# Scarce Data in Intelligent Technical Systems: Causes, Characteristics, and Implications

^{*}

## Abstract

**:**

## 1. Introduction

- A closer look into the causes and implications of scarce data is provided. A typology is presented which categorises the subtypes of scarce data.
- An overview of data augmentation, transfer learning, and information fusion methods is given.
- A combination of machine learning and fusion techniques is discussed and further research efforts in this area are motivated.

## 2. A Typology of Scarce Data

**Definition**

**1**

**Definition**

**2**

- Sensors are not available or limited in their functionality. They are technically infeasible, too costly, or not obtainable. The engineering effort to design and plan sensor systems is too complex or too expensive. The sensors’ properties are limited, for example, their sampling rate or operating range.
- The observation period or sampling size is insufficient. Observations do not cover certain concepts or phenomena (Data does not capture the Black Swan [18]). The operation of a sensor is too costly, takes too much time, or is destructive.
- Blind ignorance of human engineers prevents all potential data from being obtained. Missing knowledge about real-world phenomena or the availability of sensors limits the amount of data gathered.

**Undersampled:**Data points always represent only a sample of a distribution or the characteristics of a phenomenon. Sensors only provide a window into the real world. Their observations are a fragmented representation. A phenomenon is undersampled if there is insufficient data available to make sound and significant findings about its characteristics. Due to undersampled data, information remains partially hidden. The aleatoric uncertainty of a phenomenon can only be described inadequately. Figure 2 illustrates two cases of undersampling using a scatter plot in a one-dimensional and a two-dimensional feature space.

**Non-representative:**Data or information is non-representative when only certain parts or subconcepts of a phenomenon are observable or represented in the data. Other subconcepts may be very well represented. Take, for example, a bi-modal distribution of a phenomenon’s characteristics. One of the modes may be very well sampled, whereas the other is absent in the data. In extreme cases, complete concepts are missing. In less extreme cases, subconcepts may merely be undersampled. Data in which subconcepts are undersampled are often also referred to as biased. The observation of industrial machines (condition monitoring) often produces non-representative data. Machines are specifically built to run as smoothly and faultlessly as possible. Consequently, data obtained during normal operation is often available in abundance. In contrast, data on fault states or unusual operating conditions are often rare. Reducing this kind of epistemic uncertainty is difficult in practice since running a machine in fault states is either costly or infeasible. Figure 3 shows the multi-modal and condition monitoring examples as a form of non-representative data.

**Low-dimensional:**Real-world processes can only be observed by a finite number of sensors. Data may be incomplete due to missing data sources – in this case, the data space is too low-dimensional. A low-dimensional space may be insufficient to handle the aleatoric uncertainty of the phenomenon at hand. Figure 4 illustrates a case where data is scarce with respect to the number of available sources.

**Sparse:**Sparse data is caused by sensors or data sources which do not provide data continuously. For example, data is missing over certain time periods or data from different sources cannot be synchronised with each other. Missing data can be caused by defective sensors. This leads to data gaps. Take, for instance, data which is organised in a two-dimensional table. Its rows represent data instances and its columns are data sources. Sparse data is then characterised by missing entries throughout this table (think of a sparse matrix).

**Without Context:**Context is needed to extract information and knowledge from data. Roughly speaking, context is itself information that surrounds the phenomenon of interest and its data-generating process [34]. Context aids in understanding the phenomenon. It can be provided by domain knowledge. Examples of context are labels in classification applications or maps in applications of autonomous driving. Context, and specifically labels, are often costly to produce or provide. If in large datasets only a fraction of data instances are labelled, then the problem relates to undersampled data.

**Drifting/Shifting:**The effectiveness of machine learning algorithms relies heavily on the assumption that training and test data are taken from the same or at least similar distributions [35]. In reality, concepts and phenomena often drift in their distribution over time, e.g., data clusters move through feature space. As a consequence, models which have learned from training data are outdated as soon as significant drift occurs. Adaptation or retraining is usually necessary. Because the drifting data distribution over time is not known, drift is categorised as a form of incomplete information.

