Procedure: Morris screening method¶
The Morris method, also know as the Elementary Effect (EE) method, utilises a discrete approximation of the average value of the Jacobian (matrix of the partial derivatives of the simulator output with respect to each of the simulator inputs) of the simulator over the input space. The motivation for the method was screening for deterministic computer models with moderate to large numbers of inputs. The method relies on a one-factor-at-a-time (OAT) experimental design where the effects of a single factor on the output is assessed sequentially. The individual randomised experimental designs are known as trajectories. The method’s main advantage is the lack of assumptions on the inputs and functional dependency of the output to inputs such as monotonicity or linearity.
Algorithm¶
The algorithm involves generating \(R\) trajectory designs, as described below. Each trajectory design is used to compute the expected value of the elementary effects in the simulator function locally. By averaging these a global approximation is obtained. The steps are:
Rescale all input variables to operate on the same scale using either standardisation or linear scaling as shown in the procedure page for data standardisation (ProcDataPreProcessing). Otherwise different values of the step size (see below) will be needed for each input variable.
Each point in the trajectory differs from the previous point in only one input variable by a fixed step size, \(\Delta\). For \(k\) variables, each trajectory has \(k+1\) points, changing each variable exactly once. The start point for the trajectory is random, although this can be modified to improve the algorithm. See the discussion below.
Compute the elementary effect for each input variable \(1,\ldots,k\): \(EE_i(x)=\frac{f(x+\Delta e_i)-f(x)}{\Delta}\). \(e_i\) is the unit vector in the direction of the \(i^{th}\) axis for \(i=1,\ldots,k\). Each elementary effect is computed with observations at the pair of points \(x, x+\Delta e_i\) that differ in the \(i^{th}\) input variable by the fixed step size \(\Delta\).
Compute the moments of the elementary effects distribution for each input variable:
\[\begin{split}\mu_i &= \sum_{r=1} ^{R} \frac{EE_i(x_r)}{R}, \\ \mu^{*}_i &= \sum_{r=1} ^{R} \left|\frac{EE_i(x_r)}{R}\right|, \\ \sigma_i &= \sqrt{\sum_{r=1}^{R} \frac{(EE_i(x_r) - \mu_i)^2}{R}}.\end{split}\]
The sample moment \(\mu_i\) is an average effect measure, and a high value suggests a dominant contribution of the \(i^{th}\) input factor in positive or negative response values (i.e. typically linear, or at least monotonic). The sample moment \(\mu^{*}_i\) is a total effect measure; a high value indicates large influence of the corresponding input factor. \(\mu_i\) may prove misleading due to cancellation effects (that is on average over the input space the output goes up in response to the input as much as it comes down), thus to capture main effects \(\mu^{*}_i\) should be used. Non-linear and interaction effects are estimated with \(\sigma_i\). The total number of model runs needed in the Morris’s method is \((k+1)R\).
An effects plot is constructed by plotting \(\mu_i\) or \(\mu^*_i\) against \(\sigma_i\). This plot is a visual tool to detecting and ranking effects.
An example on synthetic data demonstrating the Morris method is provided at the example page ExamScreeningMorris.
Setting the parameters of the Morris method¶
There is interest in undertaking input screening with as few simulator runs as possible, but as the number of input factors \(k\) is fixed, the size of the experiment required is controlled by the number of trajectory designs \(R\). Usually small values of \(R\) are used; for instance, in Morris (1991) the values \(R=3\) and \(R=4\) were used in the examples. A value of \(R\) between 10 and 50 is mentioned in the more recent literature (see References). A larger value of \(R\) may improve the quality of the global estimates at the price of extra runs. For a reasonably high dimensional input space, with more than say 10 inputs, it would seem unwise to select \(R\) less than 10, since coverage of the space, and thus global effects estimates require something close to space filling. It is likely for large \(k\) the number of trajectory designs will need some dependency on \(R\).
The step size \(\Delta\) is selected in such a way that all the simulator runs lie in the input space and the elementary effects are computed with reasonable precision. The usual choice of \(\Delta\) in the literature is determined by the input space considered for experimentation, which is a \(k\) dimensional grid constructed with \(p\) uniformly spaced values for each input. The number \(p\) is recommended to be even and \(\Delta\) to be an integer multiple of \(1/(p-1)\). Morris (1991) suggests a value of \(\Delta = p/2(p-1)\) that ensures good coverage of the input space with few trajectories. One value for \(\Delta\) is generally used for all the inputs, but the method can be generalised to instead use different values of \(\Delta\) and \(p\) for every input.
Extending the Morris method¶
In Morris’s original proposal, the starting points of the trajectory designs were taken at random from the input space grid. Campolongo (2007) proposed generating a large number of trajectories, selecting the subset that maximise the distance between trajectories in order to cover the design space. Another option is to use a Latin Hypercube design or a Sobol sequence to select the starting points of the trajectories.
A potential drawback of OAT runs in the Morris’s method is that design points fall on top of each other when projected into lower dimensions. This disadvantage becomes more apparent when the design runs are to be used in further modelling after discarding unimportant factors. An alternative is to construct a randomly rotated simplex at every point from which elementary effects are computed (Pujol, 2009). The computation of distribution moments \(\mu_i,\mu^*_i,\sigma_i\) and further analysis is similar as the Morris’s method, with the advantage that projections of the resulting design do not fall on top of existing points, and all observations can be reused in a later stage. A potential disadvantage of this approach is the loss of efficiency in the computation of elementary effects.
Lastly, it is possible to modify the standard Morris algorithm to minimize the number of simulator runs required by employing a sequential version of the algorithm. Details can be found in Boukouvalas et al (2010).
References¶
Morris, M. D. (1991, May). Factorial sampling plans for preliminary computational experiments. Technometrics, 33 (2), 161–174.
Boukouvalas, A., Gosling, J.P. and Maruri-Aguilar, H., An efficient screening method for computer experiments. NCRG Technical Report, Aston University (2010)
Saltelli, A., Chan, K. and Scott, E. M. (eds.) (2000). Sensitivity Analysis. Wiley.
Francesca Campolongo, Jessica Cariboni, and Andrea Saltelli. An effective screening design for sensitivity analysis of large models. Environ. Model. Softw., 22(10):1509–18, 2007.
Francesca Campolongo, Jessica Cariboni, Andrea Saltelli, and W. Schoutens. Enhancing the Morris Method. In Sensitivity Analysis of Model Output, pages 369–79, 2004.
Gilles Pujol. Simplex-based screening designs for estimating metamodels. Reliability Engineering & System Safety, 94:1156–60, 2009.