A Bayesian analysis of industrial lifetime data with Weibull distributions

HAL Id: inria-00115528

https://hal.inria.fr/inria-00115528v3

Submitted on 22 Nov 2006 (v3), last revised 23 Nov 2006 (v4)

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A Bayesian analysis of industrial lifetime data with Weibull distributions

Nicolas Bousquet

To cite this version:

Nicolas Bousquet. A Bayesian analysis of industrial lifetime data with Weibull distributions. [Research

Report] 2006, pp.24. �inria-00115528v3�

a p p o r t

d e r e c h e r c h e

Thème COG

A Bayesian analysis of industrial lifetime data with Weibull distributions

Nicolas Bousquet

N° ????

Novembre 2006

Unité de recherche INRIA Futurs

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” HKMCgiM

(a, b, β) > 0

“Y” bZiH0L?^`L?MaŒZ

l

[η ^k ] = b ^k/β Γ(a − k/β)

Γ(a) ∀ aβ > k > 0,

o`ausL@^YsYM2,

d [η] = (bβ/aβ + 1) ^1/β

–EGIHKb‘XdeoL?b‘’Vkb‘X]’^ŒXrMzs~ŒVXi]Co’‘MEZig+oauXrd•^`g\Lo"Zib‘^`a?±•b5–M`–

X ∼ GIG(a, b, β) ⇒

∀c > 0

^“

cX ∼ GIG (a, bc ^β , β)

²'” Hub‘]+H Lo™7MCX'bZbaŒZ\Mg\MCXrZib‘aK_Zi^·g\MpKg\MCXiMaŒZtpKg\b^7gtbaYd•^7giLoZib‘^`auX'^7a o0X\]o`’M puo`g\o`L?MZiMCgC– ,·^7giMC^"}`Mgz“Œbcd ”œMt]^`auXib‘sKMgIZiHuMtgiMCpuog+oL?MZ\gib‘¤CoZib‘^`a

µ = η ^−β

“K”œM'^7~YZ\o`bapKg\b^7goauspq^ŒX[Z\Mg\b^7g

µ|β ∼ G(a, b), µ|β, t _n ∼ G a + r, b +

n

X

i=1

t i

! .

GIHWŽuX˜ZiHKMkpq^ŒX[Z\Mg\b^7gœ]^7L?pKŽYZ\oZib‘^`a£± XrMCM

§

7²ŸbX˜Lo`sYMkMCo7Xrb‘Mgz–9.š^"”œMC}`Mgz“7—#MCgi_7Mg˜oaqs1TWŽKa°±rƒ67² sK^aK^`ZhpKg\^›

}WbsYMœošL@MzoaKb‘aK_šZi^Z\HKMhHWVWp=Mg\puog+oL?MZ\Mg+X

a

^oaus

b

–GIHKbXE]+HK^`b]M˜g\MLo`bauXE]^7aŒZig\^"}`Mg+Xib‘o`’yXib‘au]M

π(µ|β) = π(µ)

^–

GIHKMCa0”œM'pKg\^`p=^7XiMo`ao`’cZ\Mg\auo"Z\b}7Mt]+HK^`b]M7–

TWŽKpupq^ŒXrM°ZiHuoZZ\HKM "ŒŽuo`a7Z\bcZ+o"Z\b}7M£MYp=MgiZ^`pubaKb‘^`a ]oa ~qM¢giMCpKgiMzXrMCaŒZiMCs ~WV o ¥z¦¨§©ªu«¬h ¡Mb‘~KŽK’‘’tXio`L@pu’M

˜

x _m = (˜ x 1 , . . . , x ˜ m )

^–˜mMZ

π ^J (η|β) ∝ η ⁻¹

~=MkZ\HKM']^`aqsYbcZ\b^7auo’ 7M9giMCVYXhpKg\b‘^`gz–­;dE^`ŽKgI]^7ausYbZib‘^`auo`’=pugib‘^`g#^`a

η

]Coa~qMtg\MpKg\MCXiMaŒZ\MCsž~WV1ZiHKMtp=^7XrZiMCgib‘^`g

π ^J (η|β, x ˜ _m )

“K”œM'^7~YZ\o`ba







a = m,

b = b(a, β) =

a

P

i=1

˜ x ^β _i .

GIHWŽuX˜ZiHKM']+HK^`b]Mš^d

b

o`Xœotd•ŽKaq]Zib‘^`a1^d

β

o`pKpqMzog+XŸZ\^~qMauoZiŽKg+o’5–E—#MzXrbsYMCXC“

a

Z\o`™`MzXŸZiHuMšL?MCo`aKbau_^d9Z\HKMXib‘¤M

^`dEoO}Œb‘giZiŽuo`’9X\oL?pK’‘M'VWbMC’‘sYb‘aK_OZiHKMtX\oL?M'baKd•^`g\L?oZib‘^`ao`XœZiHKM'MWp=MgiZ ^`pKb‘aKb‘^`a– .šMau]M

a

bX#aK^"” o'%+«"¬¦§i«©¦21>

HWVWpqMCgipqog+oL?MZ\Mg Xrb‘au]MœZiHuM—#oyV7MCXib‘o`aoauo`’VYXrZ ]Coa@L?^WsKŽK’‘oZiM#ZiHuMIXrZig\Mau_ZiH@^dqZiHKMIpKg\b‘^`gªu«> u©5¦¨©;«©¦¨¥E^`pubaKb‘^`a

Z\HKgi^7ŽK_`HZ\HKb‘XXibL?pK’‘MHWVWpqMCpuog+oL?MZ\Mgz–#GIHub‘XkXio`L@pu’MXib‘¤MbXšo`aMzo`XiVb‘aŒZiMCgipKg\MZ+o~K’‘Mtpuo`g\o`L@MZiMCgoaus·o_7^W^Ws

d•^Y]ŽqXœp=^`b‘aŒZœd•^7g#oOsYbXi]ŽuXiXib‘^`až~=MZ[”#MMCažZiHuMo`auo’‘VYX[ZIoausžZiHKMMYp=MgiZC–  ¢M'pKg\^`p=^7XiMšXi^`L?MksKMdeoŽu’cZ]o`’b‘~Kg+o"Z\b^7a

