Phase I Control Charts (with Emphasis on Distribution-Free Methods)

Description

Statistical methods for retrospectively detecting changes in location and/or dispersion of univariate and multivariate variables. Data values are assumed to be independent, can be individual (one observation at each instant of time) or subgrouped (more than one observation at each instant of time). Control limits are computed, often using a permutation approach, so that a prescribed false alarm probability is guaranteed without making any parametric assumptions on the stable (in-control) distribution.

Details

The main functions are:

• shewhart and mshewhart: univariate and multivariate Shewhart-type control charts based either on the original observations or on a rank transformation. These functions are particularly useful for detecting isolated shifts in the mean and/or variance of subgrouped observations. Functions shewhart and mshewhart also allow the simultaneously use of two control charts originally designed to detect separately location and scale shifts. In particular, note that when more than one critical values are needed, the false alarm probability is “balanced” between the separate control charts as discussed by Capizzi (2015).

• changepoint and mchangepoint: univariate or multivariate control charts useful for detecting sustained (and other patterned) mean and/or variance shifts. The control statistic is based on a generalized likelihood ratio test computed under a Gaussian assumption. However, the control limits are computed by permutation. An optional preliminary rank transformation can be used to improve the performance in the case of nonnormal data.

• rsp and mphase1: the univariate and multivariate methods introduced by Capizzi and Masarotto (2013) and (2017) to detect multiple isolated or step shifts in individual or subgrouped data.

The use of distribution-free control limits is emphasized. However, the package also includes some functions for computing normal-based control limits. As noted in the individual help pages, these limits can also be suitable for some non-normal distributions (e.g., applying a multivariate rank.-transformation, normal-based control limits mantain the desired false alarm probability in the class of the multivariate elliptical distributions). Nevertheless, their use is not generally recommended.

The data should be organized as follows:

• Univariate control charts: an nxm matrix, where n and m are the size of each subgroup and the number of subgroups, respectively. A vector of length m is accepted in the case of individual data, i.e., when n=1.

• Multivariate control charts: a pxnxm array, where p denotes the number of monitored variables. A p x m matrix is accepted in the case of individual data.

Functions phase1Plot and mphase1Plot can be used for plotting the data.

Author(s)

Giovanna Capizzi and Guido Masarotto (maintainer: Giovanna Capizzi <giovanna.capizzi@unipd.it>).

References

G. Capizzi (2015) “Recent advances in process monitoring: Nonparametric and variable-selection methods for Phase I and Phase II (with discussion)”. Quality Engineering, 27, pp. 44–80, doi:10.1080/08982112.2015.968046.

G. Capizzi and G. Masarotto (2013), “Phase I Distribution-Free Analysis of Univariate Data”. Journal of Quality Technology, 45, pp. 273–284, doi:10.1080/00224065.2013.11917938.

G. Capizzi and G. Masarotto (2017), Phase I Distribution-Free Analysis of Multivariate Data, Technometrics, 59, pp. 484–495, doi:10.1080/00401706.2016.1272494.

G. Capizzi and G. Masarotto (2018), “Phase I Distribution-Free Analysis with the R Package dfphase1”. Frontiers in Statistical Quality Control 12, eds. S. Knoth and W. Schmid, pp. 3–19, Springer, doi:10.1007/978-3-319-75295-2_1

See Also

shewhart, shewhart.normal.limits, mshewhart, mshewhart.normal.limits, changepoint, changepoint.normal.limits, mchangepoint, mchangepoint.normal.limits, rsp, mphase1, phase1Plot, mphase1Plot.

A simulated dataset

Description

This simulated data set consists in 50 subgroups, each with 5 observations, on 4 variables.

There is an isolated location shift involving only the first variable at time t=10 and a step shift, involving the third and fourth variables, starting from t=31. The in-control distribution is Student's t with 3 degrees of freedom, zero mean and such that cov(X_i,X_j)=0.8^{|i-j|}.

See the example for the exact code used to simulate the data.

Usage

data(Student)

Format

A 4x5x50 array.

