A Bayesian analysis of industrial lifetime data with Weibull distributions

HAL Id: inria-00115528

https://hal.inria.fr/inria-00115528v4

Submitted on 23 Nov 2006

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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] RR-6025, INRIA. 2006, pp.24. �inria-00115528v4�

a p p o r t

d e r e c h e r c h e

Thème COG

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

A Bayesian analysis of industrial lifetime data with Weibull distributions

Nicolas Bousquet

N° 6025

Novembre 2006

Unité de recherche INRIA Futurs

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b‘aYd–^`g\Lo"Z\b^7ao`‚q^7ŽYZIZiHKMtŽKaq]MgiZ\o`ba{Z[V ^`d

β bXoœ"o“b‘’‘o`‚K’M7˜

x t¨% 1"®‘­ 1“¬¬)«

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β”YZ\HKMV]+HK^W^7XiM€d–^7gIZiHKMX\]o`’M puo`g\o`L?MZiMCg

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η|β ∼ GIG(a, b, β)

• bZiH¸sYMCauXrbZ[V

f(η|a, b, β) = b^aβ Γ(a)

1 η^aβ+1exp

− b η^β

1[0;+∞[(η) ^²z7³

• HKMCgiM

(a, b, β)>0”Y• bZiH0L?^`L?Ma{Z

l

[η^k] = b^k/βΓ(a−k/β)

Γ(a) ∀ aβ > k >0,

o`ausŸL@^YsYM2,

d[η] = (bβ/aβ+ 1)^1/β˜EGIHKb‘Xdeo“L?b‘’Vkb‘X]’^{XrM—sŸ‚{VŸXi]Co“’‘MEZig+o“auXrd–^`g\Lo"Zib‘^`a?²–b5˜M`˜

X ∼ GIG(a, b, β)⇒

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µ=η^−β”K•žM'^7‚YZ\o`bapKg\b^7go“auspq^{X[Z\Mg\b^7g

µ|β ∼ G(a, b), µ|β,t_n ∼ G a+r, b+

n

X

i=1

ti

! .

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a^o“aus b˜GIHKbXE]+HK^`b]Mšg\MLo`bauXE]^7a{Zig\^"`Mg+Xib‘o`’œXib‘au]M

π(µ|β) =π(µ)^˜

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η

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π^J(η|β,x˜_m)”K•žM'^7‚YZ\o`ba







a = m,

b = b(a, β) =

a

P

i=1

˜ x^β_i.

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bo`Xžotd–ŽKaq]Zib‘^`a1^“d

β o`pKpqM—o“g+X¡Z\^‚qMŸauo“ZiŽKg+o“’5˜E™#M—XrbsYMCXC”

aZ\o`›`M—X¡ZiHuM€L?MCo`aKbau_^“d9Z\HKMŸXib‘¦M

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abX#aK^"• o'%+­"®¨©i­“«¨21>

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d–^Y]ŽqXžp=^`b‘a{Zžd–^7g#oOsYbXi]ŽuXiXib‘^`a ‚=MZ[•#MMCa ZiHuMŸo`auo“’‘VYX[ZIo“aus ZiHKMŸMYp=MgiZC˜ ¢¤M'pKg\^`p=^7XiM€Xi^`L?MksKMdeo“Žu’cZ]o`’b‘‚Kg+o"Z\b^7a

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§^!u˜

¯[sYM—o“’‘’V7”

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4

XiXiŽKL?b‘aK_

˜

x_m ^i.i.d.∼ W(ηe, βe) ^{• HKMCgiM} (ηe, βe)

o`giMŸpKg\b^7g MCXrZib‘L?o“Zib‘^`auXC”Y•#MŸ_7MZ

b|β ∼ G(a, η_e^−β), µ|β, b ∼ G(a, b).

