# Peculiarities and Applications of Stochastic Processes with Fractal Properties

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractal Process Model

## 3. Mathematical Model of Inertial Sensor Errors in Navigation Systems with Respect to the Chaotic Indicator

**Models of sensor errors based on fractal Wiener process**

**Sensor Error Measurement Model**

#### 3.1. Experimental Confirmation of Fractal Properties of Gyroscope Zero Drift Process

#### 3.2. Modeling of Accelerometer Bias

^{10}.

## 4. Navigation and Motion Control

**Mathematical Models of Object Movement for Trajectory Tracking Problems**

**Models of object movement based on the FWP**

**Measurement model for trajectory tracking**

## 5. Fractal Process Filtering

_{i}are independent, then the FWP does not have this property and is not a Markov process, except in cases where H = 0.5 when the FWP is consistent with the CWP. In such cases, KF only gives the optimal solution for this value of H parameter and is not optimal with other H parameter values.

## 6. Information and Telecommunication System

- (1)
- Pre-processing of the traffic time series consisting of statistical data sampling with the purpose of generating the TS we are interested in;
- (2)
- Estimation of fractality indicators using different methods such as R/S methods and wavelet analyses;
- (3)
- Identification of the mathematical models of TS by approximation to known models, or through synthesizing the structures and parameters of mathematical TS models.

