Available online at www.sciencedirect.com
ScienceDirect Procedia CIRP 12 (2013) 498 – 503
8th CIRP Conference on Intelligent Computation in Manufacturing Engineering
Vertical density profile monitoring using mixed-effects model B.M. Colosimoa, M. Menesesb,*, Q. Semeraroa a Dipartimento di Meccanica, Politecnico di Milano, via La Masa 1- 20156 Milano, Italy Escuela de Ingeniería en Producción Industrial, Instituto Tecnológico de Costa Rica, Cartago, Costa Rica *Corresponding author. Tel.: +506-2550-9211, fax: +506-2550-9255. E-mail address:
[email protected],
[email protected] b
Abstract Profile monitoring is a recent field of research in Statistical Process Control (SPC) literature, which is attracting the interests of many researchers. This approach is used where process data follow a profile and the stability of this functional relationship is checked over time. We consider nonparametric mixed effect models for functional data to model the profile. Then, multivariate control charting is applied to identify mean shifts or shape changes in the profile. A real case study dealing with density measurements along the particleboard thickness (usually referred to as Vertical Densit y Profile -VDP) is taken as reference throughout the paper. Performance of the nonparametric approach is computed for a set of outof-control scenarios. Our main conclusion is that nonparametric methods represent a flexible and effective solution to complex profile monitoring. © 2013 The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. © 2012 The Authors. Published by Elsevier B.V. Selection and/or peer-review under responsibility of Professor Roberto Teti. Selection and peer review under responsibility of Professor Roberto Teti Keywords: Profile monitoring; Statistical Process Control (SPC); Quality; Functional Data; Nonparametric mixed-effects; Vertical Density Profile.
1. Introduction In many industrial applications, the process or product quality cannot be simply modeled as a random variable but it is instead related to a functional form (i.e. a function which relates a response to one or more location variables). Examples of functional data include calibration studies, geometric specification and process signal monitoring. If we suppose that functional data are related to the quality of the product or process, then we can monitor the stability of this profile over time. In fact, profile monitoring concerns detecting deviations of the profile from a nominal or in-control pattern in order to identify the time when the process goes out of control. In profile monitoring literature one can find studies focusing on simple linear profiles or complex profiles, using both parametric and nonparametric methods [1]. Nonparametric profile monitoring is useful when the profile is too complicated to be parametrically specified. This class of methods does not require one to specify a functional form of the in-control (baseline) profile. In nonparametric regression, the aim is to define a smoothed curve computed as a weighted average of the
observed data. In the literature, several nonparametric regression models have been proposed. For monitoring purposes, Zou et al. (2008) [2] and Qiu et al. (2010) [3] used local linear kernel smoothing combined with a EWMA (Exponentially Weighted Moving Average) chart, Zou et al. (2007) [4] coupled the approach with a MEWMA (Multivariate EWMA). Mixed-effects models ([5], [6]) have been recently shown to be very useful for profile monitoring ([3], [7], [8], [9], [10] and [11]). These models provide a flexible framework in which population profile characteristics are modeled as a mixture of an overall pattern, characterizing all the profiles, and individual curve variations accounting for profile-to-profile variability. Taking as a reference the nonparametric mixed-effects models presented in [3], our work presents a new approach for solving the model selection issue and for coupling the nonparametric model with control charting. We will use as reference real process data related with particleboard panels, where the density ( y)(kg / m3 ) measured along the panel thickness ( x)(mm) is the profile of interest, usually referred to as Vertical Density Profile (VDP) (Figure 1).
2212-8271 © 2013 The Authors. Published by Elsevier B.V. Open access under CC BY-NC-ND license. Selection and peer review under responsibility of Professor Roberto Teti doi:10.1016/j.procir.2013.09.085
499
B.M. Colosimo et al. / Procedia CIRP 12 (2013) 498 – 503
yi
1000 3
density (Kg/m )
explains the nonparametric mixed-effect model and the associated control charts. Section 4 presents the performance of the proposed method and section 5 describes the conclusions.
fitting
900
2. Real case 800 700 600 0
5
10 location (mm)
15
Figure 1: Vertical Density Profile and local polynomial fitting.