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4

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te

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T

^`d#]^7L?pq^7aKMa{Z

P <sup>So`ausr¢</sup> Huo“Zb‘XŸZ\HKMpKg\^`‚uo`‚Kb’‘bZ[V

α d–^`g'Z\HKM ]^7L?pq^7aKMa{Z

P Zi^deo`b’š‚=Md–^7giM@Z\bL?M

te

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„7˜I^“dEZiHKM

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P(T < te) =α∈]0,1[^$

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[T]^$

u˜I^“dšL?^YsYM ,

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a”±Z\HKM ]+HK^`b]M@^“d

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M^˜

t¨% 1"®‘­ 1“¬¬)«

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te∈IR^+∗ a >0 ^{­ *} α∈]0,1]^{7 1"«}

b1(a, β) =

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t^β_e, b2(a, β) =

Γ(a)

Γ(1 + 1/β)Γ(a−1/β) β

t^β_e, b3(a, β) = aβ+ 1

β−1 t^β_e.

1"«

Bi i= 1,2,3^«)#¨ “¬)%$./=©¨21“©+ 16®ª®c¨ >7 1"©I­"®ª® % 1"¨%$81

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(i) P(T ≤te|B1) = α, (ii) [T|B2] = te

¨

a > β_l⁻¹ (iii) d[T|B3] = te

¨

βl>1⁷

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¨ q«©§œ­“®

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d–^`g

i = 1,2^˜ ¢£M}pKg\^`p=^7XiM Z\^¸giM—sYŽu]Mo`a{V}MYpqMCgrZ^7pKb‘aKb^7a}Z\^ rsKb‘X\]g\MZ\M ?Xip=MC]bq]Co"Zib‘^`aqX

i = 1, . . . , P ZiHKMCa¤Z\^¥XrMC’M—]Z]^`ŽKpK’‘MCX

(ai, bi)^˜

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µ|β ∼ G

P

X

i=1

P γiai,

P

X

i=1

bi(P γiai, β)

!

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• HKb]+H1bX#ZiHuMŸp=^7XrZiMCgib‘^`gI]^7L?baK_d–gi^7L XiŽu]]MCX\Xrb‘`MŸ™#oœV7MCXib‘o`abaKd–Mg\Mau]MCX#^`a{b‘giZiŽuo`’9X\o“L?pK’‘MCXC˜ ¢¤MCb_7H{Z\X

γi >0

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PP

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

β 7→b(a, β)˜¡lhXrp=MC]b‘o`’’‘V`”WbZo“pKp=MCo`g\Xh^`‚WWb^7ŽuXr’‘V@ZiHuo“Z

b1(a, β) ²ªZ\H{ŽqX#o`aWV?]^`aW`M

XiŽKL ^`d#XiM`MCg\o`’

b1(a, β)^³ rL@b‘L?b‘]CXOZiHKM?_`MC^`L?MZ\gib]o`’‚=MHqoœ{b‘^`g'^`d

β 7→Pa

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¯°aaKMWZ u_7ŽKgiM—X•#MhsYbXrpu’‘oœV'Xr^7L?MhsYMaqXrbZib‘MCX

m(t)^`d{Z\HKMhpugib‘^`gL?o`gi_7baqo“’`sYbXrZig\b‚KŽKZib‘^`a

M”œ• HKMCgiM

te= 100^” [βl, βr] = [1.1,5]^” (p, q) = (1.5,1.5) ^o“aqs a= 2˜¡´Eb_7ŽKg\M?„ŸXiHK^"•X

m(t)^{• HuMa} te

bXIXrŽu]C]M—XiXib7M’‘V]^`aqXrbsYMg\MCs

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m(t) ^{• HKMCa} te

b‘XZ\HKM

α ^{^`g+sYMg}

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• HKb]+H!bX?ŽuXiŽuo“’‘’‘V pqMCg\]Mb‘`MCs b‘X]^`HKMCgiMCa7Z@• bcZ\H!Z\HKM¸]’o`X\Xrb]o`’ž‚=MHqoœ{b‘^`g@^“d€d–giM"7ŽuMa{ZibX[ZM—X[Z\bLo"Z\b^7auX@^“dZ\HKM

puo`g\o`L?MZiMCg\X'²eŽuXrb‘aK_?ŽKaK›WaK^"• a0sKo“Z\o7³(¡bcd

β b‘XI^"7Mg\MCXrZib‘Lo"ZiM—sZ\HKMa

η bXIŽKausYMCgiM—X[Z\bLo“ZiMCs”uo“aus]^7aW`Mg+XiM’‘V`˜

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t

m(t)

