  The moving average (MA)-type scheme, also known as the smoothing method, has
been well established within the multivariate statistical process monitoring
(MSPM) framework since the 1990s. However, its theoretical basis is still
limited to smoothing independent data, and the optimality of its equally or
exponentially weighted scheme remains unproven. This paper aims to weaken the
independence assumption in the existing MA method, and then extend it to a
broader area of dealing with autocorrelated weakly stationary processes. With
the discovery of the non-optimality of the equally and exponentially weighted
schemes used for fault detection when data have autocorrelation, the essence
that they do not effectively utilize the correlation information of samples is
revealed, giving birth to an optimally weighted moving average (OWMA) theory.
The OWMA method is combined with the Hotelling's $T^2$ statistic to form an
OWMA $T^2$ control chart (OWMA-TCC), in order to detect a more challenging type
of fault, i.e., intermittent fault (IF). Different from the MA scheme that puts
an equal weight on samples within a time window, OWMA-TCC uses correlation
(autocorrelation and cross-correlation) information to find an optimal weight
vector (OWV) for the purpose of IF detection (IFD). In order to achieve a best
IFD performance, the concept of IF detectability is defined and corresponding
detectability conditions are provided, which further serve as selection
criteria of the OWV. Then, the OWV is given in the form of a solution to
nonlinear equations, whose existence is proven with the aid of the Brouwer
fixed-point theory. Moreover, symmetrical structure of the OWV is revealed, and
the optimality of the MA scheme for any IF directions when data exhibit no
autocorrelation is proven.