}"o’‘ŽKMzX#d•^`g

a

<sup>o`aus</sup> "7ŽuMCXrZib^7aKb‘aK_XrŽK_7_`MzX[Z\b^7auX#ba

§

^!u–

­[sYMzo’‘’V7“

π(µ|β)

Huo7Xtoauo"Z\ŽKg\o`’ HKbMCg\o`g\]+Hub‘]Co’ŸX[Z\giŽq]ZiŽugiM7–

4

XiXiŽKL?b‘aK_

˜

x _m ^i.i.d. ∼ W(η e , β e )

^{” HKMCgiM}

(η e , β e )

o`giMpKg\b^7g MCXrZib‘L?oZib‘^`auXC“Y”#M_7MZ

b|β ∼ G (a, η _e ^−β ), µ|β, b ∼ G (a, b).

.š^"”œMC}`Mgz“9d•^`gXib‘L@pu’b]bZ[V°g\MCo7Xr^7auX”œM?pKg\Md•MCg'Zi^£]+HK^W^7XiM?o·sYMZ\Mg\L@b‘aKbX[ZMYpKgiMzXiXib‘^`a°^`d

b

b‘a¢d•Žuau]Z\b^7a¡^`dœZ\HKM oy}"ob‘’o~K’‘Mkpugib‘^`g XipqMz]bq]oZib‘^`auXC–˜GIHKbXIb‘XsK^`aKM'bX#ZiHKMtaKMWZpuog+o_7g\o`pKH–

ˆ"‰ †->‰Y‡ ‡ ¶ E‰ ‡•‡C‰K‡ ¶°¶ ¶ ˆy‡ » »

GIHKM·MYpqMCgrZ^`pubaKb‘^`a bX ªu«> u©5¦¨©;«©¦¨¥?” HKMCa HKMZ\o’‘™YX?ba Z\Mg\L?Xž^dk’‘bd•MZib‘L?M`– ,°oaWV!oŽKZiHK^7g\X?b‘a —IoyV7MCXib‘o`a

’‘bcZiZiMCg\oZiŽKg\MŽuXiM@MYpqMCgrZ^`pKb‘aKb‘^`auXk^`a°ZiHKM@}"o’‘ŽKMCXk^dhZiHKM@puog+oL?MZiMg}`Mz]Z\^`gO±XrMCM@—Io7]+Huo·ƒ6`‚u“=mbj[^7b `€`€ž^`g

  bXiXiM©œ«¬7h`€`€7²–E­;ZI]Coaž~=MŽKausYMCg\XrZ\o`ausKo~u’Mk” HKMabcZ ]^7au]MCgiauX˜ZiHKMMYp=^`aKMCaŒZibo’=L@^YsYMC’“YXibaq]MkZiHKML?MCoa

’‘bcd•MZib‘L?MObXkZiHKM@puog+oL?MZiMgz–'—œŽKZŽuXiŽuo`’’‘V0ZiHKM@MYp=MgiZsY^WMzXkau^Z'™Œau^"”SZiHKM?X[Z+o"Z\b‘XrZib]o`’L?^YsYM’Ÿo`aus0ZiHuM@pugib‘^`g

b‘aYd•^`g\Lo"Z\b^7a¡^7a puog+oL?MZ\Mg+X']^`g\g\MCXipq^7ausKX'L?^`g\M?Zi^£o[Zig+oauXrd•^`g\Lo"Z\b^7a O^d#ZiHuMžMYp=MgiZ™WaK^"” ’MzsY_`M?~WV£Z\HKM

—IoyV`MzXrboaoauo`’VYXrZC–

GIHWŽuXC“7TWb‘aKpKŽKg\”#o`’’ošo`aus?TW^`aK_?±[ƒ 7‚7²9Z\HKMaTWb‘aKpKŽK_7pKŽKg\”#o`’’o@±[ƒ6 Œ²]^7auXib‘sYMCgiMzs'ZiHqo"Z˜oa@MYp=MgiZE”Io`X o~u’M

Z\^@XipqMzo™o~=^`ŽKZœZ\HKMkL?MzsYb‘o`a1’‘bcd•MZib‘L?M

m = η(log 2) ^1/β

* HKMkbXIXrŽupKpq^ŒXrMzs?Zi^@_`b‘}`MoaMCXrZib‘L?oZib‘^`až^`dZiHuMpugib‘^`g L?MCo`ažl

[m]

–Ÿ³EbcYb‘aK_

π(β)

“WZ\HKMV^7~YZ\o`ba1oO]^7L@pu’MZiMšpugib‘^`gœ^`až~=^Z\H1puo`g\o`L@MZiMCg\XC–­®a1^`ŽKg#baqsYŽuXrZig\b‘o`’=]^`aŒZiMŒZz“

Z\^~qMšL@^7giMš_`MauMg+o’5“”œMk]^`aqXrbsYMg˜ZiHuoZœoaMWp=MgiZœ]Coa_`b‘}`Mo`aMCXrZib‘Lo"Zib‘^`a^`do"ŒŽuoaŒZibZ[VO]+Hqog+o`]Z\Mg\b¤Cbau_kZ\HKM

+«"§`¦ =«¬"¦+©5§¦ªY©¦21>

M

^d ’‘bcd•MZib‘L@M

T

” bZiH·sYMaqXrbZ[V

m W (t) = Z

IR

Z

IR

f W (t|µ, β)π(µ|β)π(β ) dµdβ.

