derivation of the distribution function for the tampered brownian motion ...
DERIVATION OF THE DISTRIBUTION FUNCTION FOR THE TAMPERED BROWNIAN MOTION PROCESS MODEL
Conference on Applied Statistics in Defense George Mason University, Fairfax VA 22030 22 October 2015 Arthur Fries Institute for Defense Analyses, Alexandria, VA 22311 [email protected]
DERIVATION OF THE DISTRIBUTION FUNCTION FOR THE TAMPERED BROWNIAN MOTION PROCESS MODEL
Conference on Applied Statistics in Defense George Mason University, Fairfax VA 22030 22 October 2015 Arthur Fries Institute for Defense Analyses, Alexandria, VA 22311 [email protected]
No official endorsement by IDA or any of its sponsors is intended or should be inferred. The content of this briefing and the views expressed herein are my sole responsibility.
Brownian Motion Process (BMP) • Stochastic Process • B(0) = 0 • Stationary and independent increments • B(t2) – B(t1) ~ N(η(t2 – t1),δ2(t2 – t1)), 0 ≤ t1 < t2 • η = drift parameter • δ2 = diffusion parameter
• Assume Positive Drift • η>0
20 BMP Realizations (η = 1, δ = 0.05 )
B(t)
t
Source: http://www.fil.ion.ucl.ac.uk/~wpenny/bdb/sp.pdf
BMP-Induced Distributions
B(t0) ~ N(ηt0,δ2t0) B(t)
t
t = t0
20 BMP Realizations (η = 1, δ = 0.05 )
T~ IG(ξ/η, ξ2/δ2)
B(T) = ξ>0 B(t)
t
Inverse Gaussian (IG) Distribution • First passage time of BMP(η, δ) wrt critical boundary ξ • Modeling of reliability, fatigue life, degradation processes
• 𝑔 𝑡; 𝜇, 𝜆 = • • • •
𝜆 −𝜆 𝑡−𝜇 2 𝑒𝑥𝑝 2𝜋𝑡 3 2𝜇2 𝑡
Mean = μ = ξ/η, Shape parameter = λ = ξ2/δ2 Spans spectrum of shapes, including long-tailed Exponential family, with well-developed statistical inference procedures Closed-form MLEs for regression parameters (μ-1 = α + βx)
• 𝐺 𝑡; 𝜇, 𝜆 = 𝛷
𝜆 𝑡 𝑡 𝜇
−1
• 𝛷(·) denotes N(0,1)
+ 𝑒𝑥𝑝
2𝜆 𝜇
𝛷 −
𝜆 𝑡 𝑡 𝜇
+1
IG Pdf Shapes
Source: https://upload.wikimedia.org/wikipedia/commons/a/a5/PDF_invGauss.svg
BMP or BMP*?
• B(t;η,δ) is only generally increasing • May be inappropriate for modeling of some processes
• B*(t;η,δ) ≡ sup0<s≤t B(s;η,δ) is nondecreasing • Wiener Maximum Process (Singpurwalla 2006)
• IG derives from either representation
Step-Stress Partial Accelerated Life Testing • Step-Stress ALT
• PALT • Two stages only • No censoring
Tampered BMP • 𝐵 𝑡 =
𝐵1 𝑡 , 𝑡 ≤ 𝜏 𝐵1 𝜏 + 𝐵2 𝑡 − 𝜏 , 𝑡 > 𝜏
• B1(t) = B1(t;η1,δ)
• B2(t) = B2(t;η2,δ)
independent
• η2 > η1
• System fails when B(t) first attains a critical threshold value ξ
Statistical Literature • Bhattacharyya, G.K. 1987. Parametric Models and Inference Procedures for Accelerated Life Tests. in Proceedings of the 46th Session, Tokyo, Japan, September 8-16, Bulletin of the International Statistical Institute LII, 4: 145-162. • Mentioned, in a conceptual sense, the Tampered BMP
