# Theoretical Framework for Determination of Linear Structures in Multidimensional Geodynamic Data Arrays

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## Abstract

**:**

## 1. Introduction

## 2. Review

## 3. Materials and Methods

#### 3.1. Discrete Perfect Sets

**Definition**

**1.**

**Statment**

**1.**

**Definition**

**2.**

**Statment**

**2.**

**Consequence**

**1.**

**Statement**

**3.**

**Remark**

**1.**

**Definition**

**3.**

**Statement**

**4.**

- 1.
- If$A\subseteq B$, then$A\left(\alpha \right)\subseteq B\left(\alpha \right)$.
- 2.
- If P, Q densities on X and${P}_{A}\left(x\right)\le {Q}_{A}\left(x\right)\phantom{\rule{4pt}{0ex}}\forall x\in X,A\subseteq B$, then${A}_{P}\left(\alpha \right)\le {A}_{Q}\left(\alpha \right)$.
- 3.
- If$\alpha <\beta $, then$A\left(\beta \right)\subseteq A\left(\alpha \right)$.
- 4.
- If$A\subseteq X\subset Y$and measure P are set to Y, then$A\left(\alpha \right|X)\subseteq A(\alpha \left|Y\right)$.

#### 3.2. Complete $\mathbb{DPS}$: Scheme and Algorithms

**Definition**

**4.**

**Remark**

**2.**

- increasing$${A}^{n}\uparrow {A}^{\infty}\leftrightarrow A\subset {A}^{1}\subset \dots ={A}^{\infty}$$
- and decreasing$${A}^{\infty}\downarrow A\left(\alpha \right)\leftrightarrow {A}^{\infty}\supset {A}^{2\infty}\supset \dots =A\left(\alpha \right)$$

#### 3.3. Simple DPS: Scheme and Algorithms

**Remark**

**3.**

**Definition**

**5.**

**Statement**

**5.**

#### 3.3.1. Topological Retreat

**Statement**

**6.**

**Definition**

**6.**

#### 3.3.2. Parameter Selection: Localization Radius r

#### 3.3.3. Parameter Selection: Density Level α

#### 3.3.4. Quality Criterion

#### 3.4. Density

**Definition**

**7.**

**Statement**

**7.**

- 1.
- If P and Q are densities on X and$R=R({y}_{1},{y}_{2}):[-1,1]\times [-1,1]\to [-1,1]$nondecreasing mapping, then superposition$$R{(P,Q)}_{A}\left(x\right)=R({P}_{A}\left(x\right),{Q}_{A}\left(x\right))$$
- 2.
- If ¬ fuzzy negation on$[-1,1]$, then the superposition$$\neg {P}_{A}\left(x\right)=\neg \left({P}_{\overline{A}}\left(x\right)\right)$$
- 3.
- If n is a fuzzy comparison on${\mathbb{R}}^{+}$, then the superposition$$C{(P,Q)}_{A}\left(x\right)=n({Q}_{\overline{A}}\left(x\right),{P}_{A}\left(x\right))$$

**Consequence**

**2.**

- 1.
- R-connection P and$\neg P$will be the density on X$$R{(P,\neg P)}_{A}\left(x\right)=R({P}_{A}\left(x\right),\neg {P}_{A}\left(x\right))$$
- 2.
- If ⊤ (⊥, ${M}_{p}$) is t-norm (t-co-norm, generalized averaging operator) [7], then superpositions $\top ({P}_{A}\left(x\right),{Q}_{A}\left(x\right))$, $\perp ({P}_{A}\left(x\right),{Q}_{A}\left(x\right))$, ${M}_{p}({P}_{A}\left(x\right),{Q}_{A}\left(x\right))$ will be densities on X.
- 3.
- If$\lambda \in [0,1]$, then$\lambda $-connection$\lambda {P}_{A}\left(x\right)+(1-\lambda ){Q}_{A}\left(x\right)$will be density on X.
- 4.
- A “fuzzy comparison”$CP$will be the density on X:$$C{P}_{A}\left(x\right)=C{(P,P)}_{A}\left(x\right)=n({P}_{\overline{A}}\left(x\right),{P}_{A}\left(x\right))$$

#### 3.4.1. The Logical Densities Calculus

**Remark**

**4.**

**Example**

**1.**

- 1.
- Scheme$\neg DPS:P\to DPS(\neg P)$
- 2.
- Scheme$CDPS:P\to DPS\left(CP\right)$
- 3.
- Scheme$(\lambda ,1-\lambda )DPS:P\to DPS(\lambda P+(1-\lambda )\neg P)$