^±7²

4

a°baŒZiMCauXib}7M’bZrZ\Mg+o"Z\ŽKgiMsKMCo’X” bZiH£ZiHKM<"ŒŽKMzX[Z\b^7aKb‘aK_ž^dŸMWp=MgiZ\XO±eXiMM<oauMCXi™ŒHqoH£€7€>!@d•^`gogiMC}WbMC”š²–

TW^7L?MsYbX\]ŽuX\Xrb‘^`a!ZiMz]+HKaKb#"7ŽuMCX?o`XZiHKM ¦ %©5¦21 L?MZiHu^Ws ±Q'o`grZ\HW”#o`bcZ\M ©«"¬7 `€`€ `²tHqoy}`M1~=MMCapugi^7pq^ŒXrMzs

” bZiH0XrŽq]]MzXiXhZi^@^`~KZ\ob‘ažd•g\^`L MYpqMCgrZ+X "ŒŽuo`aŒZibZ\o"Z\b}7Mšb‘aYd•^`g\Lo"Z\b^7ao~=^`ŽKZœZ\HKM'~qMCHuoy}Wb^7gœ^`doOXrZiŽqsYbMzsXiVYX[Z\ML0–

­®a ^7ŽKgO]Co`XiM`“Z[”œ^0"ŒŽKMzX[Z\b^7auXog\MžMzXiXiMaŒZibo’3*+N#oa V`^7Ž _`b‘}`Mžo·g\MpugiMzXrMCa7Z+o"Z\b}7M@}"o`’ŽKM

t e

^dIZiHKM1’bd•MZ\bL?M

T

^`d#]^7L?pq^7aKMaŒZ

P

<sup>xSo`ausr</sup>  HuoZb‘XZ\HKMpKg\^`~uo`~Kb’‘bZ[V

α

d•^`g'Z\HKMž]^7L?pq^7aKMaŒZ

P

Zi^deo`b’˜~=Md•^7giM@Z\bL?M

t e

x

±•ZiHKM]^`aŒZiMŒZ ^`dE~=^Z\H?"ŒŽKMzX[Z\b^7auX sYbC=MCg * ZiHuM%ug\XrZ ^`aKM]^`au]Mg\auX§¬¦e«¦¨¬¦¨©oausZiHuMXrMz]^`aqs1^7aKMtsYMCo`’‘XI” bZiH

ªY§i«¦¨¬¦¨© $¯XrMCMtd•^`gkbauXrZ\o`au]M@moy” ’‘MCX\Xš€7€`€Œ²– GIHWŽuX“=”#MOo`X\XiŽKL?MtZiHuoZZiHuMOMWp=MgiZk]Coa·~=MOXi^`’‘b‘]bcZ\MCs0Z\^ž_`b‘}`M

o’‘bd•MZib‘L?MO}"o`’ŽKM

t e

oausZ\HKM */)$%¦ %\«©¦21> ^`d

t e

” bcZ\H£g\MCXipqMz]ZšZ\^

M

–šGIHKbXk}yo`’ŽuMO]o`a·~qMOpqMCg\]Mb‘}`Mzs·o`X Z\HKM MzX[Z\bLo"Z\b^7a

ƒ7–I^dEZiHKM

α

^7g\sKMgIpqMCg\]MaŒZib‘’‘M`“Kauo`L?M’‘V

P(T < t e ) = α ∈]0, 1[

^$

K–I^d MYpqMz]Z+o"Zib‘^`al

[T ]

^$

u–I^d˜L?^YsYM ,

d [T ]

±eHK^"”#M}`MCgC“=XrMC}`MCg\o`’pKg\b}"o"Z\MsYbXi]ŽuX\Xrb‘^`auXš” bcZ\H£g\M’‘b‘o`~Kb’‘bZ[VXipqMz]bo’‘b‘XrZ\X]^7au]’‘ŽusYMZi^žZ\HKM }`Mg\Vž”#MCo`™pKgi^7~uo~ub’‘bcZ[VžZiHuoZšoabausKŽuX[Z\gibo’¯MYpqMCgrZš]CoagiMzo’‘’VžXipqMz]bd•Vo@L?^YsYMz²–

GIHKg\^`Žu_`H sYbXi]ŽuXiXib‘^`a“ Xrp=MC]bcd•VWb‘aK_£pKg\MC]b‘XiM’‘V£Z\HKMžauoZiŽKg\Mž^`d o`a MYp=MgiZ^`pKb‘aKb‘^`a bXMzXiXiMaŒZibo’5–0³K^`gObaqX[Z+oau]M`“

Xi^`L?MtoŽYZ\HK^`g+XIHuoy}7Mau^Zib]MzsZiHuoZ ZiHKMtMWp=MgiZ\X ZiMaqs1Z\^_`b‘}`Mto@L?MCsYboa0}"o’‘ŽKMo’ZiHu^`ŽK_7H1Z\HKMVŽKausYMCg\XrZ\o`a7Z

t e

o7X o?L@Mzoa ±TK]+HKbMCgiMCa¢ƒK“Ymo`aKaK^"V1o`ausf˜gi^Y]Co`]]b‘o?€7€Kƒy²–˜GIHKMXipqMz]bq]oZib‘^`a0]o`a~=Mtg\Mb‘aYd•^`g+]Mzs1~WVXi^`L?M

^7pqMCa "ŒŽKMzX[Z\b^7auX o`X

ƒ7–I  HKMa0Z\HKMto_`M'^`d o@_`g\^`Žup1^dŸ]^7L?pq^7aKMaŒZ\XIg\MCo7]+HKMCX

t e

“K” Hub‘]+HpKg\^`p=^`giZib‘^`a

1 − α

bX X[Z\b’‘’b‘a0ŽuXrMx

K–I­[XIZiHKMt}"o`’ŽKM

t e

o?L?MCoa^d puo7X[Z M}7MaŒZ\XC“WZ\o`™WbaK_?b‘aŒZi^o`]C]^`Žua7Z MŒZ\giMCL?M}"o`’ŽKMzXI^`gIaK^`Zkx

u–I­[XIZiHKMt}"o`’ŽKM

t e

ZiHKM'deo`b’‘ŽKg\M'Zib‘L?Mt” HKb‘]+HHuo7XI~qMCMa0^`~uXiMg\}`MzsZ\HKMtL?^7XrZ#d•g\M-"ŒŽKMCaŒZi’‘V1b‘a1Z\HKMtpuo`XrZkx

