\\(\\{(0, 0), (100, 0.5), (250, 0.75), (500, 0.9),
+The article–delta confidence function, \\(C_{A\Delta}\\), is
+piecewise-defined such that confidence increases at an exponential rate as
+\\(\frac{\Delta}{A}\\) increases, until the value of
+\\(C_{A\Delta}\\) reaches the "suspected" violation threshold, at
+which point confidence increases at a decreasing rate, with
+\\(\lim_{\frac{\Delta}{A} \to 1}C\_{A\Delta}(A, \Delta)=1\\)
+holding true. The exact coefficients used are shown below:
+
+$$C_{A\Delta}(A, \Delta)=\begin{cases} \ln\frac{1}{1-\frac{\Delta}{A}} &
+\frac{\Delta}{A} \le 0.52763 \\[0.5em]
+-0.8939(\frac{\Delta}{A})^2+1.8948\frac{\Delta}{A}-0.0009 &
+\frac{\Delta}{A} \gt 0.52763 \end{cases}$$
+
+A graph can be viewed [here](/static/article-delta_confidence_function.pdf),
+with the x-axis indicating \\(\frac{\Delta}{A}\\) and the y-axis
+indicating confidence. The background is colored red, yellow, and green when a
+violation is considered suspected, possible, or not present, respectively.
+
+The delta confidence function, \\(C_{\Delta}\\), is also
+piecewise-defined. A number of confidence values were derived experimentally,
+and the function was extrapolated from there such that
+\\(\lim_{Δ \to +\infty}C\_{\Delta}(\Delta)=1\\). The reference
+points were \\(\\{(0, 0), (100, 0.5), (250, 0.75), (500, 0.9),
(1000, 0.95)\\}\\). The function is defined as follows:
$$C_{\Delta}(\Delta)=\begin{cases} \frac{\Delta}{\Delta+100} & \Delta\leq
@@ -49,13 +59,13 @@ reference points were \\(\\{(0, 0), (100, 0.5), (250, 0.75), (500, 0.9),
\frac{\Delta-50}{\Delta} & \Delta\gt500 \end{cases}$$
A graph can be viewed [here](/static/delta_confidence_function.pdf), with the
-background colored red, yellow, and green when a violation is considered
-suspected, possible, or not present, respectively.
+x-axis indicating \\(\Delta\\). The background coloring is the
+same as before.
Now that we have these two definitions, we can define the primary confidence
function, \\(C\\), as follows:
-$$C(A, \Delta) = \max(\tfrac{\Delta}{A}, C_{\Delta}(\Delta))$$
+$$C(A, \Delta) = \max(C_{A\Delta}(A, \Delta), C_{\Delta}(\Delta))$$
By feeding \\(A\\) and \\(\Delta\\) into
\\(C\\), we get our final confidence value.
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