Adding Uncertainty and Units to Quantity Types in Software Models

Business Informatics GroupInstitute of Software Technology and Interactive Systems TU WienFavoritenstraße 9-11/188-3, 1040 Vienna, Austriaphone: +43 (1) 58801 - 18804 (secretary), fax: +43 (1) 58801 - 18896office@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 WimmerBusiness Informatics Group, TU Wien, Austria

Antonio VallecilloAtenea, Universidad de Málaga, Spain

Tanja Mayerhofer

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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

phi.start = 0 rad (rotation angle)w.start = 10 rad/s (angular velocity)

(Coupled Clutches Example of Modelica Standard Library)

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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

Measure

value : Real

What kind of value is measured?In which unit is the value measured?What is the uncertainty of the measurement method?

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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

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

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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)

B. N. Taylor and A. Thompson. The International System of Units (SI). NIST, 2008. http://www.nist.gov/pml/pubs/sp811/.

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Units and Dimensions Representation of Units

Any unit can be derived from the base units: … where represents a base unit and its exponent

Hence, any unit can be defined by the exponents of the base units:

Examples

𝑀𝑒𝑡𝑒𝑟 (𝑚 )=𝑚1∗𝑘𝑔0∗ 𝑠0∗𝐴0∗𝐾 0∗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∗𝐾 0∗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/.

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Units and Dimensions Conversion Between Units

Conversion of quantity values from base units to derived units Multiply the numerical value of the quantity value with conversion factor Add an offset to the resulting numerical value

Definition:

Examples:

Conversion factors and offsets can be defined relative to the base units: :, : Examples:

,

,

,

𝑥𝑘𝑚=(𝑥 ∗1000+0 )𝑚273.15) K0) m/s

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Units and Dimensions Model-Based Representation

Domain Model

Example Instances

Unitname : Stringsymbol : Stringdimensions : Real [8]conversionFactor : Real [8]offset : Real [8]

m : Unitname = "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 : Unitname = "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>

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 expressed as a standard

deviation Representation: or

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: with mean , standard deviation

Interval : Uniform or rectangular distribution is assumed with ,

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Measurement Uncertainty Model-Based Representation

Domain Model

Example Instances

UReal

x : Realu : Real

1 : UReal

x = 10.0u = 0.0014

2 : UReal

x = 2.0u = 0.02

1 0±0.001 2±0.02

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Quantities Model-Based Representation

Domain Model

Example Instance:

Unitname : Stringsymbol : Stringdimensions : Real [8]conversionFactor : Real [8]offset : Real [8]

UReal

x : Realu : Real

Quantityvalue unit

m : Unitname = "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 = 10.0u = 0.001

q : Quantityvalue unit

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Example

Measuretime : Quantityposition : Quantity

startSectionMeasure

/duration : Quantity/distance : Quantity/avgVelocity : Quantity/avgAcceleration : Quantity

end

duration = end.time – start.timedistance = end.position – start.positionavgVelocity = distance / durationavgAcceleration = (end.velocity – start.velocity) / duration

velocity : Quantity

Start A B C N…

Measure M0 M1 M2 M3 MN

S1 S2 S3

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Unit Operations

Unit

isBaseUnit() : BooleanisDerivedUnit() : BooleanisUnitless() : BooleanisDimensionless() : Boolean

isCompatibleWith(Unit u) : Booleanequals(Unit u) : Boolean

multiplyUnits(Unit u) : UnitdivideUnits(Unit u) : UnitpowerUnits(Real s) : Unit

Query nature of unit

Combine units

Compare units

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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

Examples𝑀𝑒𝑡𝑒𝑟 (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 compatiblecompatible

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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

𝑀𝑒𝑡𝑒𝑟 (𝑚 )= ⟨1 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ⟩𝑆𝑞𝑢𝑎𝑟𝑒 𝑀𝑒𝑡𝑒𝑟 (𝑚2 )= ⟨2 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ⟩

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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

𝑀𝑒𝑡𝑒𝑟 (𝑚 )= ⟨1 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ⟩𝑆𝑞𝑢𝑎𝑟𝑒 𝑀𝑒𝑡𝑒𝑟 (𝑚2 )= ⟨2 ,0 ,0 ,0 ,0 ,0 ,0 ,0 ⟩

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Measurement Uncertainty Operations

UReal

add(r : UReal) : URealminus(r : UReal) : URealmultiply(r : UReal) : URealdivideBy(r : UReal) : URealpower(s : Real) : UReal…

lessThan(r : UReal) : BooleanlessThanOrEquals(r : UReal) : BooleangreaterThan(r : UReal) : Boolean…

Arithmetic operations

Comparison operations

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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

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.

