This document discusses representing uncertainty and units in software models. It proposes a type system to represent measurement uncertainty and units, as well as an algebra of operations to perform computations with uncertain data and units. Implementations are proposed for Java, OCL, and UML. The key contributions are a kernel representation for quantities that includes uncertainty, a computational kernel for operating on quantities, and domain models and examples to demonstrate representing quantities, units, and uncertainty in software models.

1. Business Informatics Group
Institute of Software Technology and Interactive Systems
TU Wien
Favoritenstraße 9-11/188-3, 1040 Vienna, Austria
phone: +43 (1) 58801 - 18804 (secretary), fax: +43 (1) 58801 - 18896
office@big.tuwien.ac.at, www.big.tuwien.ac.at
Business Informatics Group
Adding Uncertainty and Units to Quantity Types
in Software Models
Tanja Mayerhofer, Manuel Wimmer
Business Informatics Group, TU Wien, Austria
Antonio Vallecillo
Atenea, Universidad de Málaga, Spain
Tanja Mayerhofer

2. Motivation
Uncertainty and Units in Engineering Disciplines
 Engineers naturally think about uncertainty associated with measured values
and units of values
 Examples
 A machinist estimates that the length of a produced part lies with probability 0.5
in the interval [10.07 mm, 10.15 mm]
 A calibration certificate states that with 99 percent confidence the resistance of
a standard resistor is 10.000742 Ω ± 129 μΩ
 Uncertainty and units are explicitly defined in models and considered in
model-based simulations
2
phi.start = 0 rad (rotation angle)
w.start = 10 rad/s (angular velocity)
(Coupled Clutches Example of Modelica Standard Library)

3. Uncertainty and Units in Software Engineering
 Very limited support for representing uncertainty and units in software models
 No support for considering such properties in model-based simulations
Problem
 Movement towards cyber-physical systems demands for the accurate
representation of properties of physical entities in software models
 Important future application domain: Internet of Things
 Analysis of huge amount sensor data
Motivation
3
Measure
value : Real
What kind of value is measured?
In which unit is the value measured?
What is the uncertainty of the
measurement method?

4. Contributions
1. Type system for representing measurement uncertainty and units
 Kernel representation for quantities
2. Algebra of operations for performing computations with uncertain data
and units
 Computational kernel for computing quantities
3. Implementations for Java, OCL, UML
4

5. Quantities
Definition: Quantity Kind (Dimension)
 Any observable property of any object that can be measured and quantified
numerically.
 Examples: Length, mass, time, force, energy, power, electric charge
Definition: Quantity
 Observable property of a particular object that can be measured and
quantified numerically.
 Examples: Length, mass, speed, temperature of a particular object
Definition: Quantity Value
 Magnitude of a quantity expressed as a product of a number and a unit.
 Example: Velocity of 3.5 m/s
5

6. Units and Dimensions Systems of Units
International System of Units (SI)
 Base dimensions: Length, Mass, Time, Electric Current, Thermodynamic
Temperature, Amount of Substance, Luminous Intensity
 Base units: Meter (m), Kilogram (kg), Second (s), Ampere (A), Kelvin (K), Mole
(mol), Candela (cd)
 Derived dimensions: 90 dimensions derived from the base dimensions
e.g., Area, Volume, Velocity
 Derived units: 90 units derived from the base units
e.g., Square Meter (m²), Cubic Meter (m³), Meter per Second (m/s)
Other Systems of Units
 Centimeter-Gram-Second System (CGS)
 Imperial System
 United States Customary System (USCS, USC)
6B. N. Taylor and A. Thompson. The International System of Units (SI). NIST, 2008. http://www.nist.gov/pml/pubs/sp811/.