In the literature, different approaches for modelling VDPs have been proposed. In particular, Winistorfer et al. (1996) [12] considered spline regression, Xu and Winistorfer (1996) [13] presented a fitting technique based on Fourier analysis while Walker and Wright (2002) [14] used additive models, which include a nonparametric component to model the form of the other sources of variation in the model. With respect to VDP monitoring, most of the approaches presented in the literature were applied to VDP data presented in [14]. Young et al. (1999) [15] proposed a multivariate control chart for monitoring few features of the profile. This method does not allow one to fully investigate the VDP shape changes. Williams et al. (2007) [16] considered nonlinear regression function for monitoring VDP. Chang and Yadama (2010) [17] used splines, supported by wavelets, to develop a profile monitoring method. Fan et al. (2011) [18] proposed to break the complete profile into several segments for monitoring them separately. Wei et al. (2011) [19] proposed a nonparametric location scale model aimed at screening the shapes of the profiles. The model was built to detect local shifts, local shape distortion and overall shape deviation of VDP. Horng et al. (2009) [20] provided a monitoring schemes based on nonparametric regression. Their method first applies p-spline to each profile and then uses PCA to propose the monitoring schemes based on PC scores. Starting from the approach proposed by Qiu et al. (2010) [3], we develop a new nonparametric monitoring procedures based on mixed effects model. In particular, we focus our attention to problems concerning the model selection (for local polynomial fit) and the use of multivariate control charting for profile changes monitoring. The proposed methods allows us to monitor departures from the basic profile shape, with respect to five possible out-of-control scenarios. The remainder of the paper is organized as follows. In Section 2 we present the real case study. Section 3
Particleboard manufacturing is a complex process, which consists of bonding together the wood particles by adding synthetic adhesives and then pressing them at high pressures and temperatures ([21], [22]). The resulting product has mechanical properties (like bending strength, internal bond and surface soundness) that make them suitable for wide applications in housing and furniture. The most important characteristic of particleboard quality is related to the VDP. There have been a large number of research reports published to describe the correlation between VDP and physical and mechanical properties ([21], [23], and [24]). The VDP curve is influenced by process conditions and changes of its shape affect the mechanical characteristics of the final product. Thus, it is important to monitor VDP with time to avoid an excessive rate of nonconforming products. 2.1. Data set description The j th VDP profile is described by the vector
y j ( y1 j ,...yij ,..., ynj )T with j 1,...,J of densities measured at a set of n equally spaced locations x ( x1,...xi ,...,xn )T . Figure 1 represents a generic VDP, where yij is the density (kg / m3 ) measured at thickness
(mm) i observed on the j th panel profile. In our real case study, the VDP sample was measured by using a xray profilometer [25] that takes measurements at fixed depths across the thickness of the board. The reference sample is a set of J 263 profiles made of n 189 observations and each measurement is 0.09 mm apart. The VDP data are correlated because the density measurements are taken at close intervals along the vertical depth of the particleboard. 3. Methods 3.1. The nonparametric mixed effect model According to [3] we assume the nonparametric mixed-effects model yij
g ( xij )
f j ( xij )
ij
, j 1,...,J , i 1,...,n
(1)
where yij is the observed response (density) of profile
j at the location point xij along the thickness; g ( xij )
500
B.M. Colosimo et al. / Procedia CIRP 12 (2013) 498 – 503
models the population mean function (the fixed effect) and f j ( xiij ) represents the individual curve deviation from g ( xij ) (the random effects) curve. The
ij
the measurement errors with mean 0 and variance
2 j
.
