Talk

Sampling Plans for Control-Inspection Sampling Schemes with Applications to Photovoltaics Ansgar Steland RWTH Aachen University

WISQ 2013, CSIRO, Sydney, Australia

Outline

1

Introduction

2

Related Work

3

Applications in Photovoltaics

4

Two-Stage Acceptance Sampling

5

Approximations of the OC curve

6

Independent Sampling

7

Dependent Sampling (Spatial-Temporal)

8

Computational Aspects

9

Extensions to Functional Data

10

Simulations

A. Steland

Sampling Plans for Control-Inspection Schemes

Introduction

Let X ∼ F be a quality measurement of a produced item of a lot (e.g. power output of a solar panel). Denote σ 2 = Var(X )

µ = E (X ),

Basic question: Should the lot be rejected or accepted? The item is non-conforming (out-of-spec, a defect), if X ≤τ τ : one-sided specification limit, usually τ = µ(1 − ε),

A. Steland


Introduction • If there were no randomness (X = µ), then non-conforming ⇔ µ − τ ≤ 0 The lot should be rejected if and only if µ − τ ≤ 0. • Since µ is unknown, it should be replaced by its unbiased canonical estimator n 1X Xi . µ bn = X n = n i=1

based on a control sample X1 , . . . , Xn of size n. • Control sample: Cross-Sectional at one time point (time of delivery). • Thus, the lot should be rejected if and only if X n − τ ≤ c, √ where the control limit c(= c 0 σ n) accounts for the estimation error. A. Steland


Introduction • The fraction of non-conforming items (quality level) p = P(X ≤ τ ) is usually regarded as the key parameter in quality control and insurance. • Note: p determines the costs (of replacement) in case of total sampling. • Fix 0 < AQL < RQL < 1 (AQL: acceptable quality level, RQL: rejectable quality level). • Represent operating characteristic (= probability of acceptance) OC (p) = P(X − τ ≤ c) as a function of p. • OC (p), p < AQL: Producer risk • OC (p), p > RQL: Consumer risk Goal: Determine (n, c) such that both risks are under statistical control. A. Steland


Introduction

Problems in practice • Data are often non-normal, parametric models fail. • Optimal plan depends on unknown F • Way out in photovoltaics: Additional samples from the production line. Demands from practice • Inspection at a later time point. Sampling plan? • How to use control sample at inspection time? • Dependent sampling: Sample in batches/local cluster. • If the items form a spatial grid (as in the case of a solar plant), it makes sense to sample local batches of neighboring PV modules. • Highdimensional data: Measurement curves, Images, ...

A. Steland


Related work • Gaussian case: Liebermann and Resnikoff (1955), Bruhn-Suhr and Krumbholz (1991). • Feldmanna and Krumbholz (2002): Double sampling plans for Gaussian and exponential data. • K¨ ossler and Lenz (1997): Lack of robustness • K¨ ossler (1995): Pareto-type tail approximation, ML estimation • Willrich (2012): Bayesian approach. • Steland and Zaehle (2009), Meisen, Pepelyshev and Steland (2012): R Arbitrary unknown d.f. F with x 4 dF (x) < ∞, estimation based on sample quantiles from X10 , . . . , Xm0 ∼ F ([• − δ]/η). • Hermann and Steland (2010): Smoothed quantiles, kernel estimator with cross-validated bandwidth. • Pepelyshev, Golyandina and Steland (2012): SSA quantile estimation, data-adaptive selector of tuning constant, uniform error bounds. • Pepelyshev, Rafajlowicz and Steland (2013): Bernstein-Durrmeyer polynomial estimator with GPS-selector. • Avellan, Pepelyshev and Steland (201): Two-sided specification limits. A. Steland


Two-Stage Acceptance Samling Let Tn = Tn (X1 , . . . , Xn ) be a statistic such that the lot is accepted for large values of Tn . Operating characteristic OC (p) = P(Tn > c|p),

p ∈ [0, 1].

Given 0 < AQL < RQL < 1 and error probabilities α, β, a sampling plan (n, c) is called valid, if OC (p) ≥ 1 − α,

for all p ≤ AQL,

(1)

and OC (p) ≤ β,

A. Steland

for all p ≥ AQL.


