Conway chained arrow notation - Revision history

77.161.72.182: /* Graham's number */

2014-11-22T13:51:09Z

Graham's number

← Older revision		Revision as of 15:51, 22 November 2014
Line 1:		Line 1:
	~~In [[Vapnik–Chervonenkis theory\|statistical learning theory]], or sometimes [[computational learning theory]], the~~ '''~~VC dimension~~''' ~~(for~~ '''~~Vapnik–Chervonenkis dimension~~''') is ~~a measure of~~ the ~~[[membership function (mathematics)\|capacity]]~~ of a ~~[[statistical classification]] [[algorithm]]~~, ~~defined as the [[cardinality]] of the largest set of points~~ that ~~the algorithm can [[Shattering (machine learning)\|shatter]]~~. ~~It is~~ a ~~core concept in [[Vapnik–Chervonenkis theory]], and was originally defined by [[Vladimir Vapnik]] and [[Alexey Chervonenkis]]~~.		Each next step to this fact game''s success is that can it produces the impression that it''s a multiplayer game. I believe it''s a fantasy considering you don''t do what has necessary directly with different player. You don''t fight and explore in unison like you would in Wow, of play to protect against another player even for with a turn-by-turn justification comparable to Chess. Any time you raid another player''s village, that do player is offline combined with you could at the type of same time just be raiding a random computer-generated village.<br><br>If you are purchasing a game to a child, appear for an individual that allows several individuals carry out together. Gaming are generally a singular activity. Nonetheless, it's important to guide your youngster to be societal, and multiplayer deviate of clans trucos matches can do that. They allow siblings and buddies to all seated and laugh and strive together.<br><br>Interweaving social trends form a deep net in which we are all trapped. When Each Tygers of Pan Tang sang 'It's lonely on the top bar. Everyones trying to do we in', these people funded much from clash of clans hack tool no survey. A society without conflict of clans hack tool no survey is being a society with no knowledge, in that it is very good.<br><br>Battle of Clans is definitely a popular sport designed to end up being dabbled in on multiple systems, the car . iOS and also google android. The overall game is ach intriguing but presently and also comes a spot each morning legend, where the character gets trapped because within not enough gems. However, this problem currently is able to easily be resolved.<br><br>Give the in-online game songs the opportunity. If, nonetheless, you might nevertheless be annoyed by using this tool soon after one moment approximately, don't be frightened to mute the television set or personal computer but play some audio of the very own. You'll find a far more pleasurable game playing experience performing this and therefore are way more unlikely to get a good frustration from actively game play.<br><br>If you enjoyed this article and you would certainly such as to get more information pertaining to [http://prometeu.net hack clash of clans android] kindly visit our web site. Ought to you perform online multi-player game titles, don't forget about the strength of color of voice chat! A mic or pair of headphones is a very simple expenditure, and having the capability to speak within order to your fellow athletes would have a lot of rewards. You are able to create more vibrant connections with the avid gamers community and stay an far more successful group person when you will definitely be able connect out made some noise.<br><br>As your village grows, anyone could have to explore uncharted territories for Gold and Teak wood which are the a couple of key resources you can expect to require at start of the video games ( addititionally there is almost certainly Stone resource, that you [http://Dict.Leo.org/?search=discover discover] later inside i would say the game ). Through your exploration, you in many cases can expect to stumble after many islands whereby that villages happen to be held captive under BlackGuard slavery and you results from free Gold choices if they are vacant.

	~~Informally~~, ~~the capacity of a classification model is related~~ to ~~how complicated it can~~ be~~. For example~~, ~~consider the [[Heaviside step function\|thresholding]]~~ of a ~~high-[[degree~~ of ~~a polynomial\|degree]] [[polynomial]]: if~~ the ~~polynomial evaluates above zero, that point is classified as positive, otherwise as negative~~. ~~A high-degree polynomial can be wiggly~~, ~~so it can fit a given set~~ of ~~training points well. But one can expect that the classifier will make errors on other points, because it is too wiggly. Such a polynomial has a high capacity~~. A ~~much simpler alternative~~ is ~~to threshold~~ a ~~linear function. This function may not fit the training set well~~, ~~because~~ it ~~has a low capacity. This notion of capacity~~ is ~~made more rigorous below~~.

	~~== Shattering ==~~

	~~A classification model~~ <~~math~~>f<~~/math> with some parameter vector <math>\theta</math~~> is ~~said to ''shatter''~~ a ~~set of data points <math>(x_1,x_2,\ldots,x_n)</math> if, for all assignments of labels~~ to ~~those points~~, ~~there exists a <math>\theta</math> such that~~ the ~~model <math>f</math> makes no errors when evaluating that set of data points~~.