GIHKMCa“]^`ausYbZib‘^`aqo’‘V0Zi^ZiHKMž]+Hu^`b]M@^d

a

“¯Z\HKMž]+HK^`b]M@^d

b

bX_`b‘}`MCa¢b‘a°Z\HKM@d•^`’‘’^"” b‘aK_pKg\^`p=^7XibcZ\b^7a£d•g\^`L Z\HKM

™WaK^"” ’‘MCsY_7Mk^`dZ\HKMtLog\_`b‘auo’¯sYbXrZig\b~KŽKZib‘^`a

M

^–

t¦% 1"¬‘« 1ªª)©

@ƒ`–9©

t e ∈ IR ^+∗

a > 0

^{« *}

α ∈]0, 1]

^{7 1"©}

b 1 (a, β) =

(1 − α) ^−1/a − 1 −1

t ^β _e , b 2 (a, β) =

Γ(a)

Γ(1 + 1/β)Γ(a − 1/β) β

t ^β _e , b 3 (a, β) = aβ + 1

β − 1 t ^β _e .

1"©

B i

i = 1, 2, 3

^©)#¦ ª)%$./=§¦21§+ 16¬¨¬c¦ >7 1"§I«"¬¨¬ % 1"¦%$81B

π(β)

^{+ª)% ©}u«"©qª/&/

(π(β)) ⊂ [β l , β r ]

(i) P (T ≤ t e |B 1 ) = α, (ii)

[T |B 2 ] = t e

¦

a > β _l ⁻¹

(iii)

d [T |B 3 ] = t e

¦

β l > 1

⁷

)§ © ( /)%©®«"©5¦21 °¦t©®«6&

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TWMCM 4 pKp=MaqsYbc 4 –

{^`Zib]M'Z\Huo"ZoaMYp=MgiZ]oa_`b‘}`ML?^`g\MšZ\Huoa0^7aKMXip=MC]bq]Co"Zib‘^`a–˜³K^`gIb‘auXrZ\oaq]M`“YHuM]oa_`b‘}`Mto5%§ $¦¦¨¬¦¨©

¦ q©§¥y«¬

[t e1 , t e2 ]

^oaus pugi^7~uo~Kb‘’‘bcZ\bMzX

(α 1 , α 2 )

^{XiŽu]+H ZiHuoZ}

P (T ≤ t ei ) = α i

d•^`g

i = 1, 2

^–  ¡M°pKg\^`p=^7XiM Z\^·giMzsYŽu]Mo`aŒV°MYpqMCgrZ^7pKb‘aKb^7a°Z\^ rsKb‘X\]g\MZ\M ?Xip=MC]bq]Co"Zib‘^`aqX

i = 1, . . . , P

ZiHKMCa¢Z\^£XrMC’Mz]Z]^`ŽKpK’‘MCX

(a i , b i )

^–

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µ|β ∼ G

P

X

i=1

P γ i a i ,

P

X

i=1

b i (P γ i a i , β)

!

±!Œ²

” HKb]+H1bX#ZiHuMp=^7XrZiMCgib‘^`gI]^7L?baK_d•gi^7L XiŽu]]MCX\Xrb‘}`M—#oyV7MCXib‘o`abaKd•Mg\Mau]MCX#^`a}Œb‘giZiŽuo`’9X\oL?pK’‘MCXC–  ¢MCb_7HŒZ\X

γ i > 0

±XrŽu]+HZiHqo"Z

P P

i=1 γ i = 1

²Oo`giMKYMCs b‘a!d•Žuau]Z\b^7a ^`dkZ\HKM0giMC’‘oZib‘}`MZig\ŽuXrZžbaZiHKM£Xip=MC]bq]Co"Zib‘^`aqX– moZiMCg@b‘a Z\HKMžogiZib]’‘M`“o`a¡o`pKpK’‘b‘]Co"Z\b^7a¢” b‘’’hMYML?pK’‘bcd•V ±eo`aus¢]’‘o`gibd•VK²šZiHKbX]+HK^7b‘]M`–?­;Zb‘X'g\Mb‘aYd•^7g\]MCs¢~ŒV£ZiHKMž]^`aW}`MWbZ[V

pKg\^`p=MgiZib‘MCXœ^dEopKpK’‘b]o"Z\b^7auX

β 7→ b(a, β)

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b 1 (a, β)

±¨Z\HŒŽqX#o`aWV?]^`aW}`M

XiŽKL ^`d#XiM}`MCg\o`’

b 1 (a, β)

^² rL@b‘L?b‘]CXOZiHKM?_`MC^`L?MZ\gib]o`’~=MHqoy}Œb‘^`g'^`d

β 7→ P a

i=1 x ˜ ^β _i

–O­;ZaKMMzsKX'Xr^7L@M?”#Mb‘_`HŒZ[V ]Co’]ŽK’o"Z\b^7auX'±•aK^`Z_`b‘}`MaHKMCgiMy²h~uŽYZš]o`aMzo`Xib’‘Vž~=MMa0aWŽKL?Mg\b]o’‘’‘V1]+HKMC]+™7MCs¯–

­®aaKMWZ u_7ŽKgiMzX”#MhsYbXrpu’‘oyV'Xr^7L?MhsYMaqXrbZib‘MCX

m(t)

^`dŒZ\HKMhpugib‘^`gL?o`gi_7baqo’`sYbXrZig\b~KŽKZib‘^`a

M

“y” HKMCgiM

t e = 100

^“

[β l , β r ] = [1.1, 5]

^“

(p, q) = (1.5, 1.5)

^oaqs

a = 2

–Ÿ³Eb_7ŽKg\M?ƒXiHK^"”X

m(t)

