Exact formulas for the variance of several balance indices under the Yule model

One of the main applications of balance indices is in tests of null models of evolutionary processes. The knowledge of an exact formula for a statistic of a balance index, holding for any number $n$ of leaves, is necessary in order to use this statistic in tests of this kind involving trees of any size. In this paper we obtain exact formulas for the variance under the Yule model of the Sackin, the Colless and the total cophenetic indices of binary rooted phylogenetic trees with $n$ leaves.     

Allopatry increases the balance of phylogenetic trees during radiation under neutral speciation

The shape of a phylogenetic tree is defined by the sequence of speciation events, represented by its branching points, and extinctions, represented by branch interruptions. In a neutral scenario of parapatry and isolation by distance, species tend to branch off the original population one after the other, leading to highly unbalanced trees. In this case the degree of imbalance, measured by the normalized Sackin index, grows linearly with species richness. Here we claim that moderate values of imbalance for trees with large number of species can occur if the geographic distribution involves more than one deme (allopatry) and speciation is parapatric within demes. The combined values of balance (normalized Sackin index) and species richness provide an estimate of how many demes were involved in the process if it happened in such neutral scenario. We also show that the spatial division in demes moderately slows down the diversification process, portraying a neutral mechanism for structuring the branch length distribution of phylogenetic trees.     

Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees

The stationary birth-only, or Yule-Furry, process for rooted binary trees has been analysed with a view to developing explicit expressions for two fundamental statistical distributions: the probability that a randomly selected leaf is preceded by N nodes, or “ancestors”, and the probability that two randomly selected leaves are separated by N nodes. For continuous-time Yule processes, the first of these distributions is presented in closed analytical form as a function of time, with time being measured with respect to the moment of “birth” of the common ancestor (which is essentially inaccessible to phylogenetic analysis), or with respect to the instant at which the first bifurcation occurred.The second distribution is shown to follow in an iterative manner from a hierarchy of second-order ordinary differential equations.For Yule trees of a given number n of tips, expressions have been derived for the mean and variance for each of these distributions as functions of n, as well as for the distributions themselves.In addition, it is shown how the methods developed to obtain these distributions can be employed to find, with minor effort, expressions for the expectation values of two statistics on Yule trees, the Sackin index (sum over all root-to-leaf distances), and the sum over all leaf-to-leaf distances.     

\sqrt 2 Rule for Controlling the Tree Pattern in Forest Cut

A forest’s productivity can be optimized by the application of rules derived from monopolized circles. A monopolized circle is defined as a circle whose center is a tree and whose radius is half of the distance between the tree itself and its nearest neighbor. Three characteristics of monopolized circle are proved. (1) Monopolized circles do not overlay each other, the nearest relationship being tangent. (2) “Full uniform pattern” means that the grid of trees (a×b=N) covers the whole plot, so that the distance between each tree in a row is the same as the row spacing. The total monopolized circle area with a full uniform pattern is independent on the number of trees and $\frac{\pi }{4}$ times the plot area. (3) If a tree is removed, the area of some trees’ monopolized circle will increase without decreasing the monopolized circles of the other trees. According to the above three characteristics, “uniform index” is defined as the total area of monopolized circles in a plot divided by the total area of the monopolized circles, arranged in a uniform pattern in the same shaped plot. According to the definition of monopolized circle, the distribution of uniform index $(L) = \frac{{\chi ^2 (2n)}}{{2\pi n}}$ for a random pattern and $E(L) = \frac{1}{\pi }$ the variance of L is $D(L) = \frac{1}{{n\pi ^2 }}$ . It is evident that E(L) is independent on N and the plot area; hence, L is a relative index. L can be used to compare the uniformity among plots with different areas and the numbers of trees. In a random pattern, where L is equivalent to the tree density of the plot in which the number of trees is 1 and the area is π, the influence of tree number and plot area to L is eliminated. When n→∞, D(L)→0 and $L \to \frac{1}{\pi } = 0.318$ it indicates that the greater the number of tree is in the plots, the smaller the difference between the uniform indices will be. There are three types of patterns for describing tree distribution (aggregated, random, and uniform patterns). Since the distribution of L in the random pattern is accurately derived, L can be used to test the pattern types. The research on Moarshan showed that the whole plot has an aggregated pattern; the first, third, and sixth parts have an aggregated pattern; and the second, fourth, and fifth parts have a random pattern. None of the uniform indices is more than 0.318 (1/∏), which indicates that uniform patterns are rare in natural forests. The rules of uniform index can be applied to forest thinning. If you want to increase the value of uniform index, you must increase the total area of monopolized circles, which can be done by removing select trees. “Increasing area trees” are the removed trees and can increase the value of the uniform index. A tree is an increasing area tree if the distance between the tree and its second nearest neighbor is $\sqrt 2 $ times longer than that between the tree itself and its first nearest neighbor, which is called the $\sqrt 2 $ rule. It was very interesting to find that when six plots were randomly separated from the original plot, the proportion of increasing area trees in each plot was always about 0.5 without exception. In random pattern, the expected proportion of increasing area trees is about 0.35–0.44, which is different from the sampling value of 0.5. The reason is very difficult to explain, and further study is needed. Two criteria can be used to identify which trees should be removed to increase the uniform index during forest thinning. Those trees should be (1) trees whose monopolized circle areas are on the small side and (2) increasing area trees, which are found via the $\sqrt 2 $ rule.     