Afterwards, according with [3] and [6], we simultaneously estimate g () and f j () by combining linear mixed effects models and local polynomial techniques. For any fixed point s , g (s) and f j (s) at
yij
where zij
zijT b j
and f j (s)
[1, ( xij s)]T ,
(2)
is a deterministic two-
dimensional coefficient vector and b j is a twodimensional vector of random effects with b j ~ N (0, D) ,
D( x) E(b j bTj ) . In order to find the
where D
and b j , the usual procedure based on the local log-likelihood is used. The resulting estimators are given by unknown coefficient vectors
J j 1
bj with 1/ 2 j
Zj
1
Z Tj
j
Zj
T j
2 j
Z KjZj T
( z1 j ,...,znj ) T j
1/ 2 j
2 j
J j 1
D
1
,
1
Z Tj
j
yj
(3)
T j
Z K j yj
j
1
K Vj K
1/ 2 j
(4) , Vj
diag{Kh x1 j
s ,..., ,
K h xnj s } , K h ( ) K ( / h) / h , where K
is a
K
ZjD Z K
and
Kj
1/ 2 j
Zj
1ij
(5)
Here yij is the point observed at the j th profile,
j 1,2,...,JJ at location xij (i 1,...,n) . Then, we compute the generalized cross-validation (GCV) for each hk used J
GCV (hk )
xij can be approximated via Taylor expansion using a first-degree polynomial
g(s) zijT
G xij , hk
n
j 1i 1
yij G( xij , hk )
1 d (hk ) n * J
2
2
(6)
where G( xij , hk ) is the estimated fixed-effects (Eq. 5) and d (hk ) are the degrees of freedom due to this estimation. Because the fit G() is linear in y , we compute the effective degrees of freedom as a trace of the smoother matrix [26]. Once the n h bandwidths have been used and the corresponding GCV computed, we select the best one as h1 min nh GCV (hk ) . Next, we use h1 to compute
G( xij , h1) 1ij
and
recompute
the
residuals
yij G( xij , h1) which will be used in the second
step. It is important to point out that the value G( xij , h1) is inappropriately estimated because it is affected by the random effects f j ( xiij ) , which have not been considered in Eq. 5. Step 2: we generate n h possible bandwidth values and we fit the mixed-effects model to the residual as
1ij
G( xij , hk )
f j ( xij , hk )
2ij
(7)
symmetric kernel function centered in 0 and h is the bandwidth used by the local polynomial technique.
where
3.2. Model estimation
estimator G( xij , h1) . Eventually, we compute again
We used the Epanechnikov Kernel function, 1 a 1 , which is K (a) 0.75(1 a 2 ) with commonly used in the local smoothing literature because of its good properties [26]. Similarly to what presented in [6], we used a two-steps data driven procedure to select the bandwidths. In fact, two different bandwidths can be used to describe the fixed- and the random-effects curves, since the patterns of the two curves can be very different. The two steps procedure is described as follows. Step1: for the population bandwidth, we generate n h possible values of hk , k 1,...,nh , for each bandwidth hk , we fit the following fixed-effects model starting from the sample of J profiles
GCV (hk ) for each bandwidth and select h2 min h GCV (hk ) . To select the bandwidth, in the first step we generate nh 100 possible values of hk , uniformly spaced in the interval ranging from hmmin 0.01 to hmax range ( x ) / 8 . The range for the VDP profiles is x 0.12,17.04 mm . We fit Eq. 5 starting from the sample of J 263 profiles and then compute the GCV for each hk . The obtained value of h1 is 0.41 . Using Eq. 7 and similar procedure from step 1, the value of the bandwidth in step 2 resulted to be h2 =0.54. Figure 1 shows an example of the local polynomial fitting for a VDP. After selecting the bandwidths, the iterative procedure suggested in [3] and [6] was used to estimate the
G( xiij , hk ) accounts for the bias of the previous
501
B.M. Colosimo et al. / Procedia CIRP 12 (2013) 498 – 503
parameters of Eqs.3 and 4. This procedure allows one to take into account the correlation structure of the data.
computed in Phase I. Then we compare the T j 2 and the residual variance 2j with their respective control limits computed in Phase I.
3.3. Control charts for VDP monitoring Each monitoring tool needs two distinct phases. Phase I (the design phase) is used to evaluate the process stability and estimate the in-control parameters while Phase II (the operative phase) is aimed at monitoring the data to quickly detect process changes.
4. Performance evaluation An alternative model of the VDP, developed in [27] was used to simulate out-of-control scenarios. According to this procedure, the j th VDP can be represented as:
In the design phase, our objective is to fit the model yj
yij
g ( xij )
f j ( xij , h2)
2ij
(8)
where g ( xij ) G xij , h1 G xij , h2 and h1 and h2 are the selected bandwidths. We assume that the predicted individual or randomeffect f j ( x, h2) is a good indicator of the process variability, where x ( x1 ,...xi ,...,xn )T is the location vector and is assumed to be constant for all the profiles. T 2 control Considering Eq.8 chart is built on the predicted values of the randomeffects:
f jT ( x)
T j2
1 f
f j ( x)
(9)
T
1 where f j is the f j ( x1 , h2),..., f j ( xn , h2) , f inverse of the variance-covariance matrix given by f
1 2 ( J 1)
J 1 j 1
fj
1
fj fj
1
fj
T
([7], [9]).