## 3. An Overview of Methods for Working with Scarce Data

#### 3.1. Transfer Learning

#### 3.2. Data Augmentation

#### 3.3. Information Fusion

#### 3.3.1. Dempster-Shafer Theory

#### 3.3.2. Fuzzy Set Theory

#### 3.3.3. Possibility Theory

#### 3.4. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DST | Dempster-Shafer theory of evidence |

FST | Fuzzy set theory |

PosT | Possibility theory |

ProbT | Probability theory |

## References

- Sharp, M.; Ak, R.; Hedberg, T. A survey of the advancing use and development of machine learning in smart manufacturing. J. Manuf. Syst.
**2018**, 48, 170–179. [Google Scholar] [CrossRef] [PubMed] - Carvalho, T.P.; Soares, F.A.; Vita, R.; Francisco, R.D.P.; Basto, J.P.; Alcalá, S.G. A systematic literature review of machine learning methods applied to predictive maintenance. Comput. Ind. Eng.
**2019**, 137, 106024. [Google Scholar] [CrossRef] - Lei, Y.; Yang, B.; Jiang, X.; Jia, F.; Li, N.; Nandi, A.K. Applications of machine learning to machine fault diagnosis: A review and roadmap. Mech. Syst. Signal Process.
**2020**, 138, 106587. [Google Scholar] [CrossRef] - Babbar, R.; Schölkopf, B. Data scarcity, robustness and extreme multi-label classification. Mach. Learn.
**2019**, 108, 1329–1351. [Google Scholar] [CrossRef][Green Version] - Wang, Q.; Farahat, A.; Gupta, C.; Zheng, S. Deep time series models for scarce data. Neurocomputing
**2021**, 456, 504–518. [Google Scholar] [CrossRef] - Shu, J.; Xu, Z.; Meng, D. Small Sample Learning in Big Data Era. arXiv
**2018**, arXiv:1808.04572. [Google Scholar] - Qi, G.J.; Luo, J. Small Data Challenges in Big Data Era: A Survey of Recent Progress on Unsupervised and Semi-Supervised Methods. IEEE Trans. Pattern Anal. Mach. Intell.
**2022**, 44, 2168–2187. [Google Scholar] [CrossRef] - Adadi, A. A survey on data–efficient algorithms in big data era. J. Big Data
**2021**, 8, 24. [Google Scholar] [CrossRef] - Andriyanov, N.A.; Andriyanov, D.A. The using of data augmentation in machine learning in image processing tasks in the face of data scarcity. J. Phys. Conf. Ser.
**2020**, 1661, 012018. [Google Scholar] [CrossRef] - Hutchinson, M.L.; Antono, E.; Gibbons, B.M.; Paradiso, S.; Ling, J.; Meredig, B. Overcoming data scarcity with transfer learning. arXiv
**2017**, arXiv:1711.05099. [Google Scholar] - Chen, Z.; Liu, Y.; Sun, H. Physics-informed learning of governing equations from scarce data. Nat. Commun.
**2021**, 12, 6136. [Google Scholar] [CrossRef] [PubMed] - Vecchi, E.; Pospíšil, L.; Albrecht, S.; O’Kane, T.J.; Horenko, I. eSPA+: Scalable Entropy-Optimal Machine Learning Classification for Small Data Problems. Neural Comput.
**2022**, 34, 1220–1255. [Google Scholar] [CrossRef] [PubMed] - Bhouri, M.A.; Perdikaris, P. Gaussian processes meet NeuralODEs: A Bayesian framework for learning the dynamics of partially observed systems from scarce and noisy data. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci.
**2022**, 380, 20210201. [Google Scholar] [CrossRef] - Dubois, D.; Liu, W.; Ma, J.; Prade, H. The basic principles of uncertain information fusion. An organised review of merging rules in different representation frameworks. Inf. Fusion
**2016**, 32, 12–39. [Google Scholar] [CrossRef] - Ayyub, B.M.; Klir, G.J. Uncertainty Modeling and Analysis in Engineering and the Sciences; Chapman & Hall/CRC: Boca Raton, FL, USA, 2006. [Google Scholar]
- Lohweg, V.; Voth, K.; Glock, S. A possibilistic framework for sensor fusion with monitoring of sensor reliability. In Sensor Fusion; Thomas, C., Ed.; IntechOpen: London, UK, 2011. [Google Scholar]