Examples

data(Student)
mphase1(Student)
#
# Replication of the simulation
#
# Generation of the in-control observations
set.seed(1)
m <- 50
n <- 5
p <- 4
df <- 3
Sigma <- outer(1:p,1:p,function(i,j) 0.8^abs(i-j))
Sigma
xnorm <- crossprod(chol(Sigma),matrix(rnorm(p*n*m),p))
xchisq <- sqrt(rchisq(n*m,df)/(df-2))
x <- array(sweep(xnorm,2,xchisq,"/"),c(p,n,m))
# Then, we add an isolated shift at time 10
# (only for the first variable)
x[1,,10] <- x[1,,10]+1
# and, a step shift starting at time 31
# (only for the third and fourth variable)
x[3:4,,31:50] <- x[3:4,,31:50] + c(0.50,-0.25)
dimnames(x)<-list(paste("X",1:4,sep=""),NULL,NULL)
identical(x,Student)

Detection of a sustained change-point in univariate and multivariate data

Description

changepoint (univariate data) and mchangepoint (multivariate data) test for the presence of a sustained location and/or dispersion shift. Both functions can be applied to individual and subgrouped observations.

changepoint.normal.limits and mchangepoint.normal.limits precompute the corresponding control limits when the in-control distribution is normal.

Usage

changepoint(x, subset, score = c("Identity", "Ranks"), only.mean = FALSE,
  plot = TRUE, FAP = 0.05, seed = 11642257, L = 10000, limits = NA)

mchangepoint(x, subset, score = c("Identity", "Signed Ranks", "Spatial Signs",
  "Spatial Ranks", "Marginal Ranks"), only.mean = FALSE,
  plot = TRUE, FAP = 0.05, seed = 11642257, L = 10000, limits = NA) 

changepoint.normal.limits(n, m, score = c("Identity", "Ranks"),
  only.mean = FALSE, FAP = 0.05, seed = 11642257, L = 100000)

mchangepoint.normal.limits(p, n, m, score = c("Identity", "Signed Ranks", "Spatial Signs",
  "Spatial Ranks", "Marginal Ranks"), only.mean = FALSE,
  FAP = 0.05, seed = 11642257, L = 100000)

Arguments

Details

After an optional rank transformation (argument score), changepoint and mchangepoint compute, for \tau=2,\ldots,m, the normal likelihood ratio test statistics for verifying whether the mean and dispersion (or only the mean when only.mean=TRUE) are the same before and after \tau. See Sullivan and Woodall (1999, 2000) and Qiu (2013), Chapter 6 and Section 7.5.

Note that the control statistic is equivalent to that proposed by Lung-Yut-Fong et al. (2011) when score="Marginal Ranks" and only.mean=TRUE.

As suggested by Sullivan and Woodall (1999, 2000), control limits proportional to the in-control mean of the likelihood ratio test statistics are used. Further, when plot=TRUE, the control statistics divided by the time-varying control limits are plotted with a “pseudo-limit” equal to one.

When only.mean=FALSE, the decomposition of the likelihood ratio test statistic suggested by Sullivan and Woodall (1999, 2000) for diagnostic purposes is also computed, and optionally plotted.

Value

changepoint and mchangepoint return an invisible list with elements

changepoint.normal.limits and mchangepoint.normal.limits return a numeric vector containing the control limits.

Note

1. When limits is NA, changepoint and mchangepoint compute the control limits by permutation. The resulting control charts are distribution-free.

2. Pre-computed limits, like those computed using changepoint.normal.limits and mchangepoint.normal.limits, are recommended only for univariate data when score=Ranks. Indeed, in all the other cases, the resulting control chart will not be distribution-free.

3. However, note that, when score is Signed Ranks, Spatial Signs, Spatial Ranks the normal-based control limits are distribution-free in the class of all multivariate elliptical distributions.

Author(s)

Giovanna Capizzi and Guido Masarotto.

References

A. Lung-Yut-Fong, C. Lévy-Leduc, O. Cappé O (2011) “Homogeneity and change-point detection tests for multivariate data using rank statistics”. arXiv:11071971, https://arxiv.org/abs/1107.1971.

P. Qiu (2013) Introduction to Statistical Process Control. Chapman & Hall/CRC Press.

J. H. Sullivan, W. H. Woodall (1996) “A control chart for preliminary analysis of individual observations”. Journal of Quality Technology, 28, pp. 265–278, doi:10.1080/00224065.1996.11979677.

J. H. Sullivan, W. H. Woodall (2000) “Change-point detection of mean vector or covariance matrix shifts using multivariate individual observations”. IIE Transactions, 32, pp. 537–549 doi:10.1080/07408170008963929.

