Rand index

The adjusted Rand index (86) measures the extent of agreement between two different cluster structures obtained for the same set of data points. This is a useful measure when comparing two methods producing a different number of clusters. The Rand index is simply the proportion of agreement between the two methods and is defined as... 

N is the total number of genes in the data set. The index ranges from 0 (when the two cluster results are completely different) to 1 (when the two methods agree completely). The expectation of the Rand index for two random partitions is not a constant and it is preferable to use the adjusted Rand index, which corrects the Rand index for the case of random partitions of the data. The adjusted Rand index is given by... 

Rand index - Expected value of Rand index Maximum value of Rand index - Expected value of Rand index... 

To obtain values for the expected and maximum values, the hypergeometric distribution is assumed. If Uij is the number of genes that are common to clusters i and j from each method, the adjusted Rand index can be calculated from the following formula ... 

The upper bound of the adjusted Rand index is unity and it takes on the value of zero when the similarity between the two clustering methods matches the expectation of the hypergeometric distribution. A higher value of the adjusted Rand index indicates a greater similarity of membership between the clusters of the two methods being compared. 

Let us consider two partitions of the same n products. For instance, these could be the partitions of the same stimuli operated by two subjects in the course of a free sorting task. The Rand index (Rand, 1971) is defined by ... 

Notwithstanding its intuitive appeal, the Rand index suffers from several pitfalls. In particular, it highly depends on the number of groups in the two partitions and the number of stimuli at hand. So much so that, in some situations, it can take high values even for two random partitions (Courcoux et al., 2014 Santos and Embrechts, 2009 Youness and Saporta, 2(X)4). In order to cope with this problem, an adjusted Rand index (ARI) was proposed as a form of the Rand index that is corrected for the grouping of the stimuli by chance (Albatineh and Mihalko, 2006 Hubert and Arabie, 1985). 

Krieger, A. M. and Green, P. E. (1999). A generalized Rand-Index method for consensus clustering of separate partitions of the same data base. Journal of Classification, 16, 63-89. 

Santos, J. M. and Embrechts, M. (2009). On the use of the adjusted Rand index as a metric for evaluating supervised classification. Lecture Notes in Computer Science, 5769, 175-184. 

Solomon, K. D. and S. C. Abraham, 1979, The Index of Harm A Measure for Comparing Occupational Risk across Industries, Rand report R-2409-RC, June. 

Chemicals or toxic activities not specifically referenced in the text may be found either in The Merck Index (b-76MH05oo> or in Bowman and Rand s Textbook of Pharmacology (b-80MI10500>. Source references may be obtained from these works by using the chemical name adopted in the text. 

The Ethyl Corp. (I, p. 173) has a Remington-Rand installation for indexing the properties of fuel and lubricant additives. Both chemical structure and additive properties are coded on a single card. The use of the selective sorter, which will permit sorting in 12 adjacent columns simultaneously, and well-organized prefiling of the cards have made this an extremely workable file without the... 

Anton P, Silberglitt R, Schneider R. The Global Technology Revolution Bio/Nano/Materials Trends and their Synergies with Information Technology by 2015, Rand Corporation 2001. Santa Monica, California (http //www.rand.org/pubs/monograph reports/MR1307/ index.html). 

Octave and Matiab provide many convenient built-in functions. Some that we have used in this text include the matrix exponential, expm incomplete gamma function, gammai pseudo-random number generators, rand and randm and several others. Type help -i and select the menu item Functi on Index for a complete list of Octave s built-in functions,... 

Augerson W., A Review of the Scientific Literature as it Pertains to Gulf War Illnesses. Vol. 5 Chemical and Biological Warfare Agents (Santa Monica, CA Rand Corporation, 2000). http //www.rand.Org/pubs/monograph reports/MR1018.5/index.html... 

However, in order to satisfy the requirements of Hop and Hos/ the parameters i s and stand on the following conditions (1) Us=[m+(l/2)]. and Vp=mn iot s=100% and 7p=0) (2) Ds=m/rand t =[m-(l/2)] r(for r]s=0 and =100%), where m is a positive integer. Under these conditions, the values of related parameters m, ffrT, and Ni are listed in Table 1. In order to fulfill the required TIR inside the substrates, only conditions at m=l are valid. However, the feasibility of fabricating these elements is usually limited by the finite refractive index modulation strength n of a recording material. Therefore, an alternative design method is described below to overcome this drawback. 

Tab. 9.1. Composition, HLB temperature (Thlb)i droplet radius rand polydispersity index (pol.) for the system water—Ci2E04—hexadecane at 25 °C.

In addition to the x notation, other coefficients encountered are the rand c/tensors. These are an alternative to the x tensor, and differ in the factors involving the medium refractive index and simple numerical factors. The r tensor describes the linear electro-optic effect where the changes in refractive index are related to by... 

Average pore radius estimated from the radius rand the number distribution of pores. Average pore radius estimated from the radius rand the volume distribution of pores. Average radius of gyration estimated from the radius rand the volume distribution of pores. Refractive index measured at 633nm using spectroscopic ellipsometry. 

---