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| | Choosing a hunting knife generally is a troublesome decision for you; because [http://www.decodiams.com/forum/index.php?a=member&m=867934 opinions] range as to which varieties of sporting knives are best for hunting. Some sportsmen choose their folding pocket knives , while others say only a hard and fast-blade looking knife can work better. Whereas [http://www.thebestpocketknifereviews.com/ Best Folding Knives Under 50] the kind of knife you carry is essentially a matter of your personal selection, it’s sensible to think about the [http://wiki.3dmayka.ru/index.php/Best_Pocket_Knife_For_Whittling professionals] and cons of each type of knife and select the software that best fits your needs and budgets. Here, we focus on the strengths and weaknesses of folding & fastened-blade knives<br><br>Case pocket knives have quite a lot of makes use of. They will come in handy if you find yourself tenting or mountaineering. You need to use them to peel fruits or greens and clean fish or deer. They will also be used to [http://Wiki.Cubeworldru.ru/Best_Pocket_Knives cut ropes] or [http://procamp.uni-leipzig.de/doku.php?id=pocket_knife_reviews_2013 bandages]. These knives are protected [http://www.thebestpocketknifereviews.com/ best Pocket knives 2013] to hold and handle and that's the reason they are preferred by most. Case pocket knives may also function tools to remove rust, paint, glue and many others, or to be used in place of a screw driver or scissors. Artists of wood craft use these knives to carve intricate designs.<br><br>It comes with general length of 8.078″ (205mm) while blade uses amazing CPM-M4 American alloy from crucible and lengths about three.438″ (87mm). It weighs about 5.5 oz (155gm). In case you are wanting a knife that can easily fit into your pocket and will make it easier to for tenting; your search ends right [http://en.wikipedia.org/wiki/Blade gerber bear grylls] here. This knife is an ideal match for camping and may actually come useful to cut rope, branches of trees, meat and for survival situation. Spyderco itself says that this knife just isn't made for showmen or fingernail cleaners, the truth is it is made for utilizing on ground<br><br>Steel is basically a mix (alloy) of iron and carbon. Metal makers can add different metals to the alloy to enhance/change the properties of the metal. For example, chromium is usually added to create stainless-steel and enhance corrosion resistance. The commonest additions to the alloy in knife blades embody Vanadium, Tungsten, Manganese, Nickel, Cobalt and Molybdenum. The amount [http://en.wiktionary.org/wiki/pocketknife kiwi knives] of carbon within the alloy is very important as a result of excessive carbon steel is far more durable than low carbon steel and this is an important consider a knife blade. The next are the characteristics that each additive brings to the steel.<br><br>How a couple of good Pocket Knife ? Bizarre? I do not suppose so. Simply think about it for a minute. It is a present that is carried around in a pocket (or a purse or pocket book) more often than not. So when you take a look at it like that, then your considered most of the time. How cool is that? And there are various totally different kinds of pocket knives for many completely different individuals. Let's start with Case Knives and Buck Knives They really make some cool (and high quality) pocket knives that can show you how to match that tough to seek out particular person.<br><br>The most effective pocket knife for you depends upon what your supposed use for the pocket knife is. If you're planning on utilizing it for hunting, you will need a unique knife than should you had been planning on using it for on a regular basis carry. This guide will walk you through what you need to search for in an all-around great pocket knife that will last you for years in a wide range of tasks. These are pocket knives that you may easily and legally ( see what knives are legal in your state ) carry in your pocket day in and day trip. A Knife’s Brand Issues.<br><br>There are lots of companies that manufacture such merchandise on-line and offline at very reasonable prices. However, be sure that it's important to choose the dependable supplier. Now with the assistance of the web you can simply place your order on-line simply by following certain easy steps. Knives On-line will definitely meet your expectations and these should not very expensive. Nearly every of knife is offered on-line for purchase. Be certain that always purchase the very best featured knife and select a model according to your wants. So you'll be able to select any knife as per your choice. About the Writer. |
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| '''Energy distance''' is a [[statistical distance]] between [[probability distributions]]. If X and Y are independent random vectors in '''''R'''''<sup>d</sup> with [[cumulative distribution function]]s F and G respectively, then the energy distance between the distributions F and G is defined to be
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| : <math> D(F, G) = 2\mathbb E\|X - Y\| - \mathbb E\|X - X'\| - \mathbb E\|Y - Y'\| \geq 0,</math> | |
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| where X, X' are independent and identically distributed (iid), Y, Y' are iid, <math> \mathbb E </math> is expected value, and || . || denotes the [[Euclidean norm|length]] of a vector. Energy distance characterizes the equality of distributions: D(F,G) = 0 if and '''only if''' X and Y are identically distributed.
