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The convenience of discovering alternative blades is a real profit you'll be able to almost fill up for life for twenty dollars. Not having to sharpen it. Did I mention I hate sharpening knives? This software does not come with a sheath or belt clip as it is designed for carrying in your pocket, bag or pack.<br><br>There are completely different knife works obtainable out there as we speak and among the best are those of the Smoky Mountain Knife Works. We've got the widest options of knives at Smokey Mountain that can assist you to resolve on which model or design to choose [http://www.foxnews.com/us/2013/09/18/student-suspended-for-10-days-for-accidentally-bringing-pocket-knife-to/ personalized pocket knives]. Relying on your funds, taste and needs, Smoky Mountain knife works have limitless choices to cater to your liking. With different functions for various occasions, Smoky Mountain Knife works have it covered. No clip, no bail, only a lanyard hole. I hate lanyards. 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| In [[statistics]], '''sufficient dimension reduction (SDR)''' is a paradigm for analyzing data that combines the ideas of [[dimension reduction]] with the concept of [[sufficient statistic|sufficiency]].
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| Dimension reduction has long been a primary goal of [[regression analysis]]. Given a response variable ''y'' and a ''p''-dimensional predictor vector <math>\textbf{x}</math>, regression analysis aims to study the distribution of <math>y|\textbf{x}</math>, the [[conditional distribution]] of <math>y</math> given <math>\textbf{x}</math>. A '''dimension reduction''' is a function <math>R(\textbf{x})</math> that maps <math>\textbf{x}</math> to a subset of <math>\mathbb{R}^k</math>, ''k'' < ''p'', thereby reducing the [[dimension (vector space)|dimension]] of <math>\textbf{x}</math>.<ref name="Cook & Adragni:2009">Cook & Adragni (2009) [http://rsta.royalsocietypublishing.org/content/367/1906/4385.full ''Sufficient Dimension Reduction and Prediction in Regression''] In: ''Philosophical Transactions of the Royal Society A: Physical, Mathematical and Engineering Sciences'', 367(1906): 4385–4405</ref> For example, <math>R(\textbf{x})</math> may be one or more [[linear combinations]] of <math>\textbf{x}</math>.
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| A dimension reduction <math>R(\textbf{x})</math> is said to be '''sufficient''' if the distribution of <math>y|R(\textbf{x})</math> is the same as that of <math>y|\textbf{x}</math>. In other words, no information about the regression is lost in reducing the dimension of <math>\textbf{x}</math> if the reduction is sufficient.<ref name="Cook & Adragni:2009" />
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| == Graphical motivation ==
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| In a regression setting, it is often useful to summarize the distribution of <math>y|\textbf{x}</math> graphically. For instance, one may consider a [[scatter plot]] of <math>y</math> versus one or more of the predictors. A scatter plot that contains all available regression information is called a '''sufficient summary plot'''.
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| When <math>\textbf{x}</math> is high-dimensional, particularly when <math>p\geq 3</math>, it becomes increasingly challenging to construct and visually interpret sufficiency summary plots without reducing the data. Even three-dimensional scatter plots must be viewed via a computer program, and the third dimension can only be visualized by rotating the coordinate axes. However, if there exists a sufficient dimension reduction <math>R(\textbf{x})</math> with small enough dimension, a sufficient summary plot of <math>y</math> versus <math>R(\textbf{x})</math> may be constructed and visually interpreted with relative ease.
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| Hence sufficient dimension reduction allows for graphical intuition about the distribution of <math>y|\textbf{x}</math>, which might not have otherwise been available for high-dimensional data.
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| Most graphical methodology focuses primarily on dimension reduction involving linear combinations of <math>\textbf{x}</math>. The rest of this article deals only with such reductions.
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| == Dimension reduction subspace ==
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| Suppose <math>R(\textbf{x}) = A^T\textbf{x}</math> is a sufficient dimension reduction, where <math>A</math> is a <math>p\times k</math> [[matrix (mathematics)|matrix]] with [[rank (linear algebra)|rank]] <math>k\leq p</math>. Then the regression information for <math>y|\textbf{x}</math> can be inferred by studying the distribution of <math>y|A^T\textbf{x}</math>, and the plot of <math>y</math> versus <math>A^T\textbf{x}</math> is a sufficient summary plot.