**Remark**

**5.**

## 4. Results

#### 4.1. SDPS Algorithm

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

#### 4.2. MDPS Algorithm

**Example**

**7.**

- 1.
- Suppose that X is a uniform finite grid. The nodes at the edge X have a lower density S than the central nodes, although space X looks equally homogenous in both cases.
- 2.
- If in a full-sphere${D}_{A}(x,r)$all points other than x, are concentrated on the circle${C}_{A}(x,r)$and there are many of them, then the density${S}_{A}\left(x\right)$is significant, regardless ther-isolation x.

**Remark**

**6.**

#### 4.3. FDPS Algorithm

**Remark**

**7.**

**Example**

**9.**

#### 4.4. GDPS Gluing: Scheme and Algorithms

**Task**

**1.**

- as a schema if we are talking about a dependency given above;
- as the GDPS algorithm when it comes to the operation of the DPS scheme (Definition 6–Section 3.3.3) on density $G\left(P\right)$ with parameter $\gamma $: $\mathrm{GDPS}=\mathrm{DPS}(G,\gamma )$. Its $\mathrm{GDPS}\left(Y\right)$ result solves problem Task 1 by “gluing” local data ${U}_{\alpha}$ in a certain way. This explains the name of the algorithm (gluing).

#### 4.5. LDPS Algorithm

#### 4.5.1. Initial Data and Designations

- $xOy$—fixed orthogonal coordinate system on $\Pi $,
- ${x}_{\phi}{O}^{*}{y}_{\phi}$—loose orthogonal coordinate system on $\Pi $, obtained by moving coordinate origin O to point ${O}^{*}=({x}^{*},{y}^{*})$ and turning the axes ${x}_{\phi},{y}_{\phi}$ by the angle $\phi \in [0,\pi ]$,
- Relation of coordinates$$\begin{array}{c}{x}_{\phi}=\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}cos\phi (x-{x}^{*})+sin\phi (y-{y}^{*})\hfill \\ {y}_{\phi}=-sin\phi (x-{x}^{*})+cos\phi (y-{y}^{*})\hfill \end{array},$$
- Q is a finite-state array in $\Pi $: $Q=\left\{q\right\}=\left(\right)open="\{"\; close="\}">{q}_{i}{|}_{i=1}^{N}$,
- $\mathfrak{L}$—the property of local linearity in Q.

#### 4.5.2. Quantification $\mathfrak{L}$

#### 4.5.3. Search for Global Linear Structures

- the first of them with the selected parameters of local linearity r and h constructs its quantification ${L}_{Q}\left(q\right|r,h)$ at each point q of space Q, and the best corridor ${K}_{Q}\left(q\right|r,h)$, where the estimate ${L}_{Q}\left(q\right|r,h)$ is reached;
- the second stage includes constructing the basis for application of the GDPS scheme, namely coverage ${U}_{\alpha}\left(L\right)=\{{K}_{Q}\left(q\right|r,h):{L}_{Q}\left(q\right|r,h)\ge \alpha \}$ for a given level of local linearity $\alpha $;
- the third stage is GDPS scheme working on ${U}_{\alpha}\left(L\right)$ data. Its result will represent linear structures in Q. On space$$Y=Supp{U}_{\alpha}\left(L\right)=\{q\in Q:{L}_{Q}\left(q\right|r,h)\ge \alpha \}$$$${G}_{B}\left(y\right)=\frac{|{D}_{B}(y,r)\cap {K}_{Q}\left(q\right|r,h)|}{|{K}_{Q}\left(q\right|r,h)|}.$$
- the fourth stage is their filtering by solidity using the MDPS algorithm with a level of $\epsilon $.

**Example**

**10.**

- At the first stage with the chosen parameters at each blue point q a linear corridor${K}_{Q}\left(q\right|r,h)$is constructed with parameters$r=1.09$and$h=0.44$, then its separability${L}_{Q}\left(q\right|r,h)$is calculated. Figure 12b,c show corridors in green with centers at black points. Their separability equals to$0.64$ and $0.5$, respectively. In the second instance, it proves to be insufficient to overcome the second stage.
- Second stage. The separability level α is assumed to be$0.6$. Figure 13a shows the points y in red, that passed this selection and formed the basis Y—the first half for application of the GDPS scheme. The second half GDPS scheme: corridors${K}_{Q}\left(q\right|r,h)$are shown for the already familiar point on the left (Figure 13b) and at some point on the right (Figure 13c). It can be seen from the figures that the relative density of red dots in the left corridor is higher than in the right one. This circumstance will help the left point overcome the third stage and move into the lower linear structure, while the right point will not stand the test with GDPS operation and will not be included in the final result.
- Third scheme. The GDPS scheme operation on the red points from the Y. Its result Z is shown in Figure 14a. It needs to be filtered.

