mqr.spc.util.

solve_arl

mqr.spc.util.solve_arl(h4, p, lmda, N=20)

Find the in-control ARL of an MEWMA chart.

Follows the modified Gauss-Legendre method in [1] to evaluate:

\[L(\alpha) = 1 + \int_0^h L(u)/\lambda^2 f(u/\lambda^2 | \eta(\alpha)) du\]

where

\[\begin{split}\begin{gather} h = h_4 \lambda / (2 - \lambda),\text{ and} \\ \eta(\alpha) = \alpha ((1 - \lambda) / \lambda)^2. \end{gather}\end{split}\]

The algorithm makes a change of variables (\(\alpha \rightarrow \alpha^2\)), which improves its performance.

Parameters:

h4float

Upper control limit on MEWMA chart.

pint

Dimension of monitored vector.

lmdafloat

Decay factor for EWMA statistic.

Nint, optional

Number of quadrature sample points (Gauss-Legendre).

Returns:

float

References

[1]

Knoth, S. (2017). ARL numerics for MEWMA charts. Journal of Quality Technology, 49(1), 78-89.