Residuals 2ij s are used to compute their variance 2j , which is monitored using a traditional control chart. In order to design the two control charts, a first type error of
(1
1
'
) is set for each of the
'
0.05) . two charts ( Eventually, the empirical distributions of the monitored 2 statistics, T j and 2j are computed to define the control limits. In particular, we used bootstrapping to generate replications of the VDP as the mean of 7 profiles randomly selected among the first set of J 263 profiles. The empirical distributions of the two monitored statistics were computed using 20000 replications. In Phase II (or the operating phase of control charts), for each new profile y j we use Eq.8 to estimate f j ( x, h2) and
2 j
, by subtracting the overall fixed-effect profile
g ( x ) (known from the design stage). Then we use Eq.9 to compute the Hotel
T j 2 , using
f
10 k 1
Bk ( j ; x) c j
1 1 a1 j B - a2 j B2 a3 j B3 a4 j B4
j
(10)
where j 1,2,...,J , Bk ( j ; x) are B-spline basis of order 4 with 6 internal knots and the second part of the model represents an autoregressive error model of order 4, i.e., an AR(4) and B is a Backshift operator. The vector of in-control parameters is given by T c1 j ,...,c10 j , a1 j ,...,a4 j (6.82, 6.96, 6.49, 6.42, 6.37, 6.32,
6.38, 6.47, 6.81, 6.85, 1.24, 0.35, 0.02, 0.10) 2 j
and
5
1.48 10 . This model is used to simulate incontrol VDP profiles. Five realistic scenarios defined in Table 1 were used to evaluate performance of the proposed monitoring procedure. These cases were generated by opportunely perturbing the parameters to introduce changes in the general pattern (scenario A, B and C of Table 1) or in the autocorrelation structure of the profiles (scenarios D and E of Table 1). 4.1. Results The performance of the procedures were computed using the average run length ( ARL1 ) required to detect the specific out-of-control conditions assumed in the out-of-control scenarios. A set of 10000 run lengths were collected for each scenario before computing the ARL and the standard deviation (in parenthesis) in Table 2. We simulated different sizes of the shift to compute the operation characteristic curve (OC) given by the II type error , which is related to ARL1 as ARL1 1 / 1 . For changes of the B-spline components, we suppose that the in-control coefficients c change according to a shift size , where c indicates the coefficients involved in the scenario A, B or C of Table 1. For the first three 18 c. scenarios the shift is
B.M. Colosimo et al. / Procedia CIRP 12 (2013) 498 – 503
A
B
C
Coeff.
Change Values
c2 , c3 c8 , c9
-1, -2 -2, -1
c 4 , c5 c6 , c7 c8
1.5, -1 -1, -1 1.5
c1 , c2 c3 , c 4 c5 , c 6 c7 , c8 c9 , c10
-1.5,-1, -1, 1, 1.4, 1.4, 1,-1, -1,-1.5
D
x 1.25 a1 j , a2 j
E
a3 j , a 4 j
x 0.8
Process change description, cause (C) and consequence (Cs) Increased drop in the density between the surface layers and the middle layer. C: high temperature in the central area of the press or low speed pressing. Cs: reduce the value of SS. Increase of the density change between the surface layers and the middle layer. C: excessive moisture on the superficial layer, low press residence time or high speed pressure on entrance press. Cs: lack of cohesion of the middle layer of the particles (poor IB). Density gradient reduced (reduction of outer layers density and increase in inner layer density). C: excessive time spent in the press or poor surface moisture of the faces. Cs: facilitate the improvement of the IB but causes a deterioration in BS. Increasing variance of the noise term. C: change in particle wood size. Cs: higher 'roughness' of the profile that can affect the board strength. Reduced intensity of the autocorrelation coefficients (alteration of particles wood size). Apply the remarks made in D.
to be large. With reference to the mean perturbations (scenarios A, B and C), the OC curves (Figures 2-4) show that performances of the nonparametric method is not sensible to small shifts. For change in the variable component, the monitoring procedure is efficient in detecting small and large shift, specially for scenario E. A 1,0 0,8 0,6 ß
Table 1: Five out-of-control scenarios (BS: bending strength, IB: internal bond, SS: surface soundness ).
