper and lower bound gives a suitable approximation for <math>Q(x)</math>:
:
* Tighter bounds and approximations of <math>Q(x)</math> can also be obtained by optimizing the following expression <ref name = "Borjesson"/>
:: <math> \tilde{Q}(x) = \frac{\phi(x)}{(1-a)x + a\sqrt{x^2 + b}}. </math>
:For <math>x \geq 0</math>, the best upper bound is given by <math>a = 0.344</math> and <math>b = 5.334</math> with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by <math>a = 0.339</math> and <math>b = 5.510</math> with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by <math>a = 1/\pi</math> and <math>b = 2 \pi</math> with maximum absolute relative error of 1.17%.
*The [[Chernoff bound]] of the ''Q''-function is
Line 60 ⟶ 77:
::<math>Q(x)\leq e^{-\frac{x^2}{2}}, \qquad x>0</math>
*Improved exponential bounds and a pure exponential approximation are <ref>
::<math>Q(x)\leq \tfrac{1}{4}e^{-x^2}+\tfrac{1}{4}e^{-\frac{x^2}{2}} \leq \tfrac{1}{2}e^{-\frac{x^2}{2}}, \qquad x>0</math>
Line 66 ⟶ 83:
:: <math>Q(x)\approx \frac{1}{12}e^{-\frac{x^2}{2}}+\frac{1}{4}e^{-\frac{2}{3} x^2}, \qquad x>0 </math>
*The above were generalized by Tanash & Riihonen (2020),<ref>{{cite journal |doi=10.1109/TCOMM.2020.3006902|title=Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials|journal=IEEE Transactions on Communications|year=2020|last1=Tanash|first1=I.M.|last2=Riihonen|first2=T.|volume=68|issue=10|pages=6514–6524|arxiv=2007.06939|s2cid=220514754}}</ref> who showed that <math>Q(x)</math> can be accurately approximated or bounded by
*A tight approximation of <math>Q(x)</math> for <math>x \in [0,\infty)</math> is given by Karagiannidis & Lioumpas (2007)<ref>Karagiannidis, G. K., & Lioumpas, A. S. [http://users.auth.gr/users/9/3/028239/public_html/pdf/Q_Approxim.pdf ''An improved approximation for the Gaussian Q-function'']. 2007. Communications Letters, IEEE, 11(8), pp. 644-646.</ref> who showed for the appropriate choice of parameters <math>\{A, B\}</math> that▼
::<math>\tilde{Q}(x) = \sum_{n=1}^N a_n e^{-b_n x^2}.</math>
:In particular, they presented a systematic methodology to solve the numerical coefficients <math>\{(a_n,b_n)\}_{n=1}^N</math> that yield a [[minimax approximation algorithm|minimax]] approximation or bound: <math>Q(x) \approx \tilde{Q}(x)</math>, <math>Q(x) \leq \tilde{Q}(x)</math>, or <math>Q(x) \geq \tilde{Q}(x)</math> for <math>x\geq0</math>. With the example coefficients tabulated in the paper for <math>N = 20</math>, the relative and absolute approximation errors are less than <math>2.831 \cdot 10^{-6}</math> and <math>1.416 \cdot 10^{-6}</math>, respectively. The coefficients <math>\{(a_n,b_n)\}_{n=1}^N</math> for many variations of the exponential approximations and bounds up to <math>N = 25</math> have been released to open access as a comprehensive dataset.<ref>{{cite journal |doi=10.5281/zenodo.4112978|title=Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]|url=https://zenodo.org/record/4112978|website=Zenodo|year=2020|last1=Tanash|first1=I.M.|last2=Riihonen|first2=T.}}</ref>
▲*
:: <math>f(x; A, B) = \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x} \approx \operatorname{erfc} \left(x\right).</math>▼
▲: <math>f(x; A, B) = \frac{\left(1 - e^{-Ax}\right)e^{-x^2}}{B\sqrt{\pi} x} \approx \operatorname{erfc} \left(x\right).</math>
: The absolute error between <math>f(x; A, B)</math> and <math>\operatorname{erfc}(x)</math> over the range <math>[0, R]</math> is minimized by evaluating
:: <math>\{A, B\} = \underset{\{A,B\}}{
: Using <math>R = 20</math> and numerically integrating, they found the minimum error occurred when <math>\{A, B\} = \{1.98, 1.135\},</math> which gave a good approximation for <math>\forall x \ge 0.</math>
: Substituting these values and using the relationship between <math>Q(x)</math> and <math>\operatorname{erfc}(x)</math> from above gives
▲: <math> Q(x)\approx\frac{\left( 1-e^{-1.4x}\right) e^{-\frac{x^{2}}{2}}}{1.135\sqrt{2\pi}x}, x \ge 0. </math>
:: <math> Q(x)\approx\frac{\left( 1-e^{\frac{-1.98x} {\sqrt{2}}}\right) e^{-\frac{x^{2}}{2}}}{1.135\sqrt{2\pi}x}, x \ge 0. </math>
'''Inverse ''Q'''''▼
