# Construction and Simulation of Composite Measures and Condensation Model for Designing Probabilistic Computational Applications

## Abstract

**:**

## 1. Introduction

#### Motivation

- A generalized computational model of composite discrete measures on arbitrary smooth functions is formulated in real and complex metric spaces.
- It is illustrated that the model can operate on linear, non-linear and arbitrary smooth functions.
- The operational modes and properties of the measures on z-plane are constructed.
- The construction of composite measures on a real 2-D surface is proposed.
- The concept of condensation measure of uniform contraction map and the associated properties are presented.

## 2. Related Work

## 3. Formulation of Model

#### 3.1. Composite Measures in Real Metric Space

#### 3.2. Composite Measures in Complex Metric Space

#### 3.3. Properties of Composite Measure in $\sigma -$ Algebra

## 4. Computing Probability-Metric Product

## 5. Discrete Measure on 2-D Real Surface

#### Contraction and Condensation Measure

## 6. Computational Evaluation

#### 6.1. Evaluation on Linear Smooth Curve

#### 6.2. Evaluation on Non-Linear Curves

#### 6.3. Evaluation in z-Plane

#### 6.4. Measure on Real Surface

## 7. Comparative Analysis

## 8. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Hassin, Y.; Peleg, D. Distributed Probabilistic Polling and Applications to Proportionate Agreement. In Lecture Notes in Computer Science; Wiedermann, J., van Emde Boas, P., Nielsen, M., Eds.; Springer: Berlin, Germany, 1999; Volume 1644, pp. 402–411. [Google Scholar]
- Benjelloun, O.; Sarma, A.; Halevy, A.; Widom, J. ULDBs: Databases with uncertainty and lineage. In Proceedings of the 32nd International Conference on Very Large Data Bases, Seoul, Korea, 12–15 September 2006; pp. 953–964. [Google Scholar]
- Jampani, R.; Xu, F.; Wu, M.; Perez, L.; Jermaine, C. MCDB: A monte carlo approach to managing uncertain data. In Proceedings of the 2008 ACM SIGMOD International Conference on Management of Data, New York, NY, USA, 9–12 June 2008. [Google Scholar]
- Kent, P.; Jensen, R.K.; Kongsted, A. A comparison of three clustering methods for finding subgroups in MRI, SMS or clinical data: SPSS TwoStep Cluster analysis, Latent Gold and SNOB. BMC Med. Res. Methodol.
**2014**, 14, 113. [Google Scholar] [CrossRef] [PubMed] - Norman, G. Analyzing Randomized Distributed Algorithms. In Validation of Stochastic Systems; Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.P., Siegle, M., Eds.; Springer: Berlin, Germany; Volume 2925, pp. 384–418.
- Wigderson, A. Do probabilistic algorithms outperform deterministic ones? In Lecture Notes in Computer Science; Larsen, K.G., Skyum, S., Winskel, G., Eds.; Springer: Berlin, Germany, 1998; Volume 1443, pp. 212–214. [Google Scholar]
- Calafiore, G.C.; Dabbene, F.; Tempo, R. Randomized algorithms for probabilistic robustness with real and complex structured uncertainty. IEEE Trans. Autom. Control
**2000**, 45, 2218–2235. [Google Scholar] [CrossRef] [Green Version] - Chung, K.M.; Pettie, S.; Su, H.H. Distributed Algorithms for the Lovasz Local Lemma and Graph Colouring. In Proceedings of the ACM Symposium on Principles of Distributed Computing, Paris, France, 15–18 July 2014. [Google Scholar]
- Jibrin, S.; Boneh, A.; Caron, R.J. Probabilistic algorithms for extreme point identification. J. Interdiscip. Math.
**2007**, 10, 131–142. [Google Scholar] [CrossRef] - Richard, M.K. An Introduction to Randomized Algorithms. Discret. Appl. Math.
**1991**, 34, 165–201. [Google Scholar] - Dubhashi, D.; Grable, D.A.; Panconesi, A. Near-optimal, distributed edge colouring via the nibble method. Theor. Comput. Sci.
**1998**, 203, 225–251. [Google Scholar] [CrossRef] - Barenboim, L.; Elkin, M.; Pettie, S.; Schneider, J. The Locality of Distributed Symmetry Breaking. J. ACM
**2016**, 63. [Google Scholar] [CrossRef] - Lassaigne, R.; Peyronnet, S. Probabilistic verification and approximation. Ann. Pure Appl. Logic
**2008**, 152, 122–131. [Google Scholar] [CrossRef] - Repovš, D.; Savchenko, A.; Zarichnyi, M. Fuzzy Prokhorov metric on the set of probability measures. Fuzzy Sets Syst.
