Generalized lifting

2012-12-03T15:53:42Z

83.244.157.210: /* Definition */

In [[probability theory]] and [[information theory]], '''adjusted mutual information''', a variation of [[mutual information]] may be used for comparing clusterings.<ref name="vinh-icml09">{{cite doi|10.1145/1553374.1553511}}</ref> It corrects the effect of agreement solely due to chance between clusterings, similar to the way the [[adjusted rand index]] corrects the [[Rand index]]. It is closely related to [[variation of information]]:<ref>{{cite doi|10.1016/j.jmva.2006.11.013}}</ref> when a similar adjustment is made to the VI index, it becomes equivalent to the AMI.<ref name="vinh-icml09" /> The adjusted measure however is no longer metrical.<ref name="vinh-jmlr10">{{Citation
| title = Information Theoretic Measures for Clusterings Comparison: Variants, Properties, Normalization and Correction for Chance
| journal = The Journal of Machine Learning Research
| pages = 2837–54 | volume = 11 | issue = oct | year = 2010
| last1 = Vinh | first1 = Nguyen Xuan | last2 = Epps | first2 = Julien | last3 = Bailey | first3 = James
| url=http://jmlr.csail.mit.edu/papers/volume11/vinh10a/vinh10a.pdf}}</ref>

==Mutual Information of two Partitions==
Given a set ''S'' of ''N'' elements <math>S=\{s_1, s_2,\ldots s_N\}</math>, consider two [[partition of a set|partitions]] of ''S'', namely <math>U=\{U_1, U_2,\ldots, U_R\}</math> with ''R'' clusters, and <math>V=\{V_1, V_2,\ldots, V_C\}</math> with ''C'' clusters. It is presumed here that the partitions are so-called ''hard clusters;'' the partitions are pairwise disjoint:
:<math>U_i\cap U_j = V_i\cap V_j = \varnothing</math>
for all <math>i\ne j</math>, and complete:
:<math>\cup_{i=1}^RU_i=\cup_{j=1}^C V_j=S</math>
The [[mutual information]] of cluster overlap between ''U'' and ''V'' can be summarized in the form of an ''R''x''C'' [[contingency table]] <math>M=[n_{ij}]^{i=1 \ldots R}_{j=1 \ldots C}</math>, where <math>n_{ij}</math> denotes the number of objects that are common to clusters <math>U_i</math> and <math>V_j</math>. That is,

:<math>n_{ij}=\left|U_i\cap V_j\right|</math>

Suppose an object is picked at random from ''S''; the probability that the object falls into cluster <math>U_i</math> is:
:<math>P(i)=\frac{|U_i|}{N}</math>
The [[entropy]] associated with the partitioning ''U'' is:
:<math>H(U)=-\sum_{i=1}^R P(i)\log P(i)</math>
''H(U)'' is non-negative and takes the value 0 only when there is no uncertainty determining an object's cluster membership, ''i.e.'', when there is only one cluster. Similarly, the entropy of the clustering ''V'' can be calculated as:
:<math>H(V)=-\sum_{j=1}^C P'(j)\log P'(j) </math>
where <math>P'(j)=|V_j|/n</math>. The [[mutual information]] (MI) between two partitions{{citation needed|date=September 2011}}:
:<math>MI(U,V)=\sum_{i=1}^R \sum_{j=1}^C P(i,j)\log \frac{P(i,j)}{P(i)P'(j)}</math>
where ''P(i,j)'' denotes the probability that a point belongs to both the cluster <math>U_i</math> in ''U'' and cluster <math>V_j</math> in ''V'':
:<math>P(i,j)=\frac{|U_i \cap V_j|}{N}</math>
MI is a non-negative quantity upper bounded by the entropies ''H''(''U'') and ''H''(''V''). It quantifies the information shared by the two clusterings and thus can be employed as a clustering similarity measure.

