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```					                                                                                                                                                                               An Application of Dirichlet Process Isotonic Regression
to the Study of Radiation Eﬀects in Spaceborne Microelectronics
Marian Farah, Athanasios Kottas, and Robin D. Morris
Department of Applied Mathematics and Statistics, University of California, Santa Cruz

Motivation & Background                                                                                                                                                                                        Bayesian Semi-Parametric Model                                                                                                                                                                                                                              Interpolation/Extrapolation
• This work is concerned with the vulnerability of spaceborne microelectronics to single event                                                                                                                                                                                                                  µci
p(ci|µi) = i exp(−µi), µi = fiσG(ℓi)                                                                                                                                                                                                                          ˜
• Interpolation is based on the posterior predictive distribution p θ|data ,
upset (SEU), a change of state caused by ions or electromagnetic radiation (e.g. solar wind)                                                                                                                                                                                                                   ci!
striking a sensitive node.                                                                                                                                                                                                                                                                                                                                                                                                                                                                          ˜     ˜        ˜
where θ = θ1, . . . , θM are the new cross-section values corresponding to
G ∼ DP α, G0(ℓi; w, s)
• The number of upsets, c, depends on the linear energy transfer (LET), ℓ, the limiting cross-                                                                                                                                                                                                                                            s                                                                                                                                                                            ˜    ˜        ˜                   ˜            ˜
unobserved LET values, ℓ = ℓ1, . . . , ℓM , such that ℓi ≤ ℓ1 ≤ · · · ≤ ℓM ≤
section of interaction, σ, and the ﬂuence, or the strength of the radiation ﬁeld, f .                                                                                                                                                                                                                                    ℓi
G0(ℓi; w,s) = 1 − exp −                                                                                                                                                                           ℓi+1 for i = 1, . . . , N − 1.
w
• The                                number of upsets is assumed to be monotonically increasing with LET at a given orbit.                                                                                                                                                                                                       ind
p(σ),p(α), p(w), p(s) ∼ Gamma                                                                                                                                                                                                        ˜
p θ|data =                                                                   ˜
p θ|θ p (θ, σ, ψ|data) dθdσdψ
• Devices   1 and 2 have distinct ℓi for each ci. Let θi = G(ℓi),                                                                                                                                                     ˜
• p θ|θ is the density of:
i = 1, . . . , N , θ = (θ1, . . . , θN ), and ψ = (w, s). Then, the joint
posterior, p(θ, σ, α, ψ|data), is proportional to
θi + (θi+1 − θi)ω1 , θi + (θi+1 − θi)ω2 , . . . , θi + (θi+1 − θi)ωM ,
                               
N                    N
N
ci                                Γ (α)
σ   i=1 ci            θi exp −σ           fiθi ×     N +1
˜ ˜             ˜ ˜
with (ω1, ω2, . . . , ωM ) ∼ Ordered Dirichlet d1, d2, . . . , dM , dM +1 ,
i=1                  i=1               i=1 Γ(di(ψ))
˜              ˜                    ˜            ˜
d1 = α G0(ℓ1; ψ) − G0(ℓi; ψ) , dm = α G0(ℓm; ψ) − G0(ℓm−1; ψ) ,     ˜
d1(ψ)−1                    d2(ψ)−1                   dN (ψ)−1                                                        dN +1(ψ)−1
θ1        − θ1)
(θ2         . . . (θN − θN −1)                                                   (1 − θN )                                                                                                                                         ˜                            ˜
m = 2, . . . , M , and dM +1 = α G0(ℓi+1; ψ) − G0(ℓM ; ψ)
× p(σ) × p(α) × p(s) × p(w)
˜
• p θ|data is then obtained using Monte Carlo integration.
where d1(ψ) = αG0(ℓ1; ψ), di(ψ) = α (G0(ℓi; ψ) − G0(ℓi−1; ψ)),
i = 2, . . . , N , and dN +1(ψ) = α (1 − G0(ℓN ; ψ)).
• Extrapolation proceeds in an analogous fashion. Here, extrapolating beyond
• Device 3 has repeated measurements for each ℓi, which results in a                                                                                                                                                observed LET values is key because it could reveal important
diﬀerent likelihood, but the joint posterior is obtained similarly.                                                                                                                                               diﬀerences in the prediction of upset rates between the parametric and the
Figure 1: Artistic rendition of the Magnetosphere (image created by NASA).                                                                                                                                                                                               • The  posterior full conditional distribution of θ is sampled through a                                                                                                                                            semiparametric models.
carefully designed slice sampler. The posterior full conditional of σ is
• Tomeasure the susceptibility of a semiconductor device to SEU, the device is exposed to                                                                                                                                                                                   given by a gamma distribution; α & the pair (w, s) are sampled with                                                                                                                                             • Distributions of upset rates at a particular orbit are obtained by inputting
high-energy particles in a particle accelerator.                                                                                                                                                                                                                           two Metropolis-Hastings steps.                                                                                                                                                                                    posterior samples of cross-section to CREME96.
• Thekey inferential objective in particle accelerator experiments is the cross-section vs. LET
curve, G(ℓ).
• Thestandard practice in the nuclear physics literature is to assume a Weibull parametric                                                                                                                                                                                                                                                                                                                                                      Results
form for G(ℓ) estimated with (weighted) least squares.

