tt strs tr r - - PowerPoint PPT Presentation
tt strs tr r sts strs r t
❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts
❚❛❣❡ ❖st❡rs❡♥ ❆❞✈❛♥❝❡❞ ❍❡r❞ ▼❛♥❛❣❡♠❡♥t
✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✴✸✵
✶ ❇❛❝❦❣r♦✉♥❞ ✷ ❉❛t❛ ✸ ❉✉r❛t✐♦♥ ♦❢ ❱✐s✐ts ✹ ❇r❡❛❦ ✺ ❋r❡q✉❡♥❝② ♦❢ ❱✐s✐ts ✻ ❈♦♠❜✐♥✐♥❣ ❋r❡q✉❡♥❝② ❛♥❞ ❉✉r❛t✐♦♥ ♦❢ ❇♦❛r ❱✐s✐ts ✼ ❈♦♥❝❧✉s✐♦♥
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✴✸✵
❇❛❝❦❣r♦✉♥❞ ✶
❛r❡ ❢❡❞ ❜② ❊❙❋
❤♦✉s❡❞
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✸✴✸✵
❇❛❝❦❣r♦✉♥❞ ✷
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✹✴✸✵
❇❛❝❦❣r♦✉♥❞ ✸
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✺✴✸✵
P✉r♣♦s❡ ♦❢ ▼♦♥✐t♦r✐♥❣ ❱✐s✐ts t♦ ❛ ❇♦❛r
r❡♣r♦❞✉❝t✐♦♥ ❝②❝❧❡
t✐♠❡ t♦ ✐♥s❡♠✐♥❛t❡ t❤❡ s♦✇
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✻✴✸✵
❘❛✇ ❉❛t❛
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✼✴✸✵
❈❤❛r❛❝t❡r✐st✐❝s ♦❢ t❤❡ ❘❛✇ ❉❛t❛
♠♦r❡ ❞✐st✐♥❝t
♦❜s❡r✈❛t✐♦♥s ❞❡❝❧✐♥❡ st❡❡♣❧②
❝❤❛♥❣❡ ✐♥ ❞✉r❛t✐♦♥ ❛♥❞ ✐♥ ❢r❡q✉❡♥❝②
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✽✴✸✵
❖✈❡r✈✐❡✇
❞✐str✐❜✉t❡❞
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✾✴✸✵
❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▼♦❞❡❧ ❉❡s❝r✐❜✐♥❣ ❉✉r❛t✐♦♥
♥❡❛r t❤❡ ❜♦❛r ♣❡r ♦❜s❡r✈❛t✐♦♥ ❤♦✉r
❞✐❞ ♥♦t ✈✐s✐ts t❤❡ ❜♦❛r ❛r❡ ❡①❝❧✉❞❡❞ ❖❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥ : ❨t = µt + ✈t, ✇❤❡r❡ ✈t ∼ ◆(✵, ❱ ) ❙②st❡♠ ❡q✉❛t✐♦♥ : µt = µt−✶ + ✇t, ✇❤❡r❡ ✇t ∼ ◆(✵, ❲ ) ◆❇✿ ❆ ❉▲▼ ✜❧t❡rs r❛✇ ❞❛t❛ ✕ ♣r♦✈✐❞❡s ❛♥ ❡st✐♠❛t❡ ♦❢ t❤❡ ✉♥❞❡r❧②✐♥❣ ❧❡✈❡❧ ✭
t✮ ❛♥❞ t❤❡ ✈❛r✐❛♥❝❡ ✭❱ ✮ ❜② ♠❡❛♥s
♦❢ ❛ ❑❛❧♠❛♥ ✜❧t❡r
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✵✴✸✵
❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▼♦❞❡❧ ❉❡s❝r✐❜✐♥❣ ❉✉r❛t✐♦♥
♥❡❛r t❤❡ ❜♦❛r ♣❡r ♦❜s❡r✈❛t✐♦♥ ❤♦✉r
❞✐❞ ♥♦t ✈✐s✐ts t❤❡ ❜♦❛r ❛r❡ ❡①❝❧✉❞❡❞ ❖❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥ : ❨t = µt + ✈t, ✇❤❡r❡ ✈t ∼ ◆(✵, ❱ ) ❙②st❡♠ ❡q✉❛t✐♦♥ : µt = µt−✶ + ✇t, ✇❤❡r❡ ✇t ∼ ◆(✵, ❲ )
t❤❡ ✉♥❞❡r❧②✐♥❣ ❧❡✈❡❧ ✭µt✮ ❛♥❞ t❤❡ ✈❛r✐❛♥❝❡ ✭❱ ✮ ❜② ♠❡❛♥s ♦❢ ❛ ❑❛❧♠❛♥ ✜❧t❡r