^{” HuMa}

t e

bXIXrŽu]C]MzXiXib}7M’‘V]^`aqXrbsYMg\MCs

o7X1Z\HKM¡pKgib‘^`g0L?MCsKb‘o`a#yL?MCo`a#yL?^YsYM7–*³b‘_`ŽugiM !XiHK^"”X1ZiHKM M}7^`’‘ŽYZib‘^`a ^d

m(t)

^{” HKMCa}

t e

b‘XZ\HKM

α

^{^`g+sYMg}

p=Mg+]MCa7Z\b’‘M`“`d•^`ghXiM}7Mg+o’W}"o’‘ŽKMCX˜^d

α

–ŸGIHKM]^`g\giMC’‘oZib‘^`a@~qMZ[”œMCMa?puo`g\o`L@MZiMCg\X b‘ausYŽu]MCs@~WVZ\HKM pKg\b^7g˜L?^YsYM’‘’b‘aK_

bX#giMC}`MCo`’Mzs~WV?ZiHKM % 1 q¥( YªY¬¨¬7^`dZiHuMpugib‘^`gIXio`L?pK’b‘aK_$Yba0³Eb_7ŽKgiM K“ %$ ^dpugib‘^`gIXrb‘LŽu’‘oZiMCs

(η, β)

}"o`’ŽKMzX o`giMžŽqXrMzs d•^`g@sYM’‘bL?bZib‘aK_£Z\HKM0]^`aW}7M¢HWŽK’‘’5–¢GIHKM1d•^`g\L ^`d ZiHKM]^7gig\M’o"Z\b^7a¡g\MLo`bauXaKMCo`gZ\HKMžd•^7giL ^d Z\HKM

Z\HKM^7giMZib]o’uj[^`b‘aŒZ]^7a!qsYMCau]MOog\MCoXiHK^"”œMzs0~WV ¡Ž!±`€`€Œ`²%$=ZiHuMb‘aŒ}7Mg+XrMsKb‘XrZig\b~uŽYZib‘^`a£^`d˜puog+oL?MZiMgš}yo`’ŽuMCX

” HKb]+H!bX?ŽuXiŽuo’‘’‘V pqMCg\]Mb‘}`MCs b‘X]^`HKMCgiMCa7Z@” bcZ\H!Z\HKM·]’o`X\Xrb]o`’œ~=MHqoy}Œb‘^`g@^dšd•giM"7ŽuMaŒZibX[ZMzX[Z\bLo"Z\b^7auX@^dZ\HKM

puo`g\o`L?MZiMCg\X'±eŽuXrb‘aK_?ŽKaK™WaK^"” a0sKoZ\o7²*Ÿbcd

β

b‘XI^"}7Mg\MCXrZib‘Lo"ZiMzsZ\HKMa

η

bXIŽKausYMCgiMzX[Z\bLoZiMCs“uoaus]^7aW}`Mg+XiM’‘V`–

t

m(t)

0 50 100 150 200 250 300

mean mode median

te=100

³Eb‘_`ŽKg\Mƒ*šMCauXibcZ\bMzX

m(t)

^`dEZ\HKMpKg\b‘^`gLog\_`b‘auo`’

sYbXrZig\b~KŽKZib‘^`a

M

±•b‘ausYMYMCsO~ŒVZiHKMXipqMz]bq]oZib‘^`aO^d

t e = 100

^²–

t

m(t)

x

m(x)

x

m(x)

0 50 100 150 200 250

te=100

10%

25%

33%

50%

66%

75%

90%

³b‘_`ŽugiM* kMauXibZib‘MCX

m(t)

^d ZiHKMpKgib‘^`gšL?o`gi_7baqo’

sYb‘XrZig\b‘~KŽYZib‘^`a

M

±ebaqsYMYMCsO~WVtZiHuMXrp=MC]bq]Co"Z\b^7a^d

α

^²–

eta

beta

0 20 40 60 80 100

0 20 40 60 80 100 120 140

³Eb_7ŽKgiM B*œNœ^`aW}`MžHŒŽu’’^`dEo@pKg\b^7gXio`L?pK’b‘aK_@^d puog+oL?MZiMg+X

(η, β)

^–

E #/ '

TW^7L?M˜oŽYZ\HK^`g+Xo`XTWbau_`pKŽKg\”Io’‘’‘oIoaqsTW^`aK_±rƒ6 `‚7²q~KŽub’stpKg\b^7g¯L?^WsKM’‘’b‘aK_7X” bcZ\HtaWŽKL?Mg\^`ŽuX¯HWVWp=Mg\puog+oL?MZ\Mg+X“

” HKb]+Ho’‘’‘^"”°Z\^L?^WsKbcd•VkZiHKMœMYp=MgiZbaKd•^`g\L?oZib‘^`a'b‘at’^Y]oZib‘^`ao`aus'ŽKau]MgiZ\ob‘aŒZ[V`–

4

XmbausK’MCVo`ausTWb‘aK_`pKŽugi”Io’‘’‘o

±rƒ6 7‚7²aK^Z\b‘]MCs¯“¯bZtHuo`X~=MMa¡^`~qXrMCgi}7MCs0Z\Huo"Z'MYp=MgiZ\XkZiMCaus£Z\^0pKgi^YsYŽq]M [’‘^Y]o"Z\b^7a°oaqs°X\]o’‘M@~Kb‘o7X –t­®a¢ZiHKbX

”#^`g\™9“^`Žug^`pubaKb‘^`a bXtZ\Huo"ZOZiHKMd•Žu’’#MYp=MgiZ™WaK^"” ’MzsY_`MbX_7b}7Ma¡ZiHugi^7ŽK_`H

t e

±eo`aus

α

^bcd

t e

bXo·p=Mg+]MCaŒZib‘’M

}"o’‘ŽKMy²–Ÿ{^@^`~Yj[Mz]Z\b}7Mk]g\bZiMg\b‘^`a1o`’’‘^"”X˜Z\HKM'—#oyV7MCXib‘o`ažoauo`’VYXrZhZ\^OL?^YsYbd•VZiHKbX#MYp=MgiZ#™WaK^"” ’‘MCsY_7M`– GIHKM^`aK’‘V

p=^7X\Xrb‘~Kb‘’bZ[VžbX#Zi^?L?^YsYŽK’o"ZiMZ\HKMtMYp=MgiZIŽuau]MCgrZ+ob‘a7Z[VžZiHKg\^`ŽK_7HžZ\HKM]+HK^`b]M^d

a

–E­®a0^7ŽKg XrMCauXrM7“WZiHKg\MMtXrZiMCpuXI^d ]Co’‘b~Kg+o"Z\b^7aXiHK^`Žu’‘s~qM]^`auXibsYMg\MCs¯–

