6.
Process or Product Monitoring and Control
6.3. Univariate and Multivariate Control Charts 6.3.4. What are Multivariate Control Charts?


Multivariate EWMA Control Chart  
Univariate EWMA model  The model for a univariate EWMA chart is given by: $$ Z_i = \lambda X_i + (1\lambda)Z_{i1}, \,\,\,\,\, i = 1, \, 2, \, \ldots, \, n \, , $$ where \(Z_i\) is the \(i\)th EWMA, \(X_i\) is the the \(i\)th observation, \(Z_0\) is the average from the historical data, and \(0 < \lambda \le 1\).  
Multivariate EWMA model  In the multivariate case, one can extend this formula to $$ Z_i = \Lambda X_i + (1\Lambda)Z_{i1} \, , $$ where \(Z_i\) is the \(i\)th EWMA vector, \(X_i\) is the the \(i\)th observation vector \(i = 1, \, 2, \, \ldots, \, n\), \(Z_0\) is the vector of variable values from the historical data, \(\Lambda\) is the \(\mbox{diag}(\lambda_1, \, \lambda_2, \, \ldots, \, \lambda_p)\) which is a diagonal matrix with \(\lambda_1, \, \lambda_2, \, \ldots, \, \lambda_p\) on the main diagonal, and \(p\) is the number of variables; that is the number of elements in each vector.  
Illustration of multivariate EWMA  The following illustration may clarify this. There are \(p\) variables and each variable contains \(n\) observations. The input data matrix looks like the following. $$ \begin{array}{cccc} X_{11} & X_{12} & \cdots & X_{1p} \\ X_{21} & X_{22} & \cdots & X_{2p} \\ \vdots & \vdots & \ddots & \vdots \\ & & & \\ X_{n1} & X_{n2} & \cdots & X_{np} \end{array} $$ The quantity to be plotted on the control chart is $$ T_i^2 = Z_i' \, \Sigma_{Z_i}^{1} \, Z_i $$  
Simplification 
It has been shown (Lowry et al.,
1992) that the \( (k,l) \)th
element of the covariance matrix of the \(i\)th
EWMA, \(\Sigma_{Z_i}\),
is
$$ \Sigma_{Z_i}(k,l) = \lambda_k \lambda_l \,
\frac{\left[ 1(1\lambda_k)^i (1\lambda_l)^i \right]}{(\lambda_k + \lambda_l  \lambda_k \lambda_l )} \,
\sigma_{k,l} \, , $$
where \(\sigma_{k,l}\)
is the \((k,l)\)th
element of \(\Sigma\),
the covariance matrix of the \(X\)'s.
If \(\lambda_1 = \lambda_2 = \cdots = \lambda_p = \lambda\), then the above expression simplifies to $$ \Sigma_{Z_i}(k,l) = \frac{\lambda}{2  \lambda} \left[ 1(1\lambda)^{2i} \right] \Sigma \, , $$ where \(\Sigma\) is the covariance matrix of the input data. 

Further simplification  There is a further simplification. When \(i\) becomes large, the covariance matrix may be expressed as: $$ \Sigma_{Z_i} = \frac{\lambda}{2  \lambda} \Sigma \, . $$ The question is "What is large?". When we examine the formula with the \(2i\) in it, we observe that when \(2i\) becomes sufficiently large such that \((1\lambda)^{2i}\) becomes almost zero, then we can use the simplified formula.  
Table for selected values of \(\lambda\) and \(i\) 
The following table gives the values of \((1\lambda)^{2i}\)
for selected values of \(\lambda\) and \(i\).


Simplified formula not required  It should be pointed out that a wellmeaning computer program does not have to adhere to the simplified formula, and potential inaccuracies for low values for \(\lambda\) and \(i\) can thus be avoided.  
MEWMA computer output for the Lowry data 
Here is an example of the application of an MEWMA control chart. To
faciltate comparison with existing literature, we used data from
Lowry et al. The data were simulated from a bivariate normal
distribution with unit variances and a correlation coefficient
of 0.5. The value for \(\lambda = 0.10\)
and the values for \(T_i^2\)
were obtained by the equation
given above. The covariance of the MEWMA vectors was obtained
by using the nonsimplified equation. That means that for each
MEWMA control statistic, the computer computed a covariance matrix,
where \(i = 1, \, 2, \, \ldots, \, 10\).
The results of the computer routine
are:
***************************************************** * MultiVariate EWMA Control Chart * ***************************************************** DATA SERIES MEWMA Vector MEWMA 1 2 1 2 STATISTIC 1.190 0.590 0.119 0.059 2.1886 0.120 0.900 0.095 0.143 2.0697 1.690 0.400 0.255 0.169 4.8365 0.300 0.460 0.199 0.198 3.4158 0.890 0.750 0.090 0.103 0.7089 0.820 0.980 0.001 0.191 0.9268 0.300 2.280 0.029 0.400 4.0018 0.630 1.750 0.037 0.535 6.1657 1.560 1.580 0.189 0.639 7.8554 1.460 3.050 0.316 0.880 14.4158 VEC XBAR MSE Lamda 1 .260 1.200 0.100 2 1.124 1.774 0.100The UCL = 5.938 for \(\alpha = 0.05\). Smaller choices of \(\alpha\) are also used. 
Sample MEWMA plot 
The following is the plot of the above MEWMA.