(2)

Two-Stage Acceptance Samling

The lot is examined at two time points t1 , t2 . t1 : time of production or delivery, t2 : time of inspection Procedure: 1

The lot/shipment is examined at time t1 .

2

If the lot is rejected, we stop.

3

If the lot is accepted, we proceed to stage 2 and inspect the lot at time t2

4

If the lot is accepted, the lot has passed.

5

If the lot is rejected, the lot needs maintenance.

A. Steland


Two-Stage Acceptance Samling Denote by Tni the control statistic used at time ti based on the sampling plan (ni , ci ), i = 1, 2. OC curves: OC1 (p) = P(Tn1 > c1 |p),

p ∈ [0, 1],

and, since the sampling plan (n2 , c2 ) is constructed given Tn1 > c1 , OC2 (p) = P(Tn2 > c2 |Tn1 > c1 , p),

p ∈ [0, 1].

A lot is accepted if and only if it is accepted at stage 1 and stage 2, The overall operating characteristic, OC (p) = P(’lot acceptance’|p), is thus given by OC (p) = OC1 (p)OC2 (p), A. Steland

p ∈ [0, 1].


(3)

Two-Stage Acceptance Samling Proposal: Control the overall OC curve. Note that OC (AQL) = (1 − α1 )(1 − α2 ) Symmetry: αi = βi reduces the problem to the proper choice of α1 , α2 . If α1 = α2 (s.th. β1 = β2 ), then select (α1 , β1 ) such that (1 − α1 )2 = 1 − α,

β12 = β.

But α1 < α2 makes sense (to me). Then α2 = 1 −

A. Steland

1−α 1 − α1


A two-stage procedure using additional data • Assume we are given an additional data set i.i.d.

X0 , X01 , . . . , X0m ∼ F0 , sampled at time t0 < t1 , with mean µ0 = E (X0 ) and variance σ02 = Var(X0 ), e.g. from the production line or historical data. • Control sample at time t1 i.i.d.

X1 , X11 , . . . , X1n1 ∼ F1 , • Inspection sample at time t2 i.i.d.

X2 , X21 , . . . , X2n2 ∼ F2 . A. Steland


A two-stage procedure using additional data Distributional assumptions   ,  F •−∆ η Xji ∼ F (•),   F d• ,

j = 0, j = 1, , j = 2.

for constants ∆ ∈ R, 0 < η < ∞ and 0 < d < ∞. Equivalently, in terms of equality in distribution, d

X0 = ∆ + ηX1 , X1 ∼ F (x) = G

x −µ , σ

d

X2 = dX1 .

A. Steland


(4)

A two-stage procedure using additional data Acceptance sampling procedure • At stage 1, i.e. at time t1 , based on a sampling plan (n1 , c1 ), we accept if √ X1 − τ Tn1 = n1 > c1 , (5) Sm P 1 X1i is the average of the observations taken where X 1 = n11 ni=1 at time t1 and one uses v u m u 1 X Sm = t (X0i − X 0 )2 m−1 i=1

if η = 1 (assumed in the sequel, otherwise use Sn , theory still applies...). A. Steland


A two-stage procedure using additional data • If the lot is accepted at time t1 , we draw the inspection sample and calculate √ DX 2 − τ Tn2 = n2 , Sm P 2 X2i . The lot is accepted at where D = 1/d and X 2 = n12 ni=1 inspection time, if Tn1 + Tn2 > c2 . (6) Rationale: – Tn1 comprises the evidence in favor of acceptance of rejection. – Even if we accept, the decision could be a close one (due to small sample). – Aggregate available sampling information available at inspection time. A. Steland


Approximations of the OC The OC curves cannot be calculated exactly → approximations. W.l.o.g. D = 1 in what follows. The standardized arithmetic averages will be denoted by ∗

Xi =

√ Xi − µ , ni σ

i = 1, 2.

The sampling plans turn out to depend on the quantile function G −1 (p) of the standardized observations Xi∗ = (Xi − µ)/σ,

i = 1, 2.

Thus, we shall assume that we have an arbitrary consistent quantile estimator Gm−1 (p) of G −1 (p) at our disposal.