Examples

data(gravel)
changepoint(gravel[1,,])
mchangepoint(gravel)
mchangepoint(gravel,score="Signed Ranks")

Colonscopy Times

Description

This data set contains the colonscopy times (minutes) for 30 subgroups of 5 patients given in Allison Jones-Farmer et al. (2009).

Usage

data(colonscopy)

Format

A 5x30 matrix.

References

L. A. Jones-Farmer, V. Jordan, C. W. Champs (2009) “Distribution-free Phase I control charts for subgroup location”, Journal of Quality Technology, 41, pp. 304–316, doi:10.1080/00224065.2009.11917784.

Examples

data(colonscopy)
phase1Plot(colonscopy)

Ferric Oxide data

Description

This data set contains 189 ferric-oxide individual measurement collected in an aluminum smelter.

Usage

data(fe)

Format

A vector of length 189.

References

M. D. Holland, D. M. Hawkins (2014) “A Control Chart Based on a Nonparametric Multivariate Change-Point Model”, Journal of Quality Technology, 46, pp 63–77, doi:10.1080/00224065.2014.11917954.

Examples

data(fe)
phase1Plot(fe)

Gravel data

Description

This data set contains 56 individual bivariate observations from a gravel-producing plant given by Holmes and Mergen (1993). There are two variables measuring the percentage of the particles (by weight) that are large or medium in size, respectively.

Usage

data(gravel)

Format

A 2x56 matrix.

References

D. S. Holmes, A. Mergen (1993) “Improving the Performance of the T^2 Control Chart”, Quality Engineering, 5, pp. 619–625, doi:10.1080/08982119308919004.

Examples

data(gravel)
mphase1Plot(gravel)

Distribution-free Phase I analysis of multivariate data

Description

Retrospective change point detection using the method described by Capizzi and Masarotto (2017).

Usage

mphase1(x, plot = TRUE, post.signal = TRUE, isolated = dim(x)[2] > 1, step = TRUE,
        alpha = 0.05, gamma = 0.5, K = min(50, round(sqrt(dim(x)[3]))),
        lmin = 5, L = 1000, seed = 11642257)

Arguments

Value

Functions mphase1 returns an object of class mphase1 containing

Author(s)

Giovanna Capizzi and Guido Masarotto.

References

G. Capizzi and G. Masarotto (2017), Phase I Distribution-Free Analysis of Multivariate Data, Technometrics, 59, pp. 484–495, doi:10.1080/00401706.2016.1272494.

See Also

postsignal.

Examples

  # A simulated in-control data from a Student's t distribution
  # with 3 degrees of freedom
  set.seed(123)
  x <- sweep(array(rnorm(5*5*50),c(5,5,50)),c(2,3),sqrt(rchisq(5*50,3)/3),"/")
  mphase1(x)
  # Reproduction of the two examples given in Capizzi and Masarotto (2016)
  data(ryan)
  mphase1(ryan)
  data(gravel)
  mphase1(gravel)

Methods for objects of class mphase1

Description

Methods print and plot allow to write to the console and plot (optionally changing the layout) the result of the Phase I analysis performed with function mphase1.

Method postsignal implements the post-signal Phase I analysis based on the adaptive LASSO described in Capizzi and Masarotto (2016). It uses the p-value and the results on the forward search contained in its first argument. Hence, it is useful for re-running the analysis with different values of alpha and/or gamma.

Usage

## S3 method for class 'mphase1'
print(x,...)
## S3 method for class 'mphase1'
plot(x,layout=c(1,p),...)
## S3 method for class 'mphase1'
postsignal(x, plot = TRUE, alpha = 0.05, gamma = 0.5,...)

Arguments

Value

An object of class mphase1. See mphase1 for the description.

Author(s)

Giovanna Capizzi and Guido Masarotto.

References

G. Capizzi and G. Masarotto (2017), Phase I Distribution-Free Analysis of Multivariate Data, Technometrics, 59, pp. 484–495, doi:10.1080/00401706.2016.1272494.

