In the present thesis work, proposed declustering models are unsupervised in nature which do not have any target and label information in the catalog. They are blindly applied for clustering and classifying the earthquake events according to the features present in the catalog itself. After label assignment based on the clustering task, supervised learning can
7.2 Future works 133 be implemented in the future with the effective use of artificial neural network techniques.
In the future, intelligence and soft computing systems can be implemented in the spatio- temporal framework which are capable of automatically assigning earthquakes to relevant seismic sources. Many and muti-optimization techniques will be another focus of research in the future for simultaneously optimize the involved parameters by correlating the resulted clusters with known faults. It will be a better choice if more features can be explored like soil behavior, type of the fault and climate change, etc. for recognizing the better pattern or cluster belonging to a seismic region. Declustering seismicity models may be helpful and can be used further for assessment of future hazard analysis, estimation of the occurrence time of large earthquakes and development of predictive models for local and global seismic sources.
Bibliography
[1] S. Nanda, K. Tiampo, G. Panda, L. Mansinha, N. Cho, and A. Mignan, “A tri-stage cluster identification model for accurate analysis of seismic catalogs,” Nonlinear Processes in Geophysics, vol. 20, no. 1, pp. 143–162, 2013.
[2] M. Ester, H.-P. Kriegel, J. Sander, and X. Xu, “A density-based algorithm for discover- ing clusters in large spatial databases with noise,” inKdd, vol. 96, no. 34, 1996, pp.
226–231.
[3] A. Rodriguez and A. Laio, “Clustering by fast search and find of density peaks,”
Science, vol. 344, no. 6191, pp. 1492–1496, 2014.
[4] NCEDC, “Northern California Earthquake Data Center,”UC Berkeley Seismological Laboratory (ANSS Dataset), 2017.
[5] M. Båth, “Lateral inhomogeneities of the upper mantle,”Tectonophysics, vol. 2, no. 6, pp. 483–514, 1965.
[6] B. Gutenberg, “The energy of earthquakes,” Quarterly Journal of the Geological Society, vol. 112, no. 1-4, pp. 1–14, 1956.
[7] F. Omori,On the after-shocks of earthquakes. The University, 1894, vol. 7,1894.
[8] T. Utsu, “Magnitude of earthquakes and occurrence of their aftershocks,”Zisin, vol. 10, pp. 35–45, 1957.
[9] T. Utsu and Y. Ogata, “The centenary of the omori formula for a decay law of aftershock activity,”Journal of Physics of the Earth, vol. 43, no. 1, pp. 1–33, 1995.
[10] P. A. Reasenberg and L. M. Jones, “Earthquake hazard after a mainshock in California,”
Science, vol. 243, no. 4895, pp. 1173–1176, 1989.
[11] P. Reasenberg and L. Jones, “Earthquake aftershocks: update,”Science, vol. 265, no.
5176, pp. 1251–1252, 1994.
[13] L. Knopoff and J. Gardner, “Higher seismic activity during local night on the raw worldwide earthquake catalogue,” Geophysical Journal of the Royal Astronomical Society, vol. 28, no. 3, pp. 311–313, 1972.
[14] J. Gardner and L. Knopoff, “Is the sequence of earthquakes in southern california, with aftershocks removed, poissonian?”Bulletin of the Seismological Society of America, vol. 64, no. 5, pp. 1363–1367, 1974.
[15] T. van Stiphout, J. Zhuang, and D. Marsan, “Seismicity declustering,”Community Online Resource for Statistical Seismicity Analysis, vol. 10, pp. 1–25, 2012.
[16] R. Uhrhammer, “Characteristics of northern and central California seismicity,”Earth- quake Notes, vol. 57, no. 1, p. 21, 1986.
[17] P. Reasenberg, “Second-order moment of central California seismicity, 1969–1982,”
Journal of Geophysical Research: Solid Earth, vol. 90, no. B7, pp. 5479–5495, 1985.
[18] H. Kanamori and D. L. Anderson, “Theoretical basis of some empirical relations in seismology,”Bulletin of the seismological society of America, vol. 65, no. 5, pp.
1073–1095, 1975.