## 5. Discussion

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

URL | Underground research laboratory |

HLRW | High-level radioactive waste |

DMA | Discrete mathematical analysis |

DPS | Discrete perfect sets |

FMS | Finite metric space |

## References

- Gvishiani, A.D.; Kaftan, V.I.; Krasnoperov, R.I.; Tatarinov, V.N.; Vavilin, E.V. Geoinformatics and systems analysis in geophysics and geodynamics. Phys. Earth
**2019**, 1, 42–60. [Google Scholar] [CrossRef] - Laverov, N.P.; Omelyanenko, B.I.; Velichkin, V.I. Geological aspects of the problem of radioactive waste disposal. Geoecology
**1994**, 6, 3–20. (In Russian) [Google Scholar] - Dzeboev, B.A.; Karapetyan, J.K.; Aronov, G.A.; Dzeranov, B.V.; Kudin, D.V.; Karapetyan, R.K.; Vavilin, E.V. FCAZ-recognition based on declustered earthquake catalogs. Russ. J. Earth. Sci.
**2020**, 20, ES6010. [Google Scholar] [CrossRef] - Gorshkov, A.I.; Soloviev, A.A. Recognition of earthquake-prone areas in the Altai-Sayan-Baikal region based on the morphostructural zoning. Russ. J. Earth. Sci.
**2021**, 21, ES1005. [Google Scholar] [CrossRef] - Belov, S.V.; Gvishiani, A.D.; Kamnev, E.N.; Morozov, V.N.; Tatarinov, V.N. Development of complex model of evolution of structural-tectonic blocks of the Earth’s crust for choosing storage sites of high level radioactive waste. Russ. J. Earth. Sci.
**2008**, 10, ES4004. [Google Scholar] [CrossRef] [Green Version] - Zadeh, L. The concept of a linguistic variable and its application to approximate reasoning. Inf. Sci.
**1975**, 8, 199–249. [Google Scholar] [CrossRef] - Averkin, A.N.; Batyrshin, I.Z.; Blishun, A.F.; Silov, V.B.; Tarasov, V.B. Fuzzy Sets in Models of Control and Artificial Intelligence; Publ. Nauka: Moscow, Russia, 1986; 312p. (In Russian) [Google Scholar]
- Agayan, S.M.; Bogoutdinov, S.R.; Krasnoperov, R.I. Short introduction into DMA. Russ. J. Earth Sci.
**2018**, 18, ES2001. [Google Scholar] [CrossRef] [Green Version] - Gvishiani, A.D.; Dzeboev, B.A.; Agayan, S.M. FCAZm intelligent recognition system for locating areas prone to strong earthquakes in the Andean and Caucasian mountain belts. Izv. Phys. Solid Earth
**2016**, 52, 461–491. [Google Scholar] [CrossRef] - Widiwijayanti, C.; Mikhailov, V.; Diament, M.; Deplus, C.; Louat, R.; Tikhotsky, S.; Gvishiani, A. Structure and evolution of the Molucca Sea area: Constraints based on interpretation of a combined sea-surface and satellite gravity dataset. Earth Planet. Sci. Lett.
**2003**, 215, 135–150. [Google Scholar] [CrossRef] - Gvishiani, A.; Soloviev, A.; Krasnoperov, R.; Lukianova, R. Automated Hardware and Software System for Monitoring the Earth’s Magnetic Environment. Data Sci. J.
**2016**, 15, 18. [Google Scholar] [CrossRef] - Agayan, S.M.; Bogoutdinov, S.R.; Dobrovolsky, M.N. On one algorithm for searching the dense areas and its geophysical applications. In Proceedings of the Materials of 15th Russian National Workshop “Mathematical Methods of Pattern Recognition, MMRO-15”, Petrozavodsk, Russia, 11–17 September 2011; Maks Press: Moscow, Russia, 2011; pp. 543–546. (In Russian). [Google Scholar]
- Agayan, S.M.; Bogoutdinov, S.R.; Dobrovolsky, M.N. Discrete Perfect Sets and Their Application in Cluster Analysis. Cybern. Syst. Anal.