ƒ7– 4 «ªY¬©-%+«"¬c¦§i«©¦21> žX[Z\Mp $Yo7]]^7g\sKbaK_Zi^OZiHKMaWŽKL~=Mg oaqs?ZiHKM'auoZiŽKg\Mš^`dXipqMz]bq]oZib‘^`auXC“7”#MpKgi^7pq^ŒXrM

Xr^7L@M@sYMdeo`ŽK’Zk}"o’‘ŽKMCXd•^`g

a

–GIHKb‘XkbXšsY^7aKMb‘a

§

!q–ƒ7–š—#MCXib‘sKMCXC“q” HKMCa

a

Hqo`Xk~qMCMa°]+HK^ŒXrMCa“u”#Mpugi^7pq^ŒXrM

ƒz€ t¦% 1"¬‘« 1ªª)©

ba

§

<sup>!u–‘ƒ`–</sup>¢o¡sKMdeoŽu’cZžL?MZ\HK^Ysd•^`gž]o`’b‘~Kg+o"Z\baK_

π(β)

^{” HKMa aK^} baKd•^`g\L?oZib‘^`a b‘Xoy}"o`b’o~K’‘M·o~=^`ŽYZ?Z\HKM

ŽKau]MCgrZ+ob‘aŒZ[Vž^d

β

^–

K–

4 % 1>+#¦ $ %+«"¬c¦§i«©¦21> žX[Z\Mp $Y~=MC]CoŽuXiM

a

]Coa1~=M'ŽKausYMCg\XrZi^W^Ysžo7X#oO}WbgiZiŽqo’9Xio`L@pu’M'Xrb‘¤M7“WbcZIb‘XIo@]_7^Œ^Ys puog+oL?MZ\Mgd•^7go sKb‘X\]ŽuX\Xib^7a ~qMZ[”œMCMa ZiHuM°MYp=MgiZoaqs ZiHKM¡—#oyV7MCXib‘o`a o`auo’‘VYX[Zz– ³Ebg+X[Z}yo`’ŽuMCXž^`d

a

pKgi^7pq^ŒXrMzs?~WVOZ\HKM'sYMdeo`ŽK’cZ ]o`’b‘~Kg+o"Z\b^7a?XrZiMCp]Coa~=MkHKb‘_`H1” HKMaZ\HKMMYpqMCgrZ#b‘Xœ}`MCgiV@pKg\MC]b‘XiM$WZiHuMšb‘a7Z\MaŒZ

b‘XIZ\^?L?o`™`MZiHuMtMYpqMCgrZ g\MCo7]ZIZi^HKbX XiM’dEg\MC]+™W’‘MCX\XraKMzXiX#oaqs]^`g\giMz]Z

a

^–

u– 4 =«¬E§%+«¬¦§\«"©5¦21 ·X[Z\Mp $ZiHKM—IoyV`MzXrboa£o`auo’‘VYX[Z']oa¢Huoy}`M@Xr^7L?MpKg\b^7gk™WaK^"” ’‘MCsK_`MOo`~q^7ŽYZZiHKM@giMC’b›

o~Kb‘’bZ[V¡^dIZiHKM1MYpqMCgrZ^7pKb‘aKb^7a– 4 HWŽK_`M’‘bZrZiMCg\oZiŽKg\MbXsYMzsYb‘]Co"Z\MCs¢Zi^·Z\HKbXoaqo’‘VWXibX±eXiMMNœ^W^`™7Mžoaus

Q^Œ^ŒXiXiMaqXš€7€Kƒd•^`go?giMC}Œb‘M”š²– GIHKb‘Xkoauo`’VYXib‘Xš]oa0~=M~uo7XrMzs^`a·^Z\HKMgkXrŽK~Kj[MC]Z\b}7M~=M’‘bMdeX±•” Hub‘]+H£o`giM

aK^ZMYpK’‘^`g\MCs!HKMg\Mz²Ooaqs Z\HKM·]^`L?puo`gibXi^`a ” bZiH ^`~Yj[Mz]Z\b}7Mb‘ausYb]oZi^`g+XC– .MCgiM7“˜bcd

a < n

oaqs!bdšZ\HKM sKo"Z+o

t _n

o`giMŽuau]MCauXr^7giMzs¯“ZiHKMžp=^7XrZiMCgib‘^`gOsYb‘XrZig\b‘~KŽYZib‘^`a bXMCauXiŽKgiMzs¡Zi^£~=M1L?^7giMžsYMCpqMCausYMCa7ZOd•gi^7L Z\HKM

^`~Yj[Mz]Zib‘}`M'sKoZ\oO™Œau^"” ’MzsY_`M7–­®a

§

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a

b‘XIsYMuaKMzs1” HKMCažZ\HKMtsKo"Z+o

t _n

^o`giM

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ZiHKM]^`auXiMauXiŽuXI^`dEXiM}`MCg\o`’9MYpqMCgrZ^7pKb‘aKb^7auX#b‘XšX[Z\ŽusYb‘MCs¯–

