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| {{Primary sources|date=May 2010}}
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| {{Expert-subject|Statistics|date=May 2010}}
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| }}
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| The '''Davies–Bouldin index (DBI)''' (introduced by David L. Davies and Donald W. Bouldin in 1979) is a metric for evaluating [[clustering algorithm]]s.<ref>{{cite doi |10.1109/TPAMI.1979.4766909 }}</ref> This is an internal evaluation scheme, where the validation of how well the clustering has been done is made using quantities and features inherent to the dataset. This has a drawback that a good value reported by this method does not imply the best information retrieval.
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| ==Preliminaries==
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| Let ''C''<sub>''i''</sub> be a cluster of vectors. Let ''X''<sub>''j''</sub> be an n dimensional feature vector assigned to cluster ''C''<sub>''i''</sub>.
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| : <math> S_i = \sqrt[q]{\frac{1}{T_i} \displaystyle\sum_{j=1}^{T_i} {|X_j-A_i|^q} } </math>
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| Here <math>A_i</math> is the [[centroid]] of ''C''<sub>''i''</sub> and ''T''<sub>''i''</sub> is the size of the cluster ''i''. ''S''<sub>''i''</sub> is a measure of scatter within the cluster. Usually the value of ''q'' is 2, which makes this a [[Euclidean distance]] function between the centroid of the cluster, and the individual feature vectors. Many other distance metrics can be used, in the case of [[manifolds]] and higher dimensional data, where the euclidean distance may not be the best measure for determining the clusters. It is important to note that this distance metric has to match with the metric used in the clustering scheme itself for meaningful results.
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| : <math> M_{i,j} = \left|\left|A_i-A_j\right|\right|_p = \sqrt[p]{\displaystyle\sum_{k=1}^{n}\left|a_{k,i}-a_{k,j}\right|^p } </math>
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| : <math> M_{i,j}</math> is a measure of separation between cluster <math>C_i</math> and cluster <math>C_j</math>. | |
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| : <math> a_{k,i} </math> is the ''k''th element of <math>A_i</math>, and there are n such elements in ''A'' for it is an n dimensional centroid.
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| Here k indexes the features of the data, and this is essentially the [[Euclidean distance]] between the centers of clusters ''i'' and ''j'' when ''p'' equals 2.
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| ==Definition== | |
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| Let ''R''<sub>''i,j''</sub> be a measure of how good the clustering scheme is. This measure, by definition has to account for ''M''<sub>''i,j''</sub> the separation between the ''i''<sup>''th''</sup> and the ''j''<sup>''th''</sup> cluster, which ideally has to be as large as possible, and ''S''<sub>''i''</sub>, the within cluster scatter for cluster i, which has to be as low as possible. Hence the Davies Bouldin Index is defined as the ratio of ''S''<sub>''i''</sub> and ''M''<sub>''i,j''</sub> such that these properties are conserved:
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| # <math> R_{i,j} \geqslant 0 </math>.
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| # <math> R_{i,j} = R_{j,i} </math>.
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| # if <math> S_j \geqslant S_k </math> and <math> M_{i,j} = M_{i,k} </math> then <math> R_{i,j} > R_{i,k} </math>.
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| # and if <math> S_j = S_k </math> and <math> M_{i,j} \leqslant M_{i,k} </math> then <math> R_{i,j} > R_{i,k} </math>.
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| : <math> R_{i,j} = \frac{S_i + S_j}{M_{i,j}} </math>
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| This is the [[symmetry]] condition. Due to such a formulation, the lower the value, the better the separation of the clusters and the 'tightness' inside the clusters.
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| : <math> D_i \equiv \max_{j : i \neq j} R_{i,j}</math>
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| If N is the number of clusters:
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| : <math> {DB} \equiv \frac{1}{N}\displaystyle\sum_{i=1}^N D_i</math> | |
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| ''DB'' is called the Davies Bouldin Index. This is dependent both on the data as well as the algorithm. ''D''<sub>''i''</sub> chooses the worst case scenario, and this value is equal to ''R''<sub>''i,j''</sub> for the most similar cluster to cluster ''i''. There could be many variations to this formulation, like choosing the average of the cluster similarity, weighted average and so on.
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| ==Explanation==
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| These conditions constrain the index so defined to be symmetric and non-negative. Due to the way it is defined, as a function of the ratio of the within cluster scatter, to the between cluster separation, a lower value will mean that the clustering is better. It happens to be the average similarity between each cluster and its most similar one, averaged over all the clusters, where the similarity is defined as ''S''<sub>''i''</sub> above. This affirms the idea that no cluster has to be similar to another, and hence the best clustering scheme essentially minimizes the Davies Bouldin Index. This index thus defined is an average over all the ''i'' clusters, and hence a good measure of deciding how many clusters actually exists in the data is to plot it against the number of clusters it is calculated over. The number ''i'' for which this value is the lowest is a good measure of the number of clusters the data could be ideally classified into. This has applications in deciding the value of ''k'' in the [[kmeans]] algorithm, where the value of k is not known apriori. The SOM toolbox contains a [[MATLAB]] implementation.<ref>{{cite web |url=http://www.cis.hut.fi/somtoolbox/package/docs2/db_index.html |title=Matlab implementation |accessdate=12 November 2011 }}</ref>
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| ==External links==
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| * http://machaon.karanagai.com/validation_algorithms.html
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| * http://citeseer.ist.psu.edu/viewdoc/summary?doi=10.1.1.17.2072
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| * http://books.google.com/books?id=HY8gB2OIqSoC
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| == Notes and references ==
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| {{Reflist}}
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| {{DEFAULTSORT:Davies-Bouldin index}}
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| [[Category:Clustering criteria]]
| |
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When you think of UGG boots will get dirty due to wet weather, it is best to use alternatives when heavy rain shoes and where the snow is soft and moist. When wearing your Ugg boots, not running on dirt, dry quickly and is difficult to stand during cleaning. You should keep away from wetlands, streams, ponds and slough. Another thing, do not wear your Ugg boots at places such as walking, can be scratched. Ugg boots are so soft you could leave a permanent scratch easily.
http://tinyurl.com/ku6vjks Where to buy your UGG boots, ask for any spray water repellent and stain resistant spray. Consider spraying your boots UGG boots with these easy solutions, so they can at least keep you away from many stains and dirt. To do this, spray evenly boots and leave for a period of 24 hours of drying. Consistent application of these solutions can provide additional protective boots.
Washing is done by hand. Most manufacturers will suggest http://tinyurl.com/ku6vjks and advise uggs to wash by hand. It is recommended to use the hand cleaner or vinegar to 1 part water. Wetting of the outside of the pocket using a sponge or clean cloth, do not pour water directly on your boots, Ugg Cardy Boots, not wet, it is essential, because they might lose their shape. It is easy to apply to areas with dirt or stains.
Let your ugg boots to dry away from direct heat. It is dry on fire or heat. You hang your boots upside down coat hanger using his ugg boots sale boot to keep Sabo Bracelet shape.Thomas let dry for 24 hours. After drying, use a soft suede brush or comb, soft brush uggs its natural feel your uniform. spray to make your selling ugg boots and a clean look shine again.