Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
Dear Mark,
Thank you very much for your advice.
I will try it.
I really appreciate your all kind advice.
Thanks a lot again.
Best regards,
Kohkichi
(11/08/19 22:28), Mark Difford wrote:
On Aug 19, 2011 khosoda wrote:
I used x10.homals4$objscores[, 1] as a predictor for logistic regression
as in the same way as PC1 in PCA.
Am I going the right way?
Hi Kohkichi,
Yes, but maybe explore the sets= argument (set Response as the target
variable and the others as the predictor variables). Then use Dim1 scores.
Also think about fitting a rank-1 restricted model, combined with the sets=
option.
See the vignette to the package and look at
@ARTICLE{MIC98,
author = {Michailides, G. and de Leeuw, J.},
title = {The {G}ifi system of descriptive multivariate analysis},
journal = {Statistical Science},
year = {1998},
volume = {13},
pages = {307--336},
abstract = {}
}
Regards, Mark.
-
Mark Difford (Ph.D.)
Research Associate
Botany Department
Nelson Mandela Metropolitan University
Port Elizabeth, South Africa
--
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Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
Dear Mark, Thank you very much for your kind advice. Actually, I already performed penalized logistic regression by pentrace and lrm in package rms. The reason why I wanted to reduce dimensionality of those 9 variables was that these variables were not so important according to the subject matter knowledge and that I wanted to avoid events per variable problem. Your answer about dudi.mix$l1 helped me a lot. I finally was able to perform penalized logistic regression for data consisting of 4 important variables and x18.dudi.mix$l1[, 1]. Thanks a lot again. One more question, I investigated homals package too. I found it has ndim option. mydata is followings; head(x10homals.df) age sex symptom HT DM IHD smoking hyperlipidemia Statin Response 1 62 M asymptomatic positive negative negative positive positive positive negative 2 82 M symptomatic positive negative negative negative positive positive negative 3 64 M asymptomatic negative positive negative negative positive positive negative 4 55 M symptomatic positive positive positive negative positive positive negative 5 67 M symptomatic positive negative negative negative negative positive negative 6 79 M asymptomatic positive positive negative negative positive positive negative age is continuous variable, and Response should not be active for computation, so, ... x10.homals4 - homals(x10homals.df, active = c(rep(TRUE, 9), FALSE), level=c(numerical, rep(nominal, 9)), ndim=4) I did it with ndim from 2 to 9, compared Classification rate of Response by predict(x10.homals). p.x10.homals4 Classification rate: Variable Cl. Rate %Cl. Rate 1 age 0.4712 47.12 2 sex 0.9808 98.08 3 symptom 0.8269 82.69 4 HT 0.9135 91.35 5 DM 0.8558 85.58 6 IHD 0.8750 87.50 7 smoking 0.9423 94.23 8 hyperlipidemia 0.9519 95.19 9 Statin 0.8942 89.42 10 Response 0.6154 61.54 This is the best for classification of Response, so, I selected ndim=4. Then, I found objscores. head(x10.homals4$objscores) D1 D2 D3 D4 1 -0.002395321 -0.034032230 -0.008140378 0.02369123 2 0.036788626 -0.010308707 0.005725984 -0.02751958 3 0.014363031 0.049594466 -0.025627467 0.06254055 4 0.083092285 0.065147519 0.045903394 -0.03751551 5 -0.013692504 0.005106661 -0.007656776 -0.04107009 6 0.002320747 0.024375393 -0.017785415 -0.01752556 I used x10.homals4$objscores[, 1] as a predictor for logistic regression as in the same way as PC1 in PCA. Am I going the right way? Thanks a lot for your help in advance. Best regards -- Kohkichi Hosoda (11/08/19 4:21), Mark Difford wrote: On Aug 18, 2011 khosoda wrote: I'm trying to do model reduction for logistic regression. Hi Kohkichi, My general advice to you would be to do this by fitting a penalized logistic model (see lrm in package rms and glmnet in package glmnet; there are several others). Other points are that the amount of variance explained by mixed PCA and MCA are not comparable. Furthermore, homals() is a much better choice than MCA because it handles different types of variables whereas MCA is for categorical variables. On the more specific question of whether you should use dudi.mix$l1 or dudi.mix$li, it doesn't matter: the former is a scaled version of the latter. Same for dudi.acm. To see this do the following: ## plot(x18.dudi.mix$li[, 1], x18.dudi.mix$l1[, 1]) Regards, Mark. - Mark Difford (Ph.D.) Research Associate Botany Department Nelson Mandela Metropolitan University Port Elizabeth, South Africa -- View this message in context: http://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-tp3750251p3753437.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. -- * 神戸大学大学院医学研究科 脳神経外科学分野 細田 弘吉 〒650-0017 神戸市中央区楠町7丁目5-1 Phone: 078-382-5966 Fax : 078-382-5979 E-mail address Office: khos...@med.kobe-u.ac.jp Home : khos...@venus.dti.ne.jp __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
On Aug 19, 2011 khosoda wrote:
I used x10.homals4$objscores[, 1] as a predictor for logistic regression
as in the same way as PC1 in PCA.