/5 @;0 <0>B:" > @1:90> :"$#

¶ ‰>•‡Š`‰>•…Eˆ"‰Y‡ » » ‡ ¶ †zŠ`‰>¶ E‰Yˆ"‰ ¶ ‡ ¶ ˆ

GIHKbX]Co’‘b~Kg+o"Z\b^7a]Coa0~=MXrMCpuog+o"Z\MCso`]]^`g+sYb‘aK_Z\^?ZiHKM™Œb‘aus^dŸXipqMz]bq]oZib‘^`a–#GIHKMt}"o’‘ŽKMt^d

a

ZiHqo"Zš”#M'pKgi^`›

p=^7XiMCsOb‘XŸZiHKM L?b‘aKbLo`’KaWŽKL~=Mg

a min

^`d=puo7X[Z˜}Wb‘grZ\Žuo’Ksuo"Z\okZ\Huo"ZhXiMMCL aKMz]MCX\X\og\V'Zi^to`X\XrMzXiXEZ\HKbX˜Xrp=MC]b=]o"Z\b^7a–

= ¶ ‰

¶

¼ ‰

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ˆ » ¼

¶

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Z[”#^@ug+X[Z#^`aKMzX\²–ŸGIHKMg\Md•^7giMšoOsYMdeo`ŽK’Z#]+HK^7b‘]MšLŽqX[Z#~qMsY^`aKM7–  ¢M'XrŽu_`_`MzX[ZhZi^O]+Hu^Œ^ŒXrM

a min = 1

d•^7ghZ\HKML@Mzoa o`aus

a min = 2

d•^7g ZiHKML?MzsYb‘o`a–˜TWp=MC]bd•VWbau_'ZiHKML?^YsYM g\M-"ŒŽKb‘giMzX baŒZ\ŽKbcZ\b}7M’‘VL?^`g\M puo7X[Z˜}Wb‘grZ\Žuo’usuo"Z\o±oausOZ\HKM ]MgiZ\ob‘a™WaK^"” ’‘MCsY_7M ^do_7bau_Œ²–E ¡MkXiŽK_`_7MCXrZ

a min = 3

Xib‘au]MkbcZ#b‘X˜ZiHKML?baKb‘Lo’qaŒŽuL~=Mg#^dsKo"Z+otZ\^sYM qaKMZ\HKM L?^YsKo’]’‘o7XiXI^`dEo?]^7aŒZib‘aŒŽu^`ŽuX sYbXrZig\b~KŽKZib‘^`a– ­®a·Go`~K’‘M@ƒ7“K”#M_7b}7MkMzX[Z\bLoZib‘^`auXI^`dZ\HKMtX[Z+oausKo`g\ssYMC}Œbo"Z\b^7a

σ

o`aus0ZiHuM &

*D$

γ

^{^`d}

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a

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X

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Z\HKMtL?MCoa0XipqMz]bq]oZib‘^`a–

a

σ γ σ γ σ γ

!

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& %

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σ

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γ

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¶ † ¶ Š )( Š`‰Y‡

» PkŽKg@”œ^7gi™Wb‘aK_£HWVWpq^`ZiHKMzXrbX@b‘XOZ\Huo"Z@ZiHKM0MWp=MgiZ?Huo`XOpqMCg\]Mb‘}`Mzs o`X@LŽq]+H!puo7X[Z

}Wb‘grZ\Žuo’hsuo"Z\o°o`XtbZObXaKMz]MzXiX\og\V·Zi^°^`~YZ+ob‘a ZiHKMpKgiMz]bXrb‘^`a

α

^{” HKMa} HuM1Xrp=MC]b=]b‘MCX

t e

o7Xo·p=Mg+]MCa7Z\b’‘M`–·³u^`g

b‘auXrZ\oaq]M`“=Z\HKM—#oyV7MCXib‘o`a£oauo`’VYXrZ]Coa°pugi^7pq^ŒXrM

a = 10

^{” HKMCa}

α = 10%

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±rƒ6Œ`²#pKg\^`p=^7XiMZi^1XiM_7L?MaŒZ

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Z\^O™WaK^"” Z\HKMtsKo"Z+o7²“WHKMt]Coa1~=M'o`Xi™`MCs1o~=^`ŽYZ HKbX#pKgi^7~uo~ub’‘bcZ[V?^`ddeob‘’ŽugiM7– {šMWZ MKoL?pK’‘Mb’‘’ŽqX[Z\g\oZiMCXœ^`Žug#”IoyV

^`dZiHubaK™Wb‘aK_u–

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(50, 80, 90, 170)

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D 1 = [0, 90)

^{« *}

D 2 = [90, ∞)

^{7 9©}

(n 1 , n 2 )

^{$?©) uª + $§$} 1B@¥C¦¨§©5ªu«"¬2`«©;«¦

D 1

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D 2

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7

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D 1

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D 1,a = [0, 50)

^{« *}

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7D *"¦%+«"©O© 11©)# ( /)§©#©u«"©I© #/%©5¦¨¥O¬¦©¦+

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D 1

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n 1 + n 2 = 6

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TWŽKpupq^ŒXrMZ\^£^`~KZ\ob‘a¡d•gi^7L o`a MYp=MgiZtZiHKMbaŒZiMCgi}"o`’

[t e1 , t e2 ]

^oaus

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(α 1 , α 2 )

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P (T ≤ t ei ) = α i

d•^7g

i = 1, 2

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a 1 , a 2

XiŽu]+H¢ZiHuoZ

a = γ 1 a 1 + γ 2 a 2

b‘XZ\HKMLo"YbLo`’ Xib‘¤Mo"ZiZ\ob‘auo`~K’M@~WV·ZiHuM?XiMpqog+o"ZiM

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(50%, 90%)

” b‘’’7_`b‘}`M Z\HKM˜Xio`L?M˜sYMdeo`ŽK’cZEXrb‘¤M

a = 10

Z\HuoatXipqMz]bq]oZib‘^`auX

(10%, 90%)

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β

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” bZiHZiHuMt ¢MCb~KŽu’’L?^YsYMC’LŽuXrZ~=M]+HKMC]+™7MCsžŽqXrb‘aK_@d•^`g\LŽK’o0±[ƒC²–˜­;d

β e = log

log(1 − α 2 ) log(1 − α 1 )

log ⁻¹ (t e2 /t e1 )

^±,`²

bX aK^ZŸb‘a

[β l , β r ]

