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By Trevor Hastie, Visit Amazon's Robert Tibshirani Page, search results, Learn about Author Central, Robert Tibshirani, , Martin Wainwright

Discover New tools for facing High-Dimensional Data

A sparse statistical version has just a small variety of nonzero parameters or weights; accordingly, it really is a lot more straightforward to estimate and interpret than a dense version. Statistical studying with Sparsity: The Lasso and Generalizations offers equipment that take advantage of sparsity to aid recuperate the underlying sign in a collection of data.

Top specialists during this swiftly evolving box, the authors describe the lasso for linear regression and an easy coordinate descent set of rules for its computation. They speak about the appliance of 1 consequences to generalized linear types and help vector machines, conceal generalized consequences equivalent to the elastic internet and workforce lasso, and assessment numerical equipment for optimization. in addition they current statistical inference equipment for equipped (lasso) types, together with the bootstrap, Bayesian tools, and lately built techniques. moreover, the ebook examines matrix decomposition, sparse multivariate research, graphical versions, and compressed sensing. It concludes with a survey of theoretical effects for the lasso.

In this age of huge information, the variety of gains measured on anyone or item will be huge and may be higher than the variety of observations. This publication indicates how the sparsity assumption permits us to take on those difficulties and extract invaluable and reproducible styles from vast datasets. information analysts, laptop scientists, and theorists will savor this thorough and updated therapy of sparse statistical modeling.

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Extra resources for Statistical Learning with Sparsity: The Lasso and Generalizations

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There are N = 7291 training images of the digits {0, 1, . . , 9}, each digitized to a 16 × 16 gray-scale image. Using the p = 256 pixels as features, we fit a 10-class lasso multinomial model. 4 shows the training and test misclassification error as a function of the sequence of λ values used. 5 we display the coefficients as images (on average about 25% are nonzero). Some of these can be identified as appropriate contrast functionals for highlighting each digit. 4 Training and test misclassification errors of a multinomial lasso model fit to the zip code data, plotted as a function of log(λ).

5) Here there are K coefficients for each variable (one per class). In this chapter, we discuss approaches to fitting generalized linear models that are based on maximizing the likelihood, or equivalently minimizing the negative log-likelihood along with an 1 -penalty minimize − β0 ,β 1 L(β0 , β; y, X) + λ β N 1 . 6) Here y is the N -vector of outcomes and X is the N × p matrix of predictors, and the specific form the log-likelihood L varies according to the GLM. 6) corresponds to the ordinary linear least-squares lasso.

For each value of λ, we create an outer loop which cycles over ∈ {1, . . , K} and computes the par˜ Then tial quadratic approximation Q about the current parameters (β˜0 , β). 3 1} . 5, the lasso penalty will select different variables for different classes. This can mean that although individual coefficient vectors are sparse, the overall model may not be. In this example, on average there are 25% of the coefficients nonzero per class, while overall 81% of the variables are used. 3) for the set of coefficients βj = (β1j , β2j , .

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