In herein presented work, the relation between number of ES iterations and convergence of the whole GPA-ES hybrid algorithm will be studied due to increasing needs to analyze and model large data sets. Evolutionary algorithms are applicable in the areas which are not covered by other artificial intelligence or soft computing techniques like neural networks and deep learning like search of algebraic model of data.
❱❖▲❯▼❊✿ ✸ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❉❡♣❡♥❞❡♥❝② ♦❢ ●P❆✲❊❙ ❆❧❣♦r✐t❤♠ ❊❢❢✐❝✐❡♥❝② ♦♥ ❊❙ P❛r❛♠❡t❡rs ❖♣t✐♠✐③❛t✐♦♥ ❙tr❡♥❣t❤ ❚♦♠❛s ❇❘❆◆❉❊❏❙❑❨∗ ❯♥✐✈❡rs✐t② ♦❢ P❛r❞✉❜✐❝❡✱ ✺✸✷ ✶✵ P❛r❞✉❜✐❝❡✱ ❈③❡❝❤ ❘❡♣✉❜❧✐❝ ✯❈♦rr❡s♣♦♥❞✐♥❣ ❆✉t❤♦r✿ ❚♦♠❛s ❇❘❆◆❉❊❏❙❑❨ ✭❊♠❛✐❧✿ t♦♠❛s✳❜r❛♥❞❡❥s❦②❅✉♣❝❡✳❝③✮ ✭❘❡❝❡✐✈❡❞✿ ✷✹✲❉❡❝✲✷✵✶✽❀ ❛❝❝❡♣t❡❞✿ ✶✸✲▼❛r✲✷✵✶✾❀ ♣✉❜❧✐s❤❡❞✿ ✸✶✲▼❛r✲✷✵✶✾✮ ❉❖■✿ ❤tt♣✿✴✴❞①✳❞♦✐✳♦r❣✴✶✵✳✷✺✵✼✸✴❥❛❡❝✳✷✵✶✾✸✶✳✷✷✻ ❆❜str❛❝t✳ ■♥ ❤❡r❡✐♥ ♣r❡s❡♥t❡❞ ✇♦r❦✱ t❤❡ r❡❧❛✲ t✐♦♥ ❜❡t✇❡❡♥ ♥✉♠❜❡r ♦❢ ❊❙ ✐t❡r❛t✐♦♥s ❛♥❞ ❝♦♥✲ ✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✇❤♦❧❡ ●P❆✲❊❙ ❤②❜r✐❞ ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❜❡ st✉❞✐❡❞ ❞✉❡ t♦ ✐♥❝r❡❛s✐♥❣ ♥❡❡❞s t♦ ❛♥❛✲ ❧②③❡ ❛♥❞ ♠♦❞❡❧ ❧❛r❣❡ ❞❛t❛ s❡ts✳ ❊✈♦❧✉t✐♦♥❛r② ❛❧✲ ❣♦r✐t❤♠s ❛r❡ ❛♣♣❧✐❝❛❜❧❡ ✐♥ t❤❡ ❛r❡❛s ✇❤✐❝❤ ❛r❡ ♥♦t ❝♦✈❡r❡❞ ❜② ♦t❤❡r ❛rt✐✜❝✐❛❧ ✐♥t❡❧❧✐❣❡♥❝❡ ♦r s♦❢t ❝♦♠♣✉t✐♥❣ t❡❝❤♥✐q✉❡s ❧✐❦❡ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❛♥❞ ❞❡❡♣ ❧❡❛r♥✐♥❣ ❧✐❦❡ s❡❛r❝❤ ♦❢ ❛❧❣❡❜r❛✐❝ ♠♦❞❡❧ ♦❢ ❞❛t❛✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✐♠❡ ❛♥❞ ❛❧❣♦r✐t❤✲ ♠✐❝ ❝♦♠♣❧❡①✐t② ✇✐❧❧ ❜❡ ❛❧s♦ ♠❡♥t✐♦♥❡❞ ❛s ✇❡❧❧ ❛s t❤❡ ♣r♦❜❧❡♠s ♦❢ ♠✉❧t✐t❛s❦✐♥❣ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ●P❆✱ ✇❤❡r❡ ❡①t❡r♥❛❧ ✐♥✢✉❡♥❝❡s ❝♦♠♣❧✐❝❛t❡ ✐♥❝r❡❛s✐♥❣ ♦❢ ●P❆ ❡✣❝✐❡♥❝② ✈✐❛ Ps❡✉❞♦ ❘❛♥✲ ❞♦♠ ◆✉♠❜❡r ●❡♥❡r❛t♦r ✭P❘◆●✮ ❝❤♦✐❝❡ ♦♣t✐✲ ♠✐③❛t✐♦♥✳ ❍②❜r✐❞ ❡✈♦❧✉t✐♦♥❛r② ❛❧❣♦r✐t❤♠s ❧✐❦❡ ●P❆✲❊❙ ✉s❡s ●P❆ ❢♦r s♦❧✉t✐♦♥ str✉❝t✉r❡ ❞❡✈❡❧♦♣♠❡♥t ❛♥❞ ❊✈♦❧✉t✐♦♥❛r② ❙tr❛t❡❣② ✭❊❙✮ ❢♦r ♣❛r❛♠❡t❡rs ✐❞❡♥t✐✜❝❛t✐♦♥ ❛r❡ ❝♦♥tr♦❧❧❡❞ ❜② ♠❛♥② ♣❛r❛♠❡✲ t❡rs✳ ❚❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ❛r❡ s✐③❡s ♦❢ ●P❆ ♣♦♣✲ ✉❧❛t✐♦♥ ❛♥❞ s✐③❡s ♦❢ ❊❙ ♣♦♣✉❧❛t✐♦♥s r❡❧❛t❡❞ t♦ ❡❛❝❤ ♣❛rt✐❝✉❧❛r ✐♥❞✐✈✐❞✉❛❧ ✐♥ ●P❆ ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡r❡ ✐s ❛❧s♦ ❧✐♠✐t ♦❢ ❊❙ ❛❧❣♦r✐t❤♠ ❡✈♦❧✉t✐♦♥✲ ❛r② ❝②❝❧❡s✳ ❚❤✐s ❧✐♠✐t ♣❧❛②s t✇♦ ❝♦♥tr❛❞✐❝t♦r② r♦❧❡s✳ ❖♥ ♦♥❡ s✐❞❡ ❜✐❣❣❡r ♥✉♠❜❡r ♦❢ ❊❙ ✐t❡r❛✲ t✐♦♥s ♠❡❛♥s ❧❡ss ❝❤❛♥❝❡ t♦ ♦♠✐t ❣♦♦❞ s♦❧✉t✐♦♥ ❢♦r ✇r♦♥❣❧② ✐❞❡♥t✐✜❡❞ ♣❛r❛♠❡t❡rs✱ ♦♥ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❊❙ ✐t❡r❛t✐♦♥s s✐❣♥✐✜❝❛♥t❧② ✐♥❝r❡❛s❡s ❝♦♠♣✉t❛t✐♦♥❛❧ t✐♠❡ ❛♥❞ t❤✉s ❧✐♠✐ts ❛♣✲ ♣❧✐❝❛t✐♦♥ ❞♦♠❛✐♥ ♦❢ ●P❆✲❊❙ ❛❧❣♦r✐t❤♠✳ ✸✵✹ ❑❡②✇♦r❞s ●❡♥❡t✐❝ Pr♦❣r❛♠♠✐♥❣ ❆❧❣♦r✐t❤♠✱ ❊✈♦✲ ❧✉t✐♦♥❛r② ❙tr❛t❡❣②✱ ❍②❜r✐❞ ❊✈♦❧✉t✐♦♥❛r② ❙②st❡♠✱ ❆❧❣♦r✐t❤♠ ❊✣❝✐❡♥❝②✱ ❖♣t✐♠✐③❛✲ t✐♦♥✳ ✶✳ ■♥tr♦❞✉❝t✐♦♥ ■♥❝r❡❛s✐♥❣ ❛♠♦✉♥t ♦❢ ❞❛t❛ t♦ ❜❡ ♣r♦❝❡ss❡❞ ❢♦r❝❡s ♥❡❡❞s ❢♦r ✐♠♣r♦✈✐♥❣ ❡✣❝✐❡♥❝② ♦❢ ❡①✐st✲ ✐♥❣ ❛❧❣♦r✐t❤♠s✳ ❲❤✐❧❡ ❛rt✐✜❝✐❛❧ ♥❡✉r❛❧ ♥❡t✇♦r❦s ❛♥❞ ❞❡❡♣ ❧❡❛r♥✐♥❣ t❡❝❤♥♦❧♦❣② ❝❛♥ ❜❡ r❡❛s♦♥❡❞ r❛t❤❡r ❛s ❞❛t❛ r❡♣r❡s❡♥t❛t✐♦♥ ♦r ✐♥t❡r♣♦❧❛t✐♦♥✱ r❡❝❛❧❧ ❛♥❞ ❛♣♣r♦①✐♠❛t✐♦♥ t♦♦❧✱ ❡✈♦❧✉t✐♦♥❛r② ❛❧✲ ❣♦r✐t❤♠s ❛r❡ ❝❛♣❛❜❧❡ t♦ tr❛♥s❢♦r♠ ❞❛t❛ ✐♥t♦ ♠♦❞❡❧s ✇❤✐❝❤ ❝❛♥ ❜❡ ✉♥❞❡rst♦♦❞ ❛♥❞ ❛♥❛❧②③❡❞ ❜② ❤✉♠❛♥s✱ s❡❛r❝❤❡❞ ❢♦r ♦♣t✐♠❛✱ ❛♥❞ s♦❧✈❡❞ ♠❛♥② ♥❡①t t❛s❦s ♦♥ t❤❡ ❜❛s❡ ♦❢ tr❛✐♥✐♥❣ ❞❛t❛✳ ■♥ t❤❡ ❛r❡❛ ♦❢ ♠♦❞❡❧ ❞❡✈❡❧♦♣♠❡♥t ❜② ❡✈♦❧✉t✐♦♥❛r② ❛❧❣♦r✐t❤♠s ❝❛❧❧❡❞ s②♠❜♦❧✐❝ r❡❣r❡ss✐♦♥✱ ●❡♥❡t✐❝ Pr♦❣r❛♠♠✐♥❣ ❆❧❣♦r✐t❤♠s ✭●P❆✮✱ ❆♥❛❧②t✐❝ Pr♦✲ ❣r❛♠♠✐♥❣ ❬✶❪ ❛♥❞ r❡❧❛t❡❞ t❡❝❤♥✐q✉❡s ❛r❡ ✉s❡❞✳ ❍②❜r✐❞ ❡✈♦❧✉t✐♦♥❛r② ❛❧❣♦r✐t❤♠s ❧✐❦❡ ●P❆✲❊❙ ❬✷❪ ✉s❡s ●P❆ ❢♦r s♦❧✉t✐♦♥ str✉❝t✉r❡ ❞❡✈❡❧♦♣♠❡♥t ❛♥❞ ❊✈♦❧✉t✐♦♥❛r② ❙tr❛t❡❣② ✭❊❙✮ ❢♦r ♣❛r❛♠❡t❡rs ✐❞❡♥t✐✜❝❛t✐♦♥ ❛r❡ ❝♦♥tr♦❧❧❡❞ ❜② ♠❛♥② ♣❛r❛♠❡✲ t❡rs✳ ❚❤❡ ♠♦st s✐❣♥✐✜❝❛♥t ❛r❡ s✐③❡s ♦❢ ●P❆ ♣♦♣✲ ✉❧❛t✐♦♥ ❛♥❞ s✐③❡s ♦❢ ❊❙ ♣♦♣✉❧❛t✐♦♥s r❡❧❛t❡❞ t♦ ❡❛❝❤ ♣❛rt✐❝✉❧❛r ✐♥❞✐✈✐❞✉❛❧ ✐♥ ●P❆ ♣♦♣✉❧❛t✐♦♥✳ ❚❤❡r❡ ✐s ❛❧s♦ ❧✐♠✐t ♦❢ ❊❙ ❛❧❣♦r✐t❤♠ ❡✈♦❧✉t✐♦♥✲ ❛r② ❝②❝❧❡s✳ ❚❤✐s ❧✐♠✐t ♣❧❛②s t✇♦ ❝♦♥tr❛❞✐❝t♦r② ❝ ✷✵✶✾ ❏♦✉r♥❛❧ ♦❢ ❆❞✈❛♥❝❡❞ ❊♥❣✐♥❡❡r✐♥❣ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥ ✭❏❆❊❈✮ ❱❖▲❯▼❊✿ ✸ r♦❧❡s✳ ❖♥ ♦♥❡ s✐❞❡ ❜✐❣❣❡r ♥✉♠❜❡r ♦❢ ❊❙ ✐t❡r❛✲ t✐♦♥s ♠❡❛♥s ❧❡ss ❝❤❛♥❝❡ t♦ ♦♠✐t ❣♦♦❞ s♦❧✉t✐♦♥ ❢♦r ✇r♦♥❣❧② ✐❞❡♥t✐✜❡❞ ♣❛r❛♠❡t❡rs ❬✸❪ ❛♥❞ ✐t ✇❛s t❤❡ ♠❛✐♥ ✐❞❡❛ ♦❢ ●P❆✲❊❙ ❤②❜r✐❞ ❛❧❣♦r✐t❤♠ ❞❡✲ ✈❡❧♦♣♠❡♥t✳ ❖♥ t❤❡ ♦♣♣♦s✐t❡ s✐❞❡ ❧❛r❣❡ ♥✉♠❜❡r ♦❢ ❊❙ ✐t❡r❛t✐♦♥s s✐❣♥✐✜❝❛♥t❧② ✐♥❝r❡❛s❡s ❝♦♠♣✉t❛✲ t✐♦♥❛❧ t✐♠❡ ❛♥❞ t❤✉s ❧✐♠✐ts ❛♣♣❧✐❝❛t✐♦♥ ❞♦♠❛✐♥ ♦❢ ●P❆✲❊❙ ❛❧❣♦r✐t❤♠✳ ■♥ t❤✐s st✉❞②✱ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ♥✉♠❜❡r ♦❢ ❊❙ ✐t❡r❛t✐♦♥s ❛♥❞ ❝♦♥✈❡r❣❡♥❝❡ ♦❢ t❤❡ ✇❤♦❧❡ ●P❆✲❊❙ ❤②❜r✐❞ ❛❧❣♦r✐t❤♠ ✇✐❧❧ ❜❡ st✉❞✐❡❞✳ ❚❤❡ ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t✐♠❡ ❛♥❞ ❛❧❣♦r✐t❤♠✐❝ ❝♦♠✲ ♣❧❡①✐t② ✇✐❧❧ ❜❡ ❛❧s♦ ♠❡♥t✐♦♥❡❞ ❛s ✇❡❧❧ ❛s t❤❡ ♣r♦❜❧❡♠s ♦❢ ♠✉❧t✐t❛s❦ ❛♥❞ ♠✉❧t✐t❤r❡❛❞ ✐♠♣❧❡✲ ♠❡♥t❛t✐♦♥ ♦❢ ●P❆ ❛s ❛❧❣♦r✐t❤♠s ✇❤❡r❡ ❡①t❡r✲ ♥❛❧ ✐♥✢✉❡♥❝❡s ❝♦♠♣❧✐❝❛t❡s ✐♥❝r❡❛s✐♥❣ ♦❢ ●P❆ ❡❢✲ ✜❝✐❡♥❝② ✈✐❛ Ps❡✉❞♦ ❘❛♥❞♦♠ ◆✉♠❜❡r ●❡♥❡r❛t♦r ✭P❘◆●✮ ❝❤♦✐❝❡ ♦♣t✐♠✐③❛t✐♦♥✳ ✷✳ ❍②❜r✐❞ ●P❆✲❊❙ ❛❧❣♦r✐t❤♠ ●P❆✲❊❙ ❤②❜r✐❞ ❛❧❣♦r✐t❤♠ ❝♦♠❜✐♥❡s ●P❆ ❛❧❣♦✲ r✐t❤♠ ❢♦r s♦❧✉t✐♦♥ str✉❝t✉r❡ ❞❡✈❡❧♦♣♠❡♥t ❛♥❞ ❊❙ ❛❧❣♦r✐t❤♠ ❢♦r ♦♣t✐♠✐③❛t✐♦♥ ♦❢ ♣❛r❛♠❡t❡rs ♦❢ ❡❛❝❤ ✐♥❞✐✈✐❞✉❛❧ ✐♥ ●P❆ ♣♦♣✉❧❛t✐♦♥✳ ❙✉❝❤ ❞❡✲ s✐❣♥ ♦❢ ❡✈♦❧✉t✐♦♥❛r② ❛❧❣♦r✐t❤♠ ♣r❡✈❡♥ts s✐t✉❛✲ t✐♦♥s ✇❤❡♥ ❣♦♦❞ str✉❝t✉r❡ s♦❧✉t✐♦♥ ✭❡✳❣✳ ✇❡❧❧ ❝♦♠♣♦s❡❞ ❡q✉❛t✐♦♥✮ ❜✉t ✇✐t❤ ✇r♦♥❣❧② ❡st✐♠❛t❡❞ ❝♦♥st❛♥ts ✭❝♦❡✣❝✐❡♥t✮ ✐s ❡❧✐♠✐♥❛t❡❞ ❢r♦♠ ♣♦♣✉✲ ❧❛t✐♦♥ ❛♥❞ r❡♣❧❛❝❡❞ ❜② ✐♥❞✐✈✐❞✉❛❧ ♦❢ ✇♦rs❡ str✉❝✲ t✉r❡ ❜✉t ❜❡tt❡r ✜tt❡❞ ❝♦♥st❛♥ts✱ ✇❤✐❝❤ ❤❛s ✇♦rs❡ ❡✈♦❧✉t✐♦♥❛r② ♣♦t❡♥t✐❛❧✳ ❆❧❣♦r✐t❤♠ ✶✳ ❙t✉❞✐❡❞ ●P❆ ❛❧❣♦r✐t❤♠✳ ✶✮ ❋❖❘ ❆▲▲ ✐♥❞✐✈✐❞✉❛❧s ❉❖ ■♥✐t✐❛❧✐③❡✭✮ ❊◆❉ ❋❖❘❀ ✷✮ ❋❖❘ ❆▲▲ ✐♥❞✐✈✐❞✉❛❧s ❉❖ ❊✈❛❧✉❛t❡✭✮ ⇒ ✜t✲ ♥❡ss ❊◆❉ ❋❖❘❀ ✸✮ ❙♦rt✭✐♥❞✐✈✐❞✉❛❧s✮❀ | ■❙❙❯❊✿ ✶ | ✷✵✶✾ | ▼❛r❝❤ ❈❆❙❊ ❛ ❉❖ ▼✉t❛t❡✭✮⇒ ♥❡✇❴✐♥❞✐✈✐❞✉❛❧s❀ ❈❆❙❊ ❜ ❉❖ ❙②♠♠❡tr✐❝❴❝r♦ss♦✈❡r✭✮ ⇒ ♥❡✇❴✐♥❞✐✈✐❞✉❛❧s❀ ❈❆❙❊ ❝ ❉❖ ❖♥❡❴♣♦✐♥t❴❝r♦ss♦✈❡r✭✮ ⇒ ♥❡✇❴✐♥❞✐✈✐❞✉❛❧s❀ ❈❆❙❊ ❞ ❉❖ ❘❡✲❣❡r❛t✐♥❣✭✮ ⇒ ♥❡✇❴✐♥❞✐✈✐❞✉❛❧s❀ ❊◆❉ ❙❊▲❊❈❚❀ ❊◆❉ ❋❖❘❀ ✻✮ ❋❖❘ ❆▲▲ ✐♥❞✐✈✐❞✉❛❧s ❉❖ ◆❡✇ ❊❙❴❛❧❣♦r✐t❤♠❴♦❜❥❡❝t❀ ✴✴❢♦r ❡❛❝❤ ●P❆ ✐♥❞✐✈✐❞✉❛❧ ♥❡✇ ✐♥❞❡♣❡♥❞❡♥t ♣❛r❛♠❡t❡r ♦♣t✐♠✐③❡r ✐s ❝r❡❛t❡❞ ❋❖❘ ❆▲▲ ❊❙❴✐♥❞✐✈✐❞✉❛❧s ❉❖ ■♥✐t✐❛❧✲ ✐③❡✭✮❊◆❉❀ ❡✈❛❧✉❛t❡✭✮ ⇒ ❊❙❴✜t♥❡ss ❋❖❘ ❆▲▲ ❊❙❴❝②❝❧❡s ❉❖ ❋❖❘ ❆▲▲ ❊❙❴✐♥❞✐✈✐❞✉❛❧s ❉❖ ❊✈❛❧✉❛t❡ ⇒ ❊❙❴✜t♥❡ss ❊◆❉ ❋❖❘❀ ❋❖❘ ❆▲▲ ❊❙❴✐♥❞✐✈✐❞✉❛❧s ❉❖ ✐♥t❡❧❧✐❣❡♥t❴❝r♦ss♦✈❡r✭✮ ⇒ ♥❡✇❴❊❙❴✐♥❞✐✈✐❞✉❛❧s ❊✈❛❧✉❛t❡ ⇒ ♥❡✇❴❊❙❴✜t♥❡ss ❊◆❉ ❋❖❘❀ ❋❖❘ ❆▲▲ ❊❙❴✐♥❞✐✈✐❞✉❛❧s ❉❖ ■❋ ♥❡✇❴❊❙❴✜t♥❡ss>❊❙❴✜t♥❡ss ❚❍❊◆ ❊❙❴✐♥❞✐✈✐❞✉❛❧ ❂ ♥❡✇❴❊❙❴✐♥❞✐✈✐❞✉❛❧❀ ✜t♥❡ss ❂ ♥❡✇❴✜t♥❡ss❀ ❊◆❉ ■❋❀ ❙♦rt✭❊❙❴✐♥❞✐✈✐❞✉❛❧s✱ ❊❙❴✜t♥❡ss✮❀ ❊◆❉ ❋❖❘❀ ❊◆❉ ❋❖❘❀ ♥❡✇❴✐♥❞✐✈✐❞✉❛❧❂❊❙❴✐♥❞✐✈✐❞✉❛❧❬✵❪❀ ♥❡✇❴✜t♥❡ss ❂ ❊❙❴✜t♥❡ss❬✵❪❀ ✼✮ ❋❖❘ ❆▲▲ ✐♥❞✐✈✐❞✉❛❧s ♥❡✇❴✜t♥❡ss