: Alternative coefficients are also available for the above 'Karagiannidis–Lioumpas approximation' for tailoring accuracy for a specific application or transforming it into a tight bound.<ref>{{cite journal |doi=10.1109/LCOMM.2021.3052257|title=Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function|journal=IEEE Communications Letters|year=2021|last1=Tanash|first1=I.M.|last2=Riihonen|first2=T.|volume=25|issue=5|pages=1468–1471|arxiv=2101.07631|s2cid=231639206}}</ref>
*A tighter and more tractable approximation of <math>Q(x)</math> for positive arguments <math>x \in [0,\infty)</math> is given by López-Benítez & Casadevall (2011)<ref>{{cite journal |doi=10.1109/TCOMM.2011.012711.100105 |url=http://www.lopezbenitez.es/journals/IEEE_TCOM_2011.pdf|title=Versatile, Accurate, and Analytically Tractable Approximation for the Gaussian Q-Function|journal=IEEE Transactions on Communications|volume=59|issue=4|pages=917–922|year=2011|last1=Lopez-Benitez|first1=Miguel|last2=Casadevall|first2=Fernando|s2cid=1145101}}</ref> based on a second-order exponential function:
:: <math> Q(x) \approx e^{-ax^2-bx-c}, \qquad x \ge 0. </math>
: The fitting coefficients <math> (a,b,c) </math> can be optimized over any desired range of arguments in order to minimize the sum of square errors (<math>a = 0.3842</math>, <math>b = 0.7640</math>, <math>c = 0.6964</math> for <math>x \in [0,20]</math>) or minimize the maximum absolute error (<math>a = 0.4920</math>, <math>b = 0.2887</math>, <math>c = 1.1893</math> for <math>x \in [0,20]</math>). This approximation offers some benefits such as a good trade-off between accuracy and analytical tractability (for example, the extension to any arbitrary power of <math>Q(x)</math> is trivial and does not alter the algebraic form of the approximation).
The inverse ''Q''-function can be related to the [[error function#Inverse functions|inverse error functions]]:
Line 85 ⟶ 121:
:<math>\mathrm{Q\text{-}factor} = 20 \log_{10}\!\left(Q^{-1}(y)\right)\!~\mathrm{dB}</math>
where ''y'' is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for [[
[[File:Q-factor vs BER.png|thumb|none|400px|Q-factor vs. bit error rate (BER).]]
Line 92 ⟶ 128:
<!-- This table was calculated in Matlab as follows:
x=0:0.1:
y = qfunc(x);
for i=1:length(x),
Line 233 ⟶ 269:
== Generalization to high dimensions ==
The ''Q''-function can be generalized to higher dimensions:<ref>{{cite journal|last1=Savage|first1=I. R.|title=Mills ratio for multivariate normal distributions|journal=Journal
:<math>Q(\mathbf{x})= \mathbb{P}(\mathbf{X}\geq \mathbf{x}),</math>
where <math>\mathbf{X}\sim \mathcal{N}(\mathbf{0},\, \Sigma) </math> follows the multivariate normal distribution with covariance <math>\Sigma </math> and the threshold is of the form
<math>\mathbf{x}=\gamma\Sigma\mathbf{l}^*</math> for some positive vector <math> \mathbf{l}^*>\mathbf{0}</math> and positive constant <math>\gamma>0</math>. As in the one dimensional case, there is no simple analytical formula for the ''Q''-function. Nevertheless, the ''Q''-function can be [http://www.mathworks.com/matlabcentral/fileexchange/53796 approximated arbitrarily well] as <math>\gamma</math> becomes larger and larger.<ref>{{cite journal|last1=Botev|first1=Z. I.|title=The normal law under linear restrictions: simulation and estimation via minimax tilting|journal=Journal of the Royal Statistical Society, Series B|volume=79|pages=125–148|date=2016|doi=10.1111/rssb.12162|arxiv=1603.04166|bibcode=2016arXiv160304166B|s2cid=88515228}}</ref><ref name="bmc17">{{cite book |chapter=Logarithmically efficient estimation of the tail of the multivariate normal distribution |last1=Botev |first1=Z. I. |last2=Mackinlay |first2=D. |last3=Chen |first3=Y.-L. |date=2017 |publisher=IEEE |isbn=978-1-5386-3428-8 |title= 2017 Winter Simulation Conference (WSC)|pages=1903–191 |doi= 10.1109/WSC.2017.8247926 |s2cid=4626481 }}
</ref>
== References ==
|
Latest revision as of 04:08, 18 June 2024
In statistics, the Q-function is the tail distribution function of the standard normal distribution.[1][2] In other words, is the probability that a normal (Gaussian) random variable will obtain a value larger than standard deviations. Equivalently, is the probability that a standard normal random variable takes a value larger than .