A protocol to compare nestedness among submatrices

Searching for nestedness has become a popular exercise in community ecology. Significance of a nestedness index is usually evaluated using z values, and finding that a matrix is nested is typically a common result. However, nestedness is not likely to be spread uniformly within a matrix of species presence/absence per site. Selected parts of the matrix may show a degree of nestedness significantly higher (or lower) than expected from the overall pattern. Here we describe a procedure to assess if a particular submatrix (i.e., a peculiar combination of rows and columns extracted from the complete matrix) is more or less nested than expected for an assortment of sites and species taken at random from the same overall matrix. The idea is to obtain several submatrices of different sizes from the same overall matrix and to calculate their z values. A regression is then performed between z values of submatrices and their sizes. A nestedness index independent of matrix size is suggested as the deviation of the z value of a particular submatrix from that expected according to the regression line. We applied our protocol to 55 matrices with different nestedness indices under various null-models and, for purpose of demonstration, we discussed in detail a single case study regarding various animal groups of the Aegean Islands (Greece). The obtained results strongly encourage further research to focus not only on the question whether a matrix is nested or not, but also on where and why nestedness is confined.     

林木分布格局多样性测度方法: 以阔叶红松林为例

林木分布格局是森林结构的重要组成部分, 直接影响森林生态系统的健康与稳定, 维持森林结构多样性被认为是保护生物多样性的最佳途径。本研究探讨了林木分布格局多样性的测度方法, 以期为揭示森林结构多样性提供理论依据。格局多样性研究的关键在于选择合适的生物多样性测度方法和具有分布属性的格局指数。本研究通过统计角尺度分布频率和Voronoi多边形边数分布频率, 运用Simpson指数分别计算角尺度多样性和Voronoi多边形边数分布多样性, 作为表达林木分布格局多样性指数的方法, 并以我国东北吉林蛟河的3个100 m × 100 m的阔叶红松(Pinus koreansis)林长期定位监测标准地为例, 分析了林木分布格局的多样性。结果表明: 无论是角尺度分布还是Voronoi多边形的边数分布都接近正态分布, 角尺度分布中随机分布林木的频数最多, 占55%以上; Voronoi多边形的类型多达10个以上, 50%以上的林木有5-6株最近相邻木。利用Simpson指数衡量林木格局多样性, 角尺度分布与Voronoi多边形的边数分布都显示出聚集分布的林分比随机分布林分的格局多样性高。研究还发现, 两种格局判定方法得出的Simpson指数值有所不同, 角尺度分布的多样性数值明显低于Voronoi多边形的边数分布的多样性数值, 主要原因是二者的等级数量不同。可见, 林木分布格局多样性研究应选择具有分布属性的格局指数, 但由于各指数反映的角度不同, 所以在分析比较不同林分格局多样性时应采用相同的分析方法。     