ZiHKMI^7g\sYMCg\XEXiHK^`ŽK’sO~qM L?^YsYbuMCs@XrŽq]+HZ\Huo"Z

β e = β 0

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[β l , β r ]

²–˜TW^7’}Wb‘aK_

(

) = β 0

” bZiHo'{MC”IZi^7aY›;nšopKHuXi^`a@o’‘_`^7gibZiHKLvbXŸXibL?pK’‘M±eXiMM —#^`ŽuX"ŒŽKMZh`€`€`‚d•^`g˜sYMZ\o`b’X\²–

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γ 1 /(1−γ 1 )

– GIHWŽuXC“zbd

γ 1 − → 1

^“

α 1

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„>=» ˆ •‡ ƒ`–%$ ¶ 9‡ ‡ ¶ Šˆ ¶ ¼e…:>e‡ ¸

α

&'(*)+

0 < α ⁰ ₁ < α ⁰ ₂ < 1

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^{'10) -2 +)}

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³

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,.-/54762*28

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0 < ρ 1

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9

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E

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2;

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(1 − α ^k ₁ ) log(1 − α ^k ₁ ) (`(α k ) − ` 0 ) /`(α k )

−(1 − α ^k ₂ ) log(1 − α ^k ₁ ) (`(α k ) − ` 0 )

^J

E

42@F9G

+)

h k = − (1 − α ^k ₁ )

(1 − α ^k ₂ )`(α k )

'

E

42@F9G

+)

α ^k+1 =

α ^k ₁ − ρh k (α ^k ₂ − α ^k ₁ ), α ^k ₂ − ρh k (α ^k ₂ − α ^k ₁ ) _1−γ ^{γ 1}

1

E 8 +

29LK*6

) -

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K–NM 11

β 0 = 3

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1©®«"¦

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α = (0.55, 0.93)

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π ^J (β| ˜ x _a ) ∝ β ^a−1 _a

Q

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β

a

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π(β )

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Z β r

β _l

π ^J (β | ˜ x _a ) log π ^J (β | ˜ x _a )

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η e

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β e

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π ^J (β| ˜ x _a )

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π ^J _a (β) = Z

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π ^J (β | ˜ x _a )

a

Y

i=1

f W (˜ x i |η e , β e ) d˜ x 1 . . . d˜ x a .

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(p, q)

bX XibL?b‘’‘o`gIZi^L?baubL?b‘¤M

log Γ(p)Γ(q)

Γ(p + q) + (p + q) log(β r − β l ) − (p + q)

^l

log β r − β β − β l

± 7²

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[.]

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π ^J _a (β)

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π(β)

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˜

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t _n

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a

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˜

n

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kMaK^`ZiM

x _m

^oa

i . i . d .

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m

VWbMC’‘sYb‘aK_ZiHKMXio`L@MsKo"Z+ob‘aYd•^`g\Lo"Z\b^7až^d

t _n

–Ÿ³K^7giLo’‘’‘V`“

sYMCaK^Z\bau_

D ^{J c ,J} ( t _n , x _m ) = Z

Θ

π _c ^J (θ| t _n ) log π _c ^J (θ| t _n )

π ^J (θ| x _m ) dθ

^±Œ²

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˜

n = arg min

m

l

X m D ^{J c ,J} ( t _n , x _m ).

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˜

n

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π ^J

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r

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r = 0

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˜

n > 0

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t _n

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˜ n = n

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π ^J

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π ^J _c

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θ n (α)

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α−

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θ

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t _n

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P θ (θ ≤ θ n (α)) = P _c ^J (θ ≤ θ n (α)| t _n ) + O(n ^−i/2 )

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θ

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θ n (α)

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i

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π ^J

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t _n

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n − r

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c 1 , . . . , c n−r

–

9©

γ

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γ ' 0.57722

^{7 ©}

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⁷

1"©

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+ π ² /6 − 1, δ(µ, β) = n −

n−r

X

i=1

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(µ, β)

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α

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t e

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  HKMCa1pKg\b‘^`g+X

π 1 (θ), . . . , π M (θ)

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π(θ) =

M

Q

i=1

π ^$ _i ⁱ (θ) Z

Θ M

Y

i=1

π ^ϕ _i ⁱ (θ) dθ ,

” HKMCgiM

P M

i=1 ϕ i = 1

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KL(π; π 1 , . . . , π M |β) =

M

X

i=1

$ i

Z

IR

π(θ) log π(θ) π i (θ) dθ

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$ i

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µ|β ∼ G

M

X

i=1

$ i a i ,

M

X

i=1

$ i b i (a i , β)

! ,

β ∼ B e M

X

i=1

$ i p i ,

M

X

i=1

$ i q i

!

” HKMCa

π i (β)

b‘XIoO—#MZ\o@sYMauXibZ[V

B e (p i , q i )

sYMuaKMzs1^7a1oO]^`L?L?^`asY^`Lob‘a

[β l , β r ]

–ŸGIHKMt]^7auXiMauXiŽuX#sYb‘XrZig\b‘~KŽY›

Z\b^7a^7a

µ

o`pKpqMzog+X o`X#Z\HKMp=^7XrZiMg\b‘^`g sYbX[Z\gib‘~KŽYZ\b^7a0]^`L?b‘aK_?d•gi^7Lo?]^7auXiMauXiŽuXI}WbgiZiŽqo’X\oL?pK’‘M'” HK^7XiMXrb‘¤M'bX Z\HKMk”#Mb‘_`HŒZ\MCsžXiŽKL ^do`’’9Xib¤CMCXC–E­;dMYpqMCgrZ+Xœ]Coažau^ZI~qM]^`aqXrbsYMg\MCs1o`XœbausKMp=MausYMCaŒZ'±eŽuXrŽqo’‘’V?” HKMCa

M > 2

^²“

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™Wb‘aus^dpugib‘^`g L?^YsYM’‘’‘baK_q–

¯EIE #$ 0žt

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π(β)

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(π(β))

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