• Luh, Y. and Storer, B. 2001. A Tampered Brownian Motion Process Model for Partial Step-Stress Accelerated Life Testing. Journal of Statistical Planning and Inference 94:1524. • Presented pdf f(t), with incorrect attribution • Established basic properties of f(t) • Continuous and may be either unimodal or bimodal • All positive integer moments exist • MLEs are unique with probability tending to 1, are strongly consistent, and are asymptotically normally distributed
Pdf Derivation
• Bhattacharyya 1987 Derived before 1987 invited paper • Luh and Storer 2001 Derived by Bhattacharyya after his 1987 invited paper • Fries, A. 2015. Derivation of the Distribution Function for the Tampered Brownian Motion Process Model. Gnedenko e-Forum International Group on Reliability electronic journal – Reliability: Theory & Applications, Vol. 10, No. 3, September 2015
Pdf Derivation • Bhattacharyya 1982. Unpublished note: problem definition • Fries 1982. Unpublished note: derivation • Bhattacharyya and Fries 1982. Unpublished draft paper
• Bhattacharyya 1987 Derived prior to 1987 invited paper • Luh and Storer 2001 Derived by Bhattacharyya after his 1987 invited paper • Fries, A. 2015. Derivation of the Distribution Function for the Tampered Brownian Motion Process Model. Gnedenko e-Forum International Group on Reliability electronic journal – Reliability: Theory & Applications, Vol. 10, No. 3, September 2015
Pdf Representation 𝑓 𝑡 =
𝜆 𝜆𝑡𝑐 2 (𝜇1 , 𝑡) 𝑒𝑥𝑝 − 2𝜋𝑡 3 2 𝜆 𝜆𝑡 2 𝑡 𝑒𝑥𝑝 − 𝑐 𝜇1 , 𝑡 + 2 𝑐 𝜏, 𝑡 3 2𝜋𝑡 2 𝜇2
• • • • • •
λ = ξ2/δ2 μi = ξ/ηi, for i = 1, 2 c(a,b) = (1/a – 1/b) for a, b ≠ 0 s(t) = q(t,∆+1,λ) – q(t,∆–1,λ) ∆ = τ∙c(μ2,μ1) q(t,a,λ) = a∙exp(½a2λc(τ,t))∙ 𝛷(a(λc(τ,t))½)
𝑡 ≤ 𝜏,
𝑠 𝑡 ,
𝑡 > 𝜏.
Derivation – Pursuit of Cdf • For a fixed t ≤ τ: 𝑃 𝑇≤𝑡 =𝑃
𝑠𝑢𝑝 𝑠 𝜖 0,𝑡
𝐵 𝑠 >𝜉 =𝑃
𝑠𝑢𝑝 𝑠 𝜖 0,𝑡
𝐵1 𝑠 > 𝜉 = 𝐺 𝑡; 𝜇1 , 𝜆
• For the non-trivial case, with a fixed t > τ, proceed by conditioning on the value of 𝐵1 𝜏 : 𝑃 𝜏 τ: 𝑃 𝜏 τ: 𝑃 𝜏 τ: 𝑃 𝜏 τ:
1 1 1 𝜂1 𝜏 − 𝜉 2 2 𝑓 𝑡 = ∙ 𝑒𝑥𝑝 − + 𝜂 𝑡−𝜏 2 2𝜋𝛿 2 𝜏 𝑡 − 𝜏 3 2𝛿 2 𝜏 𝜉 𝜉 1 1 1 1 + 𝜂 − 𝜂 + + 𝜂 − 𝜂 − 2 1 1 𝜏 −𝐼 𝑡 −𝜏 𝜏 ,− 2 𝜏 ∙ 𝐼 𝑡 − 𝜏2 𝜏 ,− 2𝛿 𝛿2 2𝛿 2 𝛿2 where ∞
𝜗 ∙ 𝑒𝑥𝑝 − 𝛼𝜗 2 + 𝛽𝜗
𝐼 𝛼, 𝛽 ≡ 0
• Collect terms
d𝜗 =
1 1− 2𝛼
𝛽 𝛽 2𝛼 2𝛼 𝜙 − 𝛽 2𝛼 𝛷 −
,
Possible TBMP Extensions • Cdf • > 2 stages • TBMP*
Possible TBMP Extensions – Cdf • Could try integrating the pdf directly • Would need: 𝑡 𝜏 𝜏 𝜇1
𝜆 𝜆𝑠 𝑒𝑥𝑝 − 2𝜋𝑠 3 2 −
𝜏 𝜇2
+ 1 exp
1 1 − 𝜇1 𝑠
λ
𝜏 2 𝜇1
−
𝜏 𝜇2
2
𝑠 1 1 + 2 − 𝜇2 𝜏 𝑠
+1
2 1 𝜏
1 −𝑠
⋅ 𝛷
𝜏 𝜇1
−
𝜏 𝜇2
+1
λ
1 1 − 𝜏 𝑠
1/2
− 𝜏 𝜇1
−
• ?