- Hüllermeier, E.; Waegeman, W. Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Mach. Learn.
**2021**, 110, 457–506. [Google Scholar] [CrossRef] - Taleb, N.N. The Black Swan: The Impact of the Highly Improbable; Incerto, Random House Publishing Group: New York, NY, USA, 2007. [Google Scholar]
- Abdar, M.; Pourpanah, F.; Hussain, S.; Rezazadegan, D.; Liu, L.; Ghavamzadeh, M.; Fieguth, P.; Cao, X.; Khosravi, A.; Acharya, U.R.; et al. A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Inf. Fusion
**2021**, 76, 243–297. [Google Scholar] [CrossRef] - Huang, Z.; Lam, H.; Zhang, H. Quantifying Epistemic Uncertainty in Deep Learning. arXiv
**2021**, arXiv:2110.12122. [Google Scholar] - Bengs, V.; Hüllermeier, E.; Waegeman, W. Pitfalls of Epistemic Uncertainty Quantification through Loss Minimisation. arXiv
**2022**, arXiv:2203.06102. [Google Scholar] - Smithson, M. Ignorance and Uncertainty: Emerging Paradigms; Cognitive science; Springer: New York, NY, USA; Heidelberg, Germany, 1989. [Google Scholar]
- Smets, P. Imperfect Information: Imprecision and Uncertainty. In Uncertainty Management in Information Systems: From Needs to Solutions; Motro, A., Smets, P., Eds.; Springer: New York, NY, USA, 1997; pp. 225–254. [Google Scholar]
- Bosu, M.F.; MacDonell, S.G. A Taxonomy of Data Quality Challenges in Empirical Software Engineering. In Proceedings of the 2013 22nd Australian Software Engineering Conference, Melbourne, VIC, Australia, 4–7 June 2013; pp. 97–106. [Google Scholar]
- Rogova, G.L. Information quality in fusion-driven human-machine environments. In Information Quality in Information Fusion and Decision Making; Bossé, É., Rogova, G.L., Eds.; Springer: Cham, Switzerland, 2019; pp. 3–29. [Google Scholar]
- Raglin, A.; Emlet, A.; Caylor, J.; Richardson, J.; Mittrick, M.; Metu, S. Uncertainty of Information (UoI) Taxonomy Assessment Based on Experimental User Study Results; Human-Computer Interaction. Theoretical Approaches and Design, Methods; Kurosu, M., Ed.; Springer: Cham, Switzerland, 2022; pp. 290–301. [Google Scholar]
- Jousselme, A.L.; Maupin, P.; Bosse, E. Uncertainty in a situation analysis perspective. In Proceedings of the Sixth International Conference of Information Fusion, Cairns, QSL, Australia, 8–11 July 2003; Volume 2, pp. 1207–1214. [Google Scholar]
- de Almeida, W.G.; de Sousa, R.T.; de Deus, F.E.; Daniel Amvame Nze, G.; de Mendonça, F.L.L. Taxonomy of data quality problems in multidimensional Data Warehouse models. In Proceedings of the 2013 8th Iberian Conference on Information Systems and Technologies (CISTI), Lisbon, Portugal, 19–22 June 2013; pp. 1–7. [Google Scholar]
- Krause, P.; Clark, D. Representing Uncertain Knowledge: An Artificial Intelligence Approach; Springer: Dordrecht, The Netherlands, 2012. [Google Scholar]
- Huber, W.A. Ignorance Is Not Probability. Risk Anal.
**2010**, 30, 371–376. [Google Scholar] [CrossRef] - Kim, Y.; Bang, H. Introduction to Kalman Filter and Its Applications: 2. In Introduction and Implementations of the Kalman Filter; Govaers, F., Ed.; IntechOpen: London, UK, 2018. [Google Scholar]
- Calude, C.; Longo, G. The deluge of spurious correlations in big data. Found. Sci.