0 50 100 150 200 250 300

mean mode median

te=100

´Eb‘_`ŽKg\M„(€MCauXibcZ\bM—X

m(t) ^`dEZ\HKMpKg\b‘^`gLo“g\_`b‘auo`’

sYbXrZig\b‚KŽKZib‘^`a

M ²–b‘ausYMYMCsO‚{VZiHKMXipqM—]bq]o“Zib‘^`aO^“d

te= 100^³˜

t

m(t)

x

m(x)

x

m(x)

0 50 100 150 200 250

te=100

10%

25%

33%

50%

66%

75%

90%

´b‘_`ŽugiMz( kMauXibZib‘MCX

m(t) ^“d ZiHKMpKgib‘^`g€L?o`gi_7baqo“’

sYb‘XrZig\b‘‚KŽYZib‘^`a

M²ebaqsYMYMCsO‚WVtZiHuMXrp=MC]bq]Co"Z\b^7a^“d

α^³˜

eta

beta

0 20 40 60 80 100

020406080100120140

´Eb_7ŽKgiM @(žNž^`aW`M H{Žu’’^`dEo@pKg\b^7gXio`L?pK’b‘aK_@^“d puo“g+o“L?MZiMg+X

(η, β)^˜

E #/ '

TW^7L?Mšo“ŽYZ\HK^`g+Xo`XTWbau_`pKŽKg\•Io“’‘’‘oIo“aqsTW^`aK_²r„6 `x7³q‚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–VkZiHKMžMYp=MgiZbaKd–^`g\L?o“Zib‘^`a'b‘at’^Y]o“Zib‘^`ao`aus'ŽKau]MgiZ\o“b‘a{Z[V`˜

4

XmbausK’MCVŸo`ausTWb‘aK_`pKŽugi•Io“’‘’‘o

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•#^`g\›9”^`Žug^`pubaKb‘^`a bXtZ\Huo"ZOZiHKMd–Žu’’#MYp=MgiZ›WaK^"• ’M—sY_`MbX_7b7Ma£ZiHugi^7ŽK_`H

te

²eo`aus

α ^bcd te

bXo¸p=Mg+]MCa{Zib‘’M

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p=^7X\Xrb‘‚Kb‘’bZ[V bX#Zi^?L?^YsYŽK’o"ZiMŸZ\HKMtMYp=MgiZIŽuau]MCgrZ+o“b‘a7Z[V ZiHKg\^`ŽK_7H Z\HKM]+HK^`b]MŸ^“d

a˜E¯°a0^7ŽKg XrMCauXrM7”WZiHKg\MMtXrZiMCpuXI^“d ]Co“’‘b‚Kg+o"Z\b^7aXiHK^`Žu’‘s‚qM]^`auXibsYMg\MCs±˜

„7˜ 4 ­“¬Y®«-%+­"®c¨©i­“«¨21>  X[Z\Mp $Yo7]]^7g\sKbaK_Zi^OZiHKMŸaWŽKL‚=Mg o“aqs?ZiHKM'auo“ZiŽKg\M€^`dXipqM—]bq]o“Zib‘^`auXC”7•#MŸpKgi^7pq^{XrM

Xr^7L@M@sYMdeo`ŽK’Zk"o“’‘ŽKMCXd–^`g

a˜ŸGIHKb‘XkbX€sY^7aKMb‘a

§ !q˜„7˜€™#MCXib‘sKMCXC”q• HKMCa

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