3 intro edited Sep 26 '16 at 23:36 amoeba 65.5k1717 gold badges221221 silver badges274274 bronze badges If I understood correctly, you imagine a linear model where one of the predictors is categorical (e.g. college major); and you expect that for some subgroups of its levels (subgroups of categories) the coefficients might be exactly the same. So perhaps the regression coefficients for Maths and Physics are the same, but different from those for Chemistry and Biology. In a simplest case, you would have a "one way ANOVA" linear model with a single categorical predictor: $$y_{ij} = \mu + \alpha_i + \epsilon_{ij},$$ where $$i$$ encodes the level of the categorical variable (the category). But you might prefer a solution that collapses some levels (categories) together, e.g. $$\begin{cases}\alpha_1=\alpha_2, \\ \alpha_3=\alpha_4=\alpha_5.\end{cases}$$ This suggests that one can try to use a regularization penalty that would penalize solutions with differing alphas. One penalty term that immediately comes to mind is $$L=\omega \sum_{i This resembles lasso and should enforce sparsity of the $$\alpha_i-\alpha_j$$ differences, which is exactly what you want: you want many of them to be zero. Regularization parameter $$\omega$$ should be selected with cross-validation. I have never dealt with models like that and the above is the first thing that came to my mind. Then I decided to see if there is something like that implemented. I made some google searches and soon realized that this is called fusion of categories; searching for lasso fusion categorical will give you a lot of references to read. Here are a few that I briefly looked at: Gerhard Tutz, Regression for Categorical Data, see pp. 175-175 in Google Books. Tutz mentions the following four papers: Land and Friedman, 1997, Variable fusion: a new adaptive signal regression method Bondell and Reich, 2009, Simultaneous factor selection and collapsing levels in ANOVA Gertheiss and Tutz, 2010, Sparse modeling of categorial explanatory variables Tibshirani et al. 2005, Sparsity and smoothness via the fused lasso is somewhat relevant even if not exactly the same (it is about ordinal variables) Gertheiss and Tutz 2010, published in the Annals of Applied Statistics, looks like a recent and very readable paper that contains other references. Here is its abstract: Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $$L_1$$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies. I like their Lasso-like solution paths that show how levels of two categorical variables get merged together when regularization strength increases: If I understood correctly, you imagine a linear model where one of the predictors is categorical (e.g. college major); and you expect that for some subgroups of its levels (subgroups of categories) the coefficients might be exactly the same. In a simplest case, you would have a "one way ANOVA" linear model with a single categorical predictor: $$y_{ij} = \mu + \alpha_i + \epsilon_{ij},$$ where $$i$$ encodes the level of the categorical variable (the category). But you might prefer a solution that collapses some levels (categories) together, e.g. $$\begin{cases}\alpha_1=\alpha_2, \\ \alpha_3=\alpha_4=\alpha_5.\end{cases}$$ This suggests that one can try to use a regularization penalty that would penalize solutions with differing alphas. One penalty term that immediately comes to mind is $$L=\omega \sum_{i This resembles lasso and should enforce sparsity of the $$\alpha_i-\alpha_j$$ differences, which is exactly what you want: you want many of them to be zero. Regularization parameter $$\omega$$ should be selected with cross-validation. I have never dealt with models like that and the above is the first thing that came to my mind. Then I decided to see if there is something like that implemented. I made some google searches and soon realized that this is called fusion of categories; searching for lasso fusion categorical will give you a lot of references to read. Here are a few that I briefly looked at: Gerhard Tutz, Regression for Categorical Data, see pp. 175-175 in Google Books. Tutz mentions the following four papers: Land and Friedman, 1997, Variable fusion: a new adaptive signal regression method Bondell and Reich, 2009, Simultaneous factor selection and collapsing levels in ANOVA Gertheiss and Tutz, 2010, Sparse modeling of categorial explanatory variables Tibshirani et al. 2005, Sparsity and smoothness via the fused lasso is somewhat relevant even if not exactly the same (it is about ordinal variables) Gertheiss and Tutz 2010, published in the Annals of Applied Statistics, looks like a recent and very readable paper that contains other references. Here is its abstract: Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $$L_1$$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies. I like their Lasso-like solution paths that show how levels of two categorical variables get merged together when regularization strength increases: If I understood correctly, you imagine a linear model where one of the predictors is categorical (e.g. college major); and you expect that for some subgroups of its levels (subgroups of categories) the coefficients might be exactly the same. So perhaps the regression coefficients for Maths and Physics are the same, but different from those for Chemistry and Biology. In a simplest case, you would have a "one way ANOVA" linear model with a single categorical predictor: $$y_{ij} = \mu + \alpha_i + \epsilon_{ij},$$ where $$i$$ encodes the level of the categorical variable (the category). But you might prefer a solution that collapses some levels (categories) together, e.g. $$\begin{cases}\alpha_1=\alpha_2, \\ \alpha_3=\alpha_4=\alpha_5.