**2014**, 50, 176–190. [Google Scholar] [CrossRef] - Everitt, B.S. Cluster Analysis; Halsted-Heinemann: London, UK, 1980; 170p. [Google Scholar]
- Dzeboev, B.A.; Gvishiani, A.D.; Agayan, S.M.; Belov, I.O.; Karapetyan, J.K.; Dzeranov, B.V.; Barykina, Y.V. System-Analytical Method of Earthquake-Prone Areas Recognition. Appl. Sci.
**2021**, 11, 7972. [Google Scholar] [CrossRef] - Mandel, I.D. Cluster Analysis; Publ. Finansy i Statistika: Moscow, Russia, 1988; 176p. (In Russian) [Google Scholar]
- Mark, S.A.; Roger, K.B. Cluster Analysis (Quantitative Applications in the Social Sciences); SAGE Publications, Inc.: Newbury Park, CA, USA, 1984; 88p. [Google Scholar]
- Ester, M.; Kriegel, H.-P.; Sander, J.; Xu, X. A density-based algorithm for discovering clusters in large spatial databases with noise. In Proceedings of the Second International Conference on Knowledge Discovery and Data Mining (KDD-96), Portland, OR, USA, 2–4 August 1996; Simoudis, E., Han, J., Fayyad, U.M., Eds.; AAAI Press: Palo Alto, CA, USA, 1996; pp. 226–231. [Google Scholar]
- Ankerst, M.; Breunig, M.; Kriegel, H.-P.; Sander, J. OPTICS: Ordering Points To Identify the Clustering Structure. In Proceedings of the ACM SIGMOD International Conference on Management of Data, Philadelphia, PA, USA, 31 May–3 June 1999; ACM Press: New York, NY, USA, 1999; pp. 49–60. [Google Scholar] [CrossRef]
- Bojchevski, A.; Matkov, Y.; Günnemann, S. Robust Spectral Clustering for Noisy Data: Modeling Sparse Corruptions Improves Latent Embeddings. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD-17), Halifax, NS, Canada, 13–17 August 2017; Association for Computing Machinery: New York, NY, USA, 2017; pp. 737–746. [Google Scholar] [CrossRef]
- Gvishiani, A.D.; Agayan, S.M.; Bogoutdinov, S.R.; Soloviev, A.A. Discrete mathematical analysis and geological and geophysical applications. Bull. Earth Sci.
**2010**, 2, 109–125. (In Russian) [Google Scholar] - Mikhailov, V.; Galdeano, A.; Diament, M.; Gvishiani, A.; Agayan, S.; Bogoutdinov, S.; Graeva, E.; Sailhac, P. Application of artificial intelligence for Euler solutions clustering. Geophysics
**2003**, 68, 168–180. [Google Scholar] [CrossRef] [Green Version] - Agayan, S.M.; Soloviev, A.A. Highlight dense areas in metric spaces based on crystallization. Syst. Res. Inf. Technol.
**2004**, 2, 7–23. (In Russian) [Google Scholar] - Agayan, S.M.; Tatarinov, V.N.; Gvishiani, A.D.; Bogoutdinov, S.R.; Belov, I.O. Strong-Earthquake-Prone Areas FDPS algorithm in stability assessment of the Earth’s crust structural tectonic blocks. Russ. J. Earth Sci.
**2020**, 20, ES1005. [Google Scholar] [CrossRef] - Agayan, S.; Bogoutdinov, S.; Soloviev, A.; Sidorov, R. The Study of Time Series Using the DMA Methods and Geophysical Applications. Data Sci. J.
**2016**, 15, 16. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The concept of density relative to a set. A—a set of red dots, in which four points a, b, c and d are highlighted. The density S of the subset A at the selected points is equal to the number of red points included in the balls described around them. The densest point relative to A will be point c, followed by points b, d and a.