[19] J. Zhuang, Y. Ogata, and D. Vere-Jones, “Stochastic declustering of space-time earth- quake occurrences,”Journal of the American Statistical Association, vol. 97, no. 458, pp. 369–380, 2002.
[20] Y. Ogata, “Statistical models for earthquake occurrences and residual analysis for point processes,”Journal of the American Statistical association, vol. 83, no. 401, pp.
9–27, 1988.
[21] Y. Ogata and J. Zhuang, “Space–time ETAS models and an improved extension,”
Tectonophysics, vol. 413, no. 1, pp. 13–23, 2006.
[22] D. Marsan and O. Lengline, “Extending earthquakes’ reach through cascading,”Sci- ence, vol. 319, no. 5866, pp. 1076–1079, 2008.
[23] A. K. Jain, “Data clustering: 50 years beyond k-means,”Pattern recognition letters, vol. 31, no. 8, pp. 651–666, 2010.
[24] G. Weatherill and P. W. Burton, “Delineation of shallow seismic source zones using k-means cluster analysis, with application to the Aegean region,”Geophysical Journal International, vol. 176, no. 2, pp. 565–588, 2009.
[25] K. Rehman, P. W. Burton, and G. A. Weatherill, “K-means cluster analysis and seismicity partitioning for Pakistan,”Journal of seismology, vol. 18, no. 3, pp. 401–
419, 2014.
[26] A. Morales-Esteban, F. Martínez-Álvarez, A. Troncoso, J. Justo, and C. Rubio- Escudero, “Pattern recognition to forecast seismic time series,”Expert Systems with Applications, vol. 37, no. 12, pp. 8333–8342, 2010.
[27] A. Morales-Esteban, F. Martínez-Álvarez, S. Scitovski, and R. Scitovski, “A fast partitioning algorithm using adaptive Mahalanobis clustering with application to seismic zoning,”Computers & Geosciences, vol. 73, pp. 132–141, 2014.
[28] J. C. Bezdek, R. Ehrlich, and W. Full, “FCM: The fuzzy c-means clustering algorithm,”
Computers & Geosciences, vol. 10, no. 2-3, pp. 191–203, 1984.
[29] A. Ansari, A. Noorzad, and H. Zafarani, “Clustering analysis of the seismic catalog of Iran,”Computers & Geosciences, vol. 35, no. 3, pp. 475–486, 2009.
[30] I. Gath and A. B. Geva, “Unsupervised optimal fuzzy clustering,”IEEE Transactions on pattern analysis and machine intelligence, vol. 11, no. 7, pp. 773–780, 1989.
[31] F. Wang, Y. Wan, H. Cao, Z. Jin, and Q. Ren, “Application of fuzzy clustering method to determining sub-fault planes of earthquake from aftershocks sequence,”Earthquake Science, vol. 25, no. 2, pp. 187–196, 2012.
[32] A. Gvishiani, M. Dobrovolsky, S. Agayan, and B. Dzeboev, “Fuzzy-based clustering of epicenters and strong earthquake-prone areas.” Environmental Engineering &
Management Journal (EEMJ), vol. 12, no. 1, 2013.
[33] A. Gorshkov, V. Kossobokov, E. Y. Rantsman, and A. Soloviev, “Recognition of earthquake-prone areas: Validity of results obtained from 1972 to 2000,”Computa- tional Seismology and Geodynamics, vol. 7, pp. 37–44, 2005.
[34] S. J. Nanda and G. Panda, “Design of computationally efficient density-based cluster- ing algorithms,”Data & Knowledge Engineering, vol. 95, pp. 23–38, 2015.
tering,”Expert Systems with Applications, vol. 40, no. 10, pp. 4183–4189, 2013.
[36] S. Scitovski, “A density-based clustering algorithm for earthquake zoning,”Computers
& Geosciences, vol. 110, pp. 90 – 95, 2018.
[37] M. Du, S. Ding, and H. Jia, “Study on density peaks clustering based on k-nearest neighbors and principal component analysis,”Knowledge-Based Systems, vol. 99, pp.
135–145, 2016.