**2017**, 22, 595–612. [Google Scholar] [CrossRef][Green Version] - Horenko, I. On a Scalable Entropic Breaching of the Overfitting Barrier for Small Data Problems in Machine Learning. Neural Comput.
**2020**, 32, 1563–1579. [Google Scholar] [CrossRef] [PubMed] - Snidaro, L.; Herrero, J.G.; Llinas, J.; Blasch, E. Recent Trends in Context Exploitation for Information Fusion and AI. AI Mag.
**2019**, 40, 14–27. [Google Scholar] [CrossRef] - Gama, J.; Žliobaitė, I.; Bifet, A.; Pechenizkiy, M.; Bouchachia, A. A Survey on Concept Drift Adaptation. ACM Comput. Surv.
**2014**, 46, 44:1–44:37. [Google Scholar] [CrossRef] - Pan, S.J.; Yang, Q. A Survey on Transfer Learning. IEEE Trans. Knowl. Data Eng.
**2010**, 22, 1345–1359. [Google Scholar] [CrossRef] - Weiss, K.; Khoshgoftaar, T.M.; Wang, D. A survey of transfer learning. J. Big Data
**2016**, 3, 9. [Google Scholar] [CrossRef][Green Version] - Oreshkin, B.N.; Carpov, D.; Chapados, N.; Bengio, Y. Meta-learning framework with applications to zero-shot time-series forecasting. arXiv
**2020**, arXiv:2002.02887. [Google Scholar] [CrossRef] - Mihalkova, L.; Huynh, T.N.; Mooney, R.J. Mapping and Revising Markov Logic Networks for Transfer Learning; AAAI: Menlo Park, CA, USA, 2007. [Google Scholar]
- Niculescu-Mizil, A.; Caruana, R. Inductive Transfer for Bayesian Network Structure Learning. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, San Juan, Puerto Rico, 21–24 March 2007; Meila, M., Shen, X., Eds.; PMLR Proceedings of Machine Learning Research: New York City, NY, USA, 2007; Volume 2, pp. 339–346. [Google Scholar]
- Yao, S.; Kang, Q.; Zhou, M.; Rawa, M.J.; Abusorrah, A. A survey of transfer learning for machinery diagnostics and prognostics. Artif. Intell. Rev.
**2022**. [Google Scholar] [CrossRef] - Sun, Q.; Liu, Y.; Chua, T.S.; Schiele, B. Meta-Transfer Learning for Few-Shot Learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Long Beach, CA, USA, 15–20 June 2019. [Google Scholar]
- Wang, Z.; Dai, Z.; Poczos, B.; Carbonell, J. Characterizing and Avoiding Negative Transfer. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Long Beach, CA, USA, 15–20 June 2019. [Google Scholar]
- Zhang, W.; Deng, L.; Zhang, L.; Wu, D. A Survey on Negative Transfer. IEEE/CAA J. Autom. Sin.
**2022**, 9, 1. [Google Scholar] [CrossRef] - Geirhos, R.; Rubisch, P.; Michaelis, C.; Bethge, M.; Wichmann, F.A.; Brendel, W. ImageNet-trained CNNs are biased towards texture; increasing shape bias improves accuracy and robustness. In Proceedings of the International Conference on Learning Representations, New Orleans, LA, USA, 6–9 May 2019. [Google Scholar]
- Dekhtiar, J.; Durupt, A.; Bricogne, M.; Eynard, B.; Rowson, H.; Kiritsis, D. Deep learning for big data applications in CAD and PLM – Research review, opportunities and case study. Emerg. Ict Concepts Smart Safe Sustain. Ind. Syst.
**2018**, 100, 227–243. [Google Scholar] [CrossRef] - Židek, K.; Lazorík, P.; Piteľ, J.; Hošovský, A. An Automated Training of Deep Learning Networks by 3D Virtual Models for Object Recognition. Symmetry
**2019**, 11, 496. [Google Scholar] [CrossRef][Green Version] - Židek, K.; Lazorík, P.; Piteľ, J.; Pavlenko, I.; Hošovský, A. Automated Training of Convolutional Networks by Virtual 3D Models for Parts Recognition in Assembly Process. In ADVANCES IN MANUFACTURING; Trojanowska, J., Ciszak, O., Machado, J.M., Pavlenko, I., Eds.; Lecture Notes in Mechanical Engineering; Springer: Cham, Switzerland, 2019; Volume 13, pp. 287–297. [Google Scholar]
- Perez, L.; Wang, J. The effectiveness of data augmentation in image classification using deep learning. arXiv
