sttst ssts - PowerPoint PPT Presentation

❚♦♣♦❧♦❣② ♦❢ st❛t✐st✐❝❛❧ s②st❡♠s ❆ ❝♦❤♦♠♦❧♦❣✐❝❛❧ ❛♣♣r♦❛❝❤ t♦ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r② ❏✉❛♥ P❛❜❧♦ ❱✐❣♥❡❛✉① ❆r✐③tí❛ ■▼❏✲P❘● ❯♥✐✈❡rs✐té P❛r✐s ❉✐❞❡r♦t P❛r✐s✱ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✶ ✴ ✹✺

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s ●❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r② ✷ ■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ✸ ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✷ ✴ ✹✺

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s ●❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r② ✷ ■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ✸ ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✸ ✴ ✹✺

✏■❢ ❛ ❝❤♦✐❝❡ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ ✐♥t♦ t✇♦ s✉❝❝❡ss✐✈❡ ❝❤♦✐❝❡s✱ t❤❡ ♦r✐❣✐♥❛❧ ✶ s❤♦✉❧❞ ❜❡ t❤❡ ✇❡✐❣❤t❡❞ s✉♠ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ✈❛❧✉❡s ♦❢ ✶ ✳✑ ✶ ✶ ✷ ✶ ✶ ✷ ✸ ✶ ✶ ✷ ✶ ✷ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✸ ✶ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ❙❤❛♥♥♦♥ ✭✶✾✹✽✮ ❝❤❛r❛❝t❡r✐③❡❞ t❤❡ ❢✉♥❝t✐♦♥s S ✶ : = S ( n ) : ∆ n → R ✱ ❣✐✈❡♥ ❜② ✶ n � S ✶ ( p ✵ ,..., p n ) : = − p i ln p i , ✭✶✮ k = ✵ ❛s t❤❡ ♦♥❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✮✱ s✉❝❤ t❤❛t S ✶ ( ✶ / n ,..., ✶ / n ) ✐s ♠♦♥♦t♦♥✐❝ ✐♥ n ❛♥❞ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹ ✴ ✹✺

✶ ✶ ✷ ✶ ✶ ✷ ✸ ✶ ✶ ✷ ✶ ✷ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✸ ✶ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ❙❤❛♥♥♦♥ ✭✶✾✹✽✮ ❝❤❛r❛❝t❡r✐③❡❞ t❤❡ ❢✉♥❝t✐♦♥s S ✶ : = S ( n ) : ∆ n → R ✱ ❣✐✈❡♥ ❜② ✶ n � S ✶ ( p ✵ ,..., p n ) : = − p i ln p i , ✭✶✮ k = ✵ ❛s t❤❡ ♦♥❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✮✱ s✉❝❤ t❤❛t S ✶ ( ✶ / n ,..., ✶ / n ) ✐s ♠♦♥♦t♦♥✐❝ ✐♥ n ❛♥❞ ✏■❢ ❛ ❝❤♦✐❝❡ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ ✐♥t♦ t✇♦ s✉❝❝❡ss✐✈❡ ❝❤♦✐❝❡s✱ t❤❡ ♦r✐❣✐♥❛❧ S ✶ s❤♦✉❧❞ ❜❡ t❤❡ ✇❡✐❣❤t❡❞ s✉♠ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ✈❛❧✉❡s ♦❢ S ✶ ✳✑ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹ ✴ ✹✺