**Figure 2.**Application of the SDPS algorithm to the array X (

**a**). Four iterations are shown in figures (

**b**–

**e**). The result is a local $\alpha $-perfect subset of $X\left(\alpha \right)$ in X (

**e**). The green points in figures (

**b**–

**d**) show the points that did not pass the next iteration in SDPS. SDPS further splits $X\left(\alpha \right)$ into connected components ((

**f**), yellow and black subsets).

**Figure 3.**Examples of clustering a complex array (

**a**) using algorithms: MDPS (

**b**), DBSCAN (

**c**) and OPTICS (

**d**).

**Figure 5.**Application of the SDPS algorithm ($q=-2$, $\beta =-0.3$, $p=0$): (

**a**) the original array; (

**b**) the result of the first iteration ${X}_{1}(-0.3)$ containing isolated points; and (

**c**) the final result of applying SDPS.

**Figure 6.**Inverse dependence of the result of the SDPS algorithm on the parameter $\beta $: (

**a**) the result of the SDPS algorithm at $\beta =-0.35$; (

**b**) the result of work at $\beta =-0.15$; and (

**c**) the result of work at $\beta =0.05$. (In all cases $q=-2$, $p=0$).

**Figure 7.**Dependence of the result of the SDPS algorithm on the parameter q: (

**a**) the result of the SDPS algorithm at $q=-2$; (

**b**) the result of work at $q=-2.8$; and (

**c**) the result of work for $q=-3.5$ (in all cases $\beta =-0.2$, $p=0$).

**Figure 8.**The inverse nature of the dependence of the SDPS algorithm on the parameter p: (

**a**) the result of the SDPS algorithm at $p=0$; (

**b**) the result of work at $p=0.5$; and (

**c**) the result of work for $p=1$ (in all cases $q=-2$, $\beta =-0.5$).

**Figure 9.**Illustration from left to right of the quality of the SDPS algorithm at $\beta =-0.3$; $0.1$; $0.3$. Clustering in figure (

**a**) is clearly better, and in figures (

**b**,

**c**) is approximately the same. (As a fuzzy comparison, we used $n(a,b)=(b-a)/(b+1)$.

**Figure 10.**An illustration of the independence of the MDPS and SDPS algorithms: the “dumbbell” in the original array (

**a**) has a sparse handle, which MDPS highlights cleanly (

**c**); SDPS cannot do this (

**b**).

**Figure 11.**Operation of the FDPS algorithm on a regular grid (

**a**). The FDPS algorithm results in two red lines on the horizontal axis, which serve as the bases of the two most significant stochastic heights. Figure (

**b**) shows a classic choice with respect to a given level, highlighting many weak heights.

**Figure 12.**The first stage of the LDPS algorithm. (

**a**) original array; (

**b**) a corridor with a separability equal to $0.64$; and (

**c**) a corridor with a separation of 0.5. The position of the fragments of the original array shown in figures (

**b**,

**c**) is indicated by the leaders in figure (

**a**).

**Figure 13.**The second stage of the algorithm. (

**a**) The original array with the selected base (red dots); (

**b**) points from in the corridor from Figure 12b; and (

**c**) points from in the corridor from Figure 12c. The position of the fragments of the original array shown in figures (

**b**), (

**c**) is indicated by the leaders in figure (

**a**).

**Figure 14.**Result of the third and fourth stages of the LDPS algorithm. (

**a**) The result of the third stage–the work of the GDPS algorithm; (

**b**) the result of the fourth stage–the work of the MDPS algorithm; (

**c**) the result of the DBSCAN algorithm; and (

**d**) the result of the OPTICS algorithm.

**Figure 15.**Operation of the LDPS algorithm. (

**a**) The original array. (

**b**) The linear structure indicated by red dots, built by the LDPS algorithm.

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## Share and Cite

**MDPI and ACS Style**

Agayan, S.; Bogoutdinov, S.; Kamaev, D.; Kaftan, V.; Osipov, M.; Tatarinov, V.
Theoretical Framework for Determination of Linear Structures in Multidimensional Geodynamic Data Arrays. *Appl. Sci.* **2021**, *11*, 11606.
https://doi.org/10.3390/app112411606

**AMA Style**

Agayan S, Bogoutdinov S, Kamaev D, Kaftan V, Osipov M, Tatarinov V.
Theoretical Framework for Determination of Linear Structures in Multidimensional Geodynamic Data Arrays. *Applied Sciences*. 2021; 11(24):11606.
https://doi.org/10.3390/app112411606

**Chicago/Turabian Style**

Agayan, Sergey, Shamil Bogoutdinov, Dmitriy Kamaev, Vladimir Kaftan, Maxim Osipov, and Victor Tatarinov.
2021. "Theoretical Framework for Determination of Linear Structures in Multidimensional Geodynamic Data Arrays" *Applied Sciences* 11, no. 24: 11606.
https://doi.org/10.3390/app112411606