A. Steland


Approximations of the OC Assumption Q: One of the following two conditions holds. (i) Gm−1 (p) is a quantile estimator of the quantile function G −1 (p) of the standardized measurements satisfying √ d m(Gm−1 (p) − G −1 (p)) → V , as m → ∞, for some random variable V . (ii) Fm−1 (p) is a quantile estimator of the quantile function F0−1 (p) of the measurements taken at time t0 , satisfying √ d m(Fm−1 (p) − F0−1 (p)) → U, as m → ∞, for some random variable U. Notice that under condition (ii), one may construct a quantile estimator for G −1 (p) by Gm−1 (p) :=

Fm−1 (p) − X 0 , Sm

where (X 0 , Sm ) consistently estimates (µ0 , σ0 ). A. Steland


(7)

Approximations of the OC (i) Sample quantiles Fem−1 (p) = X0,(dmpe) ,

p ∈ (0, 1),

where X0,(1) ≤ · · · ≤ X0,(m) is the order statistic associated to X01 , . . . , X0m . Poor for this problem, cf. Hermann and S. (2010), Pepelyshev, Rafajlowicz and S. (2013). (ii) The Bernstein-Durrmeyer polynomial estimator of degree N ∈ N for F0−1 (p) is then defined as −1 Fm,N (p)

= (N + 1)

N X

(N)

ai Bi

(p),

p ∈ (0, 1),

i=0

R1 with coefficients ai = 0 Fem−1 (u)BiN (u) du (N) where[Bi (x) = Ni x i (1 − x)N−i for i = 0, . . . , N are the Bernstein polynomials, see Pepelyshev, Rafajlowicz and Steland (2013), JNonpSt, to appear. A. Steland


Approximations of the OC For the estimator using a data adaptively selected N, iit holds that p √ sup |Fb −1b (q) − F0−1 (q)| ≤ 2 2 log log m/ m. q

m,N

(iii) Numerically invert integrated kernel density esimator, n

1 X b fm (x) = Kh (x − X0i ), mh

x ∈ R,

i=1

This provides accurate results for sampling plan estimation, see Herrmann and Steland (2010). Here K (•) is a regular kernel, usually chosen as a density with mean 0 and unit variance, h > 0 the bandwidth and Kh (z) = K (z/h)/h, z ∈ R, its rescaled version. The associated quantile estimator is obtained by solving for given p ∈ (0, 1) the nonlinear equation Z xp ! b Fm (xp ) = fm (x) dx = p. −∞ A. Steland


Approximations of the OC

The overall OC is given by OC (p) = P(Tn1 > c1 , Tn1 + Tn2 > c2 |p),

p ∈ [0, 1].

The OC of stage 2 is a conditional one and given by OC2 (p) = P(Tn1 + Tn2 > c2 |Tn1 > c1 , p),

A. Steland

p ∈ [0, 1].


Independent Sampling Assume that the samples Xi1 , . . . , Xini , i = 1, 2, are independent i.i.d. samples. The OC approximations rely on the following asymptotic expansions. Proposition. Under independent sampling, we have ∗

Tn1 = X 1 −

√

n1 Gm−1 (p) + oP (1),

as n1 → ∞ and m/n1 = o(1), and ∗

Tn2 = X 2 −

√

n2 Gm−1 (p) + oP (1),

as n2 → ∞ and m/n2 = o(1). If a quantile estimator Fm−1 (p) of F0−1 (p) is available, both expansions still hold true with Gm−1 (p) defined by (7). A. Steland


Independent Sampling

Theorem. Under independent sampling and Assumption Q we have R √ √ 2 1 − Φ(c2 − z + ( n1 + n2 )Gm−1 (p)) e −z /2 dz OC2 (p) ≈

√ −1 c1 + n1 Gm (p)

√

2π[1 − Φ(c1 +

√ n1 Gm−1 (p))]

for any fixed p ∈ (0, 1). Here An ≈ A, if An = A + oP (1), as min(n1 , n2 ) → ∞.