[38] J. Xie, H. Gao, W. Xie, X. Liu, and P. W. Grant, “Robust clustering by detecting density peaks and assigning points based on fuzzy weighted k-nearest neighbors,”
Information Sciences, vol. 354, pp. 19–40, 2016.
[39] D. Thirumalai and R. D. Mountain, “Activated dynamics, loss of ergodicity, and transport in supercooled liquids,” Physical Review E, vol. 47, no. 1, pp. 479–489, 1993.
[40] K. Tiampo, J. Rundle, W. Klein, J. S. Martins, and C. Ferguson, “Ergodic dynamics in a natural threshold system,”Physical review letters, vol. 91, no. 23, pp. 238 501(1–4), 2003.
[41] K. Tiampo, J. Rundle, W. Klein, J. Holliday, J. S. Martins, and C. Ferguson, “Er- godicity in natural earthquake fault networks,”Physical Review E, vol. 75, no. 6, pp.
066 107(1–8), 2007.
[42] R. Clausius, “On the motive power of heat, and on the laws which can be deduced from it for the theory of heat (translated from German by Magie, WF),”The second law of thermodynamics: Memoirs by Carnot, Clausius and Thomson, 1850.
[43] A. Ben-Naim,A Farewell to Entropy: Statistical Thermodynamics Based on Informa- tion: S. World Scientific, 2008.
[44] C. E. Shannon, “A mathematical theory of communication,” Bell system technical journal, vol. 27, no. 3, pp. 379–423, 1948.
[45] T. Nicholson, M. Sambridge, and O. Gudmundsson, “On entropy and clustering in earthquake hypocentre distributions,”Geophysical Journal International, vol. 142, no. 1, pp. 37–51, 2000.
[46] C. Goltz and M. Böse, “Configurational entropy of critical earthquake populations,”
Geophysical Research Letters, vol. 29, no. 20, pp. 51(1–4), 2002.
[47] C. Frohlich and S. Davis, “Identification of aftershocks of deep earthquakes by a new ratios method,”Geophysical Research Letters, vol. 12, no. 10, pp. 713–716, 1985.
[48] A. Corral, “Long-term clustering, scaling, and universality in the temporal occurrence of earthquakes,”Physical Review Letters, vol. 92, no. 10, pp. 108 501(1–4), 2004.
[49] S. Hainzl, F. Scherbaum, and C. Beauval, “Estimating background activity based on interevent-time distribution,”Bulletin of the Seismological Society of America, vol. 96, no. 1, pp. 313–320, 2006.
[50] M. Bottiglieri, E. Lippiello, C. Godano, and L. De Arcangelis, “Identification and spatiotemporal organization of aftershocks,”Journal of Geophysical Research: Solid Earth, vol. 114, no. B03303, pp. 1–12, 2009.
[51] R. Batac and H. Kantz, “Observing spatio-temporal clustering and separation using interevent distributions of regional earthquakes,”Nonlinear Processes in Geophysics, vol. 21, no. 4, pp. 735–744, 2014.
[52] Z. Wu, “A hidden markov model for earthquake declustering,”Journal of Geophysical Research: Solid Earth, vol. 115, no. B03306, pp. 1–11, 2010.
[53] J. E. Ebel, D. W. Chambers, A. L. Kafka, and J. A. Baglivo, “Non-poissonian earth- quake clustering and the hidden markov model as bases for earthquake forecasting in california,”Seismological Research Letters, vol. 78, no. 1, pp. 57–65, 2007.
[54] J. Wang, I. G. Main, and R. M. W. Musson, “Earthquake clustering in modern seis- micity and its relationship with strong historical earthquakes around beijing, china,”
Geophysical Journal International, vol. 211, no. 2, pp. 1005–1018, 2017.
[55] T. J. Ader and J.-P. Avouac, “Detecting periodicities and declustering in earthquake catalogs using the schuster spectrum, application to himalayan seismicity,”Earth and Planetary Science Letters, vol. 377, pp. 97–105, 2013.
[56] J. Golay, M. Kanevski, C. D. V. Orozco, and M. Leuenberger, “The multipoint Morisita index for the analysis of spatial patterns,”Physica A: Statistical Mechanics and its Applications, vol. 406, pp. 191–202, 2014.