## 7. Defect Detection

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Crownover, R.M. Introduction to Fractals and Chaos; Jones and Bartlett Publishers, Inc.: Boston, MA, USA; London, UK, 1995. [Google Scholar]
- Aguirre, J.; Viana, R.L.; Sanjuan, M.A.F. Fractal structures in nonlinear dynamics. Rev. Mod. Phys.
**2009**, 81, 333–386. [Google Scholar] [CrossRef] - Nualart, D. Fractional Brownian motion: Stochastic calculus and applications. In Proceedings of the International Congress of Mathematicians, Madrid, Spain, 22–30 August 2006; pp. 1541–1562. [Google Scholar]
- Stepanov, O.A.; Motorin, A.V. Performance Criteria for the Identification of Inertial Sensor Error Models. Sensors
**2019**, 19, 1997. [Google Scholar] [CrossRef] [Green Version] - Yang, X.B.; Jin, X.Q.; Du, Z.M.; Zhu, Y.H. A novel model-based fault detection method for temperature sensor using fractal correlation dimension. Build. Environ.
**2011**, 46, 970–979. [Google Scholar] [CrossRef] - Yang, L.; Sun, Z.L.; Xu, C.; Chai, X.D. The Application of Software Reliability Based on Fractal Theory in Integrated Navigation System. In Proceedings of the 2013 International Forum on Special Equipments and Engineering Mechanics, Nanjing, China, 27–28 July 2016; pp. 118–122. [Google Scholar]
- Han, T.; Yang, Y.X.; Huang, G.W. Vertical scaling optimization algorithm for a fractal interpolation model of time offsets prediction in navigation systems. Appl. Math. Model.
**2021**, 90, 862–874. [Google Scholar] [CrossRef] - Liu, J. Fractal Network Traffic Analysis with Applications. Ph.D. Thesis, Georgia Institute of Technology, Atlanta, GA, USA, 2006; 115p. [Google Scholar]
- Radev, D.; Lokshina, I.; Radeva, S. Modeling and Simulation of Self-Similar Traffic for Wireless IP Networks. In Proceedings of the 2009 Wireless Telecommunications Symposium, Prague, Czech Republic, 22–24 April 2009. [Google Scholar] [CrossRef]
- Xianglin, Z.; Jin, S.J. A Fractal-Based Flaw Feature Extraction Method for Ultrasonic Phased Array Nondestructive Testing. In Proceedings of the International Conference on Mechatronics and Automation, Changchun, China, 9–12 August 2009. [Google Scholar] [CrossRef]
- Jovančević, I.; Pham, H.H.; Orteu, J.J.; Gilblas, R.; Harvent, J.; Maurice, X.; Brèthes, L. 3D Point Cloud Analysis for Detection and Characterization of Defects on Airplane Exterior Surface. J. Nondestruct. Eval.
**2017**, 36, 1–17. [Google Scholar] [CrossRef] [Green Version] - Kolmogorov, A.N. Wienersche Spiralen und Einige Interessante Kurven im Hilbertschen Raum. C.R. (Doklady) Acad. URSS (N.S.)
**1940**, 26, 115–118. [Google Scholar] - Benoit, B.M.; Van Ness, J.W. Fractional Brownian Motions, Fractional Noises and Applications. SIAM Rev.
**1968**, 10, 422–437. [Google Scholar] - Amosov, O.S.; Amosova, S.G.; Muller, N.V. Identification of potential risks to system security using wavelet analysis, the time-and-frequency distribution indicator of the time series and the correlation analysis of wavelet-spectra. In Proceedings of the 2018 International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon), Vladivostok, Russia, 3–4 October 2018. [Google Scholar]
- Stepanov, O.A.; Motorin, A.V. Comparison of methods for identifying sensor error models based on Allan variations and nonlinear filtering algorithms. In Proceedings of the XXI St. Petersburg International Conference on Integrated Navigation Systems, St. Petersburg, Russia, 2–7 October 2018; pp. 98–103. [Google Scholar]
- Bar-Shalom, Y. Estimation with Applications to Tracking and Navigation; Bar-Shalom, Y., Li, X.-R., Kirubarajan, T., Eds.; John Wiley & Sons: New York, NY, USA, 2001; 558p. [Google Scholar]
- Gosteva, N.D.; Litvinenko, Y.A. Research of a mathematical model of care of a two-stage float gyroscope. In Proceedings of the XV Conference of Young scientists “Navigation and Motion Control”, St. Petersburg, Russia, 2–7 October 2018; pp. 117–125. [Google Scholar]
- Amosov, O.S.; Baena, S.G. Wavelet Based Filtering of Mobile Object Fractional Trajectory Parameters. In Proceedings of the IEEE International Conference on Control and Automation (IEEE ICCA 2017), Ohrid, Macedonia, 3–6 July 2017; pp. 118–123. [Google Scholar]
- Amosov, O.S.; Amosova, S.G.; Magola, D.S. Identification of information recourses threats based on intelligent technologies, fractal and wavelet analysis. In Proceedings of the 2018 IEEE International Conference on Applied System Innovation (ICASI), Chiba, Japan, 13–17 April 2018; pp. 528–531. [Google Scholar]
- Doering, E. NI myRIO Project Essentials Guide; Rose-Hulman Institute of Technology; National Technology and Science Press, 2016; p. 249. Available online: https://download.ni.com/evaluation/academic/myRIO_project_essentials_guide__Feb_09_2016___optimized.pdf (accessed on 2 September 2021).
- MEMS Motion Sensors: Ultra-Stable Three-Axis Digital Output Gyroscope L3G4200D. Available online: https://www.mouser.com/datasheet/2/389/l3g4200d-954834.pdf (accessed on 2 September 2021).
- Abry, P.; Flandrin, P.; Taqqu, M.S.; Veitch, D. Self-similarity and long-range dependence through the wavelet lens. Theory Appl. Long-Range Depend.
**2003**, 1, 527–556. [Google Scholar] - Bardet, J.-M.; Lang, G.; Oppenheim, G.; Philippe, A.; Stoev, S.; Taqqu, M.S. Semi-parametric estimation of the long-range dependence parameter: A survey. Theory Appl. Long-Range Depend.
**2003**, 557, 557–577. [Google Scholar] - Flandrin, P. Wavelet analysis and synthesis of fractional Brownian motion. IEEE Trans. Inf. Theory
**1992**, 38, 910–917. [Google Scholar] [CrossRef] - Digital Accelerometer ADXL345. Available online: https://www.analog.com/media/en/technical-documentation/data-sheets/ADXL345.pdf (accessed on 2 September 2021).
- Kuzmin, S.Z. Digital Radar. Introduction to the Theory; KVITS Publishing House: Kiev, Ukraine, 2000; 428p. [Google Scholar]
- Amosov, O.S. Markov sequence filtering on the basis of bayesian and neural network approaches and fuzzy logic systems in navigation data processing. J. Comput. Syst. Sci. Int.
**2004**, 43, 551–559. [Google Scholar] - Amosov, O.S. Peculiarities of stochastic processes with fractal properties and their applications in problems of navigation information processing. In Proceedings of the 25th Saint Petersburg International Conference on Integrated Navigation Systems, ICINS 2018, Saint Petersburg, Russia, 28–30 May 2018; Volume 137790, pp. 1–5. [Google Scholar]
- Stepanov, O.A.; Amosov, O.S. Nonrecurrent Linear Estimation and Neural Networks. IFAC Proc. Vol.
**2004**, 37, 213–218. [Google Scholar] [CrossRef] - Lv, X.; Duan, F.; Jiang, J.J.; Fu, X.; Gan, L. Deep Metallic Surface Defect Detection: The New Benchmark and Detection Network. Sensors
**2020**, 20, 1562. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chu, H.-H.; Wang, Z.-Y. A vision-based system for post-welding quality measurement and defect detection. Int. J. Adv. Manuf. Technol.
**2016**, 86, 3007–3014. [Google Scholar] [CrossRef] - Fan, Y.; Park, U.; Udpa, L.; Ramuhalli, P.; Shih, W.; Stockman, G.C. Automated rivet inspection for aging aircraft with magneto-optic imager. Electromagn. Nondestruct. Eval. (IX)
**2005**, 25, 185–194. [Google Scholar] - Lingvall, F.; Stepinski, T. Automatic detecting and classifying defects during eddy current inspection of riveted lap-joints. NDT&E Int.
**2000**, 33, 47–55. [Google Scholar] - Amosov, O.S.; Amosova, S.G.; Iochkov, I.O. Defects Detection and Recognition in Aviation Riveted Joints by Using Ultrasonic Echo Signals of Non-Destructive Testing. In Proceedings of the 17th IFAC Symposium on Information Control Problems in Manufacturing, IFAC INCOM 2021, Budapest, Hungary, 7–9 June 2021; pp. 484–489. [Google Scholar]

