Random partitioning

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

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... 

Ahn, H., Moon, H., Fazzari, M. J., Lim, N., Chen, J. J., and KodeU, R. L. (2007). Classification by ensembles fiom random partitions of high-dimensional data. Computat. Stat. Data Anal., 51 6166-6179. 

We envisage, then, a series of energy states like so many compartments into which molecules can be placed, and the distribution among which is to be investigated. The postulate of molecular chaos deliberately assimilates the problem to that of the random partition of a large number of objects among a number of boxes. 

Even a randomly partitioning of a training set into two classes may give linear separability and convergence in the training process if the portion of one class is less than 10 % C53, 94, 251, 3103. 

Figure 48 shows f as a function of the ratio n/d. For n/d < 3 exists a non-negIigible probability for a linear separability of random partitioning into two classes. Therefore, a ratio n/d > 3 must be requested for the development of binary classifiers. 

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). 

Let k < tn. In /c-fold cross-validation (CV) m is randomly partitioned into k subsets of... 

Fork cross-validation (LOOCV). A predicting function / is calculated for each i m, using the observations from m i = 0,..., - 1, i + 1,..., m - 1 exclusively for learning. The formula for the sum of squared residuals is simplified to... 

In this example we demonstrate validation of QSPR models using a test set. The real library was randomly partitioned into learning set and test set of 80 compounds each. The structures had to be subjected to some preprocessing ... 

The landslide inventory is naturally event-based. It is not possible to use a multi-temporal landshde inventory as the traditional landslide susceptibiUty analysis. Therefore, the inventory of landsUdes triggered by the earthquake was randomly partitioned into two subsets, training dataset and testing dataset. From the application of ANN model, the landslide susceptibility map is constructed. The results of verification show 81.274% of success rate and 79.915% of prediction rate. The verification results are of high values. It shows that the ANN model can be used as a precise tool in the earthquake triggered landslide susceptibility mapping when a sufficient number of data are available. 

Now, by comparing the global formulas in case of diatomic and triatomic molecules, one can generalize (by induction) similar results for a working molecule,M, randomly partitioned in molecular fragments, sayM = (2T f). 

Zhao, L and Choi, P. (2002) Measurement of solvent-independent polymer-polymer Flory-Huggins interaction parameters with the use of non-random partitioning solvents in inverse gas chromatography. Polymer, 43, 6677-6681. 

The set of available data about catalytic materials is randomly partitioned into k parts of approximately equal size. 

The vessels are randomly partitioned into seven priority classes, and the preferred berth location is generated according to the berthing location within the template in Fig. 10. 

It is also not true that products always thermalize on the surface before they can be probed experimentally. In situations for which a gas-phase product is produced which escapes quickly from the surface, non-random partitioning of energy in products may survive. This type of information can provide distinctive clues about features of the potential energy surface such as locations of barriers that are very difficult to probe in other ways. 

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