A. Steland


Dependent Sampling • Using different modules for control sample and inspection not always meaningful. • Rely on a panel design: Draw panel of size n1 at time t1 , independent from t0 sample. • If n2 < n1 , then draw subsample of modules and remeasure them. • If n2 > n1 , then remeasure all n1 panel modules and draw n2 − n1 further modules from the lot. n1 observations from the stage 2 sample are stochastically dependent from the stage 1 observations, the others not. Thus, we are given a paired sample (X1i , X2i ),

i = 1, . . . , n2 ,

with ∗ ∗ Cov(X 1 , X 2 )

r n1 X n2 1 X n1 0 Cor(X1i , X2j ) = =√ ρ, n1 n2 n2 i=1 j=1

where

A. Steland


Dependent Sampling Proposition. Suppose that the above sampling scheme at stages 1 and 2 is applied and assume that one of the following assumptions is satisfied. (i) X01 , . . . , X0m is an i.i.d. sample with common distribution function F0 (x) = F ([x − ∆]/η) and Assumption Q holds. (i) Assumption Q’ is satisfied. Then we have

∗

X1 ∗ X2

d

→N

0 0

,Σ ,

as min(n1 , n2 ) → ∞ with n1 /n2 → λ, where the asymptotic covariance matrix is given by 1 ρ Σ= , ρ 1 with ρ =

√

λ Cor(X1 , X2 ). A. Steland


Dependent Sampling Asymptotic expansions: Theorem. Suppose that the above sampling scheme at stages 1 and 2 is applied and assume that one of the following assumptions is satisfied. (i) X01 , . . . , X0m is an i.i.d. sample with common distribution function F0 (x) = F ([x − ∆]/η) and Assumption Q holds. (i) Assumption Q’ is satisfied. Then we have ∗ √ Tn1 X1 n1 Gm−1 (p) √ + oP (1), − = ∗ n2 Gm−1 (p) Tn2 X2 as min(n1 , n2 ) → ∞ with n1 /n2 → ∞ and max(n1 , n2 )/m = o(1).

A. Steland


Dependent Sampling Theorem. Assume the above dependent sampling design and that (i) or (ii) are satisfied, (i) X01 , . . . , X0m is an i.i.d. with common distribution function F0 (x) and Assumption Q holds. (i) Assumption Q’ is satisfied. If, additionally, |ρ| ≤ ρ < 1, then we have √ √ −1 R 2 c2 −z+( n1 + n2 )Gm (p)−b ρz √ √ e −z /2 dz −1 (p) 1 − Φ c1 + n1 Gm 1−b ρ2 √ , OC2 (p) ≈ √ 2π[1 − Φ(c1 + n1 Gm−1 (p))] (8) where r n n1 γ 1X b 2 ρb = (Xji − X i )2 , , σ bj = n2 σ b1 σ b2 n i=1 P n for j = 1, 2, and γ b = n1 i=1 (X1i − X 1 )(X2i − X 2 ). A. Steland


Dependent Sampling: Spatial Batches Assume that batches of size b are drawn. Thus ni = ri b,

i = 1, 2,

where ri is the number of randomly selected batches. Write (1)

(1)

(r )

(r )

(Xi1 , . . . , Xini ) = (X1 , . . . , Xb , . . . , X1 , . . . , Xb ), (j)

where X` is the `th observation from batch j, ` = 1, . . . , b, j = 1, . . . , ri . Spatial-temporal model: Xi,(`−1)b+j = µi + B` + ij , for i = 1, 2, ` = 1, . . . , r and j = 1, . . . , b. Here {ij : 1 ≤ j ≤ b, i = 1, 2} are i.i.d. (0, σ2 ) error terms, {B` : ` = 1, . . . , r } are i.i.d. (0, σB2 ) random variables representing the batch effect. A. Steland


Dependent Sampling: Spatial Batches We have Cov(Xi ) =

ri M

[σB2 Jb + σ2 Ib ],

i=1

for i = 1, 2, where Jb denotes the (b × b)-matrix with entries 1 and Ib is the b-dimensional identity matrix. Observing that √ √ S Cov( n1 X 1 , n2 X 2 ) = √ √ n1 n2 where S is the sum of all elements of Cov(X1 ), we obtain r r √ √ r1 b 2 σB2 + r1 bσ2 r1 2 r1 2 = bσB + σ . Cov( n1 X 1 , n2 X 2 ) = √ r1 r2 b r2 r2