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| [[Without loss of generality]], only the [[vector space|space]] [[linear span|spanned]] by the columns of <math>A</math> need be considered. Let <math>\eta</math> be a [[basis (linear algebra)|basis]] for the column space of <math>A</math>, and let the space spanned by <math>\eta</math> be denoted by <math>\mathcal{S}(\eta)</math>. It follows from the definition of a sufficient dimension reduction that
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| : <math>F_{y|x} = F_{y|\eta^Tx},</math> | |
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| where <math>F</math> denotes the appropriate [[cumulative distribution function|distribution function]]. Another way to express this property is
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| : <math>y\perp\!\!\!\perp\textbf{x}\,|\,\eta^T\textbf{x},</math>
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| or <math>y</math> is [[conditional independence|conditionally independent]] of <math>\textbf{x}</math>, given <math>\eta^T\textbf{x}</math>. Then the subspace <math>\mathcal{S}(\eta)</math> is defined to be a '''dimension reduction subspace (DRS)'''.<ref name="Cook:1998">Cook, R.D. (1998) ''Regression Graphics: Ideas for Studying Regressions Through Graphics'', Wiley ISBN 0471193658 </ref>
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| === Structural dimensionality ===
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| For a regression <math>y|\textbf{x}</math>, the '''structural dimension''', <math>d</math>, is the smallest number of distinct linear combinations of <math>\textbf{x}</math> necessary to preserve the conditional distribution of <math>y|\textbf{x}</math>. In other words, the smallest dimension reduction that is still sufficient maps <math>\textbf{x}</math> to a subset of <math>\mathbb{R}^d</math>. The corresponding DRS will be ''d''-dimensional.<ref name="Cook:1998" />
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| === Minimum dimension reduction subspace ===
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| A subspace <math>\mathcal{S}</math> is said to be a '''minimum DRS''' for <math>y|\textbf{x}</math> if it is a DRS and its dimension is less than or equal to that of all other DRSs for <math>y|\textbf{x}</math>. A minimum DRS <math>\mathcal{S}</math> is not necessarily unique, but its dimension is equal to the structural dimension <math>d</math> of <math>y|\textbf{x}</math>, by definition.<ref name="Cook:1998" />
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| If <math>\mathcal{S}</math> has basis <math>\eta</math> and is a minimum DRS, then a plot of ''y'' versus <math>\eta^T\textbf{x}</math> is a '''minimal sufficient summary plot''', and it is (''d'' + 1)-dimensional.
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| == Central subspace ==
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| If a subspace <math>\mathcal{S}</math> is a DRS for <math>y|\textbf{x}</math>, and if <math>\mathcal{S}\subset\mathcal{S}_{drs}</math> for all other DRSs <math>\mathcal{S}_{drs}</math>, then it is a '''central dimension reduction subspace''', or simply a '''central subspace''', and it is denoted by <math>\mathcal{S}_{y|x}</math>. In other words, a central subspace for <math>y|\textbf{x}</math> exists [[if and only if]] the intersection <math>\cap\mathcal{S}_{drs}</math> of all dimension reduction subspaces is also a dimension reduction subspace, and that intersection is the central subspace <math>\mathcal{S}_{y|x}</math>.<ref name="Cook:1998" />
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| The central subspace <math>\mathcal{S}_{y|x}</math> does not necessarily exist because the intersection <math>\cap\mathcal{S}_{drs}</math> is not necessarily a DRS. However, if <math>\mathcal{S}_{y|x}</math> ''does'' exist, then it is also the unique minimum dimension reduction subspace.<ref name="Cook:1998" />
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| === Existence of the central subspace ===
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| While the existence of the central subspace <math>\mathcal{S}_{y|x}</math> is not guaranteed in every regression situation, there are some rather broad conditions under which its existence follows directly. For example, consider the following proposition from Cook (1998):
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| : Let <math>\mathcal{S}_1</math> and <math>\mathcal{S}_2</math> be dimension reduction subspaces for <math>y|\textbf{x}</math>. If <math>\textbf{x}</math> has [[probability density function|density]] <math>f(a) > 0</math> for all <math>a\in\Omega_x</math> and <math>f(a) = 0</math> everywhere else, where <math>\Omega_x</math> is [[convex set|convex]], then the intersection <math>\mathcal{S}_1\cap\mathcal{S}_2</math> is also a dimension reduction subspace.
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| It follows from this proposition that the central subspace <math>\mathcal{S}_{y|x}</math> exists for such <math>\textbf{x}</math>.<ref name="Cook:1998" />
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| == Methods for dimension reduction ==
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| There are many existing methods for dimension reduction, both graphical and numeric. For example, '''[[sliced inverse regression]] (SIR)''' and '''sliced average variance estimation (SAVE)''' were introduced in the 1990s and continue to be widely used.<ref name="Li:1991">Li, K-C. (1991) [http://www.jstor.org/stable/2290563 ''Sliced Inverse Regression for Dimension Reduction''] In: ''[[Journal of the American Statistical Association]]'', 86(414): 316–327</ref> Although SIR was originally designed to estimate an ''effective dimension reducing subspace'', it is now understood that it estimates only the central subspace, which is generally different.
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| More recent methods for dimension reduction include [[likelihood function|likelihood]]-based sufficient dimension reduction,<ref name="Cook & Forzani(2009)">Cook, R.D. and Forzani, L. (2009) ''Likelihood-Based Sufficient Dimension Reduction'' In: [[Journal of the American Statistical Association]], 104(485): 197–208</ref> estimating the central subspace based on the inverse third [[moment (mathematics)|moment]] (or ''k''th moment),<ref name="Yin & Cook:2003">Yin, X. and Cook, R.D. (2003) [http://www.jstor.org/stable/30042023 ''Estimating Central Subspaces via Inverse Third Moments''] In: ''[[Biometrika]]'', 90(1): 113–125</ref> estimating the central solution space,<ref name="Li & Dong:2009">Li, B. and Dong, Y.D. (2009) [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aos/1239369022 ''Dimension Reduction for Nonelliptically Distributed Predictors''] In: ''[[Annals of Statistics]]'', 37(3): 1272–1298</ref> and graphical regression.<ref name="Cook:1998" /> For more details on these and other methods, consult the statistical literature.
