| Methods | |
| ======= | |
| CUSUM parameters | |
| ---------------- | |
| .. csv-table:: CUSUM parameters | |
| :file: ../../assets/params.csv | |
| :header-rows: 1 | |
| CUSUM chart | |
| ----------- | |
| A two-sided CUSUM control chart computes the cumulative differences or | |
| deviations of individual observations from the target mean (or | |
| in-control mean, :math:`\mu_{in}`). The positive and negative cumulative | |
| sums are calculated: | |
| .. math:: | |
| \\ S_{hi}(d) = max(0, S_{hi}(d-1) + x_d - \hat{\mu}_{in} - K) | |
| \\ S_{lo}(d) = max(0, S_{lo}(d-1) - x_d + \hat{\mu}_{in} - K) | |
| where *d* denotes a unit of time, :math:`x_d` is the value of quantity | |
| being monitored at time :math:`d`, :math:`\hat{\mu}_{in}` is the | |
| in-control mean of :math:`x_d`, and :math:`K` is a "reference value" | |
| related to the magnitude of change that one is interested in detecting. | |
| :math:`S_{hi}` and :math:`S_{lo}` are the cumulative sum of positive and | |
| negative changes. To detect a change in the observed values from the | |
| in-control mean, the CUSUM scheme accumulates deviations that are | |
| :math:`K` units away from the in-control mean. Let :math:`\sigma_{in}` | |
| denote the in-control standard deviation of :math:`x_d`. |