rt ts rs r - PowerPoint PPT Presentation

❈r✐t✐❝❛❧ ❡①♣♦♥❡♥ts ♦❢ ❣r❛♣❤s ❆♣♦♦r✈❛ ❑❤❛r❡ ❙t❛♥❢♦r❞ ❯♥✐✈❡rs✐t② ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❉♦♠✐♥✐q✉❡ ●✉✐❧❧♦t ✭❯✳ ❉❡❧❛✇❛r❡✮ ❛♥❞ ❇❛❧❛ ❘❛❥❛r❛t♥❛♠ ✭❙t❛♥❢♦r❞✮ ❆▼❙ ❙❡❝t✐♦♥❛❧ ▼❡❡t✐♥❣✱ ▲♦②♦❧❛ ❖❝t♦❜❡r ✸✱ ✷✵✶✺

❘❛✐s❡ ❡❛❝❤ ❡♥tr② t♦ t❤❡ t❤ ♣♦✇❡r ❢♦r s♦♠❡ ✳ ❲❤❡♥ ✐s t❤❡ r❡s✉❧t✐♥❣ ♠❛tr✐① ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡❄ ❲♦r❦✐♥❣ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✳ ❙✉♣♣♦s❡   1 0 . 6 0 . 5 0 0 0 . 6 1 0 . 6 0 . 5 0     A = 0 . 5 0 . 6 1 0 . 6 0 . 5 .     0 0 . 5 0 . 6 1 0 . 6   0 0 0 . 5 0 . 6 1 ✷

❲♦r❦✐♥❣ ❡①❛♠♣❧❡ ◗✉❡st✐♦♥✳ ❙✉♣♣♦s❡   1 0 . 6 0 . 5 0 0 0 . 6 1 0 . 6 0 . 5 0     A = 0 . 5 0 . 6 1 0 . 6 0 . 5 .     0 0 . 5 0 . 6 1 0 . 6   0 0 0 . 5 0 . 6 1 ❘❛✐s❡ ❡❛❝❤ ❡♥tr② t♦ t❤❡ α t❤ ♣♦✇❡r ❢♦r s♦♠❡ α > 0 ✳ ❲❤❡♥ ✐s t❤❡ r❡s✉❧t✐♥❣ ♠❛tr✐① ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡❄ ✷

▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t ✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ✈❡❝t♦r ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ ■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ ❣✐✈❡♥ ❞❛t❛ ♦❢ ✳ ▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s ●r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X 1 , . . . , X p ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ✸

❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① ♦❢ t❤❡ ✈❡❝t♦r ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ ■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ ❣✐✈❡♥ ❞❛t❛ ♦❢ ✳ ▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s ●r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X 1 , . . . , X p ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ X i ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t X i ✳ ✸

■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ ❣✐✈❡♥ ❞❛t❛ ♦❢ ✳ ▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s ●r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X 1 , . . . , X p ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ X i ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t X i ✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ ♦❢ t❤❡ ✈❡❝t♦r ( X 1 , . . . , X p ) ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ Σ = ( σ jk ) p j,k =1 = (Cov( X j , X k )) p j,k =1 , ✸