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| Energy distance for statistical applications was introduced in 1985 by [[Gábor J. Székely]], who proved that for real-valued random variables this distance is exactly twice [[Harald Cramér]]'s distance:<ref>Cramér, H. (1928) On the composition of elementary errors, Skandinavisk Aktuarietidskrift, 11, 141–180.</ref>
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| : <math> \int_{-\infty}^\infty (F(x) - G(x))^2 \, dx </math>.
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| For a simple proof of this equivalence, see Székely and Rizzo (2005).<ref name=SR2005>{{Cite journal|author=Székely, G. J., Rizzo, M. L.|year=2005|title=A new test for multivariate normality|journal=Journal of Multivariate Analysis|volume= 93|issue=1|pages= 58–80|doi=10.1016/j.jmva.2003.12.002|unused_data=.}} [http://personal.bgsu.edu/~mrizzo/energy/MVN-GOF-2005.pdf Reprint]</ref> In higher dimensions, however, the two distances are different because the energy distance is rotation invariant while Cramér's distance is not. (Notice that Cramér's distance is not the same as the [[distribution-free]] [[Cramer-von-Mises criterion]].)
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| ==Generalization to metric spaces==
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| One can generalize the notion of energy distance to probability distributions on metric spaces. Let <math>(M, d)</math> be a [[metric space]] with its [[Borel sigma algebra]] <math>\mathcal{B} (M)</math>. Let <math>\mathcal{P} (M)</math> denote the collection of all [[probability measure]]s on the [[measurable space]] <math>(M, \mathcal{B} (M))</math>. If μ and ν are probability measures in <math>\mathcal{P} (M)</math>, then the energy distance of μ and ν can be defined as
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| : <math> D(\mu, \nu)= 2 \mathbb E[d(X,Y)] - \mathbb E[d(X,X')] - \mathbb E[d(Y,Y')] . </math>
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| This is not necessarily non-negative, however. If <math>(M, d^{1/2})</math> is [[isometry|isometric]] to a subset of a [[Hilbert space]], then <math> D^{1/2}</math> is a [[Metric (mathematics)|pseudometric]], and conversely.<ref>Klebanov, L. B. (2005) N-distances and their Applications, Karolinum Press,
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| Charles University, Prague.</ref> This condition is expressed by saying that <math>(M, d)</math> has negative type. Negative type is not sufficient for <math>D</math> to be a metric; the latter condition is expressed by saying that <math>(M, d)</math> has strong negative type. In this situation, the energy distance is zero if and only if X and Y are identically distributed. An example of a metric of negative type but not of strong negative type is the plane with the [[Taxicab geometry|taxicab metric]]. All Euclidean spaces and even separable Hilbert spaces have strong negative type.<ref>[http://front.math.ucdavis.edu/1106.5758 Lyons, R. (2011) Distance covariance in metric spaces]</ref>
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| In the literature on [[kernel methods]] for [[machine learning]], these generalized notions of energy distance are studied under the name of maximum mean discrepancy.<ref>Sejdinovic, D., Gretton, A., Sriperumbudur, B. and Fukumizu, K. (2012) Hypothesis testing using pairwise distances and associated kernels, Proc. of the 29th International Conference on Machine Learning, Edinburgh, Scotland, UK, 2012. [http://arxiv.org/abs/1205.0411v2]</ref>
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| ==Energy statistics==
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| A related statistical concept, the notion of '''E-statistic''' or '''energy-statistic''' was introduced by [[Gábor J. Székely]] in the 1980s when he was giving colloquium lectures in Budapest, Hungary and at MIT, Yale, and Columbia. This concept is based on the notion of Newton’s [[potential energy]].<ref name="Székely, G.J. 2002">Székely, G.J. (2002) E-statistics: The Energy of Statistical Samples, Technical Report BGSU No 02-16.</ref> The idea is to consider statistical observations as [[heavenly body|heavenly bodies]] governed by a statistical [[potential energy]] which is zero only when an underlying statistical [[null hypothesis]] is true. Energy statistics are functions of [[distances]] between statistical observations.