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| [[Principal components analysis]] '''(PCA)''' and similar methods for dimension reduction are not based on the sufficiency principle.
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| === Example: linear regression ===
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| Consider the regression model
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| : <math>y = \alpha + \beta^T\textbf{x} + \varepsilon,\text{ where }\varepsilon\perp\!\!\!\perp\textbf{x}.</math>
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| Note that the distribution of <math>y|\textbf{x}</math> is the same as the distribution of <math>y|\beta^T\textbf{x}</math>. Hence, the span of <math>\beta</math> is a dimension reduction subspace. Also, <math>\beta</math> is 1-dimensional (unless <math>\beta=\textbf{0}</math>), so the structural dimension of this regression is <math>d=1</math>.
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| The [[ordinary least squares|OLS]] estimate <math>\hat{\beta}</math> of <math>\beta</math> is [[consistent estimator|consistent]], and so the span of <math>\hat{\beta}</math> is a consistent estimator of <math>\mathcal{S}_{y|x}</math>. The plot of <math>y</math> versus <math>\hat{\beta}^T\textbf{x}</math> is a sufficient summary plot for this regression.
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| == See also ==
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| *[[Dimension reduction]]
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| *[[Sliced inverse regression]]
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| *[[Principal component analysis]]
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| *[[Linear discriminant analysis]]
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| *[[Curse of dimensionality]]
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| *[[Multilinear subspace learning]]
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| == Notes ==
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| {{Reflist}}
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| == References ==
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| {{refbegin}}
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| *Cook, R.D. (1998) ''Regression Graphics: Ideas for Studying Regressions through Graphics'', Wiley Series in Probability and Statistics. [http://www.stat.umn.edu/RegGraph/ Regression Graphics].
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| *Cook, R.D. and Adragni, K.P. (2009) "Sufficient Dimension Reduction and Prediction in Regression", [[Philosophical Transactions of the Royal Society A: Physical, Mathematical and Engineering Sciences]], 367(1906), 4385–4405. [http://rsta.royalsocietypublishing.org/content/367/1906/4385.full Full-text]
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| *Cook, R.D. and Weisberg, S. (1991) "Sliced Inverse Regression for Dimension Reduction: Comment", [[Journal of the American Statistical Association]], 86(414), 328–332. [http://www.jstor.org/stable/2290564 Jstor]
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| *Li, K-C. (1991) "Sliced Inverse Regression for Dimension Reduction", [[Journal of the American Statistical Association]], 86(414), 316–327. [http://www.jstor.org/stable/2290563 Jstor]
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| {{refend}}
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| == External links ==
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| * [http://www.stat.umn.edu/~dennis/SDR/ Sufficient Dimension Reduction]
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| [[Category:Multivariate statistics]]
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| [[Category:Dimension reduction]]
| |
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To do a run down on the professional’s and cons of this blade. The size and weight are huge pluses. It clips well and rides with a fairly low pocket profile. It’s so small and light-weight that it really works properly as a pocket knife. So long as you alter the blades commonly they continue Best Brand Pocket Knives to be sharp. The convenience of discovering alternative blades is a real profit you'll be able to almost fill up for life for twenty dollars. Not having to sharpen it. Did I mention I hate sharpening knives? This software does not come with a sheath or belt clip as it is designed for carrying in your pocket, bag or pack.
There are completely different knife works obtainable out there as we speak and among the best are those of the Smoky Mountain Knife Works. We've got the widest options of knives at Smokey Mountain that can assist you to resolve on which model or design to choose personalized pocket knives. Relying on your funds, taste and needs, Smoky Mountain knife works have limitless choices to cater to your liking. With different functions for various occasions, Smoky Mountain Knife works have it covered. No clip, no bail, only a lanyard hole. I hate lanyards. If the knife was larger this would be a problem, as it is, I am not bent off form about it.
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Dimension, shape, interval, material involving knife edge – Sure, size, shape, size as well as materials involving blade are essential while deciding on a knife. A new knife that provides blade with best material that's corrode free and sturdy makes high demand. Nobody will cherish a blade that gets corrosion easily and never robust enough to do its process. One of many key characteristics of an excellent survival knife sharpener is that it needs to be simply portable and be operable with out electricity. It wants to really sharpen a blade and not just hone it and in some circumstances you’ll need a sharpener that can deal with serrated edges.
By carefully weighing all of these categories, patrons can be certain that they discover the proper knife for his or her wants. Knives will be purchased in all completely different shapes and sizes, from jack knives and pen knives to multipurpose Swiss Army style knives. After studying about the totally different varieties of knives, guests can find out about particular products that meet their wants. Those that need to learn extra a few specific type or brand of knife can read via blog posts on the web site. Blog posts embrace articles about one of the best costly pocket knives, the very best ceramic pocket knives, and even the best pink pocket knives.
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