Am I going the right way?
Hi Kohkichi,
Yes, but maybe explore the sets= argument (set Response as the target
variable and the others as the predictor variables). Then use Dim1 scores.
Also think about fitting a rank-1 restricted model, combined with the sets=
option.
See the vignette to the package and look at
@ARTICLE{MIC98,
author = {Michailides, G. and de Leeuw, J.},
title = {The {G}ifi system of descriptive multivariate analysis},
journal = {Statistical Science},
year = {1998},
volume = {13},
pages = {307--336},
abstract = {}
}
Regards, Mark.
-
Mark Difford (Ph.D.)
Research Associate
Botany Department
Nelson Mandela Metropolitan University
Port Elizabeth, South Africa
--
View this message in context:
http://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-tp3750251p3755163.html
Sent from the R help mailing list archive at Nabble.com.
__
R-help@r-project.org mailing list
https://stat.ethz.ch/mailman/listinfo/r-help
PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
and provide commented, minimal, self-contained, reproducible code.
Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
Pooling nominal with numeric variables and running pca on them sounds like conceptual nonsense to me. You use PCA to reduce the dimensionality of the data if the data are numeric. For categorical data analysis, you should use latent class analysis or something along those lines. The fact that your first PC captures only 20 percent of the variance indicates that either you apply the wrong technique or that dimensionality reduction is of little use for these data more generally. The first step should generally be to check the correlations/associations between the variables to inspect whether what you intend to do makes sense. HTH, Daniel khosoda wrote: Hi all, I'm trying to do model reduction for logistic regression. I have 13 predictor (4 continuous variables and 9 binary variables). Using subject matter knowledge, I selected 4 important variables. Regarding the rest 9 variables, I tried to perform data reduction by principal component analysis (PCA). However, 8 of 9 variables were binary and only one continuous. I transformed the data by transcan of rms package and did PCA with princomp. PC1 explained only 20% of the variance. Still, I used the PC1 as a predictor of the logistic model and obtained some results. Then, I tried multiple correspondence analysis (MCA). The only one continuous variable was age. I transformed age variable to age_Q factor variable as the followings. quantile(mydata.df$age) 0% 25% 50% 75% 100% 53.00 66.75 72.00 76.25 85.00 age_Q - cut(x17.df$age, right=TRUE, breaks=c(-Inf, 66, 72, 76, Inf), labels=c(53-66, 67-72, 73-76, 77-85)) table(age_Q) age_Q 53-66 67-72 73-76 77-85 26272526 Then, I used mjca of ca pacakge for MCA. mjca1 - mjca(mydata.df[, c(age_Q,sex,symptom, HT, DM, IHD,smoking,DL, Statin)]) summary(mjca1) Principal inertias (eigenvalues): dimvalue % cum% scree plot 1 0.009592 43.4 43.4 * 2 0.003983 18.0 61.4 ** 3 0.001047 4.7 66.1 ** 4 0.000367 1.7 67.8 - Total: 0.022111 The dimension 1 explained 43% of the variance. Then, I was wondering which values I could use like PC1 in PCA. I explored in mjca1 and found rowcoord. mjca1$rowcoord [,1] [,2][,3] [,4] [1,] 0.07403748 0.8963482181 0.10828273 1.581381849 [2,] 0.92433996 -1.1497911361 1.28872517 0.304065865 [3,] 0.49833354 0.6482940556 -2.4314 0.365023261 [4,] 0.18998290 -1.4028117048 -1.70962159 0.451951744 [5,] -0.13008173 0.2557656854 1.16561601 -1.012992485 . . [101,] -1.86940216 0.5918128751 0.87352987 -1.118865117 [102,] -2.19096615 1.2845448725 0.25227354 -0.938612155 [103,] 0.77981265 -1.1931087587 0.23934034 0.627601413 [104,] -2.37058237 -1.4014005013 -0.73578248 -1.455055095 Then, I used mjca1$rowcoord[, 1] as the followings. mydata.df$NewScore - mjca1$rowcoord[, 1] I used this NewScore as one of the predictors for the model instead of original 9 variables. The final logistic model obtained by use of MCA was similar to the one obtained by use of PCA. My questions are; 1. Is it O.K. to perform PCA for data consisting of 1 continuous variable and 8 binary variables? 