A. Steland


Computational Aspects Explicit solution for stage 1 −1 (Φ (α1 ) − Φ−1 (1 − β1 ))2 n1 = , (Gm−1 (AQL) − Gm−1 (RQL))2 √ n1 −1 (Gm (AQL) + Gm−1 (RQL)). c1 = − 2

(9) (10)

Plan for stage 2 has to calculated numerically by solving R

√ c1 + n1 Gm−1 (AQL)

2 √ √ 1 − Φ(c2 − z + ( n1 + n2 )Gm−1 (AQL)) e −z /2 dz √ = 1−α2 √ 2π[1 − Φ(c1 + n1 Gm−1 (AQL))]

and 1 √ 2π

R

√ c1 + n1 Gm−1 (RQL)

2 √ √ 1 − Φ(c2 − z + ( n1 + n2 )Gm−1 (RQL)) e −z /2 dz = β2 , √ 1 − Φ(c1 + n1 Gm−1 (RQL))

A. Steland


Computational Aspects Algorithm 1 Select ε > 0. 2 Calculate (n , c ) using (9) and (10). 1 1 3 Perform a grid search minimization of the OC curve over (n, c) ∈ {(n0 , c 0 ) : c 0 = 1, . . . , c ∗ (n0 ), n0 = 1, . . . , 200}, where c ∗ (n0 ) = min{1 ≤ c 00 ≤ 60 : (OC (AQL) − (1 − α2 ))2 + (OC (RQL) − β2 )2 ≤ ε}. → (n∗ , c ∗ ). 4 Use the grid-minimizer (n ∗ , c ∗ ) as a starting value for numerically solving the nonlinear equations up to an error bound ε for the sum of squared deviations from the target. → (n2∗ , c2∗ ). 5 Put n = dn e and minimize numerically the nonlinear 2 2 equations with respect to c2 up to an error bound ε for the sum of squared deviations from the target. Denote the minimizer by c2∗ . 6 Output (n , c ) = (n , c ∗ ). 2 2 2 2 A. Steland


Extensions to Functional Data Highdimensional data ... distributed sensors images movies ... Collaborative project on photovoltaics (PV): PVScan

A. Steland


Extensions to Functional Data We observe L2 random functions (e.g. measurement curves) t ∈ [a, b], j = 1, . . . , ni , i = 0, 1, 2. Rb Inner product: (f , g ) = a f (t)g (t) dt Reduction of Dimensionality: Use t0 sample to calculate FPCA Covariance operator Xij (t),

C (t, s) = E (X01 (t)X01 (s)),

t, s ∈ [a, b],

is well defined and allows the spectral representation ∞ X C (t, s) = λk ϕk (t)ϕk (s) k=1

λ1 ≥ λ2 ≥ · · · : ordered eigenvalues. ϕ1 (t), ϕ2 (t), . . . : eigenfunctions, i.e. (λk , ϕk ) satisfies Z b (C ϕk )(t) := C (t, s)ϕk (s) ds = λk ϕk (t). a A. Steland


Extensions to Functional Data

Estimator: m

X bm (t, s) = 1 C (X0i (t) − X 0 (t))(X0i (s) − X 0 (s)). m i=1

bk , ϕ Calculate a solution (λ bk ) of the empirical equations Z bk ϕ bm (t, s)ϕ λ bk (t) = C bk (s) ds.

A. Steland


Extensions to Functional Data Noting the expansions Xij (t) =

∞ X

(ϕk , Xij )ϕk (t),

j = 1, . . . , ni ,

k=1

suggests the approximation Xij (t) ≈

d X

θbijk ϕ bk (t),

k=1

where θbijk = (ϕ bk , Xij ) = Define the values

Rb a

ϕ bk (s)Xij (s) ds.