√2 Rule for Controlling the Tree Pattern in Forest Cut

A forest's productivity can be optimized by the application of rules derived from monopolized circles.A monopolized circle is defined as a circle whose center is a tree and whose radius is half of the distance between the tree itself and its nearest neighbor.Three characteristics of monopolized circle are proved.(1) Monopolized circles do not overlay each other,the nearest relationship being tangent.(2)"Full uniform pattern"means that the grid of trees (a×=N) covers the whole plot,so that the distance between each tree in a row is the same as the row spacing.The total monopolized circle area with a full uniform pattern is independent on the number of trees and π/4 times the plot area.(3) If a tree is removed,the area of some trees'monopolized circle will increase without decreasing the monopolized circles of the other trees.According to the above three characteristics,"uniform index"is defined as the total area of monopolized circles in a plot divided by the total area of the monopolized circles,arranged in a uniform pattern in the same shaped plot.According to the definition of monopolized circle,the distribution of uniform index (L) = x2(2n)/2πn for a random pattern and E(L)=1/π;the variance of L is D(L)=1/nπ2.It is evident that E(L) is independent on N and the plot area;hence,L is a relative index.L can be used to compare the uniformity among plots with different areas and the numbers of trees.In a random pattern,where L is equivalent to the tree density of the plot in which the number of trees is 1 and the area is π,the influence of tree number and plot area to L is eliminated.When n→∞,D(L)→0 and L→1/π= 0.318;it indicates that the greater the number of tree is in the plots,the smaller the difference between the uniform indices will be.There are three types of patterns for describing tree distribution (aggregated,random,and uniform patterns).Since the distribution of L in the random pattern is accurately derived,L can be used to test the pattern types.The research on Moarshan showed that the whole plot has an aggregated pattern;the first,third,and sixth parts have an aggregated pattern;and the second,fourth,and fifth parts have a random pattern.None of the uniform indices is more than 0.318 (1/Ⅱ),which indicates that uniform patterns are rare in natural forests.The rules of uniform index can be applied to forest thinning.If you want to increase the value of uniform index,you must increase the total area of monopolized circles,which can be done by removing select trees."Increasing area trees"are the removed trees and can increase the value of the uniform index.A tree is an increasing area tree if the distance between the tree and its second nearest neighbor is √2 times longer than that between the tree itself and its first nearest neighbor,which is called the √2 rule.It was very interesting to find that when six plots were randomly separated from the original plot,the proportion of increasing area trees in each plot was always about 0.5 without exception.In random pattern,the expected proportion of increasing area trees is about 0.35-0.44,which is different from the sampling value of 0.5.The reason is very difficult to explain,and further study is needed.Two criteria can be used to identify which trees should be removed to increase the uniform index during forest thinning.Those trees should be (1) trees whose monopolized circle areas are on the small side and (2) increasing area trees,which are found via the √2 rule.     

On optimal Bayesian classification and risk estimation under multiple classes

A recently proposed optimal Bayesian classification paradigm addresses optimal error rate analysis for small-sample discrimination, including optimal classifiers, optimal error estimators, and error estimation analysis tools with respect to the probability of misclassification under binary classes. Here, we address multi-class problems and optimal expected risk with respect to a given risk function, which are common settings in bioinformatics. We present Bayesian risk estimators (BRE) under arbitrary classifiers, the mean-square error (MSE) of arbitrary risk estimators under arbitrary classifiers, and optimal Bayesian risk classifiers (OBRC). We provide analytic expressions for these tools under several discrete and Gaussian models and present a new methodology to approximate the BRE and MSE when analytic expressions are not available. Of particular note, we present analytic forms for the MSE under Gaussian models with homoscedastic covariances, which are new even in binary classification.     

Similarity indices,sample size and diversity

Summary The effect of sample size and species diversity on a variety of similarity indices is explored. Real values of a similarity index must be evaluated relative to the expected maximum value of that index, which is the value obtained for samples randomly drawn from the same universe, with the diversity and sample sizes of the real samples. It is shown that these expected maxima differ from the theoretical maxima, the values obtained for two identical samples, and that the relationship between expected and theoretical maxima depends on sample size and on species diversity in all cases, without exception. In all cases but one (the Morisita index) the expected maxima depend strongly to fairly strongly on sample size and diversity. For some of the more useful indices empirical equations are given to calculate the expected maximum value of the indices to which the observed values can be related at any combination of sample sizes. It is recommended that the Morisita index be used whenever possible to avoid the complex dealings with effects of sample size and diversity; however, when previous logarithmic transformation of the data is required, which often may be the case, the Morisita-Horn or the Renkonen indices are recommended.     

