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Morris

Morris came up with an experimental plan that is composed of individually randomized OAT designs. The data analysis is then based on the so called elementary effects, the changes in an output due to changes in a particular input factor in the OAT design. The method is global in the sense that it does vary over the whole range of uncertainty of the input factors. The Morris method can determine if the effect of the input factor xi on the output y is negligible, linear and additive, nonlinear or involved in interactions with other input factors x~i. xi is the input factor under consideration; x~i is all input factors, except the input factor under consideration.

According to Morris the input factor x~i may be important if:

  1. f(xi + Δ, x~i) − f(x) is nonzero, then x~i affects the output.

  2. f(xi+Δ, x~i) − f(x) varies as xi varies, then xi affects the output nonlinearly.

  3. f(xi+Δ, x~i) − f(x) varies as x~i varies, then xi affects the output with interactions.

Δ is the variation size.

The input factor space is “discretized” and the possible input factor values will be restricted to be inside a regular k-dimensional p-level grid, where p is the number of “levels” of the design. The elementary effect of a given value xi of input factor Xi is defined as a finite-difference derivative approximation

       eei(x) = [ f(x1,x2,…,xi-1,xi+Δ,xi+1,…,xk) - f(x)]/Δ

for any xi between 0 and 1 − Δ where x ∈ , and Δ is a predetermined multiple of 1/(p − 1). The influence of xi is then evaluated by computing several elementary effects at randomly selected values of xi and x~i.

If all samples of the elementary effect of the i’th input factor are zero, then xi doesn’t have any effect on the output y, the sample mean and standard deviation will both be zero. If all elementary effects have the same value, then y is a linear function of xi. The standard deviation of the elementary effects will then of course be zero. For more complex interactions, due to interactions between factors and nonlinearity, Morris states that if the mean of the elementary effects is relatively large and the standard deviation is relatively small, the effects of xi on y is “mildly nonlinear”. If the opposite, the mean is relatively small and the standard deviation is relatively large, then the effect is supposed to be “strongly nonlinear”. As a rule of thumb:

  1. a high mean indicates a factor with an important overall influence on the output and
  2. a high standard deviation indicates that either the factor is interacting with other factors or the factor has nonlinear effects on the output.

To compute r elementary effects of the k inputs we need to do 2rk model evaluations. With the use of Morris randomized OAT design the number of evaluations are reduced to r(k + 1).

The main advantage of the Morris design is the relatively low computational cost. The design requires only about one model evaluation per computed elementary effect.

One drawback with the Morris design is that it only gives an overall measure of the interactions, indicating whether interactions exists, but it does not say which are the most important. Also it can only be used with a set of orthogonal input factors, i.e. correlations cannot be induced on the input factors.

In an implementation, there is a need to think about the choice of the p levels among which each input factor is varied. In Ecolego these levels correspond to quantiles of the input factor distributions, if the distributions are not uniform. For uniform distributions, the levels are obtained by dividing the interval into equidistant parts. The choice of the sizes of the levels p and realizations r is also a problem; various experimenters have demonstrated that the choice of p = 4 and r = 10 produces good results.

Reference

  • Max D. Morris. Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2):161–174, May 1991.
  • Saltelli, A. et al. Sensitivity analysis in practice. A Guide to Assessing Scientific Models. John Wiley & Sons Ltd., Chichester, 2004.