40–47, 2015.
[58] M. Baiesi and M. Paczuski, “Scale-free networks of earthquakes and aftershocks,”
Physical review E, vol. 69, no. 6, pp. 066 106(1–8), 2004.
[59] I. Zaliapin, A. Gabrielov, V. Keilis-Borok, and H. Wong, “Clustering analysis of seismicity and aftershock identification,”Physical Review Letters, vol. 101, no. 1, pp.
018 501(1–4), 2008.
[60] I. Zaliapin and Y. Ben-Zion, “Earthquake clusters in southern California I: Identifica- tion and stability,”Journal of Geophysical Research: Solid Earth, vol. 118, no. 6, pp.
2847–2864, 2013.
[61] C. Frohlich and S. D. Davis, “Single-link cluster analysis as a method to evaluate spatial and temporal properties of earthquake catalogues,”Geophysical Journal Inter- national, vol. 100, no. 1, pp. 19–32, 1990.
[62] L. Telesca, V. Lapenna, and N. Alexis, “Multiresolution wavelet analysis of earth- quakes,”Chaos, Solitons & Fractals, vol. 22, no. 3, pp. 741–748, 2004.
[63] I. A. Sarafis, P. W. Trinder, and A. M. Zalzala, “NOCEA: A rule-based evolutionary algorithm for efficient and effective clustering of massive high-dimensional databases,”
Applied Soft Computing, vol. 7, no. 3, pp. 668–710, 2007.
[64] R. Xu and D. C. Wunsch, “Survey of clustering algorithms,” 2005.
[65] S. J. Nanda and G. Panda, “A survey on nature inspired metaheuristic algorithms for partitional clustering,”Swarm and Evolutionary computation, vol. 16, pp. 1–18, 2014.
[66] Z. Huang, “Extensions to the k-means algorithm for clustering large data sets with categorical values,”Data mining and knowledge discovery, vol. 2, no. 3, pp. 283–304, 1998.
[67] J. C. Dunn, “A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters,” 1973.
[68] J. C. Bezdek, C. Coray, R. Gunderson, and J. Watson, “Detection and characterization of cluster substructure i. linear structure: Fuzzy c-lines,”SIAM Journal on Applied Mathematics, vol. 40, no. 2, pp. 339–357, 1981.
[69] Y. Kagan and L. Knopoff, “Spatial distribution of earthquakes: the two-point cor- relation function,” Geophysical Journal International, vol. 62, no. 2, pp. 303–320, 1980.
[70] Y. Y. Kagan and D. D. Jackson, “Long-term earthquake clustering,” Geophysical Journal International, vol. 104, no. 1, pp. 117–133, 1991.
[71] S. Touati, M. Naylor, and I. G. Main, “Origin and nonuniversality of the earthquake interevent time distribution,”Physical review letters, vol. 102, no. 16, pp. 168 501(1–4), 2009.
[72] C. Frohlich, “Aftershocks and temporal clustering of deep earthquakes,”Journal of Geophysical Research: Solid Earth, vol. 92, no. B13, pp. 13 944–13 956, 1987.
[73] M. Wyss and Y. Toya, “Is background seismicity produced at a stationary Poissonian rate?”Bulletin of the Seismological Society of America, vol. 90, no. 5, pp. 1174–1187, 2000.
[74] C. Frohlich, “The nature of deep-focus earthquakes,” Annual Review of Earth and Planetary Sciences, vol. 17, no. 1, pp. 227–254, 1989.
[75] D. Eberhard, “Multiscale seismicity analysis and forecasting: examples from the western Pacific and Iceland,” Ph.D. dissertation, Diss., Eidgenössische Technische Hochschule ETH Zürich, Nr. 21897, 2014.
[76] V. Prasannakumar, H. Vijith, R. Charutha, and N. Geetha, “Spatio-temporal clus- tering of road accidents: GIS based analysis and assessment,”Procedia-Social and Behavioral Sciences, vol. 21, pp. 317–325, 2011.