Examples

  data(gravel)
  u <- mphase1(gravel,plot=FALSE)
  print(u)
  plot(u,layout=c(2,1))
  postsignal(u,plot=FALSE,gamma=1)

Multivariate Shewhart-type control charts

Description

mshewhart computes, and, optionally, plots, several Shewhart-type Phase I control charts for detecting location and scale changes in multivariate subgrouped data.

mshewhart.normal.limits pre-computes the corresponding control limits when the in-control distribution is multivariate normal.

Usage

mshewhart(x, subset, stat = c("T2Var", "T2", "Var", "Depth Ranks"), score = c("Identity",
  "Signed Ranks",  "Spatial Signs", "Spatial Ranks", "Marginal Ranks"),
  loc.scatter = c("Classic", "Robust"), plot = TRUE, FAP = 0.05,
  seed = 11642257, L = 1000, limits = NA)

mshewhart.normal.limits(p, n, m, stat = c("T2Var", "T2", "Var", "Depth Ranks"),
  score = c("Identity", "Signed Ranks",  "Spatial Signs", "Spatial Ranks",
  "Marginal Ranks"), loc.scatter = c("Classic", "Robust"),
  FAP = 0.05, seed = 11642257, L = 100000) 

Arguments

Details

The implemented control statistics are

• T2Var: combination of the T2 and Var statistics described below.

• T2: Hotelling's T^2 control statistics (see Montgomery, 2009, equation 11.19, or Qiu, 2013, equation 7.7) with control limit equal to A.

• Var: normal likelihood ratio control statistics for detecting changes in the multivariate dispersion (see Montgomery, 2009, equation 11.34), with control limit equal to B.

• Depth Ranks: control statistics based on the rank of the Mahalanobis depths, proposed by Bell et. al.. As suggested Bell et al., the Mahalanobis depths are computed using the BACON estimates of the multivariate mean vector and the mean of the subgroups sample covariance matrices. An alarm is signalled if any of the statistics is greater than a positive control limit C.

The T2 and Var control statistics are computed

• score=Identical: from the original data standardized using either the classical pooled estimates of the mean vector and dispersion matrix (Montgomery, 2009, equations 11.14–11.18; Qiu, 2013, equations at page 269) or the highly robust minimum covariance determinant (MCD) estimate when argument loc.scatter is equal to Classic or Robust, respectively.

• score=Signed Ranks, Spatial Signs, Spatial Ranks, Marginal Ranks: from a “rank” transformation of the original data. In particular, see Hallin and Paindaveine (2005) for the definition of the multivariate signed ranks and Oja (2010) for those of the spatial signs, spatial ranks, and marginal ranks. Multivariate signed ranks, spatial signs and ranks are “inner” standardized while marginal ranks are “outer” standardized (see Oja (2010) for the definition of “inner” and “outer” standardization). When loc.scatter is equal to Classic, inner standardization takes into account the subgroup structure of the data imposing that the average of the within-group covariances of the transformed data is proportional to the identity matrix. Otherwise, i.e., when loc.scatter is equal to Robust, it is based on a standard Hettmansperger-Randles-like scatter estimate. Note that the T^2 control statistics based on the spatial signs corresponds to the control charts suggested by Cheng and Shiau (2015) when loc.scatter is equal to Robust.

Value

mshewhart returns an invisible list with elements:

mshewhart.normal.limits returns a numeric vector containing the control limits.

Note

1. When limits is NA, mshewhart computes the control limits by permutation. Then, the resulting control chart is distribution-free.

2. Pre-computed limits, such as those computed by using mshewhart.normal.limits, are not recommended. Indeed, the resulting control chart will not be distribution-free.

3. However, when score is Signed Ranks, Spatial Signs, Spatial Ranks or stat is Depth Ranks, the computed control limits are distribution-free in the class of all multivariate elliptical distributions.

Author(s)

Giovanna Capizzi and Guido Masarotto.

References

R. C. Bell, L. A. Jones-Farmer, N. Billor (2014) “A distribution-free multivariate Phase I location control chart for subgrouped data from elliptical distributions”. Technometrics, 56, pp. 528–538, doi:10.1080/00401706.2013.879264.

C. R. Cheng, J. J. H. Shiau JJH (2015) “A distribution-free multivariate control chart for Phase I applications”. Quality and Reliability Engineering International, 31, pp. 97–111, doi:10.1002/qre.1751.

M. Hallin and D. Paindaveine (2005) “Affine-Invariant Aligned Rank Tests for the Multivariate General Linear Model with VARMA Errors”. Journal of Multivariate Analysis, 93, pp. 122–163, doi:10.1016/j.jmva.2004.01.005.

