In data analysis, one is interested in using the information about the response variable contained in the predictors in the best way possible. This can lead to problems when the predictors are highly collinear, as it implies an inherent lower-dimensional structure in the data. One method of analyzing data of this form is to make the assumption that these structured dependencies arise due to the predictors lying on some implicit lower-dimensional manifold. This assumption helps solve the problem of reducing the dimension of the predictors in the interest of removing some redundant information, but it introduces the problem of analyzing the transformed data. In particular, making accurate predictions with the lower-dimensional data that can be interpreted in the higher-dimensional space can be difficult. The technique of weighted regression with regularization on the model parameters can help to overcome these issues. | In data analysis, one is interested in using the information about the response variable contained in the predictors in the best way possible. This can lead to problems when the predictors are highly collinear, as it implies an inherent lower-dimensional structure in the data. One method of analyzing data of this form is to make the assumption that these structured dependencies arise due to the predictors lying on some implicit lower-dimensional manifold. This assumption helps solve the problem of reducing the dimension of the predictors in the interest of removing some redundant information, but it introduces the problem of analyzing the transformed data. In particular, making accurate predictions with the lower-dimensional data that can be interpreted in the higher-dimensional space can be difficult. The technique of weighted regression with regularization on the model parameters can help to overcome these issues. |