Q-function: Difference between revisions
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== Generalization to high dimensions == |
== Generalization to high dimensions == |
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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= |
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 of Research of the National Bureau of Standards B|date=1962|volume=66|pages=93–96|zbl=0105.12601}}</ref> |
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:<math>Q(\mathbf{x})= \mathbb{P}(\mathbf{X}\geq \mathbf{x}),</math> |
:<math>Q(\mathbf{x})= \mathbb{P}(\mathbf{X}\geq \mathbf{x}),</math> |
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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 |
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 |
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<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}}</ref><ref name="bmc17">{{cite |
<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}}</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 }} |
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|title=Logarithmically efficient estimation of the tail of the multivariate normal distribution |last1=Botev |first1=Z. I. |
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|last2=Mackinlay |first2=D. |last3=Chen |first3=Y.-L. |date=2017 |publisher=IEEE|ISBN=978-1-5386-3428-8 |
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|book-title= 2017 Winter Simulation Conference (WSC) |pages=1903–1913 |location= 3th–6th Dec 2017 Las Vegas, NV, USA |
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|doi= 10.1109/WSC.2017.8247926 }} |
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</ref> |
</ref> |
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Revision as of 04:45, 14 December 2019
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
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.
Bounds and approximations
- The Q-function is not an elementary function. However, the bounds, where is the density function of the standard normal distribution,[5]
- become increasingly tight for large x, and are often useful.
- 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 Q(x):
- Tighter bounds and approximations of the Q(x) can also be obtained by optimizing the following expression [5]
- 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 [6]
- Another approximation of for is given by Karagiannidis & Lioumpas (2007)[7] 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
- A tighter and more tractable approximation of for positive arguments is given by López-Benítez & Casadevall (2011)[8] 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
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 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
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.
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Generalization to high dimensions
The Q-function can be generalized to higher dimensions:[9]
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.[10][11]
References
- ^ The Q-function, from cnx.org
- ^ a b Basic properties of the Q-function Archived March 25, 2009, at the Wayback Machine
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ Savage, I. R. (1962). "Mills ratio for multivariate normal distributions". Journal of Research of the National Bureau of Standards B. 66: 93–96. 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.
- ^ 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.