**2017**, arXiv:1712.04621. [Google Scholar] - Shorten, C.; Khoshgoftaar, T.M. A survey on Image Data Augmentation for Deep Learning. J. Big Data
**2019**, 6, 60. [Google Scholar] [CrossRef] - Feng, S.Y.; Gangal, V.; Wei, J.; Chandar, S.; Vosoughi, S.; Mitamura, T.; Hovy, E. A Survey of Data Augmentation Approaches for NLP. arXiv
**2021**, arXiv:2105.03075. [Google Scholar] - Shorten, C.; Khoshgoftaar, T.M.; Furht, B. Text Data Augmentation for Deep Learning. J. Big Data
**2021**, 8, 101. [Google Scholar] [CrossRef] [PubMed] - Bayer, M.; Kaufhold, M.A.; Reuter, C. A Survey on Data Augmentation for Text Classification. ACM Computing Surveys
**2022**, accept. [Google Scholar] [CrossRef] - Wen, Q.; Sun, L.; Yang, F.; Song, X.; Gao, J.; Wang, X.; Xu, H. Time Series Data Augmentation for Deep Learning: A Survey. In Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence, Virtual, 19–27 August 2021; International Joint Conferences on Artificial Intelligence Organization: Menlo Park, CA, USA, 2021. [Google Scholar]
- Parente, A.P.; de Souza Jr, M.B.; Valdman, A.; Mattos Folly, R.O. Data Augmentation Applied to Machine Learning-Based Monitoring of a Pulp and Paper Process. Processes
**2019**, 7, 958. [Google Scholar] [CrossRef] - Shi, D.; Ye, Y.; Gillwald, M.; Hecht, M. Robustness enhancement of machine fault diagnostic models for railway applications through data augmentation. Mech. Syst. Signal Process.
**2022**, 164, 108217. [Google Scholar] [CrossRef] - Dao, T.; Gu, A.; Ratner, A.J.; Smith, V.; de Sa, C.; Ré, C. A Kernel Theory of Modern Data Augmentation. arXiv
**2018**, arXiv:1803.06084. [Google Scholar] - Antoniou, A.; Storkey, A.; Edwards, H. Data Augmentation Generative Adversarial Networks. arXiv
**2017**, arXiv:1711.04340. [Google Scholar] - Jain, N.; Manikonda, L.; Hernandez, A.O.; Sengupta, S.; Kambhampati, S. Imagining an Engineer: On GAN-Based Data Augmentation Perpetuating Biases. arXiv
**2018**, arXiv:1811.03751. [Google Scholar] - Hall, D.; Llinas, J. Multisensor Data Fusion. In Handbook of Multisensor Data Fusion; Electrical Engineering & Applied Signal Processing Series; Hall, D., Llinas, J., Eds.; CRC Press: Boca Raton, FL, USA, 2001; Volume 3. [Google Scholar]
- Mönks, U. Information Fusion Under Consideration of Conflicting Input Signals; Technologies for Intelligent Automation; Springer: Berlin/Heidelberg, Germany, 2017. [Google Scholar]
- Bloch, I.; Hunter, A.; Appriou, A.; Ayoun, A.; Benferhat, S.; Besnard, P.; Cholvy, L.; Cooke, R.; Cuppens, F.; Dubois, D.; et al. Fusion: General concepts and characteristics. Int. J. Intell. Syst.
**2001**, 16, 1107–1134. [Google Scholar] [CrossRef][Green Version] - Dubois, D.; Everaere, P.; Konieczny, S.; Papini, O. Main issues in belief revision, belief merging and information fusion. In A Guided Tour of Artificial Intelligence Research: Volume I: Knowledge Representation, Reasoning and Learning; Marquis, P., Papini, O., Prade, H., Eds.; Springer: Cham, Switzerland, 2020; pp. 441–485. [Google Scholar]
- Denœux, T.; Dubois, D.; Prade, H. Representations of uncertainty in artificial intelligence: Probability and possibility. In A Guided Tour of Artificial Intelligence Research: Volume I: Knowledge Representation, Reasoning and Learning; Marquis, P., Papini, O., Prade, H., Eds.; Springer: Cham, Switzerland, 2020; pp. 69–117. [Google Scholar]
- Shafer, G. A Mathematical Theory of Evidence; Princeton University Press: Princeton, NJ, USA, 1976. [Google Scholar]