❙❤❛♥♥♦♥ ❡♥tr♦♣② ❙❤❛♥♥♦♥ ✭✶✾✹✽✮ ❝❤❛r❛❝t❡r✐③❡❞ t❤❡ ❢✉♥❝t✐♦♥s S ✶ : = S ( n ) : ∆ n → R ✱ ❣✐✈❡♥ ❜② ✶ n � S ✶ ( p ✵ ,..., p n ) : = − p i ln p i , ✭✶✮ k = ✵ ❛s t❤❡ ♦♥❧② ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ✭✉♣ t♦ ❛ ♠✉❧t✐♣❧✐❝❛t✐✈❡ ❝♦♥st❛♥t✮✱ s✉❝❤ t❤❛t S ✶ ( ✶ / n ,..., ✶ / n ) ✐s ♠♦♥♦t♦♥✐❝ ✐♥ n ❛♥❞ ✏■❢ ❛ ❝❤♦✐❝❡ ❜❡ ❜r♦❦❡♥ ❞♦✇♥ ✐♥t♦ t✇♦ s✉❝❝❡ss✐✈❡ ❝❤♦✐❝❡s✱ t❤❡ ♦r✐❣✐♥❛❧ S ✶ s❤♦✉❧❞ ❜❡ t❤❡ ✇❡✐❣❤t❡❞ s✉♠ ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ ✈❛❧✉❡s ♦❢ S ✶ ✳✑ p ✶ p ✶ + p ✷ S ✶ ( p ✶ , p ✷ , p ✸ ) = p ✶ p ✶ + p ✷ � � p ✶ p ✷ S ✶ ( p ✶ + p ✷ , p ✸ ) + ( p ✶ + p ✷ ) S ✶ p ✷ , p ✷ p ✶ + p ✷ p ✶ + p ✷ p ✶ + p ✷ p ✸ p ✸ ✶ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✹ ✴ ✹✺

✶ ✶ ✷ ✶ ✶ ✷ ✸ ✶ ✶ ✷ ✶ ✷ ✶ ✶ ✷ ✸ ✶ ✷ ✶ ✷ ✷ ✶ ✷ ✶ ✷ ✶ ✷ ✸ ✸ ✶ ❲❤② t❤❡s❡ ❛①✐♦♠s❄ ❲❤❛t ✐s t❤❡✐r r♦❧❡ ✐♥ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②❄ ❙tr✉❝t✉r❛❧ α ✲❡♥tr♦♣② ✭❚s❛❧❧✐s α ✲❡♥tr♦♣②✮ ❍❛✈r❞❛ ❛♥❞ ❈❤❛r✈át ✐♥tr♦❞✉❝❡❞ ❛ ❣❡♥❡r❛❧✐③❡❞ ❡♥tr♦♣②✱ ❢♦r ❡❛❝❤ α > ✵✱ α �= ✶✿ � � n � p α S α ( p ✵ ,..., p n ) = K α ✶ − . ✭✷✮ i i = ✵ ■t s❛t✐s✜❡s ❛ ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛①✐♦♠s✳ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✺ ✴ ✹✺

❲❤② t❤❡s❡ ❛①✐♦♠s❄ ❲❤❛t ✐s t❤❡✐r r♦❧❡ ✐♥ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②❄ ❙tr✉❝t✉r❛❧ α ✲❡♥tr♦♣② ✭❚s❛❧❧✐s α ✲❡♥tr♦♣②✮ ❍❛✈r❞❛ ❛♥❞ ❈❤❛r✈át ✐♥tr♦❞✉❝❡❞ ❛ ❣❡♥❡r❛❧✐③❡❞ ❡♥tr♦♣②✱ ❢♦r ❡❛❝❤ α > ✵✱ α �= ✶✿ � � n � p α S α ( p ✵ ,..., p n ) = K α ✶ − . ✭✷✮ i i = ✵ ■t s❛t✐s✜❡s ❛ ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛①✐♦♠s✳ p ✶ p ✶ + p ✷ S ✶ ( p ✶ , p ✷ , p ✸ ) = p ✶ p ✶ + p ✷ � � p ✶ p ✷ S ✶ ( p ✶ + p ✷ , p ✸ ) + ( p ✶ + p ✷ ) α S ✶ p ✷ , p ✷ p ✶ + p ✷ p ✶ + p ✷ p ✶ + p ✷ p ✸ p ✸ ✶ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✺ ✴ ✹✺