Zij (t) = s

d X

2 θbijk

k=1

as an univariate summary. A. Steland


Simulations Parameters α = β = 0.1 (global error probabilities) AQL = 2% and RQL = 5%. α1 = β1 ∈ {0.03, 0.05, 0.07}, α2 = 1 − (1 − α)/(1 − α1 ) Simulation models

Model 1:

X0 ∼ F 1 = N(220, 4),

Model 2:

X0 ∼ F 3 = 0.9N(220, 4) + 0.1N(230, 8).

Model 3:

X0 ∼ F 3 = 0.2N(200, 4) + 0.6N(220, 4) + 0.2N(230, 8).

Model 4:

X0 ∼ F 4 = 0.2N(212, 4) + 0.6N(220, 8) + 0.2N(228, 6).

A. Steland


Simulations

Kernel quantile estimators using bandwidth selectors Biased cross-validation (BCV). Sheather-Johnson bandwidth selection (SJ), Sheather and Jones (1991). Golyandina-Pepeyshev-Steland method (GPS), Golyandina, Pepelyshev and Steland (2012). Indirect cross-validation (ICV), Savchuk et al (2010).

A. Steland


Simulations α1

α2

3%

7.22% 250 250 250 250 3.23% 250 250 250 250 7.22% 500 500 500 500 3.23% 500 500 500 500

7%

3%

7%

m

Type

E (n1 )

sd(n1 )

c1

sd(c1 )

E (n2 )

sd(n2

BCV SJ GPS ICV BCV SJ GPS ICV BCV SJ GPS ICV BCV SJ GPS ICV

80.35 83.37 78.82 80.18 49.50 51.40 48.71 49.62 80.78 82.65 80.11 80.94 50.25 50.60 49.11 49.96

22.12 25.21 21.87 22.96 13.69 15.50 13.29 13.56 18.69 19.85 17.32 18.75 11.51 12.11 10.50 11.23

17.46 17.53 17.35 17.40 13.69 13.77 13.64 13.70 17.30 17.36 17.30 17.30 13.64 13.58 13.56 13.61

2.11 2.37 2.13 2.17 1.68 1.82 1.64 1.65 1.76 1.86 1.65 1.76 1.39 1.45 1.29 1.35

18.47 20.42 17.72 18.58 22.82 24.43 22.31 22.98 19.21 20.38 18.73 19.39 23.55 24.17 22.71 23.37

8.52 9.99 8.30 8.90 7.81 9.30 7.56 7.85 7.22 7.71 6.62 7.31 6.96 7.47 6.47 6.93

A. Steland


Simulations α1

α2

3%

7.22% 250 250 250 250 3.23% 250 250 250 250 7.22% 500 500 500 500 3.23% 500 500 500 500

7%

3%

7%

m

Type

E (n1 )

sd(n1 )

c1

sd(c1 )

E (n2 )

sd(n2

BCV SJ GPS ICV BCV SJ GPS ICV BCV SJ GPS ICV BCV SJ GPS ICV

175.96 235.04 173.75 175.55 108.51 144.84 108.23 109.82 232.33 254.26 210.49 223.13 143.51 156.45 127.81 137.61

51.92 65.62 44.52 46.76 32.15 42.35 28.01 29.24 53.76 57.17 45.28 49.59 32.56 34.23 26.40 30.12

24.75 27.53 24.65 24.72 19.44 21.58 19.43 19.54 27.48 28.41 26.50 27.04 21.58 22.32 20.67 21.25

2.71 3.31 2.46 2.51 2.15 2.69 1.94 1.97 2.66 2.79 2.32 2.53 2.06 2.14 1.72 1.93

61.64 89.11 60.73 61.59 60.57 89.85 60.53 61.56 88.16 97.85 78.47 84.12 88.75 99.11 75.96 84.12

24.03 28.80 21.32 21.90 25.26 33.27 22.22 23.00 23.91 24.81 20.47 22.16 26.26 27.24 21.83 24.51

A. Steland


Conclusions

Concluding Remarks Extension of acceptance sampling to a Control-Inspection Scheme. At inspection time, samples are aggregated. Independent as well as dependent sampling. Stage-two sampling plan can be estimated with reasonable accuracy. Extensions to multiple time points deserves future research. Extensions to functional data deserves future research.

A. Steland


End

Thanks for your attention.

A. Steland


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