- Dempster, A.P. Upper and lower probabilities induced by a multivalued mapping. Ann. Math. Stat.
**1967**, 38, 325–339. [Google Scholar] [CrossRef] - Salicone, S.; Prioli, M. Measuring Uncertainty within the Theory of Evidence; Springer Series in Measurement Science and Technology; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Shafer, G. Dempster’s rule of combination. Int. J. Approx. Reason.
**2016**, 79, 26–40. [Google Scholar] [CrossRef] - Yager, R.R. On the dempster-shafer framework and new combination rules. Inf. Sci.
**1987**, 41, 93–137. [Google Scholar] [CrossRef] - Campos, F. Decision Making in Uncertain Situations: An Extension to the Mathematical Theory of Evidence. Ph.D. Thesis, Dissertation.Com., Boca Raton, FL, USA, 2006. [Google Scholar]
- Polikar, R. Ensemble Learning. In Ensemble Machine Learning: Methods and Applications; Zhang, C., Ma, Y., Eds.; Springer: New York, NY, USA, 2012; pp. 1–34. [Google Scholar]
- Sagi, O.; Rokach, L. Ensemble learning: A survey. WIREs Data Min. Knowl. Discov.
**2018**, 8, e1249. [Google Scholar] [CrossRef] - Dong, X.; Yu, Z.; Cao, W.; Shi, Y.; Ma, Q. A survey on ensemble learning. Front. Comput. Sci.
**2020**, 14, 241–258. [Google Scholar] [CrossRef] - Zhou, Z.H. Ensemble Learning. In Machine Learning; Springer: Singapore, 2021; pp. 181–210. [Google Scholar]
- Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef][Green Version] - Mönks, U.; Petker, D.; Lohweg, V. Fuzzy-Pattern-Classifier training with small data sets. Information Processing and Management of Uncertainty in Knowledge-Based Systems. Theory and Methods; Hüllermeier, E., Kruse, R., Hoffmann, F., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 426–435. [Google Scholar]
- Bocklisch, S.F. Prozeßanalyse mit unscharfen Verfahren, 1st ed.; Verlag Technik: Berlin, Germany, 1987. [Google Scholar]
- Bocklisch, S.F.; Bitterlich, N. Fuzzy Pattern Classification—Methodology and Application—. In Fuzzy-Systems in Computer Science; Kruse, R., Gebhardt, J., Palm, R., Eds.; Vieweg+Teubner Verlag: Wiesbaden, Germany, 1994; pp. 295–301. [Google Scholar]
- Holst, C.A.; Lohweg, V. A conflict-based drift detection and adaptation approach for multisensor information fusion. In Proceedings of the 2018 IEEE 23rd International Conference on Emerging Technologies and Factory Automation (ETFA), Torino, Italy, 1–4 September 2018; pp. 967–974. [Google Scholar]
- Holst, C.A.; Lohweg, V. Improving majority-guided fuzzy information fusion for Industry 4.0 condition monitoring. In Proceedings of the 2019 22nd International Conference on Information Fusion (FUSION), IEEE, Ottawa, ON, Canada, 2–5 July 2019. [Google Scholar]
- Holst, C.A.; Lohweg, V. A redundancy metric set within possibility theory for multi-sensor systems. Sensors
**2021**, 21, 2508. [Google Scholar] [CrossRef] - Holst, C.A.; Lohweg, V. Designing Possibilistic Information Fusion—The Importance of Associativity, Consistency, and Redundancy. Metrology
**2022**, 2, 180–215. [Google Scholar] [CrossRef] - Aizerman, M.A.; Braverman, E.M.; Rozonoer, L.I. Theoretical foundations of the potential function method in pattern recognition learning. Autom. Remote Control
**1964**, 25, 821–837. [Google Scholar] - Lohweg, V.; Diederichs, C.; Müller, D. Algorithms for hardware-based pattern recognition. EURASIP J. Appl. Signal Process.
**2004**, 2004, 1912–1920. [Google Scholar] [CrossRef][Green Version] - Hempel, A.J. Netzorientierte Fuzzy-Pattern-Klassifikation nichtkonvexer Objektmengenmorphologien. Ph.D. Thesis, Technische Universität Chemnitz, Chemnitz, Germany, 2011. [Google Scholar]
- Zadeh, L.A. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst.