\end{cases}$$ This suggests that one can try to use a regularization penalty that would penalize solutions with differing alphas. One penalty term that immediately comes to mind is $$L=\omega \sum_{i This resembles lasso and should enforce sparsity of the $$\alpha_i-\alpha_j$$ differences, which is exactly what you want: you want many of them to be zero. Regularization parameter $$\omega$$ should be selected with cross-validation. I have never dealt with models like that and the above is the first thing that came to my mind. Then I decided to see if there is something like that implemented. I made some google searches and soon realized that this is called fusion of categories; searching for lasso fusion categorical will give you a lot of references to read. Here are a few that I briefly looked at: Gerhard Tutz, Regression for Categorical Data, see pp. 175-175 in Google Books. Tutz mentions the following four papers: Land and Friedman, 1997, Variable fusion: a new adaptive signal regression method Bondell and Reich, 2009, Simultaneous factor selection and collapsing levels in ANOVA Gertheiss and Tutz, 2010, Sparse modeling of categorial explanatory variables Tibshirani et al. 2005, Sparsity and smoothness via the fused lasso is somewhat relevant even if not exactly the same (it is about ordinal variables) Gertheiss and Tutz 2010, published in the Annals of Applied Statistics, looks like a recent and very readable paper that contains other references. Here is its abstract: Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $$L_1$$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies. I like their Lasso-like solution paths that show how levels of two categorical variables get merged together when regularization strength increases: 2 added 22 characters in body edited Sep 26 '16 at 20:07 amoeba 65.5k1717 gold badges221221 silver badges274274 bronze badges If I understood correctly, you imagine a linear model where one of the predictors is categorical (e.g. college major); and you expect that for some subgroups of its $$k$$ levels (subgroups of categories) the coefficients might be exactly the same. In a simplest case, you would have a "one way ANOVA" linear model with a single categorical predictor: $$y_{ij} = \mu + \alpha_i + \epsilon_{ij},$$ where $$i=1\ldots k$$$$i$$ encodes the levelslevel of the categorical variable (the category). But you might prefer a solution that collapses some levels (categories) together, e.g. $$\begin{cases}\alpha_1=\alpha_2, \\ \alpha_3=\alpha_4=\alpha_5.\end{cases}$$ This suggests that one can try to use a regularization penalty that would penalize solutions with all differing alphas. One penalty term that immediately comes to mind is $$L=\omega \sum_{i This resembles lasso and should enforce sparsity of the $$\alpha_i-\alpha_j$$ differences, which is exactly what you want: you want many of them to be zero. Regularization parameter $$\omega$$ should be selected with cross-validation. I have never dealt with models like that and the above is the first thing that came to my mind. Then I decided to see if there is something like that implemented. I made some google searches and soon realized that this is called fusion of categories; searching for lasso fusion categorical will give you a lot of references to read. Here are a few that I briefly looked at: Gerhard Tutz, Regression for Categorical Data, see pp. 175-175 in Google Books. Tutz mentions the following four papers: Land and Friedman, 1997, Variable fusion: a new adaptive signal regression method Bondell and Reich, 2009, Simultaneous factor selection and collapsing levels in ANOVA Gertheiss and Tutz, 2010, Sparse modeling of categorial explanatory variables Tibshirani et al. 2005, Sparsity and smoothness via the fused lasso is somewhat relevant even if not exactly the same (it is about ordinal variables) Gertheiss and Tutz 2010, published in the Annals of Applied Statistics, looks like a recent and very readable paper that contains other references. Here is its abstract: Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $$L_1$$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies. I like their Lasso-like solution paths that show how levels of two categorical variables get merged together when regularization strength increases: If I understood correctly, you imagine a linear model where one of the predictors is categorical (e.g. college major); and you expect that for some subgroups of its $$k$$ levels the coefficients might be exactly the same. In a simplest case, you would have a "one way ANOVA" linear model with a single categorical predictor: $$y_{ij} = \mu + \alpha_i + \epsilon_{ij},$$ where $$i=1\ldots k$$ encodes the levels of the categorical variable. But you might prefer a solution that collapses some levels (categories) together, e.g. $$\begin{cases}\alpha_1=\alpha_2, \\ \alpha_3=\alpha_4=\alpha_5.\end{cases}$$ This suggests that one can try to use a regularization penalty that would penalize solutions with all differing alphas. One penalty term that immediately comes to mind is $$L=\omega \sum_{i This resembles lasso and should enforce sparsity of the $$\alpha_i-\alpha_j$$ differences, which is exactly what you want: you want many of them to be zero. Regularization parameter $$\omega$$ should be selected with cross-validation. I have never dealt with models like that and the above is the first thing that came to my mind. Then I decided to see if there is something like that implemented. I made some google searches and soon realized that this is called fusion of categories; searching for lasso fusion categorical will give you a lot of references to read. Here are a few that I briefly looked at: Gerhard Tutz, Regression for Categorical Data, see pp. 175-175 in Google Books. Tutz mentions the following four papers: Land and Friedman, 1997, Variable fusion: a new adaptive signal regression method Bondell and Reich, 2009, Simultaneous factor selection and collapsing levels in ANOVA Gertheiss and Tutz, 2010, Sparse modeling of categorial explanatory variables Tibshirani et al. 2005, Sparsity and smoothness via the fused lasso is somewhat relevant even if not exactly the same (it is about ordinal variables) Gertheiss