Sequential embedding statistical analysis for multidimensional tolerance limits
The invention relates to statistical processing of multi-dimensional samples according to a step-wise sequence of iterative tolerance limit definitions using rank statistics. The processing is performed in the context of defining tolerance limits for a population that are compared to multiple process limits or acceptance criteria, with the requirement that a specified fraction of the population be confirmed to fall within the stated acceptance criteria. The symmetry (or asymmetry) may be allocated and controlled by selecting the frequency of occurrence of a specific figure of merit, and its order or position, in the sequential embedding processing sequence.
1. A computational method of establishing tolerance limits for a population of a controlled process with multiple figures of merit, comprising:
defining the figures of merit;
establishing acceptance criteria for the figures of merit, wherein the acceptance criteria is a safety limit or a performance requirement for the controlled process;
establishing a required fraction of the population, γ, to satisfy the acceptance criteria;
establishing a required confidence level, β, for the tolerance limits;
confirming that the required fraction of the population, γ, satisfies the acceptance criteria prior to operation of the controlled process, comprising:
defining a number of observations, N, to comprise a sample of the population;
calculating a number of steps, K, in a sequential embedding sequence;
processing the sample according to the sequential embedding sequence such that the required fraction of the population, γ, is bounded by the tolerance limits with the required confidence level, β:
β
=
1
-
I
(
γ
,
N
-
K
+
1
,
K
)
=
∑
j
=
0
N
-
K
(
N
j
)
γ
j
(
1
-
γ
)
N
-
j
where
K=Σ i=1 p k i
wherein, β is confidence level, γ is fraction of the population bounded by the tolerance limits, N is the number of observations in the sample, k i represents the total number of steps in the sequential embedding sequence allocated to each dimension, p represents the total number of dimensions, and K represents the total steps taken in the sequential embedding sequence;
selecting the figures of merit for which upper and/or lower tolerance limits are defined for each step in the sequential embedding sequence;
randomly drawing N observations from the population to comprise the sample;
establishing upper and/or lower tolerance limits for each of the figures of merit;
(a) for each step in the sequential embedding sequence, defining the upper and/or lower tolerance limits for the figure of merit as defined in a current step of the sequential embedding sequence, comprising:
ranking the observations in the sample according to their values for the figure of merit;
defining upper and/or lower tolerance limits for the figure of merit as the value of the figure of merit for a highest and/or lowest, respectively, ranked observation in the sample; and
reducing the sample for subsequent steps of the sequential embedding sequence by discarding the most extreme observation that is used to define the tolerance limits in the current step;
(b) repeating the actions of (a) in an iterative step-wise order for the K steps in the sequential embedding sequence to define a final set of upper and/or lower tolerance limits for the multiple figures of merit defined for the population, wherein asymmetry of the multiple figures of merit is controlled by the frequency and positioning of each figure of merit in the iterative sequence steps;
confirming if the final tolerance limits satisfy the acceptance criteria; and
setting the tolerance limits of the operation of the controlled process according to the final tolerance limits that met the acceptance criteria.
2. The computational method of claim 1 , wherein the figure of merits are peak cladding temperature (PCT), maximum local oxidation (MLO) and core-wide oxidation (CWO) for a loss of coolant accident (LOCA) analysis.
3. The computational method of claim 1 , wherein the tolerance limits for a specified figure of merit is optimized by having the specified figure of merit appear more often and later relative to other figures of merit in the sequential embedding sequence.
4. The computational method of claim 1 , wherein the asymmetry is minimized by assigning equal appearances to each figure of merit in the iterative sequence steps and rotating each figure of merit in the iterative sequence steps.