[77] J. Barreal and M. Loureiro, “Modelling spatial patterns and temporal trends of wildfires in galicia (nw spain). forest systems, volume 24, issue 2, e–022, xx pages,” 2015.
[78] R. Oliveira, M. Y. Santos, and J. M. Pires, “4d+ SNN: a spatio-temporal density-based clustering approach with 4d similarity,” inProc. IEEE 13th International Conference on Data Mining Workshops (ICDMW-2013), Dec 2013, pp. 1045–1052.
[79] S. Stein and M. Liu, “Long aftershock sequences within continents and implications for earthquake hazard assessment,”Nature, vol. 462, no. 7269, pp. 87–89, 2009.
[80] D. Birant and A. Kut, “ST-DBSCAN: An algorithm for clustering spatial–temporal data,”Data & Knowledge Engineering, vol. 60, no. 1, pp. 208–221, 2007.
401, 2016.
[82] R. K. Vijay and S. J. Nanda, “A variableε-DBSCAN algorithm for declustering earth- quake catalogs,” inProc. of (Springer) International conference on Soft Computing for Problem Solving (SocProS-2017), Indian Institute of Technology, Bhubaneswar, Dec 2017, pp. 639–651.
[83] Z. I. Botev, J. F. Grotowski, and D. P. Kroese, “Kernel density estimation via diffusion,”
The annals of Statistics, vol. 38, no. 5, pp. 2916–2957, 2010.
[84] S. Wiemer, “A software package to analyze seismicity: ZMAP,”Seismological Re- search Letters, vol. 72, no. 3, pp. 373–382, 2001.
[85] A. Alexandridis, E. Chondrodima, E. Efthimiou, G. Papadakis, F. Vallianatos, and D. Triantis, “Large earthquake occurrence estimation based on radial basis function neural networks.”IEEE Trans. Geoscience and Remote Sensing, vol. 52, no. 9, pp.
5443–5453, 2014.
[86] R. K. Vijay and S. J. Nanda, “Tetra-stage cluster identification model to analyse the seismic activities of Japan, Himalaya and Taiwan,”IET Signal Processing, vol. 12, no. 1, pp. 95–103, 2017.
[87] J. Golay and M. Kanevski, “A new estimator of intrinsic dimension based on the multipoint morisita index,”Pattern Recognition, vol. 48, no. 12, pp. 4070–4081, 2015.
[88] K. Tiampo, W. Klein, H.-C. Li, A. Mignan, Y. Toya, S. Kohen-Kadosh, J. Rundle, and C.-C. Chen, “Ergodicity and earthquake catalogs: Forecast testing and resulting implications,”Pure and applied geophysics, vol. 167, no. 6-7, pp. 763–782, 2010.
[89] D. Thirumalai, R. D. Mountain, and T. Kirkpatrick, “Ergodic behavior in supercooled liquids and in glasses,”Physical Review A, vol. 39, no. 7, pp. 3563–3574, 1989.
[90] N. Cho, K. Tiampo, S. Mckinnon, J. Vallejos, W. Klein, and R. Dominguez, “A simple metric to quantify seismicity clustering,”Nonlinear Processes in Geophysics, vol. 17, no. 4, pp. 293–302, 2010.
[91] N. Cho, K. Tiampo, P. Bhattacharya, R. Shcherbakov, C. Chen, H. Li, and W. Klein,
“Declustering seismicity using the Thirumalai-Mountain metric,” inAGU Fall Meeting Abstracts, 2010.
[92] K. L. Du and M. Swamy, Search and Optimization by Metaheuristics. Springer, 2016.
[93] X. S. Yang,Nature-inspired optimization algorithms. Elsevier, 2014.
[94] S. Mirjalili, S. M. Mirjalili, and A. Lewis, “Grey Wolf Optimizer,” Advances in Engineering Software, vol. 69, pp. 46–61, 2014.
[95] S. Mirjalili, “How effective is the grey wolf optimizer in training multi-layer percep- trons,”Applied Intelligence, vol. 43, no. 1, pp. 150–161, 2015.
[96] H. Faris, I. Aljarah, M. A. Al-Betar, and S. Mirjalili, “Grey wolf optimizer: a review of recent variants and applications,”Neural computing and applications, pp. 1–23, 2018.