❙tr✉❝t✉r❛❧ α ✲❡♥tr♦♣② ✭❚s❛❧❧✐s α ✲❡♥tr♦♣②✮ ❍❛✈r❞❛ ❛♥❞ ❈❤❛r✈át ✐♥tr♦❞✉❝❡❞ ❛ ❣❡♥❡r❛❧✐③❡❞ ❡♥tr♦♣②✱ ❢♦r ❡❛❝❤ α > ✵✱ α �= ✶✿ � � n � p α S α ( p ✵ ,..., p n ) = K α ✶ − . ✭✷✮ i i = ✵ ■t s❛t✐s✜❡s ❛ ❞❡❢♦r♠❡❞ ✈❡rs✐♦♥ ♦❢ t❤❡ ❛①✐♦♠s✳ p ✶ p ✶ + p ✷ S ✶ ( p ✶ , p ✷ , p ✸ ) = p ✶ p ✶ + p ✷ � � p ✶ p ✷ S ✶ ( p ✶ + p ✷ , p ✸ ) + ( p ✶ + p ✷ ) α S ✶ p ✷ , p ✷ p ✶ + p ✷ p ✶ + p ✷ p ✶ + p ✷ p ✸ p ✸ ✶ ❲❤② t❤❡s❡ ❛①✐♦♠s❄ ❲❤❛t ✐s t❤❡✐r r♦❧❡ ✐♥ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r②❄ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✺ ✴ ✹✺

❖✉t❧✐♥❡ ■♥tr♦❞✉❝t✐♦♥ ✶ ❊♥tr♦♣② ❛♥❞ r❡❝✉rr❡♥❝❡ ❈♦♠❜✐♥❛t♦r✐❝s ●❡♥❡r❛❧✐③❡❞ ✐♥❢♦r♠❛t✐♦♥ t❤❡♦r② ✷ ■♥❢♦r♠❛t✐♦♥ str✉❝t✉r❡s ❛♥❞ t❤❡✐r ❝♦❤♦♠♦❧♦❣② ✸ ❋♦✉♥❞❛t✐♦♥s ❈♦❤♦♠♦❧♦❣② ♦❢ ❞✐s❝r❡t❡ ✈❛r✐❛❜❧❡s ❈♦❤♦♠♦❧♦❣② ♦❢ ❝♦♥t✐♥✉♦✉s ✭❣❛✉ss✐❛♥✮ ✈❛r✐❛❜❧❡s ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✻ ✴ ✹✺

❋r♦♠ t❤❡ ♣♦✐♥t ♦❢ ✈✐❡✇ ♦❢ ♣r♦❜❛❜✐❧✐t② ❛♥❞ ❝♦♠❜✐♥❛t♦r✐❝s✱ ❙❤❛♥♥♦♥ ❡♥tr♦♣② ❛♣♣❡❛rs ♥❛t✉r❛❧❧② ✐♥ t❤❡ ❛s②♠♣t♦t✐❝ ❢♦r♠✉❧❛ ✶ ✶ ✶ ✭✸✮ ✶ ✶ ✶ ▼✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥ts ❚❤❡ ♠✉❧t✐♥♦♠✐❛❧ ❝♦❡✣❝✐❡♥t � � n ! Γ ( n + ✶ ) n : = k ✶ ! ··· k s ! = Γ ( k ✶ + ✶ ) ··· Γ ( k s + ✶ ) k ✶ ,..., k s ❝♦✉♥ts t❤❡ ♥✉♠❜❡r ♦❢ ✇♦r❞s w ∈ Σ n ✱ ✇✐t❤ Σ = { σ ✶ ,..., σ s } ✱ s✉❝❤ t❤❛t σ i ❛♣♣❡❛rs k i t✐♠❡s✳ ❏✉♥❡ ✶✹✱ ✷✵✶✾ ✼ ✴ ✹✺