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| ===Testing for equal distributions===
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| Consider the null hypothesis that two random variables, X and Y, have the same [[probability distribution]]s: μ = v . For [[statistical sample]]s from X and Y:
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| : x<sub>1</sub>,…,x<sub>n</sub> and y<sub>1</sub>,…,y<sub>m</sub>,
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| the following arithmetic averages of distances are computed between the X and the Y samples:
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| : A:= (1/nm)∑|x<sub>i</sub> – y<sub>j</sub>|, B:= (1/n<sup>2</sup>)∑|x<sub>i</sub> – x<sub>j</sub>|, C:= (1/m<sup>2</sup>)∑|y<sub>i</sub> – y<sub>j</sub>|.
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| The E-statistic of the underlying null hypothesis is defined as follows:
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| : Ε<sub>n,m</sub>(X,Y):= 2A – B – C.
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| One can prove <ref name=SR2005/><ref name="Székely, G.J. 2002"/> that Ε<sub>n,m</sub>(X,Y) ≥ 0 and that the corresponding population value, E(X,Y):= D(μ,ν), is zero if and only if X and Y have the same distribution (μ=ν). Under this null hypothesis the test statistic
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| :<math> T = \frac{nm}{n+m} E_{n,m}(X,Y) </math>
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| [[Convergence of random variables|converges in distribution]] to a quadratic form of independent standard [[normal random variable]]s. Under the alternative hypothesis ''T'' tends to infinity. This makes it possible to construct a consistent [[statistical test]], the energy test for equal distributions.<ref>G. J. Szekely and M. L. Rizzo (2004). Testing for Equal Distributions in High Dimension, ''InterStat'', Nov. (5). [http://personal.bgsu.edu/~mrizzo/energy/reprint-ksamples.pdf Reprint].</ref>
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| The E-coefficient of inhomogeneity can also be introduced. This is always between 0 and 1 and is defined as
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| :<math> H = \frac{D(F_X,F_Y)}{ 2 \operatorname{\mathbb E}\|X-Y\|} =
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| \frac{2\mathbb E\|X - Y\| - \mathbb E\|X - X'\| - \mathbb E\|Y - Y'\|}
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| {2 \operatorname{\mathbb E}\|X-Y\|},</math>
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| where <math> \mathbb E </math> denotes the [[expected value]]. ''H'' = 0 exactly when ''X'' and ''Y'' have the same distribution.
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| ===Goodness-of-fit===
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| A multivariate goodness-of-fit measure is defined for distributions in arbitrary dimension (not restricted by sample size). The energy goodness-of-fit statistic is
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| :<math>
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| Q_n = n \left( \frac{2}{n} \sum_{i=1}^n \mathbb E \|x_i - X\|^\alpha - \mathbb E\|X - X'\|^\alpha - \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \|x_i - x_j\|^\alpha \right),
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| </math> | |
| where X and X' are independent and identically distributed according to the hypothesized distribution, and <math>\alpha \in (0,2)</math>. The only required condition is that X has finite <math>\alpha</math> moment under the null hypothesis. Under the null hypothesis <math>\mathbb EQ_n=\mathbb E\|X-X'\|^\alpha</math>, and the asymptotic distribution of Q<sub>n</sub> is a quadratic form of centered Gaussian random variables. Under an alternative hypothesis, Q<sub>n</sub> tends to infinity stochastically, and thus determines a statistically consistent test. For most applications the exponent 1 (Euclidean distance) can be applied. The important special case of testing [[Multivariate normal distribution|multivariate normality]]<ref name=SR2005/> is implemented in the ''energy'' package for R. Tests are also developed for heavy tailed distributions such as Pareto ([[power law]]), or [[stable distribution]]s by application of exponents in (0,1).