𝜏 𝜇2
− 1 exp
λ
𝜏 2 𝜇1
−
𝜏 𝜇2
−1
2 1 𝜏
1 − 𝑠
𝛷
𝜏 𝜇1
−
𝜏 𝜇2
−1
λ
1 1 − 𝜏 𝑠
1/2
ds
Possible TBMP Extensions – Cdf • Could try solving the integral in the BMP derivation • Would need: ∞ 0
2𝜉𝜐 1 − 𝑒𝑥𝑝 − 2 𝛿 𝜏
𝛷
𝜐2 ∕ 𝛿 2 𝑡 − 𝜏 −1 𝑡 − 𝜏 𝜐 ∕ 𝜂2
2𝜐𝜂2 𝜐2 ∕ 𝛿 2 𝑡 − 𝜏 + 𝑒𝑥𝑝 𝛷 − +1 𝛿2 𝑡 − 𝜏 𝜐 ∕ 𝜂2
• Linear (in 𝜐) arguments! • But still ?
1
𝜐 − 𝜉 + 𝜂1 𝜏 𝜙 d𝜐 𝛿 𝜏 𝛿 𝜏
General Integral ∞ 0
Φ 𝛼1 + 𝛽1 𝑠 𝜙 𝛼2 + 𝛽2 𝑠 d𝑠
• Fayed & Atiya (2014) • Related integral (via linear transformation): 𝐼 𝑎, 𝑏, 𝑥 ≡
𝑥 Φ 0
𝑎 + 𝑏𝑤 𝜙 𝑤 d𝑤
• “To our knowledge, this integral has no closed form or even a single series expansion.” • Establish representation as as an infinite series of the normalized incomplete Gamma function and the Hermite polynomial
Fayed & Atiya (2014) Result
Possible TBMP Extensions – 3+ Stages • Conditioning approach conceptually works • Encounter cumbersome analytical obstacles • Same problematic integral type appears for 3stage setting
Possible TBMP Extensions – TBMP* • Now condition on the value of 𝐵1∗ 𝜏 • 𝐵∗ 𝑡 =
𝐵1∗ 𝜏 , 𝑡 ≤ 𝜏 𝐵1∗ 𝜏 + 𝐵2∗ 𝑡 − 𝜏 , 𝑡 > 𝜏
• 𝐵1∗ (t) = 𝐵1∗ (t;η1,δ) • 𝐵2∗ (t) = 𝐵2∗ (t;η2,δ) • η2 > η1 • System fails when 𝐵∗ (t) first attains a critical threshold value ξ
Possible TBMP Extensions – TBMP* ∞
𝑃 𝜏
Conference on Applied Statistics in Defense George Mason University, Fairfax VA 22030 22 October 2015 Arthur Fries Institute for Defense Analyses, Alexandria, VA 22311 [email protected]
DERIVATION OF THE DISTRIBUTION FUNCTION FOR THE TAMPERED BROWNIAN MOTION PROCESS MODEL
Conference on Applied Statistics in Defense George Mason University, Fairfax VA 22030 22 October 2015 Arthur Fries Institute for Defense Analyses, Alexandria, VA 22311 [email protected]
No official endorsement by IDA or any of its sponsors is intended or should be inferred. The content of this briefing and the views expressed herein are my sole responsibility.
Brownian Motion Process (BMP) • Stochastic Process • B(0) = 0 • Stationary and independent increments • B(t2) – B(t1) ~ N(η(t2 – t1),δ2(t2 – t1)), 0 ≤ t1 < t2 • η = drift parameter • δ2 = diffusion parameter
• Assume Positive Drift • η>0
20 BMP Realizations (η = 1, δ = 0.05 )
B(t)
t
Source: http://www.fil.ion.ucl.ac.uk/~wpenny/bdb/sp.pdf
BMP-Induced Distributions
B(t0) ~ N(ηt0,δ2t0) B(t)
t
t = t0
20 BMP Realizations (η = 1, δ = 0.05 )
T~ IG(ξ/η, ξ2/δ2)
B(T) = ξ>0 B(t)
t
Inverse Gaussian (IG) Distribution • First passage time of BMP(η, δ) wrt critical boundary ξ • Modeling of reliability, fatigue life, degradation processes
• 𝑔 𝑡; 𝜇, 𝜆 = • • • •
𝜆 −𝜆 𝑡−𝜇 2 𝑒𝑥𝑝 2𝜋𝑡 3 2𝜇2 𝑡
Mean = μ = ξ/η, Shape parameter = λ = ξ2/δ2 Spans spectrum of shapes, including long-tailed Exponential family, with well-developed statistical inference procedures Closed-form MLEs for regression parameters (μ-1 = α + βx)
• 𝐺 𝑡; 𝜇, 𝜆 = 𝛷
𝜆 𝑡 𝑡 𝜇
−1
• 𝛷(·) denotes N(0,1)
+ 𝑒𝑥𝑝
2𝜆 𝜇
𝛷 −
𝜆 𝑡 𝑡 𝜇
+1
IG Pdf Shapes
Source: https://upload.wikimedia.org/wikipedia/commons/a/a5/PDF_invGauss.svg
BMP or BMP*?