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✵✴✸✵
❚❤❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✭❝❧❛ss ■■✮
❛♣♣r♦♣r✐❛t❡ ♠♦❞❡❧ ✇❡✐❣❤s t❤❡ ♠♦st
Pr❡s❡♥t ❝❛s❡✿ ✹ ♠♦❞❡❧s
◆♦r♠❛❧ ♠♦❞❡❧ ❖✉t❧✐❡r ♠♦❞❡❧ ▲❡✈❡❧ s❤✐❢t ♠♦❞❡❧ ❖❡str✉s ♠♦❞❡❧
✹ ♠♦❞❡❧s ❡♥t❛✐❧ ✻✹ ♠♦❞❡❧ ❝♦♠❜✐♥❛t✐♦♥s✱ ✇❤❡♥ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ❧❛st t❤r❡❡ ♦❜s❡r✈❛t✐♦♥s ✭✹✸ ✻✹✮
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✶✴✸✵
❚❤❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✭❝❧❛ss ■■✮
❛♣♣r♦♣r✐❛t❡ ♠♦❞❡❧ ✇❡✐❣❤s t❤❡ ♠♦st
❝♦♥s✐❞❡r✐♥❣ t❤❡ ❧❛st t❤r❡❡ ♦❜s❡r✈❛t✐♦♥s ✭✹✸ = ✻✹✮
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✶✴✸✵
❯♣❞❛t✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❛ s✐♠♣❧❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✶
(✐❥❦) = (▼t−✷, ▼t−✶, ▼t)
❛t(✐❥❦) = ♠t−✶(✐❥) ❘t(✐❥❦) = ❈t−✶(✐❥) + ❲t(❦) ❖♥❡ st❡♣ ❢♦r❡❝❛st ❛t t✐♠❡ t ❢t ✐❥❦ ❛t ✐❥❦ ◗t ✐❥❦ ❘t ✐❥❦ ❱t ❦
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✷✴✸✵
❯♣❞❛t✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❛ s✐♠♣❧❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✶
(✐❥❦) = (▼t−✷, ▼t−✶, ▼t)
❛t(✐❥❦) = ♠t−✶(✐❥) ❘t(✐❥❦) = ❈t−✶(✐❥) + ❲t(❦)
❢t(✐❥❦) = ❛t(✐❥❦) ◗t(✐❥❦) = ❘t(✐❥❦) + ❱t(❦)
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✷✴✸✵
❯♣❞❛t✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❛ s✐♠♣❧❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✷
♠t(✐❥❦) = ❛t(✐❥❦) + ❆t(✐❥❦) · ❡t(✐❥❦) ❈t(✐❥❦) = ❆t(✐❥❦) · ❱t(✐❥❦) ❆t(✐❥❦) = ❘t(✐❥❦) · ◗−✶
t
(✐❥❦) ❡t(✐❥❦) = ❨t − ❢t(✐❥❦)
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✸✴✸✵
❯♣❞❛t✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❛ s✐♠♣❧❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✸
♣t(✐❥❦) = ❢ (❨t|❢t(✐❥❦), ◗t(✐❥❦)) · ♣t−✶(✐❥)·♣(❦)
❝t
✱✇❤❡r❡ ❢ ✐s t❤❡ ❞❡♥s✐t② ❢✉♥❝t✐♦♥ ♦❢ ❛ ♥♦r♠❛❧ ❞✐str✐❜✉t✐♦♥ ❛♥❞ ❝t ✐s ❛ ♥♦r♠❛❧✐③✐♥❣ ❝♦♥st❛♥t s♦ t❤❛t
−2 2 4 6 0.0 0.1 0.2 0.3 0.4 Model Probability Var = 1 Var = 2.25 Var = 25
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✹✴✸✵
