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▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙t❛t✐st✐❝❛❧ ♠♦❞❡❧❧✐♥❣ ●✐❧❧❡s ●✉✐❧❧♦t ❣✐❣✉❅❞t✉✳❞❦ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶ ✴ ✸✸

❊①❛♠♣❧❡ ❊①❛♠♣❧❡ ❈♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❉❉❚ ✭❛ t♦①✐❝ ❝❤❡♠✐❝❛❧✮ ✐♥ ✶✺ ♣✐❦❡ ✜s❤ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ ✜s❤ ❛❣❡✳✳✳✳ ❙❡❡ ✜❧❡ ♣✐❦❡❴❞❛t❛✳t①t ✐♥ ❞❛t❛ ❢♦❧❞❡r✳ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✷ ✴ ✸✸

❊①❛♠♣❧❡ ❱❛r✐♦✉s q✉❡st✐♦♥s ❉♦❡s ❝♦♥❝❡♥tr❛t✐♦♥ ✐♥❝r❡❛s❡ ✇✐t❤ ❛❣❡❄ ❍♦✇ ♠✉❝❤❄ P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ✭❛❦❛ ✐♥❢❡r❡♥❝❡✮ ❍♦✇ ♠✉❝❤ ❝♦♥✜❞❡♥❝❡ s❤♦✉❧❞ ✇❡ ♣❧❛❝❡ ✐♥ t❤❡ ❛♥s✇❡r❄ ❚❡st✐♥❣ s✐❣♥✐✜❝❛♥❝❡✱ ♠♦❞❡❧ ❝❤❡❝❦✐♥❣ ❲❤❛t ✐s t❤❡ ❛✈❡r❛❣❡ ❝♦♥❝❡♥tr❛t✐♦♥ ♦❢ ❛ ✸✳✺ ♦r ✽ ②❡❛r ♦❧❞ ✜s❤❄ Pr❡❞✐❝t✐♦♥ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✸ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥ ❊♠♣✐r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ■ ❉❡✜♥✐t✐♦♥✿ ❊♠♣✐r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡ � Cov ( x, y ) = 1 ( x i − ¯ x )( y i − ¯ y ) n i ❚❡♥❞s t♦ ❜❡ ❧❛r❣❡ ✇❤❡♥ x i ❛♥❞ y i ❛r❡ ❧❛r❣❡ s✐♠✉❧t❛♥❡♦✉s❧②✳ ❍❡♥❝❡ q✉❛♥t✐✜❡s ❤♦✇ ♠✉❝❤ x ❛♥❞ y ✏❝♦✲✈❛r② t♦❣❡t❤❡r✑✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ✐s s❝❛❧❡✲❞❡♣❡♥❞❡♥t✿ Cov ( ax, y ) = aCov ( x, y ) ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✹ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥ ❊♠♣✐r✐❝❛❧ ❝♦✈❛r✐❛♥❝❡ ❛♥❞ ❝♦rr❡❧❛t✐♦♥ ■■ ❉❡✜♥✐t✐♦♥✿ ❊♠♣✐r✐❝❛❧ ❝♦rr❡❧❛t✐♦♥ ✭❛❦❛ P❡❛rs♦♥✬s ❝♦rr❡❧❛t✐♦♥ ❝♦❡❢✳✮ � i ( x i − ¯ x )( y i − ¯ y ) �� x ) 2 �� Cor ( x, y ) = i ( x i − ¯ i ( y i − ¯ y ) 2 ❈♦r✳ ❂ ❈♦✈✳ r❡s❝❛❧❡❞ ❜② st❛♥❞❛r❞ ❞❡✈✐❛t✐♦♥s ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✺ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥ ❈♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t✿ ✐♥t❡r♣r❡t❛t✐♦♥ ❛♥❞ ♣✐t❢❛❧❧s ❈♦rr❡❧❛t✐♦♥ ❢♦r ✈❛r✐♦✉s ❞❛t❛ ♣❛tt❡r♥s ✭r❡♣r✐♥❡t❞ ❢r♦♠ ✇✐❦✐♣❡❞✐❛✮ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✻ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥ ❚❤❡ ❝♦rr❡❧❛t✐♦♥ ❝♦❡✣❝✐❡♥t ρ ❞♦❡s ♥♦t t❡❧❧ t❤❡ ✇❤♦❧❡ st♦r② ❆❧❧ ❢♦✉r s❡ts ❤❛✈❡ ✐❞❡♥t✐❝❛❧ ρ ≈ 0 . 816 ✱ ❜✉t ✈❛r② ❝♦♥s✐❞❡r❛❜❧② ✇❤❡♥ ❣r❛♣❤❡❞✳ ❉❛t❛ ♣❧♦tt❡❞ ❛❜♦✈❡ ❛r❡ s②♥t❤❡t✐❝ ❞❛t❛ ♠❛❞❡ ✉♣ ❜② ❋✳ ❆♥s❝♦♠❜❡ ✐♥ ✶✾✼✸ t♦ ✐❧❧✉str❛t❡ t❤❡ ♣✐t❢❛❧❧s ❛ss♦❝✐❛t❡❞ ✇✐t❤ ρ ✳ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✼ ✴ ✸✸