• B(t;η,δ) is only generally increasing • May be inappropriate for modeling of some processes
• B*(t;η,δ) ≡ sup0<s≤t B(s;η,δ) is nondecreasing • Wiener Maximum Process (Singpurwalla 2006)
• IG derives from either representation
Step-Stress Partial Accelerated Life Testing • Step-Stress ALT
• PALT • Two stages only • No censoring
Tampered BMP • 𝐵 𝑡 =
𝐵1 𝑡 , 𝑡 ≤ 𝜏 𝐵1 𝜏 + 𝐵2 𝑡 − 𝜏 , 𝑡 > 𝜏
• B1(t) = B1(t;η1,δ)
• B2(t) = B2(t;η2,δ)
independent
• η2 > η1
• System fails when B(t) first attains a critical threshold value ξ
Statistical Literature • Bhattacharyya, G.K. 1987. Parametric Models and Inference Procedures for Accelerated Life Tests. in Proceedings of the 46th Session, Tokyo, Japan, September 8-16, Bulletin of the International Statistical Institute LII, 4: 145-162. • Mentioned, in a conceptual sense, the Tampered BMP
• Luh, Y. and Storer, B. 2001. A Tampered Brownian Motion Process Model for Partial Step-Stress Accelerated Life Testing. Journal of Statistical Planning and Inference 94:1524. • Presented pdf f(t), with incorrect attribution • Established basic properties of f(t) • Continuous and may be either unimodal or bimodal • All positive integer moments exist • MLEs are unique with probability tending to 1, are strongly consistent, and are asymptotically normally distributed
Pdf Derivation
• Bhattacharyya 1987 Derived before 1987 invited paper • Luh and Storer 2001 Derived by Bhattacharyya after his 1987 invited paper • Fries, A. 2015. Derivation of the Distribution Function for the Tampered Brownian Motion Process Model. Gnedenko e-Forum International Group on Reliability electronic journal – Reliability: Theory & Applications, Vol. 10, No. 3, September 2015
Pdf Derivation • Bhattacharyya 1982. Unpublished note: problem definition • Fries 1982. Unpublished note: derivation • Bhattacharyya and Fries 1982. Unpublished draft paper
• Bhattacharyya 1987 Derived prior to 1987 invited paper • Luh and Storer 2001 Derived by Bhattacharyya after his 1987 invited paper • Fries, A. 2015. Derivation of the Distribution Function for the Tampered Brownian Motion Process Model. Gnedenko e-Forum International Group on Reliability electronic journal – Reliability: Theory & Applications, Vol. 10, No. 3, September 2015
Pdf Representation 𝑓 𝑡 =
𝜆 𝜆𝑡𝑐 2 (𝜇1 , 𝑡) 𝑒𝑥𝑝 − 2𝜋𝑡 3 2 𝜆 𝜆𝑡 2 𝑡 𝑒𝑥𝑝 − 𝑐 𝜇1 , 𝑡 + 2 𝑐 𝜏, 𝑡 3 2𝜋𝑡 2 𝜇2
• • • • • •
λ = ξ2/δ2 μi = ξ/ηi, for i = 1, 2 c(a,b) = (1/a – 1/b) for a, b ≠ 0 s(t) = q(t,∆+1,λ) – q(t,∆–1,λ) ∆ = τ∙c(μ2,μ1) q(t,a,λ) = a∙exp(½a2λc(τ,t))∙ 𝛷(a(λc(τ,t))½)
𝑡 ≤ 𝜏,
𝑠 𝑡 ,
𝑡 > 𝜏.