	The ~~VC dimension of~~ a ~~model <math>f</math> is~~ the ~~maximum number of points that can be arranged so that <math>f</math> shatters them~~. ~~More formally~~, it is ~~<math>h'</math> where <math>h'</math> is the maximum <math>h</math> such that some data point set of [[cardinality]] <math>h</math> can~~ be ~~shattered by~~ <~~math~~>f<~~/math~~>.

	~~For example, consider a [[linear classifier\|straight line]] as~~ the ~~classification model:~~ the ~~model used by a [[perceptron]]~~. ~~The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed~~ be ~~shattered~~ using this ~~model (any 3 points that are not collinear can be shattered). However~~, ~~no set of 4 points can be shattered: by [[Radon~~'~~s theorem]], any four points can~~ be ~~partitioned into two subsets with intersecting convex hulls, so it is not possible~~ to ~~separate one~~ of ~~these two subsets from~~ the ~~other~~. ~~Thus, the VC dimension of~~ this ~~particular classifier is 3. It is important~~ to ~~remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment~~. ~~Note, only 3 of the 2~~<~~sup~~>3<~~/sup~~>~~ = 8 possible label assignments are shown for the three points.~~

	~~{\| border="0" cellpadding="4" cellspacing="0"~~
	~~\|- style="text-align:center;"~~
	~~\| style="background:#dfd;"\|~~ [~~[File~~:~~VC1~~.~~svg~~]]
	~~\| style="background:#dfd;"\| [[File:VC2~~.~~svg]]~~
	~~\| style="background:#dfd;"\| [[File:VC3.svg]]~~
	~~\| style="background:#fdd;"\| [[File:VC4.svg]]~~
	\|- ~~style="text-align:center;"~~
	~~\| colspan="3" style="background:#dfd;"\|~~ '~~''3 points shattered'''~~
	~~\| style="background:#fdd;"\| '''4 points impossible'''~~
	\|}

	~~== Uses ==~~

	~~The VC dimension has utility in statistical learning theory, because it can predict a [[probabilistic]] [[upper bound]] on~~ the ~~test error~~ of a ~~classification model.~~

	~~Vapnik <ref>Vapnik~~, ~~Vladimir. The nature~~ of ~~statistical learning theory~~. ~~springer, 2000.</ref> proved that~~ the ~~probability of the test error distancing from~~ an ~~upper bound (on data that is drawn [[Independent identically-distributed random variables\|i~~.~~i.d.]] from the same distribution as the training set) is given by~~

	~~<math>~~
	~~P \left(\text{test error} \leq \text{training error} + \sqrt{h(\log(2N/h)+1)-\log(\eta/4)\over N} \right) = 1 - \eta~~
	~~</math>~~

	~~where~~ <~~math~~>h<~~/math~~> ~~is the VC dimension of the classification model~~, and ~~<math>N</math> is~~ the ~~size~~ of ~~the training set (restriction: this formula is valid when <math>h \ll N</math>). Similar complexity bounds~~ can be derived using [[Rademacher complexity]], but Rademacher complexity can sometimes provide more insight than VC dimension calculations into such statistical methods such as those using [[kernel methods\|kernels]].

	~~In [[computational geometry]], VC dimension is one~~ of the ~~critical parameters in the size of [[E-net~~ (~~computational geometry)\|ε-nets]]~~, ~~which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all.~~

	~~==See also==~~
	*[[Sauer–Shelah lemma]], a bound on the number of sets in a set system in terms of the VC dimension

	~~== References ==~~
	~~<references/>~~
	* Andrew Moore's [http://~~www-2.cs~~.~~cmu~~.~~edu~~/~~~awm/tutorials/vcdim.html VC dimension tutorial~~]
	* Vapnik, Vladimir. "The nature of statistical learning theory". springer, 2000.
	* V. Vapnik and A. Chervonenkis. "On the ~~uniform convergence of relative frequencies of events to their probabilities." ''Theory of Probability and its Applications'', 16(2~~)~~:264–280, 1971.~~
	* A. ~~Blumer, A. Ehrenfeucht, D. Haussler~~, and ~~[[Manfred K. Warmuth\|M. K. Warmuth]]. "Learnability and the Vapnik–Chervonenkis dimension~~.~~" ''Journal of the ACM'', 36(4):929–865, 1989.~~
	* Christopher Burges Tutorial on SVMs for Pattern Recognition (containing information also for VC dimension) [http://citeseer.ist.psu.edu/burges98tutorial.html]
	* [[Bernard Chazelle]]. "The Discrepancy Method." [http://www.cs.princeton.edu/~chazelle/book.html]

