Statistics of a population

Definitions and estimators of the quantities that describe a population of drawn circuits. They are computed by spicefault.statistics, and by Dataset.statistics and Dataset.yield_report, from the samples of a dataset; nothing is simulated again. The quantities that involve faults are in reliability metrics.

1. Data and assumptions

A campaign without faults draws \(n\) circuits \(\theta_1, \dots, \theta_n\) independently from the variations and measures each one. The estimators below assume that the circuits are independent and drawn at random. They are refused for a dataset whose samples were designed, such as tolerance corners: a proportion over chosen circuits is not a probability.

A simulation that fails gives no measurement. It is counted and reported with every statistic, never dropped silently: a statistic over the successful simulations describes the circuits that could be simulated, and that is all it describes.

S1. Mean and spread

\(\bar m\) and \(s\) (with \(n - 1\)) of a measurement over the \(n\) successful samples. The interval of the mean is \(\bar m \pm z\, s / \sqrt{n}\), which holds when \(n\) is large enough for the mean to be normally distributed; it says nothing about where a single circuit falls.

S2. Quantiles

The estimate of the quantile \(\xi_q\) interpolates linearly between order statistics. Its interval is distribution-free. The number of samples below \(\xi_q\) is \(\text{Binomial}(n, q)\), so the ranks \(l < u\) with

\[P(X \le l - 1) \le \alpha/2, \qquad P(X \ge u) \le \alpha/2, \qquad X \sim \text{Binomial}(n, q)\]

give \(P(x_{(l)} \le \xi_q \le x_{(u)}) \ge 1 - \alpha\) for any continuous distribution. The interval is conservative, because the ranks are whole numbers.

When no such ranks exist the quantile is not estimable from the sample, and the estimate and its interval are reported as missing. This happens in the tails: the smallest of \(n\) values lies below \(\xi_q\) with probability \(1 - (1 - q)^n\), so 100 samples cannot bracket the 1 % quantile at 95 % confidence, and the smallest of them is not an estimate of it.

S3. Yield

With a set of specification limits, \(c_i = 1\) if circuit \(i\) meets all of them:

\[Y = P(c = 1 \mid \varnothing), \qquad \hat Y = \frac{k}{n}, \quad k = \sum_i c_i.\]

The interval is binomial: Wilson by default, Clopper-Pearson (exact, conservative) on request. The same holds for each specification alone. A value that could not be computed in a successful simulation is a violation.

Failed simulations are not known to pass or to fail. With \(n_f\) of them, the report gives \(\hat Y\) over the \(n\) successful ones and the two bounds over all \(n + n_f\): \(k / (n + n_f)\) if every failed one is counted as not compliant, and \((k + n_f) / (n + n_f)\) if every one is counted as compliant.

A finite sample bounds the yield and does not prove it. With no failure in \(n\) circuits the failure probability is below

\[1 - (1 - \gamma)^{1/n} \approx \frac{3}{n} \quad (\gamma = 0.95),\]

not zero. A half-width \(h\) for a yield near \(p\) needs about \(z^2 p (1 - p) / h^2\) circuits.

S4. Margin

For a specification with limits \([a, b]\) on a quantity of mean \(\bar m\) and spread \(s\):

\[\text{margin} = \min\!\left(\frac{\bar m - a}{s},\ \frac{b - \bar m}{s}\right),\]

with one term when a bound is open. It describes how many standard deviations separate the centre of the sample from the nearest limit. It is not a guarantee, and for a distribution far from normal it is a poor guide to the fraction beyond the limit, which is what the yield measures.

Sampling other than independent draws

The intervals of S1 to S3 assume independent samples. The circuits of a Latin hypercube or a Sobol design are not independent, and those intervals are not given for them. Their uncertainty comes from \(k\) independent replications of the whole design, with different seeds: each gives one estimate \(\hat e_i\), the estimates are independent, and

\[\bar e \pm t_{k-1}\, \frac{s_e}{\sqrt{k}}\]

is the interval (spicefault.statistics.replicated_interval), with the Student \(t\) of \(k - 1\) degrees of freedom. It assumes that the estimates are roughly normal.