If is a Gaussian random variable with mean and variance , then is standard normal and
where .
Other definitions of the Q-function, all of which are simple transformations of the normal cumulative distribution function, are also used occasionally.[3]
Because of its relation to the cumulative distribution function of the normal distribution, the Q-function can also be expressed in terms of the error function, which is an important function in applied mathematics and physics.
Definition and basic properties
[edit]Formally, the Q-function is defined as
Thus,
where is the cumulative distribution function of the standard normal Gaussian distribution.
The Q-function can be expressed in terms of the error function, or the complementary error function, as[2]
An alternative form of the Q-function known as Craig's formula, after its discoverer, is expressed as:[4]
This expression is valid only for positive values of x, but it can be used in conjunction with Q(x) = 1 − Q(−x) to obtain Q(x) for negative values. This form is advantageous in that the range of integration is fixed and finite.
Craig's formula was later extended by Behnad (2020)[5] for the Q-function of the sum of two non-negative variables, as follows:
Bounds and approximations
[edit]- The Q-function is not an elementary function. However, it can be upper and lower bounded as,[6][7]
- where is the density function of the standard normal distribution, and the bounds become increasingly tight for large x.
- Using the substitution v =u2/2, the upper bound is derived as follows:
- Similarly, using and the quotient rule,
- Solving for Q(x) provides the lower bound.
- The geometric mean of the upper and lower bound gives a suitable approximation for :
- Tighter bounds and approximations of can also be obtained by optimizing the following expression [7]
- For , the best upper bound is given by and with maximum absolute relative error of 0.44%. Likewise, the best approximation is given by and with maximum absolute relative error of 0.27%. Finally, the best lower bound is given by and with maximum absolute relative error of 1.17%.
- The Chernoff bound of the Q-function is
- Improved exponential bounds and a pure exponential approximation are [8]
- The above were generalized by Tanash & Riihonen (2020),[9] who showed that can be accurately approximated or bounded by
- In particular, they presented a systematic methodology to solve the numerical coefficients that yield a minimax approximation or bound: , , or for . With the example coefficients tabulated in the paper for , the relative and absolute approximation errors are less than and , respectively. The coefficients for many variations of the exponential approximations and bounds up to have been released to open access as a comprehensive dataset.[10]
- Another approximation of for is given by Karagiannidis & Lioumpas (2007)[11] who showed for the appropriate choice of parameters that
- The absolute error between and over the range is minimized by evaluating
- Using and numerically integrating, they found the minimum error occurred when which gave a good approximation for
- Substituting these values and using the relationship between and from above gives
- Alternative coefficients are also available for the above 'Karagiannidis–Lioumpas approximation' for tailoring accuracy for a specific application or transforming it into a tight bound.[12]
- A tighter and more tractable approximation of for positive arguments is given by López-Benítez & Casadevall (2011)[13] based on a second-order exponential function:
- The fitting coefficients can be optimized over any desired range of arguments in order to minimize the sum of square errors (, , for ) or minimize the maximum absolute error (, , for ). This approximation offers some benefits such as a good trade-off between accuracy and analytical tractability (for example, the extension to any arbitrary power of is trivial and does not alter the algebraic form of the approximation).
Inverse Q
[edit]The inverse Q-function can be related to the inverse error functions:
The function finds application in digital communications. It is usually expressed in dB and generally called Q-factor:
where y is the bit-error rate (BER) of the digitally modulated signal under analysis. For instance, for quadrature phase-shift keying (QPSK) in additive white Gaussian noise, the Q-factor defined above coincides with the value in dB of the signal to noise ratio that yields a bit error rate equal to y.