❯♣❞❛t✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❛ s✐♠♣❧❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✹
♠t =
✐
❈t =
✐
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✺✴✸✵
❯♣❞❛t✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❛ s✐♠♣❧❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✺
♣t(❥❦) =
✐ ♣t(✐❥❦)
♠t(❥❦) =
✐ ♠t(✐❥❦)♣t(✐❥❦)/♣t(❥❦)
❈t(❥❦) =
✐
· ♣t(✐❥❦)
♣t(❥❦)
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✻✴✸✵
❯♣❞❛t✐♥❣ ❊q✉❛t✐♦♥s ♦❢ ❛ s✐♠♣❧❡ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✻
t❤❡ s❛♠❡ ❛s ♥♦r♠❛❧ ❉▲▼
♣r♦❜❛❜✐❧✐t② ❛❝❝♦r❞✐♥❣ t♦ ❢♦r❡❝❛st❡❞ ❧❡✈❡❧ ❛♥❞ ✈❛r✐❛♥❝❡
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✼✴✸✵
❋✐tt✐♥❣ ❛ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ t♦ ❇♦❛r ❱✐s✐t ❉❛t❛
❤✐❣❤❡r ✈❛r✐❛♥❝❡
✐♥✢✉❡♥❝❡ t❤❡ ❜❡❧✐❡❢ ✐♥ ♦❡str✉s ♣t ✐❥❦ ❢ ❨t ❢t ✐❥❦ ◗t ✐❥❦
♣t
✶ ✐❥
♣ ❦ ❝t
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✽✴✸✵
❋✐tt✐♥❣ ❛ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ t♦ ❇♦❛r ❱✐s✐t ❉❛t❛
❤✐❣❤❡r ✈❛r✐❛♥❝❡
✐♥✢✉❡♥❝❡ t❤❡ ❜❡❧✐❡❢ ✐♥ ♦❡str✉s
❝t
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✽✴✸✵
❋✐tt✐♥❣ ❛ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ t♦ ❇♦❛r ❱✐s✐t ❉❛t❛
▼(α✶) ✖ ◆♦r♠❛❧ ♠♦❞❡❧ ✇✐t❤ ♦❜s❡r✈❛t✐♦♥❛❧ ✈❛r✐❛♥❝❡ ❢❛❝t♦r ❈❦ = ✶✱ ❞✐s❝♦✉♥t ❢❛❝t♦r δ❦ = ✵.✾✾ ❛♥❞ ✜①❡❞ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② π = π◆ ▼(α✷) ✖ ❖✉t❧✐❡r ♠♦❞❡❧ ✇✐t❤ ♦❜s❡r✈❛t✐♦♥❛❧ ✈❛r✐❛♥❝❡ ❢❛❝t♦r ❈❦ = ✷✵✱ ❞✐s❝♦✉♥t ❢❛❝t♦r δ❦ = ✵.✾✾ ❛♥❞ ✜①❡❞ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② π = π❖ ▼(α✸) ✖ ▲❡✈❡❧ s❤✐❢t ♠♦❞❡❧ ✇✐t❤ ♦❜s❡r✈❛t✐♦♥❛❧ ✈❛r✐❛♥❝❡ ❢❛❝t♦r ❈❦ = ✶✱ ❞✐s❝♦✉♥t ❢❛❝t♦r δ❦ = ✵.✵✶ ❛♥❞ ✜①❡❞ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② π = π▲ ▼(α✹) ✖ ❖❡str✉s ♠♦❞❡❧ ✇✐t❤ ♦❜s❡r✈❛t✐♦♥❛❧ ✈❛r✐❛♥❝❡ ❢❛❝t♦r ❈❦ = ✷✵✱ ❞✐s❝♦✉♥t ❢❛❝t♦r δ❦ = ✵.✾✾ ❛♥❞ ✜①❡❞ tr❛♥s✐t✐♦♥ ♣r♦❜❛❜✐❧✐t② π = ✵
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✶✾✴✸✵
❘❡s✉❧ts ♦❢ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✶
4 5 6 7 8
