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The lasso estimate thus solves the minimization of theleast-squares penalty with \(\alpha ||w||_1\) added, where\(\alpha\) is a constant and \(||w||_1\) is the \(\ell_1\)-norm ofthe coefficient vector.
LassoLars is a lasso model implemented using the LARSalgorithm, and unlike the implementation based on coordinate descent,this yields the exact solution, which is piecewise linear as afunction of the norm of its coefficients.
With a pure lasso model (i.e., mixture = 1), the Austin station predictor is selected out in each resample. With a mixture of both penalties, its influence increases. Also, as the penalty increases, the uncertainty in this coefficient decreases.
The variable importance of the penalized regression, especially for lasso and elastic net, is more or less out of the box. As discussed, these methods will set regression coefficients for irrelevant variables to zero. This provides a system for selecting important variables but it does not necessarily provide a way to rank them. Using the size of the regression coefficients is a way to rank predictor variables, however if the data is not normalized, you will get different scales for different variables. In our case, we normalized the data and we know that the variables have the same scale before they went into the training. We can use this fact and rank them based on the regression coefficients. The caret::varImp() function uses the coefficients to rank the variables from the elastic net model. Below, were going to plot the top 10 important variables which are normalized to the importance of the most important variable.
It offers precise control where and how weight is distributed. Instead of painting in hard to get areas with brushes, you can now select individual vertex and adjust the weight value. Along with all the familiar tools in edit mode like loop select, xray, lasso select, hide/unhide elements, it is never easier to modify weight values during character skinning process.
In this exercise, you will fit a lasso regression to the Gapminder data you have been working with and plot the coefficients. Just as with the Boston data, you will find that the coefficients of some features are shrunk to 0, with only the most important ones remaining.
Recall that lasso performs regularization by adding to the loss function a penalty term of the absolute value of each coefficient multiplied by some alpha. This is also known as $L1$ regularization because the regularization term is the $L1$ norm of the coefficients. This is not the only way to regularize, however.
Like the alpha parameter of lasso and ridge regularization that you saw earlier, logistic regression also has a regularization parameter:C. C controls the inverse of the regularization strength, and this is what you will tune in this exercise. A large C can lead to an overfit model, while a small C can lead to an underfit model.
Remember lasso and ridge regression from the previous chapter? Lasso used the $L1$ penalty to regularize, while ridge used the $L2$ penalty. There is another type of regularized regression known as the elastic net. In elastic net regularization, the penalty term is a linear combination of the $L1$ and $L2$ penalties: 781b155fdc