2. Is it O.K to perform transformation of age from continuous variable to factor variable for MCA? 3. Is mjca1$rowcoord[, 1] the correct values as a predictor of logistic regression model like PC1 of PCA? I would appreciate your help in advance. -- Kohkichi Hosoda __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. -- View this message in context: http://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-tp3750251p3752062.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
Dear Daniel, Thank you for your mail. Your comment is exactly what I was worried about. I konw very little about latent class analysis. So, I would like to use multiple correspondence analysis (MCA) for data redution. Besides, the first plane of the MCA captured 43% of the variance. Do you think my use of mjca1$rowcoord[, 1] in ca package for data reduction in the previous mail is O.K.? Thank you for your help. -- Kohkichi Hosoda (11/08/18 17:39), Daniel Malter wrote: Pooling nominal with numeric variables and running pca on them sounds like conceptual nonsense to me. You use PCA to reduce the dimensionality of the data if the data are numeric. For categorical data analysis, you should use latent class analysis or something along those lines. The fact that your first PC captures only 20 percent of the variance indicates that either you apply the wrong technique or that dimensionality reduction is of little use for these data more generally. The first step should generally be to check the correlations/associations between the variables to inspect whether what you intend to do makes sense. HTH, Daniel khosoda wrote: Hi all, I'm trying to do model reduction for logistic regression. I have 13 predictor (4 continuous variables and 9 binary variables). Using subject matter knowledge, I selected 4 important variables. Regarding the rest 9 variables, I tried to perform data reduction by principal component analysis (PCA). However, 8 of 9 variables were binary and only one continuous. I transformed the data by transcan of rms package and did PCA with princomp. PC1 explained only 20% of the variance. Still, I used the PC1 as a predictor of the logistic model and obtained some results. Then, I tried multiple correspondence analysis (MCA). The only one continuous variable was age. I transformed age variable to age_Q factor variable as the followings. quantile(mydata.df$age) 0% 25% 50% 75% 100% 53.00 66.75 72.00 76.25 85.00 age_Q- cut(x17.df$age, right=TRUE, breaks=c(-Inf, 66, 72, 76, Inf), labels=c(53-66, 67-72, 73-76, 77-85)) table(age_Q) age_Q 53-66 67-72 73-76 77-85 26272526 Then, I used mjca of ca pacakge for MCA. mjca1- mjca(mydata.df[, c(age_Q,sex,symptom, HT, DM, IHD,smoking,DL, Statin)]) summary(mjca1) Principal inertias (eigenvalues): dimvalue % cum% scree plot 1 0.009592 43.4 43.4 * 2 0.003983 18.0 61.4 ** 3 0.001047 4.7 66.1 ** 4 0.000367 1.7 67.8 - Total: 0.022111 The dimension 1 explained 43% of the variance. Then, I was wondering which values I could use like PC1 in PCA. I explored in mjca1 and found rowcoord. mjca1$rowcoord [,1] [,2][,3] [,4] [1,] 0.07403748 0.8963482181 0.10828273 1.581381849 [2,] 0.92433996 -1.1497911361 1.28872517 0.304065865 [3,] 0.49833354 0.6482940556 -2.4314 0.365023261 [4,] 0.18998290 -1.4028117048 -1.70962159 0.451951744 [5,] -0.13008173 0.2557656854 1.16561601 -1.012992485 . . [101,] -1.86940216 0.5918128751 0.87352987 -1.118865117 [102,] -2.19096615 1.2845448725 0.25227354 -0.938612155 [103,] 0.77981265 -1.1931087587 0.23934034 0.627601413 [104,] -2.37058237 -1.4014005013 -0.73578248 -1.455055095 Then, I used mjca1$rowcoord[, 1] as the followings. mydata.df$NewScore- mjca1$rowcoord[, 1] I used this NewScore as one of the predictors for the model instead of original 9 variables. The final logistic model obtained by use of MCA was similar to the one obtained by use of PCA. My questions are; 1. Is it O.K. to perform PCA for data consisting of 1 continuous variable and 8 binary variables? 