Raw data and filtered mean (Sow no 18) Log seconds pr. hour
(a)
0.2 0.4 0.6 0.8 1
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 01 02 03 04 05 06
P(Mα4) Model probability
Date (b)
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✵✴✸✵
❘❡s✉❧ts ♦❢ ▼✉❧t✐♣r♦❝❡ss ❉▲▼ ✲ ✷
❚r✉❡ P♦s✐t✐✈❡ ❚r✉❡ P♦s✐t✐✈❡+❋❛❧s❡ ◆❡❣❛t✐✈❡
❚r✉❡ ◆❡❣❛t✐✈❡ ❚r✉❡ ◆❡❣❛t✐✈❡+❋❛❧s❡ P♦s✐t✈❡
❞❡✜♥❡❞ ❛s ❛♥ ❛❧❛r♠✲❜❧♦❝❦ ▼♦❞❡❧ ❇❧♦❝❦ ♦❢ ✷✹ ❤♦✉rs ❇❧♦❝❦ ♦❢ ✼✷ ❤♦✉rs ❚♦t❛❧ ❚P ✶✼✻ ✾✼ ❚♦t❛❧ ❚◆ ✸✶✻✶✶✹ ✶✵✺✻✺✵ ❚♦t❛❧ ❋P ✶✽✷✷ ✶✻✶✵ ❚♦t❛❧ ❋◆ ✶✺✺ ✶✹ ❙❡♥s✐t✐✈✐t② ✵✳✺✸✷ ✵✳✽✼✹ ❙♣❡❝✐✜❝✐t② ✵✳✾✾✹ ✵✳✾✽✺
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✶✴✸✵
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✷✴✸✵
❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▼♦❞❡❧ ❉❡s❝r✐❜✐♥❣ ❋r❡q✉❡♥❝②
❖❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥ ❨t
t ❱t
t
❡
t
✇❤❡r❡
t
❋t t ❙②st❡♠ ❡q✉❛t✐♦♥
t
✶
✇t ✇❤❡r❡ ✇t ✵ ❲ ❋ ❤✐❣❤
t
✶ ✵ ❋ ❧♦✇
t
✵ ✶
✶ ✵ ✵ ✶
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✸✴✸✵
❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▼♦❞❡❧ ❉❡s❝r✐❜✐♥❣ ❋r❡q✉❡♥❝②
❖❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥ : (❨t|ηt, ❱t) ∼ P(ληt) = P(❡ηt) , ✇❤❡r❡ : ηt = ❋ ′
tθt
❙②st❡♠ ❡q✉❛t✐♦♥ : θt =
❋ ❤✐❣❤
t
✶ ✵ ❋ ❧♦✇
t
✵ ✶
✶ ✵ ✵ ✶
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✸✴✸✵
❉❡✜♥✐t✐♦♥ ♦❢ t❤❡ ▼♦❞❡❧ ❉❡s❝r✐❜✐♥❣ ❋r❡q✉❡♥❝②
❖❜s❡r✈❛t✐♦♥ ❡q✉❛t✐♦♥ : (❨t|ηt, ❱t) ∼ P(ληt) = P(❡ηt) , ✇❤❡r❡ : ηt = ❋ ′
tθt
❙②st❡♠ ❡q✉❛t✐♦♥ : θt =
❋ ❤✐❣❤
t
= ✶ ✵
t
= ✵ ✶
✶ ✵ ✵ ✶
❙❧✐❞❡ ✷✸✴✸✵
▼♦♥✐t♦r✐♥❣ t❤❡ ❋r❡q✉❡♥❝② ♦❢ ❇♦❛r ❱✐s✐ts
♦❡str✉s ✐♥❞✐❝❛t♦r = ♦❜s❡r✈❡❞ ❢r❡q✉❡♥❝②−❢♦r❡❝❛st❡❞ ❢r❡q✉❡♥❝②
❢♦r❡❝❛st❡❞ ❢r❡q✉❡♥❝②+✶
10 20 30 40 50
Raw data, 1 step forecast and filtered mean (Sow no 18) Visits per 6 hours
(a)
10 20 30 40
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 01 02 03 04 05 06
Observered Frequency − Forecasted Frequency Forecasted Frequency + + 1
Oestrus Indicator
Date 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 01 02 03 04 05 06 (b)
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✹✴✸✵