An updated consumer’s guide to evenness and related indices

Ecologists widely agree that species diversity consists of two components, richness (the number of species) and evenness (a measure of the equitability of the proportional abundances of those species). However, no consensus on an exact definition of evenness (or equitability) has emerged. Instead, numerous equitability indices have been used in the ecological literature, as different researchers have preferred indices with different mathematical properties. In this paper, I show that the phrase ‘species diversity consists of two independent components, richness and evenness’ logically leads to one particular definition of evenness (Evenness = Diversity/Richness). To facilitate accurate communication, I propose that the term ‘evenness’ be used only to refer to this phenomenon, and that other terms be used for the equitability indices that measure other things. Here I provide a review of popular equitability indices, explain what each measures in practice, and show how they relate to each other and to evenness itself. I also explore how the partitioning of diversity into richness and evenness components is related to the partitioning of diversity into alpha and beta components. Dissecting the indices makes it easier to see the conceptual differences among them. Such understanding is necessary to ensure that an appropriate index is chosen for the questions at hand, as well as to interpret the index values correctly and to assess when index values can and when they cannot be considered comparable.     

Amino acid sequencing by gas chromatography-mass spectrometry using perfluoro-dideuteroalkylated peptide derivatives: A. Gas chromatographic retention indices

The gas chromatographic properties of more than 200 O-trimethylsilylated perfluoro-dideuteroalkyl polyamino alcohols were evaluated which were obtained by LiAlD₄-reduction and O-trimethylsilylation of N-perfluoroalkyl oligopeptide methyl esters. Complex derivatized mixtures were analyzed by gas chromatography-mass spectrometry which contained between 2 nmol and 12 μmol of the original peptides. Retention indices of these derivatives can be predicted by summation of retention index increments which are assigned to each amino acid residue. Upon application of a uniform factor, the predicted and experimentally found retention index values generally agree within ±30 units up to retention index 3000. Larger compounds, especially the heptafluoro derivatives, emerge significantly earlier than expected, which makes these the most useful derivatives of complex peptides.     

A theoretical comparison of the consumer surplus and the elasticities of demand as measures of motivational strength

The price and income elasticities of demand have been used by ethologists to estimate motivational strength. The consumer surplus is an alternative measure of motivation, deriving from microeconomic theory. We made a theoretical assessment of the validity and versatility of these indices. Two factors are expected to compromise the internal validity (veracity) of the elasticity of demand indices: failure to take into account the amount that an animal is required to pay to maintain some level of consumption; and a tendency to confuse its readiness to defend a preferred consumption level with a propensity to become satiated. A third factor, expected to compromise the external validity (usefulness) of these indices, is the unrealistic assumption that a single value can be assigned to each resource. None of these problems applies to the consumer surplus index. One further factor, expected to compromise the internal validity of both the consumer surplus and price elasticity indices, is their failure to account for the effects that income has upon consumption. Overall, we conclude that the consumer surplus should be more valid, both internally and externally, than the price elasticity index. The consumer surplus should also be more externally valid than the income elasticity index, but it is unclear, on balance, which of these indices is the more internally valid. Finally, we show that both elasticity indices are considerably less versatile than the consumer surplus, owing to the assumption that a single value can be assigned to each resource. Copyright 2003 The Association for the Study of Animal Behaviour. Published by Elsevier Science Ltd. All rights reserved.      

Analyzing the (mis)behavior of Shannon index in eutrophication studies using field and simulated phytoplankton assemblages

The Shannon index of diversity H′ is a commonly used metric in ecology. The tendency of this index to show a unimodal relationship with productivity has been the subject of several studies. In the present work, the behavior of H′ and three related ecological indices (Simpson, Hill, and Evenness) was investigated using phytoplankton assemblages along a eutrophication gradient. We used both natural and simulated assemblages, whereby the comparison enabled us to assess the role of environmental ‘noise’ on index behavior. We developed simulated assemblages based on phytoplankton distributions predicted by two model types: the log series statistical model and the random fraction niche-based model. Using field data, H′ and the related Simpson index showed expected unimodal relationships with eutrophication. The same unimodal relationships were reproduced with simulated assemblages. Comparing the simulations with natural assemblages along a eutrophication gradient showed that there was much unexplained variance in the real-world data, suggesting that these diversity indices are sensitive to stochastic processes. An analysis of the simulated assemblages using relative abundance distributions suggested that increasing H′ and Simpson index values in the low range of the eutrophication gradient were due to increasing species richness, and that decreasing index values in the high range of the eutrophication gradient were due to decreasing evenness. In addition, this analysis revealed how assemblages of identical H′ values arose from contrasting community structures found in the low- and high-range of eutrophication. The high variability and non-linearity of the Shannon and Simpson indices along a eutrophication gradient suggests that these measures of diversity are inappropriate for use as water quality monitoring assessment tools. Indeed, when calculating ecological quality ratios that are employed by the European Water Framework Directive, unreliable (non-monotonic) predictions of water quality resulted.     