Cross−section (cm2/bit)

Cross−section (cm2/bit)

Cross−section (cm /bit)
x 10
−7                Device 1                                                                                 x 10
−6                  Device 2                                                                     x 10
−9             Device 3
1                                                                                                               6                                                                                                        2
• The upset rate at a particular orbit is obtained by inputting the estimated value of cross-

2
5

section to CREME96 (or Cosmic Ray Eﬀects on Micro-Electronics), a widely-used code for                                                                                                                                                                                   The semiparametric model ﬁts the                                                                 0.8

4
1.5                                                                      Data
95% Prob Intvl (SP)
modeling radiation environments to evaluate radiation eﬀects in spacecraft                                                                                                                                                                                               data better but has wider probability                                                            0.6
3                                                                                                        1                                                                    Posterior Mean (SP)

intervals between observations and in                                                            0.4
2
0.5
95% Prob Intvl (P)
Posterior Mean (P)
the extrapolation region.                                                                        0.2
1

0                                                                                                               0                                                                                                        0
0        20      40          60          80         100        120                                             0             20       40         60       80        100         120                                     0                  5             10                  15
2                                                                                                           2                                                                                                   2
LET (MeV/mg/cm )                                                                                                    LET (MeV/mg/cm )                                                                                     LET (MeV/mg/cm )

Our Approach                                                                                                                                                                                                                                                                                                                                                            x 10
6
x 10
5

250                                                                                                                  2                                                                                                             5

200                                                                                                                                                                                                                                4
• We work with a Poisson model for the upset counts and propose a semiparametric isotonic                                                                                                                                                                                                                                                                                                                                                                              1.5

Density

Density

Density
Semiparametric
regression method for count responses.                                                                                                                                                                                                                                                                                                                       150                                                                                                                                                                                                                                3
Model
The choice of model impacts the                                                     100
1
2                                                                          Parametric

• The approach is based on a Dirichlet Process (DP) prior for G(ℓ), which allows the data to                                                                                                                                                                              predicted SEU. In general, the                                                              50
0.5
1
Model

drive the shape of the cross-section-LET relationship.                                                                                                                                                                                                                   average SEU is smaller under the                                                                 0                                                                                                       0                                                                                                             0

semiparametric model                                                                             0.1       0.12      0.14    0.16
SEU/device/day
0.18        0.2       0.22    0.24                                       0           0.5        1        1.5
SEU/device/day
2   2.5       3
−6
x 10
3.5                                  2           2.5    3       3.5
SEU/device/day
4        4.5       5
x 10
5.5
−5

• We apply our methods to data obtained from three particle accelerator experiments corre-
sponding to three diﬀerent experimental scenarios.                                                                                                                                                                                                                                                                                 Figure 3: Upper panels: The posteriors of cross-section vs. LET curve under the semiparametric (SP) and parametric (P) models.
x 10
−8             Device 1                                                    x 10
−6                   Device 2                                                                x 10
−9              Device 3
Lower panels: The distribution of the SEU’s under the semiparametric and parametric models
7                                                                            3.5                                                                                              1.8

1.6
6                                                                             3

Discussion
Cross−section (cm2/bit)

Cross−section (cm2/bit)

Cross−section (cm2/bit)

1.4
5                                                                            2.5
1.2

4                                                                             2                                                                                                1

3                                                                            1.5                                                                                              0.8
• Wehave presented a Bayesian semiparametric method for modeling the cross-section vs. LET curve, G(ℓ), for the prediction of distributions of SEU. Our
2                                                                             1
0.6                                                             method relaxes the parametric assumption of a Weibull ﬁt and enables accurate quantiﬁcation of uncertainty due to the functional form of G(ℓ).
0.4

1                                                                            0.5
0.2                                                                    the model to the data from devices 1 & 2 is challenging, because only the left tail of the distribution associated with the cross-section-LET curve
• Fitting
0
0        20   40      60
LET (MeV/mg/cm )
80
2
100   120
0
0        10   20   30   40   50   60
LET (MeV/mg/cm )
2
70   80   90   100
0
0        2   4       6     8
2
LET (MeV/mg/cm )
10   12    14         has been observed. This results in weak identiﬁability between w and σ. We resolve this issue by placing a highly informative prior on σ.
• We  are currently conducting a formal model comparison between the semiparametric and parametric models using a cross-validation posterior predictive
Figure 2: Experimental data for three devices. Observed cross-section equals to count/ﬂuence.
criterion. In future work, we will extend the model to account for ﬂuence uncertainty.

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