7. Units and Dimensions Representation of Units
 Any unit can be derived from the base units:
𝐵1
𝑒1
∗ 𝐵2
𝑒2
… 𝐵𝑛
𝑒 𝑛
where 𝐵𝑖 represents a base unit and 𝑒𝑖 its exponent
 Hence, any unit can be defined by the exponents 𝑒𝑖 of the base units:
𝑒1, 𝑒1, … , 𝑒 𝑛
 Examples
7
𝑀𝑒𝑡𝑒𝑟 𝑚 = 𝑚1 ∗ 𝑘𝑔0 ∗ 𝑠0 ∗ 𝐴0 ∗ 𝐾0 ∗ 𝑐𝑑0 ∗ 𝑚𝑜𝑙0 ∗ 𝑟𝑎𝑑0 = 1, 0, 0, 0, 0, 0, 0, 0
𝑆𝑞𝑢𝑎𝑟𝑒 𝑀𝑒𝑡𝑒𝑟 𝑚2
= 𝑚2
∗ 𝑘𝑔0
∗ 𝑠0
∗ 𝐴0
∗ 𝐾0
∗ 𝑐𝑑0
∗ 𝑚𝑜𝑙0
∗ 𝑟𝑎𝑑0
= 2, 0, 0, 0, 0, 0, 0, 0
𝑀𝑒𝑡𝑒𝑟 𝑝𝑒𝑟 𝑆𝑒𝑐𝑜𝑛𝑑 𝑚/𝑠 = 𝑚1 ∗ 𝑘𝑔0 ∗ 𝑠−1 ∗ 𝐴0 ∗ 𝐾0 ∗ 𝑐𝑑0 ∗ 𝑚𝑜𝑙0 ∗ 𝑟𝑎𝑑0 = 1, 0, −1, 0, 0, 0, 0, 0
R. Hodgson, P. J. Keller, J. Hodges, and J. Spivak. QUDT – Quantities, Units, Dimensions and Data Types Ontologies.
TopQuadrant, Inc. and NASA AMES Research Center, 2014. http://qudt.org/.

8. Units and Dimensions Conversion Between Units
 Conversion of quantity values from base units 𝐵𝑖 to derived units 𝐷𝑖
 Multiply the numerical value of the quantity value with conversion factor 𝑐𝑓
 Add an offset 𝑜 to the resulting numerical value
 Definition: 𝑥 𝐷𝑖 = (𝑥 ∗ 𝑐𝑓𝑖 + 𝑜𝑖) 𝐵𝑖
 Examples:
 Conversion factors and offsets can be defined relative to the base units:
𝑐𝑓𝑖: 𝑐𝑓1, 𝑐𝑓1, … , 𝑐𝑓𝑛 , 𝑜𝑖: 𝑜1, 𝑜2, … , 𝑜 𝑛
 Examples:
8
𝐾𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟 𝑘𝑚 : 𝑐𝑓 = 1000, 1, 1, 1, 1, 1, 1, 1 , 𝑜𝑓 = 0, 0, 0, 0, 0, 0, 0, 0
𝐶𝑒𝑙𝑐𝑖𝑢𝑠 °𝐶 : 𝑐𝑓 = 1, 1, 1, 1, 1, 1, 1, 1 , 𝑜𝑓 = 0, 0, 0, 0, 273.15, 0, 0, 0
𝐾𝑖𝑙𝑜𝑚𝑒𝑡𝑒𝑟 𝑝𝑒𝑟 𝐻𝑜𝑢𝑟 𝑘𝑚/ℎ : 𝑐𝑓 = 1000, 1, 3600, 1, 1, 1, 1, 1 , 𝑜𝑓 = 0, 0, 0, 0, 0, 0, 0, 0
𝑥 𝑘𝑚 = 𝑥 ∗ 1000 + 0 𝑚
𝑥 °𝐶 = (𝑥 ∗ 1 + 273.15) K
𝑥 𝑘𝑚/ℎ = (𝑥 ∗
1000
3600
+ 0) m/s

9. Units and Dimensions Model-Based Representation
 Domain Model
 Example Instances
9
Unit
name : String
symbol : String
dimensions : Real [8]
conversionFactor : Real [8]
offset : Real [8]
m : Unit
name = "Meter"
symbol = "m"
dimensions = <1,0,0,0,0,0,0,0>
conversionFactor = <1,1,1,1,1,1,1,1>
offset = <0,0,0,0,0,0,0,0>
km/h : Unit
name = "Kilometer per Hour"
symbol = "km/h"
dimensions = <1,0,-1,0,0,0,0,0>
conversionFactor = <1000,1,3600,1,1,1,1,1>
offset = <0,0,0,0,0,0,0,0>

