Appendix: normal quantile (NQ) plots
Appendix: normal quantile (NQ) plots
NQ plot is an alternative view of a distribution that facilitates easy
comparison to Gaussian distributions, and enables detailed analysis of
the distribution’s tails (very useful in high-sigma analysis).
figure below shows three views of the same distribution. The only
difference between each view is the y-axis. Each view has different
advantages and disadvantages. Crucially, we can transform among the
views in a fairly straightforward fashion.
The top plot in the
figure has the PDF (probability density function) on the y-axis. This is
the plot that we’ve used in the examples. The area under the PDF must
integrate to 1.0. The area under the curve at a given range of x-values
is the probability of that range of x-values. The PDF curve is intuitive
because its y-value for a range of x values is proportional to the
number of samples seen in that range of x-values. “Discretized” PDFs –
histograms – are intuitive and easy to compute.
the PDF view has disadvantages. First, lower-probability regions run
extremely close to the x-axis, and have such small values that it is
difficult to distinguish one low-probability value from another (e.g.
1e-4 from 1e-5), because they are both hugging the x-axis. This is
troublesome for high-sigma analysis. Second, the “best behaved” or
“typical” distributions, namely Gaussian distributions, have a highly
nonlinear bell-shaped curve. This means that small nonlinear distortions
of that curve are difficult to identify, and in general the nonlinear
curve is hard to work with. Compare this to many subdomains of
electrical engineering, from circuit analysis to control theory, where
the “best behaved” models or systems are linear or linearized models;
and linear techniques are key analysis and design tools. Nonlinear
bell-shaped curves are not amenable to such analysis.
plot in the figure has the CDF (cumulative distribution function) on the
y-axis. A CDF value at x is the area under the PDF from ∞ to x. That
the y-axis can be viewed as the probability p that up to a given
x-value can occur. CDFs are highly useful for computing yield: CDF
values are equal to yield values for “≤” specs; and equal to 1-yield
when for “≥” specs. Accordingly, one may inspect CDF plots to see the
tradeoff between yield values and spec values. To go backwards from CDF
to PDF, one takes the derivative.
However, CDFs have the same
disadvantages of PDFs: hard to distinguish among low-probability
regions, and Gaussian distributions have highly nonlinear curves which
Consider a view of a distribution that
overcomes some of the disadvantages described, and keeps key benefits.
Specifically, consider a view where:
- The “best behaved”
distributions (Gaussian) are linear curves, and the larger the deviation
from Gaussian the more nonlinear the curve. Different types of
deviations indicate different nonlinearities.
- One can directly see the tradeoff between yield and performance.
- One can easily distinguish among different low-probability values.
such a view of a distribution exists: these are NQ plots. An example NQ
plot is shown in the bottom of Figure 4.18. An NQ plot is like a CDF,
but the y-axis is warped “just right” such that if the underlying
distribution is Gaussian, then the NQ curve would be linear. The
specific function to warp with is the “normal quantile” function, also
known as the “inverse CDF of the Gaussian”, or “probit” function. It
takes in CDF value (a probability p), and outputs a normal quantile (NQ)
value. The NQ function is not available in closed form, so must be
numerically computed as:
where is the “inverse error function” and is typically provided in software packages and libraries.
“normal quantile” value has a specific interpretation that aids
intuition: the number of standard deviations away from the mean, if the
distribution was Gaussian. Sigma is a unit for yield that is often
simpler to talk about than percent yield or probability of failure; and
one can build intuition about the units. Specifically for intuition: 3
sigma is about 1 failure in 1000, 4 sigma is about 1 in 50 thousand, 5
sigma is about 1 in a million, and 6 sigma is about 1 in a billion.
Certain circuit types of circuits typically have particular target
ranges of sigma values. For example, non-replicated circuits like analog
circuits are typically 3-4 sigma; and bitcells are typically around 6
About the author
McConaghy is co-founder and Chief Technology Officer of Solido Design
Automation Inc. He was a co-founder and Chief Scientist of Analog Design
Automation Inc., which was acquired by Synopsys Inc. in 2004. Prior to
that, he did research for the Canadian Department of National Defense.
Trent has about 40 peer-reviewed technical papers and patents granted /
pending. In 2001, Trent was awarded the 2001 Outstanding Young Alumni
Award by the University of Saskatchewan. He received his PhD degree in
Electrical Engineering from the Katholieke Universiteit Leuven, Belgium,
in 2008. He received a Bachelor’s in Engineering (with great
distinction), and a Bachelor’s in Computer Science (with great
distinction), both from the University of Saskatchewan, Canada, in 1999.
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