==Adjustment for chance==
Like the [[Rand index]], the baseline value of mutual information between two random clusterings does not take on a constant value, and tends to be larger when the two partitions have a larger number of clusters (with a fixed number of set elements ''N'').
By adopting a [[Hypergeometric distribution|hypergeometric]] model of randomness, it can be shown that the expected mutual information between two random clusterings is:
:<math>\begin{align} E\{MI(U,V)\} = &
\sum_{i=1}^R \sum_{j=1}^C
\sum_{n_{ij}=(a_i+b_j-N)^+}^{\min(a_i, b_j)}
\frac{n_{ij}}{N}
\log \left( \frac{ N\cdot n_{ij}}{a_i b_j}\right) \times \\
& \frac{a_i!b_j!(N-a_i)!(N-b_j)!}
{N!n_{ij}!(a_i-n_{ij})!(b_j-n_{ij})!(N-a_i-b_j+n_{ij})!} \\
\end{align}</math>
where <math>(a_i+b_j-N)^+</math>
denotes <math>\max(0,a_i+b_j-N)</math>. The variables <math>a_i</math> and <math>b_j</math> are partial sums of the contingency table; that is,
:<math>a_i=\sum_{j=1}^Cn_{ij}</math>
and
:<math>b_j=\sum_{i=1}^Rn_{ij}</math>

The adjusted measure<ref name="vinh-icml09"/> for the mutual information may then be defined to be:
:<math> AMI(U,V)= \frac{MI(U,V)-E\{MI(U,V)\}} {\max{\{H(U),H(V)\}}-E\{MI(U,V)\}}
</math>.

The AMI takes a value of 1 when the two partitions are identical and 0 when the MI between two partitions equals to that expected by chance.

==References==
<references />

==External links==
* [http://sites.google.com/site/vinhnguyenx/softwares Matlab code for computing the adjusted mutual information]

[[Category:Information theory]]
[[Category:Machine learning]]

Polyphase matrix

2012-12-03T15:43:32Z

83.244.157.210: /* Invertibility/perfect reconstruction */

:''In other countries, use [[Kv (flow factor)]]''

The '''flow coefficient''' of a device is a relative measure of its efficiency at allowing [[fluid]] flow. It describes the relationship between the [[pressure drop]] across an [[:wikt:orifice|orifice]], valve or other assembly and the corresponding [[volumetric flow rate|flow rate]].

Mathematically the flow coefficient can be expressed as:

<math>C_v = F \sqrt{\dfrac{SG}{\Delta P}}</math>

where: 
Cv = Flow coefficient or flow capacity rating of valve. 
F = Rate of flow (US gallons per minute). 
SG = [[Specific gravity]] of fluid (Water = 1). 
ΔP = Pressure drop across valve (psi).

In more practical terms, the ''flow coefficient'' Cv is the volume (in US gallons) of water at 60°F that will flow per minute through a valve with a pressure drop of 1 psi across the valve.

The use of the flow coefficient offers a standard method of comparing valve capacities and sizing valves for specific applications that is widely accepted by industry. The general definition of the flow coefficient can be expanded into equations modeling the flow of liquids, gases and steam as follows:

[[Coefficient of discharge]] is the ratio of actual flow rate to theoretical discharge.

For gas flow in a pneumatic system the Cv for the same assembly can be used with a more complex equation.<ref> Swagelok Cv Flow Coefficient
[http://www.swagelok.com/downloads/webcatalogs/EN/MS-06-84.PDF "Swagelok CV Flow Coefficient"]</ref><ref>[http://generant.com/cvcalc.aspx "Generant Cv Calculator"]</ref> Absolute pressures (psia) must be used for gas rather than simply differential pressure.

For air flow at room temperature, when the outlet pressure is less than 1/2 the absolute inlet pressure, the flow becomes quite simple (although it reaches sonic velocity internally). With Cv = 1.0 and 200 psia inlet pressure the flow is 100 standard cubic feet per minute (scfm). The flow is proportional to the absolute inlet pressure so that the flow in scfm would equal the Cv flow coefficient if the inlet pressure were reduced to 2 psia and the outlet were connected to a vacuum with less than 1 psi absolute pressure (1.0 scfm when Cv = 1.0, 2 psia input).

==References==
{{Reflist}}

==External links==
*[http://hydraulik.empass.biz/index.html#druckverlustkv_en Online-Calculation of pressure drop with flow coefficient]
*[http://www.novanumeric.com/samples.php?CalcName=Valves Online Cv Calculator For Multiple conditions]

{{DEFAULTSORT:Flow Coefficient}}
[[Category:Fluid dynamics]]

formulasearchengine - User contributions [en]

Generalized lifting

Polyphase matrix