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Register a new userThe Case against Generally Weighted Moving Average (GWMA) Control Charts
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Added by
Sven Knoth
Publication date
2021
fields
Mathematical Statistics
and research's language is
English
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We argue against the use of generally weighted moving average (GWMA) control charts. Our primary reasons are the following: 1) There is no recursive formula for the GWMA control chart statistic, so all previous data must be stored and used in the calculation of each chart statistic. 2) The Markovian property does not apply to the GWMA statistics, so computer simulation must be used to determine control limits and the statistical performance. 3) An appropriately designed, and much simpler, exponentially weighted moving average (EWMA) chart provides as good or better statistical performance. 4) In some cases the GWMA chart gives more weight to past data values than to current values.
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We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive, $$ P(M_n %max^n_{i=1} X_i leq x) = sum_{j=1}^infty beta_{jx} u_{jx}^{n} approx B_{x} u_{1x}^{n} $$ where %${X_i}$ is a moving average of order 1 with positive correlation, and ${ u_{jx}}$ are the eigenvalues (singular values) of a Fredholm kernel and $ u_{1x}$ is the eigenvalue of maximum magnitude. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. % there are more terms, and $$P(M_n <x) approx B_{x} (1+ u_{1x})^n.$$ For the continuous case the integral equations for the left and right eigenfunctions are converted to first order linear differential equations. The eigenvalues satisfy an equation of the form $$sum_{i=1}^infty w_i(lambda-theta_i)^{-1}=lambda-theta_0$$ for certain known weights ${w_i}$ and eigenvalues ${theta_i}$ of a given matrix. This can be solved by truncating the sum to an increasing number of terms.
We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables are discrete. When the correlation is positive, $$ P(M_n max^n_{i=1} X_i leq x) = sum_{j=1}^infty beta_{jx} u_{jx}^{n} approx B_{x} r{1x}^{n} $$ where ${ u_{jx}}$ are the eigenvalues of a certain matrix, $r_{1x}$ is the maximum magnitude of the eigenvalues, and $I$ depends on the number of possible values of the underlying random variables. The eigenvalues do not depend on $x$ only on its range.
A multivariate control chart is designed to monitor process parameters of multiple correlated quality characteristics. Often data on multivariate processes are collected as individual observations, i.e. as vectors one at the time. Various control charts have been proposed in the literature to monitor the covariance matrix of a process when individual observations are collected. In this study, we review this literature; we find 30 relevant articles from the period 1987-2019. We group the articles into five categories. We observe that less research has been done on CUSUM, high-dimensional and non-parametric type control charts for monitoring the process covariance matrix. We describe each proposed method, state their advantages, and limitations. Finally, we give suggestions for future research.
We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension $d$. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for $d=2$. The classical result of Sedgewick and Vitter [Algorithmica 86] shows that for weighted unit-disk graphs in the plane the $A^*$ search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstras algorithm. Specifically, if the $n$ corresponding points of a weighted unit-disk graph $G$ are picked from a unit square uniformly at random, and the connectivity radius is $rin (0,1)$, $A^*$ finds a shortest path in $G$ in $O(n)$ expected time when $r=Omega(sqrt{log n/n})$, even though $G$ has $Theta((nr)^2)$ edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to $A^*$. Specifically, we show that, if we are able to report the set of all $k$ points of $G$ from an arbitrary rectangular region of the plane in $O(k + t(n))$ time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in $O(1/r^2 + t(n))$ expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all $r=Omega(sqrt{log n/n})$ and $O(sqrt{n})$ expected bound for $r=Omega(n^{-1/4})$ after only near-linear preprocessing of the point set.
Let ${s_n}_{ninmathbb{N}}$ be a decreasing nonsummable sequence of positive reals. In this paper, we investigate the weighted Birkhoff average $frac{1}{S_n}sum_{k=0}^{n-1}s_kphi(T^kx)$ on aperiodic irreducible subshift of finite type $Sigma_{bf A}$ where $phi: Sigma_{bf A}mapsto mathbb{R}$ is a continuous potential. Firstly, we show the entropy spectrum of the weighed Birkhoff averages remains the same as that of the Birkhoff averages. Then we prove that the packing spectrum of the weighed Birkhoff averages equals to either that of the Birkhoff averages or the whole space.
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