回归，机器学习分类

Question

I have a classification and regression question on machine learning. First question, the following dataset http://it.tinypic.com/view.php?pic=oh3gj7&s=8#.VIjhRDGG_lF

Can we say, the data set is linearly separable? In order to apply a linear model for classi cation, a transformation of the input space is not needed for this dataset, or is not possible for this dataset? My answer is no, but I am not sure for the second, I am not sure a transformation is possible for the dataset.

Second question about regression probl: Give the following data set f : R -> R http://it.tinypic.com/view.php?pic=madsmr&s=8#.VIjhVjGG_lE

Can we say that : A linear model for regression can be used to learn the function associated to this data set ? Given this data set, it is not possible to determine an optimal con guration of the linear model?

I am reading the book of Tom Mitchell Machine learning, and Pattern Recognition and Machine Learning Bishop, but I still have trouble giving the right answer. Thanks in advance.

Answer 1

Neither of this datasets can be modeled using linear classification/regression.

In case of the "input data transfromation" if only dataset is consistent (there are no two exact same points with two different labels) there always exists transformation after which data is linearly separable . In particular one can construct it with:

phi(x) = 1 iff label of x is "1"

in other words, you map all positive samples to "1" and negatives to "0", so your data is now trivialy linearly separable. Or simply map your N points into N unit vectors in R^N space in such a way that i'th point is mapped to [0 0 0 ... 1 ... 0 0 0]^T where this "1" appears at i'th place. Such dataset is trivialy linearly separable for any labeling.