Values
[edit]The Q-function is well tabulated and can be computed directly in most of the mathematical software packages such as R and those available in Python, MATLAB and Mathematica. Some values of the Q-function are given below for reference.
|
|
|
|
Generalization to high dimensions
[edit]The Q-function can be generalized to higher dimensions:[14]
where follows the multivariate normal distribution with covariance and the threshold is of the form for some positive vector and positive constant . As in the one dimensional case, there is no simple analytical formula for the Q-function. Nevertheless, the Q-function can be approximated arbitrarily well as becomes larger and larger.[15][16]
References
[edit]- ^ "The Q-function". cnx.org. Archived from the original on 2012-02-29.
- ^ a b "Basic properties of the Q-function" (PDF). 2009-03-05. Archived from the original (PDF) on 2009-03-25.
- ^ Normal Distribution Function – from Wolfram MathWorld
- ^ Craig, J.W. (1991). "A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations" (PDF). MILCOM 91 - Conference record. pp. 571–575. doi:10.1109/MILCOM.1991.258319. ISBN 0-87942-691-8. S2CID 16034807.
- ^ Behnad, Aydin (2020). "A Novel Extension to Craig's Q-Function Formula and Its Application in Dual-Branch EGC Performance Analysis". IEEE Transactions on Communications. 68 (7): 4117–4125. doi:10.1109/TCOMM.2020.2986209. S2CID 216500014.
- ^ Gordon, R.D. (1941). "Values of Mills' ratio of area to bounding ordinate and of the normal probability integral for large values of the argument". Ann. Math. Stat. 12: 364–366.
- ^ a b Borjesson, P.; Sundberg, C.-E. (1979). "Simple Approximations of the Error Function Q(x) for Communications Applications". IEEE Transactions on Communications. 27 (3): 639–643. doi:10.1109/TCOM.1979.1094433.
- ^ Chiani, M.; Dardari, D.; Simon, M.K. (2003). "New exponential bounds and approximations for the computation of error probability in fading channels" (PDF). IEEE Transactions on Wireless Communications. 24 (5): 840–845. doi:10.1109/TWC.2003.814350.
- ^ Tanash, I.M.; Riihonen, T. (2020). "Global minimax approximations and bounds for the Gaussian Q-function by sums of exponentials". IEEE Transactions on Communications. 68 (10): 6514–6524. arXiv:2007.06939. doi:10.1109/TCOMM.2020.3006902. S2CID 220514754.
- ^ Tanash, I.M.; Riihonen, T. (2020). "Coefficients for Global Minimax Approximations and Bounds for the Gaussian Q-Function by Sums of Exponentials [Data set]". Zenodo. doi:10.5281/zenodo.4112978.
- ^ Karagiannidis, George; Lioumpas, Athanasios (2007). "An Improved Approximation for the Gaussian Q-Function" (PDF). IEEE Communications Letters. 11 (8): 644–646. doi:10.1109/LCOMM.2007.070470. S2CID 4043576.
- ^ Tanash, I.M.; Riihonen, T. (2021). "Improved coefficients for the Karagiannidis–Lioumpas approximations and bounds to the Gaussian Q-function". IEEE Communications Letters. 25 (5): 1468–1471. arXiv:2101.07631. doi:10.1109/LCOMM.2021.3052257. S2CID 231639206.
- ^ Lopez-Benitez, Miguel; Casadevall, Fernando (2011). "Versatile, Accurate, and Analytically Tractable Approximation for the Gaussian Q-Function" (PDF). IEEE Transactions on Communications. 59 (4): 917–922. doi:10.1109/TCOMM.2011.012711.100105. S2CID 1145101.
- ^ Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal of Research of the National Bureau of Standards Section B. 66 (3): 93–96. doi:10.6028/jres.066B.011. Zbl 0105.12601.
- ^ Botev, Z. I. (2016). "The normal law under linear restrictions: simulation and estimation via minimax tilting". Journal of the Royal Statistical Society, Series B. 79: 125–148. arXiv:1603.04166. Bibcode:2016arXiv160304166B. doi:10.1111/rssb.12162. S2CID 88515228.
- ^ Botev, Z. I.; Mackinlay, D.; Chen, Y.-L. (2017). "Logarithmically efficient estimation of the tail of the multivariate normal distribution". 2017 Winter Simulation Conference (WSC). IEEE. pp. 1903–191. doi:10.1109/WSC.2017.8247926. ISBN 978-1-5386-3428-8. S2CID 4626481.