Slides presented on June 27, 2018 at ECMFA in Toulouse, France. Paper available for download from: https://link.springer.com/chapter/10.1007%2F978-3-319-92997-2_4

1. Expressing Measurement Uncertainty
in OCL/UML Models
Toulouse, June 27, 2018
M. F. Bertoa1, N. Moreno1, G. Barquero1, L. Burgueño1, J.Troya2 and A. Vallecillo1
1 Atenea Research Group, Univ. Málaga, Spain
2 ISA Group, Univ. Sevilla, Spain
ECMFA 2018

2. Motivation
Uncertainty in Engineering Disciplines
 Engineers naturally think about:
 uncertainty associated with measured values
 Uncertainty is explicitly defined in their models
and considered in model-based simulations
2

3. Motivation
However the situation is not the same when modeled in software! 
3

4. Measurement Uncertainty in Software Models
Some Attempts 
context RectangleSw::area() : Real = h*w
4

5. Uncertainty in Software Engineering
 Very limited support for representing uncertainty in software models
 No support for considering such properties in model-based simulations
 Not part of their type systems!
Motivation
Measure
value : Real
What is the uncertainty of the
measurement method?
How is this uncertainty propagated when
making calculations with uncertain
values?
5

6. Abstraction vs Precision
“Being abstract is something profoundly different from being
vague... The purpose of abstraction is not to be vague, but to
create a new semantic level in which one can be absolutely
precise.”
Edsger Dijkstra
6

7. (Magnitudes and) Uncertainty Representation of Uncertainty
Definition: Standard Uncertainty [GUM]
 Uncertainty of the result of a measurement 𝑥𝑥 expressed as a standard
deviation 𝑢𝑢 of the possible variation of the values of 𝑥𝑥.
 Representation: 𝑥𝑥 ± 𝑢𝑢 or 𝑥𝑥, 𝑢𝑢
 Examples:
[GUM] JCGM 100:2008. Evaluation of measurement data – Guide to the expression of uncertainty in measurement.
http://www.bipm.org/utils/common/documents/jcgm/JCGM_100_2008_E.pdf
Normal distribution: (𝑥𝑥, 𝜎𝜎) with mean 𝑥𝑥, standard deviation 𝜎𝜎
Interval 𝑎𝑎, 𝑏𝑏 : Uniform or rectangular distribution is assumed
(𝑥𝑥, 𝑢𝑢) with 𝑥𝑥 =
𝑎𝑎+𝑏𝑏
2
, 𝑢𝑢 =
(𝑏𝑏−𝑎𝑎)
2 3
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8. Measurement Uncertainty Operations
 Computations with uncertain values have to respect the propagation of
uncertainty (uncertainty analysis)
Two Methods for Computing Aggregated Uncertainty
 Normal or Uniform distribution: Analytical (closed-form) solutions
 General case: Using samples (SIPMath Std.)
A. Vallecillo, C. Morcillo, and P. Orue. Expressing Measurement Uncertainty in Software Models. In Proc. of
QUATIC 2016, pages 1–10, 2016.
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9. Our proposal
 Extension of the OCL/UML
 Primitive types
 Collections
Ongoing
work
9

10. Our proposal
 Extension of the OCL/UML
 Primitive types
 extension based on subtyping
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11. UReal
 UReal are pairs (x, u)
 x : Real represents the (expected, estimated or actual) value
 u : Real represents the uncertainty, expressed as a standard deviation of the
possible variation of the values of 𝑥𝑥.
 Real numbers correspond to pairs (x, 0.0)
 Example:
(2.0, 0.3)
(2.5, 0.25)
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12. UReal
 Constants
 UReal(3.0, 0.05), UReal (0.0, 0.0), UReal (-1.0, 0.003)
 Operations
 Arithmetic
 Trigonometric
 Comparisons
 Conversions
 …
 Examples (in OCL):
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13. UReal - Operations
 Operations. Comparisons
 Should return probabilities, rather than crisp Boolean values
a=(2.0, 0.3)
b=(2.5, 0.25)
d=(1, 0.5)
c=(1, 0.75)
a<b  (true, 0.893)
a=b  (true, 0.106)
a>b  (true, 1.11∙10-16 )
c<d  (true, 0.152)
c=d  (true, 0.754)
c>d  (true, 0.094) 13

