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

## Multivariate EWMA Charts

Multivariate EWMA Control Chart
Univariate EWMA model The model for a univariate EWMA chart is given by: $$Z_i = \lambda X_i + (1-\lambda)Z_{i-1}, \,\,\,\,\, i = 1, \, 2, \, \ldots, \, n \, ,$$ 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 $$Z_i = \Lambda X_i + (1-\Lambda)Z_{i-1} \, ,$$ 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 $$i$$.

$$2i$$
$$1 - \lambda$$ 4 6 8 10 12 20 30 40 50
0.9 0.656 0.531 0.430 0.349 0.282 0.122 0.042 0.015 0.005
0.8 0.410 0.262 0.168 0.107 0.069 0.012 0.001 0.000 0.000
0.7 0.240 0.118 0.058 0.028 0.014 0.001 0.000 0.000 0.000
0.6 0.130 0.047 0.017 0.006 0.002 0.000 0.000 0.000 0.000
0.5 0.063 0.016 0.004 0.001 0.000 0.000 0.000 0.000 0.000
0.4 0.026 0.004 0.001 0.000 0.000 0.000 0.000 0.000 0.000
0.3 0.008 0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.2 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
0.1 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

Simplified formula not required It should be pointed out that a well-meaning 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 non-simplified 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:

*****************************************************
*      Multi-Variate 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     0.100


The 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.