Tree-based networks: characterisations,metrics, and support trees

Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree T, called a base tree, and adding arcs between arcs of T. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network N, we characterise the support trees of N, that is, the tree-based embeddings of N. We use this characterisation to give an explicit formula for the number of support trees of N when N is binary. This formula is written in terms of the components of a bipartite graph.

     

Integration of genomic information into sport horse breeding programs for optimization of accuracy of selection

Reliable selection criteria are required for young riding horses to increase genetic gain by increasing accuracy of selection and decreasing generation intervals. In this study, selection strategies incorporating genomic breeding values (GEBVs) were evaluated. Relevant stages of selection in sport horse breeding programs were analyzed by applying selection index theory. Results in terms of accuracies of indices (r_TI) and relative selection response indicated that information on single nucleotide polymorphism (SNP) genotypes considerably increases the accuracy of breeding values estimated for young horses without own or progeny performance. In a first scenario, the correlation between the breeding value estimated from the SNP genotype and the true breeding value (= accuracy of GEBV) was fixed to a relatively low value of r_mg = 0.5. For a low heritability trait (h² = 0.15), and an index for a young horse based only on information from both parents, additional genomic information doubles r_TI from 0.27 to 0.54. Including the conventional information source ‘own performance’ into the before mentioned index, additional SNP information increases r_TI by 40%. Thus, particularly with regard to traits of low heritability, genomic information can provide a tool for well-founded selection decisions early in life. In a further approach, different sources of breeding values (e.g. GEBV and estimated breeding values (EBVs) from different countries) were combined into an overall index when altering accuracies of EBVs and correlations between traits. In summary, we showed that genomic selection strategies have the potential to contribute to a substantial reduction in generation intervals in horse breeding programs.     

黄土高原油松天然纯林结构特征的研究

结构是森林群落的基本特征,决定着群落的功能和发展方向。该研究采用结构参数角尺度(W)、混交度(M)、大小比数(U)以及样地纵剖面图分析了两块长宽均为60 m×60 m的油松天然林的空间结构特征,同时采用胸径(DBH)、树高(TH)和冠幅面积(CA)分布的直方图以及Shannon-Wiener多样性指数分析了它的非空间结构特征。结果表明:油松天然林种间隔离程度很低(=0.019),几乎为油松纯林,个体大小分化均匀(=0.478),整体呈随机分布(=0.485)。小树(TH≤5 m)个体相对较少,而大树(TH10 m)占多数且其树高分布集中。平均树高多样性THD=2.35,胸径集中分布在14~34 cm,58.4%~62.8%的树冠面积分布在20~40 m~2。林下油松幼苗更新丰富,但分布不均。这些特征表明成熟的油松天然林群落结构不稳定,可能趋向衰退并将逐渐被其他阔叶树种代替。     

N90 index: A new approach to biodiversity based on similarity and sensitive to direct and indirect fishing impact

An important effort has been made to develop diversity indices suitable to monitor the loss of biodiversity due to anthropogenic impacts in an accurate and comprehensible way. Here, N₉₀, a diversity index based on the species’ contribution to the similarity between samples in a group, is presented. N₉₀ uses the results of the classic Similarity Percentage analysis and a jack-knife routine to calculate the average and a dispersion value of the number of species contributing up to the ninety percent of the similarity in a group of samples. N₉₀ is applied to two groups of samples subjected to contrasting levels of bottom trawl fishing pressure using time series of experimental bottom trawl surveys of the Balearic Islands. The results are compared to those obtained using more ‘traditional’ diversity indices such as species richness, Shannon–Wienner, Simpson, Pielou, and Margalef diversity indices. The N₉₀ diversity index displayed a clear response to fishing pressure with significantly lower values in impacted communities, while the ‘traditional’ diversity indices showed almost null sensitivity to fishing pressure. In addition, N₉₀ also detects indirect fishing impacts by fluctuating in response to environmental variation in impacted areas, making this index sensitive to the synergies between climate and fishing impact at community level. The application of the N₉₀ diversity index to the case study shows that it may be an alternative to ‘traditional’ diversity indices when trying to monitor fishing impacts and the effects of environmental changes. Its units, number of species, and the corresponding summary list of species facilitate the interpretability of the results, improving the communication to managers and stakeholders.     

The equivalence of two phylogenetic biodiversity measures: the Shapley value and Fair Proportion index

Most biodiversity conservation programs are forced to prioritise species in order to allocate their funding. This paper contains a mathematical proof that provides biological support for one common approach based on phylogenetic indices. Phylogenetic trees describe the evolutionary relationships between a group of taxa. Two indices for computing the distinctiveness of each taxon in a phylogenetic tree are considered here—the Shapley value and the Fair Proportion index. These indices provide a measure of the importance of each taxon for overall biodiversity and have been used to prioritise taxa for conservation. The Shapley value is the biodiversity contribution a taxon is expected to make if all taxa are equally likely to become extinct. This interpretation makes it appealing to use the Shapley value in biodiversity conservation applications. The Fair Proportion index lacks a convenient interpretation, however it is significantly easier to calculate and understand. It has been empirically observed that there is a high correlation between the two indices. This paper shows the mathematical basis for this correlation and proves that as the number of taxa increases, the indices become equivalent. Consequently in biodiversity prioritisation the simpler Fair Proportion index can be used whilst retaining the appealing interpretation of the Shapley value.     

Comparison of methods for the demarcation between earlywood and latewood in tree rings of Norway spruce

The precise demarcation between earlywood and latewood is important for the detailed analysis of intra-annual tree ring features. Different techniques based on visual assessment, wood anatomy analysis and X-ray densitometry have been developed and are currently used for this purpose. Depending on the chosen method, tree species and environmental conditions, the results can significantly vary. Thus, it is important to determine the technique optimal for a particular research. Here, we investigated Norway spruce (Picea abies) tree rings to examine the agreement among the following demarcation methods: (1) direct visual assessment, (2) Mork’s index (anatomical definition of the transition from earlywood to latewood based on cell wall-lumen ratio) and (3) fixed and floating density thresholds applied to intra-ring density profiles. The aim was to modify both the Mork’s criterion and density thresholds on the basis of reference values given by visual identification of earlywood/latewood transition. A total of 231 tree rings were analysed by all methods. Our results showed that the usage of floating threshold (defined for each ring separately based on density profiles) is more reliable in comparison with fixed threshold (the same threshold value used for all tree rings and samples). Statistical analysis revealed the best correspondence between visual identification of earlywood/latewood transition and demarcation based on the standard Mork’s index and the floating density threshold derived as 80 % of maximum latewood density. In terms of Mork’s index calibration, the results showed that to determine latewood cells in Norway spruce trees growing in temperate conditions, it is sufficient to use an index value equal to 0.83. The results are applicable for the studied spruce population growing in a temperate climate. The methodology itself, however, is universal and can help to calibrate criteria for earlywood-latewood demarcation under specific conditions.     

EXACT INDICES, CRITERIA TO SELECT FROM MINIMUM LENGTH TREES

Abstract— Modifications of the consistency, retention and rescaled consistency indices are introduced. These apply to particular transformations of a character state rather than all of the transformations of a character. For example, if one observes relatively many losses in a character state over a suite of minimum length trees, a low weight is applied to the transformation to a loss; however, these observations infer nothing on the probability of the character state being gained independently. If the same character state shows few or no convergent gains on the suite of minimum length cladograms, then gains receive a relatively high weight. Conversely, if for a particular character state, convergent gains are common and losses rare, the transformation to a loss is given a higher weight than the transformation to a gain. For multistate characters, each possible transformation is weighted independently. Three indices are proposed, i.e. the exact consistently index, the exact retention index and the exact rescaled consistency index. The consistency index is modified to deal with characters with unknown entries.
The methods outlined not only select (or generate) preferred tree topologies but they also choose character optimizations, even for trees of identical topology.