▼♦t✐✈❛t✐♦♥ ❢r♦♠ ❤✐❣❤✲❞✐♠❡♥s✐♦♥❛❧ st❛t✐st✐❝s ●r❛♣❤✐❝❛❧ ♠♦❞❡❧s✿ ❈♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ st❛t✐st✐❝s ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s✳ ▲❡t X 1 , . . . , X p ❜❡ ❛ ❝♦❧❧❡❝t✐♦♥ ♦❢ r❛♥❞♦♠ ✈❛r✐❛❜❧❡s✳ ■♥ ❛ ✈❡r② ❧❛r❣❡ ✈❡❝t♦r✱ ✐t ✐s r❛r❡ t❤❛t ❛❧❧ t❤❡ ✈❛r✐❛❜❧❡s ❞❡♣❡♥❞ str♦♥❣❧② ♦♥ ❡❛❝❤ ♦t❤❡r✳ ▼❛♥② ✈❛r✐❛❜❧❡s ❛r❡ ✐♥❞❡♣❡♥❞❡♥t ♦r ❝♦♥❞✐t✐♦♥❛❧❧② ✐♥❞❡♣❡♥❞❡♥t✳ ❱❛r✐❛❜❧❡s ✐♥❞❡♣❡♥❞❡♥t ♦❢ X i ❛r❡ ♦❢ ♥♦ ✉s❡ t♦ ♣r❡❞✐❝t X i ✳ ❚❤❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐① Σ ♦❢ t❤❡ ✈❡❝t♦r ( X 1 , . . . , X p ) ❝❛♣t✉r❡s ❧✐♥❡❛r r❡❧❛t✐♦♥s❤✐♣s ❜❡t✇❡❡♥ ✈❛r✐❛❜❧❡s✿ Σ = ( σ jk ) p j,k =1 = (Cov( X j , X k )) p j,k =1 , ■♠♣♦rt❛♥t ♣r♦❜❧❡♠✿ ❊st✐♠❛t❡ Σ ❣✐✈❡♥ ❞❛t❛ x 1 , . . . , x n ∈ R p ♦❢ ( X 1 , . . . , X p ) ✳ ✸

✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ ❤❛s r❛♥❦ ❛t ♠♦st ✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ ✱ s♠❛❧❧ ✑ ♣r♦❜❧❡♠s✱ ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ ✳ ▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ ✈❛r✐❛❜❧❡s✳ ❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ n 1 ( x j − x )( x j − x ) T . � S := n − 1 j =1 ✹

■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ ✱ s♠❛❧❧ ✑ ♣r♦❜❧❡♠s✱ ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ ✳ ▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ ✈❛r✐❛❜❧❡s✳ ❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ n 1 ( x j − x )( x j − x ) T . � S := n − 1 j =1 S ✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ S ❤❛s r❛♥❦ ❛t ♠♦st n ✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ✹

▼♦❞❡r♥ ❛♣♣r♦❛❝❤✿ ❈♦♥✈❡① ♦♣t✐♠✐③❛t✐♦♥✿ ♦❜t❛✐♥ s♣❛rs❡ ❡st✐♠❛t❡ ♦❢ ✭❡✳❣✳✱ ♣❡♥❛❧✐③❡❞ ❧✐❦❡❧✐❤♦♦❞ ♠❡t❤♦❞s✮ ❲♦r❦s ✇❡❧❧ ❢♦r ❞✐♠❡♥s✐♦♥s ✉♣ t♦ ❛ ❢❡✇ t❤♦✉s❛♥❞s✳ ❉♦❡s ♥♦t s❝❛❧❡ t♦ ♠♦❞❡r♥ ♣r♦❜❧❡♠s ✇✐t❤ ✈❛r✐❛❜❧❡s✳ ❈♦✈❛r✐❛♥❝❡ ❡st✐♠❛t✐♦♥ ❈❧❛ss✐❝❛❧ ❡st✐♠❛t♦r ✭s❛♠♣❧❡ ❝♦✈❛r✐❛♥❝❡ ♠❛tr✐①✮✿ n 1 ( x j − x )( x j − x ) T . � S := n − 1 j =1 S ✐s ♣♦s✐t✐✈❡ s❡♠✐❞❡✜♥✐t❡✳ S ❤❛s r❛♥❦ ❛t ♠♦st n ✳ ❉❡♥s❡ ♠❛tr✐① ✭♥♦ ❣r❛♣❤✐❝❛❧ str✉❝t✉r❡✮✳ ■♥ ♠♦❞❡r♥ ✏❧❛r❣❡ p ✱ s♠❛❧❧ n ✑ ♣r♦❜❧❡♠s✱ S ✐s ❦♥♦✇♥ t♦ ❜❡ ❛ ♣♦♦r ❡st✐♠❛t♦r ♦❢ Σ ✳ ✹