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Introduction to High-Dimensional Statistics by Christophe Giraud

By Christophe Giraud

Ever-greater computing applied sciences have given upward push to an exponentially becoming quantity of information. this day monstrous info units (with very likely millions of variables) play a major position in nearly each department of recent human task, together with networks, finance, and genetics. notwithstanding, examining such facts has awarded a problem for statisticians and knowledge analysts and has required the advance of recent statistical tools able to keeping apart the signal from the noise.

Introduction to High-Dimensional Statistics is a concise consultant to cutting-edge versions, options, and ways for dealing with high-dimensional information. The ebook is meant to reveal the reader to the foremost suggestions and concepts within the most basic settings attainable whereas warding off pointless technicalities.

Offering a succinct presentation of the mathematical foundations of high-dimensional data, this hugely available text:

  • Describes the demanding situations with regards to the research of high-dimensional data
  • Covers state of the art statistical equipment together with version choice, sparsity and the lasso, aggregation, and studying theory
  • Provides targeted routines on the finish of each bankruptcy with collaborative options on a wikisite
  • Illustrates thoughts with basic yet transparent useful examples

Introduction to High-Dimensional Statistics is acceptable for graduate scholars and researchers attracted to studying sleek information for large information. it may be used as a graduate textual content or for self-study.

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Example text

Fortunately, the useful information usually concentrates around low-dimensional structures, and building on this feature allows us to circumvent this curse of the dimensionality. This book is an introduction to the main concepts and ideas involved in the analysis of the high-dimensional data. Its focus is on the mathematical side, with the choice to concentrate on simple settings in order to avoid unessential technical details that could blur the main arguments. 2 References The book by Hastie, Tibshirani, and Friedman [73] is an authoritative reference for the data scientist looking for a pedagogical and (almost) comprehensive catalog of statistical procedures.

The first term (I − ProjSm ) f ∗ 2 is a bias term that reflects the quality of Sm for approximating f ∗ . The second term dm σ 2 is a variance term that increases linearly with the dimension of Sm . In particular, we notice that enlarging Sm reduces the first term but increases the second term. The oracle model Smo is then the model in the collection {Sm , m ∈ M }, which achieves the best trade-off between the bias and the variance. 6). It follows from the decomposition Y − fm = (I − ProjSm )( f ∗ + ε) that E Y − fm 2 =E (I − ProjSm ) f ∗ = (I − ProjSm ) f ∗ 2 2 + 2 (I − ProjSm ) f ∗ , ε + (I − ProjSm )ε 2 + (n − dm )σ 2 = rm + (n − 2dm )σ 2 .

1 for details on the LDA). Clearly, the plane S2 is more useful than V2 for classifying strains of bacteria according to their pathogenicity. This better result is simply due to the fact that V2 has been computed independently of the classification purpose, whereas S2 has been computed in order to solve at best this classification problem. ,n . Classical results carefully describe the asymptotic behavior of estimators when n goes to infinity (with p fixed), which makes sense in such a setting. 11 Iconic example of classical statistics: n = 100 observations (gray dots) for estimating the p = 2 parameters of the regression line (in black).

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