unattended svm parameters fitting for monitoring nonlinear profiles
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Unattended SVMparameters fitting formonitoring nonlinear
profilesEmilio L. Cano (University of Castilla-La Mancha)Javier M. Moguerza (Rey Juan Carlos University)
Mariano Prieto (ENUSA Industrias Avanzadas)
Statistical Process Control
Assignable causes of variation may befound and eliminated
Walter A. Shewhart
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Statistical Process Control (cont.)
Observation
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Statistical Process Control (cont.)
Observation
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Nonlinear profiles
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X−Location
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One function per sample (instead of a data point)
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Nonlinear profiles
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X−Location
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One function per sample (instead of a data point)
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Illustrative example
I Engineered woodboards
I Data set of 50 boards
I Sample of 5 boards per shift
I Quality characteristic: density
I Total measurements: 500
I Every 0.001 in along the board
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Illustrative example (cont.)
Data
library(SixSigma)
str(ss.data.wbx)
## num [1:500] 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 ...
str(ss.data.wby)
## num [1:500, 1:50] 58.4 58 58.2 58.4 57.9 ...
## - attr(*, "dimnames")=List of 2
## ..$ : NULL
## ..$ : chr [1:50] "P1" "P2" "P3" "P4" ...
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Illustrative example (cont.)
plotProfiles(profiles = ss.data.wby,
x = ss.data.wbx)
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Profiles
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Support Vector Machines (SVM)
SVM regression modelGiven response y, input space x:
y = r(x) + δ
r(x) feature space (higher dimension than x)
r(x): non-linear, high dimensional transformation of x input vectorThen, a linear combination over the feature space is the predictionmodel:
f(x, ω) =∑j
ωj ,gj(x)
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Support Vector Machines (SVM) (cont.)
SVM regression parametersRegression estimates are obtained minimizing the ε-intensive lossfunction. This function contains two input parameters: ε and C(regularization parameter)
Details: Vapnik, V. (1998). Statistical learning theory. New York: Wiley;Vapnik, V. (1999). The nature of statistical learning theory (2nd ed).Berlin: Springer.
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Support Vector Machines (SVM) (cont.)
Parameters selection
I C trade off between model complexity and deviations larger thanε in optimization
I ε controls the width of the ε-insensitive zone
I Several practical approaches (cross validation, experts opinion,. . . )
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(unattended) Parameters selection
Regularization parameter C
C = max{|y + 3σy|, |y − 3σy|}
max(c(abs(mean(y) + 3*sd(y)), abs(mean(y) - 3*sd(y))))
ε parameter
ε = 3σ
√log n
n
3*par.sigma*sqrt(log(nrowprofiles)/nrowprofiles)
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(unattended) Parameters selection (cont.)
Input noise level σAn approximation using polynomials
mloess <- loess(y ~ x)
yhat <- predict(mloess, newdata = x)
deltas <- y - yhat
par.sigma <- sd(deltas)
Details: Cherkassky, V and Ma, Y (2004). Practical selection of SVMparameters and noise estimation for SVM regression. NeuralNetworks, 17(1), 113-126
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Regularization of nonlinear profiles via SVM
P1.smooth <- smoothProfiles(
profiles = ss.data.wby[, "P1"],
x = ss.data.wbx)
plotProfiles(profiles = cbind(P1.smooth,
ss.data.wby[, "P1"]),
x = ss.data.wbx)
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Regularization of nonlinear profiles via SVM(cont.)
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Profiles
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Smoothed prototype and confidence bands
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Profiles
Apply control charts philosophy to complex processes
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Smoothed prototype and confidence bands(cont.)
wby.phase1 <- ss.data.wby[, 1:35]
wb.limits <- climProfiles(profiles = wby.phase1[, -28],
x = ss.data.wbx,
smoothprof = TRUE,
smoothlim = TRUE)
wby.phase2 <- ss.data.wby[, 36:50]
wb.out.phase2 <- outProfiles(profiles = wby.phase2,
x = ss.data.wbx,
cLimits = wb.limits,
tol = 0.8)
plotProfiles(wby.phase2,
x = ss.data.wbx,
cLimits = wb.limits,
outControl = wb.out.phase2$idOut,
onlyout = FALSE)
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