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Quantity Operations

Quantity

compatibleUnits(q : Quantity) : Boolean

convertTo(u : Unit) : QuantityconvertToSIUnits() : QuantityconvertFromSIUnits() : Quantity

add(q : Quantity) : Quantityminus(q : Quantity) : Quantitymultiply(q : Quantity) : QuantitydivideBy(q : Quantity) : Quantity…

lessThan(q : Quantity) : BooleanlessThanOrEquals(q : Quantity) : BooleangreaterThan(q : Quantity) : Boolean…

Arithmetic operations

Comparison operations

Unit conversion operations

Unit comparison

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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

5𝑚+1𝑘𝑚=5𝑚+1000𝑚=1005𝑚

UReal operation aggregates uncertainty

Unit operation converts units of summands

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Example

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.timedistance = end.position – start.positionavgVelocity = distance / durationavgAcceleration = (end.velocity – start.velocity) / duration

M0 : Measuretime = 0.0 ± 0.0014 s position = 0.0 ± 0.001 kmvelocity = 0.0 m/s

M1 : Measuretime = 10.0 ± 0.0014 s position = 10.0 ± 0.001 mvelocity = 2.0 ± 0.02 m/s

Start A B C N…

Measure M0 M1 M2 M3 MN

S1 S2 S3

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Quantities Static Type Checking

Domain Model

Unitname : Stringsymbol : Stringdimensions : Real [8]conversionFactor : Real [8]offset : Real [8]

UReal

x : Realu : Real

Quantityvalue unit

Mass Time ElectricCurrent

ThermodynamicTemperature

AmountOfSubstance

LuminosityIntensity Angle

LinearAcceleration ForcePower

Length

LinearVelocity Resistance…

Quantity types for base dimensions

Quantity types for derived dimensions

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Quantities Static Type Checking

Domain Model

Unitname : Stringsymbol : Stringdimensions : Real [8]conversionFactor : Real [8]offset : Real [8]

UReal

x : Realu : Real

Quantityvalue unit

Mass Time ElectricCurrent

ThermodynamicTemperature

AmountOfSubstance

LuminosityIntensity Angle

LinearAcceleration ForcePower

Length

LinearVelocity Resistance…

Quantity types for base dimensions

Quantity types for derived dimensions

Length

add(l : Length) : Lengthminus(l : Length) : Lengthmultiply(l : Length) : Areamultiply(l : Area) : VolumedivideBy (t : Time) : LinearVelocity…

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Example

duration = end.time – start.timedistance = end.position – start.positionavgVelocity = distance / durationavgAcceleration = (end.velocity – start.velocity) / duration

Start A B C N…

Measure M0 M1 M2 M3 MN

S1 S2 S3

Measuretime : Timeposition : Length

startSectionMeasure

/duration : Time/distance : Length/avgVelocity : LinearVelocity/avgAcceleration : LincearAcceleration

end velocity : LinearVelocity

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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

USE Tool: https://sourceforge.net/projects/useocl/ MagicDraw: http://www.nomagic.com/products/magicdraw.html Eclipse Papyrus UML: https://eclipse.org/papyrus/

Java ExampleLength 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

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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

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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

Thank you!

Questions?

Manuel Wimmerwimmer@big.tuwien.ac.at

Antonio Vallecilloav@lcc.uma.es

Tanja Mayerhofermayerhofer@big.tuwien.ac.at

Contact

Business Informatics Group

Addendum:

Implementation of Quantity Types for fUML

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fUML Implementation

Types

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fUML Implementation

Operations: Realized as function behaviors (syntactical elements)

…

……

fUML Implementation

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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+

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fUML Implementation

Example

Classes

Behavior

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fUML Implementation

Example

Values