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`.