By Trevor Hastie, Visit Amazon's Robert Tibshirani Page, search results, Learn about Author Central, Robert Tibshirani, , Martin Wainwright
Discover New equipment for facing High-Dimensional Data
A sparse statistical version has just a small variety of nonzero parameters or weights; for this reason, it truly is a lot more uncomplicated to estimate and interpret than a dense version. Statistical studying with Sparsity: The Lasso and Generalizations offers tools that take advantage of sparsity to assist recuperate the underlying sign in a suite of data.
Top specialists during this quickly 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 applying of ℓ1 consequences to generalized linear types and help vector machines, conceal generalized consequences akin to the elastic web and crew lasso, and evaluate numerical tools for optimization. additionally they current statistical inference equipment for equipped (lasso) versions, together with the bootstrap, Bayesian equipment, and lately built techniques. furthermore, the publication examines matrix decomposition, sparse multivariate research, graphical types, and compressed sensing. It concludes with a survey of theoretical effects for the lasso.
In this age of huge facts, the variety of positive aspects measured on someone or item could be huge and may be better than the variety of observations. This publication exhibits how the sparsity assumption permits us to take on those difficulties and extract important and reproducible styles from titanic datasets. info analysts, machine scientists, and theorists will have fun with this thorough and up to date therapy of sparse statistical modeling.
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Extra info for Statistical Learning with Sparsity: The Lasso and Generalizations
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If we now include a third predictor x3 = x2 into the mix, an identical copy of the second, then for any α ∈ [0, 1], the vector ˜ β(α) = (β1 , α · β2 , (1 − α) · β2 ) produces an identical fit, and has 1 norm ˜ β(α) 1 = β 1 . Consequently, for this model (in which we might have either p ≤ N or p > N ), there is an infinite family of solutions. In general, when λ > 0, one can show that if the columns of the model matrix X are in general position, then the lasso solutions are unique. To be precise, we say the columns {xj }pj=1 are in general position if any affine subspace L ⊂ RN of dimension k < N contains at most k + 1 elements of the set {±x1 , ±x2 , .
Ex. 6). 15). Ex. 5 Uniqueness of fitted values from the lasso. For some λ ≥ 0, suppose that we have two lasso solutions β, γ with common optimal value c∗ . (a) Show that it must be the case that Xβ = Xγ, meaning that the two solutions must yield the same predicted values. ) (b) If λ > 0, show that we must have β (Tibshirani2 2013). 1 = γ 1. Ex. 6 Here we use the bootstrap as the basis for inference with the lasso. 2. Use the nonparametric bootstrap, sampling features and outcome values (xi , yi ) with replacement from the observed data.
B) If λ > 0, show that we must have β (Tibshirani2 2013). 1 = γ 1. Ex. 6 Here we use the bootstrap as the basis for inference with the lasso. 2. Use the nonparametric bootstrap, sampling features and outcome values (xi , yi ) with replacement from the observed data. Keep the bound t fixed at its estimated value from the original lasso fit. Estimate as well the probability that an estimated coefficient is zero. ˆ for each bootstrap replication. (b) Repeat part (a), but now re-estimate λ Compare the results to those in part (a).