**2011**, 175, 96–104. [Google Scholar] [CrossRef] - Hoyrup, M.; Rojas, C. Computability of probability measures and Martin-Löf randomness over metric spaces. Inf. Comput.
**2009**, 207, 830–847. [Google Scholar] [CrossRef] [Green Version] - Tian, Y.; Yin, Z.; Huang, M. Missing data probability estimation-based Bayesian outlier detection for plant-wide processes with multisampling rates. Symmetry
**2018**, 10, 475. [Google Scholar] [CrossRef] - Shao, S.; Zhang, X.; Li, Y.; Bo, C. Probabilistic single-valued (interval) neutrosophic hesitant fuzzy set and its application in multi-attribute decision making. Symmetry
**2018**, 10, 419. [Google Scholar] [CrossRef] - Ercan, S. On the statistical convergence of order α in paranormed space. Symmetry
**2018**, 10, 483. [Google Scholar] [CrossRef] - Breugel, F.; Worrell, J. A behavioural pseudometric for probabilistic transition systems. Theor. Comput. Sci.
**2005**, 331, 115–142. [Google Scholar] [CrossRef] - Kurtz, T.G.; Manber, U. A probabilistic distributed algorithm for set intersection and its analysis. Theor. Comput. Sci.
**1987**, 49, 267–282. [Google Scholar] [CrossRef] - Hind, I.; Ali, D. Probabilistic distributed algorithm for uniform election in polyo-triangular grid graphs. In Proceedings of the 2014 Second World Conference on Complex Systems (WCCS), Agadir, Morocco, 10–12 November 2014. [Google Scholar]
- Anashin, V.; Khrennikov, A. Applied Algebraic Dynamics; Walter de Gruyter: Berlin, Germany, 2009. [Google Scholar]
- Eifler, L. Open mapping theorems for probability measures on metric spaces. Pac. J. Math.
**1976**, 66, 89–97. [Google Scholar] [CrossRef] [Green Version] - Parthasarathy, K.R. Probability measures in a metric space. In Probability and Mathematical Statistics; Academic Press: New York, NY, USA, 1967; pp. 26–55. [Google Scholar]
- Horvath, A.G. Normally Distributed Probability Measure on the Metric Space of Norms. Acta Math. Sci.
**2013**, 33, 1231–1242. [Google Scholar] [CrossRef] [Green Version] - Vovk, V.; Shen, A. Prequential randomness and probability. Theor. Comput. Sci.
**2010**, 411, 2632–2646. [Google Scholar] [CrossRef] - Hertling, P.; Weihrauch, K. Randomness spaces. In Lecture Notes in Computer Science; Larsen, K.G., Skyum, S., Winskel, G., Eds.; Springer: Berlin, Germany, 1998; Volume 1443, pp. 796–807. [Google Scholar]
- Edalat, A. The Scott topology induces the weak topology. In Proceedings of the 11th Annual IEEE Symposium on Logic in Computer Science, Washington, DC, USA, 27–30 July 1996. [Google Scholar]
- Gács, P. Uniform test of algorithmic randomness over a general space. Theor. Comput. Sci.
**2005**, 341, 91–137. [Google Scholar] [CrossRef] - Calude, C.S.; Hertling, P.H.; Jurgensen, H.; Weihrauch, K. Randomness on full shift spaces. Chaos Solitons Fractals
**2001**, 12, 491–503. [Google Scholar] [CrossRef] - Willis, G.A. Probability measures on groups and some related ideals in group algebras. J. Funct. Anal.
**1990**, 92, 202–263. [Google Scholar] [CrossRef]

Models | Locally Compactness | Haar Measurability Condition | Norm Closure | Amenability |
---|---|---|---|---|

PMG | Yes | Yes | Finite | Integrable (convergent to 0) |

DCM | Yes | No | Finite | Summable (convergent in positive interval) |

LM | No | No | Possibly infinite | Integrable (may not be convergent) |

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**MDPI and ACS Style**

Bagchi, S.
Construction and Simulation of Composite Measures and Condensation Model for Designing Probabilistic Computational Applications. *Symmetry* **2018**, *10*, 638.
https://doi.org/10.3390/sym10110638

**AMA Style**

Bagchi S.
Construction and Simulation of Composite Measures and Condensation Model for Designing Probabilistic Computational Applications. *Symmetry*. 2018; 10(11):638.
https://doi.org/10.3390/sym10110638

**Chicago/Turabian Style**

Bagchi, Susmit.
2018. "Construction and Simulation of Composite Measures and Condensation Model for Designing Probabilistic Computational Applications" *Symmetry* 10, no. 11: 638.
https://doi.org/10.3390/sym10110638