❘❡s✉❧ts ♦❢ ❉●▲▼
▼♦❞❡❧ ❇❧♦❝❦ ♦❢ ✷✹ ❤♦✉rs ❇❧♦❝❦ ♦❢ ✼✷ ❤♦✉rs ❚♦t❛❧ ❚P ✶✻✽ ✾✻ ❚♦t❛❧ ❚◆ ✸✶✸✵✸✾ ✶✵✷✻✷✵ ❚♦t❛❧ ❋P ✹✽✾✼ ✹✻✹✵ ❚♦t❛❧ ❋◆ ✶✻✸ ✶✺ ❙❡♥s✐t✐✈✐t② ✵✳✺✵✽ ✵✳✽✻✺ ❙♣❡❝✐✜❝✐t② ✵✳✾✽✺ ✵✳✾✺✼
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✺✴✸✵
❈♦♠❜✐♥✐♥❣ ❋r❡q✉❡♥❝② ❛♥❞ ❉✉r❛t✐♦♥ ♦❢ ❇♦❛r ❱✐s✐ts
♠✉❧t✐♣r♦❝❡ss ❉▲▼ ❛s ❛ ♣r✐♦r ♣r♦❜❛❜✐❧✐t② ♦❢ ♦❡str✉s
✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❞✉r❛t✐♦♥ ♠♦❞❡❧ P(♦❡str✉s|+) = P(+|♦❡str✉s) P(+) · P(♦❡str✉s) = s❡♥s✐t✐✈✐t② · P(▼α✹) s❡♥s✐t✐✈✐t② · P(▼α✹) + (✶ − s♣❡❝✐✜❝✐t②) · (✶ − P(▼α✹))
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✻✴✸✵
❈♦♠❜✐♥✐♥❣ ❋r❡q✉❡♥❝② ❛♥❞ ❉✉r❛t✐♦♥ ♦❢ ❇♦❛r ❱✐s✐ts ✕ ❘❡s✉❧ts
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✼✴✸✵
❈♦♠❜✐♥✐♥❣ ❋r❡q✉❡♥❝② ❛♥❞ ❉✉r❛t✐♦♥ ♦❢ ❇♦❛r ❱✐s✐ts ✕ ❘❡s✉❧ts
❇❧♦❝❦ ♦❢ ✷✹ ❤♦✉rs ▼♦❞❡❧ ❈♦♠❜✐♥❡❞ ❉✉r❛t✐♦♥ ❋r❡q✉❡♥❝② ❚♦t❛❧ ❚P ✶✾✺ ✶✼✻ ✶✻✽ ❚♦t❛❧ ❚◆ ✸✶✹✷✵✾ ✸✶✻✶✶✹ ✸✶✸✵✸✾ ❚♦t❛❧ ❋P ✸✼✷✼ ✶✽✷✷ ✹✽✾✼ ❚♦t❛❧ ❋◆ ✶✸✻ ✶✺✺ ✶✻✸ ❙❡♥s✐t✐✈✐t② ✵✳✺✽✾ ✵✳✺✸✷ ✵✳✺✵✽ ❙♣❡❝✐✜❝✐t② ✵✳✾✽✽ ✵✳✾✾✹ ✵✳✾✽✺ ❊rr♦r ❘❛t❡ ✵✳✾✺ ✵✳✾✶✷ ✵✳✾✻✼
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✽✴✸✵
❈♦♠❜✐♥✐♥❣ ❋r❡q✉❡♥❝② ❛♥❞ ❉✉r❛t✐♦♥ ♦❢ ❇♦❛r ❱✐s✐ts ✕ ❘❡s✉❧ts
❇❧♦❝❦ ♦❢ ✷✹ ❤♦✉rs ▼♦❞❡❧ ❈♦♠❜✐♥❡❞ ❉✉r❛t✐♦♥ ❋r❡q✉❡♥❝② ❚♦t❛❧ ❚P ✶✾✺ ✶✼✻ ✶✻✽ ❚♦t❛❧ ❚◆ ✸✶✹✷✵✾ ✸✶✻✶✶✹ ✸✶✸✵✸✾ ❚♦t❛❧ ❋P ✸✼✷✼ ✶✽✷✷ ✹✽✾✼ ❚♦t❛❧ ❋◆ ✶✸✻ ✶✺✺ ✶✻✸ ❙❡♥s✐t✐✈✐t② ✵✳✺✽✾ ✵✳✺✸✷ ✵✳✺✵✽ ❙♣❡❝✐✜❝✐t② ✵✳✾✽✽ ✵✳✾✾✹ ✵✳✾✽✺ ❊rr♦r ❘❛t❡ ✵✳✾✺ ✵✳✾✶✷ ✵✳✾✻✼
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✽✴✸✵
❈♦♥❝❧✉s✐♦♥
t❤❛♥ ❞✉r❛t✐♦♥ ❛❧♦♥❡
❖t❤❡r ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ♦❡str✉s ✐♥ t❤❡ ✐♥❞✐✈✐❞✉❛❧ s♦✇ s❤♦✉❧❞ ❜❡ ✐♥❝❧✉❞❡❞
❋❛rr♦✇✐♥❣ r❛t❡ ♦❢ t❤❡ ❤❡r❞ Pr❡❣♥❛♥❝② t❡st ♦❢ t❤❡ ✐♥❞✐✈✐❞✉❛❧ s♦✇ ❆❝t✐✈✐t② ♠❡❛s✉r❡♠❡♥ts❄ ❉❛② ♦❢ r❡♣r♦❞✉❝t✐♦♥ ❝②❝❧❡ ❱✐s✉❛❧ ♦❜s❡r✈❛t✐♦♥s❄
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✾✴✸✵
❈♦♥❝❧✉s✐♦♥
t❤❛♥ ❞✉r❛t✐♦♥ ❛❧♦♥❡
s❤♦✉❧❞ ❜❡ ✐♥❝❧✉❞❡❞
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✷✾✴✸✵
❚❛❣❡ ❖st❡rs❡♥ ✖ ❉❡t❡❝t✐♦♥ ♦❢ ❖❡str✉s ❜② ▼♦♥✐t♦r✐♥❣ ❇♦❛r ❱✐s✐ts ✖ ✷✷✴✾ ✷✵✵✾ ❙❧✐❞❡ ✸✵✴✸✵