[97] J. Sun, W. Xu, and B. Feng, “A global search strategy of quantum-behaved particle swarm optimization,” inProc. of IEEE International conference on Cybernetics and Intelligent Systems (CIS-2004), vol. 1, Dec 2004, pp. 111–116.
[98] J. Sun, W. Fang, X. Wu, V. Palade, and W. Xu, “Quantum-behaved particle swarm optimization: analysis of individual particle behavior and parameter selection,”Evolu- tionary computation, vol. 20, no. 3, pp. 349–393, 2012.
[99] J. Sun, B. Feng, and W. Xu, “Particle swarm optimization with particles having quantum behavior,” inProc. of IEEE Congress on Evolutionary Computation (CEC- 2004), vol. 1, June 2004, pp. 325–331.
[100] K. Srikanth, L. K. Panwar, B. Panigrahi, E. Herrera-Viedma, A. K. Sangaiah, and G.- G. Wang, “Meta-heuristic framework: quantum inspired binary grey wolf optimizer for unit commitment problem,” Computers & Electrical Engineering, vol. 70, pp.
243–260, 2018.
[101] J. Kennedy, “Particle swarm optimization,”Encyclopedia of machine learning, pp.
760–766, 2010.
[102] S. Tinti and F. Mulargia, “Effects of magnitude uncertainties on estimating the param- eters in the gutenberg-richter frequency-magnitude law,”Bulletin of the Seismological Society of America, vol. 75, no. 6, pp. 1681–1697, 1985.
[103] R. Vijay and S. J. Nanda, “Declustering of an earthquake catalog based on ergodicity using parallel grey wolf optimization,” inProc. of IEEE Congress on Evolutionary Computation (CEC-2017), San Sebastian, Spain, June 2017, pp. 1667–1674.
no. 1, pp. 87–101, 2015.
Appendix A
Euclidean and Haversine formula
The Haversine formula [104] is considered the irregularity having in the surface of an earth. It gives the “as-the-crow-flies” distance where earth surface is assumed nearly a spherical. It is the shortest distance along a great circle on a sphere from their longitudes and latitudes. It is easy to implement and a nice balance of accuracy over complexity in spherical trigonometry.
A geometric difference between the Euclidean and Haversine distance is given in Fig. A.1.
The haversine formula is given by
∆σ=2×r×arcsin[
q
(sin2(∆φ/2) +cos(φA)×cos(φB)×sin2(∆λ/2))] (A.1) DAB=r×∆σ (A.2)
(a) (b)
Fig. A.1.Distance calculation based on: (a) Laws of Haversines (Spherical Trigonometry) (b) Laws of Cosines (Plane Trigonometry).
X =r×cos(φ)∗cos(λ),Y =r×cos(φ)∗sin(λ), Z=r×sin(φ) (A.3) whereris an approximate radius of the earth (6371 Km).φ andλ represents the longitude and latitude in radians. Herez-coordinate does not correspond to an altitude, it is the altitude at North Pole and in the north south direction at equator. The Euclidean distanceDAB∈R3 between two pointAandBis determined as follows:
DAB= q
[(XA−XB)2+ (YA−YB)2+ (ZA−ZB)2] (A.4) or
DAB= q
(φB−φA)2+ (λB−λA)2 (A.5)
Appendix B
Benchmark Functions for Optimization
The information of benchmark test functions which are used to analyze the performance of proposed quantum grey wolf optimizer and its comparison with other algorithms (described in Chapter 6). Here, a Table B.1. highlights the description of each function and their pictorial view is shown in Fig.B.1.
Table B.1 Information about the 2D benchmark test functions
Num Func Name Representationf(x1,x2) Range Global min
F1 Bukin N. 6 100
q
|x2−0.01x21|+0.01|x1+10| x∈[−15,−5] x∗= (−10,1) x2∈[−3,3] f(x∗) =0 F2 Cross-in-Tray −0.0001
sin(x1)sin(x2)exp
100−
√x21+x22 π
+1 0.1
xi∈[−10,10] x∗= (±1.349,±1.349) f(x∗) =−2.0626 F3 De Jong N. 5
0.002+
25
∑
j=1
"
j+
d
∑
j=1
(xi−ai,j)6
#−1
−1
xi∈[−65.536, x∗= (0,0) 65.536] f(x∗) =1 F4 Drop-wave
−1+cos 12.