0,4 0,2 0,0 0
1
2
3
4
5
6
7
8
9
l
Figure 2: Probability of type II error for the monitoring approach for shifts in the scenario A. B 1,0 0,8 0,6 ß
502
0,4 0,2
Table 2: scenario ARL1 (standard errors)
ARL1 results in all the simulated scenarios listed in Tab.1. A
B
C
D
E
1.0118 (0.0011)
3.2244 (0.0265)
1.0000 (0.0000)
1.0140 (0.0012)
1.0021 (0.0005)
0,0 0
1
2
3
4
5
6
7
8
9
l
Figure 3: Probability of type II error for the monitoring approach for shifts in the scenario B C 1,0 0,8
ß
0,6 0,4 0,2 0,0 0
1
2
3
4
5
6
7
8
9
l
Figure 4: Probability of type II error for the monitoring approach for shifts in the scenario C D 1,0 0,8 0,6 ß
For example, in the scenario A the shifts are defined by 1 8 c2 , c3 , c8 , c9 1 8 1, 2, 2, 1 1 / 8 0.125, 0.250, 0.250, 0.125 . We will use l in the next figures to indicate the times that we apply the shift in each case, l 0 meaning no changes in the in-control coefficients, l 1 the in-control coefficients change in , l 2 the in-control coefficients change in 2 and so on until l 9 . Figures 2 to 4 show the OC curves for scenarios A, B and C. For changes of the small-scale components, shifts are obtained in the following way. For the scenario D the noise term increase by 1 7 0.25 0.0357 , using the same number l . Therefore, for l 1 the standard deviation changes to 1.0357 , for l 2 to 1.0714 and so on until l 8 . Finally, for scenario E we simulate a reduction of the autocorrelation coefficients a1 j , a2 j , a3 j , a4 j . Figures 5 and 6 plot the OC curves for the scenarios D and E, respectively Table 2 shows that the monitoring procedure is effective in detecting out-of-control states. It detects the out of control in the first sample for each scenario except for case B. The OC curve of scenario B (Figure 3) shows that the method detects the change only when they start
0,4 0,2 0,0 0
1
2
3
4
5
6
7
8
9
l
Figure 5: Probability of type II error for the monitoring approach for shifts in the scenario D.
B.M. Colosimo et al. / Procedia CIRP 12 (2013) 498 – 503
E 1,0
[8]
0,8
[9]
0,4
[10]
ß
0,6
0,2
[11]
0,0 0
1
2
3
4
5
6
7
l
Figure 6: Probability of type II error for the monitoring approach for shifts in the scenario E.
[12] [13]
5. Conclusions We have presented a performance study of one promising approach for complex profile monitoring with autocorrelated errors. The nonparametric procedure based on mixed effects modeling, local polynomial smoothing and multivariate control chart for random curves, is effective in detecting changes in both the mean and the correlation structure of a complex profile. The nonparametric regression technique denotes a great flexibility in modeling the response, since no specific functional form has to be defined. Moreover, it takes naturally into account the correlation structure of the data. The application to VDP demonstrates that the approach can be implemented conveniently in industrial applications, being able to accommodate within-profile correlation. Acknowledgements This work was partially developed within the research Strutture Ibride per Meccanica ed project STIMA Aerospazio, co-funded by Regione Lombardia. The authors acknowledge the support of Instituto Tecnológico de Costa Rica and the costarican MICIT and CONICIT institutions.