14. UInteger and UUnlimitedNatural
 UInteger
 Pairs (n, u)
 n : Integer represents the (expected, estimated or actual) value
 u : Real represents the uncertainty, expressed as a standard deviation of the possible
variation of the values of n.
 Integer numbers correspond to pairs (n, 0.0)
 UInteger operations
 Behavior defined by lifting the operation to UInteger and then projecting the result if
needed
 UUnlimitedNatural
 non-negative Integer or a special unlimited value (*)
 (n, u) where n:Integer, u:Real, n≥0
 The uncertainty of * is always 0.0
 Operations not involving special value * are defined by lifting them to UInteger
 Comparison operations need to consider the particular case of special value *,
lifting the operation to the supertype if this value is not involved
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15. UBoolean
 UBooleans are pairs (b, c)
 where b:Boolean and c:Real, c ϵ [0, 1]
 c represents the confidence that the actual value of the value is indeed b
 Canonical form: (true, c)
 Equivalence relation: (b, c) = (not b, 1 - c)
 Constants
 UBoolean(true, 0.999), UBoolean(false, 0.001)
 Operations
 Redefined basic operations: and, or, not
 Redefined secondary operations: implies, equivalent, xor
 Kept equals (=) and distinct (<>) w/o uncertainty and,
 Added operations uEquals():UBoolean and uDistinct():UBoolean
 Conversion operations: toBoolean() and toBooleanC(c:Real)
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16. UBoolean
 Operations
 Two implementations:
 Assuming all values are independents: Analytical specification:
 When no assumption can be made about the independence: Monte-Carlo simulation
method:
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17. Collections
 Extension based on the extended operators for the primitive datatypes
 uForAll(), uExists(), uIncludes(), uExcludes(), uSelect(), …
 Examples:
17

18. Trains
* Note that in the implementation in USE the uncertain types are included as basic primitive data types, as well as their native operations

19. Trains
 Train arrival time: T = 44.560 ± 10.581
 User arrival time: M= 40.045 ± 5.704
 Their diference: T-M = 4.515 ± 12.374
 M + 3 <= T  (true; 0.887)
 The probability that the user catches the train is 0.887
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20. MARTE and SysML
 Compatibility problems when combining MARTE and SysML models
NOTE: Of course, simulation tools (Modelica, Matlab/Simulink) and mathematic languages (Mathematica) provide
support for units, dimensions and uncertainty, but they are at a different abstraction level
 MARTE has stereotypes to decorate values with information
about the units and with measurement uncertainty
(“precision”)
 However:
 It is simply “decorative information”: no type checking, no
operations for aggregating uncertainty values
 SysML 1.4 provides the QUDV (Quantities, Units,
Dimensions) and ISO 80000 library with all units and
dimensions.
 However:
 No support for dealing with measurement uncertainty
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21. Conclusions and future work
 Extension of OCL/UML datatypes to capture and manipulate properties of physical
systems, in particular measurement uncertainty
 Implementations available for Java and USE
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22. Conclusions and future work
 Extension of OCL/UML datatypes to capture and manipulate properties of
physical systems, in particular measurement uncertainty
 Implementations available for Java and USE
 Ongoing and Future Work
 Cover the rest of the UML/OCL datatypes: String and Enum
 Mathematical properties
 Mappings from our specifications to simulation languages (Modelica, Simulink)
 Further validation of our proposal:
 case studies
 expressiveness
 applicability
22

23. Expressing Measurement Uncertainty
in OCL/UML Models
Toulouse, June 27, 2018
M. F. Bertoa1, N. Moreno1, G. Barquero1, L. Burgueño1, J.Troya2 and A. Vallecillo1
1 Atenea Research Group, Univ. Málaga, Spain,
2 ISA Group, Univ. Sevilla, Spain
ECMFA 2018
Thanks!