**Figure 2.**Gyroscope drift according to X coordinate (

**a**) and the Hurst parameter with the gyroscope drift for 1 h (

**b**).

**Figure 4.**R.m.s. of estimation errors with different Hurst parameters (H.). (

**a**) H = 0.1, (

**b**) H = 0.5, (

**c**) H = 0.9.

**Figure 5.**Scheme of the organization’s telecommunication system with incoming and outgoing Internet traffic.

**Figure 6.**Characteristics of information and telecommunication systems: (

**a**,

**c**) show the analysis of 2-month, and weekly, series of intensity, respectively, in bytes; (

**b**,

**d**)—in packages.

**Figure 7.**Echo signals of ultrasonic flaw detector during non-destructive testing of rivet joints: (

**a**) is a normal rivet; (

**b**) defect in the middle of the rivet at the 80% undercut; (

**c**) defect in the middle of the rivet at the 50% undercut; (

**d**) defect at the rivet cap base at the 50% undercut.

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

© 2021 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**

Amosov, O.S.; Amosova, S.G.
Peculiarities and Applications of Stochastic Processes with Fractal Properties. *Sensors* **2021**, *21*, 5960.
https://doi.org/10.3390/s21175960

**AMA Style**

Amosov OS, Amosova SG.
Peculiarities and Applications of Stochastic Processes with Fractal Properties. *Sensors*. 2021; 21(17):5960.
https://doi.org/10.3390/s21175960

**Chicago/Turabian Style**

Amosov, Oleg Semenovich, and Svetlana Gennadievna Amosova.
2021. "Peculiarities and Applications of Stochastic Processes with Fractal Properties" *Sensors* 21, no. 17: 5960.
https://doi.org/10.3390/s21175960