10. Measurement Uncertainty Representation of Uncertainty
 It is impossible to know, estimate or measure values with complete precision
 The value of a quantity is only complete when it is accompanied by a
statement about the associated uncertainty
Definition: Standard Uncertainty [GUM]
 Uncertainty of the result of a measurement 𝑥 expressed as a standard
deviation 𝑢
 Representation: 𝑥 ± 𝑢 or (𝑥, 𝑢)
 Examples:
10[GUM] JCGM 100:2008. Evaluation of measurement data – Guide to the expression of uncertainty in measurement.
Joint Committee for Guides in Metrology, 2008.
Normal distribution: (𝑥, 𝜎) with mean 𝑥, standard deviation 𝜎
Interval 𝑎, 𝑏 : Uniform or rectangular distribution is assumed
(𝑥, 𝑢) with 𝑥 =
𝑎+𝑏
2
, 𝑢 =
(𝑏−𝑎)
2 3

11. Measurement Uncertainty Model-Based Representation
 Domain Model
 Example Instances
11
UReal
x : Real
u : Real
1 : UReal
x = 10.0
u = 0.0014
2 : UReal
x = 2.0
u = 0.02
10 ± 0.001 2 ± 0.02

12. Quantities Model-Based Representation
 Domain Model
 Example Instance: 𝐿𝑒𝑛𝑔𝑡ℎ 10 ± 0.001 𝑚
12
Unit
name : String
symbol : String
dimensions : Real [8]
conversionFactor : Real [8]
offset : Real [8]
UReal
x : Real
u : Real
Quantityvalue unit
m : Unit
name = "Meter"
symbol = "m"
dimensions = <1,0,0,0,0,0,0,0>
conversionFactor = <1,1,1,1,1,1,1,1>
offset = <0,0,0,0,0,0,0,0>
ur : UReal
x = 10.0
u = 0.001
q : Quantityvalue unit

13. Example
13
Measure
time : Quantity
position : Quantity
start
SectionMeasure
/duration : Quantity
/distance : Quantity
/avgVelocity : Quantity
/avgAcceleration : Quantity
end
duration = end.time – start.time
distance = end.position – start.position
avgVelocity = distance / duration
avgAcceleration = (end.velocity – start.velocity) / duration
velocity : Quantity
Start A B C N…
Measure M0 M1 M2 M3 MN
S1 S2 S3

14. Unit Operations
14
Unit
isBaseUnit() : Boolean
isDerivedUnit() : Boolean
isUnitless() : Boolean
isDimensionless() : Boolean
isCompatibleWith(Unit u) : Boolean
equals(Unit u) : Boolean
multiplyUnits(Unit u) : Unit
divideUnits(Unit u) : Unit
powerUnits(Real s) : Unit
Query nature of unit
Combine units
Compare units

15. Unit Operations
isCompatibleWith(Unit u) : Boolean
 Checks whether two units are compatible for being combined or compared
 Two units are compatible, if they are of the same dimension
 Specification: self.dimensions = u.dimensions
 Examples
15
𝑀𝑒𝑡𝑒𝑟 𝑚 = 1, 0, 0, 0, 0, 0, 0, 0
𝑆𝑞𝑢𝑎𝑟𝑒 𝑀𝑒𝑡𝑒𝑟 𝑚2
= 2, 0, 0, 0, 0, 0, 0, 0
𝑀𝑖𝑙𝑒 𝑚𝑙 = 1, 0, 0, 0, 0, 0, 0, 0
not compatible
not compatible
compatible

16. Unit Operations
multiplyUnits(Unit u) : Unit
 When two quantity values are multiplied, their units have to be multiplied too
 Specification:
result.dimensions = self.dimensions -> sum(u.dimensions)
 Examples
16
𝑀𝑒𝑡𝑒𝑟 𝑚 = 1, 0, 0, 0, 0, 0, 0, 0
𝑆𝑞𝑢𝑎𝑟𝑒 𝑀𝑒𝑡𝑒𝑟 𝑚2
= 2, 0, 0, 0, 0, 0, 0, 0
𝑀𝑒𝑡𝑒𝑟 𝑚 ∗ 𝑀𝑒𝑡𝑒𝑟 𝑚 = 1, 0, 0, 0, 0, 0, 0, 0 + 1, 0, 0, 0, 0, 0, 0, 0 = 2, 0, 0, 0, 0, 0, 0, 0
= Square Meter 𝑚2