**1978**, 1, 3–28. [Google Scholar] [CrossRef] - Solaiman, B.; Bossé, É. Possibility Theory for the Design of Information Fusion Systems; Information Fusion and Data Science; Springer: Cham, Switzerland, 2019. [Google Scholar]
- Dubois, D.; Prade, H. Practical methods for constructing possibility distributions. Int. J. Intell. Syst.
**2016**, 31, 215–239. [Google Scholar] [CrossRef][Green Version] - Wang, G.; Li, W.; Aertsen, M.; Deprest, J.; Ourselin, S.; Vercauteren, T. Aleatoric uncertainty estimation with test-time augmentation for medical image segmentation with convolutional neural networks. Neurocomputing
**2019**, 338, 34–45. [Google Scholar] [CrossRef] - Diez-Olivan, A.; Del Ser, J.; Galar, D.; Sierra, B. Data fusion and machine learning for industrial prognosis: Trends and perspectives towards Industry 4.0. Inf. Fusion
**2019**, 50, 92–111. [Google Scholar] [CrossRef] - Blasch, E.; Sullivan, N.; Chen, G.; Chen, Y.; Shen, D.; Yu, W.; Chen, H.M. Data fusion information group (DFIG) model meets AI+ML. In Signal Processing, Sensor/Information Fusion, and Target Recognition XXXI; Kadar, I., Blasch, E.P., Grewe, L.L., Eds.; SPIE: Bellingham, WA, USA, 2022; Volume 12122, p. 121220N. [Google Scholar]
- Holzinger, A.; Dehmer, M.; Emmert-Streib, F.; Cucchiara, R.; Augenstein, I.; Del Ser, J.; Samek, W.; Jurisica, I.; Díaz-Rodríguez, N. Information fusion as an integrative cross-cutting enabler to achieve robust, explainable, and trustworthy medical artificial intelligence. Inf. Fusion
**2022**, 79, 263–278. [Google Scholar] [CrossRef] - Holst, C.A.; Lohweg, V. Feature fusion to increase the robustness of machine learners in industrial environments. at-Automatisierungstechnik
**2019**, 67, 853–865. [Google Scholar] [CrossRef] - Kondo, R.E.; de Lima, E.D.D.; Freitas Rocha Loures, E.D.; Santos, E.A.P.D.; Deschamps, F. Data Fusion for Industry 4.0: General Concepts and Applications. In Proceedings of the 25th International Joint Conference on Industrial Engineering and Operations Management—IJCIEOM, Novi Sad, Serbia, 15–17 July 2019; Anisic, Z., Lalic, B., Gracanin, D., Eds.; Springer International Publishing: Cham, Switzerland, 2020; pp. 362–373. [Google Scholar]
- Denœux, T.; Masson, M.H. Dempster-Shafer Reasoning in Large Partially Ordered Sets: Applications in Machine Learning. In Integrated Uncertainty Management and Applications; Huynh, V.N., Nakamori, Y., Lawry, J., Inuiguchi, M., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 39–54. [Google Scholar]
- Hui, K.H.; Ooi, C.S.; Lim, M.H.; Leong, M.S. A hybrid artificial neural network with Dempster-Shafer theory for automated bearing fault diagnosis. J. Vibroengineering
**2016**, 18, 4409–4418. [Google Scholar] [CrossRef][Green Version] - Peñafiel, S.; Baloian, N.; Sanson, H.; Pino, J.A. Applying Dempster–Shafer theory for developing a flexible, accurate and interpretable classifier. Expert Syst. Appl.
**2020**, 148, 113262. [Google Scholar] [CrossRef] - Dubois, D.; Prade, H. From possibilistic rule-based systems to machine learning—A discussion paper. In Scalable Uncertainty Management; Davis, J., Tabia, K., Eds.; Springer International Publishing: Cham, Switzerland, 2020; pp. 35–51. [Google Scholar]