D. C. Montgomery (2009) Introduction to Statistical Quality Control, 6th edn. Wiley.

H. Oja (2010) Multivariate Nonparametric Methods with R. An Approach Based on Spatial Signs and Ranks. Springer.

P. Qiu (2013) Introduction to Statistical Process Control. Chapman & Hall/CRC Press.

Examples

data(ryan)
mshewhart(ryan)
mshewhart(ryan,subset=-10)
mshewhart(ryan,subset=-c(10,20))
mshewhart(ryan,score="Signed Ranks")
mshewhart(ryan,subset=-10,score="Signed Ranks")
mshewhart(ryan,subset=-c(10,20),score="Signed Ranks")

Plot of Phase 1 data

Description

phase1Plot and mphase1Plot plot univariate or multivariate Phase 1 observations, organized as required by the dfphase1 package.

Usage

phase1Plot(x)

mphase1Plot(x, layout = c(1, p))

Arguments

Author(s)

Giovanna Capizzi and Guido Masarotto.

Examples

  x <- matrix(rt(5*20,5),5)
  x[,10] <- x[,10]+3
  phase1Plot(x)
  # a data set with many variables
  x <- array(rnorm(20*5*50),c(20,5,50))+10*(1:20)
  mphase1Plot(x)
  # it is better to organize the plot on two pages
  if (interactive()) old <- grDevices::devAskNewPage(TRUE)
  mphase1Plot(x,c(2,5,2))
  if (interactive()) grDevices::devAskNewPage(old)

Distribution-Free Phase I Analysis of Univariate Data based on Recursive Segmentation and Permutation

Description

rsp implements the Phase I method described in Capizzi and Masarotto (2013).

Usage

rsp(y, plot = TRUE, L = 1000, seed = 11642257, alpha = 0.05,
    maxsteps = min(50, round(NROW(y)/15)), lmin = max(5, min(10, round(NROW(y)/10))))

Arguments

Value

A list with elements

Author(s)

Giovanna Capizzi and Guido Masarotto.

References

G. Capizzi, G. Masarotto (2013), “Phase I Distribution-Free Analysis of Univariate Data”. Journal of Quality Technology, 45, pp. 273-284, doi:10.1080/00224065.2013.11917938.

Examples

# Individual observations with a transient level change
set.seed(112233)
level <- c(rep(0,20),rep(3,10),rep(0,20))
x <- level+rt(50,4)
rsp(x)
# Individual observations with a scale step change
scale <- c(rep(1,25),rep(3,25))
x <- scale*rt(50,4)
rsp(x)
data(fe)
rsp(fe)
data(colonscopy)
rsp(colonscopy)

Ryan data

Description

This data set contains the data given in Table 9.2 by Ryan (2011, p. 323). The sample comprises 20 subgroups, each with 4 observations, on two quality characteristics X_1 and X_2. According to Ryan (2011), the 10th and 20th subgroups are out-of-control.

Usage

data(ryan)

Format

A 2x4x20 array.

References

T. P. Ryan (2011), Statistical Methods for Quality Improvement, 3rd ed., Wiley.

Examples

data(ryan)
mphase1Plot(ryan)

Univariate Shewhart-type control charts

Description

shewhart computes, and, optionally, plots, Shewhart-type Phase I control charts for detecting changes in location and scale of univariate subgrouped data.

shewhart.normal.limits pre-computes the corresponding control limits when the in-control distribution is normal.

Usage

shewhart(x, subset, 
         stat = c("XbarS", "Xbar", "S", 
                  "Rank", "lRank", "sRank",
                  "Lepage", "Cucconi"),
         aggregation = c("mean", "median"), 
         plot = TRUE, 
         FAP = 0.05,
         seed = 11642257, 
         L = 1000, 
         limits = NA)

shewhart.normal.limits(n, m, 
                       stat = c("XbarS", "Xbar", "S", 
                                "Rank", "lRank", "sRank", 
                                "Lepage", "Cucconi"),
                       aggregation = c("mean", "median"), 
                       FAP = 0.05,
                       seed = 11642257, 
                       L = 100000)

Arguments

Details

The implemented control charts are:

• XbarS: combination of the Xbar and S control charts described in the following.

