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Regret Indices and Capability Quantification

Cumulative Capability Curves

A graphical Cumulative Capability (CC) curve can be drawn for a set of observed X data once we have specified the target value(s), T, the regret functional form, R(X), and a ąuality standard for expected regret, ER. This plot depicts the Cumulative Probability Distribution Function of the performance index:

CC( I) = Prob[ random index < fixed I value 1    [13]

With I₀ denoting a fixed numerical value for an index, we can rewrite [13] in several ways:

CC( I₀ ) = Prob[ X | I(X) < I₀ ]

= Prob[ X | R(X) < ER • I₀ ]

When the regret function is 2-to-l, we also have

CC( I₀ ) = Prob[ L₀ < X < U₀ ]    [14]

where R( L₀) = R( U₀ ) = ER * I₀-

In other words, when regret is 2-to-l, the CC curve ąuantifies process "yields" (conformance fractions) corresponding to all X-characteristic intervals with eąual, maximum regret at their two end-points. Furthermore, when regret is symmetric-about-the-target as well as 2-to-l, each of the above end-point pairings is of the special form L₀ = T - Hq and U₀ = T + Hq for some non-negative half-width, Hq.

Cumulative capability always assumes its minimum numerical value at I = R = 0:

CC(0)= Prob[I = 0]    [15]

This minimum CC may be zero; CC( 0 ) will be strictly positive only when the process distribution (empirical or theoretical) places an "atom" of probability exactly on the target value, X = T.

The proposed plotting convention for a CC curve is that CC(I) values be displayed on the vertical axis (with 0 at the bottom and 1 at the top) and that the corresponding I values be displayed along the horizontal axis (with I = 0 at