uncertainty-datatypes 1.0.13

Uncertainty Datatypes Library

uTypes Uncertainty Python Library

uTypes is a Python library that supports a set of uncertain primitive datatypes in Python, including ubool, sbool, uint, ufloat, uenum and ustr. They extend their corresponding Python types (bool, int, float, enum and str) with uncertainty. The uTypes library implements linear error propagation theory in Python.

Uncertain numerical values, ufloat and uint, are represented by pairs (x,u) where x is the numerical (nominal) value and u is its associated uncertainty. For example, ufloat(3.5,0.1) represents the uncertain real number 3.5 +/- 0.1, and uint(30,1) represents the uncertain integer 30 +/- 1.

This representation of uncertainty for numerical values follows the "ISO Guide to Measurement Uncertainty" (JCMG 100:2008), where values are represented by the mean and standard deviation of the assumed probability density function representing how measurements of ground truth values are distributed. For example, if we assume that the values of a variable X follow a normal distribution N(x, σ), then we set u = σ. If we can only assume a uniform or rectangular distribution of the possible values of X, then x is taken as the midpoint of the interval, x = (a + b)/2, and its associated standard deviation as u = (b - a)/(2 * sqrt(3)).

Type ubool extends type bool by using propabilities instead of the traditional logical truth values (True, False), and by replacing truth tables by probability expressions. Thus, an ubool value is expressed by a probability representing the degree of belief (i.e., the confidence) that a given statement is true. For example, ubool(0.7) means that there is a 70% chance of an event occurring. Python bool values True and False correspond to ubool(1.0) and ubool(0.0), respectively. ubool values can be used instead of bool values, by projecting the probability using a certainty threshold.

Type sbool provides an extension of ubool to represent binomial opinions in Subjective Logic. They allow expressing degrees of belief with epistemic uncertainty, and also trust. A binomial opinion about a given fact X by a belief agent A is represented as a quadruple sbool(b,d,u,a) where

• b is the degree of belief that X is True
• d is the degree of belief that X is False
• u is the amount of uncommitted belief, also interpreted as epistemic uncertainty.
• a is the prior probability in the absence of belief or disbelief.

These values are all real numbers in the range [0,1] and satisfy that b+d+u=1. The "projected" probability of a binomial opinion is defined as P=b+au.

Type ustr can be used to represent Python strings with uncertainty. I.e., type ustr extends type str, adding to their values a degree of confidence on the contents of the string. This is useful, for example, when rendering strings obtained by inaccurate OCR devices or texts translated from other languages if there are doubts about specific words or phrases. Therefore, values of type ustr are pairs (s,c), where s is the nominal string and c the associated confidence (a real number between 0 and 1). To calculate the confidence of a string s, the Levenshtein distance is normally used. For example, ustr('hell0 world!',0.92) means that we do not trust at most one of the 12 characters of the string. Values of Python type str are embedded into ustr values as ustr(s,1.0).

Finally, type uenum is the embedding supertype for Python type enum that adds uncertainty to each of its values. A value of an uncertain enumeration type is not a single literal, but a set of pairs {(l₁,c₁),...,(l_n,c_n)}, where {c₁,...,c_n} are numbers in the range [0, 1] that represent the probabilities that the variable takes each literal as its value, and c₁+...+c_n=1.

All related operations and Mathematical functions on these datatypes are supported. Check the uTypes User Guide for details.

Main features

The uTypes library provides a simple implementation of uncertainty for Python primitive datatypes, and implements linear error propagation theory in Python. Our goal was to support the basic mechanisms for the expression and propagation of uncertainty, in a lightweight and efficient manner.

A distinguishing feature of the uTypes library is that comparison operators return ubool values. This is not supported by the rest of the related uncertainty libraries, such as the uncertainties package, "soerp" or "mcerp".

Another distinctive feature of uTypes library is that it naturally incorporates Subjective logic (type sbool) into the type system, as a natural extension of probabilistic logic (type ubool). This enables the seamless combination of different types of uncertainties under the same library, and in particular the representation of both second-order uncertainty and trust. The type embedding mechanisms used in uTypes allow operations to be closed in the algebra of types, and that the extended operations work as expected when values of original types are used.

Correlations between expressions are not automatically taken into account in uTypes. This saves keeping track at all times of all correlations between quantities (variables and functions), improving the performance of the calculations. However, this implies that, by default, we assume that variables are independent. Among other things, this means that users are expected to simplify numerical expressions as much as possible in order to avoid duplication of uncertain variables.

In any case, should there be a need to deal with dependent variables, uint and ufloat mathematical operations allow specifying the correlation between them.