√
x21+x22
0.5.(x21+x22+2)
xi∈[−5.12,5.12] x∗= (0,0) f(x∗) =−1
F5 Egg holder −(x2+47)sin
q
x2+x21+47
xi∈[−5.12,5.12] x∗= (512,404.231)
−x1sinp
|x1−(x2+47)|
f(x∗) =−959.640 F6 Goldstein-Price [1+ (x1+x2+1)2(19−14x1+3x21−14x2 x1,x2∈[−2,2] x∗= (0,−1)
+6x1x2+3x22)][30+ (2x1−3x2)2(18−32x1 f(x∗) =3 +12x21+4x2−36x1x2+27x22)]
F7 Schaffer N. 1
0.5+ sin
2(x21+x22)2−0.5 (1+0.001(x21+x22))2
xi∈[−5.12,5.12] x∗= (0,0) f(x∗) =0 F8 Schaffer N. 2
0.5+ sin
2(x21−x22)2−0.5
(1+0.001(x21+x22))2
xi∈[−5.12,5.12] x∗= (0,1.2531) f(x∗) =0.0015 F9 Three Hump Camel 2x21−1.05x41+x66+x1x2+x22 x1,x2∈[−5,5] x∗= (0,0)
f(x∗) =−1.031 F10 Six Hump Camel 4x21−2.1x41+x361+x1x2−4x22+4x42 x1,x2∈[−5,5] x∗= (±0.089,±0.712)
f(x∗) =−1.031
1 d
F11 Ackely Func −a.exp −b s
1 d
d
∑
i=1
x2i
!
−exp
1 d
d
∑
i=1
cos(c.xi)
+a+exp(1) xi∈[−32,32] x∗= (0, ...0) f(x∗) =0
F12 Alpine N. 1 ∑d
i=1
|xisin(xi) +0.1xi| xi∈[0,10] x∗= (0, ...0) f(x∗) =0
F13 Griewank’s Func 1+ ∑d
i=1 x2i 4000−∏d
i=1
cos(√xi
i) xi∈[−600,600] x∗= (0, ...0) f(x∗) =0
F14 Hyper-ellipsoid ∑d
i=1
i.xi2 xi∈[−10,10] x∗= (0, ...0) f(x∗) =0
F15 Powell Sum Func ∑d
i=1
|xi|i+1 xi∈[−1,1] x∗= (0, ...0) f(x∗) =0
F16 Quartic Func ∑d
i=1
i.x4i+random[0,1) xi∈[−1.28,1.28] x∗= (0, ...0) f(x∗) =0
F17 Rastrigin Func 10d+ ∑d
i=1
(x2i−10 cos(2πxi)) xi∈[−5.12,5.12] x∗= (0, ...0) f(x∗) =0
F18 Rosenbrock d−1∑
i=1
[b(xi+1−x2i)2+ (a−xi)2] xi∈[−5,5] x∗= (1, ...1) f(x∗) =0
F19 Schwefel 2.20 ∑d
i=1
|xi| xi∈[−100,100] x∗= (0, ...0) f(x∗) =0
F20 Schwefel 2.21 max
i=1,...,d|xi| xi∈[−100,100] x∗= (0, ...0)
f(x∗) =0
F21 Schwefel 2.22 ∑d
i=1
|xi|+∏d
i=1
|xi| xi∈[−100,100] x∗= (0, ...0) f(x∗) =0
F22 Schwefel 2.23 ∑d
i=1
x10i xi∈[−10,10] x∗= (0, ...0) f(x∗) =0
F23 Sphere Function ∑d
i=1
x2i xi∈[−5.12,5.12] x∗= (0, ...0) f(x∗) =0 F24 Xin-She Yang N.3 exp
−∑d
i=1
(xi/β)2m
−2 exp
−∑d
i=1
x2i d
∏
i=1
cos2(xi) xi∈[−2π,2π] x∗= (0, ...0) f(x∗) =−1
0 50
2 100
-5 F(x1, x2)150
x2 0 200
x1 -10 -2
-15
(a)Bukin N.6 (b)Cross-in-Tray (c)De-Jong Func (d)Drop Wave
(e)Egg holder
0 2 2 4
1 2
F(x1, x2) 105
6
1
x2 8
0
x1 10
0
-1 -1
-2 -2
(f)Goldstein-Price (g)Schaffer N.1 (h)Schaffer N.2
0 5 500 1000
5 F(x1, x2)1500
x2 0 2000
x1 0 -5 -5
(i)3-Hump-Camel
-2000 5 0
5 2000
F(x1, x2) 4000
x2 0
x1 6000
0 -5 -5
(j)6-Hump-Camel
0 5
20 10 F(x1, x2)
20 15
x2 0 20
x1 0
-20 -20
(k)Ackely Func
0 10 5
10 F(x1, x2)