2. Is it O.K to perform transformation of age from continuous variable to factor variable for MCA? 3. Is mjca1$rowcoord[, 1] the correct values as a predictor of logistic regression model like PC1 of PCA? I would appreciate your help in advance. -- Kohkichi Hosoda __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. -- View this message in context: http://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-tp3750251p3752062.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. __ R-help@r-project.org mailing
Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
On Aug 17, 2011 khosoda wrote: 1. Is it O.K. to perform PCA for data consisting of 1 continuous variable and 8 binary variables? 2. Is it O.K to perform transformation of age from continuous variable to factor variable for MCA? 3. Is mjca1$rowcoord[, 1] the correct values as a predictor of logistic regression model like PC1 of PCA? Hi Kohkichi, If you want to do this, i.e. PCA-type analysis with different variable-types, then look at dudi.mix() in package ade4 and homals() in package homals. Regards, Mark. - Mark Difford (Ph.D.) Research Associate Botany Department Nelson Mandela Metropolitan University Port Elizabeth, South Africa -- View this message in context: http://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-tp3750251p3752168.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
On Aug 18, 2011; Daniel Malter wrote: Pooling nominal with numeric variables and running pca on them sounds like conceptual nonsense to me. Hi Daniel, This is not true. There are methods that are specifically designed to do a PCA-type analysis on mixed categorical and continuous variables, viz dudi.mix and dudi.hillsmith in package ade4. De Leeuw's homals method takes this a step further, doing amongst other things, a non-linear version of PCA using any type of variable. Regards, Mark. - Mark Difford (Ph.D.) Research Associate Botany Department Nelson Mandela Metropolitan University Port Elizabeth, South Africa -- View this message in context: http://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-tp3750251p3752516.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
Re: [R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
Dear Mark, Thank you very much for your mail. This is what I really wanted! I tried dudi.mix in ade4 package. ade4plaque.df - x18.df[c(age, sex, symptom, HT, DM, IHD, smoking, DL, Statin)] head(ade4plaque.df) age sex symptom HT DM IHD smoking hyperlipidemia Statin 1 62 M asymptomatic positive negative negative positive positive positive 2 82 M symptomatic positive negative negative negative positive positive 3 64 M asymptomatic negative positive negative negative positive positive 4 55 M symptomatic positive positive positive negative positive positive 5 67 M symptomatic positive negative negative negative negative positive 6 79 M asymptomatic positive positive negative negative positive positive x18.dudi.mix - dudi.mix(ade4plaque.df) x18.dudi.mix$eig [1] 1.7750557 1.4504641 1.2178640 1.0344946 0.8496640 0.8248379 0.7011151 0.6367328 0.5097718 x18.dudi.mix$eig[1:9]/sum(x18.dudi.mix$eig) [1] 0.19722841 0.16116268 0.13531822 0.11494385 0.09440711 0.09164866 0.07790168 0.07074809 0.05664131 Still first component explained only 19.8% of the variances, right? Then, I investigated values of dudi.mix corresponding to PC1 of PCA. Help file say; l1 principal components, data frame with n rows and nf columns li row coordinates, data frame with n rows and nf columns So, I guess I should use x18.dudi.mix$l1[, 1]. Am I right? Or should I use multiple correpondence analysis because the first plane explained 43% of the variance? Thank you for your help in advance. Kohkichi (11/08/18 18:33), Mark Difford wrote: On Aug 17, 2011 khosoda wrote: 1. Is it O.K. to perform PCA for data consisting of 1 continuous variable and 8 binary variables? 