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| == Applications ==
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| Applications include
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| * [[Hierarchical clustering]] (a generalization of Ward's method)<ref>Székely, G. J. and Rizzo, M. L. (2005) Hierarchical Clustering via Joint Between-Within Distances: Extending Ward's Minimum Variance Method, Journal of Classification, 22(2) 151–183</ref><ref>{{Cite journal|author=Varin, T., Bureau, R., Mueller, C. and Willett, P.|year=2009|title=Clustering files of chemical structures using the Szekely-Rizzo generalization of Ward's method|journal=Journal of Molecular Graphics and Modelling|volume=28|issue=2|pages=187–195|doi=10.1016/j.jmgm.2009.06.006| url=http://www.sciencedirect.com/science/article/B6TGP-4WNXV3R-2/2/53ad7cb172888e652efe6fa9ab7213de|pmid=19640752|unused_data=.}} [http://eprints.whiterose.ac.uk/10328/1/Willett_10328.pdf "eprint"].</ref>
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| * Testing multivariate normality<ref name=SR2005/>
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| * Testing the multi-sample hypothesis of equal distributions,<ref>M. L. Rizzo and G. J. Székely (2010). DISCO Analysis: A Nonparametric Extension of Analysis of Variance, Annals of Applied Statistics Vol. 4, No. 2, 1034–1055. [http://arxiv.org/pdf/1011.2288 PDF]</ref><ref>Szekely, G. J. and Rizzo, M. L. (2004) Testing for Equal Distributions in High Dimension, InterStat, Nov. (5). [http://personal.bgsu.edu/~mrizzo/energy/reprint-ksamples.pdf Reprint].</ref><ref>
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| {{Cite journal| author = Ledlie, Jonathan and Pietzuch, Peter and Seltzer, Margo,
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| | title = Stable and Accurate Network Coordinates,
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| | booktitle = Proceedings of the 26th IEEE International Conference on Distributed Computing Systems,
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| | series = ICDCS '06,
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| | year = 2006
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| | isbn = 0-7695-2540-7
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| | pages = 74–83,
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| | doi = 10.1109/ICDCS.2006.79
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| | publisher = IEEE Computer Society,
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| | address = Washington, DC, USA
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| | pmid = 1154085| issue = 3| journal = Sovetskaia meditsina| unused_data = .
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| }} [http://www.eecs.harvard.edu/~syrah/nc/icdcs06.pdf PDF]</ref>
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| * [[Change detection|Change point detection]]<ref>{{Cite journal|author= Albert Y. Kim, Caren Marzban, Donald B. Percival, Werner Stuetzle
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| |title= Using labeled data to evaluate change detectors in a multivariate streaming environment
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| |journal= Signal Processing
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| |volume= 89|issue= 12|year=2009|issn= 0165-1684|doi= 10.1016/j.sigpro.2009.04.011|pages= 2529–2536|unused_data= .}}
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| [http://www.sciencedirect.com/science/article/B6V18-4W4JDK7-7/2/67f1359392f5707961680dba01fd06cf]
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| [http://faculty.washington.edu/dbp/PDFFILES/tr534.pdf Preprint:TR534].</ref>
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| * Multivariate independence:
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| ** [[distance correlation]],<ref>Székely, G. J., Rizzo M. L. and Bakirov, N. K. (2007). "Measuring and testing independence by correlation of distances", ''The Annals of Statistics'', 35, 2769–2794. [http://arxiv.org/pdf/0803.4101 PDF]</ref>
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| ** [[Brownian covariance]].<ref>Székely, G. J. and Rizzo, M. L. (2009). "Brownian distance covariance", ''The Annals of Applied Statistics'', 3/4, 1233–1308. [http://arxiv.org/pdf/1010.0297 PDF]</ref>
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| * Scoring rules:
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| :Gneiting and Raftery<ref>
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| {{Cite journal|title= Strictly Proper Scoring Rules, Prediction, and Estimation
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| |author= T. Gneiting and A. E. Raftery
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| |year= 2007 |journal=Journal of the American Statistical Association
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| |volume=102 |issue=477 |doi = 10.1198/016214506000001437|pages= 359–378|unused_data= .}}
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| [http://www.stat.washington.edu/people/raftery/Research/PDF/Gneiting2007jasa.pdf Reprint]</ref> apply energy distance to develop a new and very general type of proper scoring rule for probabilistic predictions, the energy score.
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| Applications of energy statistics are implemented in the open source '''''energy''''' package<ref name=energy>energy: R package version 1.2-0. [http://CRAN.R-project.org/package=energy].</ref> for [[R (programming language)|R]].
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:E-Statistic}}
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| [[Category:Statistical distance measures]]
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| [[Category:Statistical inference]]
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| [[Category:Multivariate statistics]]
| |
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