17. Unit Operations
divideUnits(Unit u) : Unit
 When a quantity value is divided by another quantity value, their units have to
be divided too
 Specification:
result.dimensions = self.dimensions -> minus(u.dimensions)
 Examples
17
𝑀𝑒𝑡𝑒𝑟 𝑚 = 1, 0, 0, 0, 0, 0, 0, 0
𝑆𝑞𝑢𝑎𝑟𝑒 𝑀𝑒𝑡𝑒𝑟 𝑚2 = 2, 0, 0, 0, 0, 0, 0, 0
Square Meter 𝑚2
𝑀𝑒𝑡𝑒𝑟 𝑚
= 2, 0, 0, 0, 0, 0, 0, 0 − 1, 0, 0, 0, 0, 0, 0, 0 = 1, 0, 0, 0, 0, 0, 0, 0
= Meter 𝑚

18. Measurement Uncertainty Operations
18
UReal
add(r : UReal) : UReal
minus(r : UReal) : UReal
multiply(r : UReal) : UReal
divideBy(r : UReal) : UReal
power(s : Real) : UReal
…
lessThan(r : UReal) : Boolean
lessThanOrEquals(r : UReal) : Boolean
greaterThan(r : UReal) : Boolean
…
Arithmetic operations
Comparison operations

19. Measurement Uncertainty Operations
 Computations with uncertain values have to respect the propagation of
uncertainty (uncertainty analysis)
Methods for Computing Aggregated Uncertainty
 Normal or Uniform distribution: Analytical solutions
 General case: Monte Carlo simulations
19
A. Vallecillo, C. Morcillo, and P. Orue. Expressing Measurement Uncertainty in
Software Models. In Proceedings of the 10th International Conference on the Quality
of Information and Communications Technology (QUATIC), pages 1–10, 2016.

20. Quantity Operations
20
Quantity
compatibleUnits(q : Quantity) : Boolean
convertTo(u : Unit) : Quantity
convertToSIUnits() : Quantity
convertFromSIUnits() : Quantity
add(q : Quantity) : Quantity
minus(q : Quantity) : Quantity
multiply(q : Quantity) : Quantity
divideBy(q : Quantity) : Quantity
…
lessThan(q : Quantity) : Boolean
lessThanOrEquals(q : Quantity) : Boolean
greaterThan(q : Quantity) : Boolean
…
Arithmetic operations
Comparison operations
Unit conversion operations
Unit comparison

21. Quantity Operations
add(q : Quantity) : Quantity
 Quantity values of compatible units can be added
 For the addition, the summands have to be converted to the same unit
 Specification:
pre: self.compatibleUnits(q.unit)
post: result.value = self.value.add(q.convertTo(self.unit).value)
and result.unit = self.unit
 Example
 Special care has to be taken for units with an offset (e.g., °C, °F)
 Units with offsets are affine, i.e., non-multiplicative
 Require affine conversions that allows multiplication and addition
21
5 𝑚 + 1 𝑘𝑚 = 5 𝑚 + 1000 𝑚 = 1005 𝑚
UReal operation aggregates uncertainty
Unit operation converts units of summands

22. Example
22
startS1 : SectionMeasure
/duration = 10.0 ± 0.0019799 s
/distance = 10.0 ± 0.0014142 m
/avgVelocity = 1.0000000392 ± 0.000489 m/s
/avgAcceleration = 0.200000008 ± 0.0632468 m/s²
end
duration = end.time – start.time
distance = end.position – start.position
avgVelocity = distance / duration
avgAcceleration = (end.velocity – start.velocity) / duration
M0 : Measure
time = 0.0 ± 0.0014 s
position = 0.0 ± 0.001 km
velocity = 0.0 m/s
M1 : Measure
time = 10.0 ± 0.0014 s
position = 10.0 ± 0.001 m
velocity = 2.0 ± 0.02 m/s
Start A B C N…
Measure M0 M1 M2 M3 MN
S1 S2 S3

23. Quantities Static Type Checking
 Domain Model
23
Unit
name : String
symbol : String
dimensions : Real [8]
conversionFactor : Real [8]
offset : Real [8]
UReal
x : Real
u : Real
Quantityvalue unit
Mass Time Electric
Current
Thermodynamic
Temperature
AmountOf
Substance
Luminosity
Intensity Angle
Linear
Acceleration ForcePower
Length
Linear
Velocity Resistance…
Quantity types for base dimensions
Quantity types for derived dimensions