	~~[[Category:Dimension]]~~
	~~[[Category:Statistical classification]]~~
	~~[[Category:Computational learning theory]]~~
	~~[[Category:Measures of complexity]]~~

en>Magioladitis: /* Properties */ fix syntax

2014-01-12T14:18:11Z

Properties: fix syntax

← Older revision		Revision as of 16:18, 12 January 2014
Line 1:		Line 1:
	~~We understand~~ the ~~famous BMI or Body Mass Index Calculation so commonly used~~ to ~~assess whether you're usual~~, ~~obese or obese? Well~~, ~~what do we actually learn about~~ it~~? Do we think~~ it~~'s important?~~<br><br>~~My point~~ is ~~I recognize I~~'~~m not overweight plus I understand I am~~ a ~~healthy person on my cholesterol and blood pressure~~, so there ~~are details~~ that ~~give to my weight~~ a ~~calculator~~ can'~~t account for.~~<br><br>~~Don~~'~~t worry about~~ the ~~non-integer exponent~~. ~~Any off-~~the~~-shelf scientific calculator may handle~~ it. ~~I wouldn't use~~ the LI for ~~clinical purposes--even though it's more reasonable than BMI~~.<br><br>~~Wondering what a weight should be?~~ ~~Try Mayo Clinics~~ [~~http~~:~~//safedietplansforwomen~~.~~com/bmi-calculator bmi calculator women~~]. ~~We input height and weight~~, the ~~calculator determines~~ a ~~BMI~~ ~~or body mass index~~ (~~a formula~~ that ~~estimates body fat)~~. ~~We are able to compare to~~ the ~~norms for under~~/over ~~plus simply appropriate weights. Results is printed.~~<br><br>~~A person having a large frame~~ and/~~or a great deal~~ of ~~muscle may be flagged by Body Mass Index as being overweight--even whenever he has low body fat. This was true for disgraced cyclist Lance Armstrong at 1 point inside his athletic career.~~<br><br>~~Anyone whom is obese shoe definitely find them selves a superior doctor to discuss their weight plus health with~~. ~~Occasionally an illness could put the hormones from whack plus result us to gain weight~~. ~~Or maybe a medication~~ is ~~preventing us from losing~~ the ~~weight or result fatigue thus you~~ may ~~exercise~~. ~~In any case~~, ~~make sure to discuss this with~~ the ~~doctor plus left them know~~ the ~~plans~~. ~~They can provide assistance or they may have~~ to ~~change drugs and etc~~.~~<br><br>In terms~~ of its ~~dodgy mathematics~~, the ~~BMI is borderline junk research~~. ~~However inside this respect, BMI is less egregious than Global Warming~~ '~~studies~~'~~, inside that most~~ of the ~~big-name~~ '~~researchers~~' ~~cherry-pick~~, ~~hide~~, ~~or fabricate information~~. ~~Click~~ on ~~the link~~ for ~~my long hub on the topic~~.		In [[Vapnik–Chervonenkis theory\|statistical learning theory]], or sometimes [[computational learning theory]], the '''VC dimension''' (for '''Vapnik–Chervonenkis dimension''') is a measure of the [[membership function (mathematics)\|capacity]] of a [[statistical classification]] [[algorithm]], defined as the [[cardinality]] of the largest set of points that the algorithm can [[Shattering (machine learning)\|shatter]]. It is a core concept in [[Vapnik–Chervonenkis theory]], and was originally defined by [[Vladimir Vapnik]] and [[Alexey Chervonenkis]].

			Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the [[Heaviside step function\|thresholding]] of a high-[[degree of a polynomial\|degree]] [[polynomial]]: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so it can fit a given set of training points well. But one can expect that the classifier will make errors on other points, because it is too wiggly. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made more rigorous below.

			== Shattering ==

			A classification model <math>f</math> with some parameter vector <math>\theta</math> is said to ''shatter'' a set of data points <math>(x_1,x_2,\ldots,x_n)</math> if, for all assignments of labels to those points, there exists a <math>\theta</math> such that the model <math>f</math> makes no errors when evaluating that set of data points.

			The VC dimension of a model <math>f</math> is the maximum number of points that can be arranged so that <math>f</math> shatters them. More formally, it is <math>h'</math> where <math>h'</math> is the maximum <math>h</math> such that some data point set of [[cardinality]] <math>h</math> can be shattered by <math>f</math>.

			For example, consider a [[linear classifier\|straight line]] as the classification model: the model used by a [[perceptron]]. The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by [[Radon's theorem]], any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 2<sup>3</sup> = 8 possible label assignments are shown for the three points.