2. Is it O.K to perform transformation of age from continuous variable to factor variable for MCA? 3. Is mjca1$rowcoord[, 1] the correct values as a predictor of logistic regression model like PC1 of PCA? Hi Kohkichi, If you want to do this, i.e. PCA-type analysis with different variable-types, then look at dudi.mix() in package ade4 and homals() in package homals. Regards, Mark. - Mark Difford (Ph.D.) Research Associate Botany Department Nelson Mandela Metropolitan University Port Elizabeth, South Africa -- View this message in context: http://r.789695.n4.nabble.com/How-to-use-PC1-of-PCA-and-dim1-of-MCA-as-a-predictor-in-logistic-regression-model-for-data-reduction-tp3750251p3752168.html Sent from the R help mailing list archive at Nabble.com. __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. -- * 神戸大学大学院医学研究科 脳神経外科学分野 細田 弘吉 〒650-0017 神戸市中央区楠町7丁目5-1 Phone: 078-382-5966 Fax : 078-382-5979 E-mail address Office: khos...@med.kobe-u.ac.jp Home : khos...@venus.dti.ne.jp __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
[R] How to use PC1 of PCA and dim1 of MCA as a predictor in logistic regression model for data reduction
Hi all, I'm trying to do model reduction for logistic regression. I have 13 predictor (4 continuous variables and 9 binary variables). Using subject matter knowledge, I selected 4 important variables. Regarding the rest 9 variables, I tried to perform data reduction by principal component analysis (PCA). However, 8 of 9 variables were binary and only one continuous. I transformed the data by transcan of rms package and did PCA with princomp. PC1 explained only 20% of the variance. Still, I used the PC1 as a predictor of the logistic model and obtained some results. Then, I tried multiple correspondence analysis (MCA). The only one continuous variable was age. I transformed age variable to age_Q factor variable as the followings. quantile(mydata.df$age) 0% 25% 50% 75% 100% 53.00 66.75 72.00 76.25 85.00 age_Q - cut(x17.df$age, right=TRUE, breaks=c(-Inf, 66, 72, 76, Inf), labels=c(53-66, 67-72, 73-76, 77-85)) table(age_Q) age_Q 53-66 67-72 73-76 77-85 26272526 Then, I used mjca of ca pacakge for MCA. mjca1 - mjca(mydata.df[, c(age_Q,sex,symptom, HT, DM, IHD,smoking,DL, Statin)]) summary(mjca1) Principal inertias (eigenvalues): dimvalue % cum% scree plot 1 0.009592 43.4 43.4 * 2 0.003983 18.0 61.4 ** 3 0.001047 4.7 66.1 ** 4 0.000367 1.7 67.8 - Total: 0.022111 The dimension 1 explained 43% of the variance. Then, I was wondering which values I could use like PC1 in PCA. I explored in mjca1 and found rowcoord. mjca1$rowcoord [,1] [,2][,3] [,4] [1,] 0.07403748 0.8963482181 0.10828273 1.581381849 [2,] 0.92433996 -1.1497911361 1.28872517 0.304065865 [3,] 0.49833354 0.6482940556 -2.4314 0.365023261 [4,] 0.18998290 -1.4028117048 -1.70962159 0.451951744 [5,] -0.13008173 0.2557656854 1.16561601 -1.012992485 . . [101,] -1.86940216 0.5918128751 0.87352987 -1.118865117 [102,] -2.19096615 1.2845448725 0.25227354 -0.938612155 [103,] 0.77981265 -1.1931087587 0.23934034 0.627601413 [104,] -2.37058237 -1.4014005013 -0.73578248 -1.455055095 Then, I used mjca1$rowcoord[, 1] as the followings. mydata.df$NewScore - mjca1$rowcoord[, 1] I used this NewScore as one of the predictors for the model instead of original 9 variables. The final logistic model obtained by use of MCA was similar to the one obtained by use of PCA. My questions are; 1. Is it O.K. to perform PCA for data consisting of 1 continuous variable and 8 binary variables? 2. Is it O.K to perform transformation of age from continuous variable to factor variable for MCA? 3. Is mjca1$rowcoord[, 1] the correct values as a predictor of logistic regression model like PC1 of PCA? I would appreciate your help in advance. -- Kohkichi Hosoda __ R-help@r-project.org mailing list https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