❉❡s❝r✐❜✐♥❣ ❛ ❧✐♥❡❛r ♣❛tt❡r♥ ❙tr✉❝t✉r❡ ❡✛❡❝t ✭❛❦❛ ❙✐♠♣s♦♥✬s ♣❛r❛❞♦①✮ ◗✉✐❝❦ ✐♥t❡r♣r❡t❛t✐♦♥ ♦❢ ρ ✈❛❧✉❡s ❝❛♥ ❧❡❛❞ t♦ ❡rr♦♥❡❛♦✉s ❝♦♥❝❧✉s✐♦♥s✳ ❙❡❡ ❛❧s♦ ❙✐♠♣s♦♥✬s ❡✛❡❝t ♦♥ ✇✐❦✐♣❡❞✐❛ ❢♦r ❢✉rt❤❡r ❞❡t❛✐❧ ❛♥❞ ❡①❛♠♣❧❡s✳ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✽ ✴ ✸✸

❚❤❡ s✐♠♣❧❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ❚❤❡ s✐♠♣❧❡ r❡❣r❡ss✐♦♥ ♠♦❞❡❧ ◆♦t❛t✐♦♥✿ x 1 , ..., x n ❛❣❡s ♦❢ t❤❡ ✈❛r✐♦✉s ✐♥❞✐✈✐❞✐❞✉❛❧s✱ y 1 , ..., y n ❉❉❚ ❝♦♥❝❡♥tr❛t✐♦♥s ❚❤❡ s✐♠♣❧❡ ❧✐♥❡❛r ♠♦❞❡❧ Y i = ax i + b + ε i t❤❡ x i s ❛r❡ ❞❡t❡r♠✐♥✐st✐❝ ✈❛r✐❛❜❧❡s ✭❛ s♦♠❡❤♦✇ ❛r❜✐tr❛r② ♠♦❞❡❧❧✐♥❣ ❝❤♦✐❝❡✮ a ❛♥❞ b ❛r❡ ✉♥❦♥♦✇♥ ❞❡t❡r♠✐♥✐st✐❝ ❝♦❡✛✜❝✐❡♥ts t❤❡ ε i ✬s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t r❡❛❧✐s❛t✐♦♥s ♦❢ ❛ N (0 , σ 2 ) ✈❛r✐❛❜❧❡ ✭♠❛❞❡ ❢♦r ❝♦♥✈✐❡♥✐❡♥❝❡✱ ❝♦♥s✐st❡♥❝② ✇✐t❤ ❞❛t❛ ❤❛s t♦ ❜❡ ❝❤❡❝❦❡❞✱ s❡❡ ❧❛t❡r✮ ❚❤❡r❡ ❛r❡ t❤r❡❡ ♣❛r❛♠❡t❡rs ✐♥ t❤✐s ♣r♦❜❧❡♠✿ ( a, b, σ ) = θ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✾ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ▲❡❛st sq✉❛r❡ ❡rr♦r ❡st✐♠❛t✐♦♥ ❋♦r ❛r❜✐tr❛r② ✈❛❧✉❡s a ❛♥❞ b ✱ e i = y i − ax i − b ♠❡❛s✉r❡s t❤❡ ❡rr♦r ♠❛❞❡ ❜② t❤❡ ❧✐♥❡❛r ♠♦❞❡❧ ♦♥ ♦❜s✳ i ❆ ❣♦♦❞ ♠♦❞❡❧ s❤♦✉❧❞ ②✐❡❧❞ ❧♦✇ ❡rr♦rs ♦♥ ❛❧❧ ♦❜s❡r✈❛t✐♦♥s ❉❡✜♥✐t✐♦♥✿ t❤❡ ▲❡❛st ❙q✉❛r❡ ❡st✐♠❛t♦r ❚❤❡ ✉♥❦♥♦✇♥ ♣❛r❛♠❡t❡rs ( a, b ) ❛r❡ ❡st✐♠❛t❡❞ ❛s � ( a, b ) LS t❤❛t ❥♦✐♥t❧② ♠✐♥✐♠✐③❡ � i ( y i − ax i − b ) 2 ✳ ■♥ ♠❛t❤ st②❧❡✿ � � ( y i − ax i − b ) 2 ( a, b ) LS = Argmin a,b i ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✵ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ❈♦♥♥❡❝t✐♦♥ ✇✐t❤ ❧❡❝t✉r❡ ♦♥ ❙t❛t✐st✐❝❛❧ ❊st✐♠❛t✐♦♥ ✭❧❡❝t✉r❡ ✶✮ ❚❤❡ ✈❡❝t♦r � ( a, b ) LS ✐s ❛♥ ❡st✐♠❛t❡ ♦❢ ( a, b ) → � ❚❤❡ ♣r♦❝❡❞✉r❡ ❉❛t❛ − ( a, b ) LS ✱ ✐✳❡✳ t❤❡ ❣❡♥❡r✐❝ ♣r♦❝❡ss ❛ss♦❝✐❛t✐♥❣ ❛♥ ❡st✐♠❛t❡ t♦ ❛ ❞❛t❛s❡t ✐s ❛♥ ❡st✐♠❛t♦r ❚❤✐s ♣r♦❝❡❞✉r❡ ♦r ❢✉♥❝t✐♦♥ ✐s ❞❡t❡r♠✐♥✐st✐❝ ✐♥ t❤❡ s❡♥s❡ t❤❛t t❤❡ s❛♠❡ ❞❛t❛s❡t ✇✐❧❧ ②✐❡❧❞ t❤❡ s❛♠❡ ❡st✐♠❛t❡ ■♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ t❤✐s ❝♦✉rs❡✱ ❉❛t❛ = ( x i , y i ) i =1 ,...,n ✇✐t❤ Y i = ax i + b + ε i ❛♥❞ ✇❤❡r❡ ε i ✐s ❛ r❛♥❞♦♠ ✈❛r✐❛❜❧❡ ❉❛t❛ ❛r❡ r❛♥❞♦♠ t❤❡r❡❢♦r❡ � ( a, b ) LS s❤♦✉❧❞ ❜❡ s❡❡♥ ❛s r❛♥❞♦♠✳ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✶ ✴ ✸✸

P❛r❛♠❡t❡r ❡st✐♠❛t✐♦♥ ❘❡♠❛r❦s ❛♥❞ q✉❡st✐♦♥s ♦♥ t❤❡ ▲❡❛st ❙q✉❛r❡ ♣r✐♥❝✐♣❧❡ ❲❤❛t ✐❢ ✇❡ ❛tt❡♠♣t t♦ ♠✐♥✐♠✐③❡ � | y i − ax i − b | ❄ ❲❤❛t ✐❢ ✇❡ s✇❛♣ x ❛♥❞ y ❄ ❲❤❛t ✐❢ t❤❡ ❞❛t❛ ♣♦✐♥ts ❛r❡ ❛♣♣r♦①✐♠❛t❡❧② ❧♦❝❛t❡❞ ♦♥ ❛ ❝✐r❝❧❡❄ ❲❤② t❤❡ sq✉❛r❡❞ ❡rr♦r❄ ●✐❧❧❡s ●✉✐❧❧♦t ✭❣✐❣✉❅❞t✉✳❞❦✮ ▲✐♥❡❛r r❡❣r❡ss✐♦♥ ❙❡♣t❡♠❜❡r ✶✼✱ ✷✵✶✸ ✶✷ ✴ ✸✸