Derivation – Pursuit of Cdf • For a fixed t ≤ τ: 𝑃 𝑇≤𝑡 =𝑃
𝑠𝑢𝑝 𝑠 𝜖 0,𝑡
𝐵 𝑠 >𝜉 =𝑃
𝑠𝑢𝑝 𝑠 𝜖 0,𝑡
𝐵1 𝑠 > 𝜉 = 𝐺 𝑡; 𝜇1 , 𝜆
• For the non-trivial case, with a fixed t > τ, proceed by conditioning on the value of 𝐵1 𝜏 : 𝑃 𝜏 τ: 𝑃 𝜏 τ: 𝑃 𝜏 τ: 𝑃 𝜏 τ:
1 1 1 𝜂1 𝜏 − 𝜉 2 2 𝑓 𝑡 = ∙ 𝑒𝑥𝑝 − + 𝜂 𝑡−𝜏 2 2𝜋𝛿 2 𝜏 𝑡 − 𝜏 3 2𝛿 2 𝜏 𝜉 𝜉 1 1 1 1 + 𝜂 − 𝜂 + + 𝜂 − 𝜂 − 2 1 1 𝜏 −𝐼 𝑡 −𝜏 𝜏 ,− 2 𝜏 ∙ 𝐼 𝑡 − 𝜏2 𝜏 ,− 2𝛿 𝛿2 2𝛿 2 𝛿2 where ∞
𝜗 ∙ 𝑒𝑥𝑝 − 𝛼𝜗 2 + 𝛽𝜗
𝐼 𝛼, 𝛽 ≡ 0
• Collect terms
d𝜗 =
1 1− 2𝛼
𝛽 𝛽 2𝛼 2𝛼 𝜙 − 𝛽 2𝛼 𝛷 −
,
Possible TBMP Extensions • Cdf • > 2 stages • TBMP*
Possible TBMP Extensions – Cdf • Could try integrating the pdf directly • Would need: 𝑡 𝜏 𝜏 𝜇1
𝜆 𝜆𝑠 𝑒𝑥𝑝 − 2𝜋𝑠 3 2 −
𝜏 𝜇2
+ 1 exp
1 1 − 𝜇1 𝑠
λ
𝜏 2 𝜇1
−
𝜏 𝜇2
2
𝑠 1 1 + 2 − 𝜇2 𝜏 𝑠
+1
2 1 𝜏
1 −𝑠
⋅ 𝛷
𝜏 𝜇1
−
𝜏 𝜇2
+1
λ
1 1 − 𝜏 𝑠
1/2
− 𝜏 𝜇1
−
• ?
𝜏 𝜇2
− 1 exp
λ
𝜏 2 𝜇1
−
𝜏 𝜇2
−1
2 1 𝜏
1 − 𝑠
𝛷
𝜏 𝜇1
−
𝜏 𝜇2
−1
λ
1 1 − 𝜏 𝑠
1/2
ds
Possible TBMP Extensions – Cdf • Could try solving the integral in the BMP derivation • Would need: ∞ 0
2𝜉𝜐 1 − 𝑒𝑥𝑝 − 2 𝛿 𝜏
𝛷
𝜐2 ∕ 𝛿 2 𝑡 − 𝜏 −1 𝑡 − 𝜏 𝜐 ∕ 𝜂2
2𝜐𝜂2 𝜐2 ∕ 𝛿 2 𝑡 − 𝜏 + 𝑒𝑥𝑝 𝛷 − +1 𝛿2 𝑡 − 𝜏 𝜐 ∕ 𝜂2
• Linear (in 𝜐) arguments! • But still ?
1
𝜐 − 𝜉 + 𝜂1 𝜏 𝜙 d𝜐 𝛿 𝜏 𝛿 𝜏
General Integral ∞ 0
Φ 𝛼1 + 𝛽1 𝑠 𝜙 𝛼2 + 𝛽2 𝑠 d𝑠
• Fayed & Atiya (2014) • Related integral (via linear transformation): 𝐼 𝑎, 𝑏, 𝑥 ≡
𝑥 Φ 0
𝑎 + 𝑏𝑤 𝜙 𝑤 d𝑤
• “To our knowledge, this integral has no closed form or even a single series expansion.” • Establish representation as as an infinite series of the normalized incomplete Gamma function and the Hermite polynomial
Fayed & Atiya (2014) Result
Possible TBMP Extensions – 3+ Stages • Conditioning approach conceptually works • Encounter cumbersome analytical obstacles • Same problematic integral type appears for 3stage setting
Possible TBMP Extensions – TBMP* • Now condition on the value of 𝐵1∗ 𝜏 • 𝐵∗ 𝑡 =
𝐵1∗ 𝜏 , 𝑡 ≤ 𝜏 𝐵1∗ 𝜏 + 𝐵2∗ 𝑡 − 𝜏 , 𝑡 > 𝜏
• 𝐵1∗ (t) = 𝐵1∗ (t;η1,δ) • 𝐵2∗ (t) = 𝐵2∗ (t;η2,δ) • η2 > η1 • System fails when 𝐵∗ (t) first attains a critical threshold value ξ
Possible TBMP Extensions – TBMP* ∞
𝑃 𝜏