The derivatives of mathematical expressions are not automatically handled by the uTypes library. Again, this saves keeping track of the value of derivatives and automatically obtaining them, something that also impacts performance. Other unsuported features include automatic handling of arrays of uncertain numbers, or higher-order analysis to error propagation.

In case derivatives are needed, there are other libraries that provide these features.

• For example, the uncertainties package provides full support for uncertainty progagation, variable correlation, derivatives, and integration with the NumPy package for scientific computation in Python. Most uncertainty calculations are performed analytically.

• soerp is another uncertainty calculation package for Python that provides higher-order approximations of uncertainty. In particular, it supports a second-order analysis to error propagation. Advanced mathematical functions, similar to those in the standard math module can also be evaluated directly.

• mcerp provides a stochastic calculator for Monte Carlo methods that uses latin-hypercube sampling to perform non-order specific error propagation (or uncertainty analysis).

The problem is that these implementations are sometimes too slow, e.g., when used in iterative methods, and their comparison operations are not expressive enough -- that is, the return crisp boolean values. The uTypes package tries to address these limitations.

In summary, the uncertain datatypes provided by the uTypes library is well suited for applications that require the basic mechanisms for the propagation of uncertainty, efficient computation, and a closed algebra of datatypes. In particular, the comparison of two uncertain numeric values returns a probability, i.e., a ubool value, and subjective logic is implemented as a natural extension of probabilistic logic, and in turn of Boolean logic: bool <: ubool <: sbool.

Installation

Use the package manager pip to install foobar.

pip install uncertainty-datatypes

_{Note: pip3 may be used instead of pip}

Usage

Import all the datatypes and functions using:

from uncertainty.utypes import *

The companion uTypes User Guide provides details about all supported datatypes and its associated operations.

Contributing

Pull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.

License

MIT Licence

Copyright (c) 2023 Atenea Research group.

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

Version control

The uTypes library was initially developed in Java. This is the first version of this Python library (July 2023).

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References and further information

The following papers contain all the details about these datatypes:

• Manuel F. Bertoa, Loli Burgueño, Nathalie Moreno, Antonio Vallecillo. "Incorporating measurement uncertainty into OCL/UML primitive datatypes" Softw. Syst. Model. 19(5):1163-1189, 2020. https://doi.org/10.1007/s10270-019-00741-0
• Paula Muñoz, Loli Burgueño, Victor Ortiz, Antonio Vallecillo. "Extending OCL with Subjective Logic" J. Object Technol. 19(3): 3:1-15, 2020. https://doi.org/10.5381/jot.2020.19.3.a1

Examples of applications of the uncertainty datatypes presented here can be found in the following papers:

• Jean-Marc Jézéquel, Antonio Vallecillo. "Uncertainty-aware Simulation of Adaptive Systems" ACM Transactions on Modeling and Computer Simulation, 33(3):8:1-8:19, 2023. https://doi.org/10.1145/3589517
• Lola Burgueño, Paula Muñoz, Robert Clarisó, Jordi Cabot, Sébastien Gérard, Antonio Vallecillo. "Dealing with Belief Uncertainty in Domain Models" ACM Trans. Softw. Eng. Methodol. 32(2):31:1-31:34, 2023. https://doi.org/10.1145/3542947
• Francisco J. Navarrete, Antonio Vallecillo. "Introducing Subjective Knowledge Graphs" In Proc. of EDOC 2021. pp. 61-70, 2021. https://doi.org/10.1109/EDOC52215.2021.00017
• Nathalie Moreno, Manuel F. Bertoa, Loli Burgueño, Antonio Vallecillo. "Managing Measurement and Occurrence Uncertainty in Complex Event Processing Systems" IEEE Access 7:88026-88048, 2019. https://doi.org/10.1109/ACCESS.2019.2923953
• Victor Ortiz, Loli Burgueño, Antonio Vallecillo, Martin Gogolla. "Native Support for UML and OCL Primitive Datatypes Enriched with Uncertainty in USE" In Proc. of OCL@MoDELS 2019:59-66, 2019. https://ceur-ws.org/Vol-2513/paper5.pdf
• Nathalie Moreno, Manuel F. Bertoa, Gala Barquero, Loli Burgueño, Javier Troya, Adrián García-López, Antonio Vallecillo. "Managing Uncertain Complex Events in Web of Things Applications". In Proc. of ICWE 2018:349-357, 2018. https://doi.org/10.1007/978-3-319-91662-0_28
• Loli Burgueño, Manuel F. Bertoa, Nathalie Moreno, Antonio Vallecillo. "Expressing Confidence in Models and in Model Transformation Elements" In Proc. of MoDELS 2018: 57-66, 2018. https://doi.org/10.1145/3239372.3239394

For more information, please visit our research group's websites:

• Atenea Research Group
• Atenea: Uncertain UML/OCL Types
• Atenea: Subjective Logic

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