Refinement adaptive measures

The application of the Morris method to evaluate TM22 energy calculation models has been enhanced by a number of adaptive refinement measures.

1. Re-scale input factor ranges: In the original method, the input factors considered are of uniform range, between [0, 1], with their values assumed in the set (0, 1/(n − 1),2/(n − 1), ..., 1). In TM22 calculation models, input factor ranges have different lengths and ranges of magnitude, thus two re-scaling input ranges are used to match the corresponding small power input factor ranges to the Morris method requirements.

Discrete input range: the initial continuous input range is stratified to create the new discrete one where the single input values are equally spaced between them. To do this, each of the input factors, Xj, are assigned a uniform distribution with lower, xj mi n , and

upper, xj max, boundaries, and the sampling stratification, or subdivision, is performed

in accordance with the established degree of freedom, n, with a constant distance be- tween values ofγ = (xj max− xj mi n)/(n − 1). The resultant new stratified range for the

input factor Xj is: (xj mi n, xj mi n+ γ, ..., xj mi n+ i γ, ..., ...xj max), that can by represented by:

(xj 1, xj 2, ..., xj i, ..., xj n), where xj i is the ith-sample point for the discrete input range Xj.

Normalized trajectory range: the discrete input range is normalised to allow the use of the Euclidean distance for the calculation of the trajectory between two points. In order to re-scale the discrete input range to the range [0, 1], and maintain a constant step

∆ = 1/(n − 1) between sampling points, as is required by the Morris method, the feature scale process is used. According to this process, the resulting normalised trajectory range for the input factor Xj is: (

xj 1−xj mi n xj max−xj mi n, ...,

xj i−xj mi n xj max−xj mi n, ...,

xj n− j xj mi n

xj max−xj mi n), that can be represented

by: (tj 1, ..., tj i, ..., tj n), where tj i is the ith-sample point from the normalised trajectory range

Xj.

2. Morris’ experimental design reduces the number of model evaluations needed by per- forming r evaluations of the elementary effect for randomly chosen sample points for each input factor Xj, resulting in a total design-cost of O(r ∗ (k + 1)), with k being the

number of input variables [69]. However, this design does not guarantee equal probability sampling for each input factor and has been shown, in some cases, to give misleading information [68].

The proposed method overcomes this problem through a systematic evaluation of all the sampled points in the input experimental spaceΩ. This increases the computational cost of the new experimental design to beO(n ∗ (k + 1)), where n is the size of input range for each input parameter (k). This extra computational cost is not of significance to the study due to the improvements in cpu operational speed since Morris’ patent for ”Non- Intrusive appliance monitor apparatus” in 1989 [111], and the relatively low number of input parameters under consideration. This change provides equal probability sampling for each input factor.

3. Standardise the elementary effect: The differences, in terms of unit and magnitude, of the small power input factors are not considered in the original Morris method and this issue can lead to misinterpretation of results. To overcome it, Sin et al. [63] pro- posed the use of a non-dimensional standardised elementary effect (se). For the input

Xj, F sej = sej 1, ...sej i, ...sej n. This new elementary effect is calculated by multiplying

each element eej i by an standardisation factor. This factor is obtained by dividing the

standard deviations of the discrete input factor range values, Xj = xj 1, ...xj i, ...xj n, by

the standard deviations of the output values obtained for each of those inputs values, yj= yj 1(xj 1), ...yj i(xj i), ...yj n(xj n). According to this, the se for the it hvalue of the discrete

input factor range, Xj, is given by Equation 3.4:

sej i= ej i

σXj

σyj

(3.4)

This SEE distribution re-scaling allows for comparison of common input factors across different calculation models.

4. The monotonicity: The use of the estimated meanµ, as calculated in Equation 4.7 to detect the overall influence of the input factors, could be prone to failing in the identification of a factor of considerable influence on the model. If the standardised elementary effect distribution of the input factor Xj, given by F sej = (sej ,1, ...sej ,i, ..sej ,r), contains both

positive and negative elements, i.e., if the model is non-monotonic, some effects may neutralise each other, producing a low value forµ, even for an important factor. The use of the mean in the absolute value for the F seej, given in Equation 3.5, addresses

the immediate problem, although it introduces another by eliminating the information contained in the sign of the effect [68].

µ∗ j = Pr 1 ¯ ¯F se_j ¯ ¯ r (3.5)

To attend to this issue, a new estimation factor (Φj =

¯

¯µ_j− µ∗_j ¯

formation on the sign of the effect is proposed. This enhancement measure comes at no noticeable computational cost and allows the analysis of the monotonocity of the model, i.e., ifΦ has a low value, the output function is monotonic, if it has a high value this indicates non-monotonicity.

5. The skewness: The difference between the median and the mean of the standard elementary effect distribution also contains relevant information, indicating the degree of dispersion (spread) and skewness in the data, and pinpointing outliers. In probability statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. In accordance with the notion of non-parametric, the skew for the input Xj is given by Equation 3.26.

ξj=

µj− mj

σj

(3.6)

whereµj is the mean, mj is the median, andσj is the standard deviation for the sej

distribution.