[14] [15] [16]
[17]
[18] [19] [20]
[21] [22] [23] [24]
[25]
References [1] Woodall, W.H., 2007. Current research on profile monitoring. Produção 17, p. 420. [2] Zou Ch., Tsung, F., Wang, Z., 2008. Monitoring profiles based on nonparametric regression methods. Technometrics 50, p. 512. [3] Qiu, P., Zou, Ch., Wang, Z., 2010. Nonparametric profile monitoring by mixed effects modeling. Technometrics 52, p. 265. [4] Zou,C., Tsung,F., Wang,Z.,2007.Monitoring general linear profiles using multivariate EWMAschemes.Technometrics49, p.395. [5] Laird, N.M., Ware, J.H., 1982. Random Effects Model for Longitudinal Data. Biometrics 38, p. 936. [6] Wu, H., Zhang, J.-T., 2002. Local polynomial mixed effect models for longitudinal data. American Statistical Association, Theory and Methods 97, p. 883. [7] Abdel-Salam, A.-S., Birch, J.B., 2010. Profile monitoring analysis with fixed and random effects using nonparametric and
[26] [27]
503
semiparametric methods. Technical Report No. 10-2 2010, Department of Statistics, Virginia Tech, 2010. Mosesova, S.A., Chipman, H.A., Mackay R.J., 2006. Profile Monitoring Using Mixed-Effects Models. Technical report RR06-06, University of Waterloo, Ontario, Canada, 2006. Jensen, W.A., Birch, J.B., 2009. Profile Monitoring via Nonlinear Mixed Models. Journal of Quality Technology 41, p. 8 Jenses, W.A., Birch, J.B., Woodall, W.H., 2008. Monitoring correlation within linear profiles using Mixed Models. Journal of Quality Technology 40, p.167. Horng,J.-J.,Huang,H.-L.,Lin,S.-H., Tsai, M.-Y., 2009. Monitoring nonlinear profiles with random effects by nonparametric regression.Communications in Statistics 38, p.1664. Winistorfer, P.M., Young, T.M., Walker, E., 1996. Modelling and Comparing Vertical Density Profiles. Wood and Fiber Science 28, p.133. Xu, W., Winistorfer, P.M., 1996. Fitting an equation to vertical density profile using Fourier analysis. Holz als Roh-und Werkstoff 54, p.57. Walker, E., Wright, S., 2002. Comparing curves using additive models. Journal of Quality Technology 34, p.118. Young, T.M., Winistorfer, P.M., Wang, S., 1999. Multivariate Control Charts of MDF and OSB Vertical Density Profile Attributes. Forest Products Journal 49, p. 79. Williams, J.D, Woodall, W.H., Birch, J.B., 2007. Statistical monitoring of nonlinear product and process quality profiles. Quality and Reliability Engineering International 23, p.925. Chang, S.I., Yadama, S., 2010. Statistical process control for monitoring non-linear profiles using wavelets filtering and Bspline approximation. International Journal of Production Research 48, p.1049. Fan, S.-K., Yao, N.-Ch., Chang, Y.-J., Jen, Ch.-H, 2011. Statistical monitoring of nonlinear profiles by using picewise linear approximation. Jounal of Process Control 21, p.1217. Wei, Y., Zhao, Z., Lin, D., 2012. Profile control chart based on nonparametric L-1 regression methods. The Annals Applied Statistics, 6, 1, p. 409. Horng, J-J., Huang, H-L., Lin, S-H., Tsai, M-Y., 2009. Monitoring nonlinear profiles with random effects by nonparametric regression. Communications in Statistics, p.1664. Thoemen, H., Irle, M., Sernek, M., 2010. Wood-Based Panels An introduction for specialists .Brunel University Press, London. Maloney, T.M., 1996. The family of wood composite materials. Forest Products Journal 46, p.19. Maloney, T.M., 1993. Modern particleboard and dry-process fiberboard manufacturing. Miller Freeman, San Francisco. Kelly, M.W, 1997. Critical review of relationships between processing parameters and physical properties of particleboard. Gen. Tech. Rep. FPL.10, Forest Service, Forest Products Laboratory, Department of Agriculture, Madison, WI, USA, 1977. Doriguzzi, S., 2008. Monitoraggio statistico del profilo verticale di densità nella produzione di pannelli truciolati. Tesi Laurea. Anno accademico 2007-2008. Fan, J., Gijbels I., 1996. Local Polynomial Modeling and Its Applications . Chapman & Hall, London. Colosimo, B.M., Doriguzzi, S., Meneses, M., Semeraro, Q., 2011. Functional Data Modeling and Monitoring applied to Particleboard Manufacturing, Proceedings of the ASMDA, Rome, Italy.