**Figure 1.**A typology of data and information imperfection with a detailed subcategorisation of incompleteness. The typology is based on the work of Smets’ [23]. It recognises incompleteness as one of three major sources of imperfection – besides inconsistency and imprecision. Imprecision captures deficiencies that prevent unambiguous statements from being made based on individual data points. Inconsistency refers to situations in which a piece of information is contradictory to existing knowledge or with other information sources. Incompleteness is lacking, absent, or non-complete data and information.

**Figure 2.**Two examples showcasing undersampled data: (

**a**) an ill-represented one-dimensional distribution and (

**b**) an ill-represented two-dimensional distribution. A two-dimensional scatter plot showcasing undersampled data. The plots show the distributions of phenomena in feature space (red). The distributions are unknown and represent the aleatoric uncertainty of the phenomena. In both examples, the sampled data points (blue) are insufficient to draw conclusions about the distributions. The missing data points are a form of epistemic uncertainty.

**Figure 3.**Two cases of non-representative data. In (

**a**) a bi-modal distribution is shown (red, unknown). One mode is very-well sampled; the second is missing in the data. Plot (

**b**) shows a multi-class classification problem, in which certain classes are missing in the data. Such missing data can, for example, be due to unseen fault states of a machine.

**Figure 4.**A classification example in which the addition of a new data source allows us to distinguish two classes perfectly (

**b**). In the two-dimensional space shown in (

**a**), the aleatoric uncertainty prevents a clear separation of classes. Low-dimensional data is still a form of epistemic uncertainty as it is unknown how the class distributions evolve with new sources.

**Figure 5.**Probability theory versus Dempster-Shafer’s theory in a condition monitoring example. The basic propositions are h: the monitored object is healthy and ${f}_{1}$, ${f}_{2}$: the object is in one of two fault states. The distribution modelled with ProbT (

**a**) is ambiguous since it cannot distinguish between ignorance (epistemic uncertainty) and well-informed uncertainty (aleatoric uncertainty). Using DST (

**b**), it turns out that the expert or model is indeed partly ignorant. This is expressed by $m\left(\right\{{f}_{1},{f}_{2}\left\}\right)=0.4$ (a fault occurred but it is unknown which one) and by $m(\Omega )=0.2$ (nothing is known).

**Figure 6.**A continuous probability (

**a**) and a continuous possibility distribution (

**b**). The probability distribution models a random phenomenon quantitatively; the possibility of distribution of incomplete information qualitatively. The following applies: ${\int}_{x\in \Omega}p\left(x\right)=1$, ${\int}_{x\in \Omega}\pi \left(x\right)\ge 1$, and $\pi \left(x\right)\ge p\left(x\right)$.

**Table 1.**Taxonomies of uncertainty, imperfection, ignorance, and quality which address the topic of data or information incompleteness (in the sense of missing data or information, i.e., scarcity). Incompleteness is recognised as the main concept of imperfection throughout the referenced works. However, a categorisation of the various kinds of missing data or information is not carried out.

Authors | Focus | Builds | Relies on | Details Subcategories of Incompleteness |
---|---|---|---|---|

upon | Incompleteness | |||

Smithson [22] | Ignorance | - | yes | Partially. Incompleteness is subcategorised into Uncertainty (including Vagueness, Probability, Ambiguity) and Absence. Absence of information is not further detailed. |

Smets [23] | Imperfection | - | yes | No |

Krause and Clark [29] | Uncertainty | - | yes | No |

Ayyub and Klir [15] | Ignorance | [22] | yes | Partially. Similar to Smithson. |

Bosu and MacDonell [24] | Data Quality | - | yes | No |

Rogova [25] | Information Quality | [23] | yes | No |

Raglin et al. [26] | Uncertainty | - | yes | No |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Holst, C.-A.; Lohweg, V.
Scarce Data in Intelligent Technical Systems: Causes, Characteristics, and Implications. *Sci* **2022**, *4*, 49.
https://doi.org/10.3390/sci4040049

**AMA Style**

Holst C-A, Lohweg V.
Scarce Data in Intelligent Technical Systems: Causes, Characteristics, and Implications. *Sci*. 2022; 4(4):49.
https://doi.org/10.3390/sci4040049

**Chicago/Turabian Style**

Holst, Christoph-Alexander, and Volker Lohweg.
2022. "Scarce Data in Intelligent Technical Systems: Causes, Characteristics, and Implications" *Sci* 4, no. 4: 49.
https://doi.org/10.3390/sci4040049