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en>Cranberry Products: /* Definition using equivalence classes */ sp

2014-10-29T14:02:50Z

Definition using equivalence classes: sp

← Older revision		Revision as of 16:02, 29 October 2014
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	~~In the field~~ of ~~[[machine learning]], the goal of [[statistical classification]]~~ is ~~to use an object~~'s ~~characteristics to identify which class (or group) it belongs~~ to. A '~~''linear classifier''' achieves this by making~~ a ~~classification decision based on~~ the ~~value~~ of a ~~[[linear combination]] of the characteristics. An object's characteristics are also known as [[Features (pattern recognition)\|feature values]]~~ and are ~~typically presented to the machine in~~ a ~~vector called a [[feature vector]].~~		The use of ecigs is really a training that's getting increasingly popular, and several individuals have taken up using ecigs as a cleaner option to smoking. The reason being e-cigs don't lead to ash, plus they do not cause a person’s household, outfits, and automobile stinking terrible. Among the greatest elements of needs to utilize e cigs involves buying a beginner kit, and there are a amount of points someone should be aware of about buying a starter kit. This includes investing in a somewhat better beginner kit, and there are certainly a few points a person should have contained in their starter kit.<br><br>Investing in a somewhat better kit<br><br>One of many most typical mistakes that persons make when first starting to smoke ecigs is the fact that they tend to buy the cheapest ecig that is offered to them. This may create a person certainly not taking up the practice, looked after ensures that a person WOn't ever purchase a great e cig once they do use up the practice. For this reason it's advisable to purchase a better one, that seems just how someone want it to, and it's also even a good strategy to buy one that may be slowly upgraded overtime.<br><br>What ought to be within the kit<br><br>An excellent starter kit will have things like a charger that is simple to use. The kit also needs to contain instructions on the best way to load the e-cigarette with juice, a magazine to clean it with, also it also needs to possess a some extra ejuice or cartridges. Visit [http://ciglites.weebly.com/ where to buy electronic cigarettes].

	~~== Definition ==~~
	~~[[Image:Svm separating hyperplanes.png\|thumb\|right\|In this case, the solid and empty dots can~~ be ~~correctly classified by any number~~ of ~~linear classifiers~~. ~~H1 (blue) classifies them correctly, as does H2 (red). H2 could be considered "better"~~ in ~~the sense that it is also furthest from both groups.~~
	~~H3 (green) fails to correctly classify the dots.]]~~

	~~If the input feature vector to the classifier is~~ a ~~[[real number\|real]] vector <math>\vec x</math>~~, ~~then the output score is~~

	~~:<math>y = f(\vec{w}\cdot\vec{x}) = f\left(\sum_j w_j x_j\right),</math>~~

	~~where <math>\vec w</math> is~~ a ~~real vector of weights and ''f'' is~~ a ~~function that converts the [[dot product]] of the two vectors into the desired output~~. ~~(In other words,~~ <~~math~~>~~\vec{w}~~<~~/math~~> is a ~~[[one-form]] or [[linear functional]] mapping~~ <~~math~~>~~\vec x~~<~~/math~~> ~~onto '''R'''.) The weight vector <math>\vec w</math> is learned from a set~~ of ~~labeled training samples. Often ''f'' is a simple function~~ that ~~maps all values above a certain threshold~~ to the ~~first class and all other values~~ to the ~~second class. A more complex ''f'' might give the probability~~ that ~~an item belongs~~ to ~~a certain class~~.

	~~For~~ a ~~two-class classification problem~~, ~~one can visualize the operation of~~ a ~~linear classifier as splitting~~ a ~~high-dimensional input space with~~ a ~~[[hyperplane]]: all points on~~ one ~~side of the hyperplane are classified as "yes"~~, ~~while the others are classified as "no".~~

	~~A linear classifier is often used in situations where the speed of classification is an issue~~, ~~since~~ it is often the fastest classifier, especially when <math>\vec x</math> is sparse. Also, linear classifiers often work very well when the number of dimensions in <math>\vec x</math> is large, as in [[document classification]], where each element in <math>\vec x</math> is typically the number of occurrences of a ~~word in a document (see [[document-term matrix]]). In such cases, the classifier should~~ be ~~well-[[regularization (machine learning)\|regularized]].~~

	~~==Generative models vs~~. ~~discriminative models==~~
	~~There are two broad classes of methods for determining the parameters of a linear classifier~~ <~~math~~>~~\vec w~~<~~/math>.<ref~~>T. Mitchell, Generative and Discriminative Classifiers: Naive Bayes and Logistic Regression. Draft Version, 2005 [http://www.cs.cmu.edu/~tom/mlbook/NBayesLogReg.pdf download]</ref><ref>A. Y. Ng and M. I. Jordan. On Discriminative vs. Generative Classifiers: A comparison of logistic regression and Naive Bayes. in NIPS 14, 2002. [http://www.cs.berkeley.edu/~jordan/papers/ng-jordan-nips01.ps download]</ref> Methods of the first class model [[conditional probability\|conditional density functions]] <math>P(\vec x\|{\rm class})</math>. Examples of such algorithms include:
	* [[linear discriminant analysis\|Linear Discriminant Analysis (or Fisher's linear discriminant)]] (LDA)—assumes [[normal distribution\|Gaussian]] conditional density models
	* [[Naive Bayes classifier]] with multinomial or multivariate Bernoulli event models.

	~~The second set of methods includes [[discriminative model]]s, which attempt~~ to ~~maximize the quality of the output on a [[training set]]. Additional terms in~~ the ~~training cost function can easily perform [[regularization (machine learning)\|regularization]] of the final model. Examples of discriminative training of linear classifiers include~~
	* [[Logistic regression]]—maximum likelihood estimation of <~~math~~>~~\vec w~~<~~/math~~> ~~assuming that the observed training set was generated by~~ a ~~binomial model~~ that ~~depends on the output of the classifier.~~
	* [[Perceptron]]—an algorithm that attempts to ~~fix all errors encountered in the training set~~
	* [[Support vector machine]]—an algorithm that maximizes the [[Margin (machine learning)\|margin]] between the decision hyperplane and the examples in the training set.

	~~'''Note:''' Despite its name, LDA does not belong~~ to the ~~class of discriminative models in this taxonomy. However, its name makes sense when we compare LDA~~ to the other main linear [[dimensionality reduction]] algorithm: [[principal components analysis\|Principal Components Analysis]] (PCA). LDA is a [[supervised learning]] algorithm that utilizes the labels of the data, while PCA is an [[unsupervised learning]] algorithm that ignores the labels. To summarize, ~~the name is~~ a ~~historical artifact (see,<ref>R.O. Duda, P.E. Hart, D.G. Stork, "Pattern Classification", Wiley, (2001). ISBN 0-471-05669-3</ref> p. 117).~~

	~~Discriminative training often yields higher accuracy than modeling the conditional density functions. However, handling missing data is often easier~~ with ~~conditional density models.~~

	~~All of the linear classifier algorithms listed above can be converted into non-linear algorithms operating on a different input space <math>\varphi(\vec x)</math>~~, ~~using the [[kernel trick]].~~

	~~== See~~ also ==
	* [[Winnow (algorithm)]]
	* [[Quadratic classifier]]
	* [[Support Vector Machines]]

	~~== Notes ==~~
	~~<references/>~~
	~~See~~ also:
	~~# Y. Yang, X. Liu, "A re-examination of text categorization", Proc. ACM SIGIR Conference, pp. 42–49, (1999)~~. [http://~~citeseer~~.~~ist~~.~~psu.edu~~/~~yang99reexamination.html paper @ citeseer~~]
	~~# R. Herbrich, "Learning Kernel Classifiers: Theory and Algorithms," MIT Press, (2001)~~. ~~ISBN 0-262-08306-X~~

	~~{{DEFAULTSORT:Linear Classifier}}~~
	~~[[Category:Classification algorithms]]~~
	~~[[Category:Statistical classification]]~~

en>Jason Quinn: /* {-1,1} */ fixed up a bad text in the example

2014-01-19T16:01:54Z

{-1,1}: fixed up a bad text in the example

← Older revision		Revision as of 18:01, 19 January 2014
Line 1:		Line 1:
	~~The use~~ of ~~ecigs~~ is ~~really a training that~~'s ~~getting increasingly popular, and several individuals have taken up using ecigs as a cleaner option~~ to ~~smoking~~. ~~The reason being e-cigs don~~'~~t lead to ash, plus they do not cause~~ a ~~person’s household, outfits, and automobile stinking terrible. Among~~ the ~~greatest elements~~ of ~~needs~~ to ~~utilize e cigs involves buying~~ a ~~beginner kit~~, and ~~there are a amount of points someone should~~ be ~~aware~~ of ~~about buying a starter kit~~. ~~This includes investing~~ in a ~~somewhat better beginner kit~~, and ~~there are certainly~~ a ~~few points a person should have contained in their starter kit~~.<br><br>~~Investing in~~ a ~~somewhat better kit~~<br><br>~~One~~ of ~~many most typical mistakes~~ that ~~persons make when~~ first ~~starting~~ to ~~smoke ecigs is~~ the ~~fact~~ that ~~they tend~~ to ~~buy~~ the ~~cheapest ecig that~~ is ~~offered to them~~. ~~This may create a person certainly not taking up~~ the ~~practice~~, ~~looked after ensures that~~ a ~~person WOn't ever purchase~~ a ~~great e cig once they do use up~~ the ~~practice~~. ~~For this reason it's advisable to purchase~~ a ~~better one~~, ~~that seems just how someone want it to~~, and ~~it's also even a good strategy to buy one that may be slowly upgraded overtime~~.<br><br>~~What ought~~ to ~~be within~~ the ~~kit~~<br><br>~~An excellent starter kit will have things like~~ a ~~charger~~ that ~~is simple~~ to ~~use~~. ~~The kit also needs~~ to ~~contain instructions on~~ the ~~best way~~ to ~~load~~ the e-~~cigarette~~ with ~~juice,~~ a ~~magazine to clean it with~~, also it also ~~needs to possess a some extra ejuice or cartridges~~. ~~Visit~~ [http://~~ciglites~~.~~weebly~~.~~com~~/ [~~http~~:~~//ciglites.weebly.com/ where to buy electronic cigarettes~~]].		In the field of [[machine learning]], the goal of [[statistical classification]] is to use an object's characteristics to identify which class (or group) it belongs to. A '''linear classifier''' achieves this by making a classification decision based on the value of a [[linear combination]] of the characteristics. An object's characteristics are also known as [[Features (pattern recognition)\|feature values]] and are typically presented to the machine in a vector called a [[feature vector]].

			== Definition ==
			[[Image:Svm separating hyperplanes.png\|thumb\|right\|In this case, the solid and empty dots can be correctly classified by any number of linear classifiers. H1 (blue) classifies them correctly, as does H2 (red). H2 could be considered "better" in the sense that it is also furthest from both groups.
			H3 (green) fails to correctly classify the dots.]]

			If the input feature vector to the classifier is a [[real number\|real]] vector <math>\vec x</math>, then the output score is

			:<math>y = f(\vec{w}\cdot\vec{x}) = f\left(\sum_j w_j x_j\right),</math>

			where <math>\vec w</math> is a real vector of weights and ''f'' is a function that converts the [[dot product]] of the two vectors into the desired output. (In other words, <math>\vec{w}</math> is a [[one-form]] or [[linear functional]] mapping <math>\vec x</math> onto '''R'''.) The weight vector <math>\vec w</math> is learned from a set of labeled training samples. Often ''f'' is a simple function that maps all values above a certain threshold to the first class and all other values to the second class. A more complex ''f'' might give the probability that an item belongs to a certain class.

			For a two-class classification problem, one can visualize the operation of a linear classifier as splitting a high-dimensional input space with a [[hyperplane]]: all points on one side of the hyperplane are classified as "yes", while the others are classified as "no".

			A linear classifier is often used in situations where the speed of classification is an issue, since it is often the fastest classifier, especially when <math>\vec x</math> is sparse. Also, linear classifiers often work very well when the number of dimensions in <math>\vec x</math> is large, as in [[document classification]], where each element in <math>\vec x</math> is typically the number of occurrences of a word in a document (see [[document-term matrix]]). In such cases, the classifier should be well-[[regularization (machine learning)\|regularized]].

			==Generative models vs. discriminative models==
			There are two broad classes of methods for determining the parameters of a linear classifier <math>\vec w</math>.<ref>T. Mitchell, Generative and Discriminative Classifiers: Naive Bayes and Logistic Regression. Draft Version, 2005 [http://www.cs.cmu.edu/~tom/mlbook/NBayesLogReg.pdf download]</ref><ref>A. Y. Ng and M. I. Jordan. On Discriminative vs. Generative Classifiers: A comparison of logistic regression and Naive Bayes. in NIPS 14, 2002. [http://www.cs.berkeley.edu/~jordan/papers/ng-jordan-nips01.ps download]</ref> Methods of the first class model [[conditional probability\|conditional density functions]] <math>P(\vec x\|{\rm class})</math>. Examples of such algorithms include:
			* [[linear discriminant analysis\|Linear Discriminant Analysis (or Fisher's linear discriminant)]] (LDA)—assumes [[normal distribution\|Gaussian]] conditional density models
			* [[Naive Bayes classifier]] with multinomial or multivariate Bernoulli event models.

			The second set of methods includes [[discriminative model]]s, which attempt to maximize the quality of the output on a [[training set]]. Additional terms in the training cost function can easily perform [[regularization (machine learning)\|regularization]] of the final model. Examples of discriminative training of linear classifiers include
			* [[Logistic regression]]—maximum likelihood estimation of <math>\vec w</math> assuming that the observed training set was generated by a binomial model that depends on the output of the classifier.
			* [[Perceptron]]—an algorithm that attempts to fix all errors encountered in the training set
			* [[Support vector machine]]—an algorithm that maximizes the [[Margin (machine learning)\|margin]] between the decision hyperplane and the examples in the training set.

			'''Note:''' Despite its name, LDA does not belong to the class of discriminative models in this taxonomy. However, its name makes sense when we compare LDA to the other main linear [[dimensionality reduction]] algorithm: [[principal components analysis\|Principal Components Analysis]] (PCA). LDA is a [[supervised learning]] algorithm that utilizes the labels of the data, while PCA is an [[unsupervised learning]] algorithm that ignores the labels. To summarize, the name is a historical artifact (see,<ref>R.O. Duda, P.E. Hart, D.G. Stork, "Pattern Classification", Wiley, (2001). ISBN 0-471-05669-3</ref> p. 117).

			Discriminative training often yields higher accuracy than modeling the conditional density functions. However, handling missing data is often easier with conditional density models.

			All of the linear classifier algorithms listed above can be converted into non-linear algorithms operating on a different input space <math>\varphi(\vec x)</math>, using the [[kernel trick]].

			== See also ==
			* [[Winnow (algorithm)]]
			* [[Quadratic classifier]]
			* [[Support Vector Machines]]

			== Notes ==
			<references/>
			See also:
			# Y. Yang, X. Liu, "A re-examination of text categorization", Proc. ACM SIGIR Conference, pp. 42–49, (1999). [http://citeseer.ist.psu.edu/yang99reexamination.html paper @ citeseer]
			# R. Herbrich, "Learning Kernel Classifiers: Theory and Algorithms," MIT Press, (2001). ISBN 0-262-08306-X

			{{DEFAULTSORT:Linear Classifier}}
			[[Category:Classification algorithms]]
			[[Category:Statistical classification]]

en>R'n'B: Fix links to disambiguation page Infinite

2012-08-22T23:52:52Z

Fix links to disambiguation page Infinite

New page

The use of ecigs is really a training that's getting increasingly popular, and several individuals have taken up using ecigs as a cleaner option to smoking. The reason being e-cigs don't lead to ash, plus they do not cause a person’s household, outfits, and automobile stinking terrible. Among the greatest elements of needs to utilize e cigs involves buying a beginner kit, and there are a amount of points someone should be aware of about buying a starter kit. This includes investing in a somewhat better beginner kit, and there are certainly a few points a person should have contained in their starter kit. Investing in a somewhat better kit One of many most typical mistakes that persons make when first starting to smoke ecigs is the fact that they tend to buy the cheapest ecig that is offered to them. This may create a person certainly not taking up the practice, looked after ensures that a person WOn't ever purchase a great e cig once they do use up the practice. For this reason it's advisable to purchase a better one, that seems just how someone want it to, and it's also even a good strategy to buy one that may be slowly upgraded overtime. What ought to be within the kit An excellent starter kit will have things like a charger that is simple to use. The kit also needs to contain instructions on the best way to load the e-cigarette with juice, a magazine to clean it with, also it also needs to possess a some extra ejuice or cartridges. Visit [http://ciglites.weebly.com/ [http://ciglites.weebly.com/ where to buy electronic cigarettes]].