Size

Models used in the Software Engineering domain vary in the scope they cover and the level of elaboration they use in their descriptions. One may argue that ISO9001 is broader and less detailed than CMMI-Dev or that CMMI-Dev is broader than ISO15939 [707] in what relates to measurement and analysis and the latter is more detailed than CMMI-Dev within the measurement scope.

Narrowing the scope often results in increased detail of descriptions. Increased detail is achieved by using a greater number of components at a similar level of elabo-ration or further elaborating by creating new levels of elaboelabo-ration. An objective mea-surement of this perception would help to clarify the subjective nature of these asser-tions. Defining an elaboration hierarchy of model descriptions (relating model archi-tectural components) provides the scope boundary crucial for performing an evaluation of shared scope and level of detail in descriptions.

As stated previously, comparing models scope size requires the use of a normaliz-ing size reference. Effective comparison implies that a shared characteristic is found to derive a relative measure of size. Scope shared will provide the link to objectively com-pare scope´s size and the scope of the reference model will be used for this purpose.

The number of architectural components, at specific levels of elaboration, covering a shared portion of scope, will be used to measure detail of descriptions. The following concepts are relevant to characterize the attribute of size of a model.

Shared Scope

Shared Scope is a measure of the degree of coverage of the mapped modelM_Prelative to a reference modelM_R. It can be expressed using the following notation:

(M_P,M_R,ϕ),∀ϕ∈ℜ (3.1)

The coverage factorϕ is the result of a mapping performed at a chosen level of granularity, e.g. ISO9001 is covered by a factor ofϕby CMMI-Dev when shall state-ments are compared to CMMI-Dev practices or, considering as reference a full imple-mentation of CMMI-Dev, one can attain ϕcoverage of ISO9001 requirements. The following concepts are enunciated regarding shared scope.

1. Mapping function

A coverage function F establishes the degree of coverage between a pair of model components. It receives as input a component element C_R of M_R at a desired level of elaboration L_Ry and a component element C_P of the mapped modelM_P at the desired level of elaborationL_Px.

ϕ=F(C_P,C_R) (3.2)

The mapping function often assumes the use of a categorical ordinal scale that characterizes the level of coverage that one component has over the mapped component. The scale can be recoded to assume values that translate the mean-ing of each category into a discrete numeric scale, where the recodedϕassumes values between [0,1] where 0 represent no mapping and 1 a total coverage.

2. Component mapping

A componentC_iof a modelM_P at a desired level of granularityL_Pxis compared using a mapping functionF to every componentC_jofM_R at the desired level of detailL_Ry.

(C_i,C_j,ϕ_j),i∈[0..n_x]and j∈[0..n_y] (3.3) wheren_x is the number of components ofM_P atL_Px, n_y is the number of com-ponents of M_R at L_Ryandϕ_j is the coverage factor resulting from applying F.

The value ofϕ_jis the coverage obtained forC_iofM_Pby eachC_jofM_R. 3. Component coverage

3.3 Attributes of size and complexity for improvement models 55 A component is covered if exists at least one component inM_Rthat shares some scope with a component of M_P otherwise is not covered. The highest value obtained in a mapping is the maximum coverage obtained for that component of M_P considering all components ofM_R.

Then, a componentC_iof a modelM_Pis said to be covered byM_R by a factor of ϕ_c if,∈C_j ∃M_R that:

(C_i,C_j,ϕ_j),ϕ_j>0 (3.4) and ϕ_c =max

ϕ_j

j ∈[1..n] where n is the number of elements that satisfy equation 3.4 for eachC_i.

4. Scope coverageS_cof a mapped modelM_P

The shared portion of the mapped model is obtained by identifying architectural components with shared scope. These include components that satisfyϕc>0 in equation 3.4.

S_c= ∑ⁿ_i=1ϕc(i)

n∗ϕ_max (3.5)

In equation 3.5, n is the total number of component maps considered in the mapping andϕ_max is the maximum possible coverage for each component.

5. Scope coverage of a reference modelM_R

Concerning the reference model, it is only feasible to say that a portion of its scope is shared with the mapped model. A component of a mapped model may be covered totally by a component of the reference model, but the opposite may not occur. The reflexive property does not apply in this setting. Still, an approxi-mate measure of scope coverage can be derived considering the cardinality of the set of components from the reference model identified as result of the mapping, where:

Θ= N

N_T (3.6)

where, Nis the number of components ofM_R atL_Ryreferenced in the mapping process,N_tis the total number of components ofM_RatL_RyandΘis the reference factor.

Elaboration of Descriptions

Elaboration of descriptions uses a measure of the number of architectural components of M_R that are referenced in the mapping process for a scope bounded by each ar-chitectural component of M_P e.g., a requirement from ISO9001 maps to x practices of CMMI-Dev. The level of elaboration of a component C_i from M_P regarding M_R is given by the cardinality β_i of the set of componentsS_R from M_R that satisfies the condition in equation 3.4, where,nis the number of components ofM_PatL_Px.

(C_i,β_i,S_R),i∈[0..n] (3.7) Elaboration of descriptions ofM_P, (M_Pd) is given by the central tendency mean of frequencies of βi (nis the number of components of M_P at L_Px, f_i is the number of occurrences of each different group ofβ_iand f_t the total number of occurrences of f_i’s.

M_Pd= ∑ⁿ_i=1β_i∗f_i

f_t (3.8)

Values ofβin equation 3.7 vary from 0, indicating that a mapped component with any degree of coverage is absent ofM_R. The opposite scenario occurs when a compo-nent maps to all compocompo-nents ofM_R. Large values ofβindicate significant differences in the elaboration of descriptions for a shared scope. The inverse relation is also con-sidered for the mapped components of M_R. It measures the cardinality of the set of componentsC_iofM_P that reference a componentC_jofM_R. The relation is expressed as follows: the level of elaboration of a componentC_jofM_RregardingM_Pis given by the cardinality γ_j of the set of componentsC_i fromM_P that satisfies condition set in equation 3.4.

(C_i,γj),j∈[0..n] (3.9) In equation 3.9, n is the number of components of M_R at L_Ry. Elaboration of descriptions ofM_R, (M_Rd) is given by the central tendency mean of frequencies ofγi, wherenis the number of components ofM_R atL_Ry, f_iis the number of occurrences of each different group ofγ_iand f_tis the number of total occurrences of f_i’s.

M_Rd= ∑ⁿ_i=1γ_i∗f_i

f_t (3.10)

The difference of elaboration in models descriptions is given by the elaboration factorE_f:

E_f = M_Pd

M_Rd (3.11)

3.3 Attributes of size and complexity for improvement models 57 WhenE_f approximates 1 it indicates the overall shared scope is described on aver-age by the same number of architectural components at the chosen level of granularity.

It is expectable that, in the assumption of similar levels of detail not every component map translates to a one to one relation, some variation may occur. But, if the level of elaboration is similar between models both measures are expected to have the same order of magnitude on an average case. Values greater than one indicate thatM_P is less elaborated thenM_R for the shared scope at specific levels of elaboration. Values inferior to one indicate more elaboration inM_P.

The notion of balance in elaboration of descriptions for a shared scope is given by the value of standard deviation for equation 3.8 and equation 3.10. High values indicate considerable differences in elaboration of descriptions within a shared scope.

Standard deviation ofE_f aims to translate an evaluation of homogeneity between mod-els descriptions.