• Xbar: chart based on plotting the subgroup means with control limits

  \hat{\mu}\pm A\frac{\hat{\sigma}}{\sqrt{n}}

  where \hat{\mu} (\hat{\sigma}) denotes the estimate of the in-control mean (standard deviation) computed as the mean or median of the subgroup means (standard deviations).

• S: chart based on plotting the (unbiased) subgroup standard deviations with lower control limit B_1\hat{\sigma} and upper control limit B_2\hat{\sigma}.

• Rank: combination of the lRank and sRank control charts described in the following.

• lRank: control chart based on the standardized rank-sum control statistic suggested by Jones-Farmer et al. (2009) for detecting changes in the location parameter. Control limits are of the type \pm C.

• sRank: chart based on the standardized rank-sum control statistic suggested by Jones-Farmer and Champ (2010) for detecting changes in the scale parameter. Control limits are of the type \pm D.

• Lepage: chart based on the Lepage control statistic suggested by Li et al. (2019) for detecting changes in location and/or scale. There is only a upper control limit equal to E.

• Cucconi: chart based on the Cucconi control statistic suggested by Li et al. (2020) for detecting changes in location and/or scale. There is only a upper control limit equal to E.

Value

shewhart returns an invisible list with elements

shewhart.normal.limits returns a numeric vector containing the limits.

Note

1. If argument limits is NA, shewhart computes the control limits by permutation. The resulting control chart are distribution-free.

2. Pre-computed limits, such as those computed using shewhart.normal.limits, are not recommended when stat is XbarS, Xbar or S. Indeed, the resulting control chart will not be distribution-free.

3. When stat is Rank, lRank, sRank, Lepage or Cucconi the control limits computed by shewhart.normal.limits are distribution-free in the class of all univariate continuous distributions. So, if user plan to apply rank-based control charts on a repeated number of samples of the same size, pre-computing the control limits using mshewhart.normal.limits can reduce the overall computing time.

Author(s)

Giovanna Capizzi and Guido Masarotto.

References

L. A. Jones-Farmer, V. Jordan, C. W. Champs (2009) “Distribution-free Phase I control charts for subgroup location”, Journal of Quality Technology, 41, pp. 304–316, doi:10.1080/00224065.2009.11917784.

L. A. Jones-Farmer, C. W. Champ (2010) “A distribution-free Phase I control chart for subgroup scale”. Journal of Quality Technology, 42, pp. 373–387, doi:10.1080/00224065.2010.11917834

C. Li, A. Mukherjee, Q. Su (2019) “A distribution-free Phase I monitoring scheme for subgroup location and scale based on the multi-sample Lepage statistic”, Computers & Industrial Engineering, 129, pp. 259–273, doi:10.1016/j.cie.2019.01.013

C. Li, A. Mukherjee, M. Marozzi (2020) “A new distribution-free Phase-I procedure for bi-aspect monitoring based on the multi-sample Cucconi statistic”, Computers & Industrial Engineering, 149, doi:10.1016/j.cie.2020.106760

D. C. Montgomery (2009) Introduction to Statistical Quality Control, 6th edn. Wiley.

P. Qiu (2013) Introduction to Statistical Process Control. Chapman & Hall/CRC Press.

Examples

# A simulated example
set.seed(12345)
y <- matrix(rt(100,3),5)
y[,20] <- y[,20]+3
shewhart(y)
shewhart(y, stat="Rank")
shewhart(y, stat="Lepage")
shewhart(y, stat="Cucconi")
# Reproduction of the control chart shown
# by Jones-Farmer et. al. (2009)
data(colonscopy)
u <- shewhart.normal.limits(NROW(colonscopy),NCOL(colonscopy), 
                            stat="lRank", FAP=0.1, L=10000)
# In Jones-Farmer et al. (2009) is estimated as 2.748
u
shewhart(colonscopy,stat="lRank",limits=u)
# Examples of control limits for comparisons
# with Li et al. (2019) and (2020) but
# using a limited number of Monte Carlo
# replications
# Lepage: in Li et al. (2019) is estimated as 11.539
shewhart.normal.limits(5, 25, stat="Lepage", L=10000)
# Cucconi: in Li et al. (2020) is estimated as 0.266
shewhart.normal.limits(5, 25, stat="Cucconi", L=10000)