10
8
x2 5 15
6
x1 4 2
0 0
(l)Alpine N.1
0 5 0.5
5 1
F(x1, x2) 1.5
x2 0
x1 0 2
-5 -5
(m)Griewank’s (n)Hyper-ellipsoid (o)Powell Sum (p)Quartic Func
0 5 20
5 40
F(x1, x2) 60
x2 0
x1 0 80
-5 -5
(q)Rastrigin Func
0 5 2 4
5 104
F(x1, x2)6
x2 0 8
x1 0 -5 -5
(r)Rosenbrock (s)Schwefel 2.20 (t)Schwefel 2.21
0 100 2000 4000
50 100
F(x1, x2) 6000
50
x2 8000
0
x1 10000
-50 -50 0
-100 -100
(u)Schwefel 2.22 (v)Schwefel 2.23
0 5 10 20
5 F(x1, x2)30
x2 40
0
x1 0 50
-5 -5
(w)Sphere Func (x)Xin-She-Yang N
Fig. B.1.2D representation of benchmark functions used in the study.
Publications from the Thesis work
Peer Reviewed International Journals and Conference Proceedings
J1 R. K. Vijay and S. J. Nanda, “Tetra-stage cluster identification model to analyse the seismic activities of Japan, Himalaya and Taiwan,”IET Signal Processing, vol.12, no.1, pp. 95–103, 2017.
J2 R. K. Vijay and S. J. Nanda, “Weighted Density Peak Space-Time Clustering for Classification of Seismic events,”Journal of Pattern Analysis & Applications, Springer.
(Under review)
J3 R. K. Vijay and S. J. Nanda,“Shared Nearest Neighborhood Intensity Based Decluster- ing Model for Analysis Spatio-temporal Seismicity,”IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol.1, no.5, pp. 1–9, 2019.
J4 R. K. Vijay and S. J. Nanda,“A Quantum Grey Wolf Optimizer based Declustering model in Ergodic framework,”Elsevier Journal of Computational Science, vol.36, pp.
101019, 2019.
C1 R. K. Vijay and S. J. Nanda. “Declustering of an Earthquake Catalog Based on Ergodicity using Parallel Grey Wolf Optimization,”Proceeding of IEEE Congress on Evolutionary Computation (IEEE-CEC-2017), pp. 1667–1674, Sen-Sebastian, Spain, 05thto 8th June 2017.
C2 R. K. Vijay and S. J. Nanda, “A Variable ε-DBSCAN Algorithm for Declustering Earthquake Catalogs,”Proceeding of 7th International Conference on Soft Comput- ing for Problem Solving (Springer-SocProS-2017), pp. 639–651, Indian Institute of Technology Bhubaneswar, 23thto 24th Dec 2017.
C3 R. K. Vijay and S. J. Nanda, “Shared Nearest Neighbor Based Classification of Earth- quake Catalogs in Spatio-temporal Domain,” Proceeding of IEEE 4thInternational Conference on Computing, Control and Automation (IEEE-ICCUBEA-2018), pp. 1–6, Pimpri Chinchwad College of Engineering, Pune, 17thto 19thAug 2018.