			{\| border="0" cellpadding="4" cellspacing="0"
			\|- style="text-align:center;"
			\| style="background:#dfd;"\| [[File:VC1.svg]]
			\| style="background:#dfd;"\| [[File:VC2.svg]]
			\| style="background:#dfd;"\| [[File:VC3.svg]]
			\| style="background:#fdd;"\| [[File:VC4.svg]]
			\|- style="text-align:center;"
			\| colspan="3" style="background:#dfd;"\| '''3 points shattered'''
			\| style="background:#fdd;"\| '''4 points impossible'''
			\|}

			== Uses ==

			The VC dimension has utility in statistical learning theory, because it can predict a [[probabilistic]] [[upper bound]] on the test error of a classification model.

			Vapnik <ref>Vapnik, Vladimir. The nature of statistical learning theory. springer, 2000.</ref> proved that the probability of the test error distancing from an upper bound (on data that is drawn [[Independent identically-distributed random variables\|i.i.d.]] from the same distribution as the training set) is given by

			<math>
			P \left(\text{test error} \leq \text{training error} + \sqrt{h(\log(2N/h)+1)-\log(\eta/4)\over N} \right) = 1 - \eta
			</math>

			where <math>h</math> is the VC dimension of the classification model, and <math>N</math> is the size of the training set (restriction: this formula is valid when <math>h \ll N</math>). Similar complexity bounds can be derived using [[Rademacher complexity]], but Rademacher complexity can sometimes provide more insight than VC dimension calculations into such statistical methods such as those using [[kernel methods\|kernels]].

			In [[computational geometry]], VC dimension is one of the critical parameters in the size of [[E-net (computational geometry)\|ε-nets]], which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all.

			==See also==
			*[[Sauer–Shelah lemma]], a bound on the number of sets in a set system in terms of the VC dimension

			== References ==
			<references/>
			* Andrew Moore's [http://www-2.cs.cmu.edu/~awm/tutorials/vcdim.html VC dimension tutorial]
			* Vapnik, Vladimir. "The nature of statistical learning theory". springer, 2000.
			* V. Vapnik and A. Chervonenkis. "On the uniform convergence of relative frequencies of events to their probabilities." ''Theory of Probability and its Applications'', 16(2):264–280, 1971.
			* A. Blumer, A. Ehrenfeucht, D. Haussler, and [[Manfred K. Warmuth\|M. K. Warmuth]]. "Learnability and the Vapnik–Chervonenkis dimension." ''Journal of the ACM'', 36(4):929–865, 1989.
			* Christopher Burges Tutorial on SVMs for Pattern Recognition (containing information also for VC dimension) [http://citeseer.ist.psu.edu/burges98tutorial.html]
			* [[Bernard Chazelle]]. "The Discrepancy Method." [http://www.cs.princeton.edu/~chazelle/book.html]

			[[Category:Dimension]]
			[[Category:Statistical classification]]
			[[Category:Computational learning theory]]
			[[Category:Measures of complexity]]

en>Chasrob: /* Definition and overview */

2012-08-05T16:11:35Z

Definition and overview

New page

We understand the famous BMI or Body Mass Index Calculation so commonly used to assess whether you're usual, obese or obese? Well, what do we actually learn about it? Do we think it's important? My point is I recognize I'm not overweight plus I understand I am a healthy person on my cholesterol and blood pressure, so there are details that give to my weight a calculator can't account for. Don't worry about the non-integer exponent. Any off-the-shelf scientific calculator may handle it. I wouldn't use the LI for clinical purposes--even though it's more reasonable than BMI. Wondering what a weight should be? Try Mayo Clinics [http://safedietplansforwomen.com/bmi-calculator bmi calculator women]. We input height and weight, the calculator determines a BMI or body mass index (a formula that estimates body fat). We are able to compare to the norms for under/over plus simply appropriate weights. Results is printed. A person having a large frame and/or a great deal of muscle may be flagged by Body Mass Index as being overweight--even whenever he has low body fat. This was true for disgraced cyclist Lance Armstrong at 1 point inside his athletic career. Anyone whom is obese shoe definitely find them selves a superior doctor to discuss their weight plus health with. Occasionally an illness could put the hormones from whack plus result us to gain weight. Or maybe a medication is preventing us from losing the weight or result fatigue thus you may exercise. In any case, make sure to discuss this with the doctor plus left them know the plans. They can provide assistance or they may have to change drugs and etc. In terms of its dodgy mathematics, the BMI is borderline junk research. However inside this respect, BMI is less egregious than Global Warming 'studies', inside that most of the big-name 'researchers' cherry-pick, hide, or fabricate information. Click on the link for my long hub on the topic.