24. Quantities Static Type Checking
 Domain Model
24
Unit
name : String
symbol : String
dimensions : Real [8]
conversionFactor : Real [8]
offset : Real [8]
UReal
x : Real
u : Real
Quantityvalue unit
Mass Time Electric
Current
Thermodynamic
Temperature
AmountOf
Substance
Luminosity
Intensity Angle
Linear
Acceleration ForcePower
Length
Linear
Velocity Resistance…
Quantity types for base dimensions
Quantity types for derived dimensions
Length
add(l : Length) : Length
minus(l : Length) : Length
multiply(l : Length) : Area
multiply(l : Area) : Volume
divideBy (t : Time) : LinearVelocity
…

25. Example
25
duration = end.time – start.time
distance = end.position – start.position
avgVelocity = distance / duration
avgAcceleration = (end.velocity – start.velocity) / duration
Start A B C N…
Measure M0 M1 M2 M3 MN
S1 S2 S3
Measure
time : Time
position : Length
start
SectionMeasure
/duration : Time
/distance : Length
/avgVelocity : LinearVelocity
/avgAcceleration : LincearAcceleration
end velocity : LinearVelocity

26. Available Implementations
 Java: Reference implementation
 OCL (USE Tool):
 Specification of operations with
preconditions and postconditions
 Support for imperative use of
operations (SOIL)
 UML (Papyrus, MagicDraw):
 Support for specifying quantities and
computations with quantities
 Proof-of-concept prototype for executing
computations with quantities with fUML
 Download: https://github.com/moliz/moliz.quantitytypes
Implementation
26
USE Tool: https://sourceforge.net/projects/useocl/ MagicDraw: http://www.nomagic.com/products/magicdraw.html
Eclipse Papyrus UML: https://eclipse.org/papyrus/
Java Example
Length initialPosition = new Length(0, 0.001, Units.Meter);
Length finalPosition = new Length(10, 0.001, Units.Meter);
Length distance = finalPosition.minus(initialPosition);
USE OCL Example
!new UReal(’ip’)
!ip.x : = 0.0
!ip.u := 0.001
!new Quantity(’initialPosition’)
!initialPosition.value := ip
...
!distance := finalPosition.minus(initialPosition)
Papyrus UML Example

27. Summary
1. Type system for representing quantities
+ Allows explicit representation of measurement uncertainty and units
+ Enables static model-level checks of quantity compatibility
2. Algebra of operations for performing computations with quantities
+ Allows computations with uncertain values
+ Supports automatic conversions between units
+ Enables domain experts to use the most appropriate units
+ Supports model-level simulations that consider data uncertainty
and units
3. Implementations for Java, OCL, UML
+ Reference implementation and formal specifications for operations on quantities
+ Showcase of integrating quantities with modeling language on the example of UML
27

28. Ongoing and Future Work
 Implementation
 Evolve fUML proof-of-concept implementation to full implementation
 Alf implementation (textual action language for fUML)
 Full integration with Papyrus and MagicDraw
 Eclipse OCL implementation
 Refinement of the conceptual model of quantity types
 Different kinds of uncertainty (e.g., interval, different probability distributions)
 Different kinds of units (e.g., length units, time units, etc.)
 Representation of quantities
 Useable representation of quantities
 Integration with existing standards, e.g., MARTE and SysML
28

29. Thank you!
Questions?
Manuel Wimmer
wimmer@big.tuwien.ac.at
Antonio Vallecillo
av@lcc.uma.es
Tanja Mayerhofer
mayerhofer@big.tuwien.ac.at
Contact

30. Business Informatics Group
Addendum:
Implementation of Quantity Types for fUML

31. fUML Implementation
31
Types

32. fUML Implementation
32
Operations: Realized as function behaviors (syntactical elements)
…
…
…

33. fUML Implementation
33
Operation Implementations
 fUML execution model is extended with Execution classes for all quantity operations
(built-in extension mechanism)
 Operations of Execution classes are implemented with Java
Execution
+ execute() : void
OpaqueBehaviorExecution
+ doBody(input : ParameterValue [*], output : ParameterValue [*]) : void
+ execute() : void
BasicBehaviors
QuantityFunctions
…
QuantityMinusExecution
doBody(input : ParameterValue [*], output : ParameterValue [*]) : void+

34. fUML Implementation
34
Example
Classes
Behavior

35. fUML Implementation
35
Example
Values