and Tutz 2010, published in the Annals of Applied Statistics, looks like a recent and very readable paper that contains other references. Here is its abstract: Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $$L_1$$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies. I like their Lasso-like solution paths that show how levels of two categorical variables get merged together when regularization strength increases: If I understood correctly, you imagine a linear model where one of the predictors is categorical (e.g. college major); and you expect that for some subgroups of its levels (subgroups of categories) the coefficients might be exactly the same. In a simplest case, you would have a "one way ANOVA" linear model with a single categorical predictor: $$y_{ij} = \mu + \alpha_i + \epsilon_{ij},$$ where $$i$$ encodes the level of the categorical variable (the category). But you might prefer a solution that collapses some levels (categories) together, e.g. $$\begin{cases}\alpha_1=\alpha_2, \\ \alpha_3=\alpha_4=\alpha_5.\end{cases}$$ This suggests that one can try to use a regularization penalty that would penalize solutions with differing alphas. One penalty term that immediately comes to mind is $$L=\omega \sum_{i This resembles lasso and should enforce sparsity of the $$\alpha_i-\alpha_j$$ differences, which is exactly what you want: you want many of them to be zero. Regularization parameter $$\omega$$ should be selected with cross-validation. I have never dealt with models like that and the above is the first thing that came to my mind. Then I decided to see if there is something like that implemented. I made some google searches and soon realized that this is called fusion of categories; searching for lasso fusion categorical will give you a lot of references to read. Here are a few that I briefly looked at: Gerhard Tutz, Regression for Categorical Data, see pp. 175-175 in Google Books. Tutz mentions the following four papers: Land and Friedman, 1997, Variable fusion: a new adaptive signal regression method Bondell and Reich, 2009, Simultaneous factor selection and collapsing levels in ANOVA Gertheiss and Tutz, 2010, Sparse modeling of categorial explanatory variables Tibshirani et al. 2005, Sparsity and smoothness via the fused lasso is somewhat relevant even if not exactly the same (it is about ordinal variables) Gertheiss and Tutz 2010, published in the Annals of Applied Statistics, looks like a recent and very readable paper that contains other references. Here is its abstract: Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $$L_1$$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies. I like their Lasso-like solution paths that show how levels of two categorical variables get merged together when regularization strength increases: Bounty Ended with 100 reputation awarded by shadowtalker occurred Sep 26 '16 at 16:10 1 answered Sep 26 '16 at 15:24 amoeba 65.5k1717 gold badges221221 silver badges274274 bronze badges If I understood correctly, you imagine a linear model where one of the predictors is categorical (e.g. college major); and you expect that for some subgroups of its $$k$$ levels the coefficients might be exactly the same. In a simplest case, you would have a "one way ANOVA" linear model with a single categorical predictor: $$y_{ij} = \mu + \alpha_i + \epsilon_{ij},$$ where $$i=1\ldots k$$ encodes the levels of the categorical variable. But you might prefer a solution that collapses some levels (categories) together, e.g. $$\begin{cases}\alpha_1=\alpha_2, \\ \alpha_3=\alpha_4=\alpha_5.\end{cases}$$ This suggests that one can try to use a regularization penalty that would penalize solutions with all differing alphas. One penalty term that immediately comes to mind is $$L=\omega \sum_{i This resembles lasso and should enforce sparsity of the $$\alpha_i-\alpha_j$$ differences, which is exactly what you want: you want many of them to be zero. Regularization parameter $$\omega$$ should be selected with cross-validation. I have never dealt with models like that and the above is the first thing that came to my mind. Then I decided to see if there is something like that implemented. I made some google searches and soon realized that this is called fusion of categories; searching for lasso fusion categorical will give you a lot of references to read. Here are a few that I briefly looked at: Gerhard Tutz, Regression for Categorical Data, see pp. 175-175 in Google Books. Tutz mentions the following four papers: Land and Friedman, 1997, Variable fusion: a new adaptive signal regression method Bondell and Reich, 2009, Simultaneous factor selection and collapsing levels in ANOVA Gertheiss and Tutz, 2010, Sparse modeling of categorial explanatory variables Tibshirani et al. 2005, Sparsity and smoothness via the fused lasso is somewhat relevant even if not exactly the same (it is about ordinal variables) Gertheiss and Tutz 2010, published in the Annals of Applied Statistics, looks like a recent and very readable paper that contains other references. Here is its abstract: Shrinking methods in regression analysis are usually designed for metric predictors. In this article, however, shrinkage methods for categorial predictors are proposed. As an application we consider data from the Munich rent standard, where, for example, urban districts are treated as a categorial predictor. If independent variables are categorial, some modifications to usual shrinking procedures are necessary. Two $$L_1$$-penalty based methods for factor selection and clustering of categories are presented and investigated. The first approach is designed for nominal scale levels, the second one for ordinal predictors. Besides applying them to the Munich rent standard, methods are illustrated and compared in simulation studies. I like their Lasso-like solution paths that show how levels of two categorical variables get merged together when regularization strength increases: