Cardinal characteristic of the continuum - Revision history

Deltahedron: /* Further reading */ Jech (2003)

2014-05-18T19:06:15Z

Further reading: Jech (2003)

← Older revision		Revision as of 21:06, 18 May 2014
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	~~The~~ '~~''Dunn index (DI)''' (introduced by J~~. ~~C. Dunn) in 1974~~ is ~~a metric for evaluating [[clustering algorithm]]s~~.~~<ref>{{cite doi\|10.1080/01969727308546046}}</ref> This is part of a group of validity indices including the [[Davies–Bouldin index]]~~, ~~in that it is an internal evaluation scheme~~, ~~where the result is based~~ on ~~the clustered data itself~~. ~~As do all other such indices~~, ~~the aim is to identify sets~~ of ~~clusters that~~ are ~~compact, with a small variance between members~~ of the ~~cluster,~~ and ~~well separated~~, ~~where the means of different clusters~~ are ~~sufficiently far apart~~, as ~~compared to~~ the ~~within cluster variance~~. ~~For a given assignment of clusters, a higher Dunn index indicates better clustering~~. ~~One of the drawbacks of using~~ this, ~~is the computational cost as the number of clusters~~ and ~~dimensionality of the data increase~~.		Footwear is one of essential items to top off one's outfit. There is certainly not to say about fashion trend we all leave aside place. Nowadays, shoes not only protect your feet from cold or obstacle, they play more important role on fashion conscious. For most people, both of them are crucial. If you are searching for one kind of shoes which can enable you to go in the fashion trend and also comfy to wear in winter, Ugg shoes are your preferred choice.<br><br>If you would like to continue receiving dog-related articles, for instance ugg news, as well as advice, please click the Subscribe Button. It's free and anonymous. Appreciate reading and thank you for sharing this story with others.<br><br>The original ugg boots were popular among surfers, hikers and outdoor types. Although these even now some of UGG's customers, these days you understand all types of people appreciating these comfortable sheepskin boot styles. UGG now gets a varied collecting styles. Their boots coming from useful to chic. The following are some designs of ugg boots that imagine that find comfortable.<br><br>All the footwear whether shoes or boots consists of variety of designs. Usually a person has numerous pairs of shoes and boots each in your specific purpose and gathering. While working in either in public venues or in a muddy area there is definitely of likelihood to can get shoes dull. Therefore it is good to have cheap boots you need to type of outcome. Due to the fact are positioned on many stores and web stores.<br><br>You even can choose boots using products. In reality, there are many other shoes brands can supply the same style boots. For instance, Bearpaw is another very artist and as soon as you various, good quality UGG alternatives by reviewing the companies. Naturally, prices be cheaper. If you need to learn much more about such shoes brands, you can attempt to find online find the brands soon you find the one you prefer best. Nevertheless, you still need to keep it in mind that you get a number of discount uggs sale alternatives with various prices. Means positivity . choose, it's not necassary to only love the charge, you ought to still view the quality.<br><br>It always be summer or winter, girls never care; they wear what they wish to utilize. And uggs is really a such footwear that girls can't stop wearing the kids. You can find girls wearing them not simply for casual wear but also for party wear insanely. Girls pair mainly because with skin tight denim skirts supply them a smashing and ravishing visual appearance. Many girls love these boots but know as opposed to them love these boots (Great confusion and stress!). If you always be one who're not in love with these boots, i quickly suggest of which you wear them once inside your lifetime to feel the structure and comfort of these boots (I bet you adore wearing them again).<br><br>If you have any sort of concerns concerning where and ways to make use of [http://horizonafrica.com/img/ ugg outlet], you can call us at the web site.

	~~==Preliminaries==~~
	~~There are many ways to define the size or diameter~~ of ~~a cluster. It could be the distance between the farthest two points inside a cluster~~, ~~it could be the mean of~~ all ~~the pairwise distances between data points inside the cluster, or it could as well be the distance~~ of ~~each data point~~ from ~~the cluster centroid~~. ~~Each~~ of ~~these formulations are mathematically shown below:~~

	~~Let ''C''~~<~~sub~~>~~''i''~~<~~/sub~~> be a ~~cluster~~ of ~~vectors. Let ''x''~~ and ~~''y'' be any two n dimensional feature vectors assigned to the same cluster ''C''<sub>''i''</sub>.~~

	~~: <math> \Delta_i = \underset{x , y \~~in ~~C_i}{\text{max}} d(x,y) </math> , which calculates the maximum distance~~.

	~~: <math> \Delta_i = \dfrac{1}{\|C_i\| \|C_i - 1\|} \underset{x , y \~~in ~~C_i, x \neq y}{\sum} d(x,y) </math> , which calculates the mean distance between all pairs.~~

	~~: <math> \Delta_i = \dfrac{\underset{x \~~in ~~C_i}{\sum} d(x,\mu)}{\|C_i\|} , \mu = \dfrac{\underset{x \~~in ~~C_i}{\sum} x}{\|C_i\|} </math> , calculates distance~~ of ~~all the points from the mean.~~

	~~This can also be said about the intercluster distance, where similar formulations~~ can ~~be made, using either the closest two data points, one in each cluster, or the farthest two, or the distance between the centroids and so on~~. ~~The definition~~ of the ~~index includes any such formulation,~~ and ~~the family of indices so formed are called Dunn-like Indices~~. ~~Let~~

	: <~~math~~> ~~\delta(C_i,C_j)~~ <~~/math> be this intercluster distance metric, between clusters ''C''<sub>''i''</sub> and ''C''<sub>''j''</sub~~>.

	~~==Definition==~~
	~~With the above notation~~, if there are ~~''m'' clusters~~, ~~then the Dunn Index for the set~~ is ~~defined~~ as:

	~~: <math> DI_m = \underset{ 1 \leqslant i \leqslant m}{\text{min}} \left\{ \underset{ 1 \leqslant j \leqslant m~~, ~~j \neq i}{\text{min}} \left\{ \frac{\delta(C_i~~,~~C_j)}{ \underset{ 1 \leqslant k \leqslant m}{\text{max}} \Delta_k} \right\} \right\} </math>~~.

	~~==Explanation==~~
	~~Being defined in this way, the ''DI'' depends on ''m''~~, the ~~number of clusters in~~ the ~~set~~. ~~If the number of clusters is not known apriori~~, ~~the ''m'' for which the ''DI'' is the highest can be chosen as the~~ number of ~~clusters~~. ~~There is also some flexibility when~~ it ~~comes~~ to the ~~definition of ''d(x~~,~~y)'' where any of~~ the ~~well known metrics can~~ be ~~used~~, ~~like [[Manhattan distance]] or [[Euclidean distance]] based on the geometry of the clustering problem~~. ~~This formulation has~~ a ~~peculiar problem, in~~ that ~~if one of the clusters is badly behaved, where the others are tightly packed, since the denominator contains a~~ '~~max' term instead of an average term,~~ the ~~Dunn Index~~ for ~~that set of clusters will be uncharacteristically low~~. ~~This is thus some sort of~~ a ~~worst case indicator,~~ and ~~has~~ to be ~~used keeping that~~ in ~~mind. There are ready implementation~~ of ~~Dunn index in some vector based programming languages like [[MATLAB]], [[R~~ (~~programming language~~)~~]] and [[Apache Mahout]].<ref>{{cite web\|url=http://www.mathworks~~.~~com/matlabcentral/fileexchange/27859-dunns-index \|title=MATLAB implementation of the Dunn Index \|accessdate=5 December 2011}}~~<~~/ref~~> <~~ref~~>{{cite web\|last=Lukasz\|first=Nieweglowski\|title=Package ‘clv’\|url=http://cran.r-project.org/web/packages/clv/clv.pdf\|work=R project\|publisher=CRAN\|accessdate=2 April 2013}}</ref> <ref>{{cite web\|title=Apache Mahout\|url=http://mahout.apache.org/\|publisher=Apache Software Foundation\|accessdate=9 May 2013}}</ref>

	~~== Notes~~ and ~~references ==~~
	~~<references/>~~

	~~==External links==~~
	* http://~~www.sciencedirect~~.com/~~science~~/~~article/pii/S0031320303002838~~
	* http://ieeexplore.ieee.~~org/xpl/freeabs_all.jsp?arnumber=499469~~
	* http://machaon.karanagai.com/validation_algorithms.html

	~~[[Category:Clustering criteria]]~~

en>ChrisGualtieri: /* Examples */General Fixes using AWB

2013-10-26T16:18:07Z

Examples: General Fixes using AWB

← Older revision		Revision as of 18:18, 26 October 2013
Line 1:		Line 1:
	<br><br>~~Can XXR 521 rims make my car look better? If this describes~~ the ~~question you embark asking~~, ~~compared to article will provide you with~~ the ~~details you have to have a better idea if these wheels are befitting for your stay~~ on. ~~Just so you know~~, ~~these rims are~~ to ~~be able to update vehicle's appearance clear~~ that and ~~also essentially~~ the ~~aggressive and gutsy image~~. ~~Furthermore~~, ~~their low weight~~ and ~~race inspired construction will admit above average handling and road sensation~~.~~<br><br>A minor inconvenience my partner and i noted was that it's what I consider end up being~~ a ~~slight design defect~~. ~~The stainless trim on~~ the ~~glass door wraps around~~ inside ~~the oven~~, ~~meaning~~ it ~~can heat moving upward. It doesn't help make~~ the ~~handle itself get hot~~, ~~but you to~~ be ~~weary~~ of ~~not to touch the trim that gets exposed for~~ the ~~inner stove~~. ~~I don~~'~~t consider this to viewed as major flaw, but it~~'~~s something that in order to to keep in mind created the furnace.~~<br><br>~~Racing requires extreme use, such as hard cornering which wheels should~~ be ~~able to control~~. ~~Divert column is one of the challenges you may face competition, but may possibly~~ be ~~prevented if your use of wheel racing Tenzor. Contain~~ the ~~opportunity to withstand severe driving a racing without damage so that it will be worth difficult earned money.~~<br><br>~~Currently this Underwire Mio design by Calvin Klein comes in three exquisite solid colors in a piece brazilian bikini~~. ~~They offer first and foremost regular [http~~:/~~/www~~.~~5leji.blogspot.com~~/~~2012/05/matte-black-wraps.html تجليد أسود مطفي]~~ which ~~will is~~ the ~~ideal color for virtually any thin seem~~. ~~Also offered is a taupe color may call Mink~~, ~~which looks absolutely stunning for an added formal investigation~~. ~~The last color offered~~ can be ~~a striking Turquoise which suites a bolder taste incredibly well~~. ~~This exquisite underwire swimwear comes from a full selection~~ of ~~sizes from four to sixteen. It retails for around one hundred dollars~~, ~~therefore you~~ are ~~persistent to search around online you should find it much less expensive than this~~.<br><br>~~The VAIO X~~'~~s touchpad was a decently sized 2.1 x 1.6 inches~~ and ~~offered little chaffing~~. ~~However, like~~ the ~~keyboard~~, ~~it would have been a smidgen larger especially considering it~~'~~s capable of recognizing multitouch gestures~~, ~~like pinch and zoom. Two mouse buttons below likewise small but responsive.~~<br><br>~~An accurate thermometer is suggested~~. ~~You will use it to time your cooking~~, ~~to decide when~~ in ~~order to fuel or adjust~~ the ~~temperature and then determine as soon as the food is done~~. If the ~~thermometer that provides your smoker~~ is not ~~accurate (~~which ~~many seem~~ to be ~~not~~, ~~irrespective~~ of ~~how much obtain~~ a ~~for~~ the ~~smoker)~~, ~~get one separately. Its worth~~ the ~~amount.<br><br>There~~ are ~~extensive reasons to consider an electric barbeque grill and not charcoal or gas grills~~, the ~~first benefit being that you won~~'~~t have to concern yourself any igniters~~, ~~filling~~ the ~~gas tank or the soot from charcoal~~. ~~There isn't mess the new Dimplex electric grill additionally it only wants~~ a ~~few minutes~~ to ~~connect~~, ~~switch as well as have~~ the ~~grill ready for barbequing~~.		The '''Dunn index (DI)''' (introduced by J. C. Dunn) in 1974 is a metric for evaluating [[clustering algorithm]]s.<ref>{{cite doi\|10.1080/01969727308546046}}</ref> This is part of a group of validity indices including the [[Davies–Bouldin index]], in that it is an internal evaluation scheme, where the result is based on the clustered data itself. As do all other such indices, the aim is to identify sets of clusters that are compact, with a small variance between members of the cluster, and well separated, where the means of different clusters are sufficiently far apart, as compared to the within cluster variance. For a given assignment of clusters, a higher Dunn index indicates better clustering. One of the drawbacks of using this, is the computational cost as the number of clusters and dimensionality of the data increase.

			==Preliminaries==
			There are many ways to define the size or diameter of a cluster. It could be the distance between the farthest two points inside a cluster, it could be the mean of all the pairwise distances between data points inside the cluster, or it could as well be the distance of each data point from the cluster centroid. Each of these formulations are mathematically shown below:

			Let ''C''<sub>''i''</sub> be a cluster of vectors. Let ''x'' and ''y'' be any two n dimensional feature vectors assigned to the same cluster ''C''<sub>''i''</sub>.

			: <math> \Delta_i = \underset{x , y \in C_i}{\text{max}} d(x,y) </math> , which calculates the maximum distance.

			: <math> \Delta_i = \dfrac{1}{\|C_i\| \|C_i - 1\|} \underset{x , y \in C_i, x \neq y}{\sum} d(x,y) </math> , which calculates the mean distance between all pairs.

			: <math> \Delta_i = \dfrac{\underset{x \in C_i}{\sum} d(x,\mu)}{\|C_i\|} , \mu = \dfrac{\underset{x \in C_i}{\sum} x}{\|C_i\|} </math> , calculates distance of all the points from the mean.

			This can also be said about the intercluster distance, where similar formulations can be made, using either the closest two data points, one in each cluster, or the farthest two, or the distance between the centroids and so on. The definition of the index includes any such formulation, and the family of indices so formed are called Dunn-like Indices. Let

			: <math> \delta(C_i,C_j) </math> be this intercluster distance metric, between clusters ''C''<sub>''i''</sub> and ''C''<sub>''j''</sub>.

			==Definition==
			With the above notation, if there are ''m'' clusters, then the Dunn Index for the set is defined as:

			: <math> DI_m = \underset{ 1 \leqslant i \leqslant m}{\text{min}} \left\{ \underset{ 1 \leqslant j \leqslant m, j \neq i}{\text{min}} \left\{ \frac{\delta(C_i,C_j)}{ \underset{ 1 \leqslant k \leqslant m}{\text{max}} \Delta_k} \right\} \right\} </math>.

			==Explanation==
			Being defined in this way, the ''DI'' depends on ''m'', the number of clusters in the set. If the number of clusters is not known apriori, the ''m'' for which the ''DI'' is the highest can be chosen as the number of clusters. There is also some flexibility when it comes to the definition of ''d(x,y)'' where any of the well known metrics can be used, like [[Manhattan distance]] or [[Euclidean distance]] based on the geometry of the clustering problem. This formulation has a peculiar problem, in that if one of the clusters is badly behaved, where the others are tightly packed, since the denominator contains a 'max' term instead of an average term, the Dunn Index for that set of clusters will be uncharacteristically low. This is thus some sort of a worst case indicator, and has to be used keeping that in mind. There are ready implementation of Dunn index in some vector based programming languages like [[MATLAB]], [[R (programming language)]] and [[Apache Mahout]].<ref>{{cite web\|url=http://www.mathworks.com/matlabcentral/fileexchange/27859-dunns-index \|title=MATLAB implementation of the Dunn Index \|accessdate=5 December 2011}}</ref> <ref>{{cite web\|last=Lukasz\|first=Nieweglowski\|title=Package ‘clv’\|url=http://cran.r-project.org/web/packages/clv/clv.pdf\|work=R project\|publisher=CRAN\|accessdate=2 April 2013}}</ref> <ref>{{cite web\|title=Apache Mahout\|url=http://mahout.apache.org/\|publisher=Apache Software Foundation\|accessdate=9 May 2013}}</ref>

			== Notes and references ==
			<references/>

			==External links==
			* http://www.sciencedirect.com/science/article/pii/S0031320303002838
			* http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=499469
			* http://machaon.karanagai.com/validation_algorithms.html

			[[Category:Clustering criteria]]

en>ShelfSkewed: fix isbn